There are 3 times as many boys as girls in a room. If 4 boys and 4 girls leave the room, then there will be 5 times as many boys as girls in the room. In total, how many boys and girls were in the room originally?
(A) 15
(B) 20
(C) 24
(D) 32
(E) 40
Solution 1

Originally there are 3 times as many boys as girls, so then for every 3 boys there is 1 girl and $3+1=4$ children in the room.

That is, the number of boys in the room is $\frac{3}{4}$ of the number of children in the room.

Next we consider each of the 5 possible answers, in turn, to determine which represents the total number of children in the room originally.

If the original number of children in the room is 15 (as in answer (A)), the number of boys is $\frac{3}{4} \times 15=\frac{45}{4}=11.25$.

Since it is not possible to have 11.25 boys in the room, then we know that 15 is not the correct answer.

If the original number of children in the room is 20 (as in answer (B)), the number of boys is $\frac{3}{4} \times 20=\frac{60}{4}=15$.

If the number of boys in the room was originally 15 , then the number of girls was $20-15=5$. Next we must check if there will be 5 times as many boys as girls in the room once 4 boys and 4 girls leave the room.

If 4 boys leave the room, there are 11 boys remaining. If 4 girls leave the room, there is 1 girl remaining and since there are not 5 times as many boys as girls, then 20 is not the correct answer.

If the original number of children in the room is 24 (as in answer (C)), the number of boys is $\frac{3}{4} \times 24=\frac{72}{4}=18$.

If the number of boys in the room was originally 18 , then the number of girls was $24-18=6$.

If 4 boys leave the room, there are 14 boys left and if 4 girls leave the room, then there are 2 girls left.

Since there are not 5 times as many boys as girls, then 24 is not the correct answer.

If the original number of children in the room is 32 (as in answer (D)), the number of boys is $\frac{3}{4} \times 32=\frac{96}{4}=24$.

If the number of boys in the room was originally 24 , then the number of girls was $32-24=8$. If 4 boys leave the room, there are 20 left and if 4 girls leave the room, then there are 4 left. Since there are 5 times as many boys as girls, then we know that the original number of children is 32 .

(Note: We may check that the final answer, 40, gives 30 boys and 10 girls originally and when 4 boys and 4 girls leave the room there are 26 boys and 6 girls which again does not represent 5 times as many boys as girls.)

## Solution 2

Originally there are 3 times as many boys as girls, so if there are $x$ girls in the room, then there are $3 x$ boys.

If 4 boys leave the room, there are $3 x-4$ boys remaining.

If 4 girls leave the room, there are $x-4$ girls remaining.

At this point, there are 5 times as many boys as girls in the room.

That is, $5 \times(x-4)=3 x-4$ or $5 x-20=3 x-4$ and so $5 x-3 x=20-4$ or $2 x=16$ and so $x=8$.

Therefore, the original number of girls in the room is 8 and the original number of boys is $3 \times 8=24$.

The original number of students in the room is $24+8=32$.

ANSwER: (D)

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