Question

If the numerator of a fraction is decrease by 2, its value becomes 1/3, if the denominator increase by 1, its value becomes 1/2 . Find the fraction

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given a fraction
• If number is decreased by 2 its value becomes 1/3
• If denominator is increased by 1 its value become 1/2

To Find:

• We have to find the original fraction

Solution:

Let the numerator = x

And Denominator = y

Therefore

$\boxed{\sf{Original \: Fraction = \dfrac{x}{y} } }$

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$\underline{\bold{\large \mathtt\purple{According \: to \: the \: Question:}} }$

When numerator is decreased by 2 the value of fraction becomes 1/3

$\implies \sf{\dfrac{x-2}{y} = \dfrac{1}{3} }$

Cross Multiplying the Terms

$\implies \sf{3 \: (x-2) = y }$

$\implies \boxed{\sf{3x - 6= y} }$ ------------------ ( 1 )

When denominator is increased by 1 the value of fraction becomes 1/2

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

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

Putting value of y from Equation ( 1 )

$\implies \sf{2x = (3x - 6) + 1}$

$\implies \sf{2x = 3x - 5}$

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

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Putting x = 5 in Equation 1 we get

$\implies \sf{y = (3 \times 5) - 6}$

$\implies \sf{ y = 15 - 6}$

$\implies \boxed{\sf{y = 9}}$

Hence original fraction is 5/9

$\large\boxed{\sf{Original \: Fraction = \dfrac{5}{9}}}$

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$\huge\mathfrak{Verification:}$

1 ) When numerator is decreased by 2 the value fraction becomes 1/3

$\implies \sf{ Fraction = \dfrac{5-2}{9}}$

$\implies \sf{\dfrac{3}{9}}$

$\implies \boxed{\sf{\: Ratio = \dfrac{1}{3} \:} }$

2 ) When denominator is increased by 1 the value of fraction become 1/2

$\implies \sf{ Fraction = \dfrac{5}{9+1}}$

$\implies \sf{ \dfrac{5}{10}}$

$\implies \boxed{\sf{\: Ratio = \dfrac{1}{2} \:} }$

Hence Verified !!!

Answer 2

$\huge{\boxed{\rm{\red{Answer:-}}}}$

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${\red{\bold{\underline{\underline{Given:-}}}}}$

• $\bold{If \ the \ numerator \ of \ a \ fraction \ decrease \ by \ 2 \ it \ becomes \ 1/3}$
• $\bold{If \ the \ denominator \ increase \ by \ 1 \ it \ becomes \ 1/2}$

${\green{\bold{\underline{\underline{To\:find\::-}}}}}$

• $\bold{The \ fraction}$

${\purple{\bold{\underline{\underline{Solution:-}}}}}$

$\bold{Let \ the \ numerator \ be \ x \ and \ the \ denominator \ be \ y}$

⛬The fraction will be $\mathsf{ \dfrac{x}{y}}$

$\bold{According \ to \ the \ Q}$

$\mathsf{ \dfrac{x-2}{y}}$=$\mathsf{ \dfrac{1}{3}}$

$\implies$$\bold{3x-6=y}$

$\implies$$\bold{3x-y=6}$........(i)

$\bold{Again,}$

$\mathsf{ \dfrac{x}{y+1}}$=$\mathsf{ \dfrac{1}{2}}$

$\implies$$\bold{2x=y+1}$

$\implies$$\bold{2x-y=1}$.........(ii)

$\bold{Substracting \ (ii) \ from \ (i) \ we \ get}$

$\implies$$\bold{x=5}$

$\bold{Putting \ the \ value \ of \ x \ in \ eq \ (ii)}$

$\bold{\: \: \: \: \: \: \: \: \: \:(2×5)-y=2}$

$\implies$$\bold{(-y)=2-10}$

$\implies$$\bold{(-y)=(-8)}$

$\implies$$\bold{y=8}$

$\bold{Hence \ value \ of \ x \ is \ 5 \ and \ value \ of \ y \ is \ 8}$

⛬The fraction will be $\mathsf{ \dfrac{5}{8}}$

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$\huge\bold\green{Hope\:this\:help\:you}$