Question

The denominator of a fraction is one more than twice it's numerator. If the sum of the fraction and it's reciprocal is 34/15 ,find the fraction.

Answer 1

Let the numerator be x
and denominator be y

y = 1 + 2x .....( i )

Fraction :- x / y
reciprocal fraction :- y / x


$\frac{x}{y} + \frac{y}{x} = \frac{34}{15}$

$\frac{ {y}^{2} + {x}^{2} }{xy} = \frac{34}{15}$

$\frac{ {(1 + 2x)}^{2} + {x}^{2} }{x(1 + 2x)} = \frac{34}{15}$

$\frac{1 + 4 {x}^{2} + 4x + {x}^{2} }{x + 2 {x}^{2} } = \frac{34}{15}$

$\frac{5 {x}^{2} + 4x + 1}{x + 2 {x}^{2} } = \frac{34}{15}$
75x² + 60x + 15 = 34x + 68x²

75x² - 68x² + 60x - 34x + 15 = 0

7x² + 26x + 15 = 0

7x² + 21x + 5x + 15 = 0

7x ( x + 3 ) + 5 ( x + 3 ) = 0

( 7x + 5 ) ( x + 3 ) = 0


• ( 7x + 5 ) = 0

x = - 5/7 ....... ( neglected )

• ( x + 3 ) = 0

x = - 3

Putting the value of x in i

y = 1 + 2x

y = 1 - 6

y = - 5

$the \: required \: fraction = \frac{x}{y} = \frac{ - 3}{ - 5} = \frac{3}{5}$