Money problems are percentage problems with units
Nothing in this topic is new mathematics. What makes it examinable is that the numbers carry units and meanings, and the marks go to answers that respect them: money to two decimal places, currency conversions in the right direction, and a final answer that is a plausible amount for the thing being described.
The recurring calculations are earnings, discount, tax and profit, and each is a percentage of something specific — which is the part to get right before any arithmetic.
| Term | Meaning | Percentage of what |
|---|---|---|
| Gross pay | earnings before deductions | — |
| Net pay | what is actually received | gross minus deductions |
| Commission | a share of sales made | the value sold |
| Discount | a reduction offered | the marked price |
| Profit | selling price minus cost price | usually the COST price |
| Loss | cost price minus selling price | usually the cost price |
| Sales tax | added to the price | the pre-tax price |
Percentage profit is of the cost price
An item bought for Rs 400 and sold for Rs 500 makes Rs 100 profit. The percentage profit is 100 ÷ 400 = 25%, dividing by what it cost. Dividing by the selling price gives 20%, which is a different quantity called the margin. Unless a question says otherwise, profit and loss percentages are of the cost price.
Currency conversion
An exchange rate is a rate like any other, and the direction is decided by the units. If 1 US$ = 280 PKR, then converting dollars to rupees multiplies by 280 and converting rupees to dollars divides.
The check is to ask whether the answer should be a bigger or a smaller number. A rupee is worth far less than a dollar, so any amount expressed in rupees should have more digits. If your answer to "convert $50 to rupees" is smaller than 50, the operation was the wrong way round.
A tourist changes Rs 84 000 into dollars at 1 US$ = 280 PKR, spends $210, and changes the rest back at 1 US$ = 275 PKR. How many rupees does she end with?
- Rupees to dollars: divide.
84 000 ÷ 280 = $300.Fewer dollars than rupees, as expected. - After spending:
300 − 210 = $90. - Dollars back to rupees: multiply, at the new rate.
90 × 275 = Rs 24 750.The rate has changed, and using the old one is the trap in this question. - She has lost Rs 450 relative to the original rate, purely through the change in exchange rate.Worth stating — questions frequently ask for this comparison as the final part.
Rs 24 750
Time, timetables and the 24-hour clock
Time is the one quantity in the syllabus that is not decimal, and that is the whole difficulty. There are 60 minutes in an hour, so 2 hours 30 minutes is 2.5 hours, not 2.30.
The 24-hour clock removes am/pm ambiguity: 14:35 is unmistakable where 2:35 is not. To find a duration, work forward from the start time in whole hours first and then the minutes, rather than subtracting the two clock readings directly — 14:10 minus 09:45 is not 5:35.
A train leaves at 09:47 and arrives at 13:22. Find the journey time, and the average speed if the distance is 245 km.
- From 09:47 to 13:47 is exactly 4 hours.Move whole hours first — it avoids borrowing across the 60-minute boundary.
- But arrival is 13:22, which is 25 minutes earlier, so subtract 25 minutes.Going past and coming back is easier than counting minutes forward from 09:47.
- Journey time
= 3 hours 35 minutes. - As a decimal:
35 ÷ 60 = 0.5833, so3.5833hours.Speed needs decimal hours. Using 3.35 would give a wrong answer. - Speed
= 245 ÷ 3.5833 = 68.4km/h.
3 hours 35 minutes; 68.4 km/h
2 hours 30 minutes is 2.5, not 2.30
Minutes are sixtieths, not hundredths. Divide the minutes by 60 before using a time in any speed, rate or cost calculation. This single conversion accounts for most of the wrong answers in the topic, and the error is invisible because 2.30 looks entirely plausible.
Exponential growth and decay
A quantity grows exponentially when it is multiplied by the same factor each period, rather than having the same amount added. Compound interest is the standard example, and so are population growth, bacterial cultures, and depreciation of a car.
The formula is the compound interest formula in general clothing. A growth of 8% per period multiplies by 1.08 each time; a decay of 8% multiplies by 0.92.
A car bought for Rs 2 400 000 depreciates by 18% per year. Find its value after 4 years, and after how many years it first falls below half its original value.
- Depreciation of 18% means a multiplier of
0.82each year.100% − 18% = 82%. - After 4 years:
2 400 000 × 0.82⁴ = 2 400 000 × 0.45212 = Rs 1 085 000(3 s.f.).The multiplier raised to the power of the number of years. - Half the original value is Rs 1 200 000, so we need
0.82ⁿ < 0.5.Working with the multiplier alone avoids carrying the large number through. - Try values:
0.82³ = 0.551,0.82⁴ = 0.452.Trial is perfectly acceptable here; logarithms would also work. - So it falls below half during the fourth year.After three years it is still above half; after four it is below.
About Rs 1 085 000 after 4 years, and it drops below half during year 4.
Before you leave this chapter
- Percentage profit and loss are of the COST price unless stated otherwise.
- Currency: multiply or divide as the units require, then check the answer is the expected size.
- 2 hours 30 minutes is 2.5 hours. Divide minutes by 60 before any rate calculation.
- For a duration, move whole hours first and adjust the minutes afterwards.
- Exponential growth multiplies by (1 + r) each period; decay by (1 − r).
Compound growth is a geometric sequence
Money in a compound-interest account forms a geometric sequence: each year's balance is the previous one multiplied by a fixed factor. Recognising that connects this chapter to sequences and explains why the totals behave as they do.
The balances after 0, 1, 2, 3 … years are P, Pr, Pr², Pr³ … which is a geometric progression with first term P and common ratio r. The compound interest formula is nothing more than the nth term of that sequence.
Choose Geometric with r above 1. Each bar is larger than the last by the same factor, and the running total curves upward — which is exactly what a compound-interest balance does year on year.
Interest paid more often than yearly
A rate of 12% per year compounded monthly means 1% each month for 12 months, so the multiplier is 1.01¹² rather than 1.12. That comes to 1.1268, slightly more than the annual figure — because interest starts earning interest sooner. Always check the period the rate applies to before choosing the exponent.