Section 6, arithmetic

Profit, Loss & Discount

Profit-loss is one of CAT’s most reliable sources: most years carry a markup-discount chain, a two-scenario question or a buy-and-sell chain. All three reduce to multiplying factors on an assumed cost price of 100.

How CAT asks it

A shop marks up and discounts; a trader sells at a loss and would have gained by selling for more; goods pass from A to B to C and sometimes back. Offers (“buy 3 get 1 free”), a dishonest dealer’s light weights and part-stocks sold at different margins also recur. Recent papers add cost structures (fixed + per-unit) and price–demand trade-offs.

Concepts you must know

  • Profit = SP − CP. Profit% is on CP unless stated: (SP − CP) ÷ CP × 100.
  • Discount = MP − SP. Discount% is on MP (marked price).
  • SP = CP × (1 + profit%/100) = MP × (1 − discount%/100).
  • Markup m% then discount d% gives profit% = m − d − md/100.
  • Successive discounts d₁ and d₂ equal one discount of d₁ + d₂ − d₁d₂/100.
  • “Buy x, get y free” is a discount of y ÷ (x + y) on the goods taken.
  • Dishonest dealer charging for the true weight but giving less: profit% = (true − given) ÷ given × 100.
  • Two items sold at the same SP, one at x% profit and one at x% loss, always give an overall loss of x²/100 percent.
  • A profit of p% on SP equals p ÷ (100 − p) × 100% on CP.
  • The relation that links markup, discount and profit:
MPCP=100+profit %100−discount %\frac{MP}{CP} = \frac{100 + \text{profit \%}}{100 - \text{discount \%}}

Thought process

  1. Assume CP = 100 unless real numbers are given.
  2. Write the chain CP → MP → SP with a factor on each arrow.
  3. In two-scenario questions, subtract: the gap between the two selling prices equals the gap between the two profit percentages, applied to CP.
  4. In chains of people, track one person’s cash: money in minus money out.

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Markup + discount: find the profit, or the markup needed for a target profit (the MP/CP formula).
  2. 2Two scenarios: “8% loss at one price; 10% profit at ₹360 more” means 18% of CP = 360.
  3. 3Chains and buy-backs: A sells to B, B to C, or B sells back to A. Multiply factors, or track cash.
  4. 4Offer arithmetic: successive discounts, “buy n get m free”, a discount plus extra quantity.
  5. 5Dishonest dealer: false weights, sometimes combined with a markup or a discount.
  6. 6Partial sales: part of the stock at a profit and the rest at a loss. Use a weighted average or alligation.
  7. 7Same SP, opposite margins: always a net loss of x²/100 percent.
  8. 8Costs and demand: fixed + variable cost and break-even; price cuts that raise sales, then maximise revenue or profit (a quadratic, Section 13).

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1, markup + discount

    A shopkeeper marks goods 40% above cost and gives a 15% discount. What is his profit percent?

    Show solution

    1.40 × 0.85 = 1.19. Answer: 19%.

  2. Example 2, required markup

    A trader wants a 20% profit after giving a 20% discount. How far above cost should he mark his goods?

    Show solution

    MP/CP = 120/80 = 1.5. Answer: 50% above cost.

  3. Example 3, two scenarios

    A seller loses 8% at a certain price. Had he sold for ₹360 more, he would have gained 10%. Find the cost price.

    Show solution

    Moving from −8% to +10% is 18% of CP = 360. Answer: ₹2,000.

  4. Example 4, buy-back chain

    Asha buys a bicycle for ₹5,000 and sells it to Bina at a 20% profit. Bina sells it back to Asha at a 10% loss, and Asha then sells it to Charu for ₹6,000. Find Asha’s total gain and her profit percent on the final sale.

    Show solution

    Bina’s buy-back price is 6,000 × 0.9 = 5,400. Asha’s cash: −5,000 + 6,000 − 5,400 + 6,000 = +1,600. The final sale earned 600 on 5,400. Answer: ₹1,600 in total; 11.11% on the final sale.

  5. Example 5, offer + profit

    Shirts have a marked price of ₹1,000. The store offers “buy 4, get 1 free” plus 10% off the bill, and still makes a 20% profit. Find the cost price of one shirt.

    Show solution

    5 shirts cost the customer 4 × 1,000 × 0.9 = 3,600, so SP per shirt = 720. CP = 720 ÷ 1.2. Answer: ₹600.

  6. Example 6, dishonest dealer

    A grocer gives 900 g for every 1 kg he charges for, and also prices his goods 10% above cost. What is his profit percent?

    Show solution

    Factor = 1.1 × 1000/900 = 1.2222. Answer: 22.22%. Without the markup it would be 100/900 = 11.11%.

  7. Example 7, partial sale

    A trader sells one-third of his stock at a 20% profit and the rest at a 5% loss. What is the overall result?

    Show solution

    (1/3)(20) + (2/3)(−5) = 6.67 − 3.33. Answer: 3.33% profit.

  8. Example 8, same SP

    Two phones sell at ₹9,900 each, one at a 10% profit and the other at a 10% loss. What is the overall result?

    Show solution

    Loss = 10²/100 = 1%. Check: the CPs are 9,000 and 11,000 (total 20,000) against SPs totalling 19,800. Answer: 1% loss (₹200).

  9. Example 9, price–demand

    A café sells 200 cups a day at ₹50, and every ₹1 cut in price adds 10 cups a day. Which price maximises revenue?

    Show solution

    With a cut of x rupees, revenue = (50 − x)(200 + 10x) = 10(50 − x)(20 + x). The two brackets always add to 70, so their product peaks when they are equal: 50 − x = 20 + x, giving x = 15. Answer: ₹35 (350 cups, revenue ₹12,250).

Traps

  • Taking profit% on SP, or discount% on CP.
  • Adding successive discounts: 20% then 10% is 28%, not 30%.
  • Double-counting in buy-back chains; track one person’s cash instead.
  • “Buy 3 get 1 free” is 25% off, not 33.33%.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    MP is 25% above CP and the discount is 12%. What is the profit percent?

    Show answer

    10%: 1.25 × 0.88 = 1.10.

  2. 2

    A trader makes 25% profit after a 10% discount. What would his profit be at a 20% discount?

    Show answer

    11.11%: MP/CP = 125/90, and 125/90 × 0.8 = 100/90.

  3. 3

    Selling at ₹720 gives a 10% loss. At what price should it be sold for a 25% profit?

    Show answer

    ₹1,000: CP = 800.

  4. 4

    What single discount equals successive discounts of 20% and 10%?

    Show answer

    28%.

  5. 5

    What is the effective discount in “buy 3, get 2 free”?

    Show answer

    40%: 2/5.

  6. 6

    A dealer sells at a 10% loss but gives 800 g for every 1 kg. What is his profit or loss percent?

    Show answer

    12.5% profit: 0.9 × 1000/800 = 1.125.

  7. 7

    A man buys 12 pens for ₹10 and sells 10 pens for ₹12. What is his profit percent?

    Show answer

    44%: CP ₹0.833 and SP ₹1.20 per pen, and 1.2 ÷ 0.833 = 1.44.

  8. 8

    Two items with the same CP are sold at a 25% profit and a 15% loss. What is the overall result?

    Show answer

    5% profit: (25 − 15)/2.

  9. 9

    A workshop has fixed costs of ₹20,000 a month plus ₹30 per unit, and sells at ₹50 per unit. How many units must it sell to break even?

    Show answer

    1,000 units: each unit contributes ₹20 towards ₹20,000.

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