Partnership Problems: Types, Profit Sharing, and Solved Examples

Partnership problems in aptitude tests have one core rule: profit splits in the ratio of each partner’s capital-time product, not just their capital.

Get that distinction right and the calculation becomes mechanical. Miss it, and every mid-year entry or unequal-time problem goes wrong from the first step.

What a Partnership Problem Tests

A partnership begins when two or more people invest capital in a shared business and agree to divide profits and losses. Each partner’s contribution is defined by two quantities: the amount invested (capital) and the duration of that investment (time period). In a test question, you are asked to find the profit ratio or calculate one partner’s share from a given total profit.

Partners fall into two categories. Active or working partners participate in day-to-day management. Sleeping or silent partners only invest capital and take no part in operations. This distinction matters in one specific scenario: when a working partner draws a management fee or salary before the capital-based distribution begins. That mechanism is Case 3 below.

One common point of confusion: a partnership problem in aptitude is a calculation exercise about ratios and time, not a question about business law. Whether a partner is “working” or “sleeping” only matters when the problem explicitly says one partner receives a salary or management fee before profit is split.

The Three Profit-Sharing Cases

Every partnership aptitude question fits one of three cases.

Case 1: All partners invest for the same time period. Profit ratio equals the capital ratio directly. If A invests twice what B invests, A gets twice the profit. No time calculation needed.

Case 2: Partners invest for different time periods. Profit ratio equals the ratio of each partner’s capital multiplied by their time period. This is the capital-time product. A partner who invests a smaller amount but for a longer period can receive more profit than one who invested a larger amount for a shorter duration. Time and work problems use the same multiplicative logic: a worker active for longer contributes more total work, just as a partner invested for longer contributes more to the total capital deployed.

Case 3: One partner is a working partner with a fee. Deduct the management fee or salary from the total profit first. The remaining profit is then distributed among all partners in the ratio of their capital-time products. The working partner receives both the fee and their capital-based share.

These three cases cover the full range of partnership questions in campus placement aptitude tests.

Five Worked Examples

Each example below is derived from first principles. Values are chosen to produce round answers so you can verify in your head.

Example 1: Simple Capital Ratio (Case 1)

• Given: A invests ₹6,000; B invests ₹8,000; C invests ₹10,000. All invest for the full year. Total profit = ₹12,000.
• Step 1: Capital ratio = 6,000 : 8,000 : 10,000. Divide each by 2,000: ratio = 3 : 4 : 5. Total parts = 12.
• Step 2: A = (3/12) × 12,000 = ₹3,000. B = (4/12) × 12,000 = ₹4,000. C = (5/12) × 12,000 = ₹5,000.
• Verify: 3,000 + 4,000 + 5,000 = 12,000. Correct.

Example 2: Capital-Time Product (Case 2)

• Given: A invests ₹6,000 for 5 months; B invests ₹8,000 for 3 months; C invests ₹5,000 for 4 months. Total profit = ₹7,400.
• Step 1: Compute products: A = 6,000 × 5 = 30,000; B = 8,000 × 3 = 24,000; C = 5,000 × 4 = 20,000.
• Step 2: Ratio = 30,000 : 24,000 : 20,000. Divide each by 2,000: ratio = 15 : 12 : 10. Total parts = 37.
• Step 3: A = (15/37) × 7,400 = ₹3,000. B = (12/37) × 7,400 = ₹2,400. C = (10/37) × 7,400 = ₹2,000.
• Verify: 3,000 + 2,400 + 2,000 = 7,400. Correct.
• Speed note: Notice that A invested the most capital but B invested the second-most capital for the shortest time, so B ends up receiving less than A despite a higher capital amount. This is the trap the capital-time product prevents.

Example 3: Partner Joins Mid-Year (Case 2)

• Given: A invests ₹20,000 at the start of the year. B joins after 4 months with ₹25,000. Total profit at year end = ₹33,000.
• Step 1: A was invested for all 12 months. B was invested for 12 minus 4 = 8 months.
• Step 2: Capital-time products: A = 20,000 × 12 = 2,40,000; B = 25,000 × 8 = 2,00,000.
• Step 3: Ratio = 2,40,000 : 2,00,000. Divide each by 40,000: ratio = 6 : 5. Total parts = 11.
• Step 4: A = (6/11) × 33,000 = ₹18,000. B = (5/11) × 33,000 = ₹15,000.
• Verify: 18,000 + 15,000 = 33,000. Correct.
• Common trap: Students who skip Step 1 and assign B a 12-month period get a ratio of 20,000 : 25,000 = 4 : 5, which produces A = ₹14,667 and B = ₹18,333. That is wrong because B was not invested for the full year.

Example 4: Working Partner Takes a Management Fee (Case 3)

• Given: A invests ₹30,000; B invests ₹20,000. A manages the business and receives 20% of total profit as a management fee. Total profit = ₹25,000.
• Step 1: A’s management fee = 20% × 25,000 = ₹5,000. Remaining profit = 25,000 - 5,000 = ₹20,000.
• Step 2: Capital ratio = 30,000 : 20,000 = 3 : 2. Total parts = 5.
• Step 3: A’s capital share = (3/5) × 20,000 = ₹12,000. B’s capital share = (2/5) × 20,000 = ₹8,000.
• Step 4: A’s total = 5,000 + 12,000 = ₹17,000. B’s total = ₹8,000.
• Verify: 17,000 + 8,000 = 25,000. Correct.

Example 5: Partner Increases Investment Mid-Year (Case 2)

• Given: X, Y, Z invest in ratio 3 : 2 : 1. After 6 months, X doubles his investment. Total period = 12 months. Total profit = ₹42,000.
• Step 1: Use unit values: X = 3 units, Y = 2 units, Z = 1 unit initially.
• Step 2: X’s capital-time product: (3 × 6) + (6 × 6) = 18 + 36 = 54 units.
• Step 3: Y’s product: 2 × 12 = 24 units. Z’s product: 1 × 12 = 12 units.
• Step 4: Ratio = 54 : 24 : 12. Divide by 6: ratio = 9 : 4 : 2. Total parts = 15.
• Step 5: X = (9/15) × 42,000 = ₹25,200. Y = (4/15) × 42,000 = ₹11,200. Z = (2/15) × 42,000 = ₹5,600.
• Verify: 25,200 + 11,200 + 5,600 = 42,000. Correct.
• Key step: X’s investment for the second half of the year is 6 units (double the original 3), not 3 plus some fraction. Doubling means the new amount is twice the original.

Four Errors That Cost Marks

Campus tests use these traps deliberately. Identifying them before the test removes the pressure during it.

• Equal capital does not equal equal profit. If two partners invest the same amount for different durations, they do not split profit equally. Always check whether time periods match before using a simple capital ratio.
• Multiply capital by time, do not add them. The capital-time product is capital × time. A common error is to compute (capital + time) or to use just the capital and ignore the time difference entirely.
• Count only the months each partner was actually invested. In Example 3, B joined after 4 months and was therefore invested for 8 months, not 12. An off-by-one error here shifts the entire ratio and every individual share.
• Management fee comes off the top before the ratio is applied. In Example 4, both A and B receive their capital shares from the remaining ₹20,000, not from the full ₹25,000. Applying the capital ratio to the full profit before deducting the fee overpays both partners and makes the numbers fail the total check.

Partnership Questions in Campus Tests

Partnership is a standard topic in the quantitative aptitude sections of major placement tests.

• TCS NQT Numerical Ability: straightforward two- or three-partner problems, usually Case 1 or Case 2. These are high-confidence questions when you apply the capital-time method correctly; budget 60 to 90 seconds each.
• AMCAT Quantitative: similar difficulty and format. Expect one or two partnership questions per test, often alongside ratio-and-proportion and percentage questions in the same section.
• Mu Sigma MuApt: speed is the primary constraint. Recognising the case type on the first read saves 20 to 30 seconds per question. Most Mu Sigma partnership problems fit Case 1 or Case 2.
• D.E. Shaw preliminary quant round: harder format. Expect multi-step problems that combine a mid-year entry with a management fee, or a partnership with a percentage change in capital. These use Case 2 and Case 3 together, which is the reason Example 5 pairs with Example 4 in preparation.

For most tests, two to four partnership problems appear per test section. If a problem requires more than three steps and you have not identified the case type in the first read, mark it and return on the second pass.

The capital-time product step in Example 3 (computing A’s and B’s products independently before forming the ratio) is the same verification habit that prevents errors in AI pipelines: isolate each component, check it, then combine. TinkerLLM, at ₹299 on tinkerllm.com, applies that same step-by-step checking to live model calls.