Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio should these be mixed to get an alloy 15 times as heavy as water?

### Concept & Rule of Alligation The rule of alligation is used to find the ratio in which two ingredients at given prices (or weights) must be mixed to produce a mixture at a given price (or weight). $$ \frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \frac{\text{Price of Dearer} - \text{Mean Price}}{\text{Mean Price} - \text{Price of Cheaper}} $$ ### Step-by-Step Solution 1. Identify the given values relative to the standard reference (water): - Weight of Gold ($w_1$) = 19 - Weight of Copper ($w_2$) = 9 - Mean weight of Alloy ($w_m$) = 15 2. Apply the alligation method: Gold (19) .............................. Copper (9) .................... Alloy (15) .................... (15 - 9) = 6 ......................... (19 - 15) = 4 3. The resulting ratio of Gold to Copper is the ratio of these differences: $$ \text{Ratio} = 6 : 4 $$ 4. Simplify the ratio: $$ 6 : 4 = 3 : 2 $$ ### Exam Strategy & Shortcut Draw the alligation cross directly on rough paper: subtract the mean from the larger value ($19 - 15 = 4$) and put it under the smaller value, then subtract the smaller value from the mean ($15 - 9 = 6$) and put it under the larger value. Read the ratio straight across: $6 : 4 \implies 3 : 2$. ### Common Pitfall Mixing up the order of the ratio. Always trace back the difference to the correct starting element. The difference placed under "Gold" corresponds to Gold's share. ### Final Answer Therefore, the correct answer is **3 : 2**.