# Types of Fractions and Equivalent Fractions

> Proper and improper fractions, mixed numbers, equivalent fractions and writing fractions in lowest terms.

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

### Story time: Pizza night
The Nguyen family orders pizzas cut into 4 slices each. Dad eats 3 slices, so he ate $\frac{3}{4}$ of a pizza. That is a **proper fraction**: less than a whole pizza.

Big brother Tran eats 7 slices. That is $\frac{7}{4}$ of a pizza, more than one whole. He ate one whole pizza and 3 more slices, which is $1\frac{3}{4}$ pizzas.

$\frac{7}{4}$ and $1\frac{3}{4}$ are the same amount of pizza, just written two ways.

**In short:** A proper fraction is less than 1, an improper fraction is 1 or more, and a mixed number has a whole part and a fraction part, like $2\frac{1}{3}$. Equivalent fractions are equal fractions, made by multiplying or dividing the top and bottom by the same number.

## Before you start

- A fraction has a **numerator** (top: how many parts you have) and a **denominator** (bottom: how many equal parts make a whole).
- Times tables and division with remainders, like $13 \div 4 = 3$ remainder 1.
- **Factors** of a number divide it exactly. Factors of 12: 1, 2, 3, 4, 6, 12.

## Three types of fraction

*Diagram: Seven quarters shown as one whole bar of four quarters plus three more quarters*
*Seven quarters make one whole and three quarters more.*

| Type | What it means | Examples |
|---|---|---|
| **Proper fraction** | top is smaller than bottom, so less than 1 | $\frac{2}{5}$, $\frac{3}{10}$, $\frac{1}{7}$ |
| **Improper fraction** | top is bigger than (or equal to) bottom, so 1 or more | $\frac{13}{10}$, $\frac{7}{4}$, $\frac{29}{12}$ |
| **Mixed number** | a whole number and a proper fraction | $1\frac{1}{3}$, $2\frac{7}{8}$, $6\frac{1}{10}$ |

## Improper fraction to mixed number

### Key idea: Divide the bottom into the top
Divide the numerator by the denominator. The answer is the whole number, and the **remainder** goes over the same denominator.

$\frac{13}{4}$: $13 \div 4 = 3$ remainder 1, so $\frac{13}{4} = 3\frac{1}{4}$.

### Worked example: More practice
$\frac{26}{5} = 5\frac{1}{5}$ because $26 \div 5 = 5$ remainder 1.

$\frac{39}{15} = 2\frac{9}{15} = 2\frac{3}{5}$. Always simplify the fraction part.

## Mixed number to improper fraction

### Key idea: Multiply, then add
Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.

$2\frac{3}{5}$: $2 \times 5 + 3 = 13$, so $2\frac{3}{5} = \frac{13}{5}$.

### Worked example: More practice
$1\frac{4}{7} = \frac{7 + 4}{7} = \frac{11}{7}$

$5\frac{2}{7} = \frac{35 + 2}{7} = \frac{37}{7}$

## Equivalent fractions

**Equivalent** fractions look different but have the **same value**.

$$\frac{2}{5} = \frac{4}{10} = \frac{8}{20}$$

### Key idea: Same to top and bottom
Multiply (or divide) the numerator and denominator by the **same number**. The value does not change.

$\frac{5}{9} = \frac{\square}{54}$: $9 \times 6 = 54$, so the top is $5 \times 6 = 30$.

## Lowest terms

A fraction is in **lowest terms** (or simplest form) when no number except 1 divides both the top and bottom.

To simplify, divide top and bottom by their **highest common factor** (HCF), the biggest number that divides both.

### Worked example: Simplifying
$\frac{14}{63}$: HCF of 14 and 63 is 7. $\frac{14 \div 7}{63 \div 7} = \frac{2}{9}$.

$\frac{32}{72}$: HCF is 8. Answer: $\frac{4}{9}$.

$\frac{45}{72}$: HCF is 9. Answer: $\frac{5}{8}$.

### Exam tip: Small steps are fine
If you cannot spot the HCF, divide by any common factor, then do it again. $\frac{144}{200} \to \frac{72}{100} \to \frac{36}{50} \to \frac{18}{25}$.

### Challenge: Selective level: comparing fractions
Which is bigger, $\frac{5}{8}$ or $\frac{7}{12}$? Make the bottoms the same: $\frac{15}{24}$ and $\frac{14}{24}$. So $\frac{5}{8}$ is bigger.

$\frac{11}{13} = \frac{44}{\square}$: the top was multiplied by 4, so the bottom is $13 \times 4 = 52$.

If a certain number is added to the denominator of $\frac{91}{128}$, the answer is $\frac{7}{11}$. Since $91 = 7 \times 13$, the new denominator must be $11 \times 13 = 143$. The number added is $143 - 128 = 15$.

**Mind map**

```text
Fractions
  Types
    Proper: top smaller
    Improper: top bigger
    Mixed: whole + fraction
  Improper to mixed
    Divide top by bottom
    Remainder over bottom
  Mixed to improper
    Whole x bottom + top
  Equivalent
    Same to top and bottom
    $\frac{2}{5} = \frac{8}{20}$
  Lowest terms
    Divide by the HCF
```

## Quick summary

- Proper fractions are less than 1. Improper fractions are 1 or more. Mixed numbers have a whole part and a fraction part.
- Improper to mixed: divide top by bottom, and the remainder goes over the bottom.
- Mixed to improper: whole times bottom, plus top, over the bottom.
- Equivalent fractions: multiply or divide top and bottom by the same number.
- Lowest terms: divide top and bottom by their highest common factor.

## Quiz

Try these questions from the chapter quiz.

1. Which is an improper fraction?
   - A. $\frac{3}{10}$
   - B. $\frac{13}{10}$
   - C. $1\frac{3}{10}$
   - D. $\frac{1}{7}$
   - Answer: B. $\frac{13}{10}$. The top (13) is bigger than the bottom (10).

2. $\frac{13}{4} = $
   - A. $3\frac{1}{4}$
   - B. $4\frac{1}{3}$
   - C. $3\frac{3}{4}$
   - D. $13\frac{1}{4}$
   - Answer: A. $3\frac{1}{4}$. $13 \div 4 = 3$ remainder 1.

3. $2\frac{3}{5} = $
   - A. $\frac{6}{5}$
   - B. $\frac{23}{5}$
   - C. $\frac{13}{5}$
   - D. $\frac{11}{5}$
   - Answer: C. $\frac{13}{5}$. $2 \times 5 + 3 = 13$, so $\frac{13}{5}$.

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