1. Cost, Selling & Marked Price
Three prices, three roles — mixing them up is the single most common source of wrong answers in this chapter:
Cost Price (CP): what the seller paid to acquire the item
Marked Price (MP): the "sticker" price, before any discount
Selling Price (SP): what the buyer actually pays, after discount
Relationship:
SP = MP - Discount
Profit = SP - CP (if positive; Loss = CP - SP if negative)
Profit % is ALWAYS calculated on CP, never on SP or MPThat last line is the rule most worth burning in: profit and loss percentages are always relative to cost price. A question stating "20% profit" means SP = CP × 1.20, full stop — never SP = MP × 1.20 or anything computed from the selling price itself.
An item has CP = 400. It's marked up to MP = 600, then sold
at a 20% discount on MP.
SP = MP - 20% of MP = 600 - 120 = 480
Profit = SP - CP = 480 - 400 = 80
Profit % = 80/400 x 100 = 20% (on CP, per the rule above)2. Successive Discounts
"30% off, plus an extra 10% off" is Week 3's successive-percentage-change formula, applied to two discounts — and just like that formula, the discounts don't simply add.
A jacket marked at 2,000 gets "30% off, then an extra 10% off."
WRONG: treating this as a flat 40% off
2,000 x 0.60 = 1,200 -- INCORRECT
RIGHT: apply Week 3's successive-change formula
a = -30, b = -10
Net % = -30 + (-10) + (-30 x -10)/100 = -40 + 3 = -37%
2,000 x (1 - 0.37) = 2,000 x 0.63 = 1,260
The two discounts together are equivalent to a single 37%
discount, NOT 40% -- the store keeps 3% more than the naive
sum suggests.This is exactly why "30% off, extra 10% off" (37% effective) is genuinely less generous than an advertised flat "37% off" would sound, and less generous than a naive shopper's mental math of "that's basically 40% off." Retailers use successive discounts partly because the compounding effect keeps the true discount smaller than it appears.
3. The False-Average Trap
A shopkeeper sells two items at the same selling price — one at a 20% profit, one at a 20% loss. Total effect: a net profit, a net loss, or exactly break-even? The intuitive "20% profit and 20% loss cancel out" answer is wrong.
Both items sold at SP = 120 each.
Item A: sold at 20% PROFIT -> CP_A = 120 / 1.20 = 100
Item B: sold at 20% LOSS -> CP_B = 120 / 0.80 = 150
Total CP = 100 + 150 = 250
Total SP = 120 + 120 = 240
Net result: 240 - 250 = -10 -> a LOSS of 10, not break-even
Loss % = 10/250 x 100 = 4%
The general shortcut, worth deriving once: whenever two items are sold at the
same selling price, one at x% profit and one at
x% loss, there's always a net loss of
x²/100 percent — never a break-even, and never a profit. Here,
20²/100 = 4%, matching the worked answer exactly. This mirrors Week 3's successive-
percentage-change asymmetry: the loss is calculated relative to a larger CP than the
profit's CP, so the two never truly offset.
If the question instead gives you equal COST prices with one profit and one loss, the two percentages genuinely do partially offset differently — don't reach for x²/100 automatically. Check which value is held constant (SP or CP) before picking a shortcut; the two setups are not interchangeable.
4. Dishonest Dealer & Faulty Weight
A shopkeeper claims to sell "at cost price" but uses a weight of 900g instead of a true 1kg. What's their actual profit percentage? This looks like a new chapter but is really Week 3's percentage machinery wearing a disguise.
Profit % = (True Weight - False Weight) / False Weight x 100
Worked: claims 1000g (1kg), actually gives 900g
Profit % = (1000 - 900) / 900 x 100
= 100/900 x 100
= 11.11%
The dealer earns 11.11% profit despite "selling at cost" --
because they charged for 1000g while delivering only 900g.The intuition: the dealer collects payment for the full (true) weight but their actual cost is only for the (smaller) false weight they hand over — so the "profit" is exactly the value of the shortchanged amount, expressed as a percentage of what they actually gave out. This is structurally identical to Section 1's CP/SP relationship, with "false weight given" playing the role of CP and "true weight charged for" playing the role of SP.
5. Practice Problems
30 problems, no calculator — Easy, Medium & Tough
Identify which price (CP, SP, or MP) each percentage is relative to before calculating. Easy problems should take under 30 seconds each; Medium under 60 seconds; Tough under 90 seconds.
Easy (1–10)
- An article is bought for Rs. 200 and sold for Rs. 250. Find the profit percentage.
- An article is bought for Rs. 500 and sold for Rs. 450. Find the loss percentage.
- Find the selling price of an article bought for Rs. 800 at a 25% profit.
- Find the selling price of an article bought for Rs. 600 at a 10% loss.
- Find the cost price of an article sold for Rs. 660 at a 10% profit.
- Find the cost price of an article sold for Rs. 480 at a 20% loss.
- A marked price of Rs. 500 gets a 10% discount. Find the selling price.
- Find the discount on an item marked Rs. 1,200 with a 15% discount.
- If CP = Rs. 150 and profit = Rs. 30, find the profit percentage.
- If CP = Rs. 400 and loss = Rs. 40, find the loss percentage.
1) (250-200)/200×100=25%. 2) (500-450)/500×100=10%. 3) 800×1.25=1,000. 4) 600×0.9=540. 5) 660/1.1=600. 6) 480/0.8=600. 7) 500×0.9=450. 8) 1200×0.15=180. 9) 30/150×100=20%. 10) 40/400×100=10%.
Medium (11–20)
- A shopkeeper marks an item 25% above a cost price of Rs. 800, then gives a 10% discount. Find the selling price and the profit percentage.
- A shopkeeper marks his goods 40% above cost price and allows a discount of 10%. Find his profit percentage.
- Successive discounts of 20% and 10% are offered on an item. Find the effective discount percentage.
- A trader professes to sell at cost price but uses a weight of 900g for 1kg. Find his profit percentage.
- By selling an article for Rs. 690, a man loses 8%. Find the cost price.
- A shopkeeper bought 100 articles for Rs. 4,000 and sold them at Rs. 48 each. Find the profit percentage.
- Find the single discount equivalent to two successive discounts of 25% and 20%.
- The profit made by selling an article for Rs. 832 equals the loss made by selling it for Rs. 448. Find the cost price.
- A dealer sold two items at Rs. 1,200 each — one at 20% profit and one at 20% loss. Find the overall profit or loss percentage.
- A shopkeeper allows a 10% discount on the marked price and still gains 8%. If the cost price is Rs. 450, find the marked price.
11) MP=800×1.25=1,000; SP=1000×0.9=900; profit=100/800×100=12.5%. 12) MP=1.4×CP; SP=1.26×CP → profit=26%. 13) -20-10+(-20×-10)/100=-30+2=-28%. 14) (1000-900)/900×100=11.11%. 15) 690/0.92=750. 16) SP=4800, profit=(4800-4000)/4000×100=20%. 17) -25-20+(-25×-20)/100=-45+5=-40%. 18) CP=(832+448)/2=640. 19) Same-SP shortcut: loss=20²/100=4%. 20) SP=450×1.08=486; MP×0.9=486 → MP=540.
Tough (21–30)
- A man buys two horses for Rs. 21,000 in total. He sells one at a 20% profit and the other at a 10% loss, and on the whole transaction, makes neither profit nor loss. Find the cost price of each horse.
- A shopkeeper allows successive discounts of 20% and 10% on the marked price and still makes a profit of 8%. If the cost price is Rs. 900, find the marked price.
- Marked price of an article is 60% above its cost price. Find the maximum discount percentage the shopkeeper can offer while still making a 20% profit.
- A trader marks up his goods by x% and then gives a discount of x%. If the resulting selling price is 84% of the cost price, find x.
- A retailer buys 40 pens at the marked price of 36 pens from a wholesaler. If he sells these pens giving a discount of 1%, find his profit percentage.
- Two successive discounts on a marked price of Rs. 4,000 are equivalent to a single discount of 32%. If the first discount is 20%, find the second discount.
- A shopkeeper sold an article at a loss of 10%. Had he sold it for Rs. 90 more, he would have gained 8%. Find the cost price.
- A merchant marks his goods 50% above cost price and then offers a discount. If he still wants to make a 20% profit, find the discount percentage he can offer.
- A dishonest dealer sells goods at cost price but uses a weight which is x% less than the true weight, gaining 25% overall. Find x.
- An article's cost price is 80% of its selling price. Find the profit percentage.
21) 0.2x=0.1(21000-x) → 0.3x=2100 → x=7,000 and 14,000. 22) MP×0.8×0.9=972 → MP×0.72=972 → MP=1,350. 23) MP=1.6CP; SP needed=1.2CP; discount=(1.6-1.2)/1.6×100=25%. 24) 100(1-(x/100)²)=84 → (x/100)²=0.16 → x=40. 25) CP for 36 pens=36m; receives 40, sells at 0.99m each=39.6m; profit=3.6/36×100=10%. 26) 80×(1-d/100)=69.5%×80/80... solve: -20+b+(-20b)/100=-32 → 0.8b=-12 → b=-15, second discount=15%. 27) CP×0.18=90 → CP=500. 28) MP=1.5CP; SP needed=1.2CP; discount=(1.5-1.2)/1.5×100=20%. 29) x/(100-x)=0.25 → x=20. 30) CP=0.8SP; profit=0.2SP; profit%=0.2SP/0.8SP×100=25%.
6. Knowledge Check
Four quick questions. Expand each to check your answer.
Q1
Why is profit/loss percentage always calculated relative to cost price, never to selling price or marked price?
Why is profit/loss percentage always calculated relative to cost price, never to selling price or marked price?
This is a fixed convention in how the terms "profit percentage" and "loss percentage" are defined — they specifically measure the gain or loss relative to what the seller originally paid (cost price), which is the seller's true reference point for whether the transaction was worthwhile. Using SP or MP as the base would answer a different, less meaningful question.
Q2
Why does "30% off, then an extra 10% off" work out to a 37% effective discount rather than a flat 40%?
Why does "30% off, then an extra 10% off" work out to a 37% effective discount rather than a flat 40%?
The second 10% discount is applied to the already-reduced price (after the first 30% off), which is a smaller number than the original price — so it removes less in absolute terms than a flat 10% of the original would. This is the same successive-percentage-change effect from Week 3, applied to discounts instead of general increases and decreases.
Q3
When two items are sold at the same selling price, one at x% profit and one at x% loss, why is the net result always a loss rather than break-even?
When two items are sold at the same selling price, one at x% profit and one at x% loss, why is the net result always a loss rather than break-even?
The item sold at a loss must have had a HIGHER cost price than the one sold at a profit, since both ended up at the same selling price. That higher-cost item's x% loss is therefore a larger absolute amount than the profitable item's x% gain, so the two never fully cancel — the net result is always a loss of exactly x²/100 percent.
Q4
Why does a dealer who charges for 1000g while actually giving only 900g earn a profit, even while claiming to "sell at cost price"?
Why does a dealer who charges for 1000g while actually giving only 900g earn a profit, even while claiming to "sell at cost price"?
The dealer's true cost is only for the 900g actually handed over, but they collect payment as if 1000g were delivered — the 100g shortfall is pure extra revenue relative to what was actually given, even though the price-per-gram matched their cost price exactly. "Selling at cost" was only true for the amount claimed, not the amount actually delivered.