Partnership

A partner's claim on the profit is not just how much money he put in, but how much money for how long. That product is called his capital-months.

profit ratio = (capital × months) for each partner

Two cases:

1. Same period for everyone — the months cancel out, so the profit is simply divided in the ratio of the capitals. (A partner who invests for the full year is a "simple" partner.)
2. Different periods — multiply each capital by its own number of months first. If A invests ₹9,000 for 8 months and B ₹6,000 for 10 months, the ratio is 72,000 : 60,000 = 6 : 5, not 9,000 : 6,000.

The commonest MPSC variant is "B joined after m months": A's money works 12 months, B's only 12 − m. Read the sentence carefully — "joined after 4 months" means B was in for 8 months, not 4.

The single most common mistake is dividing by the capitals alone when the periods differ. Guard against it by writing the months beside each capital before you form any ratio; if all the months are equal you can strike them out, and if they are not, you have already caught the trap.

To get the total profit from one partner's share, invert: total = share × (sum of all parts) ÷ (that partner's parts).

Profit ratio = capital × months · equal periods → ratio of capitals · joined after m months → works 12 − m months · total = share × total parts / own parts

A invests ₹9,000 for 8 months and B ₹6,000 for 10 months. If the profit is ₹4,400, find A's share. → Capital-months: A = 9000 × 8 = 72,000; B = 6000 × 10 = 60,000; ratio 72 : 60 = 6 : 5, i.e. 11 parts. A's share = 4400 × 6/11 = ₹2,400. Sanity check: B gets 4400 × 5/11 = ₹2,000, the two add back to ₹4,400, and 2400 : 2000 is indeed 6 : 5.