What you'll learn
Per cent means "out of a hundred", so 45% is 45 out of every 100, which is the fraction 45/100 and the decimal 0.45. This topic covers converting freely between fractions, decimals and percentages, finding a percentage of an amount, working out a score as a percentage, and handling discounts and increases. It is one of the most useful topics on the SEA Mathematics paper because it turns up constantly in shops, in test results and in news reports, and questions about it are usually set as two-step problems where the arithmetic is easy but the reading has to be careful.
Key terms and definitions
Per cent (%) — out of one hundred. The symbol % is a shorthand for "/100".
Percentage — a way of writing a part of a whole using hundredths.
Convert — to rewrite the same value in a different form, such as changing 1/4 into 25%.
Discount — an amount taken off a price, often expressed as a percentage of the original.
Sale price (reduced price) — the price left to pay after the discount has been subtracted.
Increase — an amount added on, such as a fare going up by 25%.
Original price (marked price) — the price before any discount is applied.
Core concepts
The three forms are the same number
A fraction, a decimal and a percentage are three ways of writing one value. Being able to move between them quickly is what makes this topic easy.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 3/5 | 0.6 | 60% |
Learning this table by heart saves a great deal of time in the exam.
Converting between the forms
Percentage to decimal: divide by 100, which moves the digits two places right. 45% becomes 0.45.
Decimal to percentage: multiply by 100, moving the digits two places left. 0.6 becomes 60%. Note that 0.6 is six tenths, which is sixty hundredths — this is why it is 60% and not 6%.
Percentage to fraction: write it over 100 and simplify. 20% = 20/100 = 1/5.
Fraction to percentage: make the denominator 100. For 18/25, multiply top and bottom by 4 to get 72/100, which is 72%. Where the denominator does not divide neatly into 100, convert to a decimal first.
Finding a percentage of an amount
The reliable method is to build from 10%, which is found by dividing by 10.
To find 30% of 40: 10% of 40 is 4, so 30% is 4 × 3 = 12.
Other useful building blocks:
- 50% is one half — halve the amount.
- 25% is one quarter — halve it twice.
- 5% is half of 10%.
- 1% is the amount divided by 100.
So 15% of $60 is $6 (10%) plus $3 (5%), which is $9. Building a percentage from parts is quicker and safer than any formula at this level.
Writing a score as a percentage
Turn the score into a fraction of the total, then make the total 100.
Anisa scored 18 out of 25. Since 25 × 4 = 100, multiply both numbers by 4: 72 out of 100, so 72%.
For 21 out of 30, simplify first to 7 out of 10, which is 70 out of 100, or 70%.
This is how two scores with different totals can be compared fairly. A score of 24 out of 40 and a score of 15 out of 25 are both 60%, even though the raw numbers look nothing alike.
Discounts and increases
A discount is subtracted from the original price; an increase is added to it.
A shirt costs $240 with 25% off. A quarter of $240 is $60, so $60 comes off and the shirt now costs $180.
Both figures are correct answers to different questions, and questions are deliberately written to test which one you give. "How much is taken off?" wants $60. "What is the sale price?" wants $180.
For an increase, a $12 fare rising by 25% goes up by $3 to $15.
A useful shortcut
Percentages can be swapped around: 25% of 80 gives the same answer as 80% of 25, because both work out 25 × 80 ÷ 100. Both are 20. When one way round is awkward, try the other.
Worked examples
Example 1: Finding a percentage of a quantity
There are 40 children in a class and 30% of them wear glasses. How many wear glasses?
Find 10% first: 40 ÷ 10 = 4 children.
Then 30% is 4 × 3 = 12 children.
The wrong answer of 4 is 10% rather than 30% — a step left unfinished. Building from 10% is safe, but you have to finish the multiplication.
Example 2: Discount then a further reduction
A television costs $2 400. The shop takes 30% off, and Devon has a voucher for a further $150 off the sale price. How much does he pay?
10% of $2 400 is $240, so 30% is $720.
The sale price is $2 400 − $720 = $1 680.
The voucher then applies: $1 680 − $150 = $1 530.
Two wrong answers are built into this question: $720 is the discount alone, and $1 680 forgets the voucher. Working in labelled steps makes it obvious which number answers the question.
Example 3: Comparing two scores
Riya scored 24 out of 40 in Mathematics and 15 out of 25 in English. In which subject did she do better?
Convert both to percentages. For 24 out of 40, divide both by 4 to get 6 out of 10, which is 60%. For 15 out of 25, multiply both by 4 to get 60 out of 100, which is 60%.
The two scores are exactly the same, at 60% each.
The larger raw score of 24 does not mean the better result, because the totals are different. This is precisely why percentages exist.
Common mistakes and how to avoid them
Giving the discount when the sale price was asked for. A jacket reduced by $80 does not cost $80. Fix: read the final question again before writing the answer.
Moving the decimal one place instead of two. Writing 0.6 as 6%. Fix: a percentage is hundredths, so the digits move two places.
Using the number in the percentage as the denominator. Thinking 20% is 1/20. Fix: 20% is 20/100, which simplifies to 1/5. The fraction 1/20 is only 5%.
Stopping at 10%. Finding 10% correctly and forgetting to multiply up to the percentage asked for. Fix: write down what you are building — "10% is 4, so 30% is 12".
Comparing raw scores instead of percentages. Assuming 24 out of 40 beats 15 out of 25. Fix: convert both before comparing anything.
Adding a percentage when the question says reduced. Fix: underline the words "off", "discount", "reduced" or "increase" before starting.
How parents can help at home
Percentages are the one topic where a parent can help simply by talking, because the situations are everywhere.
Read sale signs together. "30% off $2 400 — what does that save us, and what will we pay?" Asking for both numbers each time builds exactly the habit the exam tests.
Use test results. When your child brings home a score of 18 out of 25, ask what that is as a percentage. Then ask which was better, that or 21 out of 30. This makes the point of percentages obvious in a way no worksheet can.
Practise the 10% step. Ask for 10% of any number you happen to see — a bill, a price, a distance. It is just dividing by ten, it takes two seconds, and it is the foundation of every other percentage they will need.
Ask "off or after?" When your child gives an answer to a discount question, ask whether that is the amount taken off or the price now paid. Most lost marks here are the right arithmetic answering the wrong question.
Exam technique for percent and converting
Learn the conversion table by heart. Recognising instantly that 3/4 is 75% and 0.2 is 20% saves time that can be spent on the harder items.
Build percentages from 10%, 25% and 50% rather than reaching for a formula. It is faster, and every intermediate figure is a sensible number you can check.
Show the steps for two-part questions. Write "discount = $720" and "sale price = $1 680" on separate lines, then answer the question actually asked. Under time pressure, unlabelled numbers get confused with one another.
Check the direction of your answer. A discount must make the price smaller and an increase must make it larger. If your answer moves the wrong way, you have added where you should have subtracted.
In multiple-choice items, expect the discount, the sale price and the original price all to appear as options. Only one answers the question.
Quick revision summary
- Per cent means out of one hundred: 45% = 45/100 = 0.45.
- Learn the table: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%.
- Percentage to decimal: divide by 100. Decimal to percentage: multiply by 100.
- Percentage to fraction: write it over 100 and simplify.
- Find 10% by dividing by 10, then build: 30% is three lots of 10%, 5% is half of 10%.
- 50% is a half, 25% is a quarter.
- Score as a percentage: make the total 100, then read off the numerator.
- Discount is what comes off; the sale price is what is left to pay. They are different answers.
- 25% of 80 equals 80% of 25 — swap them round if it makes the sum easier.