Question

A fraction becomes 1/2 if 2 is subtracted from numerator and it becomes 1/4 when 10is added to denominator. Find the fraction.

Answer 1

Given

A fraction becomes 1/2 if 2 is subtracted from numerator and it becomes 1/4 when 10 is added to denominator.

To Find

Find the fraction

Solution

★ Let the required fraction be x/y

According to the given condition

★ A fraction becomes 1/2 if 2 is subtracted from numerator

→ x - 2/y = 1/2

→ 2(x - 2) = y

→ 2x - 4 = y

→ 2x - y = 4 ---(i)

★ It becomes 1/4 when 10 is added to denominator

→ x /y +10 = 1/4

→ 4x = y + 10

→ 4x - y = 10 ---(ii)

Subtract both the equations

→ (2x - y) - (4x - y) = 4 - 10

→ 2x - y - 4x + y = -6

→ - 2x = - 6

→ x = 6/2 = 3

Putting the value of x in eqⁿ (ii)

→ 4x - y = 10

→ 4*3 - y = 10

→ 12 - y = 10

→ 12 - 10 = y

→ y = 2

Hence, the required fraction x/y is 3/2

Answer 2

Answer:

Let the Numerator be N and Denominator be D of the required Fraction.

Given : Fraction becomes 1/2 if 2 is subtracted from numerator.

$:\implies\sf \dfrac{Numerator-2}{Denominator}=\dfrac{1}{2}\\\\\\:\implies\sf \dfrac{N - 2}{D} = \dfrac{1}{2}\\\\\\:\implies\sf 2(N - 2) = D\\\\\\:\implies\sf D = 2N - 4 \qquad...eq. \:(1)$

$\rule{100}{0.8}$

Given : Fraction becomes 1/4 when 10 is added to denominator.

$:\implies\sf \dfrac{Numerator}{Denominator+10}=\dfrac{1}{4}\\\\\\:\implies\sf \dfrac{N}{D+10} = \dfrac{1}{4}\\\\\\:\implies\sf 4N = D+10$

★ Substituting value of D from eq. ( 1)

$:\implies\sf 4N=2N-4+10\\\\\\:\implies\sf 4N-2N=6\\\\\\:\implies\sf 2N=6\\\\\\:\implies\sf N=\dfrac{6}{2}\\\\\\:\implies\sf N=3$

⠀

☯ Putting value of N in eq. ( 1) :

$:\implies\sf 4N=D+10\\\\\\:\implies\sf 4(3)=D+10\\\\\\:\implies\sf 12-10=D\\\\\\:\implies\sf D=2$

$\rule{150}{2}$

$\bf{\dag}\:\boxed{\sf Fraction=\dfrac{Numerator}{Denominator} = \dfrac{N}{D} = \dfrac{\textsf{\textbf{3}}}{\textsf{\textbf{2}}}}$