A rate compares two different units, such as km/h or $/kg, so always check what unit the question wants in the answer before you calculate. A constant rate is really direct proportion in disguise: total quantity rises in step with time, in a fixed ratio. For profit and loss, a profit is SP − CP and a loss is CP − SP, and percentage profit or loss is always found by dividing by the cost price, never the selling price: (profit ÷ CP) × 100%. Keep cost price and selling price labelled clearly and the rest is straightforward arithmetic.
Rate and profit and loss questions look like everyday arithmetic, and they are, which is exactly why students rush them and lose marks on careless slips. A rate question goes wrong when the answer is given in the wrong unit, because the student did not check which unit was actually requested. A profit and loss question gets the percentage wrong because the student divided by the selling price instead of the cost price, or forgot that a loss uses CP − SP, not SP − CP.
None of this needs new mathematics. It needs a habit of labelling your quantities clearly before you calculate, and checking which number the percentage is "of".
Rate questions
Rate = quantity ÷ time (or per unit)
Check the required unit first
Profit or loss
Profit = SP − CP
Loss = CP − SP
Percentage profit
(Profit ÷ CP) × 100%
Percentage loss
(Loss ÷ CP) × 100%
A water tank fills at a rate of 18 litres every 4 minutes. Find the rate in litres per hour.
Solution
4 minutes = 4/60 hour = 1/15 hour.18 ÷ (1/15) = 18 × 15 = 270.Quick check: 270 litres per hour means 270 ÷ 60 = 4.5 litres per minute, and 4.5 × 4 = 18 litres in 4 minutes. Matches the question, correct.
Ravi buys a bicycle for $240 and sells it for $300. Find his percentage profit.
Solution
CP = $240. Selling price SP = $300.300 − 240 = $60.(60 ÷ 240) × 100% = 25%.The step students get wrong
Percentage profit and percentage loss are always calculated using the cost price as the denominator, never the selling price. Dividing 60 by 300 instead of 240 gives 20%, which is wrong. The cost price is what the seller originally paid, so it is always the reference point for measuring the change.
A shop sells a jacket for $68, making a loss of 15%. Find the cost price.
Solution
0.85 × CP = 68.CP = 68 ÷ 0.85 = $80.Quick check: 15% of $80 is $12, and $80 − $12 = $68, matching the selling price given. This "finding the original value" style question is the same reverse thinking used in reverse percentage questions: work from what percentage the given amount represents, not straight off the percentage itself.
Some exam questions combine rate and cost in the same problem, such as a taxi fare that has a fixed charge plus a rate per kilometre. Read carefully to separate the fixed part from the per-unit part before setting up your calculation.
Example: taxi fare
A taxi charges a flag-down fare of $3.20 plus $0.55 per kilometre travelled. Find the fare for a journey of 12 km.
0.55 × 12 = $6.60.3.20 + 6.60 = $9.80.Profit and loss percentage questions often produce recurring decimals partway through. Keep the full value in your calculator's memory rather than rounding early.
On the fx-97SG CW and fx-97SG X
(60÷240)×100, rather than calculating 60 ÷ 240 first and re-typing the rounded decimal.Exam habit: write CP and SP clearly at the start of every profit and loss question before doing any arithmetic. This single habit removes almost all the careless mistakes in this topic.
Why is percentage profit always based on cost price, not selling price?
Percentage profit measures how much return the seller made compared to what they originally spent, so the cost price is the natural reference point. Some real-world contexts use selling price as the base instead, but for O-Level E-Maths, always use cost price unless the question explicitly says otherwise.
How is a profit and loss question different from a reverse percentage question?
A profit and loss question usually gives you the cost price and the selling price and asks for the percentage change. A reverse percentage question gives you the final amount and the percentage change, and asks you to find the original amount, which is closer to worked example 3 above. See our reverse percentage guide for more practice on working backwards.
Is compound interest the same as repeated profit and loss?
They use similar percentage thinking but are not the same topic. Compound interest applies a percentage increase repeatedly over several periods, while profit and loss is usually a single transaction. See our simple versus compound interest guide if a question asks about interest instead of a straightforward sale.
What if a rate question mixes units, like km and minutes, but asks for m/s?
Convert everything to the required unit before doing the division. It is easiest to convert distance and time separately first, then divide, rather than dividing first and converting the messy decimal answer afterwards.
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