Gold is $19$ times as heavy as water and copper is $9$ times as heavy as water. The ratio in which these two metals are to be mixed so that the mixed alloy becomes $15$ times as heavy as water is

We can solve this problem efficiently using the rule of alligation.
* Heavy component (Gold value) $=19$
* Lighter component (Copper value) $=9$
* Desired mean value (Alloy value) $=15$

Applying alligation format:

$$\frac{\text{Quantity of Gold}}{\text{Quantity of Copper}}=\frac{\text{Mean Value}-\text{Lighter Value}}{\text{Heavier Value}-\text{Mean Value}}$$

$$\frac{\text{Quantity of Gold}}{\text{Quantity of Copper}}=\frac{15-9}{19-15}=\frac{6}{4}=\frac{3}{2}$$

Thus, the ratio in which gold and copper must be mixed is $3:2$.