In a hall, 70% of the children were boys. After 87 more children entered the hall, the number of boys increased by 20% and the number of girls increased by 50%. How many children were there in the hall in the end?

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 = $100 \text{ units} + 29 \text{ units} = 129 \text{ units}$
$129 \text{ units} = 129 \times 3 = 387$