Question

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction?​

Answer 1

Given :

• If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1.
• It becomes 1/2 if we only add 1 to the denominator.

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 :

If 1 is added to numerator and 1 is subtracted from denominator the fraction becomes 1.

Numerator = (x + 1)

Denominator = (y - 1)

Equation :

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

$\sf{\longrightarrow{x+1=1(y-1)}}$

$\sf{\longrightarrow{x+1=y-1}}$

$\sf{\longrightarrow{x=y-1-1}}$

$\sf{\longrightarrow{x=y-2\:\:\:\:(1)}}$

Case 2 :

When 1 is added to the denominator the fraction becomes 1/2.

Numerator = x

Denominator = (y + 1)

Equation :

$\sf{\longrightarrow{\dfrac{x}{y+1}=\dfrac{1}{2}}}$

$\sf{\longrightarrow{2x=1(y+1)}}$

$\sf{\longrightarrow{2x=y+1}}$

$\sf{\longrightarrow{2x-y=1}}$

$\sf{\longrightarrow{2(y-2) -y=1}}$

$\bf{\big[From\:equation\:(1)\:x\:=\:y-2\:\big]}$

$\sf{\longrightarrow{2y-4-y=1}}$

$\sf{\longrightarrow{2y-y=1+4}}$

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

Substitute, y = 5 in equation (1),

$\sf{\longrightarrow{x=y-2}}$

$\sf{\longrightarrow{x=5-2}}$

$\sf{\longrightarrow{x=3}}$

$\large{\boxed{\bold{Numerator\:of\:fraction\:=\:x\:=\:3}}}$

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

$\large{\boxed{\bold{Fraction\:=\dfrac{x}{y}=\dfrac{3}{5}}}}$

Answer 2

Given,

• If we add 1 to the numerator and subtract 1 from the denominator then the fraction reduce to 1.
• If we add 1 to the denominator then it becomes 1/2.

To Find,

• The Fraction

Solution,

⇒Suppose the numerator be a

And, Suppose the denominator be b

According to the First Condition :-

$\implies \bold{a + 1 = b - 1}$

$\implies \boxed{\bold{b = a + 2}}$.....1)

According to the Second Condition :-

$\implies \bold{\dfrac{a}{b + 1} = \dfrac{1}{2}}$

$\implies \bold{2a = b + 1}$

Now Put value of b From Equation First :-

$\implies \bold{2a = a + 2 + 1}$

$\implies \bold{2a - a = 3}$

$\implies \boxed{\bold{ a = 3}}$

Now Put value of a in Equation First :-

$\implies \bold{b = a + 2}$

$\implies \bold{b = 3 + 2}$

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

Therefore ,

$\boxed{\bf{\purple{The \ Fraction = \dfrac{3}{5}}}}$

$\rule{200}1$