Two alloys made up of copper and tin. The ratio of copper and tin in the first alloy is 1:3 in the second alloy it is 2:5 in what ratio the two alloys should be mixed to obtain a new alloy in which the ratio of tin and copper be 8:3?

Options

A. `3:5`
B. `4:7`
C. `3:8`
D. `5:11`

Answer

To solve the problem of mixing two alloys made of copper and tin in the required ratio, we can follow these steps:

### Step 1: Understand the Ratios of Alloys
The first alloy has a ratio of copper to tin as 1:3. This means:
- For every 1 part of copper, there are 3 parts of tin.

The second alloy has a ratio of copper to tin as 2:5. This means:
- For every 2 parts of copper, there are 5 parts of tin.

### Step 2: Convert Ratios to Parts
Let's denote the amounts of the first and second alloys as \( x \) and \( y \) respectively.

From the first alloy:
- Copper = \( \frac{1}{1+3} \cdot x = \frac{1}{4}x \)
- Tin = \( \frac{3}{1+3} \cdot x = \frac{3}{4}x \)

From the second alloy:
- Copper = \( \frac{2}{2+5} \cdot y = \frac{2}{7}y \)
- Tin = \( \frac{5}{2+5} \cdot y = \frac{5}{7}y \)

### Step 3: Set Up the Equation for the New Alloy
We want the new alloy to have a ratio of tin to copper as 8:3. This means:
- Total parts = 8 + 3 = 11 parts
- Copper in the new alloy = \( \frac{3}{11} \) of the total
- Tin in the new alloy = \( \frac{8}{11} \) of the total

### Step 4: Express the Total Copper and Tin in Terms of \( x \) and \( y \)
The total copper in the new alloy can be expressed as:
\[ \frac{1}{4}x + \frac{2}{7}y \]

The total tin in the new alloy can be expressed as:
\[ \frac{3}{4}x + \frac{5}{7}y \]

### Step 5: Set Up the Ratio Equation
According to the desired ratio of tin to copper:
\[ \frac{\frac{3}{4}x + \frac{5}{7}y}{\frac{1}{4}x + \frac{2}{7}y} = \frac{8}{3} \]

### Step 6: Cross-Multiply to Solve for \( x \) and \( y \)
Cross-multiplying gives:
\[ 3\left(\frac{3}{4}x + \frac{5}{7}y\right) = 8\left(\frac{1}{4}x + \frac{2}{7}y\right) \]

Expanding both sides:
\[ \frac{9}{4}x + \frac{15}{7}y = 2x + \frac{16}{7}y \]

### Step 7: Simplify the Equation
Rearranging gives:
\[ \frac{9}{4}x - 2x + \frac{15}{7}y - \frac{16}{7}y = 0 \]
\[ \left(\frac{9}{4} - \frac{8}{4}\right)x + \left(\frac{15}{7} - \frac{16}{7}\right)y = 0 \]
\[ \frac{1}{4}x - \frac{1}{7}y = 0 \]

### Step 8: Solve for the Ratio \( x:y \)
Cross-multiplying gives:
\[ 7x = 4y \implies \frac{x}{y} = \frac{4}{7} \]

### Final Answer
The two alloys should be mixed in the ratio of **4:7**.

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