Question

A fraction becomes 2/5 when 2 is substracted from numerater and becomes 4/7 when 2 is added to denominator . Find the fraction

Answer 1

Given :

• A fraction when 2 is subtracted from the numerator becomes 2/5.
• When 2 is added to the denominator, the fraction becomes 4/7.

To Find :

• The fraction.

Solution :

Let the numerator of the fraction be x.

Let the denominator of the fraction be y.

Fraction = $\bold{\dfrac{x}{y}}$

Case 1 :

When the numerator is decreased by 2, the fraction is 2/5.

Numerator = (x-2)

Denominator = y

Equation :

$\implies$ $\sf{\dfrac{x-2}{y}=\dfrac{2}{5}}$

$\implies$ $\sf{5(x-2) =2y}$

$\implies$ $\sf{5x-10=2y}$

$\implies$ $\sf{5x-2y=10}$

$\implies$ $\sf{5x=10+2y}$

$\implies$ $\sf{x=\dfrac{10+2y}{5}\:\:\:\:(1)}$

Case 2 :

When 2 is added to the denominator of the fraction, it becomes 4/7.

Numerator = x

Denominator = (y+2)

Equation :

$\implies$ $\sf{\dfrac{x}{y+2}\:=\:\dfrac{4}{7}}$

$\implies$ $\sf{7x=4(y+2)}$

$\implies$ $\sf{7\:\Big(\dfrac{10+2y}{5}\Big)\:=\:4y+8}$

$\bold{\Big[From\:equation\:(1)\:x\:=\:\dfrac{10+2y}{5}\Big]}$

$\implies$ $\sf{\Big(\dfrac{70+14y}{5}\Big) \:=\:4y+8}$

$\implies$ $\sf{70+14y=5(4y+8)}$

$\implies$ $\sf{70+14y=20y+40}$

$\implies$ $\sf{14y-20y=40-70}$

$\implies$ $\sf{-6y=-30}$

$\implies$ $\sf{y=\dfrac{-30}{-6}}$

$\implies$ $\sf{y=\dfrac{30}{6}}$

$\implies$ $\sf{y=5}$

Substitute, y = 5 in equation (1),

$\implies$ $\sf{x=\dfrac{10+2y}{5}}$

$\implies$ $\sf{x=\dfrac{10+2(5)}{5}}$

$\implies$ $\sf{x=\dfrac{10+10}{5}}$

$\implies$ $\sf{x=\dfrac{20}{5}}$

$\implies$ $\sf{x=4}$

$\large{\boxed{\bold{Numerator\:=\:x\:=\:4}}}$

$\large{\boxed{\bold{Denominator\:=\:y\:=\:5}}}$

$\large{\boxed{\tt{\purple{Fraction\:=\:\dfrac{x}{y}\:=\:\dfrac{4}{5}}}}}$

Answer 2

Given :

• Fraction becomes 2/5 when 2 is subtracted from numerator

• Fraction becomes 4/7 when 2 is added to the denominator

To Find :

• The Fraction

Solution :

Let the numerator be x and denominator be y

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

When 2 is subtracted from the numerator :

$\longrightarrow \sf \: \dfrac{x - 2}{y} = \dfrac{2}{5}$

$\longrightarrow \sf \: 5(x - 2) = 2y$

$\longrightarrow \sf \: 5x - 10 = 2y$

$\longrightarrow \sf \: 5x - 2y =10$

$\longrightarrow \sf \: 5x = 10 + 2y$

$\longrightarrow \sf \: x = \dfrac{10 + 2y}{5} ...(i)$

When 2 is added to the denominator :

$\longrightarrow \sf \: \dfrac{x }{y +2 } = \dfrac{4}{7}$

$\longrightarrow \sf \: 7x = 4y + 8$

Put the value of x from equation i)

$\longrightarrow \sf \: 7( \dfrac{10 + 2y}{5} ) = 4y + 8$

$\longrightarrow \sf \: \dfrac{70 + 14y}{5} = 4y + 8$

$\longrightarrow \sf \: 70 + 14y = 5(4y + 8)$

$\longrightarrow \sf \: 70 + 14y = 20y + 40$

$\longrightarrow \sf \: 14y - 20y= 40 - 70$

$\longrightarrow \sf \: - 6y = - 30$

$\longrightarrow \sf \: y = \cancel\dfrac{ - 30}{ - 6}$

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

Put the value of y = 5 in equation i)

$\longrightarrow \sf \: x = \dfrac{10 + 2y}{5}$

$\longrightarrow \sf \: x = \dfrac{10 + 2(5)}{5}$

$\longrightarrow \sf \: x = \dfrac{10 + 10}{5}$

$\longrightarrow \sf \: x = \cancel\dfrac{20}{5}$

$\longrightarrow \sf \red{x = 4}$

Hence,

$\dag \: \: \large \boxed{ \purple{ \sf \: Fraction = \dfrac{x}{y} = \dfrac{4}{5} }}$