Did you know? A shop can sell a product for more than its wholesale cost but still end up losing money after delivery charges, packaging and other expenses are counted. The calculation isn’t mysterious: the answer depends on which costs belong to the question.
For Secondary 1 G3 Mathematics students in Bukit Timah, simple shop-pricing problems bring together money, number operations, ratios, unitary method, percentages, algebra and sensible checking. A learner may confidently find 20% of $50 yet hesitate when the question mentions a cost price, selling price, discount, fixed charge and final gain in the same paragraph.
This guide uses fictional shop situations to make those relationships visible. It is a teaching resource, not a description of real Bukit Timah businesses or actual commercial prices. We will distinguish a profit amount, a loss amount, a discount and an amount remaining after costs, without turning the lesson into advanced financial accounting.
Guide contents
- Quick answer: what is profit or loss?
- Four labels to write before the arithmetic
- Worked example 1: straightforward profit
- Worked example 2: straightforward loss
- Why a discount isn’t automatically a loss
- Worked example 3: marked price and discount
- Worked example 4: cost, marked price and final sale
- Two consecutive discounts are not one added discount
- Worked example 5: a shop sells several items
- Unit price is a different comparison
- Worked example 6: transport changes the result
- Fixed cost versus cost per item
- Worked example 7: find the break-even quantity
Quick answer: what is profit or loss?
When the model includes only buying and selling costs, the simple profit amount is:
Profit = selling price − total cost, provided the result is positive.
If the selling price is lower than total cost, the amount of loss is total cost − selling price, a positive number describing the shortfall.
The most important term is total cost. In one school question, it may mean only the buying price. In another, it may include an expressly stated packaging or transport expense. Include exactly the costs described; do not silently add costs or taxes that the question never specified.
A discount is different. It reduces a marked or original selling price, regardless of whether the seller eventually gains or loses money.
Four labels to write before the arithmetic
- Cost price: what the seller paid to acquire an item in the simplified model.
- Marked price: the price displayed or set before a possible discount.
- Selling price: the amount the customer actually pays for the item under the specified conditions.
- Other costs: any additional expenses that the question says should be included, such as a fixed delivery charge.
For example, if a fictional item costs $80 to buy, is marked at $100 and sells for $92 after a discount, the simple gross difference between selling and buying amounts is $12.
The $100 marked price does not enter the profit calculation directly once the actual selling price of $92 is known. It is relevant for finding the discount.
A student who subtracts cost price from marked price rather than actual selling price may have read the labels incorrectly.
Worked example 1: straightforward profit
A fictional shop buys an item for $80 and sells it for $92. Assume no other relevant costs.
Profit amount = 92 − 80 = $12.
Check the direction: the selling price exceeds the cost by twelve dollars, so the shop gains twelve dollars in this simplified transaction.
If the student writes $92 as the profit, they have confused revenue with the difference after paying for the item. Revenue is the amount received from selling; profit is the amount left after specified costs.
One useful sentence is: “The shop received $92, but $80 of that amount replaces the buying cost, leaving $12.”
Worked example 2: straightforward loss
A fictional shop buys an item for $80 but sells it for $75, with no other relevant costs.
Loss amount = 80 − 75 = $5.
The shop still received $75. A loss does not mean no money was collected; it means less was collected than the stated cost.
If the student calculates 75 − 80 = −5, this signed difference correctly signals a negative outcome. When the question specifically asks for the amount of loss, report $5 as the nonnegative shortfall and label it as loss.
The two representations describe the same situation but answer different wording.
Why a discount isn’t automatically a loss
A fictional article costs the shop $35 and has a marked selling price of $50. A customer receives a 20% discount.
Discount = 0.20 × 50 = $10. Actual selling price = 50 − 10 = $40.
The simple profit amount is 40 − 35 = $5. The seller has offered a discount yet still sold above cost.
This is a valuable distinction for students: discount compares a final price with a marked price; profit compares the final selling amount with the cost.
They are different bases. A 20% discount does not mean a 20% loss to the seller.
Worked example 3: marked price and discount
A fictional bag is marked at $75 and receives a 12% discount. Find its selling price.
First find the reduction: 12% of $75 = 0.12 × 75 = $9.
Selling price = $75 − $9 = $66.
You can also use a multiplier: after a 12% reduction, 88% remains, so 75 × 0.88 = $66.
Check by adding the discount back: 66 + 9 = 75. If the child writes $84, they have increased instead of reduced the marked price.
The discount calculation alone does not tell us whether the shop makes a profit; we have not been given its cost.
Worked example 4: cost, marked price and final sale
Suppose a fictional product costs $60 and its marked price is 25% more than its cost. It is then discounted by 20% of the marked price.
Marked price = 60 × 1.25 = $75.
After discount, selling price = 75 × 0.80 = $60.
The final selling price is exactly equal to the original cost, so the simplified model breaks even, assuming no extra costs.
A student who simply says “25% increase minus 20% decrease equals 5% gain” is making a structural error. The percentages act on different bases. Their multipliers are 1.25 and 0.80, whose product is exactly 1.00.
This is one of the clearest examples of why consecutive percentage rates cannot normally be added or subtracted.
Two consecutive discounts are not one added discount
A hypothetical jacket is marked at $200. Its price falls by 25%, then receives another 8% reduction on the already reduced price.
After the first change, price = 200 × 0.75 = $150.
After the second, price = 150 × 0.92 = $138.
The overall dollar reduction is $62, or $62/$200 × 100% = 31% of the original marked price.
Do not add 25% and 8% and claim a 33% reduction, because the second reduction uses $150 as its base, not $200.
The percentage change and reverse percentage guide shows more examples of why the denominator matters.
Worked example 5: a shop sells several items
A fictional store buys forty identical notebooks for $2.40 each and sells all forty for $3.10 each. Ignore extra costs in this model.
Total cost = 40 × 2.40 = $96.
Total revenue = 40 × 3.10 = $124.
Profit = 124 − 96 = $28.
Check by unitary reasoning. The gain per notebook is 3.10 − 2.40 = $0.70. For forty notebooks, 40 × 0.70 = $28.
The two routes agree. If one gives $28 and the other gives $280, inspect a decimal-place error before changing the whole method.
Unit price is a different comparison
Imagine a fictional shop sells a 750 g pack for $6 and a 1.2 kg pack for $8.40.
The first has unit price $6/0.75 kg = $8 per kilogram.
The second has unit price $8.40/1.2 kg = $7 per kilogram.
The larger package has the lower unit price, even though its total checkout price is higher.
This is a customer comparison rather than a shop’s profit calculation. Without knowing what the shop paid for each package, we cannot calculate its gains or losses.
Labeling the question type before calculating prevents confusing cost, selling price and unit price.
Worked example 6: transport changes the result
Suppose a fictional seller buys six notebooks for $3 each, sells them for $5 each and pays a single $10 delivery fee directly associated with this batch.
Revenue = 6 × $5 = $30.
Notebook buying cost = 6 × $3 = $18.
Total specified cost = 18 + 10 = $28. Profit amount = 30 − 28 = $2.
A student who ignores the $10 fee calculates $12, which is the difference between the revenue and notebook purchase cost alone. That may be an intermediate amount, but it is not the final gain when the task explicitly includes delivery.
Do not add delivery twice merely because the question mentions six notebooks. Here the fee is one fixed charge for the batch.
Fixed cost versus cost per item
In the previous example, the total revenue from n identical notebooks at $5 each is 5n. The cost is 3n + 10, where 10 is a fixed batch fee.
The difference is 5n − (3n + 10) = 2n − 10.
This expression can be positive, zero or negative depending on n. For n = 6, it gives 12 − 10 = $2, agreeing with the worked example.
For n = 4, it gives 8 − 10 = −$2, a two-dollar shortfall.
The formula exposes why a fixed cost matters. Selling twice as many units does not double the fixed delivery charge.
This connects directly to the formulae with two variables and real situations guide.
Worked example 7: find the break-even quantity
Using the same fictional model, revenue is 5n and total specified cost is 3n + 10.
Break-even means these two amounts are equal: 5n = 3n + 10.
Subtract 3n from both sides: 2n = 10, so n = 5.
Check: five notebooks bring in $25. Buying costs $15 and delivery costs $10, making a total cost of $25. Neither gain nor loss occurs.
Because n counts notebooks, a nonnegative whole number is required in this context. An algebraic answer such as n = 4.5 would not represent a valid number of complete notebooks sold in this simple model.
Three kinds of incorrect shop mathematics
Wrong comparison: a student subtracts cost from marked price instead of from actual selling price after discounts.
Wrong percentage base: a child treats a reduction from $75 to $60 as though its dollar amount were 20% of the $60 final price. The $15 discount was 20% of the original $75, not the final $60.
Wrong fixed-fee treatment: a student adds the $10 delivery charge once per notebook instead of once per batch, making the supposed cost far too high.
These errors have different causes. A tutor should identify the first invalid step before assigning more questions.
A realistic-looking word problem can still be a simplified model
A school question may say “A shop buys 20 items at this price and sells them at that price.” Unless it supplies other expenses, treat the difference as specified by the question.
Real businesses may also have labour, rent, utilities, payment fees, tax rules and unsold inventory. Introducing imaginary expenses to a simple classroom problem would make its answer impossible to determine.
Conversely, when the question specifically lists a transport or packaging charge to include, ignoring it is incorrect.
The mathematical skill is to honour the given model without pretending that the model includes everything about a real business.
Worked example 8: missing buying price
An item sells for $90 and the simplified gain is $15. Find its buying cost.
Selling price − cost = profit. Hence 90 − cost = 15.
Rearrange: cost = 90 − 15 = $75.
Check: 90 − 75 = $15.
If a student adds 90 and 15 to obtain $105, they have confused a stated gain with an extra expense. Work backward from the relationship rather than deciding addition must be used whenever an unknown appears.
Worked example 9: missing selling price
A fictional item costs $48 and the seller incurs a $2 extra fee directly associated with the sale. The seller wants a $10 gain under the simplified model. What selling price would produce that amount?
Total cost = 48 + 2 = $50.
Target selling price = total cost + wanted gain = 50 + 10 = $60.
Check: 60 − 50 = $10. The result follows from the complete stated cost.
This is different from adding only the $10 target to the original $48, which would ignore the explicitly included extra fee.
Find a loss after a change in selling price
A fictional item originally sells for $108 after costing $90. Its simple gain is $18.
Later, the selling price is reduced by 10% of $108. The new selling price is 108 × 0.90 = $97.20.
At the same cost of $90, the new gain is $7.20. The price was discounted, and the gain decreased, but the result is not a loss.
Now imagine a further $8 sale-specific fee not included earlier. The total cost for this alternative scenario would be $98. The $97.20 selling price would then create a shortfall of $0.80.
This changed example illustrates why a tutor must insist on a clear definition of the costs being included.
Reverse a discounted price carefully
A fictional item sells for $72 after a discount of 20% from its marked price. What was the original marked price?
The final price represents 80% of the original. Let the original be M. Then 0.80M = 72, so M = 72/0.80 = $90.
Check: 20% of $90 is $18. A $90 price reduced by $18 gives $72.
Increasing $72 by 20% would give $86.40, the wrong original, because the increase would use the discounted final amount as its base.
The same percentage rate can represent different dollar amounts when the base changes.
What about GST in Singapore?
Singapore’s prevailing GST rate is 9% from 1 January 2024, but not every word problem needs GST added. An ordinary customer-facing price from a GST-registered Singapore business is generally displayed including GST, subject to specific display exceptions.
If a classroom question gives an expressly GST-exclusive taxable base, calculating 9% on that base can be appropriate. If a given price is GST-inclusive, do not add another nine percent simply because the problem takes place in Singapore.
For an accessible worked explanation, read unit prices, GST, budgets and household bills. Official tax details should be verified from IRAS where they matter.
This page keeps its arithmetic examples deliberately labelled, avoiding invented tax treatment.
Estimation as a profit-and-loss check
If forty notebooks gain about seventy cents each in the stated simple model, the total should be around $28. An answer of $280 cannot be right without an additional factor of ten.
If a shop buys for $80 and sells for $75, the loss is small relative to the buying price. An answer of $155 as the loss is implausible because it adds the two prices rather than compares them.
Before accepting a calculator display, ask whether the final amount should be positive, zero or negative under the model. Then check whether its size fits the given prices.
Estimation is a short quality-control step, not a substitute for showing the actual calculation.
Twelve-question self-test
- A. An item costs $80 and sells for $92. Find the profit amount.
- B. An item costs $80 and sells for $75. Find the loss amount.
- C. An item is marked at $75 and discounted by 12%. Find the selling price.
- D. A product costs $35 and sells for $40 after discount. Find the gain, excluding other costs.
- E. A marked price of $100 increases from a $60 cost by what multiplier? (Treat the relation as a numerical comparison.)
- F. A fictional $60 item is marked at 25% more, then discounted by 20% of the marked price. Find the final selling price.
- G. Forty notebooks cost $2.40 each and all sell at $3.10 each. Find total revenue.
- H. Find total simple profit from Question G, excluding other costs.
- I. Six notebooks sell for $5 each, cost $3 each and incur one fixed $10 fee. Find the final gain.
- J. In the model revenue = 5n, cost = 3n + 10, find break-even n.
- K. An item sells for $90 with a simple gain of $15. Find its cost.
- L. A marked price becomes $72 after a 20% reduction. Find the original marked price.
Answers with first-principles checks
A. $12; selling price minus cost.
B. $5 loss; the price collected is five dollars short of cost.
C. $66; 75 × 0.88.
D. $5; 40 − 35.
E. 100/60 = 5/3 as a multiplier, approximately 1.667. This comparison is distinct from the later discounted example.
F. $60; 60 × 1.25 × 0.80 = 60.
G. $124; 40 × 3.10.
H. $28; buying cost was $96 and revenue was $124.
I. $2; revenue $30, specified total cost $28.
J. Five notebooks; 5n = 3n + 10 gives n = 5.
K. $75; 90 − 15.
L. $90; divide 72 by 0.80.
For E, compare the multiplication factor carefully: a $100 price is 5/3 of a $60 cost. The question deliberately asks for a multiplier rather than a named commercial percentage measure.
A five-question self-correction routine
Ask the learner to identify whether each dollar amount is a cost, marked price, discount, selling price or fixed fee.
Next identify whether the question asks for the total received, a difference after costs, or a unit price. These are different types of answer.
For percentages, mark the 100% base. For batches, decide whether a fixed fee is charged once or per item. Finally check the units and compare the sign of the result with the story.
This routine is reusable in more complex word problems because it begins with meaning.
A fifteen-minute small-group teaching sequence
Start with a cost of $80 and two hypothetical selling prices, $92 and $75. Ask students to explain which creates a gain and which a loss without a calculator.
Then introduce a marked price and a discount. Make one student calculate the discounted selling price and another calculate the difference from buying cost.
Use the final portion of the lesson for a batch problem with a fixed delivery fee. The children should express the result as a sentence, a table and then an equation.
A changed problem a few days later, with different numbers and perhaps a loss, tests whether the method transfers independently.
A parent checklist for choosing Mathematics support
A useful Secondary 1 G3 Mathematics tutor should diagnose whether a wrong result comes from decimal arithmetic, the percentage base, a missing cost or translating the story incorrectly.
The eduKateSG Secondary 1 Mathematics small-group tutorial reference describes focused three-student tutorials near Sixth Avenue MRT, using a 90-minute structure with guided working, independent application and error review. The Bukit Timah Secondary 1 G3 Mathematics tuition overview provides locality-specific context. Class availability and the child’s school sequence should be confirmed directly.
The MOE G2 and G3 Mathematics syllabus provides the framework for Secondary 1 real-world number, ratio and percentage problems. This guide uses simplified gain-and-loss amounts, not an advanced accounting syllabus.
An unsold item is not the same as a missing payment
Consider a fictional seller who buys ten identical items for $5 each, paying $50 at the start. Eight items are sold for $8 each, bringing in $64. Two items remain unsold. What can we correctly calculate from the information supplied?
The buying cost associated with the eight sold items is 8 × $5 = $40, so their simple selling-minus-buying difference is $64 − $40 = $24 before other expenses. The two unsold items still have a combined original buying cost of $10. The difference between all money received so far and the full original $50 outlay is $64 − $50 = $14.
These two numbers answer different questions. The $24 compares revenue with the acquisition cost of the eight units sold. The $14 compares collected cash with the full original outlay for all ten units, including the two not yet sold. Neither should be announced as a comprehensive final business profit without information about the remaining items and other costs.
This is a valuable reading lesson: ask exactly which costs and quantities the question tells you to include, and label the difference you have calculated. Different correct calculations can exist for different precisely worded questions.
A short comparison table prevents price confusion
Suppose a fictional product costs $50. It is marked at $80 and eventually sells for $64. Make a three-row record: buying cost $50, marked price $80, actual selling price $64. The marked-to-selling discount is $16, while the selling-minus-buying difference is $14. The same sale produces two different dollar differences because the reference amounts differ.
Now change only the buying cost to $70. The marked price and selling price remain unchanged, so the customer still received a $16 discount. But the selling-minus-buying difference becomes $64 − $70 = −$6, a six-dollar shortfall. That change demonstrates why a discount alone does not determine whether a seller gains or loses money.
Ask the student to explain the *base* of each subtraction in a full sentence. This practice transfers directly to percentage questions, where a correct numerator divided by the wrong original amount can create a believable but incorrect result.
Frequently asked questions
Does giving a discount always mean the seller loses money?
No. A product can sell below its marked price yet still sell above its buying cost.
What’s the difference between revenue and profit?
Revenue is the money collected from sales. The simple profit amount is what remains after the relevant specified costs are subtracted.
Can a fixed delivery charge change break-even?
Yes. When a cost is charged once per batch, it must be recovered across the units sold. More units can spread that fixed charge across a larger number of sales.
Why can’t I just add successive percentages?
Because the second percentage generally acts on a different base. Multipliers preserve the correct sequence.
Do we automatically add GST to a word-problem selling price?
No. Follow the stated GST-inclusive or exclusive condition and the relevant rules. Do not invent extra taxes in a problem that does not supply them.
What should I practise if word problems confuse my child?
Begin by identifying the quantities and what the question asks. Then write a simple relationship before calculating. Correct translation matters more than solving quickly.
More Secondary 1 G3 Mathematics guides
- Percentage change and reverse percentages
- Unitary method, ratios and recipe scaling
- Unit prices, GST and household budgets
- Formulae with multiple variables
- Word problems with bar models, tables and equations
The next time a question mentions a shop, don’t grab the percentage key immediately. Label the cost, marked price, selling price and any fixed fees first. Once the quantities are clear, the arithmetic becomes much more trustworthy.