Question

The sum of the numerator and denominator of a certain positive fraction is 8. If 2 is added to both the numerator and denominator, the fraction is increased by `(4)/(35)`. Find the fraction.

Answer 1

Let the denominator of a positive fraction = x
then numerator = 8 – x

∴ Fraction = `(8 - x)/x`

According to the condition.
`(8 - x + 2)/(x + 2) = (8 -x)/x + (4)/(35)`

⇒ `(10 - x)/(x + 2) = (8 - x)/x + (4)/(35)`

⇒ `(10 - x)/(x + 2) - (8 - x)/x = (4)/(35)`

⇒ `(10x - x^2 - 8x + x^2 - 16 + 2x)/(x(x + 2)) = (4)/(35)`

⇒ `(4x - 16)/(x^2 + 2x) = (4)/(35)`

⇒ 4x² + 8x = 140x - 560
⇒ 4x² + 8x - 140x + 560 = 0
⇒ 4x² - 132x + 560 = 0
⇒ x² - 33x + 140 = 0
⇒ x² - 28x - 5x + 140 = 0
⇒ x(x - 28) -5(x - 28) = 0
⇒ (x - 28)(x - 5) = 0
Either x - 28 = 0,
then x = 28, 
but it is not possible as sum of numberator and denominator is 8
or
x - 5 = 0,
then x = 5

∴ Fraction = `(8 - x)/x = (8 - 5)/(5) = (3)/(5)`.