Partnership shares — SSC CGL Quant Notes

Concepts and rules

In a partnership each partner's share of profit is proportional to capital × time. The product (capital × number of months) is called capital-months, or the partner's effective investment.

Simple partnership: all capitals stay for the same period, so profit ratio = capital ratio.

Compound partnership: capitals stay for different periods, so profit ratio = C1T1 : C2T2 : C3T3.

If a partner changes the capital during the year, add the capital-months of every stretch: for example 50,000 for 4 months then 40,000 for 8 months gives 50,000 × 4 + 40,000 × 8.

A partner who joins after k months of a 12-month year stays for 12 − k months. A partner who leaves after k months stays for k months.

Reverse relations: capital ratio = (profit ratio) ÷ (time ratio) term by term, and time ratio = (profit ratio) ÷ (capital ratio) term by term.

Working partner: a salary or a percentage of profit paid for managing the business comes out first; only the remainder is divided in the capital-months ratio. A loss is shared in the same ratio as profit.

Method and shortcuts

Step 1: for each partner write capital × months for every stretch and add. Step 2: cancel common factors (drop thousands, divide by the HCF) to get a small ratio. Step 3: share = (own parts ÷ total parts) × distributable profit.

When ratios of capitals are given as relations (A is 3 times B, B is 2/3 of C), give the common partner a value that clears the fractions, such as C = 3.

If one share is known in rupees, find the value of 1 part from it first; then every other share and the total follow at once.

For an unknown joining time, equate the capital-months ratio with the profit ratio and solve for months; then joining time = 12 − months.

When it fails: the capital-months rule assumes no salary, commission or interest on capital. If any of these are mentioned, pay them first, then use the ratio on the rest.

Example 1: Simple partnership

Question: Anil and Bimal invest ₹40,000 and ₹60,000 for a year. The profit is ₹25,000. Find each share.

Solution: Ratio = 40,000 : 60,000 = 2 : 3, so 5 parts = 25,000 and 1 part = 5,000. Anil gets ₹10,000 and Bimal gets ₹15,000.

Answer: ₹10,000 and ₹15,000

Example 2: A partner joins later

Question: Asha starts a business with ₹30,000. After 4 months Bina joins with ₹45,000. The profit at the end of the year is ₹18,000. Find Bina's share.

Solution: Asha: 30,000 × 12 = 3,60,000. Bina stays 12 − 4 = 8 months: 45,000 × 8 = 3,60,000. The ratio is 1 : 1, so Bina gets 18,000 ÷ 2 = ₹9,000.

Answer: ₹9,000

Example 3: Capital ratio from profit and time

Question: The profits of P and Q are in the ratio 5 : 3, and the periods of their investments are in the ratio 2 : 3. Find the ratio of their capitals.

Solution: Capital = profit ÷ time for each: 5/2 : 3/3 = 5/2 : 1 = 5 : 2. Check: 5 × 2 : 2 × 3 = 10 : 6 = 5 : 3.

Answer: 5 : 2

Example 4: Three partners, different periods

Question: A, B and C invest capitals in the ratio 3 : 4 : 5 for 6, 4 and 3 months respectively. Out of a profit of ₹9,800, what is B's share? (a) ₹3,600 (b) ₹3,000 (c) ₹3,200 (d) ₹3,400

Solution: Capital-months = 3 × 6 : 4 × 4 : 5 × 3 = 18 : 16 : 15, total 49 parts. B's share = 16/49 × 9,800 = 16 × 200 = ₹3,200.

Answer: (c) ₹3,200

Example 5: Capital changed during the year

Question: Ajay starts with ₹50,000. After 4 months he withdraws ₹10,000, and after 3 more months he adds ₹20,000. Vijay invests ₹60,000 for the whole year. The annual profit is ₹20,100. What is Ajay's share? (a) ₹9,300 (b) ₹10,800 (c) ₹9,600 (d) ₹9,000

Solution: Ajay has 50,000 for 4 months, 40,000 for 3 months and 60,000 for the remaining 12 − 7 = 5 months: 2,00,000 + 1,20,000 + 3,00,000 = 6,20,000.

Vijay: 60,000 × 12 = 7,20,000. Ratio = 620 : 720 = 31 : 36, total 67 parts. Ajay's share = 31/67 × 20,100 = 31 × 300 = ₹9,300.

Answer: (a) ₹9,300

Example 6: Working partner

Question: Ravi invests ₹2,00,000 for 12 months and Sohan invests ₹3,00,000 for 10 months. Ravi manages the business and first receives 10% of the profit; the rest is shared in the ratio of capital-months. If Ravi receives ₹33,000 in all, what is the total profit? (a) ₹55,000 (b) ₹66,000 (c) ₹72,000 (d) ₹60,000

Solution: Capital-months = 2,00,000 × 12 : 3,00,000 × 10 = 24 : 30 = 4 : 5. Let the profit be P. After the 10% salary, 90% of P is shared, and Ravi gets 4/9 of it, which is 40% of P.

Ravi's total = 10% + 40% = 50% of P = 33,000, so P = ₹66,000. Check: 6,600 + 4/9 × 59,400 = 6,600 + 26,400 = 33,000.

Answer: (b) ₹66,000

Common mistakes

Using the joining month as the time. A partner who joins after 4 months stays 8 months, not 4.

Sharing the whole profit by capital ratio when a working partner's salary is mentioned. Pay the salary first.

Taking only the final capital when money is added or withdrawn. Add capital-months stretch by stretch.

Multiplying instead of dividing in reverse questions. Capital ratio = profit ratio ÷ time ratio, term by term.

Mixing years and months for different partners. Convert all periods to the same unit before multiplying.

Practice questions

Q1. A and B start a business with ₹15,000 and ₹25,000. After 6 months A adds ₹10,000. Out of a yearly profit of ₹18,000, what is A's share? (a) ₹8,000 (b) ₹9,000 (c) ₹7,500 (d) ₹10,000

Q2. In a business A invests 3 times as much as B, and B invests 2/3 of what C invests, all for the same period. Out of a profit of ₹66,000, what is B's share? (a) ₹18,000 (b) ₹12,000 (c) ₹36,000 (d) ₹24,000

Q3. X starts a business with ₹84,000. Y joins some months later with ₹48,000. At the end of the year the profit is divided between X and Y in the ratio 3 : 1. After how many months did Y join? (a) 4 (b) 6 (c) 5 (d) 7

Answers

Q1. (a) A: 15,000 × 6 + 25,000 × 6 = 2,40,000; B: 25,000 × 12 = 3,00,000; ratio 4 : 5, so A = 4/9 × 18,000 = ₹8,000.

Q2. (b) Take C = 3, so B = 2 and A = 6; ratio 6 : 2 : 3 with 11 parts; B = 2/11 × 66,000 = ₹12,000.

Q3. (c) 84,000 × 12 : 48,000 × m = 3 : 1 gives 48,000 × m = 3,36,000, so m = 7 months; Y joined after 12 − 7 = 5 months.

Quick revision

Profit ratio = capital × time ratio (capital-months).

Joins after k months ⇒ stays 12 − k months.

Capital changes: add capital-months for each stretch.

Working partner's salary or commission comes out before sharing.

Capital ratio = profit ratio ÷ time ratio, term by term.