# Adding and Subtracting Fractions

> Use a common denominator to add and subtract fractions and mixed numbers, including borrowing a whole.

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

### Story time: The cake that disappeared
Julie eats $\frac{1}{3}$ of a birthday cake. Her sister Jessie eats $\frac{1}{4}$ of the same cake. Mum comes home and asks, "How much is left?"

Thirds and quarters are different sized pieces, so you cannot just add the tops. Julie cuts each third into 4 and each quarter into 3. Now everything is in twelfths: she ate $\frac{4}{12}$ and Jessie ate $\frac{3}{12}$. Together $\frac{7}{12}$, so $\frac{5}{12}$ of the cake is left.

Making the pieces the same size is the whole secret of adding and subtracting fractions.

**In short:** You can only add or subtract fractions with the same denominator (bottom number). Change them to the lowest common denominator first, then add or subtract the tops, and never add the bottoms: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.

## Before you start

- Equivalent fractions: $\frac{1}{3} = \frac{4}{12}$ (see Types of Fractions).
- Changing between mixed numbers and improper fractions: $1\frac{3}{8} = \frac{11}{8}$.
- **Multiples** of a number are its times table. Multiples of 4: 4, 8, 12, 16, ...

## Same denominator: just add the tops

$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$. The pieces are already the same size (sevenths), so count them. The bottom does **not** change.

### Watch out: Never add the bottoms
$\frac{1}{3} + \frac{1}{4}$ is **not** $\frac{2}{7}$. Two pieces of cake cannot be less than half the cake! The bottom tells you the size of the pieces. It is not something you add.

## Different denominators

### Key idea: Find a common denominator
1. Find the **lowest common denominator**: the smallest number that both bottoms divide into.
2. Change each fraction to an equivalent one with that bottom.
3. Add or subtract the tops.
4. Simplify, and change an improper answer to a mixed number.

### Worked example: Adding
$\frac{3}{5} + \frac{5}{6}$. The lowest common denominator of 5 and 6 is 30.

$\frac{18}{30} + \frac{25}{30} = \frac{43}{30} = 1\frac{13}{30}$

### Worked example: Subtracting
$\frac{2}{3} - \frac{1}{12}$. Twelve is a multiple of 3, so use 12.

$\frac{8}{12} - \frac{1}{12} = \frac{7}{12}$

### Exam tip: The bigger bottom might work
Check if the bigger denominator is already a multiple of the smaller one. For $\frac{3}{7} + \frac{3}{21}$, use 21: $\frac{9}{21} + \frac{3}{21} = \frac{12}{21} = \frac{4}{7}$.

## Mixed numbers

Add or subtract the **whole numbers first**, then the fractions.

### Worked example: Adding mixed numbers
$1\frac{3}{8} + 5\frac{1}{4}$

Wholes: $1 + 5 = 6$. Fractions: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.

Answer: $6\frac{5}{8}$

### Worked example: When the fractions add to more than 1
$3\frac{5}{6} + 2\frac{2}{3}$

Wholes: $5$. Fractions: $\frac{5}{6} + \frac{4}{6} = \frac{9}{6} = 1\frac{3}{6} = 1\frac{1}{2}$.

Answer: $5 + 1\frac{1}{2} = 6\frac{1}{2}$

## Subtracting when the second fraction is bigger

Sometimes you cannot take the second fraction from the first. Then subtract the whole numbers, and change what is left into an improper fraction.

### Worked example: Borrowing
$4\frac{1}{2} - 2\frac{7}{8}$

Wholes: $4 - 2 = 2$, which leaves $2\frac{4}{8} - \frac{7}{8}$.

Change to an improper fraction: $\frac{20}{8} - \frac{7}{8} = \frac{13}{8} = 1\frac{5}{8}$.

### Worked example: Another way to borrow
$3\frac{1}{5} - 1\frac{4}{5}$. Borrow 1 from the 3: $3\frac{1}{5} = 2\frac{6}{5}$.

$2\frac{6}{5} - 1\frac{4}{5} = 1\frac{2}{5}$

### Challenge: Selective level: word problems
There were $3\frac{1}{2}$ pizzas. John ate $\frac{3}{4}$ of a pizza and Paul ate $1\frac{1}{6}$ pizzas. How many were left?

Use twelfths: $3\frac{6}{12} - \frac{9}{12} - 1\frac{2}{12}$.

$3\frac{6}{12} - 1\frac{2}{12} = 2\frac{4}{12}$, then $2\frac{4}{12} - \frac{9}{12} = 1\frac{16}{12} - \frac{9}{12} = 1\frac{7}{12}$ pizzas.

Lilyn took $3\frac{1}{2}$ minutes on one problem and $\frac{3}{5}$ of a minute less on the next. The second took $3\frac{5}{10} - \frac{6}{10} = 2\frac{9}{10}$ minutes. Both together: $3\frac{5}{10} + 2\frac{9}{10} = 5\frac{14}{10} = 6\frac{2}{5}$ minutes.

**Mind map**

```text
Adding and subtracting fractions
  Same bottom
    Add the tops
    Bottom stays
  Different bottoms
    Lowest common denominator
    Make equivalent fractions
  Mixed numbers
    Wholes first
    Then fractions
  Borrowing
    Take 1 whole
    Turn it into parts
  Finish
    Simplify
    Improper to mixed
```

## Quick summary

- You can only add or subtract fractions with the **same denominator**.
- Use the lowest common denominator, then add or subtract the numerators.
- Never add the denominators.
- With mixed numbers, do the whole numbers first, then the fractions.
- If the second fraction is bigger, borrow a whole or change to an improper fraction.
- Always simplify the answer.

## Quiz

Try these questions from the chapter quiz.

1. $\frac{2}{9} + \frac{5}{9} = $
   - A. $\frac{7}{18}$
   - B. $\frac{7}{9}$
   - C. $\frac{10}{9}$
   - D. $\frac{3}{9}$
   - Answer: B. $\frac{7}{9}$. Same bottom, so add the tops: $\frac{7}{9}$.

2. $\frac{1}{3} + \frac{5}{6} = $
   - A. $\frac{6}{9}$
   - B. $\frac{6}{6}$
   - C. $1\frac{1}{6}$
   - D. $\frac{5}{18}$
   - Answer: C. $1\frac{1}{6}$. $\frac{2}{6} + \frac{5}{6} = \frac{7}{6} = 1\frac{1}{6}$.

3. $\frac{2}{3} - \frac{1}{12} = $
   - A. $\frac{7}{12}$
   - B. $\frac{1}{9}$
   - C. $\frac{1}{4}$
   - D. $\frac{9}{12}$
   - Answer: A. $\frac{7}{12}$. $\frac{8}{12} - \frac{1}{12} = \frac{7}{12}$.

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