In an alloy, zinc and copper are in the ratio 1:2. In the second alloy, the same elements are in the ratio 2:3. If these two alloys be mixed to form a new alloy in which two elements are in the ratio 5:8, the ratio of these two alloys in the new alloy is

Options

A. `3:10`
B. `3:7`
C. `10:3`
D. `7:3`

Answer

To solve the problem step by step, we will use the concept of alligation or mixtures.

### Step 1: Understand the Ratios in Each Alloy
- In the first alloy, the ratio of zinc to copper is 1:2.
- In the second alloy, the ratio of zinc to copper is 2:3.

### Step 2: Convert Ratios to Fractions
- For the first alloy:
- Total parts = 1 + 2 = 3
- Fraction of zinc in the first alloy = \( \frac{1}{3} \)

- For the second alloy:
- Total parts = 2 + 3 = 5
- Fraction of zinc in the second alloy = \( \frac{2}{5} \)

### Step 3: Determine the Zinc Fraction in the New Alloy
- The new alloy has a zinc to copper ratio of 5:8.
- Total parts in the new alloy = 5 + 8 = 13
- Fraction of zinc in the new alloy = \( \frac{5}{13} \)

### Step 4: Set Up the Alligation
Now we will set up the alligation to find the ratio of the two alloys in the new mixture.

- We will write the fractions of zinc in the first alloy, second alloy, and the new alloy:
- First alloy: \( \frac{1}{3} \)
- Second alloy: \( \frac{2}{5} \)
- New alloy: \( \frac{5}{13} \)

### Step 5: Calculate the Differences
- Calculate the difference between the new alloy and each of the alloys:
- Difference between new alloy and first alloy:
\[ \frac{5}{13} - \frac{1}{3} = \frac{5 \times 3 - 1 \times 13}{39} = \frac{15 - 13}{39} = \frac{2}{39} \]

- Difference between new alloy and second alloy:
\[ \frac{2}{5} - \frac{5}{13} = \frac{2 \times 13 - 5 \times 5}{65} = \frac{26 - 25}{65} = \frac{1}{65} \]

### Step 6: Set Up the Ratio
- The ratio of the two alloys in the new alloy is the inverse of the differences calculated:
\[ \text{Ratio} = \frac{1/65}{2/39} = \frac{39}{130} \]

### Step 7: Simplify the Ratio
- To simplify \( \frac{39}{130} \):
- Divide both numerator and denominator by their greatest common divisor (GCD), which is 13:
\[ \frac{39 \div 13}{130 \div 13} = \frac{3}{10} \]

### Final Answer
The ratio of the two alloys in the new alloy is \( 3:10 \).

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