Question

Numerator of a fraction is 5 more than its denominator. If 4 is added to numerator and denominator, the fraction obtained is 6/5. Find the fraction

Answer 1

Given :

• Numerator of a fraction is 5 more than its denominator.
• If 4 is added to numerator and denominator, the fraction obtained is 6/5

To Find :

• The fraction.

Solution :

Let the numerator of the fraction be x.

Let the denominator of the fraction be y.

Fraction => $\mathtt{\dfrac{x}{y}}$

✪ Case 1 :

$\mathtt{Numerator\:=\:Denominator\:+\:5}$

$\longrightarrow$ $\mathtt{x=y+5}$

$\mathtt{x-y=5}$ _____(1)

✪ Case 2 :

$\longrightarrow$ $\mathtt{\dfrac{Numerator\:+\:4}{Denominator+4}}$ = $\mathtt{\dfrac{6}{5}}$

$\longrightarrow$ $\mathtt{\dfrac{x+4}{y+4}}$ = $\mathtt{\dfrac{6}{5}}$

$\longrightarrow$ $\mathtt{5(x+4)\:=\:6(y+4)}$

$\longrightarrow$ $\mathtt{5x+20=6y+24}$

$\longrightarrow$ $\mathtt{5x-6y=24-20}$

$\mathtt{5x-6y=4}$ ______(2)

✪ Multiply equation 1 by 6,

$\longrightarrow$ $\mathtt{6\:\times\:x\:-\:6\:\times\:y\:=\:6\:\times\:5}$

$\mathtt{6x-6y=30}$ ____(3)

Solve equation 2 and equation 3 to find value of x and y.

✪ Subtract equation 3 from 2,

$\longrightarrow$ $\mathtt{5x\:-6y-(6x-6y)\:=\:4-30}$

$\longrightarrow$ $\mathtt{5x-6y\:-6x+6y\:=\:-26}$

$\longrightarrow$ $\mathtt{-x\:=\:-26}$

$\longrightarrow$ $\mathtt{x=26}$

$\large{\boxed{\rm{\red{Numerator\:=\:x\:=\:26}}}}$

Substitute x = 26 in equation 1,

$\longrightarrow$ $\mathtt{x-y=5}$

$\longrightarrow$ $\mathtt{26-y=5}$

$\longrightarrow$ $\mathtt{-y=5-26}$

$\longrightarrow$ $\mathtt{-y=-21}$

$\longrightarrow$ $\mathtt{y=21}$

$\large{\boxed{\rm{\blue{ Denominator\:=\:y\:=\:21}}}}$

✪ We have the required numerator and denominator of the fraction.

$\large{\boxed{\rm{\purple{ Fraction\:=\:{\dfrac{x}{y}\:=\:{\dfrac{26}{21}}}}}}}$