Question

The numerator of a non-zero rational number is five less than the denominator. if the denominator is increased by eight and the numerator is doubled, then again we get the same rational number. the required rational number is:

Answer 1

let the denominator be x
so the numerator will be x-5
and the rational number become (x-5)/x
now according to the question the denominator becomes x+8

Answer 2

The required rational number is 3/8.

Given:

The numerator of a non-zero rational number is five less than the denominator.

if the denominator is increased by eight and the numerator is doubled, then again we get the same rational number.

To find:

Find the rational number  

Solution:

Let the denominator of the rational number be d.

The numerator of the rational number = x - 5  

The rational number = (d - 5)/d

It is given that when the denominator is increased by 8 and the numerator is doubled, then again we get the same rational number.

=> (2(d - 5))/ (d + 8) = (d - 5)/d

Multiply g both sides by d(d + 8)

=> (d - 5)(d + 8) = 2(d - 5)d

=> d² + 3d - 40 = 2d² - 10d

Now simplify the equation

=> d² - 13d + 40 = 0

Factorize the equation

=> d² - 8d - 5d + 40 = 0

=> d (d - 8) - 5(d - 8) = 0

=> (d - 8) (d - 5) = 0

Therefore, d = 8 or d = 5.

If d = 8, then the numerator of the rational number is 3, and the rational number is 3/8.

If d = 5, then the numerator of the rational number is 0, which is not a non-zero rational number.

Therefore,

The required rational number is 3/8.

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