Profit and loss for AFCAT: cost price, marked price, discounts and false weights

Profit and loss sounds like a separate chapter, but it is really percentage with a shop attached. The cost price is the base, the selling price is the changed value, and a discount is a percentage cut on a different base, the marked price. Once you keep those two bases apart, the whole topic becomes a handful of multipliers.

AFCAT asks this topic in short, one-step or two-step questions. That makes it a good place to pick up marks quickly, as long as you avoid the one error that costs most candidates: taking the percentage on the wrong price.

The terms and their bases

• Profit % = profit ÷ CP × 100. Loss % = loss ÷ CP × 100.
• Discount % = discount ÷ MP × 100.

Unless a question clearly says "profit on selling price", profit is always a percentage of cost.

The multiplier method

A profit of p% means SP is (100 + p)% of CP. A loss of l% means SP is (100 − l)% of CP.

• SP = CP × (1 + p/100) for a profit.
• SP = CP × (1 − l/100) for a loss.
• CP = SP ÷ the same multiplier, when SP is given.

So an item sold for ₹920 at a loss of 8% has CP = 920 ÷ 0.92 = ₹1,000. No equation needed.

Why this matters: most questions give you one price and one percentage and ask for another price. With the multiplier you either multiply or divide, and you know which by asking "am I going from cost to selling, or back?"

Marked price and discount together

A seller who marks goods m% above cost and then gives a d% discount has

SP = CP × (1 + m/100) × (1 − d/100)

These are two successive percentage changes on the same item, so the net profit % follows the successive-change rule: m − d − (m × d)/100.

Marking up 40% and then giving 25% off: 40 − 25 − 1,000/100 = 40 − 25 − 10 = 5% profit. By multipliers, 1.4 × 0.75 = 1.05.

Successive discounts work the same way. Discounts of 20% and 10% equal a single discount of 20 + 10 − (20 × 10)/100 = 28%, not 30%. The second discount is taken on an already reduced price.

Three shortcuts worth knowing

1. Same selling price, same percentage gain and loss. If two items are sold at the same price, one at x% profit and the other at x% loss, the seller always makes an overall loss of x²/100 %. The loss-making item had the higher cost, so the loss in rupees is bigger than the gain.

2. False weights. A seller who claims to sell at cost price but gives less than the true weight gains

gain % = (true weight − false weight) ÷ false weight × 100

The base is the false weight, because that is what the seller actually paid for.

3. Buying and selling in counts. If the selling price of a items equals the cost price of b items, the profit % = (b − a) ÷ a × 100. The seller receives the cost of b items for giving away only a.

Worked examples

Example 1 (cost from selling price). An item sold for ₹920 gives a loss of 8%. At what price should it be sold to gain 12%?

• CP = 920 ÷ 0.92 = ₹1,000.
• New SP = 1,000 × 1.12 = ₹1,120.

Example 2 (mark-up and discount). A shopkeeper marks goods 40% above cost and allows a 25% discount. Find the profit %.

• Multipliers: 1.4 × 0.75 = 1.05.
• Profit = 5%.

Example 3 (the same-price trap). Two phones are sold for ₹990 each, one at a 10% gain and the other at a 10% loss. Find the overall gain or loss %.

• Shortcut: loss of 10²/100 = 1%.
• Check: CPs are 990 ÷ 1.1 = ₹900 and 990 ÷ 0.9 = ₹1,100, total ₹2,000. Total SP = ₹1,980. Loss = ₹20, which is 1% of ₹2,000.

Example 4 (false weight with a mark-up). A trader sells at 10% above cost price and also uses a 900 g weight in place of 1 kg. Find the real gain %.

• The customer pays for 1,000 g at 1.1 times cost, but receives only 900 g.
• Gain factor = 1.1 × 1,000/900 = 1.2222…
• Gain = 22.22% (22 2/9 %).
• With honest weights the gain would be 10%, and with the false weight alone it would be 100 ÷ 900 × 100 = 11.11%. The two combine by multiplying, not adding.

Example 5 (counts). A vendor buys pens at 12 for ₹10 and sells them at 10 for ₹12. Find the profit %.

• CP per pen = 10/12 = ₹5/6. SP per pen = 12/10 = ₹6/5.
• SP ÷ CP = (6/5) ÷ (5/6) = 36/25 = 1.44.
• Profit = 44%.

Example 6 (finding the marked price). An article costs ₹600. What price must be marked so that the seller can give a 20% discount and still make a 20% profit?

• Required SP = 600 × 1.2 = ₹720.
• SP is 80% of MP, so MP = 720 ÷ 0.8 = ₹900.

Profit on selling price

Occasionally a question says "a profit of 20% on the selling price". Convert it: if profit is 20 out of every 100 of SP, the CP is 80, so profit on cost is 20/80 = 25%. In general, r% on SP equals r ÷ (100 − r) × 100 % on CP. This is the same "more than / less than" conversion from the percentage chapter.

Common mistakes

• Profit on the wrong base. A profit of ₹100 on a selling price of ₹500 is 25% (on cost ₹400), not 20%.
• Adding discounts. 20% and 10% successive discounts are 28%, not 30%.
• Assuming equal gain and loss cancel. At the same selling price they always leave a small loss.
• Using the true weight as the base in false-weight questions. Divide by the weight the customer actually receives.

Practice set

1. CP ₹250, SP ₹300. Find the profit %.
2. The marked price is ₹1,500 and the discount is 10%. Find the SP.
3. CP ₹800 and profit 15%. Find the SP.
4. An item is sold for ₹540 at a 10% loss. Find the CP.
5. The selling price of 12 articles equals the cost price of 15 articles. Find the profit %.
6. Find the single discount equal to two successive discounts of 10% each.
7. By selling an item for ₹450, a trader loses 10%. At what price must it be sold to gain 20%?
8. A shopkeeper marks goods 25% above cost and gives a 12% discount. Find the profit %.

Answers:

1. 20%. Profit ₹50 on ₹250.
2. ₹1,350. 1,500 × 0.9.
3. ₹920. 800 × 1.15.
4. ₹600. 540 ÷ 0.9.
5. 25%. (15 − 12) ÷ 12 × 100.
6. 19%. 10 + 10 − 100/100.
7. ₹600. CP = 450 ÷ 0.9 = ₹500, and 500 × 1.2 = ₹600.
8. 10%. 1.25 × 0.88 = 1.10.

What to do next

• Write the base (CP or MP) next to every percentage in the next 20 questions you solve.
• Memorise the three shortcuts: same-price loss of x²/100, false weight on the false weight, and counts (b − a)/a.
• Revise percentage successive changes, then move on to simple interest, which is percentage applied to a principal.