Lessons / Quantitative aptitude
Quantitative aptitude · Day 2

Profit and Loss

Profit and loss are always a % of cost price, never of selling price. Put CP = 100 and most questions solve themselves.

Practise (11 questions) Present as slidesCheat sheet: Fraction to percentage table

Why this topic matters

A shopkeeper buys a shirt for Rs 400 and sells it for Rs 500. The Rs 100 extra is profit, and it is 25% because profit is always compared with what he PAID.

Profit and loss has 1 or 2 questions in nearly every paper. Almost every mistake comes from one thing: taking the percentage on the selling price instead of the cost price.

CP, SP, profit and loss

Cost price (CP): what the seller paid. Selling price (SP): what the buyer pays. If SP > CP there is a profit = SP - CP; if SP < CP there is a loss = CP - SP.

Profit % and loss % are ALWAYS calculated on the cost price: profit % = profit / CP × 100.

So SP = CP × (1 + profit%/100), and CP = SP / (1 + profit%/100).

CP400SP400+100
CP 400, SP 500: profit 100 on a base of 400 = 25%
Profit % = (SP - CP)/CP × 100
Always over CP.
SP = CP × (100 + P%)/100
CP = SP × 100/(100 + P%)
Going back from SP.
Loss: SP = CP × (100 - L%)/100
Example 1A watch bought for Rs 1,600 is sold for Rs 2,000. Find the profit percentage.
  1. Profit = 2000 - 1600 = 400.
  2. Profit % = 400/1600 × 100 = 25%.Divide by CP, not SP.
Example 2By selling a bag for Rs 690 a trader gains 15%. What did the bag cost him?
  1. SP = CP × 1.15.A 15% gain means SP is 115% of CP.
  2. CP = 690 / 1.15 = 600.

Same CP or same SP, and 'loss on both'

If two items are sold at the SAME selling price, one at x% profit and the other at x% loss, the seller always loses overall: loss % = x2/100.

Why: the loss-making item cost more than the profit-making one (you need a higher CP to end up at the same SP after a loss), so the loss is on a bigger base.

Example 1Two phones are sold for Rs 9,900 each, one at 10% profit and one at 10% loss. Overall gain or loss?
  1. CP1 = 9900/1.1 = 9000. CP2 = 9900/0.9 = 11000.Go back from SP for each phone.
  2. Total CP 20,000, total SP 19,800.
  3. Loss 200 on 20,000 = 1% (= 102/100).

Dishonest weights and 'error'

A shopkeeper who uses a 900 g weight instead of 1 kg gives 900 g but charges for 1000 g. His gain % = (error)/(true weight given) × 100 = 100/900 × 100 = 11.11%.

If he also marks up or discounts, apply those as successive changes on top.

Example 1A dealer sells at cost price but uses a weight of 960 g for 1 kg. His gain %?
  1. He gives 960 g and is paid for 1000 g.The 40 g difference is his gain.
  2. Gain % = 40/960 × 100.Base is what he actually gives (his cost).
  3. = 4.17%.

Successive profits and the 'take 100' method

A makes 10% selling to B, B makes 20% selling to C. C pays 100 × 1.1 × 1.2 = 132 for every 100 that A paid.

Whenever the question has only percentages, take CP = 100 and follow the money.

Example 1A sells to B at 20% profit, B sells to C at 25% profit. C pays Rs 225. What did A pay?
  1. A's CP × 1.2 × 1.25 = 225.Two successive profits multiply.
  2. 1.2 × 1.25 = 1.5.
  3. A's CP = 225 / 1.5 = 150.

Formula summary

Profit % = (SP - CP)/CP × 100
Base is CP.
SP = CP × (100 + P%)/100
Loss: (100 - L%).
Same SP, x% gain on one, x% loss on other
Always a LOSS of x2/100 %.
CP of a articles = SP of b articles
Profit % = (a - b)/b × 100.
False weight (sells at CP)
Gain % = error / (true - error) × 100. 900 g for 1 kg → 100/900.
Chain A→B→C
Final price = CP × (1 + p1)(1 + p2)...

Shortcuts and tricks

Chain of sales backwards

Use it when: 'A sells to B at p%, B to C at q%, C pays Rs X'.

  1. Divide X by each multiplier in turn.
ExampleA→B at 20% profit, B→C at 25% profit, C pays 1500. A's CP?
  1. 1500 / 1.25 = 1200.
  2. 1200 / 1.2 = 1000.

Dishonest dealer: loss price but false weight

Use it when: Two effects together.

  1. Price paid for 'a kilo' vs what that weight really cost.
  2. Gain = (what he gets)/(what he gave) - 1.
ExampleSells at 10% loss, uses 800 g for 1 kg. Gain %?
  1. Gets 0.9 of CP for goods worth 0.8.
  2. 0.9/0.8 = 1.125.

Common mistakes

Profit % over SP.
Profit % is over CP unless the question says otherwise.
CP = SP - 15% of SP.
CP = SP / 1.15.

Before you move on, you should be able to...

  • compute profit or loss % on the cost price
  • find CP from SP and a percentage
  • explain why equal % profit and loss at the same SP is a loss
  • solve false-weight questions

Practice questions (11)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy CP of 15 pens = SP of 12 pens. Profit %?
  1. (15 - 12)/12 × 100.

Answer: 25%

easy A dealer sells at cost price but uses a 900 g weight for 1 kg. Gain % (2 d.p.)?
  1. 100/900 × 100.

Answer: 11.11%

medium Two cycles are sold at Rs 990 each, one at 10% profit and the other at 10% loss. Net result?
  1. x2/100 = 1% loss.
  2. CPs: 900 and 1100 = 2000. SP 1980. Loss 20.

Answer: 1% loss (Rs 20)

medium By selling an article for Rs 720, a man loses 10%. At what price should he sell to gain 15%?
  1. CP = 720/0.9 = 800.
  2. 800 × 1.15 = 920.

Answer: Rs 920

medium A man buys oranges at 6 for Rs 5 and sells at 5 for Rs 6. Profit %?
  1. CP per orange 5/6, SP per orange 6/5.
  2. (6/5)/(5/6) = 36/25 = 1.44.

Answer: 44%

hard If the SP is doubled, the profit triples. Profit %?
  1. 2S - C = 3(S - C) → S = 2C.
  2. Profit = C = 100%.

Answer: 100%

hard Selling at Rs 1,040 gives the same % profit as the % loss when selling at Rs 880. CP?
  1. Equal % both sides → CP is the average: (1040 + 880)/2.

Answer: Rs 960

hard Two articles sell for Rs 1,200 each, one at 20% gain and one at 20% loss. Net result in rupees?
  1. CPs: 1200/1.2 = 1000 and 1200/0.8 = 1500.
  2. CP 2500, SP 2400.

Answer: Loss of Rs 100

hard A trader cheats 20% when buying (takes 1200 g for 1 kg) and 20% when selling (gives 800 g for 1 kg), and sells at cost price. Gain %?
  1. He pays for 1000 g and gets 1200 g.
  2. He charges for 1000 g but hands over 800 g.
  3. 1200/800 = 1.5.

Answer: 50%

hard Selling 33 m of cloth gains the selling price of 11 m. Gain %?
  1. SP(33) - CP(33) = SP(11), so CP(33) = SP(22).
  2. Gain = 11/22.

Answer: 50%

hard An article is sold at 20% profit. Bought 20% cheaper and sold Rs 75 cheaper, the profit would be 25%. CP?
  1. SP = 1.2C. New: 0.8C × 1.25 = C.
  2. 1.2C - 75 = C → C = 375.

Answer: Rs 375

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
25 minCP, SP, profit %, going back from SP; five quick examples.
20 minSame SP, one profit one loss: the x2/100 loss.
20 minDishonest weights.
15 minSuccessive profits, take 100.
25 minPractice set, timed.
5 minRecap.
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