Question

The denominator of a is one more than twice the numerator the sum of the fraction and it 216/21s reciprocal

Answer 1

Hello mate!!

Let the numerator be x
Given,
The denominator of a fraction is one more than twice it's numerator,
So, denominator = 2x + 1

The original fraction will be



Also,

the sum of the fraction and it's reciprocal is


Reciprocal of the original fraction



Their sum is 58/21

So,
the following balanced equation will be formed




Taking LCM as x(2x + 1),
we have,


Multiplying the terms,
we've



21(x² + 4x² + 4x + 1) = 58(2x² + x)

21(5x² + 4x + 1) = 58(2x² + x)

105x² + 84x + 21 = 116x² + 58x

0 = 116x² - 105x² + 58x - 84x - 21

0 = 11x² - 26x - 21

Splitting the middle term
and using the rule where ab = 11 × -21 = -231 = -33 × 7 and a + b = -33 + 7 = -26
we have,

0 = 11x² - 33x + 7x - 21

Taking out common factors,
we've,

0 = 11x(x - 3) + 7(x - 3)

0 = (11x + 7)(x - 3)

11x + 7 = 0 x - 3 = 0
11x = -7 x = 0 + 3 = 3
x = -7/11.

Since the denominator of a fraction can't be taken as -7/11 because it can't be negative

we'll take x = 3

Numerator = x = 3
Denominator = 2x + 1 = 3 × 2 + 1 = 6 + 1 = 7


Thus,
the fraction is 3/7


Hope it helpful