$a, b$ are the integer part and decimal fraction of $\sqrt7$ find integer part of $\frac{a}{b}$ 
$a, b$ are the integer part and decimal fraction of $\sqrt7$
find integer part of $\frac{a}{b}$

using calculator :
$\sqrt 7$ = 2.645
$\frac{a}{b} = \frac{2}{0.6} = \frac{20}{6} = 3.333$
integer part of $\frac{a}{b} = 3$
how to do it without calculator? if i don't know $\sqrt7$ part

$2.6^2=6.76<7<7.29=2.7^2.$
So $a=2$ and $0.6<b<0.7.$ 
So $4>10/3=2/0.6>a/b>2/0.7=20/7>2.$ 
So the integer part of $a/b$ is $2$ or $3.$
Now $$3\le a/b\iff 3b\le a\iff  3b\le 2 \iff b\le 2/3 \iff$$ $$\iff  \sqrt 7=a+b=2+b\le 2+2/3=8/3\iff$$ $$\iff 7\le (8/3)^2=64/9 =7+1/9.$$ Since we used "$\iff$" throughout, and since it is true that $7\le (8/3)^2,$ therefore $3\le a/b.$ And we already have $a/b<4$. So we have $$3\le a/b<4.$$
