A box contained some red and yellow balls. At first, the number of red balls to the number of yellow balls is 1 : 3. After $\frac{1}{4}$ of the red balls and $\frac{3}{8}$ of the yellow balls were taken out, there were 63 balls left in the box. How many yellow balls were there in the box at first? Ans : ___

1. Express the initial ratio using a common multiple of denominators to avoid fractions

Initial ratio of Red : Yellow = $1 : 3$.
Fractions taken out are $\frac{1}{4}$ for red and $\frac{3}{8}$ for yellow.
To easily calculate the remaining parts, we can multiply the initial ratio by $8$ (the LCM of $4$ and $8$).
Initial Red : Yellow = $8 : 24$ units.

2. Calculate the remaining units for red and yellow balls

Red balls taken out = $\frac{1}{4} \times 8 = 2$ units.
Red balls left = $8 - 2 = 6$ units.
Yellow balls taken out = $\frac{3}{8} \times 24 = 9$ units.
Yellow balls left = $24 - 9 = 15$ units.
Total units left = $6 + 15 = 21$ units.

3. Find the value of 1 unit and the initial number of yellow balls

$21$ units = $63$ balls
$1$ unit = $63 \div 21 = 3$ balls
Initial yellow balls = $24$ units = $24 \times 3 = 72$ balls.