Question

The numerator of a fraction is 5 less than the denominator. If both the numerator and denominator are

increased by 4, the fraction is tripled in value. Find the original fraction.

Answer 1

`(1)/(6)`

Explanation:

let the denominator of the fraction be x , then the fraction is

`(x-5)/(x)`

increasing the numerator and denominator by 4

`(x-5+4)/(x+4)` =
`(x-1)/(x+4)`

this fraction is then equal to 3 times ( triple ) the value of the fraction, so

`(x-1)/(x+4)` =
`(3(x-5))/(x)` ( cross- multiplying )

3(x - 5)(x + 4) = x(x - 1) ← expand factors on left using FOIL

3(x² - x - 20) = x(x - 1) ← distribute parenthesis on both sides

3x² - 3x - 60 = x² - x ( subtract x² - x from both sides )

2x² - 2x - 60 = 0 ( divide through by 2 )

x² - x - 30 = 0 ← in standard form

(x - 6)(x + 5) = 0

equate each factor to zero and solve for x

x - 6 = 0 ⇒ x = 6

x + 5 = 0 ⇒ x = - 5

however, x > 0 , then x = 6

original fraction

=
`(x-5)/(x)` =
`(6-5)/(6)` =
`(1)/(6)`