Question

The ratio of the numerator to the denominator of a certain fraction is three to five. If two is added to the numerator and five is subtracted from the denominator, the new fraction reduces to four-fifths. Find the original fraction.

3/5

20/25

18/30

Answer 1

Initial fraction is reducible to \[\frac{3}{5}\]
That means we can write it as \[\frac{3x}{5x}\]where \(x\) is some unknown factor. Do you agree that expression gives us the correct ratio?

Answer 2

Let x and y be the numerator and the denominator of the initial fraction, respectively.
Solve the following equations for x and y.\[\left\{\frac{x}{y}=\frac{3}{5},\frac{x+2}{y-5}=\frac{4}{5}\right\} \]x=18 and y=30
\[\frac{18}{30}=\frac{3}{5} \]and\[\frac{18+2}{30-5}=\frac{4}{5} \]

Answer 3

My approach doesn't require solving a system of equations, but is otherwise equivalent.

\[\frac{3x+2}{5x-5} = \frac{4}{5}\]Cross-multiply and solve for \(x\):
\[5(3x+2) = 4(5x-5)\]\[15x+10=20x-20\]\[30=5x\]\[x=6\]therefore, fraction is \[\frac{3(6)}{5(6)} = \frac{18}{30}\]