Mixture of milk and water has been kept in two separate containers. Ratio of milk to water in one of the containers is 5 : 1 and that in the other container is 7 : 2. In what ratio should the mixtures of these two containers be added together so that the quantity of milk in the new mixture may become `80%`
(a)`3 : 2`
(b)` 2 : 3`
(c)` 4 : 5`
(d)None of these

Options

A. `3 : 2`
B. ` 2 : 3`
C. ` 4 : 5`
D. None of these

Answer

To solve the problem, we need to find the ratio in which the mixtures from two containers should be combined so that the resulting mixture contains 80% milk.

### Step-by-Step Solution:

1. **Identify the ratios of milk and water in each container:**
- In the first container, the ratio of milk to water is 5:1. This means:
- Milk = 5 parts
- Water = 1 part
- Total = 5 + 1 = 6 parts
- Fraction of milk = 5/6

- In the second container, the ratio of milk to water is 7:2. This means:
- Milk = 7 parts
- Water = 2 parts
- Total = 7 + 2 = 9 parts
- Fraction of milk = 7/9

2. **Determine the desired fraction of milk in the new mixture:**
- We want the new mixture to contain 80% milk, which can be expressed as:
- Fraction of milk = 80/100 = 4/5

3. **Set up the equation using the allegation method:**
- We will use the formula for the allegation method:
- Let the fraction of milk in the first container be \( p_1 = \frac{5}{6} \)
- Let the fraction of milk in the second container be \( p_2 = \frac{7}{9} \)
- Let the desired fraction of milk in the mixture be \( p_m = \frac{4}{5} \)

4. **Calculate the differences:**
- Difference between \( p_1 \) and \( p_m \):
\[ p_1 - p_m = \frac{5}{6} - \frac{4}{5} \]
To calculate this, find a common denominator (30):
\[ = \frac{25}{30} - \frac{24}{30} = \frac{1}{30} \]

- Difference between \( p_2 \) and \( p_m \):
\[ p_2 - p_m = \frac{7}{9} - \frac{4}{5} \]
Again, find a common denominator (45):
\[ = \frac{35}{45} - \frac{36}{45} = -\frac{1}{45} \]

5. **Set up the ratio of the two containers:**
- The ratio of the mixtures from the two containers is given by the inverse of the differences calculated:
\[ \text{Ratio} = \frac{1/30}{1/45} = \frac{45}{30} = \frac{3}{2} \]

6. **Final Ratio:**
- Therefore, the mixtures from the two containers should be mixed in the ratio of 2:3.

### Conclusion:
The required ratio of the mixtures from the two containers is **2:3**. The correct option is (b) 2:3.

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