Profit Loss and Discount

What you will learn

A price question becomes much easier when you first name the amount that represents 100%. In this lesson you will calculate profit and loss on a defined cost, discount on a marked price, and a final result after more than one change. You will also recover an unknown starting price and combine two sales by adding their actual amounts.

This lesson uses the percentage and multiplier methods from Percentages and Successive Changes. Every cost base here is positive. All stated items are sold, and no unstated expenses are added. These are arithmetic models, not a guide to business accounts or tax calculations.

Retrieve before calculating

Try these without looking at the feedback.

1. What decimal multiplier represents a 15% increase?
2. After a 12% decrease, what percentage of the starting amount remains?
3. If ₹468 is 90% of an unknown amount, how do you recover the amount?
4. Are 10% of ₹400 and 10% of ₹600 equal amounts?

Feedback:

1. 100% + 15% = 115%, so multiply by 1.15. Multiplying by 0.15 gives only the increase.
2. 88% remains, so the multiplier is 0.88.
3. Divide ₹468 by 0.90 to obtain ₹520. Check: ₹520 × 0.90 = ₹468.
4. No. The amounts are ₹40 and ₹60. Equal rates need not represent equal money because their bases differ.

If you tried to reverse a decrease by adding the same percentage to the smaller result, revisit the reverse-base method in the previous lesson. We now attach that reasoning to named prices.

Coached block A Cost and the result of a sale

1 Keep the three prices separate

Take one seller's point of view throughout a question.

• Cost price (CP) is the purchase price paid for the item
• Total cost, denoted C in our calculations, is CP plus any additional costs that the question explicitly says to include
• Marked price (MP) is the price listed before a stated discount
• Selling price (SP) is the amount received for the sale in the question

If there are no additional included costs, C = CP. If total cost has already been given, do not add its components again. Delivery or repair costs are included only when the question specifies them. An unknown rent, wage or other expense must not be invented.

For conventional profit or loss percentage in this lesson, the defined total cost C is 100%. Compare SP with C first:

• SP > C: profit = SP − C; profit percentage = (profit ÷ C) × 100%
• SP < C: loss = C − SP; loss percentage = (loss ÷ C) × 100%
• SP = C: neither profit nor loss; the percentage is 0%

The difference is an amount in rupees. The percentage compares that amount with C, so it has no currency unit. C must be greater than zero; a zero cost does not provide a denominator for this conventional percentage.

Worked case 1 Turn a gain amount into a rate

An item has total cost ₹640 and is sold for ₹736. Find the profit amount and the conventional cost-based profit percentage.

Model: C = ₹640 is the 100% base; SP = ₹736. Since ₹736 > ₹640, there is a profit of ₹736 − ₹640 = ₹96. Profit percentage = (96 ÷ 640) × 100% = 15%.

Thus each ₹100 of cost earns ₹15 of profit in this model. Check independently: 10% of ₹640 is ₹64 and 5% is ₹32. Their sum is ₹96, and ₹640 + ₹96 = ₹736.

Error check: ₹96 is not 96%. Dividing by ₹736 would answer a different, explicitly sales-based question.

Worked case 2 Include the stated cost once

A seller purchases an item for ₹720 and pays ₹80 for delivery, which the question includes in its cost. It sells for ₹920. There are no other costs. Find the total cost and profit percentage.

Build the cost ledger before calculating a rate:

• Purchase: ₹720
• Included delivery: ₹80
• Total cost C: ₹720 + ₹80 = ₹800
• Selling price SP: ₹920

Profit = ₹920 − ₹800 = ₹120. Profit percentage = (120 ÷ 800) × 100% = 15%.

Check: ₹800 × 1.15 = ₹920. The sale repays ₹800 of defined cost and leaves ₹120. Delivery is already inside that ₹800. Do not add it a second time or add it to the sale receipt. Omitting it would overstate profit.

2 Use a factor for forward and reverse prices

Suppose a cost-based profit rate is p%. The sale returns the whole cost plus p% of that cost. Therefore SP = C × (1 + p/100). At a loss of l%, SP = C × (1 − l/100).

To recover C, undo the multiplication by dividing by the same factor:

• Profit: C = SP ÷ (1 + p/100)
• Loss: C = SP ÷ (1 − l/100)

Here p and l are the numerical rates, such as 15 or 8. For a reverse loss calculation, require 0 ≤ l < 100. A 100% loss gives SP = 0 for any positive C, so that sale price cannot identify a unique original cost. In these nonnegative-price models, loss cannot exceed 100% of cost. There is no corresponding 100% upper limit on profit.

Worked case 3 Move forward and then reverse a loss

Part A: Find SP when total cost is ₹600 and loss is 15% of cost.

The retained share is 100% − 15% = 85% of ₹600. SP = ₹600 × 0.85 = ₹510. Check by the amount route: loss = ₹600 × 0.15 = ₹90; ₹600 − ₹90 = ₹510. The sale is below the cost, as a loss requires.

Part B: A separate sale is ₹552 at an 8% loss. Find its total cost.

Now ₹552 represents 92% of the unknown cost, not 100%. 0.92 × C = ₹552, so C = ₹552 ÷ 0.92 = ₹600. Check: loss = 8% of ₹600 = ₹48, and ₹600 − ₹48 = ₹552.

Error check: adding 8% of ₹552 gives ₹596.16, which does not restore the cost. The lost 8% was measured on C, not on ₹552.

Checkpoint A Choose the base before doing the arithmetic

Cover the feedback and try both prompts.

1. An item sells for ₹690 at 15% profit on its total cost. Complete: ₹690 is ___% of cost. Find the cost.
2. Purchase price is ₹360 and an included repair costs ₹40. SP is ₹380, with no other costs. Is the result a gain or loss, and what is its percentage?

Feedback:

1. ₹690 is 115% of cost. C = ₹690 ÷ 1.15 = ₹600. Check: 15% of ₹600 is ₹90, so ₹600 + ₹90 = ₹690. Subtracting 15% of ₹690 would use the wrong base.
2. C = ₹360 + ₹40 = ₹400. The sale is ₹20 below cost, so loss = (20 ÷ 400) × 100% = 5%. Comparing only with ₹360 would wrongly report a gain.

If the sign is wrong, rebuild the cost ledger. If the direction is right but the percentage is wrong, circle the denominator and ask whether it is the defined cost.

Coached block B Marked prices and changing bases

3 A discount starts from the marked price

A discount is a reduction from MP. It tells us how much a customer pays below the listed price; it does not by itself tell us the seller's profit or loss. That requires C as well.

For a discount of d%, the amount removed is MP × d/100, and the sale retains (100 − d)% of MP:

• Discount amount = MP − SP
• Discount percentage = ((MP − SP) ÷ MP) × 100%
• SP = MP × (1 − d/100)
• MP = SP ÷ (1 − d/100) for 0 ≤ d < 100

Use MP > 0. A 100% discount produces a zero sale price; reversing that zero cannot reveal a unique MP. Do not divide by a zero factor.

Worked case 4 Discount amount and the remaining price

An item is marked at ₹1,250 with a 12% discount. Find the discount amount and SP.

Model: ₹1,250 is 100%; 12% is removed and 88% remains. Discount amount = ₹1,250 × 0.12 = ₹150. SP = ₹1,250 − ₹150 = ₹1,100. The factor route gives ₹1,250 × 0.88 = ₹1,100 as well.

Check: ₹150 + ₹1,100 = ₹1,250, and ₹150 ÷ ₹1,250 = 0.12. The ₹150 reduction does not establish a ₹150 loss. We have not been given the cost.

Worked case 5 Recover the marked price

An item sells for ₹1,020 after a 15% discount. Find MP.

The ₹1,020 is the remaining 85% of MP. Write the relationship before reversing it: SP = 0.85 × MP, so MP = ₹1,020 ÷ 0.85 = ₹1,200.

Check: 15% of ₹1,200 is ₹180; ₹1,200 − ₹180 = ₹1,020. Also MP exceeds SP, as a positive discount requires.

Error check: adding 15% of ₹1,020 gives ₹1,173. Its 15% discount would not leave ₹1,020. The original discount base was ₹1,200.

4 Follow each price arrow

A cost-based markup means increasing the cost by a stated percentage to set MP. Markup is a price-setting step; the final profit still depends on the price actually received.

With markup m% on cost and discount d% on MP: C → multiply by (1 + m/100) → MP → multiply by (1 − d/100) → SP.

The two rates usually refer to different rupee amounts. Keep both multipliers. A net factor above 1 means profit, below 1 means loss, and equal to 1 means neither, measured against the original C.

Worked case 6 Markup followed by discount

Total cost is ₹800. A seller marks the item 25% above cost, then gives a 12% discount on MP. Find MP, SP and the final cost-based profit.

First arrow, base C = ₹800: MP = ₹800 × 1.25 = ₹1,000. The markup amount is ₹200. Second arrow, base MP = ₹1,000: Discount = ₹1,000 × 0.12 = ₹120. SP = ₹1,000 × 0.88 = ₹880.

Compare the final sale with the original cost: profit = ₹880 − ₹800 = ₹80; profit percentage = (80 ÷ 800) × 100% = 10%.

Check with one combined factor: 1.25 × 0.88 = 1.10, and ₹800 × 1.10 = ₹880. Error check: 25% − 12% = 13% is not the final profit. The markup adds ₹200, while the discount removes ₹120, because they have different bases. A discount and a profit can occur in the same sale.

Worked case 7 Two successive discounts

An item marked at ₹2,000 receives a 20% discount, followed by a 5% discount on the price then remaining. Find SP and the equivalent single discount on the original MP.

After the first discount: ₹2,000 × 0.80 = ₹1,600. The second discount is 5% of ₹1,600 = ₹80, so SP = ₹1,600 × 0.95 = ₹1,520. Together the factors are 0.80 × 0.95 = 0.76. Thus 76% of the original MP remains and the equivalent single discount is 100% − 76% = 24%.

Check through amounts: the discounts are ₹400 and ₹80, totalling ₹480. Then (480 ÷ 2,000) × 100% = 24%, and ₹2,000 − ₹480 = ₹1,520.

Error check: adding 20% + 5% gives 25%, incorrectly treating the second reduction as 5% of ₹2,000. For discounts d₁% then d₂% on each current price, the retained factor is (1 − d₁/100)(1 − d₂/100). Subtract this factor from 1 to obtain the fraction removed.

5 Combine amounts before calculating a combined rate

For a batch, add all defined costs and all sale receipts separately. Overall profit = total SP − total C if positive; otherwise reverse the subtraction for the loss amount. Divide that amount by total C for the conventional percentage.

Do not take a simple average of individual profit/loss rates when their cost bases differ. The bigger cost gives its rate more influence on the total. Equal weighting of signed rates is justified for two items with equal costs; it is not a general shortcut.

Worked case 8 Two sales and one short base comparison

One item costs ₹400 and is sold at 25% profit. Another costs ₹600 and is sold at 10% loss. Both are sold, and there are no other costs. Find the overall cost-based result. Then, as a separate comparison, find the sales-based margin, defined here as total profit divided by total SP, expressed as a percentage.

Item	Defined cost	Sale calculation	Sale receipt
First	₹400	₹400 × 1.25	₹500
Second	₹600	₹600 × 0.90	₹540
Total	₹1,000	Add receipts	₹1,040

Overall profit = ₹1,040 − ₹1,000 = ₹40. Cost-based profit percentage = (40 ÷ 1,000) × 100% = 4%. Check by combining changes: the first item earns ₹100, the second loses ₹60; ₹100 − ₹60 = ₹40. Averaging 25% and −10% would give 7.5%, but the unequal costs do not justify equal weighting.

The separate sales-based margin uses the same ₹40 profit with a different denominator: Sales-based margin = (40 ÷ 1,040) × 100% = (50/13)% ≈ 3.846%. The fraction is exact; the decimal is rounded to three decimal places. It is below 4% because the same numerator is divided by the larger sales amount. Use this sales denominator only when that comparison is explicitly requested. A cost-based markup, a cost-based profit percentage and a sales-based margin must not be given interchangeable labels.

Checkpoint B Connect the arrows and check the claim

Try these before the feedback.

1. MP is ₹800 and discount is 25%. If total cost is ₹500, find SP and cost-based profit percentage.
2. A learner says “Two sales at 20% profit and 20% loss always cancel.” What information is missing?
3. With positive MP, can SP = ₹0 after a 100% discount reveal the original MP?

Feedback:

1. SP = ₹800 × 0.75 = ₹600. Profit = ₹600 − ₹500 = ₹100, which is 20% of ₹500. The 25% discount and 20% profit have different bases.
2. We need the costs of the two sales. Equal rates on unequal costs represent unequal money. Calculate each receipt and aggregate amounts; cancellation occurs for these opposite equal rates if the costs are equal.
3. No. Every positive MP multiplied by 0 gives ₹0. Division by 0 is not a reverse calculation.

Before independent practice

For every question, write the base beside the percentage before choosing an operation.

1. Define C, MP and SP; add only the explicitly included costs, once
2. Decide what is asked: an amount, a percentage or a starting price
3. Use C for conventional profit/loss, MP for discount, and SP only for an explicitly sales-based margin
4. Reverse a known factor by division; do not apply the rate to the result
5. For successive changes, follow the current base at each step
6. For multiple sales, add amounts before dividing; label the result as profit or loss
7. Substitute the answer back, check its size and keep exact fractions where decimals repeat

The following 12 questions are original, untimed learning practice. Scoring is +1 for a correct answer and 0 for a wrong or unattempted answer. There is no negative marking and no pass cutoff. This practice is not a full CBT simulation.

Independent practice

1

An item has total cost ₹450 and sells for ₹540. What is its conventional cost-based profit percentage? A. (50/3)% B. 20% C. 90% D. 120%

2

An item costs ₹750 in total and sells for ₹660. What is its cost-based loss percentage? A. 90% B. 88% C. (150/11)% D. 12%

3

The marked price is ₹900 and the discount is 15% of marked price. What is the discount amount? A. ₹135 B. ₹765 C. ₹15 D. ₹90

4

An item marked at ₹1,600 is sold at a 7.5% discount. What is its selling price? A. ₹120 B. ₹400 C. ₹1,480 D. ₹1,720

5

An item sells for ₹1,035 at a 15% profit on total cost. What is its total cost? A. ₹879.75 B. ₹900 C. ₹1,190.25 D. ₹155.25

6

An item sells for ₹828 at an 8% loss on total cost. What is its total cost? A. ₹900 B. ₹894.24 C. ₹761.76 D. ₹10,350

7

Purchase price is ₹950 and explicitly included repairs cost ₹50. The item sells for ₹1,120, with no other costs. What is its cost-based profit percentage? A. (340/19)% B. (20/3)% C. 17% D. 12%

8

After an 18% discount on marked price, an item sells for ₹1,722. What was its marked price? A. ₹1,412.04 B. ₹2,031.96 C. ₹2,100 D. ₹378

9

Total cost is ₹500. Marked price is set 30% above cost, and a 20% discount is then given on marked price. What is the final cost-based profit percentage? A. 4% B. 10% C. 26% D. 104%

10

Two successive discounts are 10% and then 15%, each applied to the price then current. What single discount on the original marked price is equivalent? A. 25% B. 5% C. 76.5% D. 23.5%

11

Two items cost ₹300 and ₹700 respectively. The first sells at 20% profit and the second at 10% loss, both on their own costs. Both are sold and there are no other costs. What is the overall cost-based result? A. 5% profit B. 1% loss C. 10% profit D. 1% profit

12

An item has total cost ₹800 and sells for ₹1,000. Choose the correctly labelled pair: cost-based profit percentage; sales-based margin, where margin means profit/SP × 100%. A. 20%; 25% B. 25%; 25% C. 25%; 20% D. 20%; 20%

Answers and explained feedback

1 · B

Profit = ₹540 − ₹450 = ₹90. The cost base is ₹450, so (90/450) × 100% = 20% (B). Check: ₹450 × 1.20 = ₹540. A uses the sales denominator: (90/540) × 100% = (50/3)%. C relabels the ₹90 amount as a percentage. D is SP as a percentage of cost, 120%, and still includes the original 100% cost.

2 · D

Loss = ₹750 − ₹660 = ₹90. Divide by cost: (90/750) × 100% = 12% (D). Check: 88% of ₹750 = ₹660. A confuses the ₹90 loss amount with a rate. B is the percentage of cost recovered by the sale, not the percentage lost. C divides the loss by SP: (90/660) × 100% = (150/11)%, which is the wrong base.

3 · A

Discount = ₹900 × 15/100 = ₹135 (A). Check: 10% is ₹90 and 5% is ₹45; together they are ₹135. B is the remaining sale price, ₹900 − ₹135 = ₹765. C treats the numerical rate 15 as rupees. D calculates only 10% and omits the extra 5%.

4 · C

A 7.5% discount retains 92.5% of MP. SP = ₹1,600 × 0.925 = ₹1,480 (C). Check: discount = ₹120 and ₹1,480 + ₹120 = ₹1,600. A is the discount amount. B treats 7.5% as 75% and retains only 25%. D adds ₹120 to MP instead of subtracting it.

5 · B

SP is 115% of cost, so C = ₹1,035/1.15 = ₹900 (B). Check: 15% of ₹900 is ₹135 and ₹900 + ₹135 = ₹1,035. A subtracts 15% of SP, giving ₹1,035 × 0.85 = ₹879.75; that uses the wrong base. C multiplies SP by 1.15 instead of reversing the factor. D is 15% of SP, ₹155.25, which is neither the cost nor the actual profit.

6 · A

The sale retains 92% of cost. C = ₹828/0.92 = ₹900 (A). Check: 8% of ₹900 is ₹72, and ₹900 − ₹72 = ₹828. B adds 8% of the smaller SP, giving ₹828 × 1.08 = ₹894.24. C applies another 8% reduction to SP. D divides by 0.08 and wrongly treats ₹828 as the loss amount rather than the remaining sale amount.

7 · D

Total cost = ₹950 + ₹50 = ₹1,000. Profit = ₹1,120 − ₹1,000 = ₹120; (120/1,000) × 100% = 12% (D). Check: ₹1,000 × 1.12 = ₹1,120. A omits repairs entirely: (170/950) × 100% = (340/19)%. B counts repairs twice: cost ₹1,050 and profit ₹70 give (20/3)%. C uses ₹170 as profit while dividing by ₹1,000, inconsistently including repairs only in the denominator.

8 · C

SP is 82% of MP. MP = ₹1,722/0.82 = ₹2,100 (C). Check: 18% of ₹2,100 = ₹378; ₹2,100 − ₹378 = ₹1,722. A applies a further 18% discount to SP. B adds 18% of SP, using the smaller amount as the base. D is the actual discount amount ₹378, not the original marked price.

9 · A

MP = ₹500 × 1.30 = ₹650; SP = ₹650 × 0.80 = ₹520. Profit = ₹20, so (20/500) × 100% = 4% (A). Check: 1.30 × 0.80 = 1.04. B subtracts the different-base rates 30 − 20. C expresses the ₹130 discount as 26% of cost, which is not the final profit rate. D is SP as 104% of cost and fails to remove the original 100%.

10 · D

The retained factor is 0.90 × 0.85 = 0.765. Thus 76.5% remains and 23.5% is removed (D). Check on an original ₹100: ₹100 becomes ₹90 and then ₹76.50; total reduction is ₹23.50. A adds the rates as though both applied to the original amount. B subtracts the rates, although both changes are reductions. C reports the retained percentage rather than the discount.

11 · B

Receipts are ₹300 × 1.20 = ₹360 and ₹700 × 0.90 = ₹630. Total SP is ₹990 against total cost ₹1,000, so loss is ₹10 = 1% of total cost (B). Check through changes: ₹60 profit − ₹70 loss = −₹10. A averages 20% and −10% despite unequal costs. C subtracts the rates directly and calls the 10-point difference a total profit rate. D has the correct magnitude but the wrong direction: receipts are below cost.

12 · C

Profit is ₹1,000 − ₹800 = ₹200. Cost-based profit = (200/800) × 100% = 25%; sales-based margin = (200/1,000) × 100% = 20%, so C is correct. Check: 25% of ₹800 and 20% of ₹1,000 both equal the same ₹200 profit. A swaps the bases. B uses cost for both percentages. D uses sales for both. Changing the denominator changes the comparison, not the profit amount.

Sources and lesson scope

The explanations, worked cases and practice questions in this lesson are original. The primary textbook sections below support the mathematical methods and terminology.

• NIOS Secondary Mathematics 211, Percentage and Its Applications: §8.5.1, Profit and Loss, printed pp.212–216 (PDF pp.10–14); §8.5.2, Discount, printed pp.216–219 (PDF pp.14–17), stopping before §8.5.3
• NCERT Ganita Prakash, Grade 8 Part II, Fractions in Disguise: §1.3, price comparisons on pp.16–18 and the opening sales-based comparison on p.19; successive discounts on p.26 and markup followed by discount on p.27. Printed and PDF page numbers match. The source's accounting and tax discussion is outside this lesson
• NCERT Mathematics Class VIII, Comparing Quantities, legacy 2024–25 reprint: §7.2 and §7.2.1, printed pp.82–83 (PDF pp.4–5), for marked-price discount and the distinction between exact calculation and estimation
• NIOS Mathematics 211, प्रतिशतता और इसके अनुप्रयोग: Hindi terminology in §8.5.1, printed pp.221–225 (PDF pp.13–17), and §8.5.2, printed pp.225–229 (PDF pp.17–21), stopping before §8.5.3

This is one lesson within Ratios Percentages and Commercial Comparisons. It covers the stated commercial percentage methods, not the full module or the full RRB mathematics syllabus. The methods here use defined costs and clearly labelled percentage bases; tax, interest and accounting detail are outside this lesson.

Analogy

Imagine one box carrying three clearly labelled cards. The cost card records what the seller spent under the question's stated rules. The marked-price card records the amount displayed to customers. The sale card records what the seller actually received.

To judge profit, place the sale card beside the cost card: the cost card supplies the 100% reference. To judge discount, place the sale card beside the marked-price card: the marked-price card supplies the 100% reference. The same sale card can be below the marked-price card yet above the cost card, so a discounted sale can still earn a profit.

The cards help you keep roles separate; they do not create a formula or imply that the listed price must exceed cost. A later discount uses the price currently reached, and a batch needs the amounts from every relevant cost and sale card. This analogy covers only the explicitly stated costs, with no hidden expenses.

Quick reference

Name the base

• CP: purchase cost; C: CP plus explicitly included costs, counted once
• MP: marked price; SP: sale receipt in the stated model
• Conventional profit/loss base: C > 0
• Discount base: MP > 0
• Sales-based margin base: SP > 0, only when explicitly requested

Calculate and reverse

• Profit = SP − C when SP > C; profit percentage = ((SP − C)/C) × 100%
• Loss = C − SP when SP < C; loss percentage = ((C − SP)/C) × 100%
• SP = C: 0% profit/loss
• At p% profit: SP = C(1 + p/100); C = SP/(1 + p/100)
• At l% loss: SP = C(1 − l/100); C = SP/(1 − l/100), with l < 100 for reversal
• Discount amount = MP − SP; discount percentage = ((MP − SP)/MP) × 100%
• At d% discount: SP = MP(1 − d/100); MP = SP/(1 − d/100), with d < 100 for reversal
• With 0 ≤ l,d ≤ 100, forward loss/discount factors are nonnegative; a 100% reduction cannot be uniquely reversed

Combine with the right denominator

• Markup m% on C followed by discount d% on MP: SP = C(1 + m/100)(1 − d/100)
• Successive discounts d₁%, d₂%: retained factor = (1 − d₁/100)(1 − d₂/100); equivalent discount = (1 − retained factor) × 100%
• Batch: add costs and receipts first, then compare total SP with total C; percentage denominator is total C
• Explicit sales-based margin = (profit/SP) × 100%; for a batch use total profit and total SP

Final check

A rupee amount is not a percentage. A discount alone does not prove a loss. Unequal cost bases do not permit a simple average of rates. Reverse by division and substitute back. Keep repeating results as exact fractions; mark any rounded decimal with ≈.