Question

 The denominator of a fraction in simplest form is greater than the numerator by 3. If 7 is added to the numerator, and 5 added to the denominator, then the fraction itself is increased by 1/2 . Find the original fraction.

Answer 1

So.. if we take the numerator to be say "a", then the denominator will be "a+3"

$\bf \cfrac{a}{a+3}\textit{ if we add }\frac{1}{2}\textit{ we get then }\cfrac{a+7}{a+3+5} \\\\\\ thus\implies \cfrac{a}{a+3}+\cfrac{1}{2}=\cfrac{a+7}{a+8}\\\\ -----------------------------\\\\ \cfrac{a+3+2a}{2(a+3)}=\cfrac{a+7}{a+8}\implies \cfrac{3a+3}{2a+6}=\cfrac{a+7}{a+8} \\\\\\ 3a^2+24a+3a+24=2a^2+14a+6a+42 \\\\\\ a^2+7a-18=0\implies (a+9)(a-2)=0 \\\\\\ a= \begin{cases} 2\implies &\frac{2}{5}\\\\ -9\implies &\frac{3}{2} \end{cases}$

either of those two fractions will do