One quantity, three notations
A fraction, a decimal and a percentage are three ways of writing the same proportion, and moving between them freely is what makes the rest of this chapter easy.
A fraction becomes a decimal by dividing. A decimal becomes a percentage by multiplying by 100. A percentage becomes a fraction by writing it over 100 and cancelling. The conversions worth knowing by sight are the common ones, because recognising them saves working every time.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/3 | 0.333… | 33⅓% |
| 1/8 | 0.125 | 12.5% |
| 2/3 | 0.666… | 66⅔% |
Which form to work in
For calculating, decimals are usually easiest — a calculator handles them directly. For exact answers, fractions are better, since 1/3 is exact and 0.333 is not. For comparing, percentages win, because everything is out of the same 100. Choosing the right form for the job is half the skill.
Percentage of, increase and decrease
Finding a percentage of an amount is a multiplication: 15% of 240 is 0.15 × 240 = 36. The word "of" means multiply, and converting the percentage to a decimal first is quicker than any other route.
For an increase or decrease, the efficient method is a single multiplier rather than finding the change and then adding it. An increase of 15% multiplies by 1.15; a decrease of 15% multiplies by 0.85. One step instead of two, and far fewer errors.
A shirt costing Rs 1200 is reduced by 15%, then the reduced price is increased by 15%. Is the final price Rs 1200?
- Decrease: multiply by
0.85, giving1200 × 0.85 = 1020.One multiplier does the whole decrease. - Increase: multiply by
1.15, giving1020 × 1.15 = 1173.The 15% is now 15% of 1020, not of 1200 — that is the whole point. - The final price is Rs 1173, not Rs 1200.A 15% rise does not undo a 15% fall, because the two percentages are of different amounts.
- Combined multiplier:
0.85 × 1.15 = 0.9775, a net fall of 2.25%.Multipliers can be combined directly, which is why they are worth using.
No — Rs 1173. A percentage increase and decrease of the same size do not cancel.
Reverse percentages
This is the topic that separates grades, and it is one idea: when you are told the price after a change, that price is not 100%.
If a price includes 20% tax, the amount you see is 120% of the original. Dividing by 1.2 recovers the original. The near-universal error is to take 20% off the final price instead, which gives a different and wrong answer — because 20% of the larger figure is more than 20% of the smaller one.
After a 12% increase, a salary is Rs 44 800. What was it before?
- The new salary is 112% of the old one, so the multiplier used was 1.12.Write down what the given figure represents as a percentage. This one step is the whole question.
- To undo a multiplication, divide:
44 800 ÷ 1.12.Not subtract 12% — that would be taking 12% of the wrong number. = 40 000.- Check:
40 000 × 1.12 = 44 800✓Always check forwards. The wrong method would have given 39 424, which fails this check immediately.
Rs 40 000
How to spot a reverse percentage question
Look for the word after, or a price described as including tax, or the phrase "in a sale". If the figure you are given is the one that has already been changed, you must divide by the multiplier. If it is the one before the change, you multiply. Deciding which of the two you have been given is the entire question.
Simple and compound interest
Simple interest is calculated on the original amount every time, so the same interest is added each year and the total grows in a straight line.
Compound interest is calculated on the amount currently there, so each year's interest is slightly larger than the last and the total grows exponentially. Nearly all real savings and loans are compound, and the difference over a long period is substantial.
- P
- the principal — the amount invested or borrowed
- R
- the rate per period, as a percentage
- T
- the number of periodsusually years
Before you leave this chapter
- Fraction → decimal by dividing; decimal → percentage by multiplying by 100.
- Use a single multiplier: increase by 15% is × 1.15, decrease is × 0.85.
- Percentage change divides by the ORIGINAL value.
- Told the amount AFTER a change? Divide by the multiplier — never subtract the percentage.
- Simple interest is on the principal each time; compound is on the running total.
Seeing the three forms as one quantity
Converting between fractions, decimals and percentages becomes automatic once you stop thinking of them as three different things. They are three notations for the same amount, and a bar divided into equal parts shows all three at once.
Choose 1/3 and look at the decimal. It never terminates, so any decimal written down is an approximation — which is exactly why an exact answer must be left as a fraction.
Rounding early destroys accuracy
Working with 0.33 instead of 1/3 introduces an error that grows with every subsequent operation. In a multi-step calculation, keep fractions or full calculator accuracy throughout and round only the final answer. Rounding at each stage is one of the few ways to lose marks while doing every step correctly.