Question

The denominator of a fraction is greater than the numerator by 8. If the numerator is increased by 17 and denominator is decreased by 1, the number obtained is 3/2, find the fraction.
Challenge to mod

Answer 1

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Required solution -

${\large{\bold{\rm{\underline{Understanding \; the \; question}}}}}$

This question says that there is a fraction given. As we know that the fraction is consist of numerator and denominator. Given that - The denominator of fraction is greater than the numerator by 8. The numerator is increased by 17 and denominator is decreased by 1, the number obtained is 3/2, we have to find the fraction.

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${\large{\bold{\rm{\underline{Given \; that}}}}}$

★ The denominator of a fraction is greater than the numerator by 8.

★ The numerator is increased by 17

★ Denominator is decreased by 1

★ The number obtained is 3/2

${\large{\bold{\rm{\underline{To \; find}}}}}$

★ Fraction

${\large{\bold{\rm{\underline{Solution}}}}}$

★ Fraction = 13/21

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${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ As it's given that the denominator of a fraction is greater than the numerator by 8. Henceforth,

${\small{\boxed{\bf{a+8 \: be \: denominator}}}}$

~ Henceforth,

${\small{\boxed{\bf{Fraction \: be \: a/a \: + \: 8}}}}$

~ No according to the question, the denominator of fraction is greater than the numerator by 8. The numerator is increased by 17 and denominator is decreased by 1, the number obtained is 3/2...! Let's do it...!

${\bf{:\implies a \: + 17/a \: + 8 - 1 \: = 3/2}}$

• (+ -) = (-)

${\bf{:\implies a \: + 17/a \: + 7 \: = 3/2}}$

• Let's do cross multiplication

${\bf{:\implies 2(a+17) \: = 3(a+7)}}$

• Let's multiply

${\bf{:\implies 2a + 34 \: = 3a + 21}}$

• Let's combine like terms

${\bf{:\implies 2a - 3a \: = 21 - 34}}$

${\bf{:\implies -1a \: = -13}}$

• (- -) = (+)

${\bf{:\implies a = 13}}$

~ Now let's find the fraction..!

Numerator -

${\small{\boxed{\bf{Fraction's \: numerator \: is \: 13}}}}$

Denominator -

${\bf{:\implies a + 8}}$

${\bf{:\implies 13 + 8}}$

${\bf{:\implies 21}}$

${\small{\boxed{\bf{Fraction's \: denominator \: is \: 21}}}}$

Henceforth, fraction..!

${\small{\boxed{\bf{Fraction \: = \dfrac{Numerator}{Denominator} \: = \dfrac{13}{21}}}}}$

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Answer 2

Answer:

Given :-

• The denominator of a fraction is greater than the numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2.

To Find :-

• What is the required fraction.

Solution :-

Let, the numerator be x

And, the denominator will be x + 8

Then, the required fraction will be $\sf \dfrac{x}{x + 8}$

According to the question,

$\implies$ $\sf \dfrac{x + 17}{x + 8 - 1} =\: \dfrac{3}{2}$

$\implies$ $\sf \dfrac{x + 17}{x + 7} =\: \dfrac{3}{2}$

By doing cross multiplication we get,

$\implies$ $\sf 3(x + 7) =\: 2(x + 17)$

$\implies$ $\sf 3x + 21 =\: 2x + 34$

$\implies$ $\sf 3x - 2x =\: 34 - 21$

$\implies$ $\sf\bold{\green{x =\: 13}}$

Hence, the required fraction will be,

$\implies$ $\sf \dfrac{x}{x + 8}$

$\implies$ $\sf \dfrac{13}{13 + 8}$

$\implies$ $\sf\bold{\red{\dfrac{13}{21}}}$

$\therefore$ The required fraction is ${\purple{\boxed{\large{\bold{\dfrac{13}{21}}}}}}$.

VERIFICATION :-

$\implies$ $\sf \dfrac{x + 17}{x + 8 - 1} =\: \dfrac{3}{2}$

By putting x = 13 we get,

$\implies$ $\sf \dfrac{13 + 17}{13 + 8 - 1} =\: \dfrac{3}{2}$

$\implies$ $\sf \dfrac{30}{21 - 1} =\: \dfrac{3}{2}$

$\implies$ $\sf \dfrac{3\cancel{0}}{2\cancel{0}} =\: \dfrac{3}{2}$

$\implies$ $\sf\bold{\pink{\dfrac{3}{2} =\: \dfrac{3}{2}}}$

$\implies$ LHS= RHS

$\mapsto$ Hence, Verified ✔