Question

The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the factor.

Answer 1

Hint: In this question, we are given two statements regarding the relationship between the numerator and denominator of the fraction. We have to determine the fraction. For this, we will suppose the numerator of the fraction as x and the denominator of the fraction as y. Then we will form two equations using the given statement. After that, we will solve them to find values of x and y and hence we will find the required fraction.

Complete step-by-step solution
Here we are given two statements regarding the relationship between the numerator and the denominator of the fraction.
Let us suppose that the numerator of the fraction is x and the denominator of the fraction is y.
So our fraction looks like $\dfrac{x}{y}$.
Now, let us form an equation using the given statement. We are given that, the sum of the numerator and the denominator of a fraction is 3 less than twice the denominator. Hence, (x+y) is 3 less than 2y.
So our equation becomes $x+y=2y-3$.
Simplifying we get: $x+y-2y=-3\Rightarrow x-y=-3\cdots \cdots \cdots \left( 1 \right)$.
Now numerator and denominator are decreased by 1, so new numerator = x-1 and new denominator = y-1. The new numerator becomes half the denominator. Therefore, x-1 becomes half of (y-1).
Our equation becomes $\left( x-1 \right)=\dfrac{1}{2}\left( y-1 \right)$.
Cross multiplying we get:
$\begin{align}
  & \Rightarrow 2\left( x-1 \right)=y-1 \\
 & \Rightarrow 2x-2=y-1 \\
 & \Rightarrow 2x-y=-1+2 \\
 & \Rightarrow 2x-y=1\cdots \cdots \cdots \left( 2 \right) \\
\end{align}$
Now let us solve equation (1) and (2) to get values of x and y. We will use elimination method for solving.
Subtracting equation (1) from (2) we get:
$\begin{align}
  & 2x-y-x+y=1-\left( -3 \right) \\
 & \Rightarrow x=1+3 \\
 & \Rightarrow x=4 \\
\end{align}$
Putting values of x in equation (1) we get:
$\begin{align}
  & \Rightarrow 4-y=-3 \\
 & \Rightarrow 4+3=y \\
 & \Rightarrow y=7 \\
\end{align}$
Hence the values of x and y are 4 and 7 respectively. So our fraction becomes $\dfrac{4}{7}$.

Note: Students can make the mistake of taking 3-2y instead of 2y-3 while forming equation (1). Make sure to decrease both the numerator and denominator by 1. Students can also solve these equations using a substitution method or cross multiplication method, values will remain the same.