Question

The denominator of a positive fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2.9; find the fraction​

Answer 1

Answer:

Step-by-step explanation:

Given,

The denominator of a positive fraction is one more than twice the numerator.

If the sum of the fraction and its reciprocal is 2.9.

To Find,

The fraction.

Solution,

Let the fraction be x/(2x + 1)

Now, According to the question,

⇒ x/(2x + 1) + (2x + 1)/x = 2.9

⇒ x² + 4x² + 1 + 4x/x(2x + 1) = 29/10

⇒ 5x² + 1 + 4x/2x² + x = 29/10

By cross multiplicaton, we get

⇒ 50x² + 10 + 40x = 58x² + 29x

⇒ 8x² - 11x - 10 = 0

⇒ x = (11 ± √121 + 320)/16

⇒ x = (11 ± √441)/16

⇒ x = (11 ± 21)/16

⇒ x = 2, - 5/8

⇒ x = 2 (Avoiding negative value)

Fraction = x/(2x + 1)

= 2/2 × 2 + 1

= 2/5

Hence, the required fraction be 2/5.

Answer 2

Given :-

The  denominator of a positive fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2.9To Find :-

Fraction

Solution :-

Let the numerator be x

Since the denominator is one more than twice

x/2x + 1

Reciprocal = 2x + 1/x

2.9 can be written as 29/10

x/2x + 1 + 2x + 1/x = 29/10

x × x + 4 × x × x + 1 + 4 × x/x(2x + 1) = 29/10

x² + 4x² + 1 + 4x/x(2x + 1) = 29/10

x² + 4x² + 1 + 4x/2x² + 1 = 29/10

10(x² + 4x² + 1 + 4x) = 29(2x² + 1)

10x² + 40x² + 10 + 40x = 58x² + 29

50x² + 10 + 40x = 58x² + 29x

58x² - 50x² + 29x - 40x = 10

8x² - 11x = 10

8x² - 11x - 10 = 0

8x² - (16x - 5x) - 10 = 0

8x² - 16x + 5x - 10 = 0

x - 2 = 0

x = 2

or

x + 5 = 0

x = -5

By putting x as 2

Fraction = 2/5