A  KEY 


TO   THE 


WESTERN  CALCULATOR, 


CONTAINING 


THE  SOLUTION  OF  ALL  THE  EXAMPLES  AND 
QUESTIONS  FOR  EXERCISE, 

WITH  REFERENCE  TO  THE  PAGES  WHERE  THEY  STAND. 
TO  WHICH  IS  ADDED, 

SOME  USEFUL  RULES, 

DESIGNED     CHIEFLY     TO     FACILITATE     THE    LABOUR    OF     TEACHERS;    AND 
ASSIST    SUCH   AS    HAVE    NOT    THE    OPPORTUNITY   OF   A    TUTOR's   AID. 


BY  JOHN  ARMSTRONG,  A.  M. 


FOURTH  EDITION,  REVISED  AND   CORRECTED. 

PITTSBURGH: 

PRINTED  AND  PUBLISHED  BY  JOHNSTON  &  STOCKTON, 
MARKET  STREET. 

1836. 


[Entered  according  to  Act  of  Congress,  in  the  year  1836.  by 
Johnston  &  Stockton,  in  the  Clerk's  Office  of  the  Western  District 
of  Pennsylvania.] 


CONTENTS. 


Numeration  -        .        . 

Addition   -        -        -        - 
Subtraction  -        -        -        - 
Multiplication  - 
Division        -        .        -        . 
Federal  Money 
Compound  Addition     - 
Compound  Subtraction    - 
Compound  Multiplication 
Compound  Division 
Reduction  ... 

Addition  of  Decimals 
Subtraction  of  Decimals 
Multiplication  of  Decimals 
Division  of  Decimals  - 
Reduction  of  Decimals    - 
Single  Rule  of  Three  Direct 
Single  Rule  of  Three  Inverse 
Double  Rule  of  Three 
Practice         .        .        .        . 
Tare  and  Tret  - 
Simple  Interest 
Compound  Interest     - 
Insurance        .        .        . 
Commission        -        -        - 
Brokage 

Buying  and  Selling  Stocks     ■ 
Rebate  or  Discount 
Bank  Discount 
Equation  of  Payments    - 
Single  Fellowship 
Compound  Fellowship  - 
Profit  and  Loss 


Page. 

Page. 

5 

Barter        .... 

77 

ib. 

Exchange      .        -        -        - 

78 

6 

Alligation  -        .        -        - 

80 

-    7 

Reduction  of  Vulgar  Fraction 

3    82 

9 

Addition  of  Vulgar  Fractions 

86 

-  12 

Subtraction  of  Vulgar  Fractions  88 

13 

IMultiplication  of  Vulgar  Frac 

ib. 

-  14 

Division  of  Vulgar  Fractions 

89 

16 

Rule  of  Three  in  Vulgar  Frac. 

ih. 

-  18 

Inverse  Proportion 

90 

23 

Involution          -         .        -     • 

ib. 

-   30 

The  Square  Root 

91 

31 

The  Cube  Root 

96 

-  ib. 

Arithmetical  Progression 

102 

32 

Geometrical  Progression 

103 

-  33 

Single  Position 

106 

-   37 

Double  Position   - 

107 

-  41 

Permutation     • 

111 

47 

Combination         .        .        . 

112 

53 

Duodecimals 

ih. 

60 

The  Carpenters'  Rule 

115 

62 

Measuring  of  Boards,  &c. 

117 

68 

Carpenters'  &  Joiners'  Work 

118 

ib. 

Bricklayers'  Work 

120 

69 

Masons'  Work 

ib 

ib. 

Plasterers'  Work 

121 

70 

Pavers'  Work 

122 

ib. 

Painters'  Work    - 

123 

71 

Glaziers'  Work 

124 

72 

Measurement  of  Ground 

ih. 

ib. 

Gauging      .        -        -      - 

126 

74 

Mechanical  Powers 

ib. 

75 

Promiscuous  Questions    -    - 

127 

ARITHnXETICAZi  »IARSS  AND  SIGNS. 


=  The  sign  of  equality,  and  is  pronounced  equal. 

+  The  sign  of  addition,  pronounced  added  to. 

—  The  sign  of  subtraction,  pronounced  subtracted  by. 

X    The  sign  of  multiplication,  pronounced  multiplied  by. 

-7-  The  sign  of  division,  ^pronounced  divided  by. 

:   : :   :  The  sign  of  proportion,  pronounced  is  to,  so  is,  to. 

ui    Sign  of  difference,  pronounced  the  difference  between. 

v/,  or  i.  The  sign  of  the  square  root. 

-^,  or  i,  The  sign  of  the  cube  root. 

72,  Denotes  that  the  number  7  is  to  be  squared. 

8^,  Denotes  that  the  number  8  is  to  be  cubed. 

•.•    Or, 

Vinculum,  and  (  )  parenthesis. 

Examples. 
124-7  =  19;  twelve  added  to  7  is  equal  to  nineteen. 
^3 — 8  =  15;  twenty-three  subtracted  by  eight,  equal  fifteen. 

9  X  8=72;  nine  multiplied  into  eight  equal  seventy-two. 
24-4-3=  8;  twenty-four  divided  by  three  equal  eight. 

Sometimes  the  division  of  one  number  by  another,  is  expressed 
in  the  manner  of  a  vulgar  fraction;  thus  V=3,  twenty-seven 
divided  by  nine  equal  three.  Multiplication  is  also  frequently 
denoted  by  this  mark  .  thus  7.12=84,  seven  into  twelve  equal 
eighty-four. 

6  :  9  ::  8  :  12;  as  6  is  to  9  so  is  8  to  12. 
A  GQ  Z;  read  the  difference  between  A  and  Z» 
^81=9,  or  8U=s9;  the  square  root  of  81  equals  9. 
10  :  100  •.•  1  :  10;  as  10  is  to  100  that  is  as  1  is  to  10. 

A  vinculum  is  placed  over  several  quantities  to  denote  that 
they  are  to  be  considered  as  one  simple  quantity ;  a  parenthesis 
is  often  used  for  the  same  purpose. 


KEY 


TO    THE 


WESTERN  CALCULATOR. 


NUMERATION. 

Page  8. 
Example  1.         16  2.        49 


3.     385        4.     2610 

7.     5600006        8.     90000305 


Page  9. 
5.  64536 


6.  253842 
9.  829006002 


1.  Five. 

2.  Seventeen.  • 

3.  Thirty-five. 

4.  Four  hundred  and  fifty-eight. 

5.  Six  thousand,  eight  hundred  and  twenty-nine. 

6.  Seventy-two  thousand,  three  hundred  and  forty-eight. 

7.  Three  hundred  and   eighty-four    thousand,   seven   hundred   and 

twenty-one. 

8.  Two  millions,  six  hundred  and  eighty-three   thousand,  two 

hundred. 

9.  Fifty  millions,  six  hundred  and  seventy-eight  thousand  and 

twenty-four. 

ADDITION. 


Page  10. 


2.  1581366 
4.  2284038452 
6.  2968194198 
.  8.  5075433465 
10.  36417723180651 
1* 


8.  2468727092 
5.  2263764650 
7.  5831333360 

9.  9750563355 
11.  3847099705067 


SUBTRACTION. 


Application. 
Page  11. 


1.  125 
23 
16 

2060 

8009574 

6 

8011804 

5.  297 
125 
754 
127 
245 


1548 


2.  129 

654  i 
8070 
10000 
4000000 

4018853 

6.  1209 

476 

8017 

3119221 

3128923 


dolls.  barrels  dolls. 

A 125  4.  60  for  480 

B  286  75    675 

C  29  220    2200 

D  672  126    1386 


1112 


481  bbls.  4741  dolls. 


7.     1811 
21 


8.     George 


1832 


SUBTRACTION. 

Page  12. 


3.  44064062638 
5.  56053355680 


1.     1818 
1777 


41 


Application. 

2.     5648 
3460 

2188 


6000 
600 

5400 


cents 
5.     145 
75 
40 

given  him    260 
given  away  154 


dolls. 

3560 
William  3240 
Samuel  2850 
Henry  2555 
Thomas      2226 


14431 


4.     5390534442 
6.        00000001 


baiTels 
3.     1260 


sells  to  A  320 
B  435 

sold  in  all  755 

unsold  505 

cents 

35  penknife 
25  slate 
64  paper 
30  apples 

154 


left 


106 


multiplicatio:n. 
MULTIPLICATION. 


2.  24639576 
4 

Case  1.  Page  13. 

3.  3675432568 

8 

98558304 

29403460544 

Page  14. 

5.  476824753 
5 

6.  964703024 
6 

7.  74020005 

8 

2384123765 

5788218144 

592160040 

8.  2901946808 
9 

9.  246354276 
11 

26117521272 

2709S97036 

Case  2. 

2. 

Mult.  68523047653 
by       2367 

479661333571 
411138285918 
205569142959 
137046095306 

162194053794651 

Page  15. 

3.  Mult.  5221     4. 
by   145 

Mult.  23430     5. 
by   230 

702900 
4686 

Mult.  3800920 
by   80750 

26105 

20884 
5221 

190046000 
2660644 
30407360 

5388900 

757045 

306924290000 

6.  Mult.  89536925 
by     735 

7.  Mult.  78965987 
by    5893 

8.  1.15 
75 

447684625 
268610775 
626758475 

236697961 
710693883 
631727896 
394829935 

575 

805 

$86.25 

65809639875 

465346561391 

9.  3950 

29 

10.  40 
12 

35550 
7900 

480  pence 

$1145,50 


MULTIPLICATION. 

Case  3.    Page  15. 

2.    Mult.  871075 
by  21           3 

2613225 
7 

3.    Mult.  2453642 
by  36             6 

'       14721852 
6 

4. 

Mult.    43102 

by  64           8 

344816 

8 

18292575 

88331112 

2758528 

5.  Mult.  23645  by 
12 

144 

6.  Mul 

it.  12071  by  99 
9 

283740 
12 

108639 
11 

3404880 

Case  4.    Page  16. 

1195029 

2.    Mult.  3600 
by       400 

3.    Mult.  44000 
by  550000 

220 
220 

4. 

Mult.  663000 
by        60000 

1440000 

39780000000 

242000000Q0 


Note. 

Ex. 

1.  200x10     = 

2.  462x100  = 

3.  879x1000= 

=2000 

=46200 

=879000 

Application. 

1.     15 
25 

2.     125 
43 

3.     32  cents 
440 

75 
30 

375 

500 

1280 
128 

dolls.  375 

dolls.   53,75 

dolls.  140,80 

4.     60x1 
40  X 
34  X 

[25= 

85= 
50= 

=75,00  wheat     5. 
=34,00  rye 
=  17,00  corn 

100        6.     32 
10        sub.  20x24=480 

1000  dimes      12x18=216 
10 

126,00  amount 

10000  cents 

DIVISION. 


9 


8.     7525 
125 

37625 
15050 
7525 

940625 


S.    4)4756394344 

1189098586 
6.     12)64381259 


Page  17. 

8.    dolls.  250  A's 
3 

750  B's 
4 

3000  C's 

DIVISION. 

Case  1. 

4.     5)97036142 


$  250  A 

750  B 

3000  C 

4000  altogether 


7. 


5365104;-!. 


19407228^ 
6)3824966 

637494-1 

9.     9)8297463813 
y21940423f 

Case  2.    Page  18. 


5.     8)37846210 

47307761 

8.     7)46825486 

66893551 


divisor  dividend     quot. 
3.    64)29687624(463869 
256  64 


4.    735)47989536925(65291886  quotient 
4410  735 


408 

384 


1855476 
2783214 


326459435' 
195875659 
457043209 


247  29687616 
192       8  rem. 


556  29687624  proef 
512 

442 
384 

584 
576 

8  rem. 


3889 
3675 

2145 

1470   47989536925  proof 

6753 
6615 

*Add  in  the  re- 

1386  mainder  as  you  multi- 
735  ply  when  working  the 

proof. 

6519 
5880 

6392 

5880 

5125 
4410 


715  rem. 


10 


DIVISION. 


Page  18. 

5.  2359)4917968967  (  2084768 
4718         2359 


19996 

18872 

11248 
9436 


18129 
16513 

16166 
M354 

20127 

18872 

1255  rem. 

7.  175296  )  19842712000  ( 
175296 


18762917 
10423845 
6254306 
4169537 

4917968967  proof 


113195 

175296 


231311 
175296 


560152 

525888 

342640 

175296 

1673440 
1577664 

957760 

876480 


679170 
1018763 
226392 
565976 
792373 
113195 

19842712000 


6.  671)5374608  (  8009 
5368       671 


6608  8018 

6039  56069 

48059 

569 


5374608 


8.  108  )  5704392  (  52818 


540 


108 


304 

422552, 

216 

528184  1 

883 

5704392 

864 

199 

108 

912 
864 


81280 

Case  4.  Page  19. 
6100)876(54    3.  8I0)2834|7     4. 

146  54  rem.      354  27  rem. 


Note. 
1|00)256|54 


16100)1370|00(85 

128 

90 

80 

1000  rem. 


3.    1|000)876|029  4.    1|0000)8010000 

or  256  quot.  54  rem.      or  876  quot.  29  rem.  or  80  quot. 

Application. 

1.    855)4275(5  boys 
4275 


Divisio^r. 


11 


2.    28)2072(74 
196 

112 
112 


Page  20. 

3.    75)45000(600 
450 


4.    8164 

10  subtract 


00 


27)8154(302 
81       . 


54 

54 


5.     Here  begin  with  the     6.     13440  dollars 

150  and  work  backwards. 

thus  5)150 


30 

12  subtract 

2)18 
9  Ans. 


1=3380  eldest  son's  share 
^=2688  2d  son's  " 

1  =2240  3d  son's  " 

1=1920  4th  son's         " 
1=1680  5th  son's         « 


11888  sum  of  5  shares 


then  13440 

11888  subtract 


7.    72084)5190048(72 
504588 


144168 
144168 


1552  6th  son's  share 

8.  Because  A  has  10  cents  per  day  more 
than  B,  and  B  10  cents  per  day  more  than 
C,  it  is  evident  that  A  will  get  in  all  8  dol- 
lars more  than  C,  and  B  4  dollars  more 
than  C,  which  together  is  12  dollars, 
then  from  228 
take    12 


Divide  by  the  number  of  persons  3)216 


9. 


half  Eagle 
half  a  dollar 
quarter  dollar 


500 
50 
25 


72  C's  share 

consequently  76  B's  share 

and  80  A's  share 

575)1437,50(250 
1150 


575  cents 


2875 
2875 


10.     Captain    6  shares     Then  5|0)455|0 

Mate        4      " 

Seamen  40      " 


No.  of  shares  50 


dolls.     91  one  seaman's  sh. 
Now  91x6=546  Capt.  share 
And  91x4=364  mate's  share 


12 


FEDERAL  MOINEY. 


FEDERAL  MONEY. 
ADDITION.    Page  22. 
2.     Dolls.  19320,43  3.     Dolls.  2045S8,00i 

SUBTRACTION. 
2.     Dolls.  325216,94^  3.     Dolls.  126723,531 

MULTIPLICATION.    Page  23. 
2.    Dolls.  15823,50  3.    Dolls.  57408,79i 

DIVISION. 
2.    Dolls.  726,55  3.    Dolls.  10343,79| 

PROMISCUOUS  QUESTIONS. 
E  ddcm 


Quest.  1.    25,0,0,0,0 
6,2,0,0,0 

8,0,0 
__7^ 

31,3,5,5,5 
Or,  $  313  554  cts. 

3.  Sugar  $  39,87i 
Coflfee  22,161 
Tea  2,12i 


Dolls. 
2.     1055  in  notes 
260       gold 
3350       silver 
250  cents 


$  4967,50 


4.     From  $  645,95| 
Take       350,00 


$  64,181 
5.     Lent  $  1000,00 


$  295,651 


Received  at  j.^^^g^ 
sundry  pay'ts.  3^^281 


6.    $  102,19 
Mult,  by  120 


Received  in  all  $  746,20i 


$  12262,80 

7.     Dolls.  4,50 
16 


Unpaid  $  253,791 

8.    $17,37^ 
132 

3474 
5211 
173766 

$  2293,50 


$  72,00 

9.    5)6022,50 
$  1204,50 


COMPOUND   ADDITION.  13 

Page  23. 

10.     18)252.90(14.05  11.    45)22.50(50  cts. 

18  225 

72  0 

72 

9Q 

Page  24. 
12.         25)15555,50(622d.  22c. 

COMPOUND  ADDITION. 

ENGLISH  MONEY.     Page  25. 
2.         £136739  6  3i  3.     £15725  11  11^ 

TROY  WEIGHT. 

Ih.    oz.  diet.  gr.  lb.    oz.  dwt.  gr. 

2.         22    7    6    6  3.     33     0     9     19 

AVOIRDUPOIS  WEIGHT.     Page  26, 

r.   cwt.qr.  Ih.  oz.   dr.  T.   cwt.qr.lb.oz.dr. 

1.     128  12  1  18  14  10  2.     2921  3  0  9  0  8 

APOTHECARIES'  WEIGHT. 

lb.     oz.    dr.  sc.    gr.  lb.     oz.    dr.  sc.     gr.' 

1.        24    4    5    2     16  2.     220    2    0    2     10 

CLOTH  MEASURE. 

Yds.    qr.  na.  E.  Fl.  gr.  na.  E.  Fr.  gr*  iia. 

1.     215    2     0  2.     173     0    0  3.     128    4    0 

E.  En.  qr.   na. 
4.        221     0    2 

LONG  MEASURE.    Page  27. 

deg.    m.  fur.  po.    yd.  ft.  in.   be.  L.  M.    fur.     yds.  ft.  in. 

1.  207  27     5     6     1     0     4     0      2.    13   2      3     167   2  10 

LAND  MEASURE.    Page  28. 

A.   ft.  P.  A.    R.  P.  A.     R.  P. 

1.     161  1  29  2.    2134  2  23  3.     1999  2  32 

CUBIC,  OR  SOLID  MEASURE. 

Co. ft.   in.  T.    ft.     in.  T.    ft.      in. 

1.     26  4  1407  2.    21  16  1542  3.    271  26  1294 


14  COMPOUND  SUBTRACTION, 

TIME.    Page  29. 

F.  77M>.  d.    h.    mi.  sec.  Y.    d.    h.  mi.  sec. 

1.     104  2  26  13  23  27  2.     14  240  4  4  33 

MOTION. 

sig.  deg.  mi.    sec.  sig.      ^        '        " 

1.      9     16     9     17  2.         10     12    45    23 

LIQUID  MEASURE.    Page  30. 

T.  hhd.  gal.   at.    pt.  T.   hhd.  gal. 

1.        20     0     15     3     1  2.         79     0     11 

DRY  MEASURE. 

hu.     P.    at.  bu.        P.  qt.  hu.     P.  gt. 

1.     175     2     3         2.     1553     3     4         3.     41920  0  7 


COMPOUND  SUBTRACTION. 

Page  31. 

T.   cwt.  or.  lb.  oz.  dr.  mi.  fur.  P.  ft.  in.  he. 

2.         156  19  0  22  2  6                 3.  124  5  34  4  7  2 

bu.    P.    qt.  pt.                                  D.  h.       mi.     sec. 

4.     53     3     4     1                      5.     88  21     44     34 

T.  hhd.  gal.  qt.  pt.             sig.  deg.  mi.  sec.  A.    R.  P. 

6.     29  2     47  3  1         7.    2  22  15  41  8.     408  2  22 


Promiscuous  Questions  in  Compound  Addition  and 
Subtraction. 


£.    s.  d. 

Ist.     6     2  4 

2d.      5  10  9i 

3d.     7     0  0 

4th.    8  10  6 

5th.    9     2  6 


Quest.  1. 

No. 

2, 

yds. 
36 
45 

I- 

1 

na. 
2 
3 

a 

3, 

48 

2 

1 

ii 

4, 

52 

0 

3 

u 

5, 

64 

2 

0 

yards  247     2     1  £36     6     licost 

Received  £22  10    6 


Note  £13  15    7i 


COMPOUND   SUBTRACTION.  15 

Page  32. 

W.  oz.  diet.  gr.  U).  oz.  dr.  sc.  gr. 

Quest.  3.  Bought  26  9  10    0  4.  Bought  6  10  6  2    0 

Wrought  up  18  0  16  10  Used      4  5  4  1  17 

Has  left    8  8  13  14  lb.  2     5  2  0    3 

yrs.     mo.  day.  hr. 
5.     Charles  born         1817     3     20     9  evening 
William  born        1816     1     15     6  morning 


Year       12       5  15 


Cwt.  qr.  lb.  yds. 

6.  1st.     18     2  14  7.    sold  to  A  5^ 

2d.     16     3  18  ^6| 

3d.    22     0  24  ^2^1 

4th.   24     1  0 


Cwt.  82     0      0 


yds.    40  sold  and  left. 

yds. 

Then  from  55 
Or,  4  Ton,  2  cwt.  take  40 

yds.  15  for  D  and  E. 
But  because  E  has  half  as  much  as  D,  and  together 
have  15;  therefore  D  has  10,  and  E  has  5. 


8. 

gals. 

gals. 

bought  1  pipe 

=  126 

sold  1  hhd. 

63 

2  hhds. 

=  126 

2  qr.  casks 

52 

(  26 
\  26 

(  26 

(pipe 

17 

3  quarter  casks 

• 

leaked    <lihd. 

11 

(cask 

5i 

330 

bought 

gallons 

148i 

148i 

sold  and  leaked 

181i  left. 

9.    Bought     E.  Fr.  qr.  na.  yds.  qr.  na. 

^       C9        3    2=14  1  2 

two  first  jg       3     2==i4  1  2 

^      ,    ,  (8       2    8=12  2  3 

two  last  |g       2    3=12  2  3 

yards  54  0  2 

40  2  0 


13    2    2  yards  left. 


16  ,  COMPOUND   MULTIPLICATION. 

COMPOUND  MULTIPLICATION. 

Case  1.    Page  33. 

T.    cwt.   qr.     lb.    oz.   dr.  lb.      oz.    diet.    gr. 

2.     146  2  3  44  6  1      3.   71  10  17  7 

hu.pc.qt.  hhd.gal.gt.pt.  deg.mi.fur.p. 

4.  199  3  0    5.  283  33  2  1    6.  67  18  6  32 

yds.  ft.  in.  be.  A.     R.  P.  bu.  pc.  qt. 

1.     149  1  4  0    8.  809  0  34    9.  538  1  2 

d.    h.  mi.  sec,  yrs.  m.  w.  d. 

10.  763  8  44  15         U.  1508  2  2  6 

Application. 

£.    s.     d.  £.     s.       d. 

Qiiesf.  1.        2     6     4  2.  1     2       6i 

5  9 


£11  11     8  £10  2      Si 

£.     s.       d.  £.    s.  d. 

0     12     9i  4.         2     4  2i 

11  12 


£7       0    8h  £26  10     6 

Case  3.    Page  34. 

Application. 

T.  ctct.  qr.     lb.     oz.     dr. 
1.         4    3     1     16     8     10  by  36 

6 


25     0     1     15     3     12  product  by  6 
6 


150    2     1       7     6  8  product  by  36 

£.  s.d.  T.  cwt.  qr.  lb. 

2.    Mult.  120  6  9  by  24  3.     Mult.  24    4    2    7  by  48 

6  12 


722  0  6  product  by  6  290  14    3    0  pro.  by  12 

4  4 


2888  2  0  product  by  24  1162  19    0    0  pro.  by  48 


COMPOUND  MULTIPLICATION.  17 

Case  3.    Page  34. 

d.  h.    mi.  sec. 
Mult.  8    4     12    5  by  29 
4 


32  16     48  20  product  by  4 

7 


228  21     38  20  product  by  28 
8    4     12    5       "  1 


days  237     1     50  25  product  by  29 
Case  4.    Page  35. 

s.    d.  £,.    s.  d. 

2.        14    6  3.     1     2    3x7 

10  10 


5     0x4  11     2    6X1 

10  10 


72     10    0  111     5    Oval,  of  100 

2  11     2    6    ''       10 

7  15     9    «     '        7 


145      0     0  value  of  200 

29      0    0     ^     of  40  £130    3    3  val.  of  117 


£174      0     0  value  of  240 

£.   s.    d. 
4.         1     2    6x5 
10 


11     5    0x7 

10 

112  10    0 
2 

225    0    0  value  of  200 

78  15    0    '      of  70 

5  12    6    '      of     5 


£309    7    6  value  of  275 
2* 


18 


COMPOUND   DIVISION. 


COMPOUND  DIVISION, 

Case  1.    Page  36. 


£.       s.      d. 

2.    Ans.  187     18    5^ 

yds.  ft.   in. 
4.     Ans.  192     0    4^ 


T.  ciot.  qr.  lb. 
3.     Ans.  15    6     0    7 

T.   hhd.  gal.  qt. 
5.    Ans.  58    2     13    3|. 


w.    d.     h.    mi.     sec. 
Ans.    3    0     16     16    53 


Case  2. 


2. 


£.      s.    d. 
11)134     18     8 


£.      s.     d. 
3.     12)984    0     0 


4)  12       5    4  quot.  by  11     12)  82     0     0  by  12 


£3 


1     4     '      by44  £6  16     8byl44 

£.     s.    d. 

4.         12)474    0     0 

6)  39  10    0  quotient  by  1 
by  7 


£6  11     8 
Case  3.    Page  37. 


£.    s.    d.       £.  s.  d. 
2.    345)409    13    9      (13    9 
345 

64 
20 

345)1293(3s. 
1035 

258 
12 

345)3105(94. 
3105 


£.    s.   d. 

;.        232    4    9 
20 

s.    d. 

524)4644(8    10^|4|- 
4192  ^^* 

452 
12 

524)5433(10(/. 
524 

193 
4 

524)772(l5r. 
524 


248  rem. 


COMPOUPH)   DIVISION,   ETC.                                       !» 

Page  37. 

£.     5.  d.    £.  s.   d. 
654)3236  12  4H  4    18  11| 
2616 

£.     s.  d.    £..  s.  d. 

5.    68)132    0    8(1    18  10 
68 

620 
20 

64 
20 

654)12412(18s. 
654 

68)1280(18s. 
68 

5872 
5232 

600 
544 

640 
12 

56 
12 

654)7684(11(Z. 
654 

68)680(10(/. 
68 

1144 
654 

490 
4 


654)1962(3$/. 
1962 

Promiscuous , Questions  for  Exercise  in  Compound  Addi- 
tion, Subtraction,  Multiplication  and  Division. 


Quest.  1. 


s.     d. 
2    5x2 
10 


lb. 

oz. 

dwt. 

^• 

4 

1 

15 

22 
11 

ingots 

4    ^x7 
10 


45     7     15 


12     1 


72  10    0  value  of  600  yards 
8    9    2'      of   70     ' 
4  10    '      of     2    ' 


£81     4    0  value  of  672  yards 
T.  cwt.  qr.  lb. 


3.    1    2    3    16 
Mult,  by  8  loads 

9    3    0    16 


T.  cwt.  qr.  lb. 
4.    8)9    3    0    16 


1    2    3    16  one  share 


20  COMPOUND  DIVISION,   ETC. 

Page  37. 

A.      R.    P. 

5.     300    2    20 

5 


1503     0    20 
3 

4509     1     20 


Page  38. 

A.      R.    P.  hu.        d.      c.      d.     c.  m. 

B       5)4509     1     20         7.    179  )  201  37i  (  1     12    5 

179        01-  1  doll.  12ic,. 


3)901     3     20 

223 

300    2    20.  179 

447 
358 


895 
895 


d.  bushels. 

7x5  9.         1000 

10  Mult,  by  101  cts. 


5  l'0x6  10000 

10  500 


2  18     4  $105,00 

3 


8  15    0  in  300  days 
1  15    0  ^     60     « 
2  11  '       5     « 


£10  12  11  in  365  days 

£.    s.      d. 
hu.    d.  c.        d.     c.  11.    3)47     12     lOi 

10.     135x2  05  =276  75  sold  for  

135x1  62i  =219  37i  prime  cost  9)15     17      Ih 

$57  37i  gain  £1     15      34 


COMPOUND  DIVISION,  ETC.  21 

Page  38. 


dolls.  dwt.  gr. 

12.     9708  13.         17    8 

2  9 


5)19416  7  16    0  wt.  of9dolls. 

5 

3883  20  eldest  son.         


3)5824  80  rem. 


oz.  39    0     Owt.  of  45  dolls. 


$  1941  60  other  sons,  each 

oz.  dwt.  gr.  cwt.    lb. 

14.         84    7    20  15.        2i=2S0 

20  Mult,  by       133  mills 

dwt.  gr.  gr.  

150)1687  (  11   6+8  over  -           $37,24 
150        11   6  standard 


187       0  0+8  grains  over 
150 

37 
24 


148 

76 

150)908(6 
900 


8  grains 
16.         35  x20  dolls.  notes=700  dolls. 


63    Eagles 
284  dollars 

=630 

=284 

642  half  dollars 

=321 

368  qr.  dollars 

=  92 

256x12^  cents 

=  32 

Deposited  Dolls. 
Checks  560+820= 

2059 
1380 

Dolls.  679 


22  COMPOUND    DIVISION,   ETC. 

Page  38.  ^ 

dolls,  cts.      dolls,  cts. 
17.         36  yds.  at  4    66=167,76  cost 

Add  29,56  gained 

Must  sell  all  for  $197,32 
dolls,  cts.       d.  c. 
4  yds.  at    2    33=  9,32 
8  yds.  at    5     50=44,00 


12  yds.  is  sold  for     $53,32 

doUs.  cts. 

Now,  from  36  yds.  which  must  sell  for     197  32 

Take  12  yds.  which  brought  53  32 


The  diff.  24  yds.  must  sell  for  $144  00 

yds.  dolls. 
But  24)  144  (6  dollars. 

cts.  d.      cts. 

18.     12ix5=0    62ibeds. 

62^X4=2     50  supper  and  breakfast 

75  for  liquor 
25   x5=l     25  for  hay 


$5     12i 
From  6  dollars  take  $5,121,  and  the  remainder 

is  87i  cts.=875  mills. 

cts.   ms. 

But  2^=25)875(35  qts.=8|  galls. 
And  6  dollars-7-5  travellers =$1,20 =120  cents. 

Page  39. 

h.   m.  Ji.     m.   minutes 

19.         12  25xby5=62     5=3725 
11  30xby  9  =  103  30=6210 


[Sum    9935  minutes 
Now         9935  minutes 
Mult,  by       75  cents  per  day 


49675 
69545 

h.     m.       dolls,  cts.  ms. 

8=480  )  745125  (  15     52    3+ 


REDUCTION. 


23 


gal.     qt.  pt. 
20.  5)1534     1     1 


5)306     3     1 
galls.     61     1     1 


22. 


A  1 
B  2 

C  6 

9 


Page  39 


h.      mi. 
21.     9)114    45 


12^    ibmin. 


dolls. 
9)180 

20  A's  share 
20x2=  40  B's 
40x3=120  C's 


4. 


6. 


REDUCTION. 

MONEY.    Page  41 

4)120506  farthings 


12)  30126  i 
2|0)  251 10  6i 
£125  10  6i 


£.       s.    d. 
480     19    9 
20 


9619 
12 


9)115437  pence 
12826*  add 


128263*  cents 

9.         720  cents 
3^5=72  subtract 

648  pence 


5.     10)260  cents 

26  subtract 

234  pence 


Page  42. 

7. 


8. 


12)4658 

2|0)  38j8  2 

£19  8  2 

9)648  pence 
72  add 

720  cents 


30.        2)24235  half  pence 
12)12117^ 
210)10019  % 
I  50  9  9i 


24 


REDUCTION. 


Page  42 


11. 

Mult. 


216  French  crowns 
99  pence  in  a  crown 

1944 
1944 


12)21384  pence 


210)  17812 
89Z.  2s. 

13.        375Z. 

8 

3)3000 
$1000 


12. 


29     17 
20 

597 
12 


7164 
i=  796  add 

7960  cents 


Or,  79  dolls.  60  cents. 


TROY    WEIGHT. 

(4)115200  grains 

24  ) 

(   6)28800 

210)  48010  pennyweights 

12)240  ounces 

lb.  20 


3.    2|0)4564|8  dwts. 
228202.  Sdwt 


rains 


(4)27260  g 

24  {    

(6)  6815 

210)11315  20 

12)  56  15  20 


4. 


30  pounds 
12 

360  oz. 
20 


7200  dwts. 
24 


172800  gr. 

lb.  oz.  dwt.   gr. 
4     8     15     20 
12 

56  ounces 
20 

1135  dwts. 
24 

4540 
2272 


416.  Soz.  IbdiL^t.  20gr.  27260  grains. 


REDUCTION. 


'40 


3  tons 
20 

60  cwt. 
4 

240  qr. 

28 


1920 
480 

6720  lb. 


Tons. 
5 
20 

100 
4 


6. 


Page  42. 

dwt.    gr. 
8         6 
24 

198  gr. 
24  spoons 

792 
396 


4752  grains 
AVOIRDUPOIS  WEIGHT. 


400  gr. 

28 


11200  lb. 
16 


179200  oz. 
16 


2.  (4)2867200  drams 

16  \ 

(4)  716800 


16 


:4)  179200  oz. 


[4)    44800 


28)     112001b. 
4)     400  qr. 
2|0)  lOjOcwt. 
5  tons. 


cwt.  qr. 
4.         1     3 
4 

7 

28 

196  lbs. 
Mult.   6banels 


1176  lbs. 


2867200  dr. 


26  RiJDUCTIOIV. 


Page  42. 


cwt.  qr.    lb.  lb. 

5.         16    2     14  6.       28)2876 


4 


66 


4)  102  20 


28  26cwt,2qr.20lb. 

532 
133 


1862  lb. 


APOTHECARIES*  WEIGHT. 

lb. 

1.         15 

12 

180  oz. 

8 


1440  dr. 
3 


4320  scr. 


Page  43. 


1h.  lb. 

2  3  3.         2i  4.     2|0)57600|0  grains 

12  12 


36  oz.  30 


3)28800  sc. 


8  8  8)  9600  dr. 

288  dr.  16)240  drams  12)1200  oz. 

15  parcels  400  lb. 

864  sc. 
20 

;   ■  ■■■  -if  - 

17280  gr,' 


REDUCTION. 


2t 


250  yards 

4' 

1000  qrs. 
4 

4000  nails. 

324  Ells  Fi 
6 

1)1944  qrs. 
486  yards 

CLOTH  MEASURE.    Page  43. 

2.         4)8642  nails 

5)2160  2  nails 
432  Ells  E.  2  na. 


4.         16  bales 
36  E.  Fl. 

96 

48 

576  E.  Fl. 
3 


4)1728  qrs. 
432  yards. 


LONG  MEASURE. 


260  miles 

8 


2080 
40 


83200 
5h 

416000 
41600 

457600  yds. 
3 


1372800  feet 
12 


16473600  inches 


mi.  fu.  P.  yds.  ft. 
11  7  38  2  2 

8 

95 

40 

3838 
5^ 

19192 
1919 

21111 
3 

63335 
12 

760020 
3 


2280060  b.  c. 


28 


REDUCTION. 

Page  43. 

3)1267200  feet 

L.  fu.  yds. ft.  in. 
4.        3    2    110    1    5 
3  miles 

!0)  422400  yards 
8)  1920  furlongs 

9 

8  fur. 

74 

60)  240  miles 

220  yds. 

4  degrees 

1490 
149 

16390 
3 

49171 
12 

--  ' 

590057  inches 

5.         360  degrees  round 
60  miles 

21600 

8  furlongs 

172800 

220  yards 

38016000 

3  feet 

114048000 

12  inches 

1368576000 

LAND  MEASURE. 

25  acres                     2. 
4 

100 
40 

4|0)17600  0  perches 
4)  4400  roods 
1100  acres 

4000  perches 

3. 


REDUCTION. 

Page 

4a 

4  00)6400  00  perches 

4.             10  acres 
160 

4  0)160  0  each  share 
4)  40  roods 
10  acres 

1600  perches 
30| 

48000 
400 

48400  yards 
9 

4^5600  feet 
144 

29 


62726400  inches 
CUBIC,  OR  SOLID  MEASURE. 


ft.     feet. 

128)  3200  (  25  cords 

Page  44. 


ton.  ft. 
20x50: 

feet. 
=  1000 

3. 

ton.  ft.    cu.  in.      cu.  in. 
30x40x1728=2073600 

TIME. 

Page  45. 

w.  d. 

8    2 

7 

58 
24 

h.m. 
6  20 

2. 

da.  h. 

365    6  in  one  year 
24 

1466 
730 

238 
116 

D. 

8766 
60 

1398 
60 

525960 
60 

83900  mi] 

31557600  sec. 

10  years 

315576000 


yrs.     days        days     hrs. 
3.        1823x3654=665850     18 

w.  da.  hr.    mi.  sec. 
4.         1 X7 X24x60x60=604800  sec. 
3* 


30 


DECIMAL  FRACTIONS, 


1. 


UaUID  MEASURE.    Page  45 

4  tuns 
4 

2. 

pints, 
8)4032 

16  hhds. 
63 

63)  504  galls. 

1008  gaUs 

8  hhds. 

8 

8064  pints 

3. 

hhd.  gal.  pt. 
38x63x8=19152 

DRY  MEASURE. 

pints 

78bu.  Zpc.  7qt. 
4 

2.       2)2196 

8)  1098 

315 

8 

4)  137  2 

2527 

34  bu.  lpc.2qt 

2 
5054  pints 


ADDITION  OF  DECIMALS. 

Page  47. 
2.        Ans.  3923400,3687078 


3.  283,604 

490008,003275 

21,05 

1,2 

6200,3476 

Sum  496512,204875 


4. 


5.        26,52 
225,005 
,0035 
844, 

22 
300^825 
,00005 

Sum  1397,55355 


6. 


125  5 
10000,000005 
15,072 
2,01 


Sum  10142,582005 


,246 

,012 

,02 

,6 

,413 

,5 


Sum  1,791 


7.        5,4 
15,04 
100,004 
6000,00004 
93880,0004 

Sum  100000,44444 


DECIMAL  FBACTIOA'S. 


31 


SUBTRACTION  OF  DECIMALS. 

Page  47. 
Ans   685,495632  3.        Ans.  8,3047 


From  45,005 
Take  23,65482 

Diff.  21,35018 


Page  48. 


6.    From  5, 

Take     ,10438 

1789562 


7.     From  2, 

Take'    ,00002 

1,99998 


From  620,2 
Take  200,002 

Diff.  420,198 


8.    From  16, 
Take      ,016 

15,984 


MULTIPLICATION  OF  DECIMALS. 


Mult.  ,385746 
by   ,00463 

1157238 
2314476 
1542984 

,00178600398 

Mult.  ,024653  . 

,00022 

49306 
49306 


Mult.  158,694 
by   23,15 

793470 
158694 
476082 
317388 


367376610 


Mult.  25,04 
,002 


8.  Mult.  645,003 
,000005 


,05008 


,003225015 


,00000542366 

Contraction  in  Multiplication  of  Decimals. 

Page  50. 


23,463  multipli 
43,2  multipli 

46926 
7039 
938           5. 

54,90a     , 

cand 

er  reversed 

3,141592  multiplicand 
8347,25    mult.  rev. 

4.    234,216  multiplic. 
543,2    mult.  rev. 

46843 

7026 

937 

117 

1570796 

62832 

21991 

1257 

94 

25 

549,23 

165,6995 


•3i  DECIMAL  FE ACTIONS. 

DIVISION  OF  DECIMALS. 

Page  50. 

3.   23,7)65321,0(2756,16  4.  64,25)234,70525(3,6 

474  19275 


1792  41955 

1659  38550 


1331  34052 

1185  32125 


1460  19275 

1422  19275 


380 
237 


3)10, 


1430  3,33331 

1422 


8  rem, 

6.     ,9)9,0  7.  ,00463),00178600398(,385746 

1389 

10  


3970 
3704 

2660 
2315 


3453 
3241 

2129 

1852 

2778 
2778 


Page  51. 


8.        2,46),2327898(,09463  9.        ,09463),2327898(2,46 

2214  18926 

"84  "4iii 


1549 


37852 


1^'^6  56778 

738  56778 

738 


DECIMAL  FEACTIONS.  33 

Page  51. 

10.     ,018),000162(,009 
162 

Contraction  in  Division  of  Decimals, 

Page  52. 


4.    1,346787)74,33373(55,193             5.    9,365407)87,076326(9,297 
67339                                                  84289 

6994 
6734 

2787 
1873 

260 
135 

914 
843 

125 
121 

71 
65 

4 
4 

6 

6. 

2,45)32,68744231(13,34 
245 

818   • 

735 

83 

73 

10 

10 

7.     6,24),0046872345(,00075 
437 

31 

31 

0 

REDUCTION  OF  DECIMALS, 

Case  1.     Page  53. 
2.    2)1,0(,5  3.     4)3,00(,75 


34 


DECrWAT.  FRACTIONS 

u 

Pige  53. 

4. 

8)7,000(,875 

5. 

25)1,00(,04. 

6. 

60)57,00(,95 

7. 
Case  2. 

15)6,00(,40  cents 

s. 
2.         19-T-20= 

£ 

=,95 

3. 

d.             s. 
3-T-12=,25 

d,  £.  cwt.qr.  qr. 

3^(12x20)=,0125       5.     4  2=18-i-(20x4)=,225  ton 

^r.    lb.     lb. 

6.  2     14=70H-(28x4)=,625cwt. 

qr.  na.  na. 

7.  3     3=15-7-(4x4)=,9375yds.- 


Or  thus, 

5.    ^r.  4 
CMJi.  20 


6.     Z&.  28 


4,5 


14         7.       na.  4 
2,5  ^r.  4 


3 
3,75 


,225  ton.  cwt.  ,625 

Case  3.     Page  54. 


yds.  ,9375 


2. 

£. 

,75 
20 

shillings 

15,00 

15  s. 

4. 

,617  cwt. 
4 

qr 

.  2,468 

28 

lb. 

13,104 
16 

oz 

1,664 
16 

dr.  10,624 
2qr.  131b.  loz.  10  dr 

3. 


lb. 


12 


ounces    8,4 
20 

dwts.  8,0       Soz.  Sdwts. 

5.  ,3375  acres 

4 


rood.  1,3500 
40 


per.  14,0000 
1  rood  14  per. 


DECDLiL  FRACTIONS.  35 


Page  54. 


iun.  days. 

G.         ,258  7.          ,761 

4  24 

hhd.  1,032  hrs.  18,264 

63  60 

sal.  2,016  mi.  15,840 
60 


1  hhd.  2  gal. 


12 


10. 


sec.  50,400 
18  hr.  15  mi.  50,4  sec. 


lb.  9.         365,25  days  in  a  year 

8.         ,7  ,3 


—  days  109,575 
oz.  8,4  24 

20  

—  hrs.  13,800 
dwt.  8,0  60 


dwt. 

min.  48,000 

109  d. 

13  h.  48  m. 

day  hr.    hours 
,41  x24=9,84 
,16 

T.  cwt. 
11.   ,17  ,19 
20 

qr.  lb, 

1,17  ,7 

hrs.  9,68 
60 

cwt.  3,59 
4 

-mm.  40,80 
60 

2,53 

28 

.sec.  48,00 

15,54 

Om.  48  sec. 

3cwt.  2qr. 

15,541b, 

36 


DECLMAL  FRACTIONS. 


Promiscuous  Questions  in  Decimal  Fractions. 
Page  55. 


Quest.  1.    Mult.     ,09 
by  ,009 

2.    ,36  ton 

20 

Prod.  ,00081 

7,20  cwt, 
4 

3.     ,9125  ounces 

20 

dwt.  18,2500 
24 

gr.  6,0000 

28,80  qr. 
28 

23040 
5760 

806,40  lb. 
16 

18  dwt  6gr. 

12902,40  oz. 

4.     315)4,00G0127  nearly 

oz.  dwt.  gr. 

,5.         2     16    20=1364  grains 
And     1  pound  =.5760  grains 

Then,  5760)1364,0(,2368 


6. 

miles 

,1392 

8 

fur. 

1,1136 

40 

per. 

4,5440 

27200 
2720 

yds. 

.  2,9920 
Ifur, 

.  4  per.  3  yds. 

11. 

1  doll. 

=100  cts. 
3 

15)300 

3)  20 

6  J  cents 

13. 

222)1,000(,004504 

7. 

4)3,00(,75 

8. 

112)6,00(,0535714 

9. 

365)109,5(,3 

10. 

,04x50x1728=3456 

hhd. 

12.          ,875 

63 

2625 
5250 

gall.  55,135 
4 

qt.  ,500 
2 

pt.  1,000 

56  gal.  1  pint 

SINGLE   RULE   OF  THEEE   DIRECT.  37 


Page  55. 

14.    365,25  days  in  a  year  15.       ,73-7-(3x,25)  that  is 
^  ,75),73(,973* 

18,2625  ,       , 

24  yr.    w.    a.     h.     m. 

16.      ,05=2    2     19     12 


730500  '  T.^ 

365250 


,5=0     0       0     30 


438,3000  hr 


60  2if?.2d.  18^.  42  mi.* 


26298,0  mi.  -4.   R.   P. 

60  18.  ,6x4x40x,02=l,92 


1577880  sec. 


T.     h.  gal. 
17.         ,4     3,     ,8  cub.  in.  ft.       in. 

4  19.     1-^(128x1728)  that  is 

"i;9  221 184)  1 ,000000(,000004 + 

63 


120,5  gall. 

4  20.     (28  deg.  48  min.)-v-360  deg. 
482^  qt.  that  is, 

2  ^^*'*         ^^- 

964  pints  21600)1728,00(,08 

SINGLE  RULE  OF  THREE  DIRECT. 

Page  57. 

J7  J  77  CfS.  lb. 

Quest.  2.  As  112  :  12,32  : :  16  :     '     ^     =1  d.  76  c. 
'  112 

lb.    cts.        lb>       cwt. 
3.     As  1  :  36  :  :  336  (=3)  :  120  dolls.  96  cts. 

yds.  yds.  yds.  yds.  yds. 
4.        23+24+25+27=99  then 
yd.    cts.    yds. 
As  1  :  72  :  :  99  :  99x72=71  dolls.  28  cts. 

*  This  answer  is  obtained  by  reckoning  12  months  to  the  year,  4 
weeks  to  the  month,  7  days  to  a  week,  &c.  But  at  52  weeks  to  the 
year,  the  answer  will  be  2w.  4d.  4h.  18mi.  And  at  365^  ^days,  it  will 
be  2w.  4d.  5h.  48  mi.  the  true  Answer. 


38  SINGLE  RULE  OF  THREE   DIRECT, 

Page  57. 

lb.  cts.       lb. 
5.  As  4  :  48  :  :  512  (4  cwt.  2  qr.  8  lb.)  :  61  dolls,  44  ctg, 

lb.   cts.        lb. 
6.     As  1  :  8  :  :  128  :  10  dolls.  24  cts. 

,     pair  d.  c.      pair 

7.  As  114  (=9^  doz.)  :  68,40  :  :  3  :  1  doll.  80  cts. 

bu.      d.  c.       bu. 
8.     As  20  :  9,60  :  :  3  :  1  doll.  44  cts. 

cts.    yd.       d.  cts. 
9,         As  75  :  1  :  :  16,50  :  22  yards. 

c.  qr.  lb.        d.  cts.       oz.  cts, 
10.  As  32080  (  =  17  3  17)  :  320,80  :  :  6  :  6 


Page  58. 

oz. 

r    •    l.fS    : 

9,7x12 


OZ.        lb.      dol.         oz.     97x1 5     ^*  ^' 
11.        As  116,4(=9,7)  :  97  :   :  1,5  :  ^^      '  =1  25 


acres    dolls.  acres 

12.     As  125,5  :  627,5  •.•1:5::  4,75  :  23  doll.  75  cents. 

gal.  dolls,  cts.    gal. 

13.  As  1,5  :  4     50  :  :  378  (=1,5  tuns)  :  1134  dolls. 

d.  cts.  d.  cts.  d.  cts.    d.  cts. 

14.  First  1  66+1  97+2  31=5  94  the  price  of  1  ream 
of  each  sort.     Then  say — 

d.  cts.  of  each  sort.  d.    CtS. 

As  5  94  :  1  :   :  528  66  :  89  reams  of  each  sort. 

lb.         T.      d.  lb.     qr.lb. 

15.    As  2240  (=1)  :  224  :  :  42  (=1  14)  :  4  dolls.  20  cts. 

d.  c.    bbl.        d.    c. 

16.  As  5  50  :  1  :  :  1402,50  :  255  barrels 

da,        d.    cts.      (j^. 

17.  As  365  :  1186,25  :  :  1  :  3  dolls.  25  cents. 


SIiVGLE  RULE  OF  THREE   DIRECT.  39 

Page  58. 

da.     d.  cts.  da. 

18.       As  1  :  2  25  : :  365  :  821  dolls.    25   cents,  the 

sum  he  spends  in  a  year. 

Now,  821  dol.  25  cts.+378  dol.  75  cts.=1200  dolls. 

T.    cwt.   qr.     Ih.  lb. 

19.     4     10     1     12=10120 

lb.  cts.  lb. 

Then,  as  112  :  1,12  : :  10120  :  101  dolls.  20  cents 

20.     4  ft.  6  in. =54  inches 
^  of  54  =27  add 

81 
27 


2187 
9 


19683  solid  inches 

cu.  in.      cts.         cu.  in. 
Then,  as  1728  :  110  : :  19883  :  12  dolls.  53  cts.  nearly 

in.      in.       in.      in. 
21.     (28+14) Xl4x3,5=2058  cubic  inches 

cu.  in.       cts.        cu.  in.    d.  cts.  m. 
Then,  as  1728  :  190  : :  2058  :  2  26  2+ 

£.  s.        s.  T.  cwt.qr.lb.        lb. 

22.  One  ton=22401b.  22  8=448 &  203  9  3  3=455815 

lb.  s.  U).  s. 

Now,  as  2240  :  448  •.•  5  : 1 : :  455815  :  91163=£4558  3^. 

d.    c.    yds.  d.      c.        yds. 

23.  As  11  25  :  5  that  is,  as  225  :  1  :  :  850  50  :  378  in  all 

And,  as  18  pieces  :  378  yds.  : :  1  piece  :  21  yards 

hf.yds.     yds.         d.  £.  s.    d.   hf.yds.d.      s. 

24.     As     25  (  =  12^)  :  450  (=  1  17  6)  : :  2  :  36=3 

ft.   ft.       ft.  in, 
•25.    As  7  :  4  : :  218  9  :  125  feet 


40  SINGLE   RULE   OF  THREE   DIRECT, 

Page  58. 

A.  R.  P. 
26.     476  3  28=76308  perches.     Then  say 

P.  d.      c.        P. 

As  76308  :  4292  32i  : :  160  :  9  dollars 


da.    cts. 
.     As  1  :  214 

da. 
: :  365  : 

d.     cts. 
781  10 

spends 

Then,  from 
take 

dolls,  cts. 

1333  00  annual  income 
781  10  yearly  expense 

$551  90  he  will 

save 

Page  59. 

bu. 

28.     As  321  : 

d.     c. 
240,75  : 

bu. 
:  1  :  75  cents 

na.    yds 

K    As  24(=U 

r.       cts. 
)  :  250  : 

na.  qr. 
:6(=1 

na. 

2)  :  62i  cents 

gal.      gah    gal. 
30.     120i  +  124+126|=37H  gallons 

gal.    d.       s.  d.        gal.      pence       £,    s.    d. 
As  1  :  66(=5  6)  ::  371:J  :  24502^  =  102  1   lOi 

mi.  da.  mi. 

31.  12x5=60  the  distance  that  A  has  gone  before  B  starts 

mi.   mi. 

16 — 12=4  miles  B  gains  on  A  per  day 

mi.  da.      mi. 
Then,  as  4  :  1  : :  60  :  15  days 

£.      d.        s.  d.  £        pence        £. 

32.  As  1  :  150(=12  6)  : :  1000  :  150000=625 

men   bbls^     men      hbls.  bbls. 

33.     As  365  :  75  : :  500  :  102fjf =102f| 

35.     This  is  properly  a  question  belonging  to  the  rule  of 
Thi'ee  Inverse,  stated  thus, 

cts        v4-  ct^ 

As  375  :  360  :  :  250  then, 
cts.     A. 

375x360     ^^^ 
-J-—— — =540  acres, 
250  cts. 


SINGLE  RULE   OF   THREE   EWERSE.  41 

Page  59. 
7ni.      krs.        mi.       deg.    mi.         mi.    mi. fur. 
35.     As  1440(=24)  :  25020(=360x69i)  : :  1  :  17     3 

SINGLE  RULE  OF  THREE  INVERSE. 

Page  60. 
men    days     men  men  days  men 

Quest.  2.     As  60  :  100  ::  20         3.     As  65  :  4  : :  5 
60  4 


2|0)600|0  5)260 

300  days  52  days 

per.      days      per.  6>^24 

4.  As  6  :  24  : :  9  (=6+3)  :  -^_=:=ie-days 

cwt.      miles         cwt.     1  v  "I  50 

5.  As  1  :  150  :  :  6  :  -^ — =25  miles 

b 

ft.  wide      yds.  long        ft.  wide  Qf)  y  QAQ 

6.  As  80    :     300    :  :    60     :         ^        =400  yards 

7.  As  80  :  30': :  «t  :  '^=^J^=Sir.4ft.  8^in. 

70  7  J       7 

foot     feet  foot        12  00 

8.  As  1  :  12  ::  ,75  :   -^=16  feet 

,75 

yd.  yds.  yds.  75v42  5 

9.  As  ,75  :  42,5  : :  1,25  :  '  ^^^  '  =25,5  yards 

men    mouths       men  10v4'S 

10.  As  10  :  4,5  :  :  15  (  =  10+5  :  ^^^^^^^=3  months 

dolls,      yrs.  dolls.       S0vl5 

11.  As  80  :  15  : :  600  :  ^^^=2  years. 

hrs.     days       hrs.        12v4 

12.  As  12  :  4  : :  16  :   -^=3  days 

16  ^  / 

days        men  days      30  V  400 

13.  As  30  :  400  : :  50  :  ^^l^f^=240  the  number  of 

50 

men  the  provisions  will  serve  fifty  days 
men    men 
And  400—240=160  men  must  depart. 
4* 


42  SINGLE  RULE  OF  THREE  INVERSE. 

Page  61. 
dolls,    ino.         dolls.        292x6 

14.     As  292  :  6  : :  806  :  -^^=-'^  mo.  5-iJf da. 

men       mo.  men  1200v<^ 

15.     As  1200  :  9  :  :  1600(= 1200 +400)  :  i-^_^=:6|mo 

Again, 
men  oz.  men  1200x14        °^- 

As  1200  :  14  :  :  1600  :  ——-  =  10*  allowance  per  day 
And  14  oz. — l0^oz.=Shoz.  diminution  required. 

rod     rod  rod       40x4 

25 


16.     As  40  :  4  :  :  25  :  — ;^-=6f  rods. 


in.         in.  in.       12x12 

17.  As  12  :  12  ::  3  :  —J_  =48  inches. 

o 

cws.    days        cws.      6x91 

18.  As  6  :  91  :  :  21  :  -^^=26  days. 

mi.  hrs.      pipe        min.       600 

19.  As  600(=10)  :  1  :  :  24  :    24-=25  pipes 

inches         ft.         yds.         ft.  in.         yd.    'Jlfiv  10 

20.  As  216(  =  18)  :  10(=30)  : :  I8{=:^k):—-^^l20yds. 

lo 

in.        in.         lb.  in.        75^208 

21.  As  ,75(=|)  :  208  : :  39  :  '— 9^=4Z6. 

o9 

da.  mi.     mi. 
22.     5x20=100  A  has  gone  before  B  starts. 

mi.    mi. 

25 — 20=5  miles  B  gains  on  A  each  day. 

mi.  da.    mi.     da. 
Now,  as  5  :  1  : :  100  :  20  B  will  overtake  A . 

da.  mi.     mi. 
And  20x25=500  the  distance  B  must  travel. 

General  Rule. 

Page  62. 

mo.  mo.        men     800x2       ™^° 

Quest,  3.  As  5  ••  2  : :  800: — Jl-=320  the  number  of 

D 

men  the  provisions  will  serve  for  five  months. 

Then,  800  men — 320  men =480  men  must  depart. 


SINGLE   RULE   OF   THREE.  43 

Page  62. 

galls,  galls,  galls. 

4.  63 — 9=54  remains 

dolls,  dolls,    dolls. 
And  119+12=131  must  sell  for 

gal.   gal.       dol.  d.    c.    m. 

Then,  as  54  :  1  : :  131  :  VV=2  42  5fJ 

mi.       mi.  lb.  lb. 

5.  As  64  :  512  that  is,  as  1  :  8  : :  225  :  1800 

dolls,    dolls.  cts.        cts. 

6.  As  1750  :  10  •.•   175  :  1   . :  8750  :  50 

Promiscuous  Questions  in  Direct  and  Inverse  Proportion. 

kmi.    mi.     Jir.       imi.        mi.       hr.  mi. 
Quest.  1.  As  5  (=2i)  :  1  : :  246  (=123)  :  49  12  going 

hmi.   mi.     hr.       hmi.    hrs.  mi.  sec. 
And,  as  7  (=3i)  :  1  : :  246  :  35     8     34f  returning 

hrs.    mi.      sec. 
But  49     12         0 
35       8       34f 


Sum  84/i.  20m.  34f  sec. 

Page  63. 

dolls.        da.  dolls. 

Quest.  2.     Stated  thus,  as  1000  :  189  : :  650  Inverse  pro, 

dolls.      dolls.         days. 
Or,  as  650  :  1000  : :  189  by  the  general  rule 

dolls,    da.  da. 

Then  1000xl89_20xl89 

^''^"  650  dolls. 13 ^-^^  ''''^'- 

cii't.   qr.     lb.  casks  lb.    cwt.     lb.       cwt.    lb. 
3.         (1       1       4)xl4=144xl4=2016       1=112 

lbs.      cts.          lbs.       12  60x2016 
Then,  as  112  :  1260  :  :  2016  :       '     "^ =226d.  80c. 

lbs.        cts.  lb.     1260     45 

And,  as  112   :  1260  : :  1  :  ——-=—=11  cts.  2i  ms. 

112       4 


44 


SINGLE  RULE   OF   THREE. 


Page  63. 


act.  qr. 
4.         (1     0 

lb.  cJiests  lb.  chest,  lb. 
14)x4=126x4=504  the  whole  weight 

Now,  as  1  lb. 
lib. 
lib. 
lib. 

dolls,  cts. 
:     80  cts.  : :  126  lb.  :  100     80 
:     90cts.  ::  1261b.  :  113     40 
:  105  cts.  :  :  126  lb.  :  132     30 
:  125  cts. : :  126  lb.  :  157     50 

§504    00  theamt. 
Then,  as  5041b.  :  504  dolls. : :  1  lb.  :  1  dollar 


5.     5oz.x  12=60  ounces  of  bread  in  a  dozen  of  rolls. 

And,  as  5  :  4  : :  QOoz.  :  4:Soz.  the  weight  of  flour  in  a 
dozen  of  rolls. 

oz.       cwt.    dolls. 
Then,  as  1792  (  =  1)  :  224  •.• 


oz. 
1   :  :  48  :  6  cts. 


dolls,  cts. 

6.     Cost  780  00 

Freight  37  70 

Other  charges   30  60 

Gain  143  00 


$991     30  cts.  must  sell  the  whole  for 


bU. 


dolls,    cts. 


Then,  as  270  :  991     30 


bbl. 
1  :  3  dolls.  672V  cts. 


7.         Haifa  ton =10  cwt.  =40  qrs. 

horses     qrs.       horses  40  V  7      7 

As     80  :  40  :  :  7  :  —^=-=3  qrs.  14  lb. 
80        /^ 


Quest.  S.  12ft=144  inches,  and  Oft.  3in.=lll  inches, 
144in.Xlll  in.=15984  inches.  In  that  distance  the  large 
wheel  will  have  made  111  revolutions  and  the  smaller  144 
turns.      But    144 — 111=33  turns  of  the  less  more  than  the 


greater  in  that  distance. 


Now,  as  33  turns   :  15984   inches 


distance  : :  1000  turns  :  484363^5-  inches;  this,  when  redu- 
ced to  miles,  is  7  miles,  5  furlongs,  34  yards,  1  foot,  7j\ 
inches. 


SINGLE  RULE  OF  THREE.  45 

Page  63. 
Quest.  9.         This  question  being  inverse,  say 

hrs.       da.  hrs.       15^18       15x3 

As  15  :  18  :  :  12  :  ^i^^-^^ll^f =22J  days. 

12  2  '' 

yds.    dolls.cts.  yds. 

10.  As  42,5  :  191,25  •.•  1  :  4,5  :  :  15  :  67  dolls.  50  cts. 
what  15  yards  cost;  but  f  of  67  dolls.  50  cts.  =45  dolls,  the 
amount  that  15  yards  sells  for. 

Now,  42*  yards  at  1  dollar,  comes  to  42  dollars  50  cents? 
whole  gain. 

d.    cts.    d.    cts.      d.    cts. 

191  25+42,50=238,75  must  get  in  all 

Subtract  45,00  received  for  15  yards. 


Diff.  $188,75  the  sum  that  the  remaining 
27^  yards  must  bring.     But 

yds.       d.    c.      yd.    d.  c. 
As  27,5  :  188,75  ::    1  :  6,86^^ 

i  1.     State  by  the  general  rule 

yds.  yds.      ft.  ft. 

As  60  :  10  (=30)  •.•  6  :  1  : :  18  :  3  feet 

rod.     rod.  rod.    A.       rod.    rods. 

12.     As  40  :  640  (=160x4)  : :  1  :  16 

rod.      rod.  rod.  rods. 

Or,  As  40  :  160  •.•  1  :  4  :  :  4  :  16 

Page  64. 

13.  Because  the  large  wheel  has  70  cogs,  and  the  small  one 
52,  the  small  wheel  will  make  70  revolutions  while  the  large  ones 
make  52.  But  70 — 52=18  revolutions  that  the  small  wheel 
will  gain  in  the  same  time.     Then  say, 

rev.       rev.  rev.         fiO  V 1 00      revolutions. 

As  18  :  52  :  :  100  :     ^'^   =288| 

18  ^ 

feet,     sec.       feet. 

14.  1142x60=68520  the  dist.  sound  goes  in  a  minute. 

pul.        feet  put.       feet  m.  f.  yds.  ft. 

Then,  as  70  :  68520  : :  20  :  19577^=3  5  145  21 

dolts. 

15.  5+4-f3+2+l=15  the  cost  of  1  yard  of  each  sort. 

dolls.  dolls,  cts.     yds. 

Then,  as  15  :  1  : :  532,50  :  35i 


46  SINGLE  EULE   OF  THREE. 

Page  64. 
16.     Stated  by  the  general  rule  it  will  be 
mo.  mo.      dolls,   dolls. 
As     1  :  12  :  :  127  :  1524 

in.        in.  lb.      .^fiv^ 

17.     As  ,5  :  36  : :  5  :  rli±f  =360  pounds. 


lb.     lb. 


in. 


18.  As  90  :  5  •.•  18  :  1  : :  36  :  2  inches 

19.  f =,8  and  i=,5  therefore  it  will  be 
yd.     dolls.         yds.       d.     c.    m. 

As  1  :  2,7118  : :  67,5  :  183  04  6i 

day    s.  d.  qr.  days. 

20.        As    1  :  16  5  3  is    ; ;  355 
12      ^^^ 

197 
4 

789 
365 


3950 
4735 

2367 

4 )  286000* 

*Here  omit  multiplying  by  the  third 

12)    72000  number,  because  you  would  immediate- 

ly  have  to  divide  by  the  same  number, 

2|0)  600|0  to  bring  farthings. 

300Z. 

yds.  qrs.  ft.  ft.      sq.ft. 

21.  11x3=33x2^=74,25  in  a  piece. 

ft.     ft.    ft.     ft.     ft.    sq.feet. 
And  2  (25-f  15)  X  101=80x10^=840. 

sq.  ft.  Q4Q       sq.  ft.    sq  ft.     sq.  ft. 

But    840 --=840—84=756  in  the  walls. 

sq.ft.  piece    sq.ft 
Then,  as  74,25  :  1  : :  756  :  10^^  pieces 

ft.     ft.   in.  ft.     in. 

22.  As  50  :  50  lOi  : :  1287  4,  that  is 

in.        in.  in.  in.  ft.     in. 

As  600  :  610i  : :  15448  :  15718iJ=1309  lOfJ 


DOUBLE  RULE   OF  THREE.  47 


Page  64. 


h.yt.    gal.  h.pt.      gal. 
SB.     As  16  (=1)  :  15  ::  100  :  93|  gallons 

DOUBLE  RULE  OF  THREE. 

Believing  that  it  will  be  acceptable  to  some,  I  here  insert  two 
rules  for  stating  the  Double  Rule  of  Three,  in  addition  to  that  in 
the  W.  Calculator. 

Rule  First.  Place  the  three  conditional  terms  in  the  follow- 
ing order:  that  which  is  the  principal  cause  of  gain,  loss  or  ac- 
tion, possesses  the  first  place;  that  which  denotes  space  of  time, 
or  distance  of  place,  the  second ;  and  that  which  is  the  gain,  loss, 
or  action  the  third  j  then  place  the  other  two  terms,  which  move 
the  question,  under  those  of  the  same  name,  and  if  the  blank 
space  falls  under  the  third,  multiply  the  three  last  terms  for  a  di- 
vidend, and  the  two  first  for  a  divisor :  but  if  the  blank  fall  under 
the  first  or  second  place,  multiply  the  first,  second,  and  last  terms 
together  for  a  dividend,  and  the  other  two  for  a  divisor;  and  the 
quotient  will  be  the  answer. 

Rule  Second.  Work  by  two  statements  of  the  Single  Rule 
ofThree. 


Direct  Proportion, 

Page  66. 

Quest.  2.        Horses  18  J     i-^,     .  .       ^60  horses 
1  orvf     I'J  bushels,     loa  J 

days      2{))  (3b  days 

Then  ^^^^^^^^^=10x3x2=60  bushels. 
18x20 

Or  thus, 
horse3     days         bu.  10^60^36 

As  18  :  20  : :  10    Then     ,,i:     =60  bushels. 
60  :  36  1^X2^ 

Or,  by  two  statin  gs, 

horses  bu.    horses 
As  18  :  10  : :  60  :  33i  bushels. 

days   bu.      days  bushels 
And,  as  20  :  33i  : :  86  :  60  as  before. 


48  DOUBLE  RULE   OP  THREE. 

Page  66. 

Stated  thus,  Or  thus, 

3.  Men     7)     no,  u      J21  men  men  days  lbs. 
days  U\    ^^  ^^'    (  3  days  7_14— 56 

21—  3 

Then  5^2^^12^=4x3x3=36  pounds 
7x14 

Or  by  two  statin gs, 
men  men        lb.        lb. 
As         7  :  21  : :  56   :  158,  and 
days  days        lb. 
As         14  :  3  : :  168  :  36  pounds. 

Stated  thus,  Or  thus, 

4.  Students  8)  dolls.  ( 12  stud.  stu.  mo.  dolls. 

I    384    }  8—  6—384 

months  6)  (10  mo.  12—10 

Then,  ??i^l^— =48x2x10=960  dollars. 
8x6 

Or  by  two  statings,  i 

stu.     stu.       dolls,      dolls. 

As     8  :  12  : :  384  :  576,  and 

mo.    mo.        dolls. 

As     6  :  10  :  :  576  :  960  dollars. 

Stated  thus,  Or  thus. 

5.  Cwt.  20^  dolls.  ^40  cwt.  '  cwt.     mi.  dolls, 

}     25      \  20—  50-^25 

miles  50)  (100  miles  40—100 

_        25x40x100     ^-    ^    ^     ,^^  ,  „ 
Then,  — ^^^r—^ — =25x2x2=100  dollars. 
20x50 

Or  by  two  statings, 
»  cwt.     cwt.      dolls,  dolls. 

As  20  :  40  : :  25  :  50,  and 
miles    miles      dolls. 

As  50  :  100  : :  50  :  100  doUars. 

Stated  thus,  Or  thus, 

6.  DoUs.  700)  dolls.  (400  dolls.  dolls,  mo.  dolls. 

\     14     \  700—  6—14 

months    6)  (60  mo.  400—60 

Then,  ^^^^^^^-=2x4x10=80  dollars. 


DOUBLE   RULE   OF   THREE.  49 

Page  66. 

Or  by  two  statings, 
dolls,     dolls.       dolls,  dolls.  mo.    mo.     dolls. 

As  700  :  400  :  :  14  :  8  and,  As  (5  :  60  :  :  8  :  SOd. 

Statement,  Or  thus, 

7.     Men  4)  rods  (8  men  men  days  rods 

}     12     \  4—  6—12 

days  6)  (24  days  8—24 

^,  12X8X24     ^     o     a      aa        i 

Then, — - — - — =2x8x6=96  rods. 
4x6 

Or  by  two  statings, 

men  men.    rods    rods  da.  days       rods 

As  4  :  8  :  :  12  :  14  and.     As  6  :  24  :  :  24  :  96  rods. 


Inverse  Proportion, 

Stated  thus, 

Quest.  2.    Men  4  inverse)  days  ( 16  men  inverse 

(     ^      \ 
Dolls.  24        )  (384  dollars. 

Or  thus, 

men     days   dolls. 
4 3 24 

16 384 

_        3x4x384     ^    ^    ^      _^, 
Then,  =3x4x1=12  days 

Or  by  two  statings, 

dolls,   dolls,     daijs  days  men   men     days   days 

As  24  :  384  : :  3  :  48  and,     As  16  :  4  : :  48  :  12 


3.    Dolls.  24)  men  (96  dollars  men  days  dolls. 

[     4     ]  4—3—24 

day  3  inverse)           (inverse  16  days.  16 — 96 

4x96x3_4x4x3 
24x16  ■"     16 
Or  by  two  statings, 

dolls,  dolls.    Tnen    men                       days  days     men  men 

As  24  :  96  : :  4  :  16  and,        As  16  : 3  : :  16  :  3 

5 


Then,  -^^^ — r^  = — ^-z —  =3  men. 


50  DOUBLE  RULE  OF   THREE. 

Page  67. 

Stated  thus,  Or  thus, 

4.  Acr.  84  i  men  ( 100  acres  men  days  acres 

[     7     ]  7—12—  84 

inv.  days  12)  (5  days  inverse  5 — 100 

^^       7x100x12     1x20x1     ^_ 

Then,  — — — r — = — - — r— =20  men 
84xo  1x1 

Or  by  two  statings, 

acres      acres     men   men  days  days      men    men 

As     84   :    100  :  :  7  :  8i  .and,     As  5  :  12  :  :  8J  :  20 

Stated  thus,  \  Or  thus, 

5.  Men  inv.  7)  days  (20  men  inv.         men  days  acres 

}     12     \  7—12—  84 

acres     84)  (100  acres  ^20—0—100 

,^,        12x7x100     1x1x5     _.      . 

Then,  — ^^^    „  .     =— r — r-  =5  days^ 
'      20x84  1x1 

Or  by  two  statings, 
acres  days       acres      days  men  m^n      days 

As  84  :  12  :  :  100  :  14f  and,     As  20  :  7  :  :  14f  :  5  days 

Stated  thus, 
6.     Inverse  200  lb.)  miles  (20200  lb.  inverse. 

40cts.^  (6060  cents. 

lbs.   miles        cts. 

Or  thus,    200—40—    40 

20200—  0—6060 

mi.      U).        cts. 

40x200x6060     ^_     .. 

Then,    t^^omu    An     ^^^  "^"^s- 
20201b  x40c. 

Or  by  two  statings, 

cts.    mi.        cts.      miles 
As     40  :  40  : :  6060  :  6060,  and 

lb.        lb.         miles 
As    20200  :  200  : :  6060  :  60  miles. 

Stated  thus,  Or  thus, 

7.     $200)  w.da.  (300  dolls.  men  w.    d.  dolls. 

[  22  6    I  5—22    6—200 

inv.  men  5)  (12  men  inv.  12 —  0 300 

Tben,    ""''^OOxT^ — ^  ^^^  ^*^^^  ^  ^^^^' 


DOUBLE  RULE   OF  THREE.  51 

Page  67. 
Or  by  two  statings, 
dolls,  dolls.      w.  d.     w.  d. 
As     200  :  300  :  :  22  6  :  34  2  and, 

men     men  w.     d.        iv.    d. 

As     12     :    5     ::     34     2     :     14     2 

Promiscuous  Questions. 

Quest.  1.        Stated  thus,  Or  thus, 

12  oxen^  acres  (24  oxen         oxen  days  acres 
8  days       10      (48  days  12—8—10 

24—48—  0 

_       10x24x48 

Tlieii, — - —  =  120  acres. 

12x3 

Or  by  two  statings, 

ox.    ox.     acres  acres  days  days  acres 

As  12  :  24  :  :  10  :  20  and,  As  S  :  48  :  :  20  :  120  acres 

lii.         li't.         wt. 
2.     8000—4500=3500  and  9  days— G  days==3  days 

4500  cwt.^  horses  ^3500  cwt. 
inverse  6  days    )      18      (3  days  inverse 

horses  days      cict. 

Or  thus,     1 8 6 4500 

0 3 3500 

Tlien,  18x3500x6     ^^  ^ 

— 7— -r — - —  =28  horses. 
4o00x3 

Or  by  two  statings, 

cwt.       cwt.       hor.   hor.  days  days  hor.    hor. 

As  4500  :  3500  : :  18  :  14  and.   As  3  :  6  :  :  14  :  28 

cwt.  hhd.  cwt.         cwt.    hbl.   cwt. 
S.  12x9  =  108,  and  2,5x50=125. 

Stated  thus,  cwt.      mi.    doUs. 

108  cwt.  (   dolls.  (125  cwt.         Or  thus,  108— 60— 100 
eOmilesi     10''^     (300  miles  125-300—     0 

Then,  i^^^^^^=578  dolls.  70^^  cents 
108x60  ^^ 

Or  by  two  statings, 

mil.        mil.  dolls,        dolls. 

As     60     :    300     :  :     100     :     500,  and 

cwt.         cwt.  dolls.      dolls,    cts. 

As     108     :     125     ::     500     :    578     70^f 


52 


DOUBLE   RULE   OF   THREE. 


Page  67. 


Then, 


.     3  yds 
5  qrs. 
1x45x4 


Ih. 
1 


45  yds. 
4    qrs. 


Or, 


yds.  qrs.  lb. 

3—5—1 
45—4—0 


=  12  yards 


3x5 

By  two  statings, 
yds.    yds.       lb.      lb. 
As  3  :  45  : :  1  :  15  and, 


qrs.  qrs. 
As  5  :  4 


lb. 
15 


lb 
12 


5.     240  miles?   days 
inverse  12  hours  ]     12 


72oVHles 
16  hi"^.  inv. 


Or  thus, 
mil.  days  hrs. 
240—12—12 
720—  0—16 


Then 


12x720x12     12x3x12 


milss 
As-240 

6. 


240x16 
By  two  statings, 


16 


=27  days. 


miles 

720 


days    days 
12 


hrs. 
36  and,  As  16 


hrs. 
12 


days 
:  36 


27  days 


16,5  feet] 

1,5  feet 
1     foot^ 


dolls,  cts. 
1     25 


Then, 


1,25x30x26x4,5 
16,5x1,5x1 


,30  feet 
26  feet 
[4,5  feet 

dolls,  cts. 

=  177  27^3^ 


8  feet)  f200  feet  long 

4  feet>   1  cord  <10    feet  high 
4  feet)  (36    feet  broad 

8x4x4 


10  yards  j   lb.  H  00  yards 
6  qrs.    (3(3  qrs. 


Then, 


3x100x3 
10x6 


=15  pound. 


9.     Inv.  24  men 

200  ft.  long 
8  ft.  high 
6  ft.  thick 


Page  68. 

C  6  men  inverse 
days  1  20  feet  long 
80     I     6  feet  high 
L  4  feet  thick 


Then. 


80x24x20x6x4 
6x200x8x6 


16  davs 


PRACTICE. 


53 


Page  68. 

10.    9  persons)  dolls.  (14  (=9+5)  persons 
5  months  ^     450    \  8  months 

Then  ^^^^^^^^=1120  dollars 
9x5 


11.      8  persons)  dolls.  02  (=8+4)  persons 
1  month  >    IH     <  6  months 
10  dollars)  (11  dollars 

lHxl2x6xll 


Then 


8x10 


dolls. 
:111 


cts. 

37i 


12.     22,5  feet  long  "^  f  45  feet  long 

17,3  feet  wide  j  days  I  34,6  feet  wide 
10,25  ft.  deep  }   2,5  \  12,3  feet  deep 
Inverse     6  men  |  |    9  men  inverse 

Inverse  12,3  hours      J  18,2  days  inverse 

2,5x45x34,6x12,3x6x12,3,  „.     ^ 

Then  -%^33ao;^^X-8;2-^^  ^^^^^^^"^ 

2,5x2x2xl2,3^J23^^^2^^y^ 


10,25 


10,25 


PRACTICE. 


d.  cts. 
2.     1,75 

250  yards 

8750 
350 


Case  1.    Page  69. 


cts. 

Or,  50 

25 


250 
125 
62,50 

$437,50 


$437,50 

*  Cancelling.  That  is,  when  the  divisor  and  dividend  are  the  contin 
ued  product  of  a  number  of  factors,  you  can  divide  both  by  the  same 
divisor;  and  the  operation  will  be  greatly  abridged.  _ 

Thus,  in  the  above  example,  I  observe  that  22,5  goes  into  45  twice, 
without  a  remainder;  that  17,3  is  contained  twice,  in  34,6;  and  that 
9X8,2  goes  once  into  6X12,3.    Now,  by  dividing  the  numerator  and 

denominator  by  these  divisors,  we  obtain         Tnok         as  above. 
5* 


10^" 


54 


PRACTICE. 

Page  69. 

3.    201  yds. 
4,20 

Or,    201  yds, 
4 

4020 
804 

20  cts.  is 

804 
*=  40,2 

$844,20 

$844,20 

4.    2210  yards 
X  1,10  cents 

cts. 
Or,    10 

1 

2210  yards 
221 

$2431,00 

2431  dollars 

5.    2,415 
X    421  yards 

Or, 

421 
2 

2415 
4830 
9660 


cts. 

20 

20 

10  ms. 

5 


1016,715 

Or,  1016  dolls.  71  cts.  5  ms 

6.    625 
25 


5 
1 

T 
_1 
20 

1 

2 


84,2 


3125 
1250 

$156,25 

7.    8275 


cts. 
26  is  4 


842 

84, 

84,2 
4,21 
2,105 

1016,715 

Or  thus, 

625 


j$156,25 


4,4  ms. 


33100 
33100 

36410,0 


8.    8275 
5 


cts. 

4  is  ^V 
4ras.iV 


Or,  364  dolls.  10  cts. 


Or, 

8275 

331 
331 

364,10 

Or,    8275 


41375  mills. 


5  mills  is  ^^ 


41,375 
Or,  41  dolls.  37'cts.  5  ms. 


PRACTICE.  I  55 

Case  2.    Page  69. 


cwt.      qr.         lb.     lb. 
4         1         14=490 

i  =  122,5  Or,  122  dolls.  50  els. 

cwt.  qr.    lb.       lb.  cwt.  qr.  lb.      lb. 
12    2     13=1413  4.     14    2    7=1631 
X  2  X  7 
lb.                


3)2826  5x8=40)11417 

$942  $-285,425 

Application. 

Page  70- 

cwt.      lb.  hhds. 
Quest.  1.     12,5x112x6x1=3150  dollars. 


2.    60  ton 

3.     12,650  feet 

X3 

X  10 

5)  180 

126,500 

1  is  1=     6,325 

ton  Eag.        36 
60x2=      120 

f  is  1=    3,1625 
I  is  i=     1,58125 

156  Eagles 

137,56875 

Or,  1560  dolls. 

Or,  137  dolls.  66  cts.  8|ras. 

Case  3. 

dolls,  cts. 
2,             10    94 
Mult,  by         17  cwt 

dolls,  cts. 
3.            13    41 
Mult,  by          5  cwt. 

7658 
1094 

67,05 
1  qr.  is  4=3,352^ 

2  qr.  is  one  half  547 

1       is  one  half  2735  70,402J 

14  lb.  is  one  half  13675 


4       is  one  seventh       3907  70  dolls.  40  cts.  2^  ms, 

1       is  one  fourth  976 


$196,0408 
Or,    196  d.  4  cts. 


2. 


3. 


PKACTICE. 

Page  70. 

dolls,  cts,  ms. 
4.         15     00     5 
X  7 


105,035 


161b.  is  i=  2,14357+ 
21b.  is  1=  ,26794 -f 
llb.isi=       ,13397+ 


107,58048+ 


107  dolls.  58  cts.  &c. 


Case  4.    Page  71. 

s,    d.  s.  d. 

6    8  is  ^  )  473  yds.  at  6  8 


Or,  473  yards 
88|  cts. 


£157  13  4 


5.  d,  s.  d. 

3  4  is  I)  397  yds.  at  3  4 

£66  3  4 


3784 
3784 
4204 


$420,44| 


Or,    397  yards 

,44^  cts.  =35.  4d. 


4. 


1588 
1588 
^=  176| 

$176,44| 


1594Z&.  at     \s,Sd. 


Or,  159,25 


22^cts.=:\s.M. 


31850 


\s.  8d.  is  -jV  )  159  5  value  at  1£.       31850 


£13  5  5 


1=     3538f 


$35,3888| 


Or,  $35  38| 


PRACTICE. 


57 


lOs.  is  I 


2isi 


Page  71. 

658  at  125 

329 
65  16 


£394  16 


Or,  658Z6. 

imcis.=l2s. 


39480 
658 

$1052,80 


H.  105.  is  ^ 


5s.  is  i 


l5.  is  i 


745 

at 

165 

372 

10 

188 

5 

37 

5 

£596  0 


Or,  745  yards 


213ic.  =  165. 


2235 
745 

1490 

1=   248  j- 

$1589,331 


105.  is 

1 

2 

5   ' 

1 

4   « 

1 

T 

8(Z.^ 

1 
77 

2    ' 

1 

4 

1    « 

1 
2- 

969  at  195,  lid 

484  10 

242  5 

193  16 

32  6 

8  1  6 

4  0  9 


£964  19  3 


Or, 
969 
265|-ci5.=195.  Ud, 


4845 
5814 
1938 
i=    .538j 

$2573,231. 


.        55. 

j- 

4 

4 
4fZ. 

1 

T 

1 

T2 
1 

¥ 

3715  at  95.  4iJ. 


928  15 
743  0 

61  18  4 
7  14  92 


Or,  3715 

125c. =95.  4id. 


18575 
7430 
3715 


£1741     8  U 


$4643,75 


58 


PRACTICE. 


9. 


105. 

1 

2 

5 

1 

4 

4 

1 
5 

Sd. 

1 
'6' 

2 

1 

"4 

! 

1 

2 

X 

-1 

2 

Page  71. 
4567  at  195.  lUd.      Or,  4567 


2283  10 

1141  15 

913  8 

152  4 


19 


9  10  Si 


£4557.  9  8i 


266-ic.  =  195.  lU^. 


27402 
27402 
9134 
i=  507| 

$12153,291- 


£. 

89 


Case  5.  Pa?e  72. 


5.  d. 
6  8x1 
6 


536  0  0 
6 


3. 


£.  5.  <Z. 
8  11  5 
9 

77  2  9 


cwt.  3216    0 

10  is^  89    6 

4  '  ^  44  13 

2qr. '  1   17  17 

L41b.  '^24 
0  11 

0 
8 
4 
4 
8 
2 

£3370  13 

2. 

£.    s. 

4.        3     17 

6 
12 

617  2  0 


46  10  0 
12 

558  0  0 
'2qr.  is  i  1  18  9 
Ulb.   '  1  0  9  8|. 
7   *  ^  0  4  10 


£560  13  3i 


2  qv.  Is  1 

1       t      C      1 

4 

2 

5 
2 

8^ 
10^ 

L6  lb.  '  ^ 
8  '    '  i 

1   '    '  1 

1 
0 
0 
0 

4 

12 

3 

1 

5|+ 
2i+ 

6J- 

£625  11  10 

^.  R.    P. 

5.  476  3  28 
£  3  75.  lid. 


i)S. 


IS  4 


25.  6<Z.  '  1 


2r. 

1 

20p. 

8 


1428 
119 

59  10 

9  18  4 

1  13  11^ 

0  16  111 

3  41 

£1619  11  IJ 


PRACTICE. 

• 

Page  72. 

d.  c. 

6. 

10,55 

640  acres 

42200 
6330 

2r. 

is  I      5275 

20p. 

i      1318| 

$6758,5931 

59 


Or,  6758d.  59c.  3|m. 


C?.    6'. 

7. 

18,50 
229  acres 

16650 

370 

< 

370 

2r.  is 

1 

925 

1.     ' 

1 

4625 

lOp' 

1 

4 

11561 

8     ' 

1 
■5 

925 

$4252,4561 

Or,  4252d.  45c.  Q^m, 

4 


d.  ds. 
8.    6,34 
12 

Ih,         7608 
7  isJ^  3961 


$76,476^ 
Oi-,  76d.  47c.  61m, 


dolls, 
9.        14 

17  cwt. 

238 

s  J        7 
'    1        35 
i        175 

=    I          875 
'    1          4371 

10. 

2r.    is  A 

1       '1 
20pcr.'  4 

Or 

125,50 
16 

^qr.     \ 

1' 

Ulb. 

7' 
34' 

200800 
6275 
31375 
15687^ 
3921| 

$2121,7341 

$251,562^ 

Or,251d.56c.2ira. 


GO 


TARE  AND   THET. 


Page  72. 


11. 


3 


5. 

i7 


19    7 

6 
5 

^qr.  is  i 

1  '      '    J- 

14Z&.    '    1 

96  17 

1  18 

0  19 

9 

6  value  of  25  cwt 
9      '     of  2qrs. 
41    '      of    1  qr. 
8i    '      of  14  lb. 

£100    5 


TARE  AND  TRET. 


Page  73. 

cwt.  qr.  lb.         lb. 

Quest.  2.  No.    1.    2     1  25  tare  21 

<      2.    2    2  0    ^    21 

'      3.    2    0  21    '    21 


Gross      7    0     18  tare  63  lb.  =2  qr.  7  lb. 
Tare  2      7 


Neat  cwt.  6    2    11 


2qrs. 
8lb. 
2' 
I' 


d.  c. 

12,50 

6 


75,00 


6,25 

892f- 
223^3^ 

$82,477if 


cwt.  qr.  lb.     lb. 
16     1     3=1823  gross 
132  tare 

1691  neat 
Mult.    ,45  cts. 


8455 
6764 

$760,95 


TARE   AND   TRET.  61 

Page  74. 

cwt.  qr.  lb.  lb.  d.  cts. 

4.     No.  1.  11  1     17  tare  112  16,80 

•     2.  12  2      0    '      74  22 


Gross  23  3     17  tare  186  *                  33,60 

1861b.=  1  2     18  336 

im.  is  \    2,40 

22  0     27  neat  8       '  A     1,20 

2       '  i       30 

bbls.  cts.            d.  cts.  1        'I        15 

18x450     =  81,00  

Ton.  Ijxl20d.=180,00  .$373,65  cost 

.$261,00  gave 

$261,00  paid. 


$112,65  due 


cwt.  qr.     lb.  dolls. 

4     2     26  9 

12  50  cwt. 


56     3       4  gross  450 

2qi\  is  A        4,50 

Sib.  is  -jif  4     0      6+         14        '  ^        1,12J 
4      '    i   2     0       3+  •        7        '  i  561 

16+ 


6     0       9  tare 


i   1 

T 


Cwt.  50    2    23  neat 


$456,343 


cwt.      lb.  cwt.  qr.        lb. 

6.    6=672  gross  7.     120  3=13524  gross 

100  tare  177  tare 


572  13347 

^1^=  22  tret  2^=     513,346  tret 


lb.  550  neat      .  12833,654 

,73  cents 

lbs.     cts.    lb.  8250  ^^  

As  550  :  8250  : :  1  :  ^^^=15      38500962 

89835578 


9368,56742 
Or,  9368  dolls.  56  cts.  Tms. 


62  SIMPLE   INTEREST. 

Page  74. 
cwt.  qr.  lb. 

W.         6    2     12  16    dollars 

9  50  cwt. 


59     1  .  24  gross  800 

Iqr.  is  ^         4 

16Z6.  is  4  8     1     27  Ulb.  '  ^        2 

1       SV       2      3  7^1-1 


9    0      2  tare 


1       '  I        0,14f 


Cwt.  50     1     22  neat 


$807,14f 


lb.    cwt.  qr.  lb.     bags     lb. 
9.     325(=2  3  17)  x27=8775  gross 
8775Z&.XtV2  =1018  tare 

7757 
7757-7-26=  298  tret 


7459  lb.=66c2r^.  2qr.  Ulbs 

lbs.       ct.      dolls,  cts. 
And  7459x32=2386,88 

SIMPLt:  INTEREST. 

Case  1.    Page  75. 

doUs.  dolls,  yr.     cts^ 

4.  200x6x2=2400=24  dollars. 

Page  76. 

dolls,  dolls,  yrs.    cts.        dolls,  cts. 

5.  1260x7x4=35280=352,80 

dolls. dolls. yrs>     cts.        dolls,  cts. 

6.  560  x8  x3= 13440= 134,40 

d.    cts.      d.         d.  cts. 
Now  134,40+560=694,40 

d.        d.  yrs.        cts.  d.    cts. 

7.  4520x5x6=135600=1356,00 

d.     cts.        d. 
And  1356,00+4520=6876  dollars. 


SIMPLE  INTEREST.  63 

Page  76. 
dolls,  d.  yrs.    cts.  d.       d.        d. 

8.    400x61x2=5200=52  dolls.      And  52+400=452 

dolls,    dolls,  yrs.     cents  d.      cts.  ms. 

9.  4925x7dx9=332437i=3324  37  5 

dolls,    dolls.        cts.        dolls,  cts. 

10.  2500x71=19375=193  75 

d.         c.        d.  d.      cts. 

And  193     75+2500=2693  75 

Case  2.    Page  77. 

dolls,  c.  ms.  d.    cts.  ms. 

4.     1260  50  5  Now  283  61  3  + 

7i  1260  50  5 


8823535  $1544  11  8  amount 

6302525 


cents  9453,7875  interest  for  1  year 
3 


28361,3625 

Or,  283  dolls.  61  cts.  3+ms.  interest  for  three  years. 

dolls,  cts.  dollars  cts.      d.     yrs.      d.       cts.  ms. 

5.     630  50         6.     3462  84x3^x4=450     1692  int. 
8  1274  646x7^x3=286     79535  int. 


50  4400  int.  for  1  year  163     37385  diff, 

5 


252  20  int.  for  5  years 

630  50  principal  '       dolls,  cts.  ms. 

Or,  163    37  3,85 


^S82  70  amount 

dolls,   cts. 
7.    3  )  3422  25 

dolls,  cts.  ms. 

.  1140  75Xrlo=68    44    5  int.  for  1  year 
68  44  5  interest  for  one  year 

1209  19  5  first  payment. 


64  SIMPLE  INTEREST. 

Question  seven  continued. 

dolls,  cts.  dolls,  cts. 

d.c.m.yr.    1140  75  d.c.m.  1140  75 

68,44,5x2=  136  89  int.  for  2  yrs.  68,44,5x3=  205  335 


$1277  64  2d  payment  3d  pay't.  1346  085 

d.     c.  ms. 
Now,  1209  19  5  first  payment 
1277  64  0  second     ' 
1346  08  5  third        ' 


$3832  92  0  whole  amount 


Case  3.    Page  78. 

4.         18  months  Then,  724  dollars 

4  per  cent.  6 


12  )  72  $43,44  iiiterest 

724,00  principal 


6  rate  for  the  time 


$767  44  amount 


5.         15  months  d.  c.  m. 

Ik  per  cent.  Then,  9  375 


240  dollars.. 


12)112^ 


375000 


9,375  rate  for  the  time      18750 


22,50000 
Or,  22  dollars  50  cents. 

6.         4  months  dolls. 

6^  per  cent.  Then,  1260 


12)26 


21 


2520 
2^  rate  for  the  time  210 


$27  30  cents. 


SEVIPLE   INTEREST.  65 


Case  4.     Page  80. 

dolls.      days      (constant  div.)d.       cts. 

5.  (12000x40)  —  5214  =  92     06  nearly 

dolls,     days     (c.  d.)      d.      cts.    ms. 

6.  (8400  x20)h- 7300=23     01     3 

d.     cts.    days     (c.  d.) 

7.  (517  90x84)-^60S-:i=7  dolls.  15  cts.  1  m. 

d.    cts.  days    (c.  d.) 

8.  (73  41  x27)-=-60=33  cts. 

dolls,  cts.  days      (c.  d.)      d.     cs-  m. 

9.  (225  24x40)-^60S3=l  48  1 

dolls,     days      (c.  d.)     d.     cm. 

10.  (i200x80)-^6083=15  78  1 

dollars  cts.     days       (c.  d.)      d.      cts.  m. 

11.  (2962  19x254)-i-60S3=123  68  8 

d.      cts.    days      (c.  d.)      d.     c.  m. 

12.  (1733  97xl02)-T-6083=29  07  5 


Cases.    Page  81. 

yrs.    d.        d.         d. 
2.     (12x6)  +  100=172     Then, 

dolls,      (lolls.         dolls.       2752x100      'Jo'ls- 
As  172  :  100  :  :  2752  :  ^^^^^^'^^  =  1600 

yrs.  d,        d.  d. 

3.         (5x6)  +  100=130  Then, 

dolls.        dolls.  dolls.       79,Svl00       doU*- 

As  130  :  100  : :  728  :  lfl^=560 

130 

yrs.    d.         d.  d. 

4.         (4x74)4-100=130  Then, 

dolls.        dolls.  dolls.         IRSSvlOO        dolls. 

As  130  :  100  :  :  1638.  :         ,il        =1260 

loO 

yrs.   d.  cts.        d.  d.    cts. 

5.         (5x5  75)+ 100=128  75     Then, 

dolls,  cts.        dolls.  dolls.         2000  V 100        ^o'^'-    c^*-      d"* 

As  128  75  :  100  : :  2000  :  Vooll  =1553  39  8^ 

l^o,7o 

6* 


66  SIMPLE   INTEREST. 


Case  6.    Page  82. 

dolls.  dolls.       dolls.  dolls.       90800    ^olls. 

2.     858  amount     As  650  :  208  : :  100  :  -— —  =32 
650  principal  ^^^ 

Then32d.-4-4yrs.=8per.  cent. 

$208  interest 

dolls.  dolls.  dolls.  dolls.       l  "I  ^900    ^°^^- 

3.     2752  amount     As  1600  :  1152  : :  100  :   -11-11=72 
1600  prin.  1^^^ 


$1152  interest.  Then  72  dolls. -^  12  yrs.=6  dollars 

dolls,  cts.  dolls.        dolls.  cU.          dolls.       22080      ^°^ 

860,80  amoi  •     -—     -....  ^^ 
640,00  prin. 


4.     860,80  amount     As  640  :  220,80  : :  100  :  — --  -=341 

640 


,80  interest.     Then  34i  d.-i-6  yrs.=5|  per  cent. 

dolls.  dolls.  dolls.  d.  filQ       dolls. 

5.     20100  amount     As  12000  :  8100  : :  100  :  -nr-=67i 
12000  prin.  ..  ^^ 


$•  8100  interest.     Then  67^  d.-i-15  yrs.=4i  per  cent. 


Case  7.     Page  83. 

s 

dolls.  dolls. 

2.    650  910  amount 

8  .     650  principal 

$52,00  int.  for  1  year       $260  whole  interest 

dolls,   year        dolls.       9^0 

Then,  as  52  :  1  : :  260  :  -^=5  years 


dolls.  dolls. 

3.     1600  2080  amount 

6  1600  principal 


$96,00  int.  for  1  year        $  480  whole  interest 

dolls,  year         dolls.       4g0 

Then,  as  96  :  1  : :  480  :  -q^=5  years 


SIMPLE   INTEREST. 


Case  8.    Page  83. 


67 


2.     640£.  3^.  M—284^  dollars* 


9 

5 


§142,26|-  interest  for  1  year 
3 


426,78|-  interest  for  3  years 
2845,22|  principal 


$3272,001-  amount* 

£.       s.   d.  £.    s.     d. 

3.     1374     19  77     5     10  in'st.  for  1  yr. 

rate  per  cent.    5|  38  12     11     '      for  1  yr. 

6870     8     9         £115  18       9  int.  for  U  yr. 
|isi        687     0  10| 


171 

15 

2^ 

■^2 

£77,29 
20 

4 

10 

s.  5,84 
12 

d.  10,18 

Case  9.     Page  85. 

2.        1820,  January.  Principal  1800  dollars 

dolls.  days 

1821— April  1.    1800  Mult,  by    455  equal  819000 
paid  700 

1822— Jan.  1.      1100  Mult.         275  equal  302500 

paid    400 

700  Mult.         181  equal  126700 

July  1,  paid  500 

200            Mult.         184  equal    36800 
214  16  6  


6,0 )  128500,0 


Amount  due  $414  16  6 

Interest  $  214,166 

*  The  answer  given  with  the  question  is  obtained  by  reckoning  $4,44 
to  be  a  pound  sterling,  instead  of  $4,44* 


OS  I^'3URA^TE. 

COMPOUND  INTEREST. 

Page  87. 

tabular  number    dolls.         d.      cts. 
3.         1,27628x1500=1914  42 

tabular  no.        d.  d.         c.    m. 

-L         2,54035x4500=11431  57  5  amount 
subtract     4500  principal 


§6931  57  5  interest 

tabular  no.      d.  d.  cts. 

5.  3,20713x650=2084    03  4,5  mills 

yrs.  yrs. 

6.  21 — 4=17  years,  on  interest.     Then, 

tabular  no.       d.  d.         cts. 

2,69277x8000=21542  16  amount 

d.        cts.    sons 
But  21542  16-7-3=7180  dollars  72  cents 


INSURANCE. 

Page  88. 

dolls.  dolls. 

1650  3.         4500 

15J  per  cent.  25     per  cent. 


8250  22500 

1650  80 

825 


$255,75 


$1125,00 


5.  100  dollars — 25  dollars=75  dollars.     Then, 

dolls,      dolls.  dolls.         450000         ^°^^^- 

As     75  :  100  : :  4500  :   — — — =6000 

7o 

6.  100  dolls.— 9  dolls. =91  dolls.     Then 

dolls.       dolls.        dolls.      56000      ^°^^^-        <='»• 

As     91  :  100  :  560  :  —---=615     384| 


2. 


BROKAGE. 

COMMISSION. 

Page  88. 

dolls. 
1260 

6  per  cent. 

3.        2550 

4  per  cent. 

|75,60                                  $102,00 

dolls.                                        dolls. 
26342                            5.         6422 

3  per  cent.                •              3  per  cent. 

69 


.$790,26  4)19266 

$48,16^ 

Page  89. 

7.         100  dollars. 
3 

dolls.        doib.      412000     ,^^„  ,  „ 

As    103  :  100  : :  4120  :  =4000  dolls 

dolls,    bar.         dollars       4000 

Then,  as  8  :  1  : :  4000  :  --—=500  barrels 

'         !  8 

BROKAGE. 

d.   cts.  dolls. 

2.         1625,50  3.         186S 

3i  per  cent.  2i  per  cent. 


4876,50  3736 

541,83i  934 


5418,33*  $46,70 

Or,  54  dolls.  18*  cts. 

4.         560  dollars 
6  per  cent. 


$33,60  cents 


70 


REBATE  OR   DISCOUNT. 


BUYING  AND  SELLING  STOCKS. 

Page  90. 

dolls,  dolls.  dolls. 

2.     1500  Or,  1500  3.     1686 

110  per  ct.     -jV^    150  x    128  per  cent. 


$1650,00  $1650  13488 

3372 
dolls.                                           1686 
4.        25000  


lOS  per  cent.  $2158,08 


$27000 


1260  6.     9254  7.    isis 

90  per  cent.  84  per  cent.  83|  per  cent. 


$1134  $7773,33  $1271,321 


REBATE  OR  DISCOUNT. 


Page  91. 

mo.     d.       mo.      d. 
2.       As  12  :  8  : :  18  :  12  int.  for  the  time     Then, 

d.        d.           d.           d.    ^           2464x100      ^^^l^^ 
As  112  :  100  ::  2464  :  2200^^^^"^^ iT^ =^'200 


112 


mo.      d.        mo.    d. 


3.       As  12  :  51  : :  8  :  3|  interest  for  the  time 

d.  d,  d.      cts.         d.       cts. 

Then,  as  103f  :  100  :  :  1857  50  :  1791  SOj^-, 

For  185750x100x3     JQJ!^-    -.^^ 
^""^  103|x3        -^^^^  ^^^^^ 


4.         (7x2)4-100=114  dollars 

dolls,      dolls.        dolls.        fi50vl4      ^o^l'-  c*'- 

Then,  as  114  :  14  : :  650  :  ^^-^^=79  82f4 
'  114        '^ 


REBATE   OR  DISCOUNT.  71 

Page  91. 

mo.      mo.        d.     d. 
5,     As  12  :  4i  : :  8   :  3=  a  rate  for  the  time 

TU  *^ino       Sn  if^n       5150x100       dolls. 

Then,  as  103  :  100  ::  51o0  :   — — =5000  p.  w. 

Again, 

dolls.        dolls. 
As  100  :  99  (=100—1)  : :  5000  :  4950 

Page  92.    Note  1st. 

mo.     mo.     dolls,  dolls. 
3.     As  12  :  18  :  :  8  :  12  rate  for  the  time 
dolls,     dolls.        dolls.      dolls. 
Then,  as  112  :  12  : :  7280  :  780  the  discount 

doUs.    mo.     cents. 
Again,  7280x9=65520  interest  at  6  per  cent. 
^=21840  interest  at  2     «     ' 


$873,60  interest  at  8  per  cent. 
Subtract  §780,00  discount 

Difference  $  93,60 

Note  2d. 

dolls.  cts, 

2.         1650x3=4950=49  dollars  50  cents, 

3.     2464  dollarsx8=19712  cents  discount 
But  2464  doUars— 19712  cents=2266  doUars  88  cents 

BANK  DISCOUNT. 

Page  93. 
days            dolls. 
„        (30+4)  x250 
'^'      gQ =1  dollar  41|  cents 

^     17x250     ,^„ 

^r,  — ~ —  =1  dollar  41f  cents 

days  dolls. 

«        (90+4)  X600     o  ^  ,,       , 

^-       60 =9  dollars  40  cents 

47  days  X600  dolls.     ^ 
^^' "30 =9  dollars  40  cents 


72  SINGLE   FELLOWSHIP. 

Page  93. 
days  doUs.  cts. 

4      (55±l)25l22i±^=13  dollars  4411  cents. 

^      32x1260  40     ^_  ,  ..       .,32        ^ 
Or, =13  dollars  44ff  cents. 

days  dolls,    cts. 

5.    (iiyi)2i?^^=28  dollars  26  cents  4  mills 
60 

^     32x2649  75     ^^  -  ,,       oa       .    i      n 
Or, =28  dollars  26  cents  4  mills 

Ox) 

EQUATION  OF  PAY3IENTS. 

Page  94. 

dolls,  mo,  dolls,    mo. 

2.    500x2=  1000  3.  200x  5=1000 

1000x5=  5000  300 X  8=2400 

1500x8=12000  500x10=5000 

mo.  da. 

3000  )       18000  (6  mo.  1000    )       8400(8  12 
18000 

4.     1400  dollars— 1000  dollars =400  dollars 
d.  d.  mo. 

Then,  as  400  :  1000  : :  1  :  21  months 

SINGLE  FELLOWSHIP. 

Page  95. 

2.    A  7500+B60b0+C45d0=]8600,  Then, 

dolls.       dolls. 
dolls,      dolls.  (7500  :  2250  A's  gaini 

As  18000  :  5400  or  10  :  3  : :    ^6000  :  1800  BV    '     \ 


<uuuu   ;   J.OWW  jj  3-     •      / 
(4500  :  1350  C's     '     ) 


d.       cts.  d.    cts.  d.      cts.      d.  d. 

3.     A  1291  23+B  500  37+C  709  40+228=2729 
d.  d.       cts.  d.         d. 

Now,  as  2729  :  2046  75  : :  100  :  75 

d.    cts.      d.    cts. 

/1291  23  :  968  4•2A^ 

Then,  as  100  :  75  ::  )  500  37  :  375  27;]  ( 

Or,    4  :  3  ::  \   709  40.;  532  05  ( 

(  228  Oa  :  171    ) 


SINGLE   FELLOWSHIP.  73 

Page  95. 

dolls.     A.  R.  P. 
dolls.     A.  R.P.  (180  :  175  2  344  ) 

4.     As  600  :  5S5  2  34  : :       {195  :  190  1   17^V^ 

(225  :  219  2  22|  ) 

Page  96, 

barrels 
bbl.       bbl.  (960  :  400) 

5.  As  2160  :  900  •.•  12  :  5  :  :     ^20  :  300[ 

(480  :  200) 

dolls.  d.         d.  dolls,     d.        d. 

6.  A  1260+B  840=2100,        And  825—275=550 

d.         d. 
d.  d.         i  1260  :  330  A'sgain) 

Now,as2100:550::J    840  :  220  B's    ■      \ 

B's  gain.  B's  stock.  d.  d. 

And,  as  220  :  840  •.•  11  :  42  :  :  275  :  1050  C's  stock. 

dolls.         dolls,  dolls. 
7.     (800x2)+40=1640  whole  gain,  And 

d.  d.  d.        d. 

A's  140+B's  260+C's  300=700 

Then,  800  dollars— 700  dollars=100  dollars  D's  stock.       ^ 

dolls,    dolls. 
,  „        ,  jj  rl40  :  287  A's  gain^ 

(lolls.       UOllS.  I  OAn    •    f^QQ    TV-i      ( 

Now,  as  800  :  1640  V  20:  41::  ^^60.  533   B.     ^       ^ 

LlOO  :  205  D's 

8.  d.  cts. 
cattle  dolls.                       (  80  :  25  60  A  pays  ] 

As  300  :  96  •.•  25  :  8  :  :    hoO  :  32  00  B    <■ 

(120  :  38  40  C     *     ] 

9.  First  24-3+5=10  dollars,  Then 

dolls, 
dolls,  dolls.  (2  :  1000  A  gets  ) 

As  10  :  5000  •.•  1  :  500  : :  ^3  :  1500  B     «     } 

(5  :  2500  C    '     S 


I 


74 


COMPOUND   FELLOWSHIP. 


Page  96. 

10.     20  dollars +85  dollars  «=  105  dollars  whole  stock 
d.        d.        d.       d. 
Now,  as  105  :  63  :  :  20  :  12  A's  gain 

d.       d.      d.  d. 

And  63— (21 +  12) =63"- 33 =30  C's  gain 

d.      d. 
dolls,  dolls.        (21  :  35  B  put  in 
Then,  as  12  :  20  :  :  ^ 

^30  :  50  C  putin 

COMPOUND  FELLOWSHIP. 

Page  97. 


d.     mo. 
2.     600x8=4800 
200 


d.      mo. 
1200x10=12000 
600 


800x8=6400 


A  11200 
1000x12=12000 
800 


1800X  4=  7200 


C  19200 

doUs. 
Now,  as  46000  :  2300 
Or  20       :       1 


600  X  6=  3600 


B  15600 

11200  A 
15600  B 
19200  C 


46000 


(11200  :  560  dolls.  A's  share 
h5600  :  780  dolls.  B's     ' 
(19200  :  960  dolls.  C's    ' 


3. 


mo.  mo.      dolls,  dolls. 
As  9  :  12  ; :  600  :  800 


Page  98. 

mo.   mo.         d. 
4.     As  8  :  12  : ;  1800  :  2700  dollars 


mt.         mi.  mi.         mi. 

5.     A  600+B  600+C  300+D  100=1600  miles 


mUes  dolls. 
As  1600  :   160  •••  10  :  1  : 


:  < 


f  600:  60  dolls.  A  pays 

600:60    <      B    ' 

300:30    '     C    ' 

L100:10     *     D    ' 


PROFIT  AND  LOSS.  75 


Page  98. 

6.     5  oxenx4l  months =22^ 

8  x5  =40 

9  x6^  =^5Si 


Sum  121 

d.    cts.       (222  :  27       '  A  jmys 
Now,  as  121   :  145  20  :  :   ko     :  48         B     '' 

[bSk  :  70  20  C     ' 

7.  4000  dollars  xl2  months=48000 
3000  '  Xi5  ^  =45000 
5000      '        X  8      '       =40000 


133000 

d.  (48000  :  240  dolls.  A's  part 

As  133000  :  665  : :   {45000  :  225      '      B's     ' 
Or,   .     200  :     1  (40000  :  200     '      C's     ' 

8.  580  dollars  x6i  months  =3770 
580  '  x8i  '  =5510 
870      '        x8|       '       =7540 


d.    cts.       (3770 
As  16820  :  263  90  : :   ^5510 


^7540 


16820 

59d.  15c.  A  lost 
86      45     B     ' 
118     30     C    ' 


PROFIT  AND  LOSS. 


Pa?e  99. 


2.  150  cents,  sold  for 
120  cents,  cost 

30  cents,  gain  on  120  cents 

cts.     cts.         d.       d. 
Then,  as  120  :  30  :  :  100  :  25  per  cent. 

yd.  cts.      yds.     d.  cts. 

3.  As  1  :  66  :  :  42  :  27  72  sold  for 

Subtract  21  00  cost 


Rem.     $  6  72  gain 


76  PROFIT   AND   LOSS. 

Page  99. 

4.  32  galls.  x6  barrels =192  gallons,  bought 

Subtract     12  gallons  leaked  out 

Rem.     180  gallons 
And  96  dollars +12  dollars =108  dollars  sold  for 
galls,      d.         gal.     cts. 
Then,  as  192  :  96  : :  1  :  50  cost  per  gallon 

g.  d.  g.      cts. 

And,  as  180  :  108  :  :  1  :  60  sold  for  per  gallon 

5.  20  cents — 17  cents  =3  cents,  loss  on  each  knife 

knife  cts.    knives       doz.       d.  cts. 
As  1   :  3  ::   1440  (  =  120)  :  43  20 

Page  100. 

6.     149  dollars  7.     100  dollars 

51       "  18      " 

yds. yd.    d.  d.  —      cts.     d.  c.  ma. 

As  100  :  $200  :  :  1  :  2  As  100  :  $S2 :  :  125  :  1  02  5 

d.      d.     cts. 
8.     As  100  :  120  : :  90  :  1     08  must  sell  for 

108  cts. — 90  cts. =18  cts.  gain  per  hat 

hat    cts.        hats  doz.  d.   cts. 

Now,  as  1  :  18  : :  240  (=20)  : :.  43  20 

9.     115  gallons  X 110  cents =126  50  cost 
Add  5  00 


$131  50  must  sell  for 
But,  as  1  doll.  :  1   gall.  ::  131  dolls.  50  cts.  :  18U  galls, 
the  number  of  gallons  that  must  be  sold.     Consequently  1311: 
gallons — 115  gallons  =  16i  gallons  of  water. 

dolls,  cts. 
10.  lb.  cwt.  cts.     134  40  cost 

448     (=4)  x25=  1 12  00  sold  for 

$22  40  whole  loss 

U).      dolls,  cts.        lb. 
Then,  as  448  :  22  40  : :  1  :  5  cents  per  lb. 


BARTER.  77 


Page  100. 


*  dolls.      doUs. 

11.  As  120  :  100  : :  1728  :  1440  whole  cost 

yards     dolls.        yd. 
Then,  as  360  :  1440  :  :  1  :  4  dollars 

12.  $4  90— $4=90  cents;  Then, 

dolls,    cts.        dolls.       dolls. 
As  4  :  90  : :  1000  :  225 

d,  d.  cts. 

13.  As  120  :  1  : :  150  :  1  25  per  bushel 

d.  cts.    bu.        d.    cts. 
Now,  as  1  25  :  1  ::  37  50  :  30  bushels 

14.  5000  galls,  at  48  cents=2400  dollars 
Interest  for  2  months  =     32      ' 


Costs  $2432  to  pay  cash 
5000  galls,  at  50  cents =$2500  in  two  months 


Diff.  $     68  gained  by  borrowing 
the  money  and  paying  the  cash, 

head  doUs.  dolls. 
15.     12  )  240  (  20  the  cost  per  head 

cts.    lb.      dolls,      lb.       C.  qr.  lb. 
Then,  as  4  :  1  : :  20  :  500=4  1  24 

BARTER. 

Page  101. 

2.     320bushelsx$l  20=$384 

Subtract     160  cash 

$224 
cts.     lb. 
Then,  as  20  :  1  : :  224  doUars  :  11201b. 

cts.      cts.         hu. 

3.  As  70  :  125  : :  28  :  50  bushels 

cts.      cts.  m.  cts. 

4.  As  250  :  319  : :  23a  :  29  9^3.,  or  30  nearly 

d.  cts,      d.  ets. 

5.  As  100  :  120  ::  1  25  :  1  50 


78  EXCHATVGE. 


Page  102. 

d.    cte.      d.    cts.         d.   cts.       9  RO  V 1  20       d.  cts.  ms. 

6.  As  2  50  :  2  80  :  :  1  20  :  ±^^^^=1  34  4 

2,50 

cts.      cts.         bu. 

7.  As  99  :  90  :  :  240  :  218/^  bushels 

cts.      cts.         cts.      d.  cts. 
And,  as  90  :  95  :  :  99  :  1  04  5  mills 


oz.      d.   cts.  oz.  lb.  oz. 

S.     As  16  :  1  28  :  :  420  (=26  4)  :  33  dolls.  60  cts. 

cts.  lb.       d.  cts.      lb. 
'  Then,  as  G  :  1  : :  33  60  :  560=5  cwt. 

cts.    cts. 
9.     As  22  :  3  (=25—22)  :  :  100  ;  13-/^ 

And  13^7^+10=23^^- 
Then,  as  100  :  123-/^  : :  28  :  34|f  cents 

10.    250  barrels  x6  dolls.  25  cts.=1562  dolls.  50  cts. 

lb.       cwt.      cts.  dolls,  cts. 

And  448  (=4)  at  30=134  40 

641b.  at  $1  75=112  00 

25  y.  at  $6       =150  00 

Cash  206  10 


$602  50 


doUs.    cts.    doUs.  cts.     dolls.  d.    bbl.         d.         IM. 

But  1562  50—602  50=960     Then,  as  8  :  1  : :  960  :  120 


EXCHANGE. 

Case  1.    Page  104. 

d.         s,        d.  s.  d.  £.        £. 

2.     As  96  (=8)  :  90  (=7  6)  : :  256  :  240 

Or,    £256 

1^- 


^---^16  subtract 


$240 


JEXCHANGE.  79 

Page  104.  ' 

d.  s.  d.        d.  s.  d.  L,  L.     s.  d. 

3.     As  1)0  (=7  6)  :  56  (=4  8)  :  :  1500  :  933  6  8 

^^^£1500x31     J-    s.    d.     ^ 


6. 


5 

^=^\JOO     u     o 

AJ 

ibvver  as 

ueioru 

s. 

As  6  : 

s.       £■     £. 

8  :  :  180  :  240 

Or, 

1 

£180 
=  60 

£240  as  before 

As 

d. 
56 

s.  d.       d. 

(=4  8)  :  72  (  = 

s. 
=6) 

£. 

::  280 

£• 

:  360 

Or,     £.280 

f  =  80  add 

£360  as 

before 

As 

d. 

72 

s.      d.        s.  d. 

(=6)  :  56  (=4  8) 
Or,     304 

1=33  15  6| 

£. 

::  304 

£•  *. 

:  236  8 

lOf 

270     4  bi 
33  15  6f 

£236     8  lOf 

Case  2.    Page  105. 

£.    5.  d.  shillings  s.         s.    dolls,  cts. 

4.  468  19  6=9379,5         Then,    9379,5^-6=1563  25 

£.    s.      pence  d.  d.     dolls,  cts. 

5.  750  10=180120    Then,  180120—90=2001  33^ 

Case  3.    Page  108. 

£.    s.  £. 

5.    540  15=540,75    Then, 

£540,75x$4,10=$2217  07i 

.      2217  dolls.  07  cts.  5  ms.     _.^_     _^_  /• 

^'     A  ^  u     in    . =540,75=540  15 

4  dolls.  10  cts.  ' 


80 


ALLIGATION. 


Page  108. 

£.    d.  cts.       d.    cts. 
7.     320x4  44=1420  80  at  par 
12^8-1=  177  60  add 


$1598  40 

Page  109. 
8.     $3259-H$4  10=:£794,878,     Then, 

£•  £•  £. 

As  108  :  100  : :  794,878    :  735,9999,  or  736  nearly 
Otherwise, 

',^3259x100  £"     s.  d. 

$4,10x108'==™^^^^'"^^^  ^^  ^^^  /A,  or  736  nearly. 

ALLIGATION. 

Case  1. 

2.  20  lbs.  at  10  cents=r  200 
30  "  at  15  "  =  450 
40    "    at  25     "    =1000 


90  cts.  1650 

lb.       cts.        lb.     cts. 
Now,  as  90  :  1650  :  :  1  :  18i 

3.  10  bushels  at  150  cents=1500 
20  "  at  160  "  =3200 
30      "       at  170       "  =5100 


60  9800 

hu.      cts.        bu.    cts. 
Now,  as  60  :  9800  :  :  1  :  163i 

Page  110. 

4.    4  ounces  at  75  cents =300 
8      «        at  60     «    =480 

12  780 

oz.     cts.       oz.    cts. 
Now,  as  12  :  780  : :  1  :  65 


ALLIGATION. 


81 


2. 


Case  2.    Page  110.    - 

1+2=3  gallons  at  3  dollars 
1       =1       ^         at  5       ' 
1        =1       '         at  6      ' 


Or, 

6  busliels  of  rye 
6         ^       of  corn 
24        '      of  oats 


Or, 

5 

1 

5 

1 

15+5=20 

4 

cts. 


cts. 


4. 


\st.  rl20   1 20  at  120 
10      110 


100    "o) 

L  80 


10 
20 


90 

80 


2d. 
100 


ri20N,    .  10+20=30  at  120 
IIOA  =20      110 

90   j  =20        90 


I  80 


20+10=30 


80 


M 


f  120> 

L  80 

100 


cts. 

cts. 

10  at  120 

4tJi.      020  \                =10  at  120 

inn   )llOx)   10+20=30       110 

^^^   )   90 >'^\20+ 10=30        90 

(   80     /             =10        80 

20      110 

\20        90 

/lO        SO 

120  X.  10+20=30  at  120  cents 

110\)]             =10       110 

90/-^  I20n 
80     ^ 

f-10=30        90 
=20         80 

6th.    (  120  =20  at  120  cents 

110\sM0+20=30       110 
90/)  =10        90 

80  //20+ 10=30        80 


100 


1th. 

100 


120 
110 


10+20=30  at  120  cents 
90r  |J20+10=30 


80 


,K  110+20=30 

'^)  120+10=30 

^-^20+10=30 


110 
90 

80 


Case  3.    Page  112. 

2.         C  48     ,,  =4 

=  4 

=  4 

'26+14+8=48  given, 

Now,  as  48  :  246tt  : :  4  :  2  bushels  of  each  sort. 


82 


3. 


Now,  as  2 


REDUCTION  OF  VULGAR  FRACTIONS. 

Page  112. 

=  2 
=  2 
=  2  quantity  given 

=  2 
22+6+4+2=^34 

2  :     10  lb.  of  alloy 

10         of  16  carats  fine 
10         of  20       *        * 
170         of24       «        ' 


lb. 
10 


134 


2. 


Case  4. 


15 
17 


122/-^ 


2 
2 
2 

5+3+2=10 


Then,  as  16  :  40 


5   \^f 


gals. 


Then,  as  6  :  70 


16 

5  oz.  of  15  carats  fine 
5        ofl7      ^         « 
5        ofl8      '         ' 


r  2 

LIO  :  25        of  22 


5  :  585  galls,  of  wine 
1  :  1  If  galls,  of  water 


6 


2. 


REDUCTION  OF  VULGAR  FRACTIONS. 

Case  1.    Page  114. 


144)  216  (  1 
144 


com.  measure  72)144(2 
144 


Now,  72-)!^ J  (=1 


(8)      (9) 
144     18 


Or, 


216     27 


.=|.  as  before 


3.    75  )  125  (  1 
75 


Then,25)//3(= 


50  )  75  (  1  (5)     (5) 

.  50  Or,  ^VT=2-f-=T  as  before 


com.  measure  25  )  50  (  2 


REDUCTION  OF  VULGAR  FRACTIONS.  83 

Page  114. 

4.  4800  )  10800  (  2     Then,  1200  )  J-J^-^W  {=^ 

9600 

(12) 

com.  measure    1200)4800(4  Or,  t^||-S  J=|  as  before 

4800 

5.  91)  117  (  1 

91 

-  Then,  13  )  ^Vt  (  =i 

26  )  91  (  3 

78 

com.  measure    13)  26  (  2 
26 

6.  9876  )  88884  (  9  Therefore,  9876  )  ^%%\%  =| 

88884 


Case  2.     Page  115. 

12x17+15     2^^ 
2.     r— =  TT   ^action  required 

4.      — =  8|.2  9  fraction  required 

5 . = 2  7  3_7  0  3  fraction  required 


Case  3 

2.  3S48-T-21=:1832\  equivalent  mixed  number 

3.  2465-T-7=352-f  equivalent  mixed  number 

4.  961  -r- 17=56  j^Y  equivalent  mixed  number 

5.  8229-r-16=514-Y*^  equivalent  mixed  number 

Case  4.    Page  116. 

2.         3x5x6=  90  first  numerator 

4x4x6=  96  second  numerator 

5x4x5=100  third  numerator 
And  4x5x6=120  common  denominator 

Therefore,  ^^q,  ^\%  and  \^^,  are  the  fractions  required 


84 


EEDUCTION   OF  VULGAR   FRACTIONS. 


Page  116. 

3.         1x5x15x9=  675  first  numera tor 
3x3xl5x  9=1215  second     * 
4x3x  5x  9=  540  third 
5x3x  5x15=1125  fourth      < 
And  3x5x]5x  9=2025  common  denominator 

Therefore,  ^VA^  lUh  //2T  ^nd  Ui^  ^re  the  frac.  required 
Case  5. 
2.         4)3     8     12     20 

3)3     2      3       5 

12       15 
Now,  4x3x2x5  =  120  common  denominator 
Tiien,  120  com.  denominator 


Divisors  <J  ,  n 
L20 


40x2=80-) 

15x3=45  'tvt 

10x4=40  ^  ^^  numerators 

6x5=30J 


TTptipp      8  0       4-5       4  0     nnfl     3  0 

Page  117. 
3:        3)3     5     15     9 


5)1 


5     3 


1113 


Now,  3x5x3=45  common  denominator 
Then,  45  com.  denominator 


Divisors  - 


3 

5 

15 

9 


15x1=15-) 
o     .     TA  !>  New  numerators 


2. 


3X4=12  [^^' 
5x5=25  J 
TTpnrp    ^i.    2  7   12    anrl  2 5 

Case  6. 
7x4x  9=252  num.         Or,      3 
8x6x10=480  denom.     7x^X^ 


7x3 


Hence,  |f  J=|J 


§{    0  10     2x2x10 
2    2 


=21 


REDUCTION  OF  VULGAR  FRACTIONS.  85 

Page  117. 

Or, 


3.     5x4x3==  60  num.  5x^X^ 5____  ^_ 

9xax4^2S8denom.       ^'0    ^    4~3x2x4     ^"^ 
Hence,  5Y^=/:i^  3    2 

4.  5x4x11=220  num.  5x^Xll_  5x11  _  ^^ 

9x7x12=756  denom.     ^^'  9    7    ^^~9x7x3~^'' 


Jrlence,  jj^  —  ^^-^ 


Cas3  7. 


2.     5x20x12'"^"^'^^'^^^  ^^"^"^^'"^^  required 


3. 


0X1X1    _         1         _   1 


4. 


13x8x^^2^      13x8x7     '^' 

7 

Page  118. 

^0X1X1   ^  1         ^  J  _ 

11X^0X24      11x6x24     "^'^^ 
6 


Case  8. 
^       1x12x20     ,40     4A.    .•  •    ^ 

300X1X1""^'^^""^    ^         required 

3.      728x1  Xl~^^^^^  fraction  required 

1x24x60 

4-     Tzui — ^ — 7-  =  rllS=T  r  fraction  required 
1584x1x1      i5«*     11  ^ 

Case  9. 

2.         12  ounces=l  pound 
3 

5)36 

7  ounces  4  pennyweights 
P^ge  119. 
3.    8  furlongs=l  mile  4.    24  houi-s=l  day 

4  3 

5)33  10)72 

6  furlongs  16  perches  7  hours  12  min. 

8 


86  ADDITION  OF  VULGAR   FRACTIONS. 

Page  119. 

5.         100  cents=l  dollar 
4 

5)400 

80  cents 

Case  10. 

3.        31  galls.  2  qts.  63  galls.=l  hhd. 

4  4 

126  quarts  252  quarts 

Wherefore,  ^|=J  is  the  fraction  required 


cwt.  qr. 
4.   6  2 
4 

.lb. 

18| 

cwt. 

20=1  ton 
4 

26 

28 

80 
28 

216 
53 

2240 
3 

746 
3 

6720  thirds  ( 

2240  tliirds  of  a  pound 
Wherefore,  ||-|§=-y  is  the  fraction  required 

Case  11.    Page  120. 
2.         20  )  17,00  (  ,85 

ADDITION  OF  VULGAR  FRACTIONS, 

Case  1. 
4+8+13+16+19 

15+25+45+55_.„_ 
<S.     ^ =OT=^* 


ADDITION  OF  VULGAR  FRACTIONS. 


87 


Case  2.    Page  121. 


Given  denominators 
2.        2)2458 

2)1254 


And  2 
4 
5 


40  com.  denominator 

20  first  numerator 
10  second    ' 

8  third        ' 

5  fourth      ' 


"40 


40 


115  2 
Then,  2x2x5x2=40  common  denominator 

wi      r      20+10+8+5  _,3_3_ 
Wherelore. jt: -«  —  i  a  « 

Given  denominators 
.     5  )  5  6  7  8  15 


3)16  7  8  3 
2)1  2  7  8  1 
117  4     1 


Then,  5 

6 

'    7 

8 

15 


840  com.  denominator 

168x4=672 
140x5=700 
120x6=720 
105x3=315 

56x8=448 


Now,  5x3x2x7x4=840  com.  denom.      2855  sum 
Wherefore,  \%'=^t%\ 


Case  3.    Page  122. 


4  of  1 -^ 

T  oi  -^ — y , 


Then,J+x\+2%= 


36  +  16+9 


60 


6  1 1    1_ 

—  SO  —  ^6  0 


Wherefore,  l+9+lJo>=lHV 


4. 


2  of  1 1 

3  01^— J 


,     108+105+40+60    3^3    ^,3 
Then,  ^V+|+i+J= ^5 =4M=2t?2o 


Lastly,  l+6+7+2^y^=16^Vo 


Case  4. 


ft.  in. 
2.     i  of  a  yard=l  6 
f  ofafoot=0  8 

Sum  2  ft.  2  in. 


h.  m. 

3.     ^  of  a  day=8    0 

i  ofanhour=0  30 


Sum  8  h.  30  mi. 


88  MULTIPLICATION  OF  VULGAR  FRACTIONS. 

Page  122. 

d.  h.  m.  yds. ft.  in, 

4.     i  of  a  week=2  8    0  5.    f  of  a  mile=1540  0  0 

i  of  a  day  =0  6     0  |  of  a  yard=       0  2  0 

h  ofanhour=0  0  30  J  of  a  foot  =       0  0  9 


Sum  2  d.  14  hrs.  30  mi.  1540y.  2ft  9  in. 

SUBTRACTION  OF  VULGAR  FRACTIONS. 

Page  123. 
11 Q 

A         11 1 11 9 1 2  1 


12 


5- 


5. 


25—24 

5 4 2  5 2  4 —r    1 

6        5         30        S'O  gQ  30 


A  1  O 9  1 

«         2  0,9 7    411 2  1_ 3  97 

"•        3  16         1^^        4'3'2        412  j^QO        — 43^ 

45 — 44 

7         15 11 45 44 _1_ 

''        16         12        4  8        4  8  ^a  4  8 

8.  l2j^^-6i=l2j^^-6J^=m 

9.  181~8iA=13^3_8i4_4i6 

10.        10^^- lJ-=z'\OU 13  5 Q4,3 

11  19  5 7_ 1Q_7_5_ 7  7_^ 1S16A 

XI.        AtJjj        J  5 ^'^T^J       TBI ^°T'6T 

w.    d.   h.  mi. 

13.        7vveeks^7    0     0    0 

9^  days  =  1     2  16  48 


Diff.  5  w.  4  d.  7  h.  12  rain. 
MULTIPLICATION  OF  VULGAR  FRACTIONS, 


Page  124. 


**•  T-^T       40 10" 


4.  |of|=3iV=A     Then,  ^3^  X  ^=2^0 

5.  7ix-l=Vxi=V=lf 

6.  I  of  9=2.7     Then,VxJ=Vi=5|l 


RULE  OF  THREE   IN   VULGAR  FRACTIONS.  89 

Page  124. 

7.       48fXl3|=243x8_3  =  6  7  2  3^672^ 

DIVISION  OF  VULGAR  FRACTIONS. 
Page  125. 

J?         3.-1-.6. SyX 2  1 7_  4      17_i.3 17v5a=85 122 

"•        8     •    7        y^e         48         16  ^*    2T    •  T  —  2T'^3"       ^"S  —  *^3 

7.       91=5-5.  &  i  of  7=^  AfoW    5i-i-l *5v2 110 <2X1 

8.     52051=26  0  26^  and  I  of  91=3 6 4 

Now     26  026_:_3  6  4 2  6  026  v     5     26  02  6 711 

jRC/LE  OF  THREE  IN  VULGAR  FRACTIONS. 

2.     As  I  ton.  :  ^^  dolls.  (  =  164i  dolls.)  :  f  ton. 
3 


^,        2f    493    0     1479     <^o»^-  <=*»• 
Then,^x— ^X^=-— -=211  28^ 


19 


dolls.  i^  j[  X20      ^o""- 

3.    |of|=2V;  Then,     As /^  :  171  :■:  1  :  ^^^^=380 

7 

yd.  dolls.       yds.        4yivf»        dolls. 

4.  As  1  :  f  :  :  42  :  5?^=35 

5.  f  of  f =yt;  Then  say, 

<^o"s-  312x15      ^o"»- 

As/y  :  312  ::  1  :       ^  ^WIO 


Page  126. 
6.     U=4,  79i=2  3 8^  and  3|=  V 

Now,  as^Jw.  :  237^^5.  :•  i^jj^,  -.  202^c««. 

it    2S8     17 
Because,  |x-^X^=  ^11^=202^3^ 


7.     As  17535  :  i  : :  60120  :  ^ 

60120 

17535x8""^'^^^^^""^ 
8* 


j5e cause,  Tzrr^rr- — :k— t^tttii  n— t 


90  INVOLUTION. 

INVERSE  PROPORTION. 

Page  126. 

2.  6icwL=\^cwt.  and  222^=^^  miles 

cwt.   cwt.      mi.      miles 
Now,  as  1  :  V^   :  :  %\^   :  145i 

581 
Because,  y  xW=7^ — -  =  145i  miles 
^       2  5       2x2 

mew   men      days      days 

3.  As  16  :  12  : :  i|3   (=37|)  :  28i-  days 

47      3 

Because,  ^|g^=i|i=28|  days 

4.  lOOf  dolls.=3f  2  dolls.  6f  mo.=2^o  months 
3|-  years  =V  years=46  months 

mo.,    mo.       <?oZZ5. 
Then,  as  46  :  2^0   :  :   so 2   .  14^2 ^  dollars. 

qnoyoo 

Because  — -  =  6j)ji  0  _  1 412  2. 

isecause,  ^g^^^^—  ^,^  — i^^ov 

/if.  /if.      /if.        ft. 

5.  As  I  :  I  : :  V   (=^0^)  :  17^  feet 

^  ^    3    41     3x41     ^,3     ^^^^ 

Because,  Jx^jX-r^-^^^^-f-  =1^t  feet 

'  7    ^     ^         7  ^  ^ 

6.  20i  yards=%i  yards     H  yard=^  yards 
Now,  as  I  yd.  :  |  yd.  : :  y-  yd.  :  34-^-  yards 

Because,  ^x^r X-c-=2|-*— 34i  yards 

INVOLUTION,  OR  RAISING  OF  POWERS. 

Page  128. 

2.  (35)4=35x35x35x35=1500625 

3.  (1,03)3=1,03x1,03x1,03=1^92727 


SQUAKE   KOOT.  91 

Page  128. 

4.     (,029)*  =,029  X, 029  X, 029,  X, 029  X, 029 
=,000000020511149* 

5.     (3)4_|x|xlXl=^ 

SQUARE  ROOT. 

Page  129. 

2.    10692'9(327  root  required        3.     4782969(2187  root  required 
9  4 

62)169  41)78 

124  41 

647)  4529  428)3729 

4529  3424 


4367)30569 
30569 


4.    43046721)6561  root  required  5.     387420489(19683  root  req. 
«jo  1 

125)  704  29)287 
625  261 

1306)  7967  386)2642 
7836  2316 


13121)  13121  3928)  32604 

13121  31424 


39363)  118089 
118089 


G.     22071204(4698  root  req.     7.     36372961(6031  root  req. 
16  ,36 

86)  607  1203)  03729 

516  3609 


929)  9112  12061)  12061 

8361  12061 


9388)  75104 
75104 

*  The  answer  given  with  the  question  is  the  4th  power  of  ,029. 


92 


SQUARE   ROOT. 
Page  129. 


8.    226874i(1506,2H-root  req.    9.    7596796(2756,228+root  req. 


25)  126 
125 


47)  359 
329 


3006)18741 
18036 


30122)  70500 
60244 


545)  3067 
2725 

5506)  34296 
33036 


301243)  1025600 
903729 


55122)  126000 
110244 


121871  remainder. 


551242)  1575600 
1102484 

5512448)  47311600 
44099584 


Page  130. 


3212016  remainder 


10.  9712,718051  (98,553+root  req.    11.  3,17218120(l,78106+root  req. 
81  1 


188)1612 
1504 

1965)10871 
9825 


19705)104680 
98525 


27)217 
189 

348)  2821 
2784 

3561)    3781 
3561 


197103)  615551 
591309 


356206)  2202000 
2137236 


24242  remainder 


12.  4795,257310(69,247+root  req. 
36 

129)1195 
1161 

1382)  3425 
2764 

13844)  66173 
55376 

138487)1079710 
969409 


64764  remainder 

13.  ,06008836(,0094  root  req. 

81 

184)  736 
736 


110301 


SaUARE  ROOT.  93 


Square  root  of  vulgar  fractions. 
Page  130. 


1 .     The  greatest  common  divisor  is  576 
576)|lfJ=|;     Then,v^|=| 


2.    The  greatest  common  measure  is  169 


mim=ih  Now,  ^^i=i 


3.  15625(125  num.     46656(216  denora. 
1  4 

22)056  41)066 

44  41 

245)1225         426)2556 
1225  2556 

That  is  \/i||||-=|f|,  root  required 

Surds. 

4.  |4J=,75;  Then,  v/,'75=,S6602 

5.  |}|=,8706739526    Then,  v/,8706739526=,9330& 

6.  ff|=,5243902439    Then,  ^/,5243902439=,72414 

Square  root  of  mixed  numbers. 
Page  131. 

1-     3734=1349     Then,  ^/il|9=V=6i 
2.  '  27-5-V--\V     Then,  v''W=V-5i 

4         943        48  4        Thpp    ,/484 22_1Q1 

Surds. 

5.  ^/7^\=v/7,818181  &c.  =2,7961  + 

6.  v/84=^8,71428571  &c.  =2,9519+ 

7.  ^8514 =v/85,9333  ^^,^  =;9j27+ 


94 


SQUARE   ROOT. 


Any  two  sides  of  a  right  angled  triangle  given  to  find 
the  third  side. 

Page  132. 

1.    36x36=1298  2.    45x45=2025 

27x27=  729  60x60=3600 


'  Sura  2025  Sum  5625 

Now,  ^/2025=45  feet  Then,  v/5625=75  yards 

3.     45x45=2025  AC  squared         4.     75x75=5625 
27x27=  729  AB  squared  45x45=2025 

Diff.  1296  BC  squared  Diff.  3600 

Then,  ^1296=36  feet  Then,  v/3600=60  feet 

To  find  the  side  of  a  square  equal  in  area  to  any  given 

superfices. 

Page  133. 

1.     x/160=12,6491106  2.     ^/2025=45 

3.     ^/750=27,3S6]279 

To  find  the  diameter  of  a  circle  of  a  given  proportion,  larger 
or  less  than  a  given  one. 

1.  4x4=16,  and  16x4=64    Then,  v/64=8  feet. ' 

2.  1002  x3=30000    Then,  ^730000=173,20508  feet. 

3.  1-^=72         Then,  v/72 =8,48528 14  inches 

The  area  of  a  circle  given  to  find  the  diameter. 

Rule.     Divide  the  given  area  by  ,7854  and  the  quotient  is  the 
square  of  the  diameter. 


SQUARE  ROOT.    '  95 


Page  133. 


1.     v/160-7-,7854=v/203,717850776674=14,272976 

Or,  by  the  rule  in  the  "Calculator." 

v/ 160= 12,6491 106 
1,12837 


885437742 
379473318 
1011928848 
252982212 
1391402166 


14,272876927722  diameter 

2.  Because  160  perches  is  an  acre,  the  diameter  is  the  same 
as  in  the  foregoing  example.  Consequently  the  half  diameter, 
or  length  of  the  halter,  will  be 

14,272976-4-2=7,136488  perches 

16^  feet 


114183808 
3568244 

feet  117,752052 
12 


inches  9,024624 
117  feet  9  inches  the  length  of  the  halter 

Application. 

Page  134. 

Quest  h     v^2d736==144men        2.    25x25=625  feet 

3.        v^l97136=444  stones 

4.  600-^-\o=V_o  =30  the  perpendicular.  Because  the  per- 
pendicular and  the  other  side  is  the  same,  viz: — 30  perches,  the 
triangle  is  right  angled.    Consequently, 


V'40»  -f  30=»==^1600-f900s=  V'^SOOssSO  petches 


96 


5.    84x84=7056 
50x50=2500 


CUBE  ROOT. 

Page  134. 

Then,  ^/9556=97,7547-f  miles 


9556 

root  required 

6.    964,5192360241(31,05671 
9 

61)  64 
61 

6205)  35192 
31025 

62106)  416736 
372636 

621127)  4410002 
4347889 

6211341)  6211341 
6211341 


-.  root  required 

'.      1030892198,4001(32107,51 
9 

62)  130 
124 

641)    689 
641 

64207)~482198 
449449 

642145)~3274940 
3210725 

6421501)  6421501 
6421501 


CUBE  ROOT. 

Page  138. 


2.        2x2x3=12.. 

square  of  4=     16 
2x4x30=  240 


complete  divisor  1456  x4= 


13824(24  root  required 

8 

5824 
5824 


3.    7x7x3=147.. 

square  of  2=       04 
7x2x30=    420 


373248(72  root  required 
343 


30248 

complete  divisor  15124x2=30248 


4.       1x1x3=3.. 

square  of  7=  49 

1x7x30=210  

4735 

complete  divisor  559x7=^3913 

17x17x3=867..  "822339 

square  of  9=      81 
17x9x30=  4590 

complete  divis.  91371x9=  822339 


5735339(179  root  required 
1 


CUBE   ROOT. 
Page  138. 


97 


.5.         4x4x3=48.. 

square  of  3=     09 
4x3x30=  360 


84604519(439  root  required 
64 

20604 


complete  divisor  5169x3=15507 


43x43x3=5547.. 

square  of  9=         81 
13x9x30=   11610 


5097519 


corap.  divisor  566391  x9=     5097519 


6.     3x3x3=27.. 

defective  divisor  2700 .. 
defective  divisor  270000 . . 
^    square  of  2=  04 

300x2x30=       18000 


27054036008(3002  root 

27 


05403600S 


complete  divisor  27018004x2=         54036008 


7.         4x4x3=48.. 

square  of  9=?=     81 

4x9x30:^:1080 


complete  divisor  5961  x9: 
lOSO 
92X2=     162 


defective  divisor*  7203 . . 
square  of  6=:  36 

49x0x30=       8820 


complete  divisor  729156x6  = 
8820 
6^x2=  72 


defective  divisor  738048 .  . 
square  of  8=  64 

496x8x30=  119040 


122615327232(4968  root 
64 


58615 
53649 

4966327  dividend 


4374936  subtraliend 


591391232  dividend 


complete  divisor  73923904x8=    591391232  subtrahend 
*  Defective  divisors,  after  the  first,  may  be  more  concisely  found  by  ad" 
dition,  thus:  To  the  last  complete  divisor,  add  the  number  which  com- 
pleted it,  with  twice  the  square  of  the  last  figure  in  the  root;  the  sura 
w^U  be  the  next  defecti/e  divisor. 


98 


CUBE  HOOT. 


Page  138, 

22069810125(2805  root 

8 

14069 
complete  divisor  1744x8=    13952 

480  

8*x2=     128  117810125  dividend 


8.         2x2x3=  12.. 

square  of  8=       64 
2x8x30=    480 


defective  divisor  235200  . . 

square  of  5=  25 

280x5x30=         42000 


complete  divisor  23562025x5=117810125  subtrahend 


219365327791(6031 
216 

~3365327 


9.         6x6x3=108.. 
defective  divisor     10800  . . 

square  of  3  r=  09 

60x3x30=  5400 

complete  divisor  1085409  x  3  =   3256227 

5400  

square  of  3x2=  18  109100791 

defective  divisor  1090827  . . 

square  of  1=  01 

603x1x30=         18090 

109100791x1=109100791 

.  I'oot  required 

673373097125(8765 
512 

l6l373 
146503 


14870097 


10.         8x8x3=192.. 

square  of  7=:       49 
8x7x30=  1680 

complete  divisor  20929  x  7 = 
1680 
twice  the  sq.  of  7=      98 

defective  divisor  22707  . . 
square  of  6=  36 

87x6x30=       15660 

complete  divisor  2286396x6=  13718376 
15660 
twice  the  sq.  of  6=  72 

defective  divisor  2302128 . . 

square  of  5=  25 

876x5x30=         131400 

complete  divisor  230344225  x5«,l  151721125 


1151721125 


CUBE  ROOT. 


99 


Page  138. 


12,977875(2,35  root 

8 

4977 
4167 


11.     2x2x3=rl2.. 

square  of  3=     09 
2x3x30=  180 

complete  divisor  1 3^9  X  3  = 
180 
twice  the  sq.  of  3=     18 

defective  divisor  1587.. 

square  of  5=  25 

23x5x30=        3450 

complete  divisor  162175x5=        810875 


810875 


12.       2x2x3=  12.. 

square  of  5=       25 
2x5x30=     300 


15926,972504(25,16+ 


complete  divisor  1525x5  = 
300 
twice  the  sq.  of  5=       50 

defective  divisor  1875  . . 
square  of  1=  01 

25x1x30=         750 

complete  divisor  188251x1  = 
750 
twice  the  sq.  of  1  =  2 

defective  divisor  189003  . . 

square  of  6=  35 

251x6x300=         45180 


7926 
7625 


301972 


188251     ■ 
113721504 


complete  divisor  18945516x6=113673096 

48408  rem. 


36155,027576(33,06 

27 

9155 

8937 


13.     3x3x3=  27.. 

square  of  3=       09 

3x3x30=     270 

complete  divisor  ::^979x3=: 
33x33x3=3267.. 
defective  divisor  326700  . . 
square  of  6rr=  36 

330x6x30=       59400 
complete  divisor  32729436x6=    196376616 

21650960  rem. 


S18027576 


100  CUBE   ROOT. 

Page  139. 

14.     3x3x3=-  27..  ,053258279(,376+root 

square  of  7=       49  27 

3x7x30=     630  — 

26258 

complete  divisor  3379x7=   23653    - 

37x37x3=4107..  2605279 

square  of  6=         36 
37x3x30=     3330 


complete  divisor  414086x6=  2484396 


120883  remainder 


15.     1  Xl  X3=3 . .         ,001906624(,124  root 
square  of  2  =  04  1  "^ 

1x2x30=     60  

906 

complete  divisor  364x2=      728 

12x12x3=432..  178624 

square  of  4=:       16  > 

12x4x30=  1440 


complete  divisor  44656  x4  =       178624 


16.  ,000000729(,009  root  required 

729 

17.  1x1x3=3..  2,(l,25+root 
square  of  2=:     04  1 

1x2x30=     60  

1000 

complete  divisor  364x2=      728 


12x12x3=432..  272000 

square  of  5=       25 
12x5x30=  180O 


complete  divisor  45025  X  5  =     225 125 


4  6875  remainder 


CUBE   ROOT.  101 

Page  139. 
To  extract  the  cube  root  of  a  vulgar  fraction. 

1-  Hb-^tII     Then,  ^^11  =f  root  required 

2-  r¥o%=TV5     Then,  ^//^=|  root  required 

3.  i4|J=A     'J'^^en,  ^^V=l  '^oot  required 

Surds. 

4.  ^^=^,57142«571  &c.  ==,829+ root  required 

5.  ^1^^,666  &c.=,873+root  required 

6.  -^1=^,555  &c.=,822+root  required 

To  extract  the  cube  root  of  a  mixed  number. 
2.     ^12^|=^3_Y_7_2^ 

Surds. 
Pag3  140. 

4.  ^7^=^7,2=1 , 93 +rcot  required 

5.  ^8i.=^8,7142857  &c.=2,057+root  required 

6.  ^9^=^9,166  &c.=2,092+root  required 

To  find  the  side  of  a  cube  that  shall  be  equal  to  any  given  solid, 
as  a  globe,  a  cone,  S^c. 

1.    ^10648=22  2.    ^383017=73 

Note. 

1.  23x3=2x2x2x3=24 

Then,  ^24=2,8845  feet=2  feet  10,614  inches 

2.  123x3=12x12x12x3=5184 

Then,  ^5184=17,306  inches 
9* 


102  AHITHMETICAL  PROGRESSION. 

Page  140. 
Application. 
Quest.  1.     As  63   :  3^  •.•  216  :  27  :  :  321b.  .  4  lb. 

2.  288x216x48=2985984     Then,  ^2985984  =  144 

3.  ^389017=73     Then,  73x73=5329 

Page  141. 

Quest.  4.     Because  there  is  two  half  feet  in  a  foot. 

2x2x2 

— - — =|-=:4  solid  half  feet  in  half  a  solid  foot 
2  ■        ^ 

And  then,  4 — 1=3  sohd  half  feet 

5.  2x2x2=8  six  inch  cubes.  And  because  there  are  three 
times  4  inches  in  a  foot  we  find  that  there  is  3x3x3=27  four 
inch  cubes  in  a  solid  foot. 

ARITHMETICAL  PROGRESSION. 

Case  1.     Page  142. 


2.  (20— Ix3)+3=(19x3)-f3=60  last  term 
Then,  (60+3)  X^  =^3x10=6  dollars  30  cents 

3.  Here  4  is  the  common  difference 


And  (100— 1X4) +4=(r>9x4)+4=400  last  term 
Then,  (400+4)xi§o  =404x50=20200 yards 
But,  20200  yards=ll  miles  3  furlongs  180  yards 

Page  143. 


4     (lU—lxlO)+20=90-f  20=110  dollars  last  term 

Then,  (110+20)xV  =  i30x5=650  dollars  the  whole 
amount.     And  650-t- 10=65  dollars  per  annwm 


5.      (1000— Ixl0)+10=:9990+10=10000   cents,   or 
100  dollars  for  the  last  acre 

And  (10000+10)X^  V°=50050  dollars,  whole  cost 


GEOMETlilCAL   PROGHESSION. 

Case  2.     Page  143. 

GO— 20     ^„      ^  , 

2. =5^=*^  the  common  diiterence 

21—1       ^^ 

T jien,  20,22,24,26,28,  &c.  are  their  ages  respectively 

Page  144. 

58-3     ss     -     ■^      ^'^    • 
3.         -; =i|.=o  miles  daily  increase 

And  (58-f  3)x  V^=Blx6=36'3  miles  whole  distance 

GEOMETRICAL   PROGRESSION. 

Page  146. 

2.         Indices     12     3    4 

Ratio       2,    4,   8,  16,  powers  of  the  ratio 
16 

In  this  question  the  250=8th  power 

number  of  terms  is  12  8=3d  power 


2048=  1 1  th  power  of  the  ratio 
X    1  the  first  term 

[progression 

2048=12th  or  last  term  of  the 
X    2  the  ratio 


40C6 
Subtract        1  the  first  term 


Ratio  2—1=1)4095 


$4095  the  sum  of  the  series 


3.     Indices 
Ratio 

1 
2, 

2 
4, 

3 

8, 

4    5 
16,  32, 
32 

powers  of  the 

ratio 

Here  the  number  of 
terras  is  15 

64 

96 

1024= 

]6= 

=10th  power 
=  lth  power 

J63b4=14th  power  of  the  ratio. 
And  because  the  first  term  of  the  progression  is  1,  the  14th  power  of 
the  ratio  is  the  last,  or  ]5th  term  of  the  progression 

Then,(i?^^-I^Mi:l=32767  shillings=1638£.  7.. 
'ratio  2—1  °  ^ 


104  GEOMETRICAL   PEOGRESSIOIS'. 

Page  146. 

4.     Indices     12     3      4 

Ratio     4,  16,  64,  256  powers  of  the  ratio 
256 


1536 
Here  the  number  of       12S0 
terms  is  12  512 


6553t)=8th  power 
64=:3d  power 


262144 
393216 


41U4304^11th  power  of  the  ratio,  which 
is  also  the  12th  or  last  term  of  the  progression. 

<'4-lQ43n4  v4^ 1  <Jo'is.   cts. 

Then,  kli^!£^^J_U551)24  05  sold  for 
4 — 1 

And  12x20=240  00  cost 
^556S4  05  gain 

5.     Indices  12     3     4     5     6       7 

Ratio  2,    4,    8,  16,32,64,  128,  powers  of  the  ratio 

128 

Here  the  number  of  1024 

terms  is  32  256 

128 


16384=14th  power 
16384=      do. 


65536 
131072 
49152 
98304 
16384 

26S435456=28th  power 
8=3d  power 


2 147483648=3 1st.     power  of  the 
ratio:   And  because  the  first  term  of  the  progression  is  1,  it  is 
also  the  32d  or  last  term  of  the  progression 
(2147483648x2)— 1      ,^^,^^^  , 

Then,  ^ ir-^-ir-^ =4294967  d.  29  cts.  5  ras. 

2 — 1  Ratio 


GEOMETRICAL   PROGRESSION. 

Page  146. 

6.    Indices  12    3    4    5 

Ratio    3,    9,  27, 81, 243  powers  of  the  ratio 
243 

729 
972 
486 


105 


59049=1 0th  power 
59049=       do 


531441 
236196 
5314410 
295245 


348678440  l=20th  power 
59049=10th  power 


31381059609 
13947137604 
313810596090 
17433922005 

205891 132094649=30th  power 
3=single  power 

617673396283947=31st  power  of  the  ratio,  or  32d 
term  of  the  progression 

Then,  #125!E?^il2i!hd=826510094425  d.  92  c. 
'  Ratio    3—1 


(432—1)  18446744073709551615 
7.  1 X-      ' 


4—1.  ~  3 

6148914691236517  dollars  20  cents  5  mills 


(3-^0-1)     2x205891132094648 

^'  -^xS^r- 2 = 

205891132094648  pins.     Which  at   1000  for  a  farthing 
amounts  to  £.214469929  5  U 

cost  of  30  yards  at  £100=  3000  0  0 


gain  £,214466929  5  3^ 


106 


SINGLE   POSITION. 


SINGLE  POSITION. 

Page  148. 

Suppose  A  paid  12  dollars 
Then      B  4       ' 

And        C  1       ' 


Now,  as  17  :  340 


17  sum 
dolls.       (12 


240  A  paid 
80  B      ' 
20  C     ' 


4.     Suppose  60 


l^|2Now,as37  :  60 


i^=10 


37 

Suppose  12 
i=  6 


*=  4  Then,  as  2  :  12 

10 
left  2 


148  :  ll^=240d. 


26f  dolls.  :  160  dolls. 


6.         Suppose  A's  age     6 
ThenB's=6xli-=9 
.\ud  C's=(6H-9)x2jV=3U 


Then,  as  46,5  :   93 


12  A's  age 
18  B's    ' 
03  C's    ' 


7.     Suppose  1  to  be  the  numbfer 
35—32 

Id 


Then,  | — §=  _^_z=-3^  difference 


Now,  as  4^^ 


1 


6  :  ^^=S0 


DOUBLE   POSITION. 


107 


Page  148. 

8.     Suppose  1  the  harness 
then        2     '    horse 
and  6     *    chaise 


9 
dolls. 
Then,  as  9  :  360  :  : 

9.     Suppose       60 

i--=20 
^  =  15 
1=12 
^=10 

57 


40  dollars  harness 
80  dollars  horse 
240  dollars  chaise 

10.     Suppose  12 
*=  4 

A  spends  8 

2 

B  spends  16 
12 


Now, 
as  57 


2. 


60 


dolls,  dolls. 
228  :  240 


B  sinks  4 


Then,  as  4  :  12 


dolls,  dolls. 
:  50  :  150 


Suppose  8 

A  spends  7 
30 

B  spends  37  per  year 

8      . 


B  spends  296  in  8  years 
8x8=  64 

B  is  indebted  232 
40 


DOUBLE  POSITION. 

Page  150. 

Again,  suppose  40 


1 5 

A  spends  35  per  year 
30 

B  spends  65  per  year 

8 

B  spends  520  in  8  yrs. 
40x8=320 


B  is  indebted  200 
40 


192  error  too  little 
192  errorx40=7680 
160  error  X  8=1280 


160  e.  too  lit. 


6400  difference  of  the  products 
192 — 160=32  the  difference  of  the  errors 


108 


DOUBLE   POSITION. 


Page  150. 
Question  2d  continued. 
Then,  6400h-32=200  dollars  their  income. 

And  200—2^0=200—25=175  dolls.  A  spends  per  annum 
Also,  175 4-30  ==-205  dollars  B  spends  per  annum 


Suppose  A's  12 
Then  B's  is  15 
And  C's    '    19 

46 
100 


Again,  suppose  A's  40 

Then  B's  is  43 

And  C's  '  47 

130 
100 


Error  54  too  little 
54x40=2160 
30x12=  360 


Error  30  too  much 


2520  sum  of  the  products 
And  54+30=84  sum  of  the  errors 

Then,  2520-t-84=:30  A's  share 
304-  3=33  B's  share 
33+  4=37  C's  share 


4. 


Suppose  A  paid  100 
then  B  '  1100 
and    C     «     1200 


Again,  suppose  A  paid 
then  B    ' 
and  C     ' 


2400 
10000 


error  7600 

EiTors  are  alike,  i.  e.  both  too  little 

7600x200=1520000 
7200x100=  720000 


Diff.  of  the  products,  800000 

7600—7200=400  difference  of  the  errors 
Then,  800000-4-400=2000  dollars  A  paid 
And  2000+1000=3000  dollars  B  paid 
Also  2000+3000=5000  dollars  C  paid 


200 
1200 
1400 

2800 
10000 


error    7200 


DOUBLE  POSITIO^^  109 


Page  150. 

5.    Suppose  first  horse  $10  Again,  sup.  first  iiorse  $20 

50  50 

2)80  2)70 

The  second  horse  30  Then,  second  horse  35 

50  50 

80  85 

10x3=30  20>^3=f30 

Error  too  little    50  error  too  little  2.j 

50x20=1000  50        ^. 

25x10=  250  25 


Diff.  of  the  prod.    750    Diff.  of  the  errors  25 
Then,  750h-25=30  dollars  first  horse 

And  — - —  =40  dollars  second  horse        • 


6.     Suppose  20  body         Again,  suppose  30  body 

Then,  %o  4.9=19  tail  Then,  %"  +9=24  mil 

9  liead  9  head 

Head  and  tail=28  Head  and  tail  3S 

•    body=20  body  SO 

error  8  too  little  error  3  too  littlt 

30  x8=240        8 — 3=5  difference-of  the  errors 
20x3=  60 

Then,  180-=-5=36  inches  the  body 

Diff.  of  pro.  180      And  3^^+9=27  '    tail 

9  '    head 

72  in.  =6  feet. 

10 


110  DOUBLE  POSITIO^^ 

Page  150. 

7.     Suppose  he  wrought  20  days,  and  was  idle  20 
Then,  20  daysx20  cents=400 
20     '    xlO     '     =200 

,  He  would  receive  200  cents 

500 

error  too  little  300 
Again,  suppose  he  wrought  25  days,  and  was  idle  1 5 

Then,  25  daysx20  cents=500 
15     '     XIO     '      =150 

He  would  receive  350 
500 

error  150  too  little 
25x300=7500 
20x  150=3000    300—150=150  diflference  of  the  errors 

Diff.ofpro.  4500 

Then,  4500-i- 1 50=30  days,  wrought 
And  40— §0=10  days,  idle 

8.     Suppose  each  had  300  dollars 
Then,    300+ so 0^300 4- 75=375 
And  (300— 225)x2=75x2=150 

Difference  225  error 

Again,  suppose  each  had  400  dollars 
Then,  400+400^4004- 100=500 
And  (400—225)  x2=175  x2=350 

Difference  150  err. 
Errors  alike. 

Now,  400x225=90000 
And  300x150=45000 


Diff.  of  the  products  4500 
And  225 — 150=:75  difference  of  the  errors 
Then,  4500-^75=600  dollars 


PERMUTATION.  Ill 

Page  150. 

i).     Suppose  the  parts  Again,  suppose  the  parts 

to  be  8  and  7  to  be  10  and  5 

Then,8x  4=  32  Then,  10 x  4=40 

7x16=112  5x16=80 

80  error  40  error 

Errors  alike. 
10x80=800  80—40=40  difference  of  the  errors 

8x40=320    Then,     480-7-40=12  the  greater 

Diff.  of  pro.  480  And  15—12=3  the  less 

'■  Page  151. 

;  10.     Suppose  1  o'clock 

22 — 13 

Then,  (^  of  ll)aQ  1  =—^3-  =A  error 

Again,  suppose  it  to  be  two  of  the  clock, 

26 — 20 
Then,  (j\  of  \0yj2  2=-^^=^  error 

Errors  unlike 
Now,  t\x2=}|,  and  y'^Xl^A 
Then,  tI+tt— f  J  ^^^  ^^^  of  the  products 
And  TT+TJ^if  ^^®  s^r"  of  the  errors 
Lastly,  f|-^i4=-5^Xi|=f|=l  hour  36  minutes.  ,• 

PERMUTATION. 
2.  changes 

1x2x3x4x5x6x7x8x9x10x11x12=479001600 

10)179001600  changes 
days  firs.       6,0)  47900160 
3'o5  6=8766)     798336(91  years 

78894 

9396 
8766 

days  w.  d. 

24)  630(26=3  5 

48 

150 
144 

6  hours. 
Ans.  91  years  3  weeks  5  days  6  hours 


1 12  C03IBINATI0N. — DUODECIMAIS- 

Page  151. 
S.     1x2x3x4x5x6x7=5040  days. 
Then.  5040-^365=13  years  295  days 

Page  152. 
'     4.     1x2x3x4x5x6x7x8x9=362880 

5.     1x2x3x4X5x6x7x8x9x10x11x12x13x14x15 
Xl6x  17x18x19x20x21x22x23x24x25x26= 
10329 1461 1266056355S4000000 

COMBINATION. 

I  11    7      ^ 

o        ^^x23x^^x^Ax^0xi9_ 

ix^x;^x^x^x0 

4x23x11x7x19=134596 

5  ^  13    5      3 

,^      ;^0X29X^^X27X^0X#X0^X23 

IX  ^X  ^X  ^X  /^X  0X  ;^x  ^~ 
X20x27x  13x5x23=5852925 

ADDITION  OF  DUODECIMALS 

Page  153. 

fett     in.    "     "'  "" 
2.     3780     9     4     7     9 

SUBTRACTION  OF  DUODECIMALS 

Page  154. 
16  feet  8  inches  1  second  8  thirds  2  fourths 

feet  in.    "    "'  feet  in.  "    '"  "" 

:).     From  35    9    2     0  4.     From  803  3  4    0  0 

Take  24  10    5    4  Take    70  3  7  10  5 


Diff.  10  10    8     8  Diff.  732  11  8     18 


DUODECIMALS. 


113 


MULTIPLICATION  OF  DUODECIMALS. 


Case  1.    Page  155 

ft.  m. 
9  5 
3  11 

ft.  in. 
3.           7  10 

8  11 

8  7     7 
28  3 


7    2    2 
62    8 


square  ft.  36  10  7         sq.  ft.  69  10    2 


4. 


A 

in." 

8  4  6 

2  7  4 

2  9  6 

4 

10  7  6 

16 

9  0 

square  feet  21  10  5  0 


5. 


feet  in. 
5     7 
1  10 

4  7  10 

5  7 


square  ft.  10    2  10 


Then,     150  cents 

10  square  feet 


2  in. 
8  " 
2  " 


1500 
25 


2,ii~ 


$15,35,^ 


6. 


yil.  in. 
7  10 
6  8 
5    4 


19 

10 
3  11 

18 
59 

2    2 
6 

77 

8    2 
3 

Then,  233  square  feet  » 
14  cents 


3262 


6  is  ^V= 


7 


$32,62/3, 


233    0    6  square  feet 
10* 


114  DUODECIMAIS. 


Case  2.    Page  156. 


ft.    in.  ft.     in.  ft.     in. 

76    7x1         3.         127    6x2        4.       184    8x7 
6  10  12 


459    6  1275  0  2216  0 

3  9  10 


1378  6  11475  0  22160  0 

in.           76  7  in.           255  0  in.          1292  8 

6isi=  38  3  6         4isi^42  6  6  is  ^=     92  4 

4  «  1=  25  6    4  


sq.  ft.   1518  10  10 


11772  6  sq.  feet  23545  0 


Page  157. 


feet  in.  feet.    in. 

6.        38  11  7.        59    9 

10  4 


389    2  239     0 

7  6 


2724    2                               1434    0 
6  in.  is  1=      19     5     6       6  in.  is  i=    29  lOl 
2  '    '  ^=       6     5  10  


9)1463  lOi 


square  feet  2750     1     4 

162  yds.  5  ft.  10^  in. 

feet  in.      feet 
8.         54  9=54,75 

feet   in. 
9.        7    6 

58  6=  58,5 

3  3 

27375 
43800 
27375 

9)3202,875 

355,875  yards 
15  cents 


1 

22 

10  6 
6 

24 

4  6 
1  10 

20 
24 

3  9    0 

4  6 

$53,38125  square  feet  44    8  3    0 


CAKPENTERS'   RULE.  115 


Page  157. 


ft.  in.     ft.  in.    ft.  in.   ft.    in.  " 

10.         (2  10)X(2  4)x(l  9)  =  11     6  10 

(2  10)x(2  6)x(l  3)=  8  10  3 

(3    6)x(2  2)x(l  8)=12    7  8 

(2  10)x(2  8)x(l  9)=13     2  8 

(2  10)x(2  6)x(l  9)  =  12    4  9  '" 

(2  ll)x(2  8)x(l  8)=12  11  6  8 


71     7     8  8 
Or,  71,6435  square  feet 

ft.  dolls.  sq.feet. 

Then,  as  40  :  20  •.•  2:1:  :  71,t)435  :  §35,82  + 

Tofind  a  skip's  tonnage. 
Page  158. 

THE  CARPENTERS''  OR  SLIDING  RULE. 

Page  160. 
1st.     To  multiply  numbers  together. 

2.  Set  1  on  B  to  12  on  A,  then  against  16  on  B  stands  192 
the  product  on  A. 

3.  Set  1  on  B  to  35  on  A,  then  against  19  on  B  stands  665 
the  product  on  A. 

4.  Set  1  on  B  to  54  on  A,  then  against  270  on  B*stands 
14580  the  product  on  A. 

2d.     Division  of  numbers  by  the  Carpenters''  rule. 

2.  Set  the  divisor  19  on  B  to  the  dividend  665  on  A,  then 
against  1  on  B  stands  35  the  quotient  on  A. 

3.  Set  27  on  B  to  396  on  A,  then  against  1  on  B  stands 
14,6  the  quotient  on  A. 


116  carpenters'  RUI£. 


Page  160. 


4.  Set  42  on  B  to  741  on  A,  then  against  1  on  B  stands 
17,6  the  quotient  on  A. 

5.  Set  24  on  B  to  7680  on  A,  then  against  1  on  B  stands 
320  the  quotient  on  A. 

3d.     To  square  numbers  hy  the  Carpenters''  rule. 

Page  161. 

2.  Having  set  1  on  C  to  10  on  D,  then  against  30  on  D 
stands  900  on  C. 

3.  Having  the  rule  set  as  above,  against  35  on  D  you  will 
find  1225  its  square  on  C. 

4.  Also  against  40  on  D  stands  1600  its  square  on  C. 

4th.     To  find  a  fourth  proportional  to  three  numbers. 

2.  Set  the  first  term  25  on  B  to  75  the  second  term  on  A, 
then  against  100  the  third  term  on  B  stands  300  the  fourth  term 
on  A. 

3.  Set  27  on  B  to  20  on  A,  then  against  73  on  B  stands 
54/y  on  A. 

5th.     To  extract  the  square  root  of  any  number  by  the 
[Carpenters''  rule. 

2.  Having  set  1  on  C  to  10  on  D,  against  529  on  C  stands 
23  its  root  on  D. 

3.  Being  set  as  in  the  last  example,  against  900  on  C  stands 
30  the  root  on  D. 

4.  The  rule  set  as  before,  against  300  on  C  stands  17,3  the 
root  on  D. 


MEASURING   OF   BOARDS,  ETC.  117 

MEASURING  OF  BOARDS  AND  TIMBER 

Page  162. 
1st,  To  find  the  superficial  content  of  a  hoard  or  plank. 

foot  in. 

l.         12  Or,  by  the  sliding  rule. 

12  6  As  12onB  :  14  on  A  :  : 

12  ft.  6  in.  on  B  :  14  ft. 

7  0  7  in.  on  A*. 
14  0 


feet  14  7 

3.     13^  +  13A  +  14*  +  18+1H=70|=2|3  inches 

283 

Bat  2  83  inches=— — -Jeet  ^  feet 
"^  4x12        ^" 


And  174  feet=V  feet 

9905x3 

"96 


Then,  Vb'  X¥  X3  cts.=^^:^  ^^^  =3d.  09iJ  cts. 


:W.     Having  the  breadth  of  a  board  or  plank  in  inches,  to 
find  hoio  much  in  length  will  make  afoot,  S^c. 

Page  163. 

2.  144-^23:^=6,26  inches 

3.  U-  yards =1944  square  inches 

Then,  1 944 -t-23= 74,7692  inches=6,2307  feet 

3d.  To  find  the  solid  content  of  square  or  four  sided  timber. 

Page  164. 

15.n.xl5in.xl8ft.^     ,^^      ^^^^ 
144  ^  ^ 

,      25in.4-9in.     ,^  .     ,        , 

3. =17  inches,  the  quarter  girt 

,_,.        17in.Xl7in.x20ft.      .^,ooox-       >: 

Then, —■- =40,1388  feet,  6z,c. 

144 

''  The  operation  by  the  sliding  rule  is  omitted  after  this,  it  being  pre- 
sumed that  the  directions  in  the  W.  Calculator  make  it  superfluous  to 
work  more  examples. 


118  carpenters'  and  joiners'  work. 

Page  164. 

,      32in.+10iii.    „,  .    ^      ^  ^      ,  , 

4.    — =5^1  inches  the  mean  breadth 

.    J  20in.+6in.     ,«  .     ,       , 
And i=ld  inches  the  mean  breadth 

Then,  ??-^^|^=2  7  3=,34  125  feet 
8 

4th.     To  find  the  solid  content  of  round  timber. 

Page  165. 

o      14+2     „  ^  . 

^.     — — 2    ^  ^^^  quarter  girt     ^ 

Then,  2x2x24=96  feet. 

Page  166. 

3.  24  inches =2  feet 

Then,  2  feet x2  feet x  IS  feet=72  feet. 

,      llft.4in.-f2ft.8  in.     ,^     ,  ^^  ^ 

4.    -^— =.r*^1^75feet 

Then,  1,75  ft.  x  1,75  ft.x21  ft.=64,3125  feet 

5.  24x(|)2=24x4  =96 
14^x(3^i*)2=:14,5x,62015625=  8,9922+ 
l^ix(6^28)2=17^25x2,4649     =42,5195 


sum  147,5117  feet 

CARPENTERS'  AND  JOINERS'  WORK 

Page  168. 
^,    (53ft.6in.)x(47ft.9in.)     53,5x47,75    ^q-  ^^f 
z. ^100  "= — 100 — ^^"^  ^^'^^^ 

_    (36ft.3in.)x(16ft.6in.)    36,25x16,5     I'f^^' 
3. — = ___=^5,98125 

Then,   5,98125  squares  x4  d.  50c.=26d.  91  c.  5,625  m. 


carpenters'  and  joiners'  work.  119 

Page  168. 

4.         35  ft.x30  ft.=1050  square  feet 

(4  ft.  6  in.)  x6  ft.=  4,5x6=27  square  ft.  fire  place 
(10ft.6in.)x8ft.=10,5x8=84      '       '   stairs 

111  feet 

Then, — — =9,39  squares 

Lastly,  9,39  squares  x3  d.  75  c.=35  d.  21  c.  2,5  ms. 

ft.  in.     ft.  in.      ft.     ft.      sq.   feet 

5    (82  6)x(12  3)     82,5 Xl2i. 

o.    — : — 10  10,b^5 

.    (91ft.9m.)x(llft.3in.)    91,75xlli     'IJ""  ^,. 
6. — — 10,32187a 

squares      dolls,  cts.      d.    cts.       mills 

Then,  10,321875x4  50=46  44  8,4375 

Page  169. 

ft.  in. 
8.    2)30  6 
15  3 


45  9=45,75  feet.    And  52 feet  8  in.=522  feet 

„,        45,75  x52|    ^^_ 

Then, — — — ^=24,095  squares 

,  Lastly,  24,095  sq.xMO  cts.=33  dolls.  73  cts.  3  mills 

ft.  in. 
9.    20  6 
i=10  3 


30  9=^30,75  feet.     And  40  feet  6  inches=40,5  feet 

T,,        30,75x40,5     ,^.^„^^ 
Then,  — '  '   =12,45375  squares 

Lastly,  12,45375  squares  x225  cts.=28  dollars  2  ceuts-f 

Page  170. 

square  fea 
11.  (137  ft.6in.)x(16  ft.  3  in.)=137,5xl6i=2234,375 

Then,   2234,375  feetH-9=248  yards  2,375  feet 


120  aiASONS'   WORK. 

Page  170. 

12.     (69  ft.  9  in.)x(6  ft.  3 in.) =69,75x64  =435,9375 

i  work=217,96875 


9)653,906-25 
square  yards    72,65625 

13.     (83ft.  8in.)x(12ft.  6in.)=83f  xl2,5=1045,83i 
(7ft.  8  in.) X (2  ft.  6  in.)xU=7|x2,5xl,5=     2S,75wind. 
(3ft.6  in.)x7  ft.xi  =3,5x7x,5-=     12.25  door 


9)1086.83^^ 

square  yards     120,7if 
Then,  120,7J4  yds.  x80  cts.=96  dolls.  60  cts.  7*4  mills 

BRICKLAYERS'  WORK 

Page  173. 
ft.  in.    ft.  in.  halfhric^. 

2.  (57  3)x(24  6)x5=57,25x24,5x5=7013a25 

Then,  3)7013,125 

9)2337,7QSi  standard  thickness 
259,745+square  yards 

Or,  2337,708*  -T-272i  =8,5866  rods 

feet   in.     feet  in. 

3.  (245  9)x(16  6)x5=245,75xl6,5x5=20274.37r. 

And  20274,375—3=6758,125  standard  thickness 

^^       6758,125    ^,      ,   „ 

'    070  of;      ^^  ^*^"^  ^  quarters  19,945  feet 

4.     (45  ft.x  V  ft.  x5)-T-3=562,5  standard  thickness 
Then,  562,5-7-272,25=2  rods  18  feet 

MASONS'  WORK. 

Page  175. 

feet  in.   feet  in. 
2.    (53  6)x(12  3) x2=53,5x  12:1x2= 1310,75  feel 

Then,  1310,75—24,75=52,9595  rods 


PL-VSTERERS'   WORK.  121 

Page  175. 
3,    (107  ft.  9  in.) x(20  ft.  6  m.)=2208  feet  10^  inches 

^   (112ft.  3in.)x(16ft.  Gin.)     112,25x16,5    "o  o^'Vo^ 
t)3  63  ' 

5.     (5  ft.  7  in.)x(l  ft.  10  in.)=54x|-|=  V^V^VV  f^et 

737x10 

Then,  VV  XSO  cts.==  — ^-=S18|  cts.r=8  dolls.  18|  cts. 


PLASTERERS'  WORK. 

Page  177.   ' 
feet    in.      feet  in. 

^      (llL^(ii?.)_l££52lH!??_1767875 


3.  (22ft.  7in.)x(13ft.ll  in.)=  tVX  W=^f!l'  feet 

45257  ^ 

But,  HII^  feet=^^^^=VAV  yards 

Then,  YJ^sV  Xl5  cts.=2  2_6^2^8  5^5  dolls.  23  cts.  8^^-^-  ms. 

4.  (20  ft. +  14  ft.  6  in.)  x2=69  feet  round  tlieroom 

69  ft.x(10  ft.  4  in.)=713  sq.  feet  in  the  walls 

(4  ft.  4  in.)  x4  ft.==     17  ft.  4  inches  fire  place 
(3  ft.  2  in.) x6  ft.  x2=     38  ft.  0      '      twa  windows 

Sum  55.ft.4in.=55i  feet 
713  ft!— 55J  =r657|  Lastly,  657|-~9=732V 

5.  (14  ft.  5  in. +  13  ft.  2  in.)x2=55  ft.  2  inches  round 

-  .   ft.  in.  " 

Then,  (55ft.2in.)x(9ft.  3in.)==510  3  6 

7ft;x4ft.  =a  28  0  0 


9)482  3  6 


Rendering  53  y.  5  ft.  3  in.  6  s. 
ft.  in.  in.     ft.  in. 

.  14  5  less  10=13  7  And  13  ft.  2  in.— 10  in.=ii2!:ft.4  in. 

_         (13ft.7in)x(12ft.4in.) 

Then,  ^^ ^F^ ^^=18  yds.  5  ft.  6  in.  4" 

11 


/\ 


122  pavers'  work. 

Page  177. 
feet    in.       feet   in. 
(105  6)X(275  5)      105,5x275/^     «SS«Ti''o3 

Then,  3228,4iJ|  sq.  yds.xl2  cts.=387  d.  41  c.  9|  m. 

feet  in.      feet 
1.     (30  ft.  6  in.+24  ft.  9  in.)x2^=110  6=110,5  round 

110,5  ft.xlO  ft.  =  1105  square  feet  in  the  walls 
30,5  '  x24| '  =  754,875  in  the  ceiling 


9)1859,875 


206,652|-  square  yards 
2  cents 


$4,13305|. 
Or,  4  dollars  13  cents  3  mills. 

PAVERS'  WORK. 

Page  178. 

(35  ft.  4  in.)x(8  ft.   3  in.)     35J  x?=5,25  «q»J^  i^^ 

2.  -  =        -        -32,31 

Then,  32,3|  sq.  yds.x28  cts.=9  dolls.  06cts.  S|  mills 

3.  (27ft.  lOin.) x(14ft.  9in.)=27f-x  14,75=410,541  s.  f. 

410,54i-~9=45,613^  square  yards 

d.    cts,  ma.  ' 

Then,45,61|ix38cts.=1733,39|fcts.=17  33  3,9-f 

4.    45  feet — 5  feet  3  inches =39  feet  9  inches 

63ft.x(5ft.  3  in.)     „       ■  , 

—^ ^=7x5i  feet=36|  sq.  yds.  broad  stones 

63f..x(39ft.9in.)^^^3^^  ft.=3784  .q.  yds.  pebbles 

doU.8.  tts. 
Then,  36|  square  yardsx36  cts.=13  28 
278i     '  <    x30    *   =83  47i 


Sum  $96  70i 


painters'  work. 


123 


PAINTERS'  WORK. 


Page  179. 

2.     (74  ft.  10  in.)x(ll  ft.  7  in.)= 

(7  ft.  6in.)x(3  ft.    9m.)  = 

(6  ft.  8m.)x(3ft.  4  in.)x5= 

(22  ft.  8m.)x(lft.  2  in.)x5= 


(6  ft.  9  in.)  X5  feet: 


ft.  in. 
866  9 

If 
10 

28  1 

6  door 

111  1 

4  windows 

132  2 

8  breaks 

1138  3 

4 

-     33  9 

0  chimney 

■frrf 

1104  6 

4=1104|| 

And  1104^?. 


.9=122||-|-  square  yards 
Then,  122|||  s.  yds.  x8i  cts.=10  dolls.  43  cts.  l^  ms.. 


3.     (20  ft.  +  14  ft.  6  in.)x2=69  feet  round 

69ft.x(10ft.4in.)=69xlC^=713  square  feet 

(4  ft.  4  in.)x4  ft.=4i  x4=5       17*  feet  fire  place 
(3  ft.  2  in.)x6ft.x2=3ix6x2=38  feet  windows 


Then, 


713— 55i     6571 


9 


9 


Sum  55? 
=73^^  square  yards.* 


Page  180. 


ft- 


4.     (24^  ft.  +  16i  ft)x2xl2|  ft.  =  1039 

(3  ft.  6  in.)  x7  feet  =     24 

(7  ft.  9  in.)x(3  ft.  6  in.)x2=     54 

24ft.x(l  ft.  3in.)x2=     60 


in. " 
1  6 

6  0  door 
3  0  w.  shut. 
0  0  breaks 


(5  ft. 

6in.)x5  feet 

1177 
=     27 

10  6 
6  0 

1150 

4  6 

9)1150| 

127f| 


Then,  1274|x6  cts.=7  dolls.  66  cts.  %  ms. 

*  The  same  as  question  4,  plasterers'  work. 


124  MEASUREiyiENT   OF  GROUND. 

GLAZIERS^  WORK. 
Page  181. 

2.  (7  ft.  3  in.)  x(3  ft.  5  in.)=24  square  feet  9  in.  3  sec. 

3.  (6  ft.  11  in.-f  5  ft.  4  in.4-4  ft.  3  in.)x(3  ft.  6  in.)x7: 
16,5  x3,5  X 7  =404,25  square  feet 

Then,  404,25  sq.  ft.xl4i  cts.=58  d.  61  cts.  6,25  ms. 


/ 16  ft.  9  iii.\     „  ^i"^'^  ^^^^ 

I    \     .^.    ^«^-     104,6875 


4.  (12ft.  6in.)x(^^^^!^^ 
Then,  104,6875  sq.ft.xlO  cts.=10  d.  46  cts.  8|  ms. 

5.  (14 ft.  6  in.)x(4  ft.  9 in.)==6S  sq.  ft.  10  in.  6  sec. 

6.  (7ft.  10in.+6ft.8ia.-j-5ft.4in.)x(3ft.  Ilin.)x9=  • 

110     47 
19|.x3jix9=-^X T^X^=*V'=699i  square  feet 

2       4 
Then,  699-J  square  feet X 14  cents=97  dollars  87|  cents 

MEASUREMENT  OF  GROUND. 
1st.     To  fad  the  content  of  a  square  piece  of  ground. 

Page  182. 

2.  35x2=70  perches 

Then,  70x70=4900  square  feet=30  A.  2.  R.  20  P. 

sq.  perches    A.  R.    P. 

3.  16^Xl6i=16,5Xl6,5=272,25=l  2  32i 

2d.     To  find  the  content  of  an  ohlong  piece  of  ground, 
called  a  parallelogram. 

Page  183. 

120  perchesX84  perches     __ 

2.    '—- 1-^^ =63  acres 

160  perches 


MEASUREMENT   OF   GROtJjVD.  125 

Page  183. 
„      240x120     28800     ,^^         ,      ^_    , 

^'  i6i3n6r-272r='''p"^^"^'^'^^^^^ 

But,  105  per.  213|=2  rods  25  perches  2134  feet 

3d.     To  find  the  content  of  a  triangular  piece  of  ground. 

Page  184. 

2.  75x*/=75x25=1875  per..=  ll  A.  2  R.35P. 

3.  120x  V=120x42=5040  per.=31  acres  2  roods 

A.  R.  dolls.  dolls,  cts. 

Then,  (31  2)x45=31,5x45=^1417  50 

4.  UOft.xV  ft.=140x35=4900 square  feet 
Then  4900-^9=544  yards  4  feet. 

4th.     To  find  the  content  of  a  piece  of  ground,  in  the  form  of 
an  oblique  parallelogram. 

Page  185. 
,2.     (S0x24)-T- 160=12  acres 

5th.     To  find  the  content  of  apiece  of  ground  hounded  hyfour 
sides,  none  of  which  are  parallel  or  equal. 

Page  186. 

^_      120x(48+24) 

2.    ^r -'=4320  perches==27  acres  • 


6th.     To  find  the  area  of  a  piece  of  ground  lying  in  a 
circle,  or  ellipsis. 

Page  187. 

perches   A.  R.    P. 

2.  30x20x,7854=471,24==2    3  31,24 

3.  160xl60x,7854=20106,24  square  feet 

.     But,  20106,24-j-9=2234,02|  square  feet. 
11* 


126  MECHANICAL  POWERS. 

GAUGING. 

Page  189. 

2.  28+1(24— 18)=18+4=22  mean  diameter 

4 
T^^"'  ?^2giM==i||6  =39- gallons 
49 

3.  28+1(36— 28)=284-5*==33J  mean  diameter 

20 
33*X38ix40     100X100X^0     2ooooo_if/T.7 
'  294         =      9x^^^       ~  ^^'^   -^^^T32J 

147 

4.  15+1(18— 15)=15+2  =  17  mean  diameter 

Then,  ^^g— -  V5¥=23^fl  gallons 

5.  18+1(22— 18)=18+2f=20|  mean  diameter 

4 
20|x20|x36    62x62  x^^     ,,3,3      ..^^g      „ 
^^^^'  -^9^ =-?^59 S^tF  =42111  gallons 


MECHANICAL  POWERS. 

1st.      THE   LEA'^ER. 

Page  190. 
As  1  inch  :  20  inches  : :  5  lb.  :  100  pounds  the  answer 

2d.      THE  ^^'BEEL  AND  AXLE. 
Page  191. 
As  60  inches  (=5  lb.)  :  6  inches  : :  101b.  :  1  lb. 


PR03USCU0US   QUESTIONS. 

PROMISCUOUS   QUESTIONS. 

Page  192. 
Quest  1.    2578+2578=5156 

2.  14&7U— 1^^17  6=14676— 3609=11007 

3.  1468— (46H-581)=1468— 1042=426 

4.  ^  of  100  cents  =33^  cents 
^of^=;iof  100  cts.=:16f     • 

Sum  50  cents 

5.  1080-7-45^24  the  number  required 

6.  (476)2 --4J6=2  2^6_w  6^952 

7.  Suppose  one  side  of  the  square  to  consist  of  100  men 

Then,  100^ -{-231=10231  the  whole  number  of  men 
And     lOP —  44=10157  the  whole  of  the  men 

74  eiTor  too  little 

Again,  suppose  the  side  of  the  square  to  have  120  men 
Then,  1202+231=14631  the  whole  number  of  men 
And     12P —  44=:14597  the  whole  number  of  men 


127 


34  error  too  little 
Errors  alike. 

Then    (120X74)-(100X34)_        _,,^  ^^^  ^i^^ 

1  hen,  ^^ 3^  4  o 

Lastly,  (137)2+231,  or  (138)2—44=19000  men 

8.  113x147—213=16611—9261=7350 

9.  2  of  |=f=i  then,  as  i  :  1260  d.  : :  1  :  5040  dolls. 

10.  3500—2500=1000  dollars 
Then,  2500  dollars)   dolls.   J 100  dollars 


8  years   \   1000    (     1  year 
40 


128  PROMISCUOUS  aUESTI0?f3. 


11.     Asf  :  210  ::  1  :  — - — =$315  elder  brothers  por. 


Page  193. 
210^x3 

Now,  315x3x2=1890  dollars 

12.  (650—130)  x400=520  d.  x4  dolls.  =20S0  dolls. 

13.  17+8+46-f  (20x2)«lll  years 

14.  1000— (350+400)=250  dollars  C's  share 

dolls,    dolls.  dolls,  dolls. 

Then,  as  250  :  500  •.•  as  1  :  2  : :  350  :  700  A  put  in 

And,  as  1  :  2  : :  400  dolls.  :  SOO  dolls.  B's  cloth  is  worth 

15.  ^  of  2720=544  dollars  the  captain's  share 
Then,  — T^(T =  tW  =13  d.  60  cts.  a  sailor's  share 

16.  As  6  :  100  : :  972  dolls.  :  16200  dolls,  her  portion 
And,  as  |  :  16200  dollars  : :  1  :  27000  dollars 

Then,  27000-T-3=9000d.int.  of  the  father's  estate  1  year 

dolls.      dolls. 
Lastly,  as  6  :  100  : :  9000  :  150000  the  father's  estate 

17.  (4ft.6in.)x(2ft.9in.)x(3ft.4in.)=41  sq.ft.  3  in. 

IQ         l_|_5__Il_?_17   aiifl   1 17__;  1 

^^-       ^i^F"—      ig      —18   ^^^  ^       T¥  — Tf 

Then,  as  tV  :  12  : :  1  :  12x18=216  feet 

Page  194. 
19.     11|^=6^4=2U  years 

1710v20 

20.  As/o(=iof|)  :  1710  : :  1  :         ^      «38Q0  dolls. 

21.  63«  x(V)^  =3969x992,25=3938240,25  product 
And  (63xV)—(63+V)=1984,5— 95,5=1890  diff. 

mi.  day     miles  da.  h.  min. 

22.  As  68  (=34x2)  :  1  : :  2000  :  2||o=29  9  52|f 


PKOillSCUOUS   QUESTIONS.  129 


Page  194. 

S8 
~"6 


lb.  times  lb.  C.qr.    lb.  885x2       ^^^     . 

23.     6:2::  SS5(=7  3  17)  :  — ^ — =29d  times 


24.  Suppose  Ijthen  l-f§=|  sum 
And,  as  4  :   1  : :  20 

25.  21x|=y=15| 

24+5 
^'-     si-F—    4y    — TO- 

28.     Suppose  the  mimber  to  be  6 

Then,  6x3 — 8=10  the  wiiole  of  his  money 
And,    6x2-|-3  =  15  the  whole  of  his  money 

5  error 
Again,  suppose  10  to  be  the  number 
Then,  10x3—8=22  his  money 
And,    10x2+3=23  his  money 

1  error 
Errors  alike. 

-/       5x10—6x1      ..      ,-    ,.,j 
Now, =4  4.^1  J  children 

29.     Inverse  100  dollars  j  year  (509  dollars  inverse 
6  dollars^      1      ^500  dollars 

Then,  M4^^='r=16  years  8  months 

30.     Suppose  the  number  to  be  100 

Then,  i|o  +  15+i°o  +  10  =  10S^  the  members 
But,  lOSi— 100=S*  error  too  little 
Again,  suppose  200  members  in  all 
Then,  2oo^i5_^2oo+i0=191|  the  members 
But,  200— 191f =Si  error  too  much 
Errors  unlike. 

^      ,     200x8^  +  100x8*     200+100      ,,,     ,.^ 


130  PROMISCUOUS   QUESTIONS. 

Page  194. 

31.  360  deg.x69i  miles=25020  miles  round  the  earth 
Then,  as  20  miles  :  1  day  : :  25020  :  2*2P_2  0^^251  days 
Lastly,  1251  days-i-3654  days=3  years  155^  days 

Page  195. 

32.  100x3i=  325 
150x4^=  675 
204x5|.-r=1173 


454  )2173(4  months  23-}^^  days 

da.   ID.     da.  iv. 
33.     As  7  :  1  : :  1  :  ^  A  can  do  alone  in  one  day 

And,  as  12  :  1  :  :  1  :  .^^  the  part  B  can  do  in  a  day 

124-7 
Then,  i4-^Tj=-^_  a=i.j  the  part  of  the  work  that  they  can 

do  in  a  day,  working  together 

Lastly,  as  J-|  work  :  1  day  : :  1  work  :  ||=4^8_  ^^ys 

34     (1,05^— l)x40Q^     ,40710042265625x400 

1,05-4        ^  ;o5 

=3256,80338125=3256  dollars  SO  cents  3  mills 

Or  thus,  by  the  Table,  page  86. 

1.34009x400=^530.036 

1.27628x400=510.512 

1.21550x400=486.200 

1.15762x400=463.048  i 

1.10250x400—441.000 

1.05000x400=400.000 


$3256.796 

mo.  dolls,  mo.  d.' 
35.     As  12  :  5  : :  4  :  If  and  100  +  lf  =  101|      Then, 

dolls.       dolls.    inOv700v:3 
As  305(=,ioi|)  :  100  ::  700  :        ^  '  =688  52  4| «■ 

*RuLE.    A= ^-——.     Here  tt=the  annuity  R=: 

R — 1 
the  ratio  or  amount  of  £.1,  or  $1,  in  one  year  as  in  compound  interest; 
f=thetime  for  which  the  annuity  is  in  arrear.     A=the  amount  of  an- 
nuity at  the  end  of  t  years. 


PR03USCU0US  aUESTlONS.  131 

Page  195. 

21-1-24 
36.    I  -f-4=-^=|J  and  l-fl  =}|  C's  parts 

7 

Then,  as  H  :  140  : :  |  :M_^i^^=^rr^=267  d1^^^^^ 

8 
dolls.  A^yl40v3  ^o''»-   cts. 

And,  as  i^  :  140 ::  ?  ;^^^      /'   =.3^360 ^305  45.5^  B paid 

37.     Suppose  the  number  to  be  12. 

Then,  12 x4-f  16=84  his  money 
And,  12x6— 12=60  his  money 

4  error  too  little 

Again,  Suppose  20  to  be  the  number  of  beggars 

Then,20x4-fl6=  96  his  money 
And  20x6—12=108  his  money 

12  error  too  much 

EiTors  unlike. 

_         12x12+20x4     _^     ^., 
Then,  — -^ =2_2^4:^14  beggars 

38.  As  18  :  1  : :  1  :  -fj  the  part  B  and  C  can  do  in  a  day 
working  together. 

And,  as  11  :  1  : :  1  :  -fj  the  part  of  the  work  they  can  do  in 
a  day,  all  working  * 

1 0 1  1 

Now,  Jf — iV=    luQ    ='m  th6  part  of  the  work  that  A  can 

do  in  a  day 

Lastly,  as  ^|^  :  1  : :  1  :  i|8^28  days  3hrs.  25f  min. 

39.  Suppose  that  one  dollar  was  the  sum  paid  by  4  meraberg 
of  Congress — or  5  merchants — or  8  lawyers — or  12  citizens. 

doll.  doll. 


Then,  As  4  : 

1  :• 

20 

5 

5 

1  : 

30 

6 

8 

:  1  : 

:  24 

:  3 

12 

:  1  : 

24 

:  2 

$16 
Tiz:  16  dolls,  would  then  be  the  whole  bill. 


132  PROMISCUOUS   QUESTIONS. 

Page  195. 


(Question  39  continued. 

7  77     ^77  45  :  60  paid  bv  the  congress 

dolls,  dolls.  )g  :  72     '       '*       merchants 

^^-h^^  16  :  192  ::  <^  .^q    ,        ,       j^^^^^^.^ 


i         i 


Citizens 


Page  196. 
40.     28  x20      =560  square  inches 
14x10x2=280      '  ' 

280=11  acres 

41.    200  acres=32000  perches=& 

40  peTches=fl.     The  rule  is,* 
2  2 

\/(f  +^)— I  the  short  side,  andv/(|  +^)4-|  tne  long  side 

Th\is,s/{'^-£^  H-32000)— %o=100  the  short  side 

And  v'(^4°%32000)  +  %°=200  the. longer  side 

42.     Suppose  the  side  of  the  square  meadow  to  be  1 ; 

Then,v/l^H-l^=N/2=  1,4142136  its  diagonal 

And,     1,4142136— 1=,4142136  what  the  diagonal  is  more 

Now  say,  as  ,4142136  :  1  : :  20  :  48,28426  the  side  of  the 
required  square  in  perches.  Lastly,  square  the  side,  and  we  get 
the  contents  2331,37  perches=14a.  2r.  llp.t 

Solution  hy  Algebra. 

*  Let  a:=  the  short- side,  a=the  difference  of  the  sides,  and 

&=the  contents  in  perches 

Then,  {x-\-a)  Xa^=the  contents  of  the  ground  in  perches. 

That  is,  J'  -\-ax=b  by  the  question. By  completing  tiie 

22 
square  we  get  2,2 +«a;H- J  =f  -{-h;  and  by  evolution  a;+J= 

Vt:  +&)  consequently  aj=\/ J  +^+2" 
t  Pat  a;=the  side  of  the  square,  and  tf=20; 

Then,(Euchd  \.n)2x^={x-\-aY=x^-^2ax  -^-x^  by  trans- 
position we  get  x^ — 2ax=ta^  and  by  completing  the  square  a:*- 

2ax+a2=2a2j  by  evolution  x — a=^'Za^  hence  .T=a+av'2. 
Now  by  restoring  the  value  of  ct  wc  get  x  the  side  of  the  square  to 
be  20-f20x/2=20+20x  1,4142136=48,284272  perches  in 
the  side  the  same  as  above,  whence  the  contents  may  be  readily 
found. 


PROMISCUOUS   aUESTIONS.  133 

Page  196. 

d.      ft.         pence  L. 

43.     As  18  :  1  : :  24000(=100):  2  4_o^oo=40^oo  square  ft. 

And,  4%o  0  -T- y  =*  V  °  X 4^=  1-^1°  feet  perpendicular 

Then,  v'l  442^(10  0  0)^  |  =^\^{U^xSS^  +  mO^)=.^\ 

n/(112x332+2502)=/j^194269  one  of  the   equal  sides; 

consequently  3^3^ ^Z 1 94269  =  106,85  &c.  the  sum  required. 

Or,  decimally 

As,  18  :  1  :  :  24000  :  1333,33  &c.  square  feet 

1333  33 
And,  —    ''    =30,303  &c.  the  perpendicular 


Then,  2  v/442+30,3032=2v^2854,27+  =106,85  feet 

44.  Half  an  acre  80  per.  x30i  =2420  square  yards 

Then,    -=3081,23  the  square  of  the  diameter. 

'  ,7854  '  ^ 

,      ,      v/3081.23     55,5     __        , 
Lastly,  - — ^—  =^  =27,75  yards 

45.  114  yards  6  feet=1032  square  feet 
Then,  1032-^28=36|-  feet 

46.  7x2^x2  . 

^ =  35 S3 1 12  mches 

3 

47.    20  feel  =240  inches 
And,  240xUxH=240xfX|=^  ^1^=375  cubic  inches 
Haifa  ton  is  1120  pounds;  also,  50  feet=600  inches 
And,  600x|-X|=2  V4?^=459|  cubic  inches 

cubic  in.      lb.  cubicin.      lb. 

Then,  as  375  :  1120  :  :  459f  :  1372 

lb.     d.  lb.         pence       L.   s.  d. 

Lastly,  as  1  :  34  : :  1372  :  4802=20  0  2 


48.  Here  x/392— 152  =  ^/1521— 225=v/1296= 

36  feet  standing 
Then,  36+39=75  feet  the  whole  height 

49.  Heie  work  backwards 

Thus,  24x9=216;    ^216=6;   6x|=8;    8^=64;  (54— 
4=60;  60-7-5  =  12;  12x8=96;  lastly,  96+7=103. 

12 


134  PROMISCUOUS   aUESTIONS. 

Page  197. 
50.     First,  because  that  -^  of  the  whole  contents  was  drawn 
out  each  time,  consequently  -j-^  of  the  parts  was  drawn. 
Hence,  500  gallons  of  wine 

^V=  50  subtract 

450 


1 
To 

^  45 

405 

rV 

=  40^ 

364i 

1 

To 

=  36/o 

1  , 
To 

=32|fi 

2952V0  galls,  wine 
And,  500—295^0^  galls,  water 

Or  thus, 
Here  the  several  portions  of  wine  drawn  off  were 
50,     remains       450 

50x450         .      ^^^     50x450     450^ 

TTx — remams  450 — - — =— — - 

500  500         500 

50x4502         .    4502       50x450"  _4503 
____remams^^^^  500^  "500^ 

50x4503         .    4503      50x450s_450^ 
— 5QQ3-remams^^  5(KP         500^ 

50x4504         .    450  4      50x4504  _450^ 

5004     ^^"^^'"'5oF"~     5004     -500T 

^       ,      4505     18452812500000     _^.  .,       ..         .    . 
Therefore,_=--^^^^-^j^^^  gallons  of  wme 

Consequently,  500— 295/o%=204i|i  galls,  of  water* 


*  Analytically.    Put  500=c,  bO=d,  5=t,  and  2;=what  wine  remains 

after  drawing  t  times 

Then,  c — d  expresses  the  quantity  of  wine  left  after  the  first 

(c — d) xd  ^  .        »    .       , 

drawing;  c  :  c — d  ::  d  : the  quantity  of  wme  drawn 

,     (c—d)xd     (c—dy^, 

out  at  the  second  drawmg,  and  c — d = the 

°'  c  c 

quantity  of  wine  left  after  the  second  drawing. 


PROimSCUOUS  QUESTIONS.  135 

Page  197. 
2x2x4      16      .        ^ 

52.     1817,  April  1st.  principal  1864  dollars 

Oct.  15th  1864  doll.  mult,  by  197  days=36720S 
Paid  225,50 


1818  Jan.  10th  1638,50  X       87       =142549,5 

Paid    1SS,86 


May  16th  1449,64  X     126       =182654,64 

Paid    585 


Aug.  1st.     864,64  x       77       =  66577,28 

610)75898|9,42 


Interest  126,49 


Balance  $991,13 


Interest  ,^126,49 


36X6^X8|_2047^_     ,, 
^3.     —.^^^—_-^-_15^^,  cords 


Continuation  of  Note  on  p.  134. 
Also,  c  : ^—  ::  d  :  ^ -^ — the    quantity     of    wme 


.2 

drawn  out  at  the  third  drawing. 


(c— cZ)2     (c—d)^xd     (c—dy        ,  .-.      f    • 

And,  ^^ ^ =^^ =3  the  quantity  oi  wme 


left  after  the  third  drawing;  and  universally 

t— 1  t—1 

c  :   - — rj- — ::  d  :  ^^       ^ —the  quantity  of  wine   drawn 

.t—2  t—l 

c  c 
.  out  at  the  t  drawing. 

t—l  t 

.     ,  c — d       (c — d)         (c — d\  ,  .^       J.     . 

And, r-  —  ^ ^=- ^=rr=the   qutintity  oi  wme 

'  t—2  t—l       t—l  ^ 

6  C  C 

remaining  after  the  t  drawing.     By  substituting  the  values  of  c, 
d,  and  t,  in  this  formula,  we  get 
mQQ 5Q\5     450* 

— rnn^      =tf:?:7—^^^wSq  galls,  of  wine  the  same  as  above 
500^*  500*  ^"°  ^ 


136  PROMISCUOUS   QUESTIONS. 

Page  197. 
days     dolls,    cts.      day 

54.  As  365  :  356   34  : :  1  :  97  cents  6f  J  mills 

cts.     doll.       dolls,    cts.         dolls,    cts. 

55.  As  181  :  1  : :  2564  95i  :  13679  76 

56.  Inverse  8  men )   days  i  64  men  inverse 

$20,50     \     30       100  dollars 
25 

Then,  ?^^§^=^^=18H  days 
$       41 

57.  Inverse  34  men  ^  months  (86  men  inverse 

1  bridge )        6       (2  bridge 

6x34x^ 

!xl 
43 


Then,  -^r--2f*^2_o^4=4j|  months 


58.     As  4  :  6  : :  7  :  10i=10,5  dollars  C  must  pay  when 
A  pays  5  dollars  and  B  7  dollars. 

And,  5+7+104=221=22,5  dollars 

dolls,   dolls.        C  5    :  $    53  33i  A  receives 
Lastly,  as  22,5  :  240  : :     {7    :$    74  G6|  B 

(105,:$  112  00     C 


Page  198. 

dolls,  cts.  ms.     dolls,  cts.  dolls,  cts. 

59.  As  100  :  7     3  : :  8564  20  :  6     25+tax 

60.  (4galls.  2  qt.)— (2  galls.  2  qt.  1  pt.)=l  gall.  3  qt.  1 
pt.aBl5  pints  gained,  or  remaining  in  the  vessel  at  the  end  of 
one  hour. 

And,  84i  gallons  is  676  pints 
Then,  as  15  pts.  :  1  hr.  : :  676  pints  :  45  hours  4  minutes 

P.      ft.        ft.  sq.feet 

61.  One  acre  160x16,5x16,5=160x272^=43560 

Then,  43560—36=1210  feet=73i  perches 

62.  50  menj  bushels  (40  men 
30  daysi       12      |90  days 


PROMISCUOUS  aUESTIONS.  137 

Page  198. 
Question  62  continued. 

^        12x40x#     12x4x3     ...     ^Q .  ,     ,    , 
Then, — ,^    ^^^  X : =  i|4^28^  bushels 

63.     Suppose 

Boy  Ix  6=  6  cents 

Women  3x  8=24     ' 

Men  6x16=96    * 

126  cents 
cents    cents  (  6  :      90  cents  the  boys  get 


Then,  as  126  :  1890  : :     )24 

(96 


360  cents  the  women  get 
1440  cents  the  men  get 


Consequently,  90 -i-  6=15  boys      ) 

360-j-  8=45  women  V 

And  1440-7-16=90  men      ) 

64.  7x4i — 8x3=3U — 24=7i  miles  apart  in  one  day 
Then,  as  1  day  :  7i  mi.  : :  13  days  :  97i  miles  apart 

65.  962x6=5772         1635x34=5722,5 
320  4081 


1282x6=7692         1226^x5=6131,25 


A  13464  B  11853,75 

A  13464 
B   11853,75 


25317,75     Then, 

dolls,  cts.  ms. 
dolls,  cts.      ( 13464        :  258,79,51  Ifff 
As  25317,75  :  486  64  : :  (11853,75  :  227,84,4l-5f|f 


Proof  $486,64 
w  52803  x23        __ 

bt).     Here  2500x6  x52S0x3"~ 
44  352 

^^^0X^^^0X/^a^0xS_44x352x8_1239O4 
^^00X0xM#X^    ~~        25         ""     25 

25 
s4956^^  hours =206  days  12  hours  9  min.  36  seconds 


138  PROMISCUOUS   aUESTIONS. 

Page  198. 
67.    45i  ft.— 9i  in. =546  in.— 9^  in.=536i  inches 
And  9i  in. — 6|  in.=2|-  inches  gained  in  a  day 
Now,  536i-f-2|=iY3x^=2 146 ^195_L  days 
That  is,  it  will  be  within  9i  inches  of  the  top  in  195^^  days. 
Consequently  it  will  get  to  the  top  in  196^1^  days. 

And    1 3.1Zs=:_2  3_   Thpn 

2 
As/^:46::l:^|5-^^720trees 

69.     Suppose  200  dolls.    Again,  Suppose  300  dollars 
94  94 

106  206 


I  of  106=  53  lent  -J  of  206=103  lent 

^  of  106=  211-  clothes  }  of  206=  41|  clothes 

94  debts  94  debts 


$16Si  sum  gone  $238-1-  sum  gone 

^\  of  200=180  ^\  of  300=270 

ll-f  error  too  much       error  31|  too  much 

Errors  alike. 

^,        200x311—300x114     6360—3540     ,,,  ^  „ 

Ihen,  ■ — -^ ^— =141  dolls. 

'  31|— llf  20 


70.        ^—i-. 


Page  199. 
4—3      . 


12       '^ 
Then,  as  j\  :  84  : :  1  :  84x12=1008 

71.  Here  3 — 2  :  4  ft.  : :  3  :  12  feet  the  semidiameter  of 
the  circle  that  the  outer  wheel  makes.  Consequently  12  x2= 
24  feet  the  diameter  of  the  outer  circle. 

And,  24 — 4x2=16  feet  the  diameter  of  the  inner  circle 
Lastly,  24x3,1416=75,3984  feet  by  the  outer  wheel 
And,  16x3,1416=50,2656  feet  by  the  inner  wheel 


PROMISCUOUS   QUESTIONS.  139 

Page  199 

72.    Here*  if*  =one  of  the  equal  sides,  if  *=half  the  sum 
of  the  sides 

gtyg 250 

And  if^— 1|5= —  ^1^  one  of  the  equal  rem. 

12^2  1252 

Then,  >/l-^Xirx^|^Xir=-^x/,V=— ^X^v/3  = 
1252  15(525  15625x1,7320508 


62  ^  36    ^  36  ~" 

751,75816  square  feet. 

Or,  by  Rule  ii,  thus;  125-f-3=i|*  is  one  of  the  equal  sides 
of  the  triangle,  and  because  the  triangle  is  equilateral,  and  con- 
sequently equiangular,  the  included  angle  of  any  two  sides  is 
60°,  whose  natural  sine  is  ,8660254 

Hence,  if^X^f^ X,«^^f 2^4=751,7581596  square  feet 

•  Here  data  is  given  to  find  the  sides. 

The  rule  for  solving  this  question,  and  all  of  a  similar  kind,  is  by  raen- 
suration,  as  follows: 

1.  From  half  the  sum  of  three  sides  subtract  each  side  severally. 

2.  Multiply  the  half  sum  and  the  three  remainders  continually  to- 
gether, and  the  square  root  of  the  product  will  be  the  area  required. 

Rule  II.  Any  two  sides  of  a  triangle  being  multiplied  together,  and 
the  product  again  by  half  the  natural  sine  of  their  included  angle,  will 
give  the  area  of  the  triangle. 

That  is,  AC  X  C  B  X  natural  sine  of  the  angle  C=t wice  area. 


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