Question

In a fraction, if the numerator is increased by 2 and the denominator is decreased by 3, then the fraction becomes 1. Instead, if the numerator is decreased by 2 and denominator is increased by 3, then the fraction becomes \[\dfrac{3}{8}\]. Find the fraction.

Answer 1

Hint: In this question, we will proceed by letting the required fraction as a variable. Then by using the given data we will get two equations. Solving them by substitution method we get the required answer.

Complete step-by-step answer:
Let the required fraction be \[\dfrac{a}{b}\].
Given that if the numerator is increased by 2 and the denominator is decreased by 3, then the fraction becomes 1 i.e., \[\dfrac{{a + 2}}{{b - 3}} = 1\]
\[
   \Rightarrow a + 2 = b - 3 \\
   \Rightarrow a = b - 3 - 2 \\
  \therefore a = b - 5................................\left( 1 \right) \\
\]
Also given that in the fraction if the numerator is decreased by 2 and the denominator is increased by 3, then the fraction becomes \[\dfrac{3}{8}\] i.e., \[\dfrac{{a - 2}}{{b + 3}} = \dfrac{3}{8}\]
\[
   \Rightarrow 8\left( {a - 2} \right) = 3\left( {b + 3} \right) \\
   \Rightarrow 8a - 16 = 3b + 9 \\
   \Rightarrow 8a = 3b + 9 + 16 \\
   \Rightarrow 8a = 3b + 25...........................\left( 2 \right) \\
\]
Substituting equation (1) in (2), we get
\[
   \Rightarrow 8\left( {b - 5} \right) = 3b + 25 \\
   \Rightarrow 8b - 40 = 3b + 25 \\
   \Rightarrow 8b - 3b = 25 + 40 \\
   \Rightarrow 5b = 65 \\
  \therefore b = \dfrac{{65}}{5} = 13 \\
\]
Substituting \[b = 13\] in equation (1), we have
\[\therefore a = 13 - 5 = 8\]
Thus, the required fraction is \[\dfrac{8}{{13}}\].

Note: A fraction represents a part of a whole or, more generally, any number of equal parts. In a fraction, the numerator denotes the upper part of the fraction and the denominator denotes the lower part of the fraction.