Question

The denominator of a rational number is greater than its numerator by 7. If the numerator is increased by 18 and the denominator is decreased by 3, the number obtained is 3/2. Find the rational number.

Answer 1

Step-by-step explanation:

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

• Denominator of a fraction is greater than its numerator by 7.

• If the numerator is increased by 18 and the denominator is decreased by 3, the number obtained is 3/2.

$\bf\underline{\underline{\blue{To\:Find:-}}}$

• The rational number.

$\bf\underline{\underline{\green{Solution:-}}}$

Case 1:-

The denominator is 7 greater than the numerator.

Let the numerator be x

The denominator = x + 7

$\rm The \: rational \: number = \dfrac{Numerator}{Denominator}$

$\rm The \: rational \: number = \dfrac{x}{x+7}$

Case 2:-

If numerator is increased by 18

The numerator = x + 18

If denominator is decreased by 3

The denominator = x + 7 - 3 = x + 4

Given, the fraction becomes 3/2

So:-

$\rm \implies \dfrac{x + 18}{x + 4} = \dfrac{3}{2}$

By cross multiplication:-

$\rm \implies 2(x + 18) = 3(x + 4)$

$\rm \implies 2x + 36= 3x + 12$

$\rm \implies 2x - 3x = 12 - 36$

$\rm \implies - x = - 24$

$\rm \implies x = 24$

The numerator = x = 24

The denominator = x + 7 = 24 + 7 = 31

$\rm The \: rational \: number = \dfrac{Numerator}{Denominator}$

$\underline{\boxed{\purple{\rm\therefore The \: rational \: number = \dfrac{24}{31}}}}$