Mathematics — Chapter 5

Partnership & Chain Rule — Capital × Time, and Direct vs Inverse Proportion

Why this chapter matters: Two of the most mechanical blocks in one-paper arithmetic live here, and both reward one memorised rule each. Partnership is a single rule — profit follows capital × time — and the chain rule is a single question — does more of this mean more, or less, of that? Our verified banks hold 124 topic-relevant items for this chapter: set 137’s 25 partnership questions and set 138’s 31 chain-rule questions, plus 68 tightly matched paper items. The sets are the anchor — they contain every shape the papers ask — and the papers mostly repeat the sets’ shapes: the “P and Q started business” profit-share question appears in three papers, the “10 men complete a job in 14 days” question in five papers, and set 138’s own flagstaff-shadow and spider-web questions reappear verbatim in FPSC papers. Learn the two rules cold and this chapter is free marks.

PART A — PARTNERSHIP: PROFIT FOLLOWS CAPITAL × TIME

1. The one rule

Profit shares are in the ratio of (capital × time). If two partners invest for the same time, their shares follow their capitals. If their capitals are equal, their shares follow their times. If both differ, multiply first, then take the ratio:

Share ratio = C₁ × T₁ : C₂ × T₂ : C₃ × T₃

“Capital-months” is the working unit: Rs. 9,000 for 12 months = 108,000 capital-months. Everything in this part is that one multiplication, followed by “parts of the profit”.

• Q3 (same time, simple ratio): Ali Rs. 22,500, Danish Rs. 35,000, profit Rs. 13,800. Ratio 22500 : 35000 = 9 : 14 (divide by 2,500). 23 parts; Danish gets 14/23 × 13,800 = Rs. 8,400. Cancel the ratio down before touching the profit — 9 : 14 is far easier than 22,500 : 35,000.
• Q20 (work backwards from one share): Investments 27,000 : 72,000 : 81,000 = 3 : 8 : 9 (divide by 9,000). C’s 9 parts are Rs. 36,000, so 1 part = 4,000 and the total profit = 20 × 4,000 = Rs. 80,000. When one share is given and the total is asked, find one part first — never set up a proportion against the unknown total.
• Q18 (find a capital from the profit ratio): Profit ratio 2 : 3 with equal time means capital ratio 2 : 3. X invested Rs. 40,000, so Y invested 3/2 × 40,000 = Rs. 60,000.

2. Different times — the capital-months table

When partners join or leave at different times, build a small table: capital × months in business, one row per partner, then reduce.

• Q5 (late joiner): Kamal Rs. 9,000 for 12 months = 108,000; Shahid Rs. 8,000 for 7 months (he joins after five months) = 56,000. Ratio 108000 : 56000 = 27 : 14. 41 parts; Shahid gets 14/41 × 6,970 = Rs. 2,380 (6,970 ÷ 41 = 170 exactly — a clean division is a good sign your ratio is right).
• Q7 (two years, late joiner): Over 24 months: A 20,000 × 24 = 480,000; B 15,000 × 24 = 360,000; C 20,000 × 18 (joins after six months) = 360,000. Ratio 4 : 3 : 3. 10 parts; B gets 3/10 × 25,000 = Rs. 7,500. The trap is C’s months: 24 − 6 = 18, not 24 and not 6.
• Q8 (early leaver): Ali 8,000 × 6 = 48,000 (leaves after six months); Kamal 4,000 × 8 = 32,000; Waleed 8,000 × 8 = 64,000. Ratio 3 : 2 : 4. 9 parts; Kamal gets 2/9 × 4,005 = Rs. 890.
• Q6 (the same rule without money): A car hired for Rs. 520, used 7, 8 and 11 hours. “Capital” here is hours: 7 : 8 : 11, 26 parts; B pays 8/26 × 520 = Rs. 160. The milkmen’s pasture question (Q2) is the same shape with cows × months: 24 × 3 = 72, 10 × 5 = 50, 35 × 4 = 140, 21 × 3 = 63; total 325 cow-months; A’s 72 parts are Rs. 720, so 1 part = Rs. 10 and the total rent = Rs. 3,250. Whenever a shared cost or gain is split by usage, the usage product (cows × months, hours, capital × months) is the ratio.

3. Capital that changes mid-term — split the timeline

If a partner withdraws or adds money part-way through, split his row into two periods and add.

• Q9 (withdrawal + top-up + late joiner): A: 16,000 × 3 + 11,000 × 9 = 48,000 + 99,000 = 147,000. B: 12,000 × 3 + 17,000 × 9 = 36,000 + 153,000 = 189,000. C: 21,000 × 6 = 126,000 (joins after six months). Total 462,000. B’s share = 189/462 × 26,400 = Rs. 10,800; C’s = 126/462 × 26,400 = Rs. 7,200; B exceeds C by Rs. 3,600. Note the check: 147 + 189 + 126 = 462, and 10,800 + 7,200 + 8,400 = 26,400 — the shares must add back to the profit.
• Q16 (fractional withdrawals): Ratio 4 : 5. After 3 months A withdraws 1/4 (keeps 3) and B withdraws 1/5 (keeps 4). A: 4 × 3 + 3 × 7 = 12 + 21 = 33. B: 5 × 3 + 4 × 7 = 15 + 28 = 43. Total 76; A gets 33/76 × 760 = Rs. 330. Work with the ratio numbers, not the rupees — the rupees are never given and never needed.
• Q14 (fractional capitals, then a withdrawal): Shares 1/2 : 1/3 : 1/4 → multiply by 12 → 6 : 4 : 3. A withdraws half after 2 months: A = 6 × 2 + 3 × 10 = 12 + 30 = 42; B = 4 × 12 = 48; C = 3 × 12 = 36. Total 126; B gets 48/126 × 378 = Rs. 144. The fraction-to-integer step (× 12, the LCM of 2, 3, 4) comes first, always.
• Q15 (a 50% increase): 7/2 : 4/3 : 6/5 → multiply by 30 → 105 : 40 : 36. A increases by 50% after 4 months: A = 105 × 4 + 157.5 × 8 = 420 + 1,260 = 1,680; B = 40 × 12 = 480; C = 36 × 12 = 432. Total 2,592; B gets 480/2,592 × 21,600 = Rs. 4,000. The .5 survives the arithmetic — do not round 157.5 to 158; 480/2,592 simplifies to 5/27 and 5/27 × 21,600 = 4,000 exactly.
• Q17 (fractions of capital AND time): A invests 1/6 of the capital for 1/6 of the time → 1/36; B 1/3 × 1/3 = 4/36; C the rest (1 − 1/6 − 1/3 = 1/2) for the whole time = 18/36. Ratio 1 : 4 : 18; 23 parts; B gets 4/23 × 4,600 = Rs. 800. C’s “rest of the capital” must be computed, not guessed: 1/2, not 1/3.

4. Ratio chains — combine before you split

When two ratios share one partner, make that partner’s number the same in both, then read off the combined ratio.

• Q1: A = 3 × B; B = 2/3 × C. Take B = 2, C = 3, A = 6 → 6 : 2 : 3. 11 parts; B gets 2/11 × 6,600 = Rs. 1,200.
• Q11: 2A = 3B and B = 4C. Take C = 1, B = 4, A = 6 (2A = 12 = 3 × 4) → 6 : 4 : 1. 11 parts; B gets 4/11 × 16,500 = Rs. 6,000.
• Q13: A : C = 2 : 1 and A : B = 3 : 2. Make A the same: A = 6 → C = 3, B = 4 → 6 : 4 : 3. 13 parts; B gets 4/13 × 157,300 = Rs. 48,400 (157,300 ÷ 13 = 12,100 exactly).
• Q10 (sums, not ratios): Total subscription Rs. 50,000; A = B + 4,000; B = C + 5,000. Let C = x: x + (x + 5,000) + (x + 9,000) = 50,000 → 3x = 36,000 → C = 12,000, B = 17,000, A = 21,000. Ratio 21 : 17 : 12; 50 parts; A gets 21/50 × 35,000 = Rs. 14,700.

5. Working backwards — find the missing capital or time

• Q22 (find B’s capital): A: 3,500 × 12 = 42,000 = 2 parts of the profit ratio 2 : 3. So 1 part = 21,000; B’s capital-months = 3 × 21,000 = 63,000 over 7 months (joins after five months) → capital = 63,000 ÷ 7 = Rs. 9,000.
• Q23 (find B’s capital from his fraction): A: 16,000 × 8 = 128,000 = 5/7 of the profit, so B’s capital-months = 2/5 × 128,000 = 51,200 over 4 months → capital = Rs. 12,800. The step students miss: convert B’s profit fraction into a ratio against A’s (2/7 vs 5/7 = 2 : 5) before multiplying.
• Q25 (find B’s time): A: 1/4 capital × 15 months = 15/4; B: 3/4 × t = 3t/4. Profit ratio A : B = 1 : 2 (B gets 2/3). So (15/4)/(3t/4) = 1/2 → 15/(3t) = 1/2 → 5/t = 1/2 → t = 10 months.
• Q21 (find the total from one share): A’s capital is 3× B’s and his time 2× B’s → capital-time ratio 6 : 1. B’s Rs. 4,000 is 1 part → total = 7 × 4,000 = Rs. 28,000.
• Q19 (find C’s capital from the shares): C’s profit share = 1,000 − 500 − 300 = Rs. 200. Profit ratio 5 : 3 : 2 = capital ratio (equal time), so C’s capital = 2/5 × 10,000 = Rs. 4,000.
• Q24 (charity first): 5% of the profit goes to charity, so only 95% is shared. A gets 3/5 of 95% = 57% of the total. Total = 855 ÷ 0.57 = Rs. 1,500. Any deduction (charity, expenses, manager’s commission) comes off the top before the ratio is applied — never after.
• Q12 (the equal-shares surprise): A invests x for 12 months = 12x; B invests 2x for 6 months = 12x; C invests 3x for 4 months = 12x. Equal capital-months → equal shares: C gets 27,000 ÷ 3 = Rs. 9,000. When the products are equal, the answer is the profit divided by the head-count — no ratio work needed.

6. Paper partnership items (3, one question in three papers)

• P and Q started business with Rs. 15,000 and Rs. 10,000; annual profit Rs. 25,000 — P’s share? Ratio 3 : 2; P gets 3/5 × 25,000 = Rs. 15,000. This single question appears in an FPSC welfare paper, an NTS LDA paper and an NTS NAB paper — the papers’ partnership block is essentially this one shape, and set 137’s Q3 is the same shape with bigger numbers.

PART B — CHAIN RULE: MORE MEANS MORE, OR MORE MEANS LESS?

7. The one question

For every quantity in the question, ask: if this goes up, does the answer go up (direct) or down (inverse)? Then multiply the starting value by each ratio in the right direction:

• Direct: multiply by new ÷ old.
• Inverse: multiply by old ÷ new.

That is the whole chain rule. The formal “compound proportion” table is just this question asked once per quantity.

8. Direct proportion — cost, weight, length, shadows

• Q1: 15 toys cost Rs. 234 → 35 toys cost 234 × 35/15 = Rs. 546. More toys, more cost: new ÷ old.
• Q7: 11.25 m of rod weighs 42.75 kg → 6 m weighs 42.75 × 6/11.25 = 22.8 kg.
• Q6 (units first): A quarter kg = 250 g costs 60 paisa → 200 g costs 60 × 200/250 = 48 paisa. Convert the units before the ratio, not after — 60 × 200/250 is clean; 60 × 0.2/0.25 invites decimal slips.
• Q8 (map scale): 0.6 cm represents 6.6 km → 80.5 cm represents 6.6 × 80.5/0.6 = 885.5 km.
• Q10 (shadows): Flagstaff 17.5 m, shadow 40.25 m → building with shadow 28.75 m: 17.5 × 28.75/40.25 = 12.5 m. Height follows shadow length directly, under the same sun.
• Q5 (approximate): 6 toys Rs. 264.37 → 5 toys: 264.37 × 5/6 = Rs. 220.31, approximately Rs. 220. “Approximate” in the stem is permission to round at the end, not during.
• Paper items (direct): Quarter kg of tomatoes costs 60 paisa → 200 g = 48 paisa (FPSC IB 2018 — the set’s Q6 with tomatoes); 16 toys cost Rs. 240 → 40 toys = Rs. 600; 16 books cost 240 → 40 books = 600; 35 kg spinach costs Rs. 28 → Rs. 16 buys 35 × 16/28 = 20 kg; flour bag 20 kg Rs. 165 → 1 kg = Rs. 8.25; shadow of a 40 m pole is 30 m → a 200 m plaza’s shadow = 200 × 30/40 = 150 m.

9. Inverse proportion — men and days

• Q2: 36 men do a job in 25 hours → 15 men take 36 × 25/15 = 60 hours. Fewer men, more hours: old ÷ new.
• Q12: 36 men in 18 days → 27 men: 36 × 18/27 = 24 days.
• Paper items (the papers’ favourite shape, 51 items): 16 men build a house in 10 days → 8 men take 20 days; 20 men construct a building in 40 days → 10 men take 80 days; 10 men complete a job in 14 days → 4 men take 35 days (four papers ask this exact question); 8 men do a work in 9 days → 6 men take 12 days; 15 men can do a work in 8 days → 10 days needs 12 men; 120 men build for 30 days → 80 men take 45 days. The method never changes: men × days is constant, so new days = old men × old days ÷ new men.

10. Compound chains — men × days × hours × output

Chain the ratios one quantity at a time, each in its own direction.

• Q3 (wages): Wages of 6 men for 15 days = Rs. 2,100 → 9 men for 12 days: wages follow men directly and days directly: 2,100 × 9/6 × 12/15 = Rs. 2,520.
• Q4 (output): 20 men build 56 m in 6 days → 35 men in 3 days build 56 × 35/20 × 3/6 = 49 m. Length follows men and days directly.
• Q16: 39 persons, 12 days, 5 hours a day → 30 persons, 6 hours a day: days follow persons inversely and hours inversely: 39 × 12 × 5/(30 × 6) = 13 days.
• Q17 (hours a day): 3 pumps × 8 hours × 2 days = 48 pump-hours fixed → 4 pumps in 1 day work 48/4 = 12 hours a day.
• Q18: 8 men reap 80 hectares in 24 days → 36 men in 30 days reap 80 × 36/8 × 30/24 = 450 hectares.
• Q21 (four quantities): 18 pumps raise 2,170 tonnes in 10 days at 7 hours a day → 16 pumps, 1,736 tonnes, 9 hours a day: days follow pumps inversely, tonnes directly, hours inversely: 10 × 18/16 × 1,736/2,170 × 7/9 = 7 days. Write the chain as one line and cancel before multiplying: 18/16 × 1,736/2,170 = 18/16 × 4/5 = 9/10; 10 × 9/10 × 7/9 = 7.
• Q23: 18 binders bind 900 books in 10 days → binders for 660 books in 12 days: 18 × 660/900 × 10/12 = 11 binders. Binders follow books directly, days inversely.
• Q28 (with an equivalence): Work = 15 × 21 × 8 = 2,520 man-hours. 3 women = 2 men, so 21 women = 14 men. Days = 2,520/(14 × 6) = 30 days.
• Q29 (double work, mixed team): Work = 3 × 10 × 7 = 210 man-hours; twice = 420. 3 men = 6 boys, so 1 man = 2 boys; 6 men + 2 boys = 7 men. Days = 420/(7 × 8) = 7.5 days.
• Q30 (two teams, one equivalence): 2 men + 7 boys in 14 days; 3 men + 8 boys in 11 days. The two teams do the same work, so find 1 man in boys: from the two equations, 1 man = 2 boys (check: team 1 = 4 + 7 = 11 boys × 14 = 154 boy-days; team 2 = 6 + 8 = 14 boys × 11 = 154 boy-days — the same total confirms it). 8 men + 6 boys = 22 boys; three times the work = 462 boy-days; 462/22 = 21 days.
• Paper wages item: Six men working 8 hours a day earn Rs. 720 per week → 8 men working 6 hours a day earn 720 × 8/6 × 6/8 = Rs. 720 — the two changes cancel exactly, a favourite examiner trick.

11. Men–women–boys equivalence

• Q19: 5 men = 9 women, so 3 men + 6 women = 3 × 9/5 + 6 = 5.4 + 6 = 11.4 women. Days = 9 × 19/11.4 = 15 days. Do not round 11.4 to 11 — 171/11.4 is exactly 15, and rounding breaks it.
• Paper items: 3 men or 6 boys in 20 days → 6 men + 8 boys: 6 men = 12 boys, so the team is 20 boys vs the original 6 boys-equivalent… worked properly: 3 men = 6 boys, so the original team = 6 boys; the new team = 6 men + 8 boys = 12 + 8 = 20 boys; days = 20 × 6/20 = 6 days. And the two-team shape: 12 men + 16 boys in 5 days, 13 men + 24 boys in 4 days → the man : boy daily-work ratio is 1 : 2 (from the two equations, 1 man = 2 boys) — the same equivalence Q30 teaches.

12. Provisions — food is man-days

A stock of food is a fixed number of man-days. Days used are gone; divide what is left by the men who remain.

• Q13 (men leave): 150 men × 45 days = 6,750 man-days. After 10 days, 1,500 man-days are eaten; 5,250 remain for 125 men: 5,250/125 = 42 days.
• Q26 (same shape, bigger numbers): 95 men × 200 days = 19,000 man-days. After 5 days, 475 eaten; 18,525 remain for 65 men: 18,525/65 = 285 days.
• Q27 (reinforcement arrives): 500 men × 27 days = 13,500 man-days. After 3 days, 1,500 eaten; 12,000 remain for 800 men (500 + 300): 12,000/800 = 15 days.
• Q14 (mixed consumers): A meal feeds 120 men or 200 children, so 1 child = 120/200 = 0.6 man-meals. 150 children eat 150 × 0.6 = 90 man-meals, leaving 30 man-meals → 30 men can be fed.
• Paper item: A garrison of 1,500 men has provisions for 6 weeks; after two weeks 450 desert: remaining food = 1,500 × 4 = 6,000 man-weeks for 1,050 men = 40/7 weeks = 5 weeks 5 days.

13. Partial work and “how many more”

• Q11 (the fraction trick): 5/8 of a job in 10 days → the remaining 3/8 takes 10 × 3/5 = 6 days. Never compute the whole job first (16 days) and subtract — the ratio 3/5 is one step.
• Q22 (how many more men): 20 men do 1/3 in 20 days → the whole job is 20 × 20 × 3 = 1,200 man-days; the remaining 800 man-days in 25 days need 32 men; more men = 32 − 20 = 12. The asked number is the extra men, not the total — the last subtraction is the whole question.
• Q25 (double the men, half the work): Some persons do a job in 12 days → twice the persons on half the work: 12 × 1/2 × 1/2 = 3 days. Two halvings, one line.
• Paper item: 40 men complete 1/3 of a work in 40 days → the whole job is 40 × 40 × 3 = 4,800 man-days; the remaining 3,200 in 50 days need 64 men; more men = 24. The set’s Q22 with new numbers — same method, same last subtraction.
• Paper item (reinforcement mid-job): Twelve men take 6 hours to finish a piece of work; after 1 hour, 8 more men join. Work done = 12 man-hours of 72; remaining 60 man-hours for 20 men = 3 more hours. Always subtract the work already done before dividing by the new team.

14. The “same number” riddles — where everything cancels

• Q15: 40 cows eat 40 bags in 40 days → one cow eats one bag in 40 days. Cows, bags and days all scale together, so the answer is the same 40.
• Q20: 7 spiders make 7 webs in 7 days → one spider makes one web in 7 days (paper item, FPSC customs inspector 2011, asks exactly this).
• Q31: 4 mat-weavers weave 4 mats in 4 days → 8 weavers in 8 days weave 4 × 8/4 × 8/4 = 16 mats (paper item, FPSC IB 2018, asks exactly this). Note the difference: Q15/Q20 ask for the time (unchanged), Q31 asks for the output (quadrupled). Read which quantity is asked.
• Q24 (inverse rounds): A rope makes 70 rounds of a cylinder of radius 14 cm → rounds of a 20 cm cylinder: rope length is fixed, so rounds follow radius inversely: 70 × 14/20 = 49 rounds.
• Q9 (rate): A loom weaves 0.128 m per second → 25 m takes 25/0.128 = 195.3, approximately 195 seconds. A rate question is a division, not a chain — but the papers file it here, so recognise it.

15. Master table — which rule, which direction

16. Rapid-fire one-liners

• Profit follows capital × time — capital-months first, ratio second, profit last.
• Same time: 22,500 : 35,000 → 9 : 14 → Danish 8,400 of 13,800.
• Late joiner: 9,000 × 12 vs 8,000 × 7 → 27 : 14 → Shahid 2,380 of 6,970.
• Early leaver: 8,000 × 6, 4,000 × 8, 8,000 × 8 → 3 : 2 : 4 → Kamal 890 of 4,005.
• Split timeline: A 147,000, B 189,000, C 126,000 → B exceeds C by 3,600 of 26,400.
• Fraction capitals: 1/2 : 1/3 : 1/4 → 6 : 4 : 3 (× 12); A halves after 2 months → B 144 of 378.
• Ratio chains: 6 : 2 : 3 → B 1,200 of 6,600; 6 : 4 : 1 → B 6,000 of 16,500; 6 : 4 : 3 → B 48,400 of 157,300.
• Working backwards: B’s capital 9,000 (42,000 = 2 parts, 7 months); B’s time 10 months (5 : t = 1 : 2); total 28,000 from B’s 4,000 at 6 : 1; C’s capital 4,000 from shares 500/300/200; charity first → total 1,500 from A’s 855.
• Equal products: x × 12 = 2x × 6 = 3x × 4 → equal shares, C 9,000 of 27,000.
• Direct: 15 toys 234 → 35 toys 546; 11.25 m 42.75 kg → 6 m 22.8 kg; 250 g 60 paisa → 200 g 48 paisa; map 0.6 cm = 6.6 km → 80.5 cm = 885.5 km; shadow 40.25 m = 17.5 m → 28.75 m = 12.5 m.
• Inverse: 36 men 25 h → 15 men 60 h; 36 men 18 days → 27 men 24 days; 10 men 14 days → 4 men 35 days.
• Compound: 6 men 15 days 2,100 → 9 men 12 days 2,520; 20 men 56 m 6 days → 35 men 3 days 49 m; 39 persons 12 days 5 h → 30 persons 6 h 13 days; 18 pumps 2,170 t 10 days 7 h → 16 pumps 1,736 t 9 h 7 days.
• Equivalence: 5 men = 9 women → 3 men + 6 women = 11.4 women → 15 days; 1 man = 2 boys (two-team check 154 boy-days) → 8 men + 6 boys, triple work → 21 days.
• Provisions: 150 men 45 days, 25 leave after 10 → 42 days; 95 men 200 days, 30 leave after 5 → 285 days; 500 men 27 days, 300 join after 3 → 15 days; meal for 120 men or 200 children, 150 children fed → 30 men.
• Partial work: 5/8 in 10 days → remaining 6 days; 1/3 in 20 days by 20 men → finish in 25 days needs 12 more men; double men, half work → 3 days.
• Riddles: 40 cows 40 bags 40 days → 1 cow 1 bag 40 days; 7 spiders 7 webs 7 days → 1 spider 1 web 7 days; 4 weavers 4 mats 4 days → 8 weavers 8 days 16 mats; rope 70 rounds at 14 cm → 49 rounds at 20 cm.

17. Most-asked facts (from our verified past-paper banks)

• Set 137 is partnership’s whole syllabus in 25 questions — same-time ratios 3 (Q1, Q3, Q20), capital-months tables 4 (Q2, Q5–Q8), split timelines 5 (Q9, Q14–Q17), ratio chains 4 (Q1, Q11, Q13, plus Q10’s sum form), working backwards 6 (Q18, Q19, Q21–Q25), the equal-products surprise 1 (Q12). The split-timeline and working-backwards families together are nearly half the set — master Sections 3 and 5 first.
• Set 138’s 31 questions split into direct proportion 8 (Q1, Q5–Q10, Q24), inverse men-days 4 (Q2, Q12, plus Q25’s double-halving), compound chains 9 (Q3, Q4, Q16–Q18, Q21, Q23, Q28, Q29), equivalence 2 (Q19, Q30), provisions 4 (Q13, Q14, Q26, Q27), partial/more-men 3 (Q11, Q22, Q25), riddles and rates 3 (Q15, Q20, Q31, Q9). Compound chains are the largest family — the one-line chain of Section 10 is the skill that pays.
• The paper items (68) are led by men-days items (51) — the papers test the inverse rule more than anything else in this chapter, and most of them are one-step: men × days constant. The “10 men, 14 days → 4 men” question appears in five papers (six entries, one paper carrying it twice); “16 men, 10 days → 8 men” in three. If you can do one-step inverse in your head, the paper block is fast.
• Verbatim set-to-paper repeats: the flagstaff shadow (set Q10 → FPSC customs inspector 2011), the spider webs (set Q20 → same paper), the mat-weavers (set Q31 → FPSC IB 2018), the quarter-kg tomatoes (set Q6 → FPSC IB 2018), the P-and-Q business question (three papers). A set question that reappears in a paper is the highest-yield item in the chapter.
• Partnership is thin in the papers (3 items, one question) — the papers test it almost entirely through the set. Do not let the thin paper count mislead you: partnership is a full set, and its shapes (especially split timelines) are exactly the kind of multi-step question that separates scores.
• Overlap note: the men-days paper items counted here are also the raw material of Mathematics Topic 8 (Time & Work). This chapter teaches them as chain-rule method; Topic 8 will teach time-and-work proper (pipes, cisterns, combined rates) and cross-link here rather than recounting these items.

18. Traps & confusions

• Joining/leaving months — count what remains, not what passed. C joins after six months of a 24-month business: 18 months, not 6. A leaves after six months of an 8-month business: 6 months, not 2. Write “in for __ months” for each partner before multiplying.
• Withdrawal means two rows, not one reduced row. A invests 16,000, withdraws 5,000 after 3 months: 16,000 × 3 + 11,000 × 9 — not 11,000 × 12. The money that was in for the first three months still earns for those three months.
• Fractions of capital AND time (Q17). C’s capital is the rest: 1 − 1/6 − 1/3 = 1/2. The most common error is taking C’s capital as 1/3 (the leftover fraction of the time fractions) instead of computing it.
• Ratio chains need a common number. A : C = 2 : 1 and A : B = 3 : 2 do not combine to 2 : 1 : 2 — scale A to 6 first (6 : 4 : 3). Combining ratios without equalising the shared partner is the single most common partnership error.
• Deductions come off the top. Charity 5% means the ratio applies to 95% of the profit. Applying the ratio to the full profit and then subtracting 5% gives a different (wrong) answer.
• Direct vs inverse — ask the question every time. More men → fewer days (inverse). More days → more wages (direct). More pumps → fewer days (inverse). More tonnes → more days (direct). If you chain without asking, you will invert exactly one ratio — and one inverted ratio is a wrong answer.
• “How many more” needs the last subtraction. Q22’s answer is 12, not 32; the paper’s 40-men item’s answer is 24, not 64. The total team is an intermediate step, never the answer, when the stem asks “more”.
• Partial work: use the fraction ratio, not the whole job. 5/8 in 10 days → remaining 3/8 takes 10 × 3/5 = 6. Computing the whole job (16 days) and subtracting (6) works too, but it is two steps where one suffices — and under time pressure the two-step version is where arithmetic slips creep in.
• “At the same rate” is a promise, not filler. It guarantees the proportionality the chain rule assumes. If a question does not promise equal rates, the chain rule does not apply.
• The riddle trap: time vs output. 40 cows / 40 bags / 40 days → 1 cow 1 bag takes 40 days (time unchanged). 4 weavers / 4 mats / 4 days → 8 weavers 8 days make 16 mats (output quadrupled). Same shape, different asked quantity, different answer — read the last word of the stem.
• Equivalence decimals must stay exact. 11.4 women is not 11 women; 157.5 capital-months is not 158. The set’s answers are exact because the decimals cancel — round them and you lose the cancellation.
• Units before ratios. Quarter kg is 250 g, not 0.25 of “something”. The 200 g question is 60 × 200/250, and the clean integer 48 confirms the conversion was right.

19. Memory tricks

• “Capital × time, then parts” — say it before every partnership question. It tells you the table to build (capital × months per partner) and the last step (parts of the profit).
• “In for how long?” — the first thing you write for each partner. Joining after five months of a twelve-month year means 7 months in; leaving after six of eight means 6 months in.
• “More means more, or more means less?” — the chain-rule question, asked once per quantity. Direct: new ÷ old. Inverse: old ÷ new. Tape it to the top of every chain-rule solution.
• “Man-days are money in the bank” — food, work and wages are all stocks measured in man-days (or man-hours). What is used is gone; what is left is divided by who remains.
• “One part first” — whenever one share is given and anything else is asked, divide the share by its parts before doing anything else. Q20’s 36,000 ÷ 9 = 4,000 unlocks the whole question.
• “Equalise the shared partner” — for ratio chains, make the common partner’s number the same in both ratios (LCM), then read across.
• “Total minus present” — for every “how many more” question, the answer is the last subtraction. Compute the full team, then subtract who is already there.
• “Same numbers, same answer” — 40/40/40 → 40; 7/7/7 → 7. If every number in the riddle matches, the time does not change.

20. Visuals (site build specs)

• DIAGRAM — The capital-months table: a worked table for Q9 (three partners, three rows, split timelines for A and B, C’s six months) with the ratio 147 : 189 : 126 reducing to 49 : 63 : 42 and the shares 8,400 / 10,800 / 7,200 — placed in Section 3 as the chapter’s worked template.
• FLOWCHART — Which direction? “Does more of this quantity mean more of the answer?” → YES = direct (× new ÷ old), NO = inverse (× old ÷ new), applied once per quantity, with Q21’s four-quantity chain as the worked leaf — placed in Section 7 as the chapter’s decision graphic.
• DIAGRAM — The man-days stock: a bar for Q13’s 6,750 man-days, a slice removed for the first 10 days (1,500), the remainder divided among 125 men to give 42 days — placed in Section 12 beside the provisions method.
• TABLE — Master table as a reference card: the Section 15 table rendered colour-banded, with the partnership rows and the direct/inverse rows highlighted — placed in Section 15 as the chapter’s revision graphic.
• DIAGRAM — The riddle pair: two panels, “40 cows / 40 bags / 40 days → 40 days” (time unchanged) beside “4 weavers / 4 mats / 4 days → 16 mats” (output quadrupled), with the asked quantity circled in each — placed in Section 14 to teach the time-vs-output distinction.

21. Practice this topic

• Practice set: 124 topic-relevant items in the verified banks serve this topic — set 137’s 25 partnership questions and set 138’s 31 chain-rule questions, plus 68 tightly matched paper items (partnership 3 — the P-and-Q business question in three papers; chain-rule men-days 51, provisions 1, direct-cost 5, shadow/scale 2, pumps-weave-spiders 5, wages 1). The site serves them as this chapter’s practice set; work set 137’s split-timeline and working-backwards families and set 138’s compound chains first, then the papers’ one-step inverse men-days items for speed. Mixture/alligation, average-weight, area and volume items that merely contain cost or weight words are excluded from this count and served by their own chapters; the men-days items are cross-linked to Mathematics Topic 8 (Time & Work) rather than recounted there.
• Book mode: ← Previous: Problems on Ages · Next: Percentage, Ratio & Average →

Goldmine Notes · Mathematics · Chapter 5 · Sources: FPSC Edition 32, Arithmetic set 137 — Partnership (25 questions) and set 138 — Chain Rule (31 questions), all answers rechecked independently (partnership Q9/Q14/Q15/Q16/Q17/Q25 and chain-rule Q19/Q28/Q29/Q30 verified by hand); verified PPSC/FPSC past-paper banks (68 tightly matched paper items, script-counted — counting script hidden_files/goldmine-notes-count-ma05b.py, raw dump hidden_files/goldmine-notes-ma05-count.txt); capital-time and direct/inverse proportion methods as standard arithmetic. Standard answers only.