In a partnership business, B’s capital was \(\frac{1}{2}\) of A’s capital. After 4 months, A withdrew \(\frac{1}{4}\) of his capital, and after 2 more months, B withdrew \(\frac{1}{2}\) of his capital. Find the ratio A : B of their profits at the end of the year.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 20 : 9
Profit is shared in the ratio of capital × months, adding the parts of the year with different capitals.
Let A’s capital be 4 units; then B’s is 2 units
A: 4 units for 4 months, then 3 units for 8 months: \(16+24=40\)
B: 2 units for 6 months, then 1 unit for 6 months: \(12+6=18\)
A : B \(=40:18=20:9\)
The ratio of their profits is 20 : 9.
Why the other options are incorrect
Option (1) 10 : 9 — Incorrect. This does not use that B’s capital is only half of A’s.
Option (3) 2 : 1 — Incorrect. This is the ratio of the starting capitals; the withdrawals change the capital × months.
Option (4) 3 : 1 — Incorrect. This takes A’s reduced capital for the whole year and B’s reduced capital for the whole year.