In bottles A and B, acid and water are in the ratios 7 : 5 and 1 : 7 respectively. The contents of A and B are mixed in the ratio 1 : 2 to get a solution in which acid and water are in the ratio a : b. Find the value of \(\frac{b-a}{a}\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\frac{8}{5}\)
The ratio totals are 12 and 8, so take volumes both divide into: 24 litres of A and 48 litres of B keep the ratio 1 : 2.
A, 24 litres: acid \(\displaystyle =\frac{7}{12}\times24=14\), water \(=10\)
B, 48 litres: acid \(\displaystyle =\frac{1}{8}\times48=6\), water \(=42\)
Acid \(=14+6=20\), water \(=10+42=52\), so \(a:b=20:52=5:13\)
\(\displaystyle \frac{b-a}{a}=\frac{13-5}{5}=\frac{8}{5}\)
The value is \(\frac{8}{5}\).
Why the other options are incorrect
Option (1) \(\frac{18}{5}\) — Incorrect. This is \(\frac{b+a}{a}\), the sum instead of the difference.
Option (2) \(\frac{5}{8}\) — Incorrect. This is \(\frac{a}{b-a}\), the answer turned upside down.
Option (4) \(\frac{8}{13}\) — Incorrect. This divides the difference by \(b\) instead of by \(a\).