Two vessels contain mixtures of milk and water in the ratio of 8 : 1 and 1 : 5…
2025
Two vessels contain mixtures of milk and water in the ratio of 8 : 1 and 1 : 5 respectively. The contents of both of these are mixed in a specific ratio into a third vessel. How much mixture must be drawn from the second vessel to fill the third vessel (capacity 26 gallons) completely in order that the resulting mixture may be half milk and half water?
- A.
10 gallons
- B.
12 gallons
- C.
14 gallons
- D.
13 gallons
Show answer & explanation
Correct answer: C
Concept: The alligation rule -- when two mixtures with different concentrations of an ingredient are combined to produce a target (mean) concentration, the ratio in which they must be mixed equals the ratio of the two concentration gaps from the target: quantity(A) : quantity(B) = (target - concentration B) : (concentration A - target).
Application:
Vessel 1 has milk : water = 8 : 1, so its milk fraction of the total volume is 8/9.
Vessel 2 has milk : water = 1 : 5, so its milk fraction of the total volume is 1/6.
The target mixture must be half milk and half water, i.e. a milk fraction of 1/2.
Converting all three fractions to a common denominator of 18: 8/9 = 16/18, 1/6 = 3/18, and 1/2 = 9/18.
By alligation, quantity(vessel 1) : quantity(vessel 2) = (9/18 - 3/18) : (16/18 - 9/18) = 6/18 : 7/18 = 6 : 7.
The third vessel's 26-gallon capacity is split in the ratio 6 : 7 (13 total parts), so the quantity drawn from the second vessel is (7/13) x 26 = 14 gallons.
Cross-check: The quantity from vessel 1 is then (6/13) x 26 = 12 gallons. Milk from vessel 1 = 12 x 8/9 = 32/3 gallons; milk from vessel 2 = 14 x 1/6 = 7/3 gallons. Total milk = 32/3 + 7/3 = 13 gallons out of 26, which is exactly half -- confirming the mixture is half milk and half water.