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.
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Answer 1

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$\large \bf \clubs \:  Given :-$

• 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.

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$\large \bf \clubs \:  To \: Find :-$

• The Original Fraction

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$\large \bf \clubs \:  Solution :-$

Let,

• Numerator of Original Fraction = x .

So,

• Denominator should be = x + 8.

Hence ,

$\pink{\bf Original \: Fraction = \dfrac{x}{x + 8} }$

When the Numerator is increased by 17 and Denominator is decreased by 1 :

$\red{ \bf New \: Fraction = \dfrac{x + 17}{x + 7} }$

According To The Given Condition :

$\sf \dfrac{\mathtt{x} + 17}{\mathtt{x} + 7} = \frac{3}{2} \\ \\ :\longmapsto \sf2(\mathtt{x} + 17) = 3(\mathtt{x} + 7) \\ \\ :\longmapsto \sf2\mathtt{x} + 34 = 3x + 21 \\ \\ :\longmapsto \sf3\mathtt{x} - 2\mathtt{x} = 34 - 21 \\ \\ \purple{ \Large :\longmapsto  \underline {\boxed{{\bf x = 13} }}}$

Hence ,

${\bf Original \: Fraction = \dfrac{13}{13 + 8} }\\ \\ \underline{ \underline{\pink{:\longmapsto\bf Original \: Fraction = \dfrac{13}{21} }}}$

$\Large\red{\mathfrak{  \text{W}hich \:\:is\:\: the\:\: required} }\\ \LARGE \red{\mathfrak{ \text{ A}nswer.}}$

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Answer 2

Let the denominator of a rational number be x, then the denominator is x + 8.

$Rational \: number = \frac{x}{x + 8}$

According to the conditions,

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

$= > \frac{x + 17}{x + 7} = \frac{3}{2}$

$= > \frac{(x + 17)}{(x + 7)} \times (x + 7) = \frac{3}{2} \times (x + 7) \\ \\ (Multiplying \: both \: sides \: by \: (x + 7).)$

$= > x + 17 = \frac{3}{2} (x + 7)$

$= > 2x(x + 17) = 2 \times \frac{3}{2} (x + 7) \\ \\ \: \: \: \: \: \: \: \: (Multiplying \: both \: sides \: by \: 2.)$

$= > 2x + 34 = 3x + 21$

$= > 2x - 3x = 21 - 34 \\ \\ (Transporting \: 3x \: to \: L.H.S. \: and \: 34 \: to \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: R.H.S.)$

$= > - x = - 13$

$= > x = 13$

$Hence, \: the \: required \: rational \: number \\ \frac{x}{x + 8} = \frac{13}{13 + 8} = \frac{13}{21}$