Question

The numerator of a fraction is 4 less than the denominator. If 1 is added to
both its numerator and denominator, it becomes 1/2.Find the fraction​

Answer 1

Answer :

$\longmapsto \boxed{\rm{\pmb { \red{ Fraction = \dfrac{3}{7}}} }}$

Given :

• The numerator of a fraction is 4 less than the denominator.

• If 1 is added to both its numerator and denominator, it becomes 1/2.

To calculate :

• Find the original fraction.

Calculation :

Let us assume the denominator as "x".

As per the provided question,

• The numerator of a fraction is 4 less than the denominator.

$\rm { \longmapsto Original \: Fraction = \dfrac{x-4}{x} }$

Also,

• If 1 is added to both its numerator and denominator, it becomes 1/2.

So,

$\rm { \longmapsto New \: Fraction = \dfrac{x-4+1}{x+1} }$

$\rm { \longmapsto \dfrac{x-4+1}{x+1} = \dfrac{1}{2} }$

$\rm { \longmapsto \dfrac{x-3}{x+1} = \dfrac{1}{2} }$

By cross multiplication,

$\rm { \longmapsto 2(x-3) = 1(x+1) }$

$\rm { \longmapsto 2x - 6= x + 1 }$

$\rm { \longmapsto 2x - x= 1 + 6 }$

$\rm { \longmapsto x= 7 }$

Henceforth,

$\rm { \longmapsto Original \: Fraction = \dfrac{x-4}{x} }$

Substituting value of x.

$\rm { \longmapsto Original \: Fraction = \dfrac{7-4}{7} }$

$\longmapsto \boxed{\rm{\pmb { \blue{ Original \: Fraction = \dfrac{3}{7}}} }}$

Verification :

As per the given question, when 1 is added to both its numerator and denominator, it becomes 1/2. So, we have to verify :

$\rm { \longmapsto \dfrac{x-4+1}{x+1} = \dfrac{1}{2} }$

LHS :

$\rm { \longmapsto \dfrac{x-4+1}{x+1} }$

$\rm { \longmapsto \dfrac{7-4+1}{7+1} }$

$\rm { \longmapsto \dfrac{3+1}{8} }$

$\rm { \longmapsto \dfrac{4}{8} }$

$\longmapsto \boxed{\rm{\pmb { \red{\dfrac{1}{2} }} }}$

RHS :

$\longmapsto \boxed{\rm{\pmb { \red{\dfrac{1}{2} }} }}$

LHS = RHS

Hence, verified!!

Answer 2

Let the denominator be = d

∴ Fraction = $\dfrac{d - 4}{d}$

As per condition,

$\dfrac{(d - 4)+1}{(d)+1} = \dfrac{1}{2}$

⇒ $\dfrac{d - 3}{d + 1} = \dfrac{1}{2}$

By cross multiplication,

⇒ $2(d - 3) = d + 1$

⇒ $2d - d = 1 + 6$

⇒ $d = 7$

By putting the values,

Fraction = $\dfrac{d - 4}{d}$ = $\dfrac{7 - 4}{7}$ = $\dfrac{3}{7}$.

∴ The fraction is 3/7.