Question

The denominator of a fraction exceeds the numerator by $2$. If $5$ be added to the numerator the fraction increases by unity. The fractions are.
A. $\dfrac{5}{7}$
B. $\dfrac{1}{3}$
C. $\dfrac{7}{9}$
D. $\dfrac{3}{5}$

Answer 1

Hint: In this problem we need to find the fraction which satisfies the given conditions. First, we will assume the numerator of the fraction as $x$. Given that the denominator of the fraction exceeds the numerator by $2$, that means the denominator is $2$ greater than $x$. From this we can write the fraction. In the fraction they mentioned that $5$ is added to the numerator then the fraction is increased to unity. From this we can make an equation in terms of $x$. Now we will solve the obtained equation by using basic mathematical operations to get the value of $x$. After having the value of $x$, we can find the required fraction.

Complete step-by-step solution:
Let us assume the numerator of the fraction is $x$.
In the problem they have mentioned that the denominator of a fraction exceeds the numerator by $2$, then the denominator will be $x+2$. Now the fraction will be
$\dfrac{x}{x+2}$.
Adding $5$ to the numerator of the above fraction, then the fraction is modified as
$\dfrac{x+5}{x+2}$.
In the problem we have given that when $5$ is added to the numerator of the fraction, then the value of the fraction is increases by unity, mathematically we can write this as
$\dfrac{x+5}{x+2}=\dfrac{x}{x+2}+1$
Simplifying the above equation by using the basic mathematical operations, then we will get
$\begin{align}
  & \dfrac{x+5}{x+2}=\dfrac{x+x+2}{x+2} \\
 & \Rightarrow \dfrac{x+5}{x+2}=\dfrac{2x+2}{x+2} \\
\end{align}$
In the above denominators on both sides of the equation are the same. So, equating the numerators to each other, then we will get
$x+5=2x+2$
Simplifying the above equation by using the basic mathematical operations, then we will have
$\begin{align}
  & 2x-x=5-2 \\
 & \Rightarrow x=3 \\
\end{align}$
We have the value of $x$ which is the numerator as $3$. Now the denominator would be $x+2=3+2=5$.
So, the required fraction is $\dfrac{3}{5}$.
Hence option – D is the correct answer.

Note: In this problem we need to consider the terminology of the given problem statement. In the problem they have mentioned that the value of fraction is increased by one so we have added one to the original fraction and equated it to the modified fraction. If they have mentioned that the fraction will become unity then we will equate the modified fraction to one and simplify the equation.