Let's denote the numerator of the fraction by [tex]\( x \)[/tex] and the denominator by [tex]\( y \)[/tex]. According to the problem, the denominator exceeds the numerator by 25. This can be expressed with the equation:
[tex]\[ y = x + 25 \][/tex]
We are also given that the value of the fraction is [tex]\(\frac{3}{8}\)[/tex], which we can write as:
[tex]\[ \frac{x}{y} = \frac{3}{8} \][/tex]
Now, we have two equations:
1. [tex]\( y = x + 25 \)[/tex]
2. [tex]\( \frac{x}{y} = \frac{3}{8} \)[/tex]
We can substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ \frac{x}{x + 25} = \frac{3}{8} \][/tex]
To solve this equation, we will cross-multiply:
[tex]\[ 8x = 3(x + 25) \][/tex]
Distribute the 3 on the right-hand side:
[tex]\[ 8x = 3x + 75 \][/tex]
Next, we will move all terms involving [tex]\( x \)[/tex] to one side of the equation:
[tex]\[ 8x - 3x = 75 \][/tex]
[tex]\[ 5x = 75 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{75}{5} \][/tex]
[tex]\[ x = 15 \][/tex]
Now that we have the numerator, we can find the denominator using the first equation:
[tex]\[ y = x + 25 \][/tex]
[tex]\[ y = 15 + 25 \][/tex]
[tex]\[ y = 40 \][/tex]
So, the numerator [tex]\( x \)[/tex] is 15 and the denominator [tex]\( y \)[/tex] is 40.
The fraction is:
[tex]\[ \frac{x}{y} = \frac{15}{40} \][/tex]
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[ \frac{15}{40} = \frac{15 \div 5}{40 \div 5} = \frac{3}{8} \][/tex]
Thus, the fraction is [tex]\( \frac{3}{8} \)[/tex], consistent with the given information. The numerator is 15 and the denominator is 40.
