Let's break down the problem step by step:

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

2. The second child took \(\frac{3}{4}\) of the remainder after the first child.

Let's assume the total number of mangoes is represented by \(x\).

1. First child took: \(\frac{1}{4}x\) mangoes.

2. After the first child, the remaining mangoes: \(x - \frac{1}{4}x = \frac{3}{4}x\).

3. Second child took: \(\frac{3}{4} \cdot \frac{3}{4}x = \frac{9}{16}x\) mangoes.

So far, the first and second children have taken a total of \(\frac{1}{4}x + \frac{9}{16}x\) mangoes.

Now, the third child took what's left:

3. Third child took: Remaining mangoes = Total mangoes - Mangoes taken by first and second child

3. Third child took: \(x - \left(\frac{1}{4}x + \frac{9}{16}x\right) = x - \frac{13}{16}x = \frac{3}{16}x\) mangoes.

Now, let's find the fraction of mangoes the third child took out of the total:

\[\text{Fraction taken by third child} = \frac{\text{Mangoes taken by third child}}{\text{Total mangoes}} = \frac{\frac{3}{16}x}{x} = \frac{3}{16}\]

So, the fraction of mangoes the third child took is \(\frac{3}{16}\),