Percentages, Profit-Loss and Discount

Overview

Percentages, Profit-Loss and Discount appears about 3 times per paper across the last four AFCAT solved papers, placing it in the highest weight band of Numerical Ability.

Percentages, profit-loss and discount together form the highest-weight cluster in AFCAT Numerical Ability. Across the last four solved papers the topic produces roughly three questions per attempt — close to nine marks under the +3/−1 scheme, or almost a tenth of the entire 100-question paper sitting in one chapter. No other Numerical Ability area gives this much return on rote-formula practice.

The chapter rewards a single skill: converting any percentage into a manageable fraction (or vice versa) inside three seconds. Once the fraction table is locked, every profit-loss problem collapses into a CP-of-100 calculation, every discount chain reduces to a single multiplier, and every consumption-reduction item becomes a one-line shortcut. The work is not algebra — it is pattern recognition and a small set of formulas applied in the correct order.

This page builds the full toolkit. You get a 24-row fraction table, the seven core percentage formulas, the eight standard profit-loss expressions, the markup-and-discount net formula, the false-weight derivation, twelve worked examples with full reasoning, and a list of AFCAT trap wordings to memorise before you sit the paper.

Why this is the highest-weight Numerical topic

Across the last four solved AFCAT papers, percentage, profit-loss and discount items together averaged 3.0 questions per paper. That is more than time-speed-distance (~2.5), more than ratio-proportion (~2.0) and roughly double algebra-basics (~1.5). For a 100-question paper with +3/−1 scoring, hitting all three of these items adds nine marks before negatives — and missing them all costs three marks of negative damage plus nine marks of forgone gain.

The other reason this cluster matters: it is the most predictable Numerical area. AFCAT recycles the same four wording patterns paper after paper:

• ‘Price of X rises by p% — by how much should consumption fall to keep expenditure constant?’
• ‘Trader marks up by m% and gives a discount of d% — find effective profit.’
• ‘Two successive discounts of d₁% and d₂% — find equivalent single discount.’
• ‘Profit of p% is calculated on selling price — find actual profit on cost.’

Each of these has a one-line formula. A candidate who has drilled the four patterns walks into the exam with three near-guaranteed marks; one who has not drilled them spends six minutes of paper time on them and still gets one wrong. That is the gap this page is designed to close.

Fraction-to-percentage table

Memorise every row. Most AFCAT items reduce to fraction arithmetic — 37.5% of 800 is faster as (3/8) × 800 = 300 than as 0.375 × 800. Lock these in week one.

Core percentage formulas

The successive-change formula deserves a second look. If a salary is increased by 20% and then reduced by 10%, the net change is 20 + (−10) + (20 × −10 / 100) = 20 − 10 − 2 = +8%, not +10%. Two equal-magnitude changes of opposite sign never cancel — they always leave a small net loss equal to (x²/100).

Percentage word-problem patterns

Four families of word problems carry almost every percentage-only item on AFCAT.

• Marks-and-passing. ‘A student scores 35% and fails by 40 marks. Pass mark is 45%.’ Let total = T. Then 0.45T − 0.35T = 40 → 0.10T = 40 → T = 400. Pass mark = 180. The structure is always two percentage statements with a gap of marks between them.
• Elections-and-voting. ‘In an election between two candidates, the winner gets 56% of valid votes and wins by 144 votes. Find total valid votes.’ Margin = 56 − 44 = 12%. So 0.12 × V = 144 → V = 1200.
• Population-and-growth. ‘Population is 12,500 and grows 4% in year one, 5% in year two.’ Final = 12500 × 1.04 × 1.05 = 13,650. Or compute the successive formula: 4 + 5 + 0.2 = 9.2%.
• Salary-and-spending. ‘A man spends 70% of his salary; salary rises 20% and expenditure rises 10%. Find % rise in savings.’ Old salary 100, spent 70, saved 30. New salary 120, spent 77, saved 43. Rise = 13/30 × 100 = 43.33%.

Set up a coordinate base — 100 for salaries, T for totals, V for votes — and convert every percentage into a direct multiplication. Never write percentage symbols in your scratch work; carry numbers only.

Profit and loss formulas — comprehensive table

CP basis vs SP basis — side by side

The same numerical profit looks different depending on which base the question chooses. Take CP = ₹80 and SP = ₹100, so profit = ₹20.

The two answers differ by 5 percentage points on a problem with identical money. AFCAT exploits this by giving you the SP-basis profit and asking for the CP-basis profit (or vice versa). The conversion formulas are:

• Profit % on CP, given profit % on SP = s: p_CP = 100s / (100 − s).
• Profit % on SP, given profit % on CP = p: s = 100p / (100 + p).

Example. If profit is 20% on SP, then profit on CP = 100 × 20 / (100 − 20) = 25%. Memorise these two conversions — they are the single most common AFCAT trap in profit-loss.

Multiple discount and markup-with-discount

When a retailer offers two or more discounts in sequence, each discount is applied to the price after the previous discount, not to the original marked price. The effective discount is always less than the sum.

• Two discounts d₁% and d₂%: Net = d₁ + d₂ − (d₁d₂ / 100).
• Three discounts d₁, d₂, d₃: Apply the two-discount formula twice. First combine d₁ and d₂ into D. Then combine D and d₃.
• Markup m% and discount d%: Net change on CP = m − d − (md/100). Positive = profit, negative = loss.

Worked walk-through. A trader marks goods 40% above cost and offers successive discounts of 10% and 5%. CP = 100. MP = 140. After 10% discount = 126. After further 5% discount = 119.7. Profit on CP = 19.7%. Using formulas: combined discount = 10 + 5 − 0.5 = 14.5%. Then effective profit = 40 − 14.5 − (40 × 14.5 / 100) = 40 − 14.5 − 5.8 = 19.7%. Both routes match.

Dishonest dealer using false weight

A shopkeeper claims to sell at cost price but uses a weight stamped ‘1000 g’ that actually delivers only 900 g. He charges the buyer for 1000 g of goods, hands over 900 g, and keeps 100 g.

Derivation. The seller paid for 900 g of goods (the true weight he hands over) but received money equal to the cost of 1000 g (because he charges the buyer for the stamped weight). Profit = cost of 100 g. Profit % on his actual outlay = (100 / 900) × 100 = 11.11%.

General formula. If the true weight is T and the false weight delivered to the buyer is F (with F < T), and the seller advertises ‘cost price’ selling, then:

Gain % = ((T − F) / F) × 100

If the seller also marks goods up by p% on top of using a false weight, the combined gain is calculated by treating his marked price as the SP and the cost of F grams as the CP, then applying profit % formula directly.

Equating two prices to find a base

Many AFCAT items withhold the cost price and give you only two selling prices with their respective profit-or-loss percentages. The trick is to write CP as a single variable, express each SP in terms of CP, and equate.

Pattern. ‘By selling an article for ₹540 a man gains 8%. At what price should he sell it to gain 20%?’

1. CP = 540 × 100 / 108 = ₹500.
2. New SP at 20% profit = 500 × 1.20 = ₹600.

Pattern. ‘The difference between selling an article at 25% profit and at 25% loss is ₹150. Find the cost price.’

1. SP at 25% profit = 1.25 CP. SP at 25% loss = 0.75 CP.
2. Difference = 1.25 CP − 0.75 CP = 0.50 CP = 150.
3. CP = ₹300.

The skill is recognising that any two SPs of the same article are linked through one CP. Write CP once, express both SPs, equate or subtract.

Application — bulk buy and resell, mixed lots

‘A trader buys 15 dozen pens at ₹120 a dozen and sells them at ₹12 each. Find profit %.’

1. Total CP = 15 × 120 = ₹1800.
2. Total SP = 15 × 12 × 12 = ₹2160.
3. Profit = ₹360. Profit % = 360 / 1800 × 100 = 20%.

‘A man buys 200 articles at ₹5 each. He sells 80% of them at a profit of 25% and the rest at cost price. Find overall profit %.’

1. Total CP = 200 × 5 = ₹1000.
2. SP of 160 articles at 25% profit = 160 × 5 × 1.25 = ₹1000.
3. SP of remaining 40 at cost = 40 × 5 = ₹200.
4. Total SP = ₹1200. Profit = ₹200. Profit % = 20%.

For mixed-lot problems, compute total CP and total SP first, then apply profit % once. Do not chase per-unit profits — they always mislead when lots are unequal.

AFCAT trap patterns to memorise

• Profit on SP, not CP. Skim-readers apply profit % directly to CP and lose three marks. Always look for the prepositions ‘of’ and ‘on’ near the percentage word.
• Consumption-reduction with wrong denominator. A 25% price rise demands a 20% consumption cut (not 25%). The denominator is (100 + x), not 100.
• Dishonest dealer ‘at cost price’. The seller’s gain % equals (T − F) / F × 100, not (T − F) / T × 100. The denominator is the false (delivered) weight.
• Two equal-magnitude successive changes. +20% followed by −20% does not return you to the starting value. The net change is always −(x²/100); here, −4%.
• Faulty discount chain. Two successive discounts of 20% and 10% give a 28% net discount, not 30%. Use the xy/100 correction.
• Two articles at equal SP, one profit and one loss of p%. The dealer always loses (p²/100)%. Equal percentages on equal SP never cancel.
• ‘Marks needed to pass’ trap. Pass mark is a percentage of total marks, not of marks obtained. Read the base in every sentence.

Speed shortcuts — the 10% method

For any mental percentage of a clean number, decompose the percentage into multiples of 10% and 1%. 23% of 480 becomes 10% + 10% + 3% = 48 + 48 + 14.4 = 110.4. The arithmetic happens in chunks small enough to handle without paper.

Steps.

1. 10% of N is N ÷ 10. Halving it gives 5%. Halving again gives 2.5%.
2. 1% of N is N ÷ 100.
3. Build the target percentage from sums of these primitives.

Worked. 17.5% of 640: 10% = 64, 5% = 32, 2.5% = 16. Sum = 112. Done in eight seconds.

Worked. 35% of 1240: 10% × 3 = 372, 5% = 62. Sum = 434. Done in ten seconds.

This becomes second nature after two weeks of daily practice and removes the urge to write long multiplications in the margin under exam pressure.

Time budget per item

AFCAT gives 72 seconds per question on average (120 minutes for 100 questions). For percentage and profit-loss items, target the following bands:

If you cross 90 seconds on any single item, mark it for review and move on. The opportunity cost of one ‘stuck’ item is two faster items lost later in the paper.

Worked AFCAT-style examples

Exam-day strategy

1. Lock the 24-row fraction table in week one. Test yourself daily until conversions are sub-three-second in both directions.
2. Before applying any profit formula, underline the words ‘on cost price’ or ‘on selling price’. Default to CP if unstated.
3. Use the successive-change formula x + y + xy/100 instead of computing two separate multiplications. It is faster and removes rounding error.
4. For consumption-reduction problems, apply the 100x/(100+x) shortcut directly. Do not assume a numerical base.
5. Compute total CP and total SP first in bulk-buy questions. Per-unit profit calculation misleads when lots are unequal.
6. On dishonest-dealer items, remember the denominator is the false (delivered) weight, not the true weight stamped on the device.
7. Aim for 30–60 seconds per item across the topic. Anything past 90 seconds is a mark-and-move decision.

Frequently asked questions

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