Partnership, in the context of SSC MTS Quant, is essentially a problem of fair division. Two or more people pool money into a business, run it for some period, and then split the profit in proportion to their contribution — not equally, but fairly.
Here's the core idea: if you put in more money, or if your money stayed in the business longer, you deserve a bigger slice of the profit. That's it. The entire chapter is built on this one principle.
Think of it like renting a shared flat. If Rohit pays ₹6,000 rent and Amit pays ₹4,000, and they get back a ₹5,000 security deposit at the end, it would be unfair to split the refund equally. Rohit should get ₹3,000 (3/5 of ₹5,000) and Amit gets ₹2,000 (2/5 of ₹5,000). Partnership works the same way.
The key variable that changes things is time. If everyone invests for the same duration, you only compare the capital amounts. If people invest for different durations, you multiply capital by time to get the effective investment (also called weighted capital), and then compare those products.
A sleeping partner (or silent partner) is someone who only contributes capital but does not actively run the business. SSC MTS questions sometimes distinguish between an active partner, who may receive a fixed salary or commission out of total profit before the remainder is split by ratio, and a sleeping partner who only gets the ratio-based share. When active partner commission is mentioned, deduct it first, then split the remainder.
The ratio of profit shares equals the ratio of effective investments. That one sentence is the entire chapter.
When all partners invest for the same duration (usually the full year), time cancels out and you only compare capital amounts.
Profit ratio = Capital₁ : Capital₂ : Capital₃ ...
Working process:
Partner A's share = (A's ratio part / Total ratio parts) × Total Profit
Look — simplifying the ratio before multiplying saves you from large-number arithmetic. Always reduce first.
This is the main variation tested. You cannot just compare capitals.
Effective Investment = Capital × Time (in same units)
Profit ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃)
Steps:
Important: Keep time in the same unit throughout. If one person invests for 3 months and another for 1.5 years, convert both to months (3 months and 18 months) before multiplying.
Sometimes the question gives you the profit ratio and asks you to find one of the missing values (a capital amount or a time period).
The logic simply runs backwards:
Profit ratio = Capital × Time
So if you know the profit ratio and one partner's capital and time, you can set up an equation for the unknown.
For two partners P and Q:
(P's capital × P's time) / (Q's capital × Q's time) = P's profit share / Q's profit share
Cross-multiply and solve for the unknown. These are the "harder" looking questions on SSC MTS, but they are straightforward algebra once you write out this ratio equation.
If an active partner earns a fixed commission (say 10% of total profit) for managing the business:
SSC MTS tends to keep this simple — the commission is usually stated as a percentage of total profit or a fixed rupee amount.
When capitals are messy numbers like ₹63,000 and ₹42,000, divide both by their GCD immediately:
GCD(63000, 42000) = 21000
63000 / 21000 = 3 42000 / 21000 = 2
Ratio = 3:2. Now you're working with small numbers for the rest of the problem. This single habit saves 20-30 seconds per question.
For three-partner effective investments like 27300 : 13650 : 40950, divide by 1365 (check: 27300/1365 = 20, 13650/1365 = 10, 40950/1365 = 30). Ratio = 20:10:30 = 2:1:3. Much easier to work with.
Before doing any profit-share calculation, reduce the capital ratio to its lowest terms. For ₹63,000 : ₹42,000, mentally strike common zeros: 63 : 42 = 9 : 6 = 3 : 2. Now the share calculation is (3/5) × 9000 = 5400 — two steps. Standard approach of computing 63000/(63000+42000) × 9000 takes four steps and risks arithmetic error. Step count: 4 steps reduced to 2 steps.
For time-weighted problems, write a 2-column grid: Partner | C × T. Fill it in, then simplify the column of products. Example — P: 9100×3=27300, Q: 6825×2=13650, R: 8190×5=40950. Now find GCD of all three (try dividing by 1350 or 13650): 27300/13650=2, 13650/13650=1, 40950/13650=3. Ratio = 2:1:3. Total parts = 6. Q's share = (1/6) × 4158 = ₹693. The grid stops you from forgetting to multiply time, which is the #1 mistake in this chapter. Standard method (no grid): ~60s. Grid method: ~35s.
When a question gives profit ratio and asks for time, set up the equation: (C₁ × T₁)/(C₂ × T₂) = Profit₁/Profit₂, then isolate the unknown. For the problem where P invests ₹4,000 for (12−m) months and Q invests ₹5,600 for 12 months, and profit ratio is 1:3: write 4000(12−m)/(5600×12) = 1/3. Cross-multiply: 12000(12−m) = 5600×12 = 67200. So 12−m = 5.6, m = 6.4. Writing the ratio equation first prevents confusion about which number goes where. Saves 1-2 minutes of trial-and-error approach.
If the profit ratio matches the capital ratio exactly, the time ratio is 1:1 (all partners invested for the same effective period). In the question with ₹18,000 : ₹24,000 = 3:4 capital ratio and 3:4 profit ratio, you can immediately conclude tA:tB = 1:1 without solving any equation. This eliminates the temptation to guess a non-trivial answer. Recognition time: under 5 seconds versus 45 seconds setting up full equations.
In a two-partner problem, once you find one partner's share, subtract from total profit to get the other's. If the question gives four options, check whether your computed value appears. If it doesn't appear but total_profit minus your_answer does appear, you likely picked the wrong partner's share — swap it. This catches the "right calculation, wrong partner" mistake in under 10 seconds without recomputing.
When you see a partnership question in the exam hall, run through this decision tree:
Step 1 — Identify the type. Are all partners investing for the same time, or different durations? If same time: skip to Step 3. If different durations: go to Step 2.
Step 2 — Compute effective investments. Multiply each capital by its time. Write them in a small grid. Then proceed.
Step 3 — Simplify the ratio. Find GCD of all capital (or effective investment) values and divide. Never skip this — it prevents large-number errors later.
Step 4 — Identify what is asked. Is the question asking for (a) a profit share, (b) a capital amount, or (c) a time period? For (a): direct fraction. For (b) or (c): set up the ratio equation and solve.
Step 5 — Active partner? If mentioned, deduct commission first, then split.
Step 6 — Sanity check. All shares must sum to total profit. If they don't, you've made an arithmetic error somewhere. Re-verify Step 3.
Total target time per partnership question: 60-90 seconds.
Why this question: The simplest possible partnership — same duration, simplify capital ratio, done. This is the template all other questions build on.
Solving path: Capitals: ₹63,000 and ₹42,000. Divide both by 21,000 (their GCD). Ratio = 3:2. Total parts = 5. Raman's share = (3/5) × 9,000 = ₹5,400. Cross-check: Sanjay gets (2/5) × 9,000 = ₹3,600. Sum = ₹9,000. Correct.
Why this question: Introduces the time-weighted (effective investment) model with three partners — the most common harder variant.
Solving path: Effective investments — P: 9100 × 3 = 27,300. Q: 6825 × 2 = 13,650. R: 8190 × 5 = 40,950. Divide all by 13,650: P = 2, Q = 1, R = 3. Total parts = 6. Q's share = (1/6) × 4,158 = ₹693. Note: dividing 4,158 by 6 gives 693 exactly. If you left the ratio as 27,300:13,650:40,950, dividing 4,158 proportionally is much harder — this is exactly why simplifying the ratio first is non-negotiable.
Why this question: A reverse problem — you must find a time period, not a profit share. Tests whether you can run the ratio equation backwards.
Solving path: Two equations: P + Q = 9,600 and Q − P = 1,600. Adding: 2Q = 11,200, so Q = 5,600 and P = 4,000. P's profit share = ₹2,400, so Q's share = 9,600 − 2,400 = ₹7,200. Profit ratio = 2,400 : 7,200 = 1:3. Now write the ratio equation: (4,000 × (12 − m)) / (5,600 × 12) = 1/3. Cross-multiply: 3 × 4,000 × (12 − m) = 5,600 × 12. So 12,000(12 − m) = 67,200. Therefore 12 − m = 5.6, giving m = 6.4 months.
Why this question: A reverse problem asking for time ratio when both capital and profit ratios are given — tests whether you recognise the 1:1 shortcut.
Solving path: Capital ratio: 18,000 : 24,000 = 3:4. Profit ratio given: 3:4. Since profit ratio = capital ratio, the time ratio must be 1:1. Formal verification: (18,000 × tA)/(24,000 × tB) = 3/4 → tA/tB = (3 × 24,000)/(4 × 18,000) = 72,000/72,000 = 1. So tA:tB = 1:1. On the actual exam, recognise this pattern in under 10 seconds and move on.
Forgetting to multiply by time. In time-weighted problems, students compute the capital ratio (e.g., 9100:6825:8190) and divide profit by that. Wrong. You must compute C × T for each partner before forming the ratio. If the durations differ even by one month, capital ratio alone is incorrect.
Using different time units for different partners. If one partner invests for 6 months and another for 1 year, both must be in the same unit — either both in months (6 and 12) or both in years (0.5 and 1). Mixing units produces a wrong ratio.
Adding commission to profit before splitting. For active-partner problems, the commission comes out of total profit first, then the remainder is split. Students sometimes split the full profit by ratio and then also add the commission on top, effectively double-counting the active partner's commission.
Answering for the wrong partner. Questions occasionally ask for Q's share after making P's share very easy to calculate. You find P's share correctly but write it as the answer to "what is Q's share?" Always re-read the question after computing.
Not simplifying the ratio. Skipping ratio simplification is not just slower — it leads to arithmetic errors when dividing larger numbers. Fractions like 27,300/81,900 are harder to simplify mentally than 2/6. Always reduce before computing shares.
Treating sleeping and active partners identically when commission is given. If the question explicitly states an active partner receives a percentage of profit as salary, that must be deducted first. Applying the capital ratio to the full profit ignores this and gives the wrong answer for both partners.