Question

If 2 is added to the denominator of a fraction we get 1/4. If instead 1 is added to the numerator we get 2 /5 what is the fraction

Answer 1

SOLUTION :-

Let,

Numerator = x

Denominator = y

According to the first condition,

$\longrightarrow\sf \dfrac{x}{y+2}=\dfrac{1}{4} \\\\\longrightarrow 4x= 1(y+2)\\\\\longrightarrow 4x=y+2\\\\\longrightarrow 4x-y=2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................(1)$

According to the second condition,

$\longrightarrow\sf \dfrac{x+1}{y}=\dfrac{2}{5}\\\\\longrightarrow 5(x+1) = 2y \\\\\longrightarrow 5x+5= 2y \\\\\longrightarrow 5x-2y=- 5 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................(2)$

Equation (1) × 2,

$\longrightarrow\sf 8x-2y = 4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................(3)$

Equation (2) - Equation (1),

$\longrightarrow \sf 3x= 9 \\\\\longrightarrow\bf x= 3$

Substitute the value of x in equation (1),

$\longrightarrow\sf 4\times 3 - y = 2 \\\\\longrightarrow 12 - y = 2 \\\\\longrightarrow -y = -10\\\\\longrightarrow\bf y = 10$

Numerator = 3

Denominator = 10

$\bf Fraction = \dfrac{3}{10}$