Partnership
Partnership is one of the most reliable scoring chapters in CAT Arithmetic because it rests on a single clean idea: when people pool money into a joint business, the profit (or loss) is shared in the ratio of each partner’s contribution to the business. The size of that contribution is not just the rupees invested — it is rupees multiplied by the number of months they stay invested. This capital×time product, often called the "capital-month" or "money-month", is the heartbeat of every partnership question. Master it and you can crack a problem in two lines; miss it and you will wrongly split profit by capital alone. CAT and the other management exams (XAT, SNAP, NMAT, IIFT) rarely ask a bare three-line partnership sum; they wrap it inside a ratio, a profit-percentage, an average, or a "who joined when" twist where a working partner first draws a salary or commission off the top. This chapter builds the chain from the ground up: forming the investment ratio, sharing profit cleanly across unequal time periods, and handling the working-versus-sleeping-partner case where a fixed amount is removed before the rest splits by capital-months. Each topic carries worked CAT-style examples, the fastest ratio method, and the traps that quietly cost marks.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Always convert to capital-months (rupees × months) before forming any ratio — split by raw cash only if all times are equal.
- Find the "value of one part" first: divide any known share by its ratio units, then scale to get totals or other shares.
- Equal profit ⇒ capitals are in the INVERSE ratio of time. Equal time ⇒ ratio is simply the capitals.
- Working partner: pull salary/commission OFF THE TOP first, then divide the remainder by the capital ratio.
- Cancel the GCD of all capital-months immediately so the final division is on small integers.
- For a mid-year change, treat each phase separately and add the capital-months: C₁·m₁ + C₂·m₂.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Splitting profit by capital alone when partners invested for different time periods (must use capital-months).
- Dividing the FULL profit by the capital ratio before removing the working partner’s salary/commission.
- Forgetting that the working partner still shares the remainder by capital — their total is salary + capital share.
- Reading a "₹X per month" salary as a one-time amount instead of multiplying by the months worked.
- Confusing equal-profit (capitals inverse to time) with equal-capital (profit inverse to time) phrasing.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Partnership when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Core sharing rules
| Simple partnership (equal time) | Profit ratio = C₁ : C₂ : C₃ (capitals) |
|---|---|
| Compound partnership (unequal time) | Profit ratio = C₁t₁ : C₂t₂ : C₃t₃ |
| One partner’s share | Share = (Your C·t / Total C·t) × Total profit |
| Equal profit ⇒ capitals are | C₁ : C₂ = t₂ : t₁ (inverse of time) |
| Capital from profit share | C₁/C₂ = (P₁/t₁) ÷ (P₂/t₂) |
Working partner & money-month
| Working partner’s pay | Salary/commission taken off the top first |
|---|---|
| Remainder to split | Total profit − salary − commission |
| Commission on profit | Commission = r% × Total profit |
| Capital-month (money-month) | Rupees × Months invested |
| Mid-year change | Sum each phase: C₁·m₁ + C₂·m₂ + … |