The numerator of a fraction is 5 less than its denominator. If 2 is subtracted from the numerator and 2 is added to the denominator, the fraction becomes $\frac{2}{5}$. Find the original fraction.
$\frac{8}{13}$
Solving the Fraction Word Problem
Let the original fraction be represented as $\frac{n}{d}$, where '$n$' is the numerator and '$d$' is the denominator.
Setting Up the Equations
- According to the first condition, the numerator is 5 less than its denominator: $n = d - 5$
- The original fraction can be written in terms of the denominator '$d$': $\frac{d-5}{d}$
- According to the second condition, if 2 is subtracted from the numerator and 2 is added to the denominator, the fraction becomes $\frac{2}{5}$. $\frac{n-2}{d+2} = \frac{2}{5}$
Solving for the Denominator
Substitute the expression for '$n$' from the first condition ($n = d - 5$) into the second equation:
$ \frac{(d-5)-2}{d+2} = \frac{2}{5} $Simplify the numerator:
$ \frac{d-7}{d+2} = \frac{2}{5} $Cross-multiply to solve for '$d$':
$ 5(d-7) = 2(d+2) $ $ 5d - 35 = 2d + 4 $Rearrange the terms to isolate '$d$':
$ 5d - 2d = 4 + 35 $ $ 3d = 39 $ $ d = \frac{39}{3} $ $ d = 13 $Finding the Numerator and the Original Fraction
Now that the denominator '$d$' is found, calculate the numerator '$n$' using the first condition ($n = d - 5$):
$ n = 13 - 5 $ $ n = 8 $The original fraction is therefore:
$ \frac{n}{d} = \frac{8}{13} $Verification
Check if the conditions are met with the fraction $\frac{8}{13}$:
- Numerator (8) is 5 less than the denominator (13). (13 - 5 = 8) - Condition 1 met.
- Subtract 2 from the numerator (8 - 2 = 6) and add 2 to the denominator (13 + 2 = 15). The new fraction is $\frac{6}{15}$.
- Simplify $\frac{6}{15}$: $\frac{6 \div 3}{15 \div 3} = \frac{2}{5}$. - Condition 2 met.
The original fraction is $\frac{8}{13}$.
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