Alloys A and B contain copper and zink in the ratio `1 :3` and `3 :5` , respectively . In What ratio A and B be mixed to get a new alloy containing copper and zink in the ratio `1 :2` ?

Options

A. `3 :4`
B. `2 :1`
C. `1 :2`
D. `4 :5`

Answer

To solve the problem of mixing alloys A and B to achieve a new alloy with a copper to zinc ratio of 1:2, we will follow these steps:

### Step 1: Define the composition of alloys A and B
Alloy A has copper and zinc in the ratio of 1:3. This means:
- Copper in Alloy A = 1 part
- Zinc in Alloy A = 3 parts

Alloy B has copper and zinc in the ratio of 3:5. This means:
- Copper in Alloy B = 3 parts
- Zinc in Alloy B = 5 parts

### Step 2: Express the quantities in terms of a variable
Let’s assume we take x parts of Alloy A and y parts of Alloy B.

For Alloy A:
- Copper from A = \( \frac{1}{4}x \) (since total parts = 1 + 3 = 4)
- Zinc from A = \( \frac{3}{4}x \)

For Alloy B:
- Copper from B = \( \frac{3}{8}y \) (since total parts = 3 + 5 = 8)
- Zinc from B = \( \frac{5}{8}y \)

### Step 3: Set up the equation for the new alloy
We want the new alloy to have a copper to zinc ratio of 1:2. Therefore, we can set up the equation:

\[ \frac{\text{Copper from A} + \text{Copper from B}}{\text{Zinc from A} + \text{Zinc from B}} = \frac{1}{2} \]

Substituting the expressions we found:

\[ \frac{\frac{1}{4}x + \frac{3}{8}y}{\frac{3}{4}x + \frac{5}{8}y} = \frac{1}{2} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 2\left(\frac{1}{4}x + \frac{3}{8}y\right) = 1\left(\frac{3}{4}x + \frac{5}{8}y\right) \]

### Step 5: Simplify the equation
Expanding both sides:

\[ \frac{1}{2}x + \frac{3}{4}y = \frac{3}{4}x + \frac{5}{8}y \]

Now, we can multiply through by 8 to eliminate the denominators:

\[ 4x + 6y = 6x + 5y \]

### Step 6: Rearranging the equation
Rearranging gives:

\[ 4x - 6x + 6y - 5y = 0 \]
\[ -2x + y = 0 \]
\[ y = 2x \]

### Step 7: Find the ratio of A to B
Now, we can express the ratio of A to B:

\[ \frac{x}{y} = \frac{x}{2x} = \frac{1}{2} \]

Thus, the ratio of Alloy A to Alloy B is 1:2.

### Final Answer
The alloys A and B should be mixed in the ratio **1:2** to achieve the desired copper to zinc ratio of 1:2.

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