Multiple choice

Ayush has a cask that contains a mixture of two acids X and Y in the ratio 7 : 5. When 9 gallons of the mixture are drawn off by Ayush and the cask is filled with acid Y, the proportion becomes X : Y : : 7 : 9. How many gallons does the cask hold initially?

  1. 32 gallons

  2. 26 gallons

  3. 30 gallons

  4. 36 gallons

  5. 40 gallons
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let initial amount be x. Acid X = 7x/12, Acid Y = 5x/12. After drawing 9 gallons, X removed = 7/12 * 9 = 5.25. Remaining X = 7x/12 - 5.25. Remaining Y = 5x/12 - 3.75. After adding 9 gallons of Y, new Y = 5x/12 - 3.75 + 9 = 5x/12 + 5.25. Ratio (7x/12 - 5.25) / (5x/12 + 5.25) = 7/9. Solving for x: 9(7x/12 - 5.25) = 7(5x/12 + 5.25) => 63x/12 - 47.25 = 35x/12 + 36.75 => 28x/12 = 84 => x = 84 * 12 / 28 = 36.

AI explanation

When 9 gallons of the mixture are drawn off, the ratio of acid X to acid Y remains 7 to 5, meaning 7/12 of the drawn amount is X. This removes 5.25 gallons of X, but since the final ratio of X to Y is 7 to 9, the final amount of X must still be 7/16 of the total volume. Assuming the initial volume is V, the equation becomes (7/12)V minus 5.25 equals (7/16)V, which simplifies to (7/48)V equals 5.25. Solving for V gives a total initial capacity of 36 gallons.

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