In an amalgam, mercury and silver are in the ratio of 5 : 6. In the second amalgam, the ratio of same elements is 3 : 8 . If equal quantities of these two amalgam are mixed to form a new amalgam, then what will be the ratio of both of these elements in the new amalgam?

Options

A. `1:4`
B. `7:8`
C. `4:7`
D. `2:7`

Answer

To solve the problem step-by-step, we will analyze the ratios of mercury and silver in the two amalgams and then combine them to find the ratio in the new amalgam.

### Step 1: Understand the ratios in the first amalgam
In the first amalgam, the ratio of mercury (M) to silver (S) is given as 5:6. This means:
- For every 5 parts of mercury, there are 6 parts of silver.

### Step 2: Calculate the total parts in the first amalgam
The total parts in the first amalgam can be calculated as:
Total parts = 5 (mercury) + 6 (silver) = 11 parts.

### Step 3: Express the quantities in terms of a variable
Let’s assume we have a total quantity of X for the first amalgam. The quantities of mercury and silver can be expressed as:
- Mercury from the first amalgam = \( \frac{5}{11} \times X \)
- Silver from the first amalgam = \( \frac{6}{11} \times X \)

### Step 4: Understand the ratios in the second amalgam
In the second amalgam, the ratio of mercury to silver is given as 3:8. This means:
- For every 3 parts of mercury, there are 8 parts of silver.

### Step 5: Calculate the total parts in the second amalgam
The total parts in the second amalgam can be calculated as:
Total parts = 3 (mercury) + 8 (silver) = 11 parts.

### Step 6: Express the quantities in terms of the same variable
Let’s assume we have the same total quantity of X for the second amalgam as well. The quantities of mercury and silver can be expressed as:
- Mercury from the second amalgam = \( \frac{3}{11} \times X \)
- Silver from the second amalgam = \( \frac{8}{11} \times X \)

### Step 7: Combine the quantities to find the new amalgam
Now, we will combine the quantities from both amalgams to find the total mercury and silver in the new amalgam.

Total mercury in the new amalgam:
\[ \text{Total Mercury} = \left( \frac{5}{11} \times X \right) + \left( \frac{3}{11} \times X \right) = \frac{5X + 3X}{11} = \frac{8X}{11} \]

Total silver in the new amalgam:
\[ \text{Total Silver} = \left( \frac{6}{11} \times X \right) + \left( \frac{8}{11} \times X \right) = \frac{6X + 8X}{11} = \frac{14X}{11} \]

### Step 8: Find the ratio of mercury to silver in the new amalgam
The ratio of mercury to silver in the new amalgam can be calculated as:
\[ \text{Ratio} = \frac{\text{Total Mercury}}{\text{Total Silver}} = \frac{\frac{8X}{11}}{\frac{14X}{11}} = \frac{8}{14} = \frac{4}{7} \]

### Final Answer
The ratio of mercury to silver in the new amalgam is **4:7**.

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