Mathematics — Chapter 7

Profit & Loss; Simple & Compound Interest — Money Arithmetic

Why this chapter matters: This is a ★ chapter — no FPSC arithmetic set exists for profit & loss or for interest, yet the papers ask them relentlessly: our verified banks hold 157 topic-relevant items for this chapter (126 paper items plus the 31 questions of the True Discount and Banker’s Discount sets). The papers repeat the same numbers across years: the “car sold for Rs. 50,000 at 17% profit” question appears in four papers, the “fan sold for Rs. 475 at 5% loss” question in three, the “table Rs. 400 sold for Rs. 600” and “shirt Rs. 500 sold for Rs. 800” drills in three each, and the “simple interest Rs. 5,400 at 12% for 3 years” loan question in two. Every question here is one of a handful of shapes, and every shape has exactly one formula. Learn the shapes and these 157 items become the fastest marks in the paper.

PART A — PROFIT & LOSS: ALWAYS DIVIDE BY THE COST PRICE

1. The one idea

Four prices, one rule. Cost Price (CP) is what the seller paid. Selling Price (SP) is what the buyer paid. Profit = SP − CP (when SP > CP); Loss = CP − SP (when CP > SP). And the rule the papers test more than any other:

Profit % and Loss % are ALWAYS calculated on the Cost Price — never on the Selling Price — unless the question says otherwise in so many words.

• Profit % = (Profit ÷ CP) × 100. Loss % = (Loss ÷ CP) × 100.
• SP = CP × (1 + profit% ÷ 100) for a gain; SP = CP × (1 − loss% ÷ 100) for a loss.
• CP = SP ÷ (1 + profit% ÷ 100) for a gain; CP = SP ÷ (1 − loss% ÷ 100) for a loss.

2. The plain drills — profit or loss in rupees, then as a percentage

The papers’ favourite opening shape is two prices and one subtraction:

• Table bought for Rs. 400, sold for Rs. 600 → profit Rs. 200, profit % = 200 ÷ 400 = 50%.
• Shirt bought for Rs. 500, sold for Rs. 800 → profit Rs. 300, profit % = 300 ÷ 500 = 60%.
• TV bought for Rs. 8,000, sold for Rs. 7,000 → loss Rs. 1,000, loss % = 1,000 ÷ 8,000 = 12.5%.
• Table bought for Rs. 200, sold for Rs. 175 → loss Rs. 25. Shirt Rs. 350 → Rs. 300 → loss Rs. 50. Copy Rs. 100 → Rs. 40 → loss Rs. 60. Table Rs. 1,000 → Rs. 800 → loss 20%. Car Rs. 90,000 → Rs. 80,000 → loss = 10,000 ÷ 90,000 = 11.11%. Gun Rs. 2,500 → Rs. 900 → loss = 1,600 ÷ 2,500 = 64%. Car Rs. 50,000 → Rs. 43,000 → loss = 7,000 ÷ 50,000 = 14%. Sweater Rs. 200 → Rs. 250 → profit Rs. 50. Shoes Rs. 300 → Rs. 370 → profit Rs. 70. Shirt Rs. 500 → Rs. 525 → profit Rs. 25. Car Rs. 44,000 → Rs. 48,000 → profit % = 4,000 ÷ 44,000 = 9.09%. Old gun Rs. 2,500 → Rs. 2,500 → profit/loss 0% — the papers do ask the zero case.
• Jewellery Rs. 84,000 → Rs. 85,500 → profit = 1,500 ÷ 84,000 = 1.79%.
• Article costing Rs. 50, selling for Rs. 48 → loss % = 2 ÷ 50 = 4%. Article sold for Rs. 25 at 25% profit on CP → CP = 25 ÷ 1.25 = Rs. 20. Toy sold for Rs. 220 at 10% profit on cost → CP = 220 ÷ 1.10 = Rs. 200. Book sold for Rs. 48 at 4% loss → CP = 48 ÷ 0.96 = Rs. 50.
• Article bought for Rs. 600, sold for Rs. 750 → profit % = 150 ÷ 600 = 25%. Shirts Rs. 80 → Rs. 100 each → 25%. Calculators Rs. 120 → Rs. 150 each → 25%. Bicycle Rs. 600 → Rs. 720 → gain % = 120 ÷ 600 = 20%. Cost Rs. 400, selling Rs. 500 → gain % = 25%.

3. The reversed question — find CP or SP from a percentage

The second shape gives one price and a percentage and asks for the other. Always divide or multiply by (1 ± rate), never add or subtract the percentage from the price:

• Fan sold for Rs. 475 at 5% loss; to gain 5%, sell at? → CP = 475 ÷ 0.95 = Rs. 500; new SP = 500 × 1.05 = Rs. 525. (Three papers ask this exact fan.)
• Trader loses 20% selling at Rs. 480; to gain 20%, sell at? → CP = 480 ÷ 0.80 = Rs. 600; new SP = 600 × 1.20 = Rs. 720.
• Article sold for Rs. 45 at 10% loss; sold at Rs. 65, gain %? → CP = 45 ÷ 0.90 = Rs. 50; gain = 15 ÷ 50 = 30%.
• Chair sold for Rs. 720 at 25% loss; to gain 25%, sell at? → CP = 720 ÷ 0.75 = Rs. 960; new SP = 960 × 1.25 = Rs. 1,200.
• Car sold for Rs. 50,000 at 17% profit; profit % if sold for Rs. 47,000? → CP = 50,000 ÷ 1.17 = Rs. 42,735.04; profit at 47,000 = 4,264.96; profit % = 4,264.96 ÷ 42,735.04 = 9.98%. Work it from the CP you just found — never from the 50,000.
• Motorbike sold for Rs. 60,000 at 18% profit; profit % if sold for Rs. 55,000? → CP = 60,000 ÷ 1.18 = Rs. 50,847.46; profit = 4,152.54; profit % = 4,152.54 ÷ 50,847.46 = 8.17% (keyed 8.1% — the method, not the rounding, is what matters).
• Book sold for Rs. 76 at 52% gain; gain % if sold for Rs. 74? → CP = 76 ÷ 1.52 = Rs. 50; gain at 74 = 24 ÷ 50 = 48%.
• Goods sold for Rs. 240 at 25% gain; gain % if sold for Rs. 216? → CP = 240 ÷ 1.25 = Rs. 192; gain at 216 = 24 ÷ 192 = 12.5%.
• A person sold his watch for Rs. 75 and his percentage of profit equals the cost price. Let CP = x: profit % = x, so SP = x × (1 + x ÷ 100) = 75 → x² + 100x − 7,500 = 0 → (x + 150)(x − 50) = 0 → CP = Rs. 50. The quadratic collapses to a factorable pair — try factors of 7,500 first.
• A merchant buys an article for Rs. 27 and sells it at a profit of 10% of the SELLING price. This is the trap that breaks the “always on CP” rule — here the question says the profit is on SP: SP − 27 = 0.10 × SP → 0.90 × SP = 27 → SP = Rs. 30. When the base is named, use the named base.
• A retailer bought a product for Rs. 300, paid 15% sales tax, sold it for Rs. 410. Total cost = 300 × 1.15 = Rs. 345; profit = 410 − 345 = Rs. 65. Tax paid on purchase is part of the cost.

4. Two articles, two rates — the combined shape

• Plot bought for Rs. 8,000, sold at 20% profit, then resold at 20% loss. First sale = 8,000 × 1.20 = 9,600; second = 9,600 × 0.80 = Rs. 7,680. Successive percentages multiply (Chapter 6, Section 3) — the net is 1.20 × 0.80 = 0.96, a 4% overall loss, not zero.
• Two clocks A and B bought for Rs. 650 total; A sold at 20% profit, B at 25% loss, same selling price. 1.20 × A = 0.75 × B and A + B = 650 → B = 1.6A → 2.6A = 650 → A = Rs. 250, B = Rs. 400 (both sell for Rs. 300). Equal SP means CPs are in the inverse ratio of the multipliers.
• A shirt sold at 25% profit; had he bought it at 25% less and sold it for Rs. 25 less, he would still have gained 25%. Let CP = x: SP = 1.25x; new CP = 0.75x, new SP = 1.25x − 25, gain 25% → 1.25x − 25 = 0.75x × 1.25 = 0.9375x → 0.3125x = 25 → CP = Rs. 80.
• A merchant has 1,000 kg of sugar, part sold at 8% profit, the rest at 18% profit, overall gain 14%. By alligation (Chapter 6, Section 13): (18 − 14) : (14 − 8) = 4 : 6 → the 18% part = 6 ÷ 10 × 1,000 = 600 kg.
• Cost price of 10 chairs equals selling price of 16 chairs; loss %? CP of 10 = SP of 16 → on 16 chairs, loss = 6 chairs’ worth on a CP base of 16 → 6 ÷ 16 = 37.5%.
• A dishonest shopkeeper gives 20% less quantity. He charges for 100 but delivers 80 → profit = 20 ÷ 80 = 25%. The base is what he delivers, not what he charges for.
• A dealer buys 22 pencils for Rs. 20 and sells 10 pencils for Rs. 11. CP each = 20 ÷ 22 = Rs. 10/11; SP each = Rs. 1.10; profit % = (1.10 − 0.9091) ÷ 0.9091 = 21%.
• Oranges bought 6 for Rs. 5, sold 5 for Rs. 6. CP each = 5/6, SP each = 6/5 → gain = (6/5 − 5/6) ÷ (5/6) = (11/30) ÷ (5/6) = 44%. Work per-unit, never with the two different lots.
• Sweets bought 14 for a rupee; how many sold for a rupee to gain 40%? CP each = 1/14; SP each must be 1.40 ÷ 14 = 0.10 → 10 sweets per rupee.
• A fruit seller buys oranges 12 for Rs. 20; sell one orange to gain 20% at? CP each = 20 ÷ 12 = Rs. 1.667; SP = 1.667 × 1.20 = Rs. 2.
• Sugar costing Rs. 9/kg mixed with 27 kg at Rs. 7/kg, mixture sold at Rs. 9.24/kg at 10% gain. Mixture CP = 9.24 ÷ 1.10 = Rs. 8.40; (9x + 7 × 27) ÷ (x + 27) = 8.40 → 0.6x = 37.8 → x = 63 kg.
• Tea at Rs. 60/kg and Rs. 65/kg mixed, sold at Rs. 68.20/kg at 10% gain. Mixture CP = 68.20 ÷ 1.10 = Rs. 62; alligation (65 − 62) : (62 − 60) = 3 : 2 → mix in the ratio 3 : 2.
• Laptop bought at 20% discount on original price, then sold at 40% profit on cost. Net on original = 0.80 × 1.40 = 1.12 → total profit on original price = 12%.
• Heater: difference in selling prices at 5% and 7.5% gains is Rs. 1.25. 2.5% of CP = 1.25 → CP = Rs. 50. A difference of percentages is a percentage of CP.

5. Marked price and trade discount

Marked Price (MP) is the tag price. Discount = MP − SP, and discount % is calculated on the Marked Price — the mirror image of profit %, which is on CP:

• Discount % = (Discount ÷ MP) × 100. SP = MP × (1 − discount% ÷ 100).
• Car worth Rs. 600,000, paid Rs. 570,000 → discount = 30,000 ÷ 600,000 = 5%.
• Perfume Rs. 250, discount 25% → sale price = 250 × 0.75 = Rs. 187.50. Gift pack Rs. 80, discount 15% → 80 × 0.85 = Rs. 68. List price Rs. 20, discount 5% → Rs. 19.
• Suitcase sale price Rs. 846 after 25% discount → original = 846 ÷ 0.75 = Rs. 1,128. Hair band bought for Rs. 32 against Rs. 40 original → discount = 8 ÷ 40 = 20%.
• 100 pens at 10% discount cost Rs. 600 net; selling expenses 15% on net cost; sell at 25% profit. Cost with expenses = 600 × 1.15 = 690; SP = 690 × 1.25 = Rs. 862.50. Expenses join the cost before the profit is applied.
• Successive discounts multiply, they do not add (Chapter 6, Section 3 again): 30%, 20%, 10% in series → 0.70 × 0.80 × 0.90 = 0.504 → single equivalent discount = 49.6%. Two successive discounts of 20% and 20% → 0.80 × 0.80 = 0.64 → 36% single discount.
• Goods marked 30% above CP, discount 15% on MP → SP = CP × 1.30 × 0.85 = 1.105 × CP → profit = 10.5%. The mark-up and the discount are on different bases — multiply, never subtract.
• Shopkeeper marks tools to make 20% profit on cost while selling at 30% below the marked price. Work backwards: required SP = CP × 1.20; MP = required SP ÷ (1 − discount rate) = CP × 1.20 ÷ 0.70. For tool kits costing Rs. 96: SP = 96 × 1.20 = 115.20; MP = 115.20 ÷ 0.70 = Rs. 164.57. Work MP = required SP ÷ (1 − discount rate), never CP × (1 + profit% + discount%).
• A man buys an article for Rs. 120 and gets 25% discount; price paid = 120 × 0.75 = Rs. 90. Discount is on the marked price, profit is on the cost price — the two bases are the whole chapter in one sentence.

PART B — SIMPLE INTEREST: INTEREST ON THE PRINCIPAL ONLY

6. The one formula

SI = (P × R × T) ÷ 100, where P = principal, R = rate % per annum, T = time in years. Amount = P + SI. Time must be in years: months ÷ 12, days ÷ 365.

• SI on Rs. 500 for 4 years at 11% = 500 × 11 × 4 ÷ 100 = Rs. 220.
• Loan at 12% simple interest; after 3 years interest paid = Rs. 5,400 → P = 5,400 ÷ (0.12 × 3) = Rs. 15,000. (Two papers ask this exact loan.)
• SI Rs. 54,000 at 12% for 3 years → P = 54,000 ÷ 0.36 = Rs. 150,000.
• Principal Rs. 15,580 at 5% for 1½ years → SI = 15,580 × 5 × 1.5 ÷ 100 = Rs. 1,168.50.
• Investment pays 8% simple interest; first-year interest Rs. 84 → P = 84 ÷ 0.08 = Rs. 1,050.
• Sum amounts to Rs. 815 in 3 years and Rs. 854 in 4 years (simple interest). One year’s interest = 854 − 815 = Rs. 39; sum = 815 − 3 × 39 = Rs. 698. Under simple interest the yearly interest is constant — the difference of two amounts is one year’s interest, whatever the years.
• Sum amounts to Rs. 1,800 in 5 years at 12% simple interest → P = 1,800 ÷ (1 + 0.60) = Rs. 1,125.
• Difference of SI from two banks on Rs. 500 for 2 years is Rs. 2.50 → 500 × 2 × ΔR ÷ 100 = 2.50 → ΔR = 0.25%.
• An amount doubles itself in 5 years at simple interest → interest = principal in 5 years → R = 100 ÷ 5 = 20% per annum. General rule: to become n times in T years at SI, R = (n − 1) × 100 ÷ T.
• A financier claims simple interest at 10% but adds the interest to the principal every six months. That is compound interest in disguise: effective yearly rate = (1 + 0.10 ÷ 2)² − 1 = 1.1025 − 1 = 10.25%. Whenever interest is “included” mid-year, the nominal simple rate is not the real rate — see Part C.
• A boy deposits Rs. 300 at 5.5% interest for one year → interest = 300 × 5.5 ÷ 100 = Rs. 16.50.

PART C — COMPOUND INTEREST: INTEREST ON INTEREST

7. The one formula

Amount = P × (1 + R ÷ 100)^T. CI = Amount − P. The difference from simple interest appears only from the second year onwards — first-year CI always equals first-year SI.

• Rs. 7,500 at 4% compound interest for 2 years → Amount = 7,500 × 1.04² = 7,500 × 1.0816 = Rs. 8,112; interest = Rs. 612.
• Rs. 1,000 invested at 12% compounded half-yearly for one year → half-yearly rate 6%, two periods: Amount = 1,000 × 1.06² = Rs. 1,123.60. (Two papers ask this exact deposit.)
• Bank offers 5% compound interest half-yearly; Rs. 1,600 deposited on 1 January and Rs. 1,600 on 1 July. January deposit runs two half-years: 1,600 × 1.025² = 1,681; July deposit runs one: 1,600 × 1.025 = 1,640; total 3,321 against 3,200 deposited → interest gained = Rs. 121.
• Sum becomes Rs. 13,380 after 3 years and Rs. 20,070 after 6 years (compound interest). Ratio over 3 years = 20,070 ÷ 13,380 = 1.5 → sum = 13,380 ÷ 1.5 = Rs. 8,920. Under CI, equal time gaps give equal multiplication factors — the ratio, not the difference, is constant.
• Sum doubles in 5 years at CI; it becomes 8 times in? Doubling every 5 years: 2 → 4 in 10 years → 8 in 15 years. General rule: if it becomes m times in T years, it becomes m^k times in kT years.
• Sum becomes 3 times in 4 years; it becomes 27 times in? 3¹ → 3³ = 27 → 3 × 4 = 12 years.
• Least complete years for a sum at 20% CI to more than double: 1.2³ = 1.728, 1.2⁴ = 2.0736 → 4 years. “More than doubled” means strictly above 2 — 3 years is not enough.
• Difference between SI and CI on Rs. 1,625 for 3 years at 4% = P × (R ÷ 100)² × (3 + R ÷ 100) = 1,625 × 0.0016 × 3.04 = Rs. 7.90. For 2 years the shortcut is simpler: difference = P × (R ÷ 100)².
• Rs. 2,550 borrowed at 4% CI, repaid in two equal yearly instalments. Each instalment x satisfies x ÷ 1.04 + x ÷ 1.04² = 2,550 → x × 1.886094 = 2,550 → x = Rs. 1,352.
• Debt of Rs. 1,025 due in 2 years at 5% CI, discharged by equal annual payments. x ÷ 1.05 + x ÷ 1.05² = 1,025 → x × 1.85941 = 1,025 → x = Rs. 551.25.
• Rs. 12,500 borrowed at 20% CI, Rs. 2,000 repaid at each year-end. Year 1: 12,500 × 1.20 − 2,000 = 13,000. Year 2: 13,000 × 1.20 − 2,000 = 13,600. Year 3: 13,600 × 1.20 − 2,000 = Rs. 14,320 still owed. Repayment questions are year-by-year bookkeeping — never apply the full 3-year factor to the original sum.
• Rs. 1,000 taken at the beginning of each year at 5% CI; accumulated amount at the end of 3 years. First loan runs 3 years: 1,000 × 1.05³ = 1,157.625; second 2 years: 1,102.50; third 1 year: 1,050 → total Rs. 3,310.13.

PART D — DISCOUNT ON BILLS: TRUE DISCOUNT & BANKER’S DISCOUNT

This companion block comes from the two FPSC arithmetic sets (True Discount 16 questions, Banker’s Discount 15) plus 3 paper items. The papers ask it rarely, but when they do, the vocabulary is the whole question.

8. The vocabulary

• Sum Due (Face Value, S) — the amount written on the bill, payable later.
• Present Worth (PW) — what the sum is worth today: PW = S × 100 ÷ (100 + R × T).
• True Discount (TD) = S − PW = SI on the Present Worth = (PW × R × T) ÷ 100.
• Banker’s Discount (BD) = SI on the Sum Due = (S × R × T) ÷ 100.
• Banker’s Gain (BG) = BD − TD = SI on the True Discount.
• The chain to remember: BD > TD, and BG = TD × R × T ÷ 100.

9. Worked set questions

• Present worth of Rs. 930 due 3 years hence at 8%. PW = 930 × 100 ÷ (100 + 8 × 3) = 93,000 ÷ 124 = Rs. 750. TD = 930 − 750 = Rs. 180.
• True discount on Rs. 120 due 6 months hence at 15% is Rs. 120 — banker’s discount? BG = 120 × 15 × 0.5 ÷ 100 = Rs. 9; BD = TD + BG = Rs. 129.
• Banker’s gain Rs. 24 on a sum due 2 years hence at 10%; present worth? BG = TD × R × T ÷ 100 → 24 = TD × 0.20 → TD = Rs. 120; PW = TD × 100 ÷ (R × T) = 120 × 100 ÷ 20 = Rs. 600.
• Banker’s gain Rs. 1 on a bill due 1 year hence at 5%; true discount? TD = 1 × 100 ÷ 5 = Rs. 20.
• True discount on a bill of Rs. 540 is Rs. 90; banker’s discount keyed Rs. 108. Check: PW = 540 − 90 = 450; BD = 108 gives BG = 18 = SI on TD 90 over the bill’s term — consistent with a rate × time product of 20 (e.g. 10% for 2 years). The method: find PW first, then BG from TD, then BD = TD + BG.

10. Master table — every formula in this chapter

11. Rapid-fire one-liners

• Profit % and Loss % are on Cost Price; Discount % is on Marked Price.
• SP = CP × (1 + profit%) for gain; SP = CP × (1 − loss%) for loss.
• Fan Rs. 475 at 5% loss → CP Rs. 500 → at 5% gain Rs. 525.
• Trader loses 20% at Rs. 480 → CP Rs. 600 → at 20% gain Rs. 720.
• Chair Rs. 720 at 25% loss → CP Rs. 960 → at 25% gain Rs. 1,200.
• Plot Rs. 8,000, +20% then −20% → Rs. 7,680 (net 4% loss, not zero).
• Clocks Rs. 650 total, equal SP → A Rs. 250, B Rs. 400.
• Oranges 6 for Rs. 5, sold 5 for Rs. 6 → gain 44%.
• Sweets 14 per rupee, 40% gain → sell 10 per rupee.
• False balance giving 20% less → profit 25%.
• CP of 10 = SP of 16 → loss 37.5%.
• Successive discounts 30%, 20%, 10% → single discount 49.6%; 20% + 20% → 36%.
• Marked 30% above CP, discount 15% → profit 10.5%.
• SI = P × R × T ÷ 100; Amount (SI) = P + SI.
• SI Rs. 5,400 at 12% for 3 years → principal Rs. 15,000.
• Amounts Rs. 815 (3 years) and Rs. 854 (4 years) at SI → yearly interest Rs. 39, sum Rs. 698.
• Doubles in 5 years at SI → rate 20%.
• CI Amount = P × (1 + R ÷ 100)^T; first-year CI = first-year SI.
• Rs. 7,500 at 4% CI for 2 years → interest Rs. 612.
• Rs. 1,000 at 12% compounded half-yearly for 1 year → Rs. 1,123.60.
• Doubles in 5 years at CI → 8 times in 15 years; 3 times in 4 years → 27 times in 12 years.
• 20% CI more than doubles in 4 years (1.2⁴ = 2.0736).
• Half-yearly compounding at nominal 10% → effective 10.25%.
• Present Worth = S × 100 ÷ (100 + RT); True Discount = S − PW; Banker’s Gain = SI on TD.
• PW of Rs. 930 due 3 years hence at 8% → Rs. 750.

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

• The two-price drills (table 400→600 = 50%; shirt 500→800 = 60%; TV 8,000→7,000 = Rs. 1,000 loss) — repeated verbatim across three papers each; they are free marks if the CP base is automatic.
• The reversed fan/trader/chair questions (475 at 5% loss → 525; 480 at 20% loss → 720; 720 at 25% loss → 1,200) — the papers’ standard test of “find CP first”.
• The car at 17% profit family (50,000 → 47,000) and the book at 52% gain family (76 → 74) — both test the same CP-first move.
• The rate-per-lot questions (oranges 6-for-5 / 5-for-6 = 44%; sweets 14-per-rupee = 10) — per-unit working is the only safe method.
• SI: Rs. 5,400 at 12% for 3 years → Rs. 15,000, and SI Rs. 500 at 11% for 4 years → Rs. 220 — the two most repeated SI stems.
• CI: Rs. 1,000 at 12% half-yearly → Rs. 1,123.60, and the doubling-time questions (double in 5 years → 8 times in 15) — the CI questions papers actually repeat.
• Discount: successive discounts (30/20/10 → 49.6%; 20/20 → 36%) and the suitcase Rs. 846 after 25% (→ Rs. 1,128) — the discount stems that recur.

13. Traps & confusions

• Profit % on SP vs on CP: the default base is CP. Only when the question says “profit of 10% of the selling price” (the Rs. 27 article → Rs. 30) do you use SP. Papers hide the base in the last words of the stem — read to the end.
• Discount % on MP vs profit % on CP: a 20% discount and a 20% profit are not the same operation and not on the same base. “Marked 30% above CP, discount 15%” gives 10.5% profit, not 15%.
• Successive changes never add: +20% then −20% is a 4% loss (plot Rs. 7,680, not Rs. 8,000); discounts 30/20/10 are 49.6%, not 60%.
• “Sold for Rs. 47,000 after 17% profit at Rs. 50,000”: the 17% belongs to the 50,000 sale. Find CP (Rs. 42,735.04) first, then the new percentage (9.98%). Dividing by 50,000 gives the wrong answer.
• SI vs CI in year one: identical. Any question where CI differs from SI must involve two or more years (or half-yearly compounding within one year).
• “Interest included every six months” means CI: a nominal 10% simple rate becomes an effective 10.25% — the word “included” is the tell.
• Equal instalments are not P ÷ n: each instalment is discounted back to today (Rs. 2,550 at 4% → Rs. 1,352 each, not Rs. 1,275). Discount every instalment separately.
• Repayment questions are year-by-year: Rs. 12,500 at 20% with Rs. 2,000 yearly repayments still owes Rs. 14,320 after three years — the debt grows because the interest (Rs. 2,500 in year one) exceeds the repayment.
• True Discount vs Banker’s Discount: TD is interest on the Present Worth; BD is interest on the Sum Due. BD is always larger; the difference (Banker’s Gain) is interest on the TD itself.
• “Passes a station” vs “crosses a platform” (Chapter 3’s ambiguity) does not arise here — but the same care applies: “discount of 25% on Rs. 120” means the buyer pays Rs. 90; “profit of 25% on Rs. 120 cost” means the seller charges Rs. 150. Same 25%, different base, different answer.

14. Memory tricks

• “Profit on CP, Discount on MP” — the two bases in one line. If you remember nothing else from this chapter, remember which price sits underneath each percentage.
• “Find CP first” — every reversed question (fan, trader, chair, car, book) starts the same way: divide the given SP by (1 ± rate). The first step is always the same; only the second step varies.
• “Multiply, never add” — successive percentages, successive discounts, and CI growth all multiply. The moment you catch yourself adding two percentages, stop.
• “SI is flat, CI climbs” — simple interest adds the same amount every year (Rs. 39 a year in the 815/854 question); compound interest multiplies by the same factor every year (×1.5 every 3 years in the 13,380/20,070 question). Difference → SI; ratio → CI.
• “Double, double again” — CI doubling times stack: double in 5 years means 4 times in 10, 8 times in 15. Count the doublings, multiply the years.
• “BD bigger, BG on TD” — Banker’s Discount exceeds True Discount, and the Banker’s Gain is just simple interest charged on the True Discount.

15. Visuals (site build specs)

• DIAGRAM — The four prices: CP → (profit/loss) → SP, with MP above SP and the discount arrow MP → SP, each percentage labelled with its base (CP for profit/loss, MP for discount) — placed in Section 1 as the chapter’s orientation graphic.
• INFOGRAPHIC — SI vs CI growth: two curves over 5 years on the same principal, SI a straight line, CI a rising curve, meeting at year one — placed in Section 7.
• TABLE — The discount chain: Sum Due → Present Worth → True Discount → Banker’s Discount → Banker’s Gain, with one worked bill (Rs. 930, 3 years, 8%) running across the row — placed in Section 8.

16. Practice this topic

• Practice set: 157 verified questions are tagged to this topic’s sub-topics (profit & loss and trade discount 93 paper items, true/banker’s discount 3 paper items + 31 set questions, simple interest 15, compound interest 12, general interest 3) — the site serves them as this chapter’s practice set. The two FPSC arithmetic discount sets (True Discount, Banker’s Discount) are served whole; the profit/loss and interest items come from the paper banks, since no FPSC set exists for them.
• Book mode: ← Previous: Percentage, Ratio & Average · Next: Time & Work; HCF & LCM; Basic Algebra →

Goldmine Notes · Mathematics · Chapter 7 · Sources: verified PPSC/FPSC past-paper banks (script-counted, see practice block), FPSC Arithmetic sets True Discount (16) and Banker’s Discount (15), Chapter 6 methods cross-referenced for alligation and successive percentages. Standard answers only; ★ topic (no FPSC profit/loss or interest set exists) built from paper items.