Practice Maths

Understanding Percentages

Key Terms

percentage
“Per cent” means out of 100. 37% = 37 out of every 100.
fraction
A way of writing part of a whole — numerator over denominator, e.g. 3⁄4 means 3 out of 4 equal parts.
decimal
A number expressed using place value, e.g. 0.75 means 75 hundredths.
convert
To rewrite a value in an equivalent form, e.g. changing a fraction into a percentage.

Conversion Reference

% → Decimal: divide by 100.   e.g. 72% → 0.72

Decimal → %: multiply by 100.   e.g. 0.45 → 45%

% → Fraction: write over 100 and simplify.   e.g. 60% = 60⁄100 = 3⁄5

Fraction → %: divide numerator by denominator, then × 100.   e.g. 3⁄8 = 0.375 × 100 = 37.5%

One quantity as % of another: (part ÷ whole) × 100

Hot Tip Always simplify fractions. Be careful with direction: to turn a % into a decimal, divide by 100 — the number gets smaller. 72% = 0.72, not 72.

Worked Example

Question: Express 18 out of 24 as a percentage.

Step 1 — Write as a fraction.
18 out of 24 = 18⁄24

Step 2 — Simplify the fraction.
18⁄24 = 3⁄4

Step 3 — Convert to a percentage.
3⁄4 × 100 = 75%

What Does "Per Cent" Actually Mean?

"Per cent" comes from the Latin per centum, meaning out of 100. So whenever you see the % symbol, you can swap it for "/100". For example, 37% simply means 37 out of every 100 — like scoring 37 out of 100 on a quiz, or a shirt being made of 37 out of 100 parts cotton.

Think of a 10 × 10 grid of 100 squares. If you shade 37 of them, that is 37%. This image makes it easy to visualise: half the grid shaded = 50%, all shaded = 100%, none shaded = 0%.

Converting Between Fractions, Decimals and Percentages

All three forms represent the same value — they are just written differently. Use these rules to switch between them:

Percentage → Decimal: divide by 100 (move the decimal point 2 places left). e.g. 72% → 0.72

Decimal → Percentage: multiply by 100 (move the decimal point 2 places right). e.g. 0.45 → 45%

Percentage → Fraction: write over 100 and simplify. e.g. 60% = 60100 = 35

Fraction → Percentage: divide numerator by denominator, then multiply by 100. e.g. 38 = 3 ÷ 8 = 0.375 → 37.5%

Common ones to memorise: 12 = 0.5 = 50%; 14 = 0.25 = 25%; 34 = 0.75 = 75%; 15 = 0.2 = 20%; 110 = 0.1 = 10%.

Percentages on a Number Line

Percentages sit on the number line just like decimals. 0% is at 0, 100% is at 1, and 50% is exactly halfway at 0.5. Values above 100% (like 150% = 1.5) are possible — they mean more than the original whole amount. A 150% price means one and a half times the original cost.

Placing percentages on a number line helps you check whether your answer is reasonable. A discount of 110% would be impossible (you can't take away more than the whole thing), so the number line catches that kind of error.

Expressing One Quantity as a Percentage of Another

If a student scored 18 out of 25 on a test, what percentage is that? Use the rule: (part ÷ whole) × 100.

18 ÷ 25 = 0.72 → 0.72 × 100 = 72%

Another example: a bag of trail mix has 80 g of nuts out of a total 200 g. Percentage of nuts = (80 ÷ 200) × 100 = 40%. This is exactly the calculation you see on nutrition labels.

Always check which number is the "whole" (the total, the denominator). The part goes on top, the whole goes on the bottom.

Percentages in Real Life

Percentages appear everywhere: GST (10% added to prices in Australia), discounts at the shops ("20% off"), nutrition labels (% of daily intake), battery levels on your phone, and interest rates on bank savings. Understanding percentages means you can make smarter decisions about money and understand the world around you.

Example: a pair of shoes costs $120 and GST of 10% is added. The tax amount = 10% of $120 = 0.10 × $120 = $12. Total price = $120 + $12 = $132.

Key tip: A very common mistake is confusing "percent to decimal" direction. Remember: to turn a % into a decimal, you divide by 100 — so the number gets smaller. If 72% becomes 72.0 in your working, something has gone wrong! The decimal should be 0.72.

Mastery Practice

  1. Convert each percentage to a fraction in simplest form. Fluency

    1. 50%
    2. 25%
    3. 75%
    4. 10%
    5. 20%
    6. 40%
    7. 5%
    8. 80%

  2. Convert each percentage to a decimal. Fluency

    1. 30%
    2. 45%
    3. 8%
    4. 12.5%
    5. 100%
    6. 3%
    7. 62.5%
    8. 1%

  3. Convert each decimal or fraction to a percentage. Fluency

    1. 0.7
    2. 0.04
    3. 0.125
    4. 0.375
    5. ½
    6. 3⁄10
    7. 7⁄20
    8. 9⁄25

  4. Express the first quantity as a percentage of the second. Understanding

    1. 15 out of 60
    2. 7 out of 20
    3. 36 out of 90
    4. 8 out of 32
    5. 45 out of 75
    6. 3 out of 8

  5. Compare these results using percentages. Understanding

    1. Mia scored 18 out of 24 on a test. Ben scored 21 out of 30. Convert both to percentages. Who performed better?
    2. A school has 480 students. 360 attend sport. What percentage of students attend sport?
    3. Arrange in ascending order (as percentages): ¾,   0.68,   71%,   17⁄25
    4. Is 0.4 greater than, equal to, or less than 42%? Show your reasoning.

  6. Apply percentage understanding to real situations. Problem Solving

    1. A class of 28 students has 7 absent one day. What percentage of the class is present?
    2. In a bag of 40 marbles, 14 are blue. Write the fraction of blue marbles, convert it to a percentage, and explain what this means.
    3. Three students compare how much of their books they have read: Alice has read ⅖, Ben has read 43%, Carla has read 0.39. Order them from most to least read.
    4. A survey of 120 people found that 84 prefer tea over coffee. A second survey of 200 people found 130 prefer tea. Which group has a higher percentage of tea drinkers?

  7. Convert each percentage to a simplified fraction and a decimal. Fluency

    1. 4%
    2. 12.5%
    3. 37.5%
    4. 120%
    5. 0.4%
    6. 16⅔%
    7. 33⅓%
    8. 87.5%

  8. Interpret these percentage statements. Understanding

    1. A shirt is 100% cotton. What does this mean?
    2. A sports drink contains 8% sugar by volume. How many mL of sugar are in a 250 mL bottle?
    3. A bank account earns 2.5% interest per year on a $400 balance. How much interest is earned in one year?
    4. A factory has a 98% quality pass rate for its products. Out of 2500 items produced, how many would you expect to fail quality checks?

  9. Order and compare. Understanding

    1. Arrange these in descending order (largest first): 0.82,   79%,   4⁄5,   17⁄20
    2. Which is larger: 7⁄12 or 58%? Show your working.
    3. Three candidates in an election received the following votes: Candidate A: 48 out of 120; Candidate B: 35%; Candidate C: 0.38 of the vote. Rank them from highest to lowest vote share.
    4. A student scores 36 out of 45 on Test 1 and 52 out of 65 on Test 2. Which is the better result as a percentage?

  10. Extended percentage investigation. Problem Solving

    1. In a school of 650 students, 52% are female. How many male students are there?
    2. A car park has 80 spaces. At 9 am, 60% are occupied. By noon, 15 more cars leave and 8 new cars arrive. What percentage of spaces are occupied at noon?
    3. Over one month, the number of visitors to a website increased from 4800 to 6000. Write the increase as a fraction of the original and express it as a percentage.
    4. In a bag of mixed lollies, 30% are red, 1⁄4 are green, 0.2 are yellow, and the rest are orange. What percentage are orange? If there are 80 lollies in total, how many are orange?
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