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 ?

Options

A. `1:1`
B. `2:1`
C. `1:2`
D. `3:2`

Answer

To solve the problem of mixing gold and copper to obtain an alloy that is 15 times as heavy as water, we can use the method of allegation. Here is the step-by-step solution:

### Step 1: Identify the weights of gold and copper relative to water.
- Gold is 19 times as heavy as water.
- Copper is 9 times as heavy as water.
- We want to create an alloy that is 15 times as heavy as water.

### Step 2: Set up the allegation method.
We will use the following format for the allegation method:

```
Gold (19) Alloy (15) Copper (9)
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(19 - 15) (15 - 9)
```

### Step 3: Calculate the differences.
- The difference between the weight of gold and the desired alloy:
\[ 19 - 15 = 4 \]
- The difference between the weight of copper and the desired alloy:
\[ 15 - 9 = 6 \]

### Step 4: Write the ratio.
Using the differences calculated:
- The ratio of gold to copper is given by the differences:
\[ \text{Ratio of Gold to Copper} = 4 : 6 \]

### Step 5: Simplify the ratio.
To simplify the ratio \(4 : 6\):
\[ \frac{4}{2} : \frac{6}{2} = 2 : 3 \]

### Step 6: Conclusion.
Thus, the ratio in which gold and copper should be mixed to obtain an alloy that is 15 times as heavy as water is:
\[ \text{Gold : Copper} = 2 : 3 \]