Question

The denominator of a rational number is greater that its 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 rational number.

Answer 1

Let numerator = x

denominator = x+8

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

ACCORDING TO QUESTION

Numerator is increased by 17 , so new numerator,
(x + 17 )

Denominator is is decreased by 1 , so new denominator will be
(x + 8 - 1)

Fraction will be

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

On cross multiplication,

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

$\mathsf{2x + 34 = 3x + 21}$

On taking like terms aside

$\mathsf{ 34 - 21 = 3x - 2x }$

$\boxed{\mathsf{13 = x}}$

So numerator is 13

Denominator will be 13 + 8 = 21

Fraction is $\mathsf{\dfrac{13}{21}}$

Answer 2

$\sf\underline{Step-by-step \: explanation}$

A fraction is written as :

$\sf\dfrac{m}{n}$

Where m is denominator and n is the numerator.

Coming to the main part,

Let the numerator be $\bold{x}$

Then, the denominator will be, $\bold{x \: + \: 8}$

$\sf{According \: to \: the \: question \: :}$

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

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

$\mathrm{Using \: cross \: multiplication \: :}$

2(x + 17) = 3(x + 7)

2x + 34 = 3x + 21

3x - 2x = 34 - 21

x = 13

So, the numerator will be;

13

[x = 13]

And the denominator will be,

21

[x + 8 = 13 + 8 = 21]

$\sf{Verification \: :}$

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

$\sf\dfrac{13 \: + \: 17}{13 \: + \: 7} \: = \: \sf\dfrac{3}{2}$

$\sf\dfrac{30}{20} \: = \: \sf\dfrac{3}{2}$

$\sf\dfrac{3}{2} \: = \: \sf\dfrac{3}{2}$

L.H.S = R.H.S

Hence Verified!

So; the fraction will be,

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