Question

Form the pair of linear equations in the following problem and find their solution (if they exist) by any algebraic method:

If we add $1$ to the numerator and subtract $1$ from the denominator, a fraction reduces to $1$. It becomes if we only add $1$ to the denominator. What is the fraction?

Answer 1

Step 1: Form the linear equations

Let us assume that the fraction be $\frac{a}{b}$.

According to the given information,

$\frac{a+1}{b-1}=1$

$\Rightarrow$ $a+1=b-1$

$\Rightarrow$ $a-b=-2\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots ..\left(i\right)$

$\frac{a}{b+1}=\frac{1}{2}$

$\Rightarrow$ $2a=b+1$

$\Rightarrow$$2a-b=1\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \left(ii\right)$

Step 2: Find the solution

When equation $\left(i\right)$ is subtracted from equation $\left(ii\right)$ we get,

$2a-b-a+b=1+2$

$\Rightarrow$ $a=3$

When $a=3$ is substituted in equation $\left(i\right)$ we get,

$3–b=-2$

$\Rightarrow$$-b=-5$

$\Rightarrow$ $b=5$

Hence, the fraction is $\frac{3}{5}$.