Two vessels A and B contain milk and water mixed in the ratio 8: 5 and 5:2 respectively. The ratio in which these two mixtures be mixed to get a new mixture containing `69(3)/(13)`% milk is:

Options

A. `2:7`
B. `3:5`
C. `5:2`
D. `5:7`

Answer

To solve the problem, we need to find the ratio in which two mixtures from vessels A and B can be combined to achieve a new mixture with a specific percentage of milk. Let's break this down step by step.

### Step 1: Understand the Ratios of Milk and Water in Each Vessel
- **Vessel A** contains milk and water in the ratio of 8:5.
- **Vessel B** contains milk and water in the ratio of 5:2.

### Step 2: Calculate the Percentage of Milk in Each Vessel
- For **Vessel A**:
- Total parts = 8 (milk) + 5 (water) = 13 parts
- Percentage of milk = (8/13) * 100 = 61.54%

- For **Vessel B**:
- Total parts = 5 (milk) + 2 (water) = 7 parts
- Percentage of milk = (5/7) * 100 = 71.43%

### Step 3: Convert the Desired Percentage of Milk
The desired percentage of milk in the new mixture is given as \( 69\frac{3}{13}\% \).
- Convert this to an improper fraction:
- \( 69\frac{3}{13} = \frac{69 \times 13 + 3}{13} = \frac{897 + 3}{13} = \frac{900}{13} \)
- Now, convert this to a fraction:
- \( \frac{900}{1300} = \frac{9}{13} \)

### Step 4: Set Up the Alligation
We will use the alligation method to find the ratio in which the two mixtures should be combined.

- Let \( P_A \) be the percentage of milk in Vessel A = \( \frac{8}{13} \)
- Let \( P_B \) be the percentage of milk in Vessel B = \( \frac{5}{7} \)
- Let \( P_M \) be the desired percentage of milk = \( \frac{9}{13} \)

### Step 5: Apply Alligation Formula
Using the alligation method:
1. Calculate the difference between \( P_A \) and \( P_M \):
- \( P_A - P_M = \frac{8}{13} - \frac{9}{13} = -\frac{1}{13} \) (we take the absolute value, so it becomes \( \frac{1}{13} \))

2. Calculate the difference between \( P_B \) and \( P_M \):
- \( P_B - P_M = \frac{5}{7} - \frac{9}{13} \)
- To calculate this, we need a common denominator. The LCM of 7 and 13 is 91.
- Convert \( P_B \) and \( P_M \):
- \( P_B = \frac{5}{7} = \frac{65}{91} \)
- \( P_M = \frac{9}{13} = \frac{63}{91} \)
- Now calculate:
- \( P_B - P_M = \frac{65}{91} - \frac{63}{91} = \frac{2}{91} \)

### Step 6: Find the Ratio
Now we can set up the ratio using the differences calculated:
- The ratio of the quantities of mixtures from vessels A and B is given by:
- Ratio = Difference from B : Difference from A
- Ratio = \( \frac{2}{91} : \frac{1}{13} \)

To simplify this ratio, we can multiply both sides by the LCM of the denominators (which is 91):
- \( 2 \times 13 : 1 \times 91 = 26 : 91 \)

### Final Step: Simplify the Ratio
The final ratio of the mixtures from vessels A and B is:
- **Ratio = 26 : 91**