Two types of alloy possess gold and silver in the ratio of 7:22 and 21:37. In what ratio should these alloys be mixed so as to have a new alloy in which gold and silver would exist in the ra tio 25 : 62?

Options

A. `13:8`
B. `8:13`
C. `13:12`
D. `6:9`

Answer

To solve the problem of mixing two types of alloys containing gold and silver in specific ratios to achieve a desired ratio, we can follow these steps:

### Step 1: Identify the Ratios of Gold and Silver in the Alloys
- **Alloy A** has gold and silver in the ratio of 7:22.
- **Alloy B** has gold and silver in the ratio of 21:37.

### Step 2: Calculate the Total Parts in Each Alloy
- For **Alloy A**: Total parts = 7 + 22 = 29.
- For **Alloy B**: Total parts = 21 + 37 = 58.

### Step 3: Identify the Desired Ratio of Gold and Silver
- The desired ratio in the new alloy (C) is 25:62.
- Total parts in the new alloy = 25 + 62 = 87.

### Step 4: Find the LCM of Total Parts
- The LCM of the total parts of the alloys and the new alloy is calculated as follows:
- LCM of 29, 58, and 87 = 174.

### Step 5: Calculate the Amount of Gold and Silver in Each Alloy
- For **Alloy A**:
- Gold = (7/29) * 174 = 42
- Silver = (22/29) * 174 = 132

- For **Alloy B**:
- Gold = (21/58) * 174 = 63
- Silver = (37/58) * 174 = 111

- For **New Alloy C**:
- Gold = (25/87) * 174 = 50
- Silver = (62/87) * 174 = 124

### Step 6: Set Up the Allegation Method
Using the allegation method, we can find the ratio in which the two alloys should be mixed.

1. **For Gold**:
- Alloy A (42) and Alloy B (63) compared to the mixture (50):
- Difference for Alloy A: 63 - 50 = 13
- Difference for Alloy B: 50 - 42 = 8

2. **For Silver**:
- Alloy A (132) and Alloy B (111) compared to the mixture (124):
- Difference for Alloy A: 124 - 111 = 13
- Difference for Alloy B: 132 - 124 = 8

### Step 7: Calculate the Ratio of the Alloys
The ratio of Alloy A to Alloy B based on the differences calculated:
- Ratio = Difference for Alloy B : Difference for Alloy A = 13 : 8.

### Final Answer
The required ratio in which the two alloys should be mixed is **13:8**.

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