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?

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jimrgrant1 jimrgrant1 · Answer 1 · 2021-03-24T10:09:47+01:00

Answer:

[tex]\frac{6}{5}[/tex]

Step-by-step explanation:

let the denominator of the original fraction be x, then

[tex]\frac{x-2}{x}[/tex] ← original fraction

Then

[tex]\frac{2(x-2)}{x+5}[/tex] ← new fraction

The product of original and reciprocal of new is

[tex]\frac{x-2}{x}[/tex] × [tex]\frac{x+5}{2(x-2)}[/tex] ← cancel (x - 2)

= [tex]\frac{x+5}{2x}[/tex] = [tex]\frac{1}{4}[/tex] ( 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

[tex]\frac{x-2}{x}[/tex] = [tex]\frac{-12}{-10}[/tex] = [tex]\frac{6}{5}[/tex]