Bottle 1 contains a mixture of milk and water in 7 : 2 ratio and Bottle 2 contains a mixture of milk and water in 9 : 4 ratio. In what ratio of volumes should the liquids in Bottle 1 and Bottle 2 be combined to obtain a mixture of milk and water in 3 : 1 ratio?
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Use alligation method for mixture problems: $\frac{\text{Volume 1}}{\text{Volume 2}} = \frac{\text{Difference of higher and desired}}{\text{Difference of desired and lower}}$.
Step 1: Understanding the Concept:
This is a mixture and alligation problem. We compare the concentration of one component (e.g., milk) in two different sources to reach a target concentration. Step 2: Key Formula or Approach:
Ratio \( \frac{V_1}{V_2} = \frac{C_2 - C_m}{C_m - C_1} \).
Concentration of milk in B1 = \( 7/9 \).
Concentration of milk in B2 = \( 9/13 \).
Target concentration in mixture = \( 3/4 \). Step 3: Detailed Explanation:
Perform alligation on milk concentrations:
1. Difference between B2 and Mean: \( |9/13 - 3/4| = |36/52 - 39/52| = 3/52 \).
2. Difference between B1 and Mean: \( |7/9 - 3/4| = |28/36 - 27/36| = 1/36 \).
3. Ratio = \( \frac{3/52}{1/36} = \frac{3}{52} \times 36 \).
4. Simplify: \( \frac{3 \times 9}{13} = 27/13 \). Step 4: Final Answer:
The ratio of volumes is 27 : 13.