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A shop buys an item for $80 and sells it for $100. Its profit is $20. That profit is 25% of the cost price, but only 20% of the selling price. Both calculations are mathematically correct. They answer different questions because their reference amounts are different.
This is the quickest honest explanation of a common school Mathematics mistake: a percentage is not just a number with a percentage sign. It is a comparison with a particular whole. When a question asks for profit as a percentage of cost price, the cost price represents 100%. Dividing by the selling price changes the question.
This guide explains that distinction slowly, with complete solutions and checks. All shop examples are fictional classroom models. Unless stated otherwise, cost means the stated purchase cost, with no additional expenses or taxes. Real business accounts can use more detailed definitions. This is a Mathematics lesson, not advice about pricing a real business.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–20
Sections 21–25
Sections 26–28
Profit equals selling price minus cost price. Profit percentage on cost equals profit divided by cost price, multiplied by 100%. Profit margin on selling price equals profit divided by selling price, multiplied by 100%. Always read the named reference amount before choosing the denominator.
For cost $80 and selling price $100, calculate profit first: $100 − $80 = $20. Then $20 ÷ $80 × 100% = 25% profit on cost. The separate calculation $20 ÷ $100 × 100% = 20% gives the profit margin on selling price. Do not swap the two labels.
The most useful sentence to write before calculating is, “The percentage is measured against ______.” If you cannot complete that sentence from the question or an established convention, clarify the intended base. A bare statement such as “the profit is twenty per cent” may need more information before it can be interpreted safely.
In the simple model used here, the cost price is the amount paid to acquire an item. The selling price is the amount received when it is sold. Profit is the excess of the selling price over the cost price. It is an amount of money before it becomes any percentage.
Suppose a fictional stationery stall pays $12 for a pack and sells that pack for $15. The $15 is not all profit: $12 replaces the money spent acquiring it. The remaining $3 is the profit in this deliberately simplified model. Writing $15 − $12 = $3 makes that separation visible.
This first subtraction matters because learners sometimes divide the whole selling price by cost and report the result as profit percentage. Here $15 ÷ $12 = 1.25, meaning that selling price is 125% of cost. The extra 25 percentage points above the original 100% represent profit on cost. Selling price percentage and profit percentage are related, but they are not interchangeable names.
Before reaching for a formula, identify which amount is being described. A labelled subtraction is often a better start than an unlabelled fraction.
The base is the quantity treated as the whole, or 100%. Every percentage comparison needs one, whether it is explicitly written or understood from the wording. In “profit as a percentage of cost”, the words “of cost” identify the base. Put cost in the denominator because you are asking how many cost-sized units the profit contains.
For a $3 profit on a $12 cost, the fraction is 3/12. It means three dollars of profit for every twelve dollars of cost. Converting this fraction to a percentage gives 25%. The dollar units cancel because numerator and denominator measure the same kind of quantity.
Now change only the denominator to $15. The fraction 3/15 equals 20%. The arithmetic has not become faulty; its meaning has changed. It now describes the profit relative to the selling price. That is why a calculator cannot decide which answer a question wants. A calculator evaluates the expression you enter. The reader must decide which quantities belong in that expression before pressing the keys.
In standard school questions about percentage profit on an item’s cost, the cost price is the starting investment in the simplified transaction. The question asks how large the gain is compared with that starting amount. This is a percentage-change comparison: starting amount, change, and new amount.
If cost is $60 and profit is $15, the gain is one quarter of the starting $60. The profit on cost is therefore 25%. The selling price is $75, which equals 125% of cost. Reading those statements together shows how the starting amount remains the reference throughout the calculation.
This convention does not mean cost must be the base of every percentage mentioned in a business problem. A question can explicitly ask for profit as a percentage of sales, for a discount on marked price, or for an expense as a percentage of revenue. The named reference overrides the temptation to apply one remembered denominator everywhere.
Learn the usual convention, but keep reading the words. Understanding the comparison is more reliable than remembering that one quantity always belongs underneath another.
Start with “profit compared with cost”. Translate “compared with” into division: profit divided by cost. Then translate the resulting fraction into a percentage by multiplying by 100%. This produces the familiar formula without asking you to memorise its order blindly.
Using P for profit and C for a positive cost price, profit percentage on cost is P/C × 100%. The requirement C > 0 is important. If the stated cost were zero, this particular percentage would not be defined by division. A positive money profit could still exist, but it would not have a finite percentage on a zero cost under this formula.
Also write P = S − C, where S is selling price. Substitution gives (S − C)/C × 100%. These are two forms of the same calculation, not two different methods with different answers. The first often makes the meaning clearer because the profit is calculated and labelled before the percentage step.
The formula is a compressed sentence. When you forget it, rebuild the sentence and you can recover the correct fraction.
The question is: an item costs $80 and sells for $100. Find the percentage profit on cost price. First identify the starting amount: cost price is $80. Then calculate the profit: $100 − $80 = $20. Only this $20 gain belongs in the numerator.
Next form the comparison requested: $20/$80 = 1/4. Convert one quarter to a percentage: 1/4 × 100% = 25%. The answer is a 25% profit on cost price. Include the phrase “on cost price” when explaining the result because it records the denominator’s meaning.
Check by rebuilding the selling price. Twenty-five per cent of $80 is $20. Adding that profit to the $80 cost gives $100, matching the stated selling price. A reverse check like this tests both the arithmetic and the interpretation of the base.
If a working line instead says 20/100 × 100% = 20%, annotate it honestly: this calculates profit as a percentage of selling price. It is not a solution to the requested cost-based question, even though the division itself is accurate.
The same transaction can support several valid percentage statements. Their labels must travel with their numbers. With cost $80, profit $20 and selling price $100, the cost-based profit percentage is 25%, the sales-based profit percentage is 20%, and selling price is 125% of cost.
| Comparison | Reference amount | Calculation | Meaning |
|---|---|---|---|
| Profit on cost | $80 cost | 20/80 × 100% = 25% | Gain relative to purchase cost |
| Profit on sales | $100 selling price | 20/100 × 100% = 20% | Profit share of sales |
| Selling price on cost | $80 cost | 100/80 × 100% = 125% | Whole selling amount relative to cost |
These statements do not contradict one another. They are different views of the same three amounts. A useful correction exercise is to keep every calculated number but repair each missing label. This teaches more than crossing out all but one number as though the others were meaningless.
In an examination, however, provide the comparison actually requested. Extra unrelated percentages can make an otherwise clear answer difficult to follow. Precision means choosing the relevant statement, not listing every possible calculation.
In this guide, profit margin means profit divided by selling price. A 20% margin says that twenty cents of every dollar of sales is the stated profit in the simplified model. A 20% profit on cost says that each dollar of cost generates twenty cents of profit. Those are different starting dollars.
Take cost $100. A 20% profit on cost gives profit $20 and selling price $120. The margin is then $20/$120 = 1/6, or 16⅔%. It is not 20%, because the selling-price base is larger than the cost-price base.
Conversely, a 20% margin with selling price $100 means profit $20 and cost $80. That is the earlier 25% profit on cost. The shared number twenty does not make the two descriptions equivalent. You must know which whole is being split into one hundred parts.
Real reports may distinguish gross profit, operating profit and net profit. This lesson does not replace those accounting definitions. Its narrower purpose is to keep the numerator and denominator of a stated percentage comparison clear.
Suppose an item costs $240 and is sold at a 15% profit on cost. Fifteen per cent of $240 is 0.15 × $240 = $36. Add that profit to cost: $240 + $36 = $276. The selling price is $276 in this model.
A shorter route uses a multiplier. Cost is 100% of itself; cost plus 15% profit is 115% of cost. Therefore selling price equals 1.15 × $240 = $276. The longer route and the multiplier route express the same relationship.
Do not calculate 15% of the unknown selling price merely because selling price is what the question asks you to find. The unknown does not determine the base. The wording “profit on cost” determines it. A question can ask for one quantity while expressing a percentage of another.
Check by subtracting: $276 − $240 = $36. Then $36/$240 × 100% = 15%. This check returns to the original percentage statement and confirms that the correct reference amount was used.
An item sells for $360 at a 20% profit on cost. The unknown cost is not 80% of $360. Instead, selling price is 120% of cost. If C represents cost, then 1.20C = 360, so C = 360/1.20 = $300.
The profit is $360 − $300 = $60. Compare profit with cost: $60/$300 = 0.20, confirming 20%. Notice that profit is only 1/6 of the selling price, not 1/5. Subtracting 20% of the selling price would remove the wrong amount.
The mistaken calculation 360 × 0.80 = 288 treats selling price as the 100% whole. It would correspond to a different statement: cost is 80% of selling price. That statement was never given. This is a reverse-percentage problem because the known amount already includes the increase.
When a percentage increase is given and the final amount is known, label the final amount with its percentage first. Writing “$360 corresponds to 120% of cost” often makes the correct division immediately apparent.
Now change the wording: selling price is $360 and profit is 20% of selling price. Here selling price really is the 100% base. Profit equals 0.20 × $360 = $72. Cost equals the remaining $288 in the simplified cost-plus-profit model.
The arithmetic 360 × 0.80 = 288 is correct for this question, although it was wrong for the preceding cost-based question. This comparison shows why a method cannot be judged by its numerical appearance alone. We must examine the statement it represents.
If instead the cost is $240 and the required margin is 20%, cost represents 80% of selling price. Write 0.80S = 240, giving S = $300. Profit is $60 and $60/$300 = 20%. Adding 20% to $240 would produce only $288 and would not meet the stated margin.
You do not need a separate memory trick for every variation. Name the 100% quantity, assign the known amount its corresponding percentage, then solve the resulting relationship. That procedure works across both bases.
If selling price is below cost price, the simplified transaction makes a loss rather than a profit. For an item costing $80 and selling for $68, loss is $80 − $68 = $12. The percentage loss on cost is $12/$80 × 100% = 15%.
Selling price is therefore 85% of cost. This gives the multiplier statement S = 0.85C. If the selling price $68 is known and the cost is unknown, divide by 0.85 to recover $80. Do not add 15% of $68, because that uses a different reference amount.
Some algebraic treatments use signed profit: S − C = −$12. Dividing by cost gives −15%, which records a negative gain. In ordinary school wording, stating a 15% loss is clearer than calling it a negative profit percentage. Use the format asked for.
The base principle has not changed. What changes is whether the difference is described as a gain or a loss. Identify that direction before choosing a positive money amount for the final sentence.
A discount question introduces another possible reference amount: the marked or listed price. Suppose marked price is $150 and a 20% discount is offered. The discount is $30 and the selling price is $120. Those calculations use $150 as the base, not the item’s purchase cost.
If cost was $100, the resulting $120 sale produces profit $20, which is a 20% profit on cost. The identical percentage numbers in this example are a coincidence of the chosen amounts. The discount and profit are separate comparisons with different bases.
Change the cost to $110 while leaving marked price and discount unchanged. Selling price remains $120, but profit becomes $10. Profit on cost is then 10/110 × 100%, approximately 9.09%. The 20% advertised discount has not changed, yet the profit percentage has.
For problems with three prices, label all three before doing arithmetic. Marked price determines the stated discount; selling price is the actual receipt in the model; cost price determines the conventional profit-on-cost calculation. Keeping those roles separate prevents the numbers from exchanging jobs.
Imagine cost $100, followed by a 50% markup on cost. Marked price becomes $150. If a 20% discount is then applied to marked price, selling price becomes 0.80 × $150 = $120. The final profit on original cost is 20%, not 30%.
Subtracting the percentage numbers, 50 − 20 = 30, ignores their different reference amounts. The markup adds $50, but the discount removes $30 because it is 20% of $150. The net gain is $20. The money changes make the reason visible.
Multipliers record the changing bases efficiently: 100 × 1.50 × 0.80 = 120. The combined multiplier is 1.20, so final selling price is 120% of original cost. Its profit component is the extra 20% above the starting 100%.
For a fuller treatment of sequential percentages, use the earlier guides through the Mathematics explainer directory. Here the essential point is that each percentage acts on its own stated or implied reference amount, not automatically on the first number in the story.
A profit amount should carry money units; a profit percentage should not be labelled with a dollar sign. In $20/$80, the dollar units cancel. The result 0.25 is a ratio, which can be expressed as 25%. Writing “$25%” mixes two different kinds of quantity.
Magnitude also gives a useful check. If the profit is less than the positive cost, profit on cost is less than 100%. If profit equals cost, it is 100%, and selling price is twice cost. If profit exceeds cost, profit on cost exceeds 100%. Such percentages are possible; they are not automatically errors.
In the same nonnegative-cost classroom model, a positive profit margin on positive sales is below 100% whenever cost is positive. Profit is then a proper part of selling price. This difference in possible ranges can help reveal a mislabelled answer.
These checks do not replace the calculation. They are quick tests of whether the answer behaves like the quantity named. A plausible-looking number still needs a justified denominator and a correct subtraction.
Try these three short questions before reading the answers. First, an item costs $50 and sells for $65. Find profit as a percentage of cost. Second, for the same transaction, find profit as a percentage of selling price. Third, a marked price of $65 is reduced by $13. Find the discount percentage on marked price.
For the first question, profit is $15 and the base is $50. The result is 15/50 × 100% = 30%. For the second, the same $15 profit is compared with $65, giving 15/65 × 100% = 300/13%, approximately 23.08%.
For the third, discount is the given $13 reduction and the base is the $65 marked price. The result is 13/65 × 100% = 20%. The $50 cost from the first question is irrelevant to that discount calculation unless the problem explicitly connects the transactions.
If you obtained the correct percentages, explain the denominator of each answer aloud. If you cannot explain it, redo the comparison sentence. Correct arithmetic is most useful when you can reproduce the reason for choosing it.
An item costs $320 and is sold at a 12.5% profit on cost. What is its selling price? Since 12.5% is one eighth, profit is $320/8 = $40. Selling price is $360. The multiplier calculation 320 × 1.125 gives the same amount.
Now reverse the question. An item sells for $360 at a 12.5% profit on cost. Cost is 360/1.125 = $320. The selling price represents 112.5% of cost, so division by 1.125 is appropriate. Subtracting 12.5% of $360 would instead give $315 and would not recover the original cost.
Finally, an item costs $320 and is sold at a 12.5% loss on cost. Selling price is 0.875 × $320 = $280. The opposite direction uses subtraction from the cost-based whole, not the reciprocal of the profit multiplier.
Compare the three equations rather than memorising the answers. Forward increase multiplies by 1.125; reversing that same increase divides by 1.125; applying a loss multiplies by 0.875. Each operation follows from a different sentence about the amounts.
A fictional item costs $180. A shop wants a 20% profit on cost after giving a 10% discount on marked price. Find the marked price. Work backwards through the relationships rather than combining 20 and 10 immediately.
The required selling price is 1.20 × $180 = $216. After the discount, that selling price is 90% of marked price. If M represents marked price, then 0.90M = 216. Thus M = 216/0.90 = $240.
Check both stages. Ten per cent of $240 is $24, leaving a selling price of $216. Subtracting the $180 cost leaves profit $36. Finally $36/$180 = 0.20, which is the required 20% profit on cost. Each percentage is checked against its own base.
The common wrong approach is to mark up by 30%, producing $234, and assume that a later 10% discount leaves 20% profit. In fact 0.90 × $234 = $210.60. That gives profit $30.60, only 17% of $180. The check exposes why adding percentage rates without their bases is unsafe.
Suppose cost is $72 and selling price is $90. Profit is $18, so profit on cost is exactly 25%. But if cost is $70 and selling price is $90, profit is $20, giving 20/70 × 100% = 200/7%, or 28.571428…%.
If the question requests two decimal places, report 28.57%. Keep the exact fraction or sufficient calculator precision until the final rounding step. Prematurely rounding 20/70 to 0.29 would produce 29%, losing accuracy before the final answer is formed.
Do not assume a recurring decimal signals a wrong base. Some correct percentage comparisons do not terminate. Check the quantities and the requested precision rather than trying to force every answer into a tidy integer.
When reversing a percentage, use the original given rate rather than a rounded intermediate rate you calculated elsewhere. A rounded rate describes an interval of possible exact ratios. Treating it as exact can slightly change the reconstructed price. In classroom questions, follow the stated rounding instruction and distinguish exact working from the rounded final statement.
A stall buys one item for $40 and sells it for $50. It buys another for $60 and sells it for $66. The first profit percentage on cost is 25%; the second is 10%. The overall profit percentage is not their simple average of 17.5%.
Total cost is $100 and total selling receipts are $116. Total profit is $16. The overall profit on total cost is therefore 16%. The second item has a larger cost base, so its lower percentage has more influence on the combined result.
An unweighted average of individual percentages is justified only under suitable equal-base conditions. If both items had cost $50, adding their profits and dividing by the $100 total cost would match the average of their cost-based profit percentages. Equal numbers of items do not by themselves guarantee equal cost bases.
Whenever a question asks for an overall rate, return to the totals relevant to that rate. Add costs, add receipts, find total profit, then form the required ratio. This method also makes any losses on individual items visible instead of hiding them behind a collection of percentages.
Two items each sell for $120. One is sold at a 20% profit on cost; the other at a 20% loss on cost. It is tempting to say the gains and losses cancel because the percentages are equal. They do not share equal cost bases.
For the profit-making item, 120% of cost is $120, so cost is $100. For the loss-making item, 80% of cost is $120, so cost is $150. Total cost is $250 while total selling receipts are $240. The combined loss is $10, or 4% of total cost.
The first item gains $20 and the second loses $30. Equal rates with different bases do not produce equal money changes. This is the same principle as the earlier markup-and-discount example, expressed across two transactions instead of successive changes to one item.
This problem is useful only after the simpler base distinction is secure. If it feels confusing, write the percentage correspondence beside each $120 first. The identical selling prices represent 120% and 80% of two different unknown costs. That is the entire reason the costs differ.
Some questions specify delivery, packaging or repair costs in addition to purchase price. Read whether the stated profit percentage uses total cost or purchase cost. If a problem defines total cost as purchase plus delivery, include both in that base. If it explicitly asks for a percentage of purchase price, retain that narrower reference.
For example, purchase cost $80 plus a stated $10 delivery expense gives total cost $90 under the defined model. Selling for $108 then produces $18 profit on that total cost, or 20%. Comparing $28 with the purchase price alone answers a different simplified comparison and omits the stated expense.
Do not introduce guessed expenses into a school question that gives none. Equally, do not ignore expenses that the question says must be included. Mathematical modelling starts by identifying which quantities the problem defines and which it deliberately leaves outside the model.
For broader checking habits, the ecosystem guide on testing whether a mathematical model makes sense is a useful next step. The connection is direct: a correct percentage formula cannot repair a wrongly defined cost quantity.
Consider this response: “Cost $80, sale $100, profit $20, profit percentage = 20/100 × 100% = 20%.” If the question asked for profit on cost, the subtraction is correct and the final division is correctly executed. The mistake lies in the chosen reference amount.
A precise correction is: “The denominator $100 measures selling price, so this is a 20% margin on sales. The question asks for cost as the base. Use $20/$80 × 100% = 25%.” This preserves the correct steps and isolates the decision that needs repair.
Telling a learner simply to “be more careful” gives no reusable instruction. Asking “Which quantity is 100% here?” provides a specific action they can repeat on the next problem. The aim is not to increase anxiety around fractions; it is to attach fractions to meanings.
After the correction, change the amounts and ask the learner to explain the base without calculating. If they can do that consistently, the conceptual error is improving even before they become faster with arithmetic. Speed should follow a stable interpretation, not replace it.
Use a small set of mixed questions rather than many identical substitutions. Include one direct profit percentage, one selling-price calculation, one reverse percentage and one discount problem. Before each calculation, write a short base label such as “100% = cost” or “100% = marked price”.
After solving, reconstruct the stated relationship. If an answer claims 30% profit on cost, calculate 30% of that cost and check that it matches the profit. If the answer is a marked price, apply the stated discount and check the resulting sale. These checks make the reasoning visible.
On another day, repeat the same structures with different amounts and without the old working beside you. Being able to reproduce a decision after a delay is stronger evidence of learning than recognising a familiar completed solution immediately after reading it.
Parents can help by asking for the comparison sentence rather than supplying a formula at once. Teachers can classify errors into subtraction, base selection, reverse multiplier and rounding. These categories suggest different repairs. A learner who chooses the right denominator but mistypes division needs different help from one who does not know what the denominator represents.
This article is organised as a G1-friendly explanation of percentage reasoning. It is not a claim that every example appears in every school’s current sequence or assessment. More involved reverse-pricing and combined-transaction questions should be selected to match the learner’s actual course and readiness.
For official course context, consult the 2027 G1 Mathematics syllabus. Use the syllabus and school instructions to establish scope; use worked examples such as these to understand the reasoning within a chosen topic. A local guide should not substitute its own promises for official requirements.
The How Mathematics Works hub connects this percentage lesson with ratio, algebra, units and mathematical modelling. Those are not unrelated extras. Ratios explain the comparison, algebra supports reversing it, units distinguish money from rates, and modelling defines which costs belong in the calculation.
If the immediate difficulty is reading the question, start with naming the whole. If the difficulty is reversing an increase, start with percentage correspondences. Choose the next lesson from the actual error rather than from the length of the worksheet.
An item is sold for $84 at a 40% profit on cost. Since $84 represents 140% of cost, one hundred per cent is $84/1.40 = $60. Profit is $24, and $24/$60 = 40%. The unknown cost is recovered by division because the known sale already includes the gain.
An item is sold for $84 with a 40% margin on selling price. Profit is 40% of $84, or $33.60. Cost is $50.40. These two questions share a selling price and a percentage number but produce different costs. Their wording assigns the number forty to different comparisons.
Finally, an item marked at $84 is discounted by 40%. The sale becomes 60% of $84, or $50.40. This matches the preceding cost numerically, but it is a selling price in a different question. Equal numbers across examples do not imply equal roles.
This set is a useful reading test. Cover the calculations and identify the role of every $84 before solving. If the roles are clear, the arithmetic follows a meaningful relationship rather than a pattern copied from the previous line.
One final diagnostic question separates memory from understanding. If both cost and selling price double, does the profit percentage change? For cost $80 and sale $100, profit is $20 and the cost-based rate is 25%. Doubling both prices gives cost $160, sale $200 and profit $40. The new ratio is $40/$160, still 25%. Money profit doubles, but its percentage stays unchanged because numerator and denominator scale together. This is why a large dollar profit need not mean a large profit percentage. Compare transactions by the requested rate only after identifying the amounts behind that rate.
When a percentage problem becomes crowded, reduce it to three questions. What changed? What quantity is the change being compared with? Is the known amount the original whole or an amount after a change? Answering those questions usually identifies the subtraction, denominator and multiplier needed.
For the central example, the change is a $20 profit. It is compared with the $80 cost. Therefore the profit is 25% on cost. If the wording instead compares it with the $100 sale, the same profit is 20% of sales. The difference is not an arbitrary school rule; it is the meaning of the two fractions.
Keep the reference amount in the final explanation whenever confusion is possible. “A 25% profit on cost” is a more complete statement than “25% profit”. Clear labels make your solution easier to check and make it harder to reuse a percentage in a context where it does not belong.
That is the habit worth carrying forward: do not ask only whether a percentage was calculated correctly. Ask whether it was calculated against the right whole. The right whole is what makes the number answer the question.
These short answers summarise the distinctions used throughout the worked examples. If two answers seem to conflict, check whether their reference amounts are the same before comparing the percentages.
Why is selling price not the denominator for ordinary profit on cost?
Because that question compares the gain with the amount spent, not with the amount received. Selling price belongs in the denominator only when the required comparison is profit as a percentage of selling price. Changing the denominator changes the statement.
Can profit percentage be above 100%?
Profit on a positive cost can exceed 100% when the profit exceeds that cost. For cost $20 and selling price $50, profit is $30, or 150% of cost. Selling price is then 250% of cost, not 150%.
Why can I not subtract the profit percentage to recover cost?
A cost-based profit percentage is measured against cost, while the known selling price includes both cost and profit. If selling price represents 125% of cost, divide by 1.25. Subtracting 25% of selling price uses the wrong whole.
What should I do when the question only says profit percentage?
Use the convention established by your course and the context, which commonly treats cost as the base in school profit-and-loss questions. In an explanation, state that assumption. If the setting is genuinely ambiguous, ask which reference amount is intended rather than pretending all percentage labels mean the same thing.
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