In what ratio should 20% ethanol solution be mixed with 40% ethanol solution to obtain a 28% ethanol solution?

Options

A. `2:3`
B. `8:5`
C. `3:2`
D. `5:8`

Answer

To solve the problem of mixing a 20% ethanol solution with a 40% ethanol solution to obtain a 28% ethanol solution, we can use the method of alligation. Here are the steps:

### Step-by-Step Solution:

1. **Identify the Concentrations**:
- Let the concentration of solution A (20% ethanol) be \( C_1 = 20\% \).
- Let the concentration of solution B (40% ethanol) be \( C_2 = 40\% \).
- The desired concentration of the mixture is \( C_m = 28\% \).

2. **Set Up the Alligation**:
- We will use the alligation method to find the ratio of the two solutions.
- Calculate the difference between the concentration of solution B and the desired concentration:
\[ C_2 - C_m = 40\% - 28\% = 12\% \]
- Calculate the difference between the desired concentration and the concentration of solution A:
\[ C_m - C_1 = 28\% - 20\% = 8\% \]

3. **Create the Alligation Diagram**:
- Place the concentrations in a diagram:
```
C_2 (40%) C_m (28%) C_1 (20%)
| | |
| | |
| | |
12% 28% 8%
```

4. **Find the Ratio**:
- The ratio of the two solutions can be determined by the differences calculated:
- The ratio of solution A to solution B is given by the differences:
\[ \text{Ratio of A to B} = \frac{C_2 - C_m}{C_m - C_1} = \frac{12\%}{8\%} = \frac{12}{8} = \frac{3}{2} \]

5. **Conclusion**:
- Therefore, the two solutions should be mixed in the ratio of 3:2.

### Final Answer:
The ratio in which the 20% ethanol solution should be mixed with the 40% ethanol solution to obtain a 28% ethanol solution is **3:2**.

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