Question

The sum of the numerator and the denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes \[\dfrac{1}{2}\]. Find the fraction.

Answer 1

Hint: To solve the question given above, we will first assume that the numerator of the given fraction is p and denominator is q. Then, with the help of data given in question, we will form two linear equations in two variables. We will find out the values of p and q by solving the equations with the help of a substitution method.

Complete step-by-step answer:
To start with, we will assume that the numerator of the given fraction is p and the denominator is q. Thus, the fraction obtained will be:
$Fraction=\dfrac{p}{q}........\left( 1 \right)$
Now, it is given in the question that the sum of the numerator and the denominator will be equal to 12. Thus, we have the following equation:
$p+q=12.........\left( 2 \right)$
Another information given in the equation is that, when the denominator is increased by 3, the new fraction becomes $\dfrac{1}{2}.$ So, in the new fraction, the numerator=p and denominator becomes=q+3. So, we have the following equation:
$\dfrac{p}{q+3}=\dfrac{1}{2}$
On cross multiplication, we will get the following equation:
$\begin{align}
  & 2p=q+3 \\
 & 2p-q=3.......\left( 3 \right) \\
\end{align}$
Now, we have got two linear equations in two variables (p and q). So to find the values of p and q from the above equations, we will use the substitution method. According to this method, we substitute the value of one variable from one equation to the other equation. Thus, we will now try to substitute the value of q from (3) to (2). Thus, we will get:
$\begin{align}
  & 2p-q=3 \\
 & \Rightarrow 2p-3=q \\
 & \Rightarrow q=2p-3 \\
\end{align}$
Now, we will substitute this value of q in equation (2). After substitution, we will get:
$\begin{align}
  & p+2p-3=12 \\
 & \Rightarrow 3p-3=12 \\
 & \Rightarrow p=5.......\left( 4 \right) \\
\end{align}$
Now, we will put values of p from (4) to (2). Thus, we will get;
$\begin{align}
  & \Rightarrow 5+q=12 \\
 & \Rightarrow q=12-5 \\
 & \Rightarrow q=7.........(5) \\
\end{align}$
Now, we have to put these values of p and q from (4) and (5) into (1) to obtain the fraction. Thus, we have:
$Fraction=\dfrac{5}{7}$

Note: The pair of linear equations in two variables which we have found in the above solution can also be solved using the elimination method. This is as shown:
$\begin{align}
  & p+q=12.........\left( 2 \right) \\
 & 2p-q=3..........\left( 3 \right) \\
\end{align}$
We will add both the equations. Thus, we will get:
$\begin{align}
  & p+q\left( 2p-q \right)=12+3 \\
 & \Rightarrow 3p=15 \\
 & \Rightarrow p=5 \\
\end{align}$
On putting this value of p in (2), we will get q=7.