Question

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. Find the fraction number.

Answer 1

$\frak{Given}\begin{cases}\sf{\;\;\; Denominator = \bf{Numerator + 8}}\\ \\ \sf{ \: \: \: \dfrac{Numerator + 17}{Numerator + 8 - 1} = \bf{\dfrac{3}{2}}}\end{cases}$

$\frak{To\;find :}$ The Original fraction?

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❍ Let's say, that the Numerator of the fraction be x and Denominator be y respectively.

Given that,

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

Therefore,

• Fraction = x/y
• Denominator, y = (x + 8).

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❭❭ Also, if the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is ³⁄₂.

Therefore,

• Numerator = (x + 17)
• Denominator = (x + 8 – 1) = (x + 7)

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$\twoheadrightarrow\sf \bigg\{\dfrac{x + 17}{x + 7}\bigg\} = \bigg\{\dfrac{3}{2}\bigg\} \\\\\\\twoheadrightarrow\sf 2\Big\{x + 17\Big\} = 3\Big\{x + 7\Big\} \\\\\\\twoheadrightarrow\sf 2x + 34 = 3x + 21 \\\\\\\twoheadrightarrow\sf 2x - 3x = 21 - 34 \\\\\\\twoheadrightarrow\sf \cancel- x = \cancel- 13 \\\\\\\twoheadrightarrow{\pink{\underline{\boxed{\pmb{\frak{x = 13}}}}}}\:\bigstar$

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Hence,

• Numerator of fraction, x = 13.
• Denominator of fraction, y = (x + 8) = (13 + 8) = 21.

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$\therefore{\underline{\sf{Hence, \: the \: original \: fraction \: is \: \frak{\dfrac{13}{21}}.}}}$

Answer 2

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 :-

Fraction

Solution :-

Let the numerator be x

So

Fraction becom x/x + 8

When numerator increased by 17 and denominator decreased by 1

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

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

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

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

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

$\sf x=13$

Therefore

$\sf Fraction = \dfrac{x}{x+8}$

$\sf Fraction = \dfrac{13}{13+8}$

$\sf Fraction = \dfrac{13}{21}$