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:

We are given two alloys, A and B, which contain zinc and copper in different proportions. We need to find the ratio in which alloy A and alloy B should be mixed to get a new alloy with a specific zinc-to-copper ratio.

1. Determine the fraction of zinc in each alloy:
In Alloy A, the ratio of zinc to copper is 1 : 1.
Fraction of zinc in Alloy A = $\frac{1}{1+1}=\frac{1}{2}$

In Alloy B, the ratio of zinc to copper is 3 : 5.
Fraction of zinc in Alloy B = $\frac{3}{3+5}=\frac{3}{8}$

In the resulting mixture, the ratio of zinc to copper is 2 : 3.
Fraction of zinc in the final mixture = $\frac{2}{2+3}=\frac{2}{5}$

2. Apply the Rule of Alligation:
Let the ratio of quantity of Alloy A to quantity of Alloy B mixed be x : y.

Using the alligation method on the proportion of zinc:
Cheaper/Smaller fraction (Alloy B) = $\frac{3}{8}=0.375$
Dearer/Larger fraction (Alloy A) = $\frac{1}{2}=0.5$
Mean fraction (Final mixture) = $\frac{2}{5}=0.4$

Now, calculate the differences:
Difference between Alloy A fraction and Mean fraction = $\frac{1}{2}-\frac{2}{5}=\frac{5-4}{10}=\frac{1}{10}$
Difference between Mean fraction and Alloy B fraction = $\frac{2}{5}-\frac{3}{8}=\frac{16-15}{40}=\frac{1}{40}$

3. Calculate the required ratio:
Ratio of Alloy A to Alloy B = (Difference for Alloy B) : (Difference for Alloy A)

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

Multiply both sides by 40 to simplify:
$\text{Required Ratio}=1:4$

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