The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2 2/6 find the fraction.

Question

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is \(2\frac{16}{21}\) find the fraction.

Answer 1

Let the numerator = x 

then denominator = 2x + 1 

Then the fraction = \(\frac{x}{2x+1}\)

Its reciprocal = \(\frac{2x+1}{x}\)

105x² + 84x + 21 = 116x² + 58x 

11x² – 26x – 21 = 0 

11x² – 33x + 7x – 21 = 0 

11x (x – 3) + 7 (x – 3) = 0 

(x – 3) (11x + 7) = 0

⇒ x – 3 = 0 (or) 11x + 7 = 0 

⇒ x = 3 (or) \(\frac{-7}{11}\)

∴ x = 3 Numerator = 3; 

Denominator = 2 × 3 + 1 = 7 

Fraction = \(\frac{3}{7}\)