Partnership — SSC CGL Quantitative Aptitude

Partnership is one rule wearing different costumes: profit is divided in the ratio of each partner's capital × time. Compute everyone's "capital-months", reduce to a ratio, and share the profit. The only complications are a working partner (who takes a salary/commission off the top first) and mid-year changes (which you split into time segments). Nothing else.

1. What SSC actually asks

Tier 1: ~1 Q · Tier 2: 1 Q. Types: equal-time (share by capital), different-time (share by capital × time), a partner joining or withdrawing mid-year, a working partner taking a commission, and reverse questions (find a capital or a duration from the profit split).

2. The one principle: capital × time

$\text{Profit share}\propto \text{Capital}\times \text{Time}$

• Same time for all partners → profit ratio = capital ratio. Invest ₹5000 and ₹7000 for a year → shares in $5:7$.
• Different times → multiply each capital by its months. A's ₹8000 for 6 months vs B's ₹6000 for 8 months: $48000:48000=1:1$ despite unequal capital.

Reduce the capital-months to lowest terms; that ratio divides every rupee of profit.

3. Working partner and charity

Some profit is removed before the capital split:

• Working (active) partner takes a salary or a percentage commission for managing; the remainder is divided by capital-months.
• Charity / reserve percentages come off the top too.

Example: profit ₹19,000, A (₹20,000) is working partner taking 10%, B has ₹30,000. A's commission $=1900$; remaining $17,100$ splits $2:3$ → A gets $6840$. A's total $=1900+6840=8740$.

4. Mid-year changes — split into segments

If a partner adds or withdraws money partway, treat each stretch as its own capital-months and add them:

• A invests ₹3000 for 4 months, then ₹4000 for the next 8 months → $3000\times 4+4000\times 8=44000$ capital-months.
• A partner joining late simply invests for fewer months: ₹60,000 from month 4 counts as $60000\times 9$.

5. Reverse questions

Given the profit split, run the rule backwards:

• Find a duration: if A (₹3000, 8 months) and B (₹4000) share profit $3:2$, then $24000:4000t=3:2\Rightarrow t=4$ months.
• Find total profit: if A's capital fraction is $\frac{3}{5}$ after 5% charity and A gets ₹855, then $0.95P\times \frac{3}{5}=855\Rightarrow P=1500$.

6. Solved PYQ-style examples

Q1. A and B invest ₹5000 and ₹7000 for the same period. If the total profit is ₹2400, B's share is… Solution. Ratio $5:7$ → B $=\frac{7}{12}\times 2400=$ ₹1400.

Q2. A puts in ₹8000 for 6 months and B ₹6000 for 8 months. A profit of ₹5000 is divided as… Solution. $8000\times 6:6000\times 8=48000:48000=1:1$ → ₹2500 each.

Q3. A invests ₹3000 for 4 months and then ₹4000 for the next 8 months; B invests ₹4000 for the whole year. The profit ratio A : B is… Solution. A $=12000+32000=44000$; B $=48000$ → $44000:48000=$ 11 : 12.

Q4. A (₹45,000) starts a business; B joins after 3 months with ₹60,000. At year end the profit is ₹14,000. B's share is… Solution. A $=45000\times 12=540000$; B $=60000\times 9=540000$ → $1:1$ → ₹7000.

Q5. A, B, C invest in the ratio 2 : 3 : 5 for equal time. Of a ₹20,000 profit, C receives… Solution. $\frac{5}{10}\times 20000=$ ₹10,000.

7. Exam protocol

1. Write each partner's capital × months; reduce to a ratio before touching the profit.
2. Equal time → capital ratio directly; don't multiply by months needlessly.
3. Working partner's commission/salary and any charity come off the top first.
4. Mid-year change → split into segments and sum the capital-months.
5. Reverse questions: set the capital-months ratio equal to the profit ratio and solve for the unknown.