To determine the fraction of mangoes the third child took, follow these steps:

1. Let the total number of mangoes be \(1\) (representing the whole basket).

2. The first child took \(\frac{1}{4}\) of the mangoes:

\[
\text{Mangoes taken by the first child} = \frac{1}{4}
\]

The remaining mangoes after the first child:

\[
\text{Remaining} = 1 - \frac{1}{4} = \frac{3}{4}
\]

3. The second child took \(\frac{3}{4}\) of the remaining mangoes:

\[
\text{Mangoes taken by the second child} = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}
\]

4. The remaining mangoes after the second child:

\[
\text{Remaining} = \frac{3}{4} - \frac{9}{16}
\]

Convert \(\frac{3}{4}\) to a fraction with a denominator of 16:

\[
\frac{3}{4} = \frac{12}{16}
\]

Now subtract:

\[
\text{Remaining} = \frac{12}{16} - \frac{9}{16} = \frac{3}{16}
\]

5. Thus, the fraction of the mangoes the third child took is:

\[
\frac{3}{16}
\]

So, the correct answer is:

A. \(\frac{3}{16}\)