Question

A fraction becomes 1/3
when 1 is subtracted from the numerator and it
becomes 1/4
when 8 is added to its denominator. Find the fraction.​

Answer 1

$\bold{\underline{\red{Given:}}}$

• A fraction become 1/3 when 1 is subtracted from the numerator .
• the fraction becomes 1/4 when 8 is added to its denominator .

$\bold{\underline{\red{To \ Find :}}}$

• The fraction

$\bold{\underline{\underline{\purple{Solution :}}}}$

⇒Suppose the numerator be x

And, the denominator be y

According to First Condition of this Question :-

$\implies \bold{\frac{x - 1}{y} = \frac{1}{3}}$

$\implies \boxed{\bold{y = 3x - 3}}$.......1) Equation

According to Second Condition of this Question :-

$\implies \bold{\frac{x}{y + 8} = \frac{1}{4}}$

$\implies \bold{y + 8 = 4x}$

Now put value of y from Equation First :-

$\implies \bold{ 3x - 3 + 8 = 4x}$

$\implies \boxed{\bold{ x = 5}}$

Now Put value of x in First Equation :-

$\implies \bold{ y = 3x - 3}$

$\implies \bold{ y = 3(5) - 3}$

$\implies \boxed{\bold{ y = 12}}$

The Fraction = $\bold{\dfrac{5}{12}}$

$\rule{200}2$

Answer 2

Given :

• Fraction becomes ⅓ when 1 is subtracted from the numerator. [1st Case]

• Fraction becomes ¼ when 8 is added to its denominator. [2nd Case]

To Find :

• Fraction.

Solution :

Let the numerator be x and denominator be y

$\sf \: Fraction = \dfrac{x}{y}$

According to 1st Case :

$\sf \longrightarrow \: \dfrac{x -1 }{y} = \dfrac{1}{3}$

$\sf \longrightarrow \: 3(x - 1) = y(1)$

$\sf \longrightarrow \: 3x - 3 = y$

$\sf \longrightarrow \: 3x - y = 3$

$\sf \longrightarrow \: 3x = 3 + y$

$\sf \longrightarrow \: y = 3x - 3....(i)$

According to 2nd Case :

$\sf \longrightarrow \: \dfrac{x }{y + 8} = \dfrac{1}{4}$

$\sf \longrightarrow \: 4x = y + 8$

Put the value of y from equation i)

$\sf \longrightarrow \: 4x = (3x-3) + 8$

$\sf \longrightarrow \: 4x = 3x-5$

$\sf \longrightarrow \: 4x -3x = -5$

$\sf \longrightarrow \: -x = -5$

$\sf \longrightarrow \red{ x = 5}$

Put the value of x = 5 in equation i)

$\sf \longrightarrow \: y = 3x - 3$

$\sf \longrightarrow \: y = 3(5) - 3$

$\sf \longrightarrow \: y = 15 - 3$

$\sf \longrightarrow \red{ y = 12}$

Hence,

$\sf \boxed{\purple{\sf\: Fraction = \dfrac{5}{12}}}$