Question

A fraction becomes $\frac{9}{11}$ if $2$ is added to both numerator and denominator. If $3$ is added to both numerator and denominator it becomes $\frac{5}{6}$, find the fraction by substitution method.

Answer 1

Step 1: Find the value of $y$

Let the Numerator of the fraction to be identified be $x$

Let the Denominator of the fraction to be identified be $y$

Hence, the fraction will be$=\frac{x}{y}$

According to the given data, we note that

If $2$ is added to both numerator and denominator the fraction becomes $\frac{9}{11}$.

In equation we can represent it as

$\frac{(x+2)}{y+2}=\frac{9}{11}$

On Cross multiplying ,

$11x+22=9y+18$

Subtracting $22$ from both sides,

$11x=9y-4$

Dividing by $11$, we get

$x=\frac{\left(9y-4\right)}{11}\to \left(1\right)$

Then, another given condition if $3$ is added to both numerator and denominator it becomes $\frac{5}{6}$,

The equation can be written as

$\frac{x+3}{y+3}=\frac{5}{6}\to \left(2\right)$

On Cross multiplying ,

$6x+18=5y+15$

Subtracting the value of $x$, we get

$6\left[\frac{9y-4}{11}\right]+18=5y+15$

Subtracting $18$ from both the sides

$$\begin{gathered} 6\frac{\left[9y-4\right]}{11}=5y-3 \\ 54-24=55y-33 \\ -y=-9 \\ y=9 \end{gathered}$$

Step 2: Find the value of $x$

Putting this value of $y$ in equation $\left(1\right)$, we get

$$\begin{gathered} x=9y-4 \\ x=\frac{81-4}{77} \\ x=\frac{77}{11} \\ x=7 \end{gathered}$$

Hence, the fraction is $\frac{7}{9}$