A barrel contains a mixture of alcohol and water in the ratio 7:4 respectively. If 12 litres of water are added to it, the ratio of alcohol to water becomes 7:6. Find the initial total quantity of the mixture in the barrel.

Correct — D. Adding water leaves the alcohol untouched, and its term is 7 in both ratios, so one part is the same number of litres throughout.

Before: alcohol 7k, water 4k
After 12 L of water: water 6k
Subtract: 6k − 4k = 2k = 12, so k = 6

Initial total = 7k + 4k = 11k
Multiply: 11 × 6 = 66 litres → option (d)

Why the others are wrong

• (a)
  33 litres — 33 litres means k = 3: alcohol 21 L, water 12 L. Adding 12 L of water gives 21 : 24 = 7 : 8, not 7 : 6. The 2-part rise would be only 6 L.
• (b)
  44 litres — 44 litres means k = 4: alcohol 28 L, water 16 L. Adding 12 L makes it 28 : 28 = 1 : 1, not 7 : 6. Here the 2-part rise is 8 L, not 12.
• (c)
  55 litres — 55 litres means k = 5: alcohol 35 L, water 20 L. Adding 12 L gives 35 : 32, while 7 : 6 would need water of 30 L. The 2-part rise is 10 L.

Adding one liquid leaves the other unchanged. When water is poured in, the alcohol is the same before and after, so anchor the working to it.

Here the alcohol term is 7 in both ratios, so the parts are directly comparable. The water's 4 parts become 6 parts, and those 2 extra parts are exactly the 12 litres added.

Key facts

• Adding water changes the water quantity and leaves the alcohol quantity unchanged.
• Here 1 part = 6 litres: 42 L of alcohol and 24 L of water at the start.
• The water added equals the rise in the water's parts: 6 − 4 = 2 parts = 12 L.

Common traps

• Dividing 12 L by the new water's 6 parts (1 part = 2 L) instead of by the 2-part increase.
• Stopping at the alcohol (42 L) or at the final total (78 L) when the question asks for the initial total.

No directly related past PYQ was found.