Answer to Question #129059 in Algebra for Damilani

Question

If the numerator and denominator of a fraction are both increased by 5, the fraction becomes 5/6 . If the numerator and denominator are both decreased by one the new fraction becomes2/3 . Find the original fraction.

Answer 1

Let $\dfrac{a}{b}$ be a fraction.

$$\dfrac{a+5}{b+5}=\dfrac{5}{6}$$

$$\dfrac{a-1}{b-1}=\dfrac{2}{3}$$

$$6(a+5)=5(b+5)$$ $$3(a-1)=2(b-1)$$

$$6a+30=5b+25$$ $$3a-3=2b-2$$

$$6a=5b-5$$ $$3a=2b+1$$

$$2(2b+1)=5b-5$$ $$a=\dfrac{1}{3}(2b+1)$$

$$b=7$$ $$a=5$$

The original fraction is $\dfrac{5}{7} .$