Question

The numerator of a fraction is 2 less than the denominator. A new fraction can be formed by adding 5 to the denominator and doubling the numerator. The product of the original fraction and the reciprocal of the new fraction is 1/4. What is the original fraction?​

Answer 1

`(6)/(5)`

Explanation:

let the denominator of the original fraction be x, then

`(x-2)/(x)` ← original fraction

Then

`(2(x-2))/(x+5)` ← new fraction

The product of original and reciprocal of new is

`(x-2)/(x)` ×
`(x+5)/(2(x-2))` ← cancel (x - 2)

=
`(x+5)/(2x)` =
`(1)/(4)` ( cross- multiply )

4(x + 5) = 2x

4x + 20 = 2x ( subtract 2x from both sides )

2x + 20 = 0 ( subtract 20 from both sides )

2x = - 20 ( divide both sides by 2 )

x = - 10

Then original fraction

`(x-2)/(x)` =
`(-12)/(-10)` =
`(6)/(5)`