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RBI Grade B Officer · Quantitative Aptitude

Profit, Loss & Discount

Trading calculations involving cost price, selling price, marked price, profit, loss and discount.

Five ideas, and every trading question in Tier 1 is one of them: profit and loss live on cost price, discount lives on marked price, successive discounts do not add, a short weight is a hidden profit, and equal selling prices at equal percent profit and loss never break even.

  • RBI Grade B Officer
  • Medium level
  • 5 concepts
  • 42 practice questions

1Profit and loss are always on cost price

Profit or loss is the difference between selling price and cost price, and the corresponding percentage is always taken on the cost price: profit % = (SP − CP)/CP × 100 and loss % = (CP − SP)/CP × 100. Dividing by SP is the single most common mistake in this chapter, and it produces a tidy wrong answer every time.

Fix CP = 100 when the question gives only percentages — the profit or discount percentages become rupee amounts directly, and you scale to the real value at the end. The arithmetic is identical whoever is reading it; only the numbers change.

Figure. Profit is the gap above the CP rule. The percentage divides that gap by CP, not by SP.

How it works

  1. Name CP and SPWrite both in rupees. If either is missing, recover it from a percentage before going further.
  2. Take the differenceSP − CP is profit; CP − SP is loss. A negative difference is a loss, not an arithmetic error.
  3. Divide by CP, then × 100The base is always CP — never SP, never MP, never the larger of the two.

Finding profit percent

A trader buys an article for Rs. 400 and sells it for Rs. 500. Find the profit percent.

  • Profit = SP − CP = 500 − 400100
  • Base is CP = 400divide by 400
  • Profit % = 100/400 × 10025%
  • Check: SP = CP × (1 + 25/100) = 400 × 1.25500

Pro tip. Profit percent is always on CP — dividing by SP here would give 100/500 × 100 = 20%, a clean wrong answer that still feels like a percentage. Whenever the options include both the true figure and the SP-based one, the trap has already been written for you.

An article bought for Rs. 240 is sold for Rs. 300. The profit percent is
  1. 25%
  2. 20%
  3. 30%

Profit = 60 on a CP of 240, so 60/240 × 100 = 25%. Dividing by SP instead gives 60/300 × 100 = 20% — the classic trap. 30% comes from taking 60 against 200, as if the CP had been invented rather than given.

2Discount is on marked price, profit is still on cost

Discount is always calculated on the marked price: SP = MP × (1 − d/100). Shopkeepers mark up above cost price to an inflated MP and then offer a discount on that MP — the customer sees the discount, and the shopkeeper still clears a profit because the markup was larger than the cut.

The two percentages sit on different bases. Markup and profit percent both use CP; discount uses MP. Mixing the bases is how a 25% markup and a 10% discount get misread as a 15% profit.

Figure. Markup lifts CP to MP; discount pulls MP down to SP. Profit is the remaining gap above the cost-base rule.

How it works

  1. Build the MP from CPMP = CP × (1 + markup%/100). This is the ticketed price before any discount.
  2. Apply the discount to MPSP = MP × (1 − d/100). Never take the discount off CP.
  3. Measure profit against CPProfit % = (SP − CP)/CP × 100 — back on the cost base, not on MP.

Markup then discount

An article costs Rs. 400. It is marked 25% above cost and then sold at a 10% discount on the marked price. Find the selling price and the profit percent.

  • MP = 400 × (1 + 25/100) = 400 × 1.25Rs. 500
  • SP = 500 × (1 − 10/100) = 500 × 0.9Rs. 450
  • Profit = 450 − 400Rs. 50
  • Profit % = 50/400 × 10012.5%

Pro tip. 25 − 10 = 15 is not the profit percent. The discount acted on 500, not on 400, so only Rs. 50 of the Rs. 100 markup survived — 12.5% on cost, not 15%.

An article marked at Rs. 800 is sold at a 25% discount. If its cost price is Rs. 500, the profit percent is
  1. 20%
  2. 25%
  3. 60%

SP = 800 × 0.75 = 600. Profit on CP is (600 − 500)/500 × 100 = 20%. Answering 25% confuses the discount with the profit; 60% treats the Rs. 300 gap as a fraction of 500 without noticing the discount has already set SP at 600.

3Successive discounts do not add

Two successive discounts of x% and y% are equivalent to a single discount of (x + y − xy/100)%, which is always less than x + y. The second cut acts on a price the first has already reduced, so part of the second percentage never touches the original marked price.

Apply the factors in either order — 0.8 × 0.9 = 0.9 × 0.8 — and the single equivalent is the same. Adding the percentages always overstates the customer's saving.

Figure. Each cut shortens the bar that remains. The second 10% removes Rs. 80, not Rs. 100 — which is why the total saving is 28%, not 30%.

How it works

  1. Write the factorsAn x% discount leaves the factor (1 − x/100); a second y% leaves (1 − y/100).
  2. Multiply, then convertNet remaining = product of the factors; net discount % = (1 − product) × 100.
  3. Or use the shortcutnet = x + y − xy/100. Same number, one line, and visibly less than x + y.

Single equivalent of two discounts

Find the single discount equivalent to two successive discounts of 20% and 10%.

  • net = x + y − xy/100 with x = 20, y = 1020 + 10 − 2
  • 20 + 10 − (20 × 10)/10030 − 2
  • Single equivalent discount28%
  • Check on Rs. 1000: 1000 × 0.8 × 0.9Rs. 720

Pro tip. The single equivalent is always less than the simple sum (30% here) because the second discount acts on a reduced price. On Rs. 1000 the customer saves Rs. 280, not Rs. 300 — and 280/1000 is exactly the 28% the formula gave.

Successive discounts of 15% and 15% are equivalent to a single discount of
  1. 30%
  2. 27.75%
  3. 22.5%

15 + 15 − (15 × 15)/100 = 30 − 2.25 = 27.75%. Adding to 30% ignores that the second 15% is charged on 85% of the marked price. 22.5% is half of 45 and has no derivation here.

4A short weight is a hidden profit

A trader who uses false weights earns a gain even when the labelled selling price equals the cost price, because the customer pays for more goods than are handed over. If the error (the shortfall) is e on a true measure T, the gain percent is e/(T − e) × 100.

Selling at cost price with a 10% short weight still yields 10/90 × 100 = 11 1/9 % profit. The denominator is what was actually given, not the true measure — that swap is the whole method.

Figure. The customer pays for the tall bar; the trader's cost is the short one. Gain percent divides the 100 g gap by 900 g, not by 1000 g.

How it works

  1. Name the true measure and the errorTrue weight T, shortfall e. What the customer receives is T − e.
  2. Treat the shortfall as free profitThe trader charged for T but spent only on T − e, so the gain on cost is e against a base of T − e.
  3. Write the percentGain % = e/(T − e) × 100. For a 10% short on 100 parts: 10/90 × 100.

Ten percent short at cost price

A dealer professes to sell at cost price but uses a weight of 900 g for a kilogram. Find the gain percent.

  • True measure T = 1000 g; error e = 100 greceives 900 g
  • Gain % = e/(T − e) × 100 = 100/900 × 10011 1/9 %
  • Same with parts: e = 10, T − e = 9010/90 × 100
  • Check in rupees: CP of 900 g at Rs. 90/kg = Rs. 81; SP = Rs. 909/81 × 100 = 11 1/9 %

Pro tip. The gain percent is larger than the shortfall percent — 11 1/9 % from a 10% short — because the base has shrunk to what was actually given. Writing 10/100 × 100 = 10% is the trap that treats the true kilogram as the cost base.

A seller uses 950 g for a kg but sells at cost price. The gain percent is
  1. 5%
  2. 5 5/19 %
  3. 50/9 %

Error = 50 g on a true 1000 g, so gain % = 50/950 × 100 = 100/19 = 5 5/19 %. Answering 5% divides by the true kilogram; 50/9 % is 100/18 and belongs to a 100 g shortfall on 900 g, not this case.

5Same selling price at plus and minus x% is a loss

Selling two items at the same selling price, one at x% profit and one at x% loss, always gives a net loss of x²/100 % — never break-even. The loss percent sits on a larger cost base than the profit percent sits on, so the rupee loss outweighs the rupee gain.

Recognise the shape before computing: equal SP, equal percent either side of cost. The answer is (x²/100)% loss regardless of the goods, and the options that say "no profit no loss" are there on purpose.

Figure. Both sell at the dashed SP line. The loss article's CP sits higher above the line than the profit article's CP sits below it — so the rupee loss wins.

How it works

  1. Recover both cost pricesFrom a common SP: CP₁ = SP/(1 + x/100) and CP₂ = SP/(1 − x/100).
  2. Add costs, add selling pricesTotal CP is larger than 2 × SP × (something under 1); total SP is just 2 × SP.
  3. Read the shortcutNet loss % = x²/100. For x = 20 the loss is 4%; for x = 10 it is 1%.

Equal SP at ±20%

Two articles are sold at Rs. 100 each. One is sold at 20% profit and the other at 20% loss. Find the net profit or loss percent on the whole.

  • CP of profit article = 100 / 1.2Rs. 250/3
  • CP of loss article = 100 / 0.8Rs. 125
  • Total CP = 250/3 + 125 = 625/3; total SP = 200loss = 25/3
  • Loss % = (25/3) / (625/3) × 100 = 25/625 × 1004%

Pro tip. The shortcut is loss % = x²/100 = 400/100 = 4%, and it matches the ledger. Equal percent either side of a common SP is never neutral — the cost of the loss article is always the larger base.

Two goods sold at the same price, one at 10% profit and one at 10% loss, give a net
  1. no profit no loss
  2. 1% loss
  3. 1% profit

Net loss % = 10²/100 = 1%. "No profit no loss" is the trap the formula exists to kill; a 1% profit would require the bases to favour the gain article, which they do not.

Notes

  • Core Definitions: Profit or loss is always calculated on the cost price (CP); profit = SP - CP and loss = CP - SP, with the corresponding percentage taken relative to CP.
  • Marked Price & Discount: Discount is always calculated on the marked price (MP), so SP = MP\times(1-\frac{d}{100}); shopkeepers mark up above CP and then offer discount on that inflated MP.
  • Successive Discounts: Two discounts x\% and y\% are equivalent to a single discount of \left(x+y-\frac{xy}{100}\right)\%, which is always less than x+y.
  • Dishonest Dealer: A trader using false weights earns gain \%=\frac{\text{error}}{\text{true value}-\text{error}}\times100; selling at cost price with a 10\% short weight still yields \frac{10}{90}\times100=11\tfrac19\% profit.
  • Equal SP Trap: Selling two items at the same SP, one at x\% profit and one at x\% loss, always gives a net loss of \frac{x^2}{100}\%, never break-even.

Formulas

  • Profit \% = \frac{SP-CP}{CP}\times100; Loss \% = \frac{CP-SP}{CP}\times100
  • SP = CP\left(1+\frac{\text{Profit}\%}{100}\right)
  • SP = MP\left(1-\frac{\text{Discount}\%}{100}\right)
  • Successive discounts: net = x+y-\frac{xy}{100} (as a discount)
  • Two articles at same SP, \pm x\%: net loss =\frac{x^2}{100}\%
  • Dishonest dealer gain \% = \frac{\text{Error}}{\text{True weight}-\text{Error}}\times100

Exam traps & shortcuts

  • Fix CP =100 to turn profit/discount percentages into direct rupee amounts, then scale to the actual value at the end.
  • For two successive discounts, combine them with x+y-\frac{xy}{100} instead of applying each one separately.
  • Recognise the same-SP profit/loss trap: the answer is always a \frac{x^2}{100}\% loss regardless of the goods.

Reference tables

Every line reconstructs from the concept that owns it. The base column is the whole subject.

Formula sheet
QuantityRelationBase
Profit %(SP − CP)/CP × 100CP
Loss %(CP − SP)/CP × 100CP
SP from profitCP × (1 + profit%/100)CP
SP from discountMP × (1 − d%/100)MP
Successive discountsx + y − xy/100MP, then reduced MP
False-weight gain %error / (true − error) × 100weight actually given
Equal SP at ±x%net loss = x²/100 %combined CP

Recap

Read only this the night before.

CP base
Profit and loss percent divide by CP, never by SP. Fix CP = 100 when the question is all percentages.
MP base
Discount comes off MP. Markup and discount on different bases do not subtract to the profit percent.
Successive
net = x + y − xy/100, always less than x + y. Multiply the remaining factors to check.
False weight
Gain % = error/(true − error) × 100. A 10% short at cost price is an 11 1/9 % gain.
Equal SP trap
Same SP at +x% and −x% is a net loss of x²/100 %. Never break-even.

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