Profit and Loss: CP, SP, Overheads and Two-Article Problems
Profit and Loss: CP, SP, Overheads and Two-Article Problems
Profit and loss is percentage applied to buying and selling, and profit percent is ALWAYS on the cost price unless the question says otherwise. NTPC 2025-26 papers asked overhead costs, two articles at the same selling price, two items with a combined cost, and "selling at a fraction of the price" twists. All of them reduce to the multiplying-factor method below.
1. Formula Box
Profit = SP − CP Loss = CP − SP Profit % = Profit/CP × 100
SP = CP × (100 + Profit %)/100 SP = CP × (100 − Loss %)/100
CP = SP × 100/(100 + Profit %) CP = SP × 100/(100 − Loss %)
| Term | Meaning | Note |
|---|---|---|
| CP (cost price) | Total money spent to own the article | Includes repair, transport, loading, registration (overheads) |
| SP (selling price) | Money received on selling | Profit or loss is measured against CP |
| Overheads | Any expense before the sale | Add to CP before computing profit % |
| Profit factor | (100 + P)/100 | 20% profit means SP = 1.2 × CP |
2. Shortcut Table for Standard Situations
| Situation | Result |
|---|---|
| Bought x for ₹a, sold y for ₹b | Profit % = [(b/y − a/x)/(a/x)] × 100 |
| CP of x articles = SP of y articles | Profit % = (x − y)/y × 100 (loss if y > x) |
| Gain equals the SP of k articles out of n sold | Profit % = k/(n − k) × 100 |
| Two articles at the same SP, r% gain on one and r% loss on the other | Always a loss of r²/100 % |
| Same SP, different gain and loss percents | Find each CP separately and compare totals |
| Combined CP T, loss a% on one and gain b% on the other, net gain g% | a × x = b × (T − x) − g × T, where x is the CP of the loss-making item |
| Selling at a fraction f of the usual SP gives profit or loss p% | Usual SP = CP × (100 ± p)/(100 × f) |
| Chain A → B → C | Final price = CP × product of all factors |
✗ Gain 10% on one and lose 10% on the other at the same SP, so no profit no loss | ✓ There is always a loss of 10²/100 = 1%, because the loss-making article cost more
✗ Profit % = profit/SP × 100 | ✓ Profit % is on CP unless the question explicitly says "on selling price"
हिंदी नोट: लाभ प्रतिशत हमेशा क्रय मूल्य (CP) पर निकाला जाता है। मरम्मत, ढुलाई और लदाई जैसे खर्च CP में जोड़कर ही लाभ प्रतिशत निकालिए। एक ही विक्रय मूल्य पर r% लाभ और r% हानि हो तो कुल r²/100 % हानि होती है।
Exam Pointer: Verified NTPC patterns: repair cost as a percent of CP with an overall loss (June 2022 CBT-2), SP of x items equal to CP of y items (June 2022 CBT-2), two articles at the same SP with unequal gain and loss (June 2025 Graduate CBT-1), new SP for a target profit from a known loss (June 2025 Graduate CBT-1, June 2026 UG CBT-1), two items with a combined CP that break even (August 2025 UG CBT-1) and selling at a fraction of the usual price (August 2025 UG CBT-1). Bought-at-one-rate and chain-selling questions are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Profit or loss with overheads (repair, transport)
[PYQ: NTPC CBT-2 14-Jun-2022 Shift-2]
EXAM LEVEL
Q. A man buys an old car for ₹2,40,000 and spends 15% of this amount on repairs. He sells the car at a loss of 5% on his total cost. Find the selling price.
Total CP = 2,40,000 × 1.15 = ₹2,76,000. SP = 2,76,000 × 0.95 = ₹2,62,200. Taking the 5% loss on ₹2,40,000 alone ignores the repair cost, which is part of CP.
Answer: ₹2,62,200
EXAMATLAS LEVEL
Q. A dealer buys 100 kg of rice at ₹40 per kg and pays ₹500 for transport and loading. 10 kg gets damaged and is sold at ₹27 per kg. At what rate per kg must he sell the rest to make a 20% profit on the whole deal?
Total CP = 4,000 + 500 = ₹4,500. Target SP = 4,500 × 1.2 = ₹5,400. Damaged rice brings 10 × 27 = ₹270, so the remaining 90 kg must bring 5,400 − 270 = ₹5,130, that is 5,130 ÷ 90 = ₹57 per kg.
Answer: ₹57 per kg
Pattern 2: Bought at one rate, sold at another
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Bananas are bought at 6 for ₹10 and sold at 4 for ₹10. Find the profit percent.
Make the rupee amount equal and compare quantities: for ₹10 he buys 6 and sells 4. Profit % = (6 − 4)/4 × 100 = 50%. Unit check: CP per banana 10/6, SP 10/4, profit = (10/4 − 10/6)/(10/6) = 1/2.
Answer: 50%
EXAMATLAS LEVEL
Q. A vendor buys lemons at 15 for ₹40 and sells them at 12 for ₹36. Find his profit percent and the number of lemons he must sell to earn a profit of ₹120.
CP per lemon = 40/15 = ₹8/3, SP per lemon = 36/12 = ₹3. Profit per lemon = 3 − 8/3 = ₹1/3. Profit % = (1/3)/(8/3) × 100 = 12.5%. For ₹120 profit he must sell 120 ÷ 1/3 = 360 lemons.
Answer: 12.5%; 360 lemons
Pattern 3: Cost price of x articles equals selling price of y articles
[PYQ: NTPC CBT-2 13-Jun-2022 Shift-1]
EXAM LEVEL
Q. The cost price of 20 articles equals the selling price of 16 articles. Find the profit percent.
Let each article cost ₹1. Then 16 articles sell for ₹20, so SP of one = 20/16 = 1.25. Profit = 25%. Formula: (20 − 16)/16 × 100 = 25%.
Answer: 25%
EXAMATLAS LEVEL
Q. A shopkeeper sells 33 metres of cloth and gains the selling price of 11 metres. Find his gain percent.
SP of 33 m = CP of 33 m + SP of 11 m, so SP of 22 m = CP of 33 m. By the previous rule, gain = (33 − 22)/22 × 100 = 50%. Writing 11/33 = 33⅓% treats the gain as a fraction of SP, which is the trap.
Answer: 50%
Pattern 4: Two articles at the same selling price, gain on one and loss on the other
[PYQ: NTPC Graduate CBT-1 5-Jun-2025 Shift-1]
EXAM LEVEL
Q. Two mobile phones are sold for ₹9,900 each, one at a 10% gain and the other at a 10% loss. Find the overall gain or loss.
CP of the first = 9,900/1.1 = ₹9,000; CP of the second = 9,900/0.9 = ₹11,000. Total CP = ₹20,000, total SP = ₹19,800, so a loss of ₹200, which is 1% (= 10²/100).
Answer: Loss of ₹200 (1%)
EXAMATLAS LEVEL
Q. A man sells two bicycles for ₹3,600 each, gaining 20% on one and losing 10% on the other. Find his overall gain or loss percent.
CPs: 3,600/1.2 = ₹3,000 and 3,600/0.9 = ₹4,000. Total CP = ₹7,000, total SP = ₹7,200, gain = ₹200. Gain % = 200/7,000 × 100 = 2 6/7%. Averaging +20 and −10 to +5% is wrong because the CPs differ.
Answer: Gain of 2 6/7%
Pattern 5: Two selling prices: new price for a target profit, profit vs loss
[PYQ: NTPC Graduate CBT-1 10-Jun-2025 Shift-1 | NTPC UG CBT-1 14-Jun-2026 Shift-1]
EXAM LEVEL
Q. By selling an article for ₹840, a shopkeeper gains 20%. At what price should he sell it to gain 35%?
CP = 840/1.2 = ₹700. New SP = 700 × 1.35 = ₹945. Direct ratio: 840 × 135/120 = 945.
Answer: ₹945
EXAMATLAS LEVEL
Q. The profit earned by selling an article for ₹1,320 is 20% more than the loss incurred by selling it for ₹1,034. Find the cost price.
Let CP = c. Profit = 1,320 − c and loss = c − 1,034. Given 1,320 − c = 1.2(c − 1,034) = 1.2c − 1,240.8, so 2.2c = 2,560.8 and c = ₹1,164. Check: profit 156, loss 130, and 156 = 1.2 × 130.
Answer: ₹1,164
Pattern 6: Two items with a combined cost: loss on one, gain on the other
[PYQ: NTPC UG CBT-1 21-Aug-2025 Shift-2]
EXAM LEVEL
Q. Two watches together cost ₹1,500. One is sold at a 10% loss and the other at a 20% gain, and there is neither gain nor loss overall. Find the cost price of the watch sold at a loss.
Let it cost x. Loss = 0.1x must equal gain = 0.2(1,500 − x), so 0.1x = 300 − 0.2x, 0.3x = 300 and x = ₹1,000. The other cost ₹500. Ratio shortcut: CPs are in the inverse ratio of the percentages, 20 : 10 = 2 : 1.
Answer: ₹1,000
EXAMATLAS LEVEL
Q. Two articles together cost ₹4,000. One is sold at a 12% loss and the other at an 18% gain, and the overall gain is 6%. Find the cost price of each.
Let the loss-making article cost x. Net gain = −0.12x + 0.18(4,000 − x) = 720 − 0.3x, and this must equal 6% of 4,000 = 240. So 0.3x = 480 and x = ₹1,600; the other costs ₹2,400. Check: loss 192, gain 432, net 240.
Answer: ₹1,600 and ₹2,400
Pattern 7: Selling at a fraction of the usual selling price
[PYQ: NTPC UG CBT-1 21-Aug-2025 Shift-3]
EXAM LEVEL
Q. By selling an article at 3/4 of its usual selling price, a shopkeeper loses 10%. Find his profit percent at the usual selling price.
(3/4) × SP = 0.9 × CP, so SP = 1.2 × CP. Profit at the usual price = 20%.
Answer: 20%
EXAMATLAS LEVEL
Q. If an article is sold at 85% of its usual selling price, the profit is 2%. Find the profit or loss percent if it is sold at 70% of its usual selling price.
0.85 × SP = 1.02 × CP gives SP = 1.2 × CP. At 70%: 0.7 × 1.2 × CP = 0.84 × CP, a loss of 16%. Subtracting 15 points from 2% to get −13% is the trap; percentages of SP and CP are on different bases.
Answer: 16% loss
Pattern 8: Chain selling (A to B to C)
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A sells an article to B at a 20% profit, and B sells it to C at a 25% profit. If C pays ₹1,500, find A's cost price.
1,500 = CP × 1.2 × 1.25 = 1.5 × CP, so A's CP = ₹1,000.
Answer: ₹1,000
EXAMATLAS LEVEL
Q. A sells a watch to B at a 10% loss, B sells it to C at a 20% profit, and C sells it to D at a 10% profit for ₹2,970. Find A's cost price.
Multiply the factors: 0.9 × 1.2 × 1.1 = 1.188. A's CP = 2,970 ÷ 1.188 = ₹2,500. Check the links: 2,500 → 2,250 → 2,700 → 2,970.
Answer: ₹2,500
60-Second Revision
- Profit % is on CP; overheads (repair, transport) are part of CP.
- CP of x = SP of y gives (x − y)/y × 100; "gain = SP of k" means SP of (n − k) = CP of n.
- Same SP with ±r%: always a loss of r²/100 %; unequal percents: find both CPs.
- Combined CP with loss and gain: gain − loss = net gain in rupees (gain = loss when there is no profit, no loss).
- Fraction f of SP gives p%: SP = CP × (100 ± p)/(100f).
- Chains: multiply all the factors.