Let's determine the value of [tex]\( a \)[/tex] using the proportion given:
[tex]\[
\frac{3}{5} = \frac{a+5}{25}
\][/tex]
To solve this proportion, we will use cross-multiplication. Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal to each other. This gives us:
[tex]\[
3 \cdot 25 = 5 \cdot (a + 5)
\][/tex]
Now, let's perform the multiplication:
[tex]\[
75 = 5 \cdot (a + 5)
\][/tex]
Next, we need to distribute the 5 on the right-hand side to both terms inside the parentheses:
[tex]\[
75 = 5a + 25
\][/tex]
Now, we will isolate [tex]\( a \)[/tex] by first subtracting 25 from both sides of the equation:
[tex]\[
75 - 25 = 5a \\
50 = 5a
\][/tex]
Finally, we solve for [tex]\( a \)[/tex] by dividing both sides of the equation by 5:
[tex]\[
a = \frac{50}{5} \\
a = 10
\][/tex]
So, the value of [tex]\( a \)[/tex] is [tex]\( \boxed{10} \)[/tex].
