In an alloy ‘A’, zinc & copper is in the ratio of 1 : 1. In the second alloy ‘B’, the same elements are in the ratio 3 : 5. If these two alloys mixed to form a new alloy in which zinc and copper is in the ratio 2 : 3, find the ratio in which alloy ‘A’ and alloy ‘B’ are mixed?

The correct option is 1 : 4.

Step-by-Step Explanation:

Step 1: Understand the given ratios and find the fraction of Zinc in each alloy.

In Alloy A, the ratio of Zinc to Copper is 1 : 1.

Total parts in Alloy A = 1 + 1 = 2.

Therefore, the fraction of Zinc in Alloy A is:

$\frac{1}{2}$

In Alloy B, the ratio of Zinc to Copper is 3 : 5.

Total parts in Alloy B = 3 + 5 = 8.

Therefore, the fraction of Zinc in Alloy B is:

$\frac{3}{8}$

In the final mixture (new alloy), the ratio of Zinc to Copper is 2 : 3.

Total parts in the new alloy = 2 + 3 = 5.

Therefore, the fraction of Zinc in the new alloy is:

$\frac{2}{5}$

Step 2: Apply the Rule of Alligation.

Let us set up the alligation method based on the fraction of Zinc in each alloy:

- Proportion of Zinc in Alloy A = $\frac{1}{2}$

- Proportion of Zinc in Alloy B = $\frac{3}{8}$

- Proportion of Zinc in the Mixture = $\frac{2}{5}$

Now, calculate the differences:

Difference for Alloy A part = Mixture - Alloy B = $\frac{2}{5}-\frac{3}{8}$

To subtract these fractions, find a common denominator (which is 40):

$\frac{2}{5}-\frac{3}{8}=\frac{16-15}{40}=\frac{1}{40}$

Difference for Alloy B part = Alloy A - Mixture = $\frac{1}{2}-\frac{2}{5}$

To subtract these fractions, find a common denominator (which is 10):

$\frac{1}{2}-\frac{2}{5}=\frac{5-4}{10}=\frac{1}{10}$

Step 3: Calculate the mixing ratio of Alloy A to Alloy B.

The ratio in which Alloy A and Alloy B are mixed is:

$\text{Ratio}=\frac{1}{40}:\frac{1}{10}$

Multiply both sides by 40 to simplify:

$\text{Ratio}=1:4$

Thus, Alloy A and Alloy B are mixed in the ratio 1 : 4.