To solve the given proportion using proportional reasoning, we aim to find the value of \(a\) in the proportion:
[tex]\[
\frac{3}{5} = \frac{a + 5}{25}
\][/tex]
Follow these steps:
1. Recognize that the two fractions set equal to each other indicates a proportion.
2. To eliminate the fractions, cross-multiply the two ratios. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and setting this equal to the product of the denominator of the first fraction and the numerator of the second fraction:
[tex]\[
3 \times 25 = 5 \times (a + 5)
\][/tex]
3. Perform the multiplication on both sides:
[tex]\[
75 = 5(a + 5)
\][/tex]
4. To isolate \(a\), first distribute the 5 on the right-hand side:
[tex]\[
75 = 5a + 25
\][/tex]
5. Subtract 25 from both sides to begin isolating the term involving \(a\):
[tex]\[
75 - 25 = 5a
\][/tex]
Simplifying this:
[tex]\[
50 = 5a
\][/tex]
6. Finally, divide both sides by 5 to solve for \(a\):
[tex]\[
a = \frac{50}{5}
\][/tex]
Simplifying this:
[tex]\[
a = 10
\][/tex]
Therefore, the value of \(a\) is:
[tex]\[
a = 10
\][/tex]
So the correct answer is [tex]\( \boxed{10} \)[/tex].
