M and N begin a business venture. M invests ₹89,000 more than N for a period…

Concept — how a partnership profit is divided: a partner’s claim on the profit is proportional to the weighted capital, that is, the amount invested multiplied by the number of months it stays in the business. If two partners’ weighted capitals are in the ratio a : b, then a total profit P is divided as P × a/(a+b) and P × b/(a+b).

Concept — recovering the two shares from a total and a gap: when the total profit P and the difference d between the two shares are known, the larger share is (P + d)/2 and the smaller share is (P − d)/2. This turns a statement about shares back into the weighted-capital ratio.

Application: let N’s investment be ₹y, so M’s investment is ₹(y + 89000). M’s money works for 7 months and N’s for 5 months.

1. Split the profit using the total and the gap: M’s share = (3175 + 635)/2 = ₹1905 and N’s share = (3175 − 635)/2 = ₹1270.

2. The weighted capitals are therefore in the same ratio as the shares: 1905 : 1270 = 3 : 2, after dividing both by 635.

3. Write the weighted capitals from the investments: 7(y + 89000) for M and 5y for N, so 7(y + 89000) : 5y = 3 : 2.

4. Cross-multiply: 2 × 7(y + 89000) = 3 × 5y, which gives 14y + 1246000 = 15y.

5. Hence y = ₹1246000 is N’s investment, and M’s investment is y + 89000 = ₹1335000.

Cross-check: with M contributing ₹1335000 for 7 months and N ₹1246000 for 5 months, the weighted capitals are 9345000 and 6230000, whose ratio is exactly 3 : 2. The profit ₹3175 then splits as 3175 × 3/5 = ₹1905 and 3175 × 2/5 = ₹1270, a gap of ₹635, exactly as the question states.

Note: the equation 14y + 1246000 = 15y has a single solution, so M’s investment is pinned to ₹1335000; amounts even a few hundred rupees away from it no longer reproduce a share gap of exactly ₹635.