# Fractions, Decimals and Percentages

> Switch between fractions, decimals and percentages, and find a percentage of an amount in your head.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/fractions-decimals-percentages/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The sale sign
Amira sees a sign in a shop window: "Everything 30% off!" She wants a jacket that costs 50 dollars.

"30% means 30 out of every 100," her mum says. "So for every 10 dollars, you save 3 dollars." The jacket has five lots of 10 dollars, so Amira saves $5 \times 3 = 15$ dollars. She pays 35 dollars.

Percentages, fractions and decimals are three ways of writing the same idea: a part of a whole.

**In short:** Fractions, decimals and percentages are three ways to write the same amount, and percent means out of 100, so $35\% = \frac{35}{100} = 0.35$. To find a percentage of an amount, build it from 10%, 5% and 1%.

## Before you start

- A **fraction** like $\frac{3}{4}$ means 3 parts out of 4 equal parts. The top is the **numerator** and the bottom is the **denominator**.
- **Equivalent fractions:** multiply or divide top and bottom by the same number, such as $\frac{1}{5} = \frac{20}{100}$.
- Decimal place value: $0.35$ is 35 hundredths.

## Percent means "out of 100"

The sign $\%$ means **per cent**, which is "out of 100". So $35\% = \frac{35}{100} = 0.35$.

### Key idea: Converting
| From | To | How |
|---|---|---|
| percentage | fraction | write over 100 and simplify: $60\% = \frac{60}{100} = \frac{3}{5}$ |
| percentage | decimal | divide by 100: $5\% = 0.05$ |
| decimal | percentage | multiply by 100: $0.8 = 80\%$ |
| fraction | percentage | make the bottom 100, or multiply by $100\%$: $\frac{1}{4} = \frac{25}{100} = 25\%$ |

## Facts worth knowing by heart

| Fraction | Decimal | Percentage |
|---|---|---|
| $\frac{1}{2}$ | $0.5$ | $50\%$ |
| $\frac{1}{4}$ | $0.25$ | $25\%$ |
| $\frac{3}{4}$ | $0.75$ | $75\%$ |
| $\frac{1}{5}$ | $0.2$ | $20\%$ |
| $\frac{1}{10}$ | $0.1$ | $10\%$ |
| $\frac{1}{20}$ | $0.05$ | $5\%$ |
| $\frac{1}{100}$ | $0.01$ | $1\%$ |
| $\frac{1}{3}$ | $0.333\dots$ | $33\frac{1}{3}\%$ |

### Worked example: Mixed numbers to percentages
$1\frac{1}{2} = 1 + \frac{1}{2} = 100\% + 50\% = 150\%$

$2\frac{1}{4} = 200\% + 25\% = 225\%$

$3\frac{3}{4} = 300\% + 75\% = 375\%$

A whole is $100\%$, so $65\% + 35\% = 100\% = 1$.

## Finding a percentage of an amount

### Key idea: Build it from easy pieces
- $10\%$: divide by 10
- $5\%$: half of $10\%$, or divide by 20
- $1\%$: divide by 100
- $50\%$: halve
- $25\%$: halve, then halve again

Then add pieces together. $15\% = 10\% + 5\%$ and $30\% = 3 \times 10\%$.

### Worked example: Three ways in
$15\%$ of 200: $10\%$ is 20, $5\%$ is 10, so $15\%$ is 30.

$30\%$ of 50: $10\%$ is 5, so $30\%$ is 15.

$5\%$ of 480: $480 \div 20 = 24$.

### Worked example: Fraction of an amount
$\frac{2}{3}$ of 90: one third is $90 \div 3 = 30$, so two thirds is 60.

## Going backwards

If you know the part, you can find the whole.

### Worked example: Finding the whole
$5\%$ of a number is 12. Since $5\%$ is $\frac{1}{20}$, the whole is $12 \times 20 = 240$.

$Z\%$ of 200 is 100. 100 is half of 200, so $Z = 50$.

### Watch out: Percent of what?
$40\%$ of a tonne is 400 kg, not 400 tonnes. A tonne is 1000 kg, and $40\%$ of 1000 is 400. Always keep the unit.

### Challenge: Selective level: careful conversions
$5\%$ of $0.4$: $0.4 \div 20 = 0.02$.

$33\%$ of 10 kg in grams: $10 \text{ kg} = 10\,000 \text{ g}$, and $1\%$ is 100 g, so $33\%$ is $3300$ g.

Which is bigger, $\frac{3}{8}$ or $36\%$? $\frac{3}{8} = 0.375 = 37.5\%$, so $\frac{3}{8}$ is bigger.

**Mind map**

```text
Fractions, decimals, percentages
  Percent
    Out of 100
    $35\% = 0.35$
  Convert
    % to decimal: ÷ 100
    Decimal to %: x 100
    Fraction: make the bottom 100
  Percent of an amount
    10%: ÷ 10
    5%: ÷ 20
    Add pieces
  Backwards
    5% is 12, so the whole is 240
```

## Quick summary

- Percent means out of 100: $35\% = \frac{35}{100} = 0.35$.
- Percentage to decimal: divide by 100. Decimal to percentage: multiply by 100.
- Learn the common ones: $\frac{1}{2} = 50\%$, $\frac{1}{4} = 25\%$, $\frac{1}{5} = 20\%$, $\frac{1}{20} = 5\%$.
- For a percentage of an amount, build it from $10\%$, $5\%$ and $1\%$.
- $5\%$ of an amount is the amount divided by 20.
- To find the whole from a part, work backwards.

## Quiz

Try these questions from the chapter quiz.

1. $0.05 = $
   - A. $50\%$
   - B. $5\%$
   - C. $0.5\%$
   - D. $0.05\%$
   - Answer: B. $5\%$. Multiply by 100: $0.05 \times 100 = 5$.

2. $20\% = \frac{\square}{5}$. What is in the box?
   - A. 1
   - B. 2
   - C. 4
   - D. 20
   - Answer: A. 1. $20\% = \frac{20}{100} = \frac{1}{5}$.

3. What is $10\%$ of 340?
   - A. 3.4
   - B. 34
   - C. 340
   - D. 30
   - Answer: B. 34. Divide by 10: 34.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/fractions-decimals-percentages/#quiz