Question

There are two vessels. In the first vessels, the ratio of milk to water is 1:2 and in the second vessel the milk and water are in the ratio 2:3. In what ratio the contents in two vessels must be mixed such that the resulting mixture will have milk and water in the ratio 5:8?

• A. 1:3
• B. 3:10
• C. 3:5
• D. 10:3
• E. Cannot be determined

Hint:

When using alligation with fractions, the final calculation can be simplified by multiplying the resulting ratio by the LCM of the denominators. In this case, the ratio was \(\frac{1}{65} : \frac{2}{39}\). The LCM of 65 and 39 is \(5 \times 13 \times 3 = 195\). Multiplying both sides by 195 gives: \((\frac{1}{65} \times 195) : (\frac{2}{39} \times 195) \rightarrow 3 : (2 \times 5) \rightarrow 3:10\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a mixture problem involving ratios. The most effective method is alligation, which is used to find the ratio in which two ingredients with different concentrations must be mixed to produce a mixture of a desired concentration. We will use the concentration (fraction) of milk.
Step 2: Detailed Explanation:
1. Find the fraction of milk in each vessel and in the final mixture.


Vessel 1: Milk:Water = 1:2. Total parts = 1+2=3. Fraction of milk = \(\frac{1}{3}\).
Vessel 2: Milk:Water = 2:3. Total parts = 2+3=5. Fraction of milk = \(\frac{2}{5}\).
Final Mixture: Milk:Water = 5:8. Total parts = 5+8=13. Fraction of milk = \(\frac{5}{13}\).
2. Apply the rule of alligation.
The ratio of the quantity of Vessel 1 to the quantity of Vessel 2 is given by: \[ \frac{\text{Quantity of Vessel 1}}{\text{Quantity of Vessel 2}} = \frac{(\text{Milk fraction in Vessel 2}) - (\text{Milk fraction in Mixture})}{(\text{Milk fraction in Mixture}) - (\text{Milk fraction in Vessel 1})} \] 3. Calculate the differences.


Numerator: \( \frac{2}{5} - \frac{5}{13} = \frac{2 \times 13 - 5 \times 5}{5 \times 13} = \frac{26 - 25}{65} = \frac{1}{65} \)
Denominator: \( \frac{5}{13} - \frac{1}{3} = \frac{5 \times 3 - 1 \times 13}{13 \times 3} = \frac{15 - 13}{39} = \frac{2}{39} \)
4. Find the ratio.
\[ \frac{\text{Quantity of Vessel 1}}{\text{Quantity of Vessel 2}} = \frac{1/65}{2/39} = \frac{1}{65} \times \frac{39}{2} = \frac{39}{130} \] Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 13: \[ \frac{39 \div 13}{130 \div 13} = \frac{3}{10} \] So, the required ratio is 3:10.
Step 3: Final Answer:
The contents of the two vessels must be mixed in the ratio 3:10. This corresponds to option (B).