Question

The numerator of a fraction is 3 less than its denominator. if the denominator is increased by 2 and the numerator is decreased by 5 the fraction becomes 28/33.find the original number

Answer 1

$\mathfrak{\large{\underline{\underline{Answer :-}}}}$

The Original Fraction is $\dfrac{61}{64}$

$\mathfrak{\large{\underline{\underline{Explanation :-}}}}$

Given :

Numerator of the Fraction = 3 less than its Denominator.

The Denominator is increased by 2 and the numerator is decreased by 5 the fraction becomes = $\dfrac{28}{33}$

To Find :

The original number

Solution :

◆ $\textbf{\small{\underline{Consider the -}}}$

• Denominator as x
• Numerator as (x - 3)

$\rule{300}{1.5}$

◆ $\textbf{\small{\underline{According to the Question -}}}$

• x + 2 = 33
• (x - 3) - 5 = 28

★ $\boxed{\frac{(x - 3) - 5 }{x + 2} = \frac{28}{33}}$

$\longrightarrow \: \dfrac{(x - 3) - 5 }{x + 2} = \dfrac{28}{33}$

$\longrightarrow \: \dfrac{x - 3 - 5 }{x + 2} = \dfrac{28}{33}$

$\longrightarrow \: \dfrac{x - 8 }{x + 2} = \dfrac{28}{33}$

$\longrightarrow \: 33(x - 8) = 28(x + 2)$

$\longrightarrow \: 33x - 264 = 28x + 56$

$\longrightarrow \: 33x - 28x = 56 + 264$

$\longrightarrow \: 5x = 320$

$\longrightarrow \: x = \dfrac{320}{5}$

$\longrightarrow \: x = 64$

Denominator = 64

$\rule{300}{1.5}$

★ Value of (x - 3)

$\longrightarrow \: (64 - 3)$

$\longrightarrow \: 61$

Numerator = 61

$\therefore$ The Original Fraction is $\dfrac{61}{64}$

$\rule{300}{1.5}$

$\mathfrak{\large{\underline{\underline{Verification :-}}}}$

$\longrightarrow \: \dfrac{61 - 5}{64 + 2} = \dfrac{28}{33}$

$\longrightarrow \: \dfrac{56}{66} = \dfrac{28}{33}$

$\longrightarrow \: \dfrac{56 \div 2}{66 \div 2} = \dfrac{28}{33}$

$\longrightarrow \: \dfrac{28}{33} = \dfrac{28}{33}$

$\therefore$ The Original Fraction is $\dfrac{61}{64}$

Answer 2

» The numerator of a fraction is 3 less than its denominator.

• Let Denominator = M

and

• Numerator = M - 3

» If the denominator is increased by 2 and the numerator is decreased by 5 the fraction becomes 28/33.

A.T.Q.

New Numberator = (M - 3) - 5

Denominator = M + 2

→ $\dfrac{(M\:-\:3)\:-\:5}{M\:+\:2}\:=\:\dfrac{28}{33}$

→ $\dfrac{M\:-\:3\:-\:5}{M\:+\:2}\:=\:\dfrac{28}{33}$

→ $\dfrac{M\:-\:8}{M\:+\:2}\:=\:\dfrac{28}{33}$

Cross-multiply them

→ 33(M - 8) = 28(M + 2)

→ 33M - 264 = 28M + 56

→ 33M - 28M = 56 + 264

→ 5M = 320

→ M = 320/5

→ M = 64

_____________________________

Now..

Numerator = M - 3

→ 64 - 3

→ 61

Denominator = M

→ 64

____________________________

We have to find the original number/fraction.

i.e.

Fraction = $\dfrac{Numerator}{Denominator}\:=\:\dfrac{M}{M\:-\:3}$

→ $\dfrac{61}{64}$

___________________________

Fraction = $\dfrac{61}{64}$

_______ [ ANSWER ]

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✡ VERIFICATION :

From above calculations we have M = 64

Put value of M in this equation :

$\dfrac{(M\:-\:3)\:-\:5}{M\:+\:2}\:=\:\dfrac{28}{33}$

=> $\dfrac{(64\:-\:3)\:-\:5}{64\:+\:2}\:=\:\dfrac{28}{33}$

=> $\dfrac{56}{66}\:=\:\dfrac{28}{33}$

=> $\dfrac{28}{33}\:=\:\dfrac{28}{33}$

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