Solving a Fraction Problem
Let's denote the numerator of the original fraction as \( n \) and the denominator as \( d \). According to the problem, the denominator is four more than the numerator, so:
\( d = n + 4 \)
The problem also states that if both the numerator and denominator are increased by one, the resulting fraction is simplified to \( \frac{5}{7} \):
\( \frac{n + 1}{d + 1} = \frac{5}{7} \)
Now we have a system of two equations to solve for the values of \( n \) and \( d \):
1) \( d = n + 4 \)
2) \( \frac{n + 1}{n + 5} = \frac{5}{7} \) (since \( d = n + 4 \), we replaced \( d \) with \( n + 4 \) in the second equation)
Next, we can cross-multiply in the second equation to solve for \( n \):
\( 7(n + 1) = 5(n + 5) \)
Expand both sides:
\( 7n + 7 = 5n + 25 \)
Subtract \( 5n \) from both sides:
\( 2n + 7 = 25 \)
Subtract 7 from both sides:
\( 2n = 18 \)
Divide by 2:
\( n = 9 \)
Now that we have \( n \), we can find \( d \) using either of the original equations. Let's use the first one:
\( d = n + 4 \)
Replace \( n \) with 9:
\( d = 9 + 4 \)
Thus:
\( d = 13 \)
The original fraction is \( \frac{n}{d} = \frac{9}{13} \).
