If \( a \) varies inversely as \( b^2 \), we can express this relationship mathematically as:
\[ a = \frac{k}{b^2} \]
where \( k \) is a constant.
From the information given, when \( b = \frac{4}{5} \), \( a = 25 \). We can use this to find the constant \( k \).
Substituting the values into the equation:
\[ 25 = \frac{k}{\left(\frac{4}{5}\right)^2} \]
Calculating \( \left(\frac{4}{5}\right)^2 \):
\[ \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]
Now substituting this back into the equation:
\[ 25 = \frac{k}{\frac{16}{25}} \]
Rearranging gives:
\[ 25 \cdot \frac{16}{25} = k \]
This simplifies to:
\[ k = 16 \]
We now have the constant \( k \). The relationship between \( a \) and \( b \) is now:
\[ a = \frac{16}{b^2} \]
Next, we need to find \( a \) when \( b = \frac{5}{4} \):
\[ a = \frac{16}{\left(\frac{5}{4}\right)^2} \]
Calculating \( \left(\frac{5}{4}\right)^2 \):
\[ \left(\frac{5}{4}\right)^2 = \frac{25}{16} \]
Substituting this back into the equation for \( a \):
\[ a = \frac{16}{\frac{25}{16}} = 16 \cdot \frac{16}{25} = \frac{256}{25} \]
Thus, the value of \( a \) when \( b = \frac{5}{4} \) is:
\[ \boxed{\frac{256}{25}} \]