Question

. The numerator of a fraction is 1 less than the denominator.
If Sara adds 1 to the numerator and adds 2 to the denominator, the fraction is 34, when expressed in its simplest form.
Find the original fraction.

Answer 1

Correct Question :

The numerator of a fraction is 1 less than the denominator. If Sara adds 1 to the numerator and adds 2 to the denominator, the fraction is 3/4, when expressed in its simplest form. Find the original fraction.

Solution :

Let the numerator of the fraction be x and denominator be y

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

Given numerator is 1 less than the denominator

Fraction = $\sf\dfrac{x-1}{x}$

Sara added 1 to the numerator and 2 to the denominator.

Fraction = $\sf\dfrac{x-1+1}{x+2}$

The Fraction becomes 3/4

$\longrightarrow \sf \: \dfrac{x - 1 + 1}{x + 2} = \dfrac{3}{4}$

$\longrightarrow \sf \: \dfrac{x}{x + 2} = \dfrac{3}{4}$

$\longrightarrow \sf \: 4 \times {x} = 3 \times (x + 2)$

$\longrightarrow \sf \: 4x= 3x+ 6$

$\longrightarrow \sf \: 4x - 3x = 6$

$\longrightarrow \sf \: x = 6$

Hence,The value of x is 6 . Let's put it in our assumption $\sf\dfrac{x-1}{x}$

$\longrightarrow\sf\dfrac{x - 1}{x}$

$\longrightarrow\sf\dfrac{6 - 1}{6}$

$\longrightarrow\sf\dfrac{5}{6} \: \rightarrow\: Original \: Fraction$

Hence,Original Fraction is 5/6

Answer 2

Given :

• The numerator of a fraction is 1 less than the denominator.
• If 1 is added to the numerator and 2 is added to the denominator, the fraction is $\sf{\dfrac{3}{4}}$ expressed in simplest form.

To Find :

• The original fraction

Solution :

Let the numerator of the fraction be x.

Let the denominator of the fraction be y.

Fraction → $\large{\sf{\red{\dfrac{x}{y}}}}$

Case 1 :

$\sf{Numerator\:=\:Denominator\:-\:1}$

Equation :

$\large{\boxed{\sf{x\:=\:y-1\:\:...(i)}}}$

Case 2 :

Numerator → ( x + 1)

Denominator → ( y + 2)

Equation :

$\blue{\implies}$ $\sf{\dfrac{(x+1)}{(y+2)}\:=\:\dfrac{3}{4}}$

$\blue{\implies}$ $\sf{4(x+1)=3(y+2)}$

$\blue{\implies}$ $\sf{4x+4=3y+6}$

$\blue{\implies}$ $\sf{4x-3y=6-4}$

$\sf{4x-3y=2\:\:...(ii)}$

From equation (i), $\sf{x=y-1}$

$\blue{\implies}$ $\sf{4(y-1) - 3y = 2}$

$\blue{\implies}$ $\sf{4y-4-3y=2}$

$\blue{\implies}$ $\sf{4y-3y=2+4}$

$\blue{\implies}$ $\sf{y=6}$

Substitute, y = 6 in equation (i),

$\blue{\implies}$ $\sf{x=y-1}$

$\blue{\implies}$ $\sf{x=6-1}$

$\blue{\implies}$ $\sf{x=5}$

Fraction :

$\large{\boxed{\sf{\red{Numerator\:=\:x\:=\:5}}}}$

$\large{\boxed{\sf{\green{Denominator\:=\:y\:=\:6}}}}$

$\large{\boxed{\sf{\purple{Fraction\:=\:\dfrac{x}{y}\:=\:\dfrac{5}{6}}}}}$