In what ratio must a mixture of 30% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 45% alcohol strength ?

Options

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

Answer

To solve the problem of mixing two alcohol solutions of different strengths to achieve a desired strength, we can use the method of allegation. Here’s a step-by-step solution:

### Step 1: Identify the strengths of the alcohol mixtures
- Let the first mixture have an alcohol strength of 30%.
- Let the second mixture have an alcohol strength of 50%.
- We want to achieve a final mixture with an alcohol strength of 45%.

### Step 2: Set up the allegation method
Using the allegation method, we will calculate the differences between the strengths:

- The difference between the higher strength (50%) and the desired strength (45%):
\[ 50\% - 45\% = 5\% \]

- The difference between the desired strength (45%) and the lower strength (30%):
\[ 45\% - 30\% = 15\% \]

### Step 3: Set up the ratio
The ratio of the two mixtures can be determined by the differences calculated above. The ratio of the first mixture (30%) to the second mixture (50%) is given by the differences:

\[ \text{Ratio} = \frac{\text{Difference from higher strength}}{\text{Difference from lower strength}} = \frac{5}{15} \]

### Step 4: Simplify the ratio
Now, simplify the ratio:
\[ \frac{5}{15} = \frac{1}{3} \]

### Conclusion
Thus, the ratio in which the 30% alcohol mixture must be mixed with the 50% alcohol mixture to obtain a 45% alcohol mixture is:
\[ \text{Ratio} = 1 : 3 \]

### Final Answer
The required ratio is **1:3**.

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