Let the numerator be xGiven,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 termand using the rule where ab = 11 × -21 = -231 = -33 × 7 and a + b = -33 + 7 = -26we 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 = 011x = -7 x = 0 + 3 = 3x = -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 = 3Denominator = 2x + 1 = 3 × 2 + 1 = 6 + 1 = 7

Thus,the fraction is 3/7