A vessel contains 400 litres of a solution in which 12% is acid. Water is to be added to it so that the acid content of the resulting mixture is more than 5% but less than 8%. The quantity of water to be added must be
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → more than 200 litres but less than 560 litres
Adding water does not change the amount of acid:
\(12\% \text{ of } 400 = 48 \text{ litres}\)
Let \(x\) litres of water be added. The mixture is then \(400 + x\) litres, and the condition is a double inequality:
\(\displaystyle 5 \lt \frac{48}{400 + x} \times 100 \lt 8\)
\(\displaystyle \frac{4800}{8} \lt 400 + x \lt \frac{4800}{5}\)
\(600 \lt 400 + x \lt 960\)
\(200 \lt x \lt 560\)
So more than 200 litres but less than 560 litres of water must be added.
Why the other options are incorrect
Option (1) more than 600 litres but less than 960 litres — Incorrect. These are the limits for the whole mixture \(400 + x\), not for the water \(x\).
Option (3) more than 240 litres but less than 600 litres — Incorrect. With 580 litres of water the acid is \(\displaystyle\frac{48}{980} \times 100\), which is less than 5%.
Option (4) more than 160 litres but less than 400 litres — Incorrect. With 180 litres of water the acid is \(\displaystyle\frac{48}{580} \times 100\), which is more than 8%.