Gold and Copper are heavier than water by $19$ and $9$ times respectively. In what proportion should they be mixed to form an alloy so that it will be heavier than water by $15$ times?

We can solve this efficiently using the method of allegation:
* Component 1 (Gold density relative to water): $19$
* Component 2 (Copper density relative to water): $9$
* Target Mean (Alloy density relative to water): $15$

Applying the allegation rule:

$$\text{Ratio of Gold : Copper}=\frac{\text{Mean}-\text{Copper Value}}{\text{Gold Value}-\text{Mean}}$$

$$\text{Ratio}=\frac{15-9}{19-15}=\frac{6}{4}=\frac{3}{2}$$

Thus, Gold and Copper should be mixed in the proportion of $3:2$.