在大厅里，70% 的儿童是男孩。又有 87 名儿童进入大厅后，男孩人数增加了 20%，女孩人数增加了 50%。最后大厅里一共有多少名儿童？

1. Express the initial number of boys and girls in terms of units.

Let the initial total number of children be $100$ units.
Initial number of boys = $70$ units
Initial number of girls = $100 - 70 = 30$ units

2. Calculate the increase in the number of boys and girls in terms of units.

Increase in boys = $20\%$ of $70$ units = $\frac{20}{100} \times 70 = 14$ units
Increase in girls = $50\%$ of $30$ units = $\frac{50}{100} \times 30 = 15$ units
Total increase = $14 + 15 = 29$ units

3. Find the value of 1 unit and calculate the final number of children.

We are given that the total increase is $87$ children.
$29 \text{ units} = 87$
$1 \text{ unit} = 87 \div 29 = 3$
Final total number of children = $100 \text{ units} + 29 \text{ units} = 129 \text{ units}$
$129 \text{ units} = 129 \times 3 = 387$