Adding & subtracting fractions
🎯 In this topic you will
- Add two fractions with different denominators.
- Subtract two fractions with different denominators.
🧠 Key Words
- common denominator
- denominator
Show Definitions
- common denominator: A shared denominator that two or more fractions are converted to so they can be easily added or subtracted.
- denominator: The bottom number in a fraction that shows how many equal parts the whole is divided into.
🍕 Sharing Pizzas
Two pizzas of equal size are delivered to a family. The cheese and tomato pizza is divided into 8 equal pieces, so each slice represents one eighth of the whole pizza.
🍕 Different Pizza Portions
The special pizza is divided into 5 pieces. Tarik eats 3 pieces of the cheese and tomato pizza and one piece of the special pizza. To find what fraction of a whole pizza Tarik eats, we use fractions to represent the portions from each pizza.
This situation shows why we sometimes need to add fractions with different denominators. In this unit, you will learn how to add and subtract fractions even when the denominators are not the same.
❓ EXERCISES
1. Copy and complete the table.
| Calculation | Common denominator | Equivalent calculation | Answer |
|---|---|---|---|
| $\frac{1}{3} + \frac{1}{6}$ | |||
| $\frac{7}{10} - \frac{1}{2}$ | |||
| $\frac{6}{5} + \frac{1}{2}$ |
👀 Show answer
$\frac{1}{3} + \frac{1}{6}$ → common denominator $6$ → $\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$
$\frac{7}{10} - \frac{1}{2}$ → common denominator $10$ → $\frac{7}{10} - \frac{5}{10} = \frac{2}{10} = \frac{1}{5}$
$\frac{6}{5} + \frac{1}{2}$ → common denominator $10$ → $\frac{12}{10} + \frac{5}{10} = \frac{17}{10} = 1\frac{7}{10}$
2. Calculate.
a. $\frac{3}{4} + \frac{2}{5}$
b. $\frac{5}{8} - \frac{1}{3}$
c. $\frac{7}{8} + \frac{3}{5}$
👀 Show answer
a. $\frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20} = 1\frac{3}{20}$
b. $\frac{5}{8} - \frac{1}{3} = \frac{15}{24} - \frac{8}{24} = \frac{7}{24}$
c. $\frac{7}{8} + \frac{3}{5} = \frac{35}{40} + \frac{24}{40} = \frac{59}{40} = 1\frac{19}{40}$
3. Find the missing fractions.
a. $\frac{7}{4} - \frac{4}{5} = \square$
b. $\square + \frac{2}{3} = \frac{13}{4}$
👀 Show answer
a. $\frac{7}{4} - \frac{4}{5} = \frac{35}{20} - \frac{16}{20} = \frac{19}{20}$
b. $\square = \frac{13}{4} - \frac{2}{3} = \frac{39}{12} - \frac{8}{12} = \frac{31}{12}$
4. Chipo and Leke work out the answer to $\frac{2}{3} + \frac{3}{5}$.
Chipo says the answer is $\frac{19}{15}$.
Leke says the answer is $1\frac{4}{15}$.
Who do you agree with? Explain your answer.
👀 Show answer
5. Calculate.
a. $\frac{3}{2} + \frac{4}{5}$
b. $\frac{11}{4} - \frac{5}{3}$
c. $\frac{9}{8} + \frac{2}{3}$
👀 Show answer
a. $\frac{3}{2} + \frac{4}{5} = \frac{15}{10} + \frac{8}{10} = \frac{23}{10} = 2\frac{3}{10}$
b. $\frac{11}{4} - \frac{5}{3} = \frac{33}{12} - \frac{20}{12} = \frac{13}{12} = 1\frac{1}{12}$
c. $\frac{9}{8} + \frac{2}{3} = \frac{27}{24} + \frac{16}{24} = \frac{43}{24} = 1\frac{19}{24}$
6. Calculate.
a. $\frac{5}{2} - \frac{3}{5}$
b. $\frac{11}{4} - \frac{5}{3}$
c. $\frac{8}{3} - \frac{4}{5}$
👀 Show answer
a. $\frac{5}{2} - \frac{3}{5} = \frac{25}{10} - \frac{6}{10} = \frac{19}{10} = 1\frac{9}{10}$
b. $\frac{11}{4} - \frac{5}{3} = \frac{33}{12} - \frac{20}{12} = \frac{13}{12} = 1\frac{1}{12}$
c. $\frac{8}{3} - \frac{4}{5} = \frac{40}{15} - \frac{12}{15} = \frac{28}{15} = 1\frac{13}{15}$
7. Leroy colours $\frac{1}{4}$ and $\frac{1}{6}$ of a circle. What fraction of the circle does he leave white?
👀 Show answer
8. Nailah’s class voted for where to go on the school outing.
$\frac{3}{4}$ of the class voted for the theme park.
$\frac{2}{9}$ of the class voted for the zoo.
The rest of the class voted for a river trip.
What fraction of the class voted for the river trip?
👀 Show answer
9. Jo plants potatoes, carrots and onions in her vegetable garden.
She plants potatoes in $\frac{2}{3}$ of her garden.
She plants carrots in $\frac{1}{4}$ of her garden.
What fraction of her garden does she plant with onions?
👀 Show answer
🧠 Think like a Mathematician
Task: Copy and complete the table of fraction additions.
| Column 1 | Column 2 | Column 3 |
|---|---|---|
| $\frac{1}{5}+\frac{1}{2}=\frac{7}{10}$ | $\frac{1}{7}+\frac{1}{2}=\frac{9}{14}$ | $\frac{1}{9}+\frac{1}{2}=\frac{11}{18}$ |
| $\frac{1}{5}+\frac{1}{3}=\frac{8}{15}$ | $\frac{1}{7}+\frac{1}{3}=\frac{10}{21}$ | $\frac{1}{9}+\frac{1}{3}=?$ |
| $\frac{1}{5}+\frac{1}{4}=\frac{9}{20}$ | $\frac{1}{7}+\frac{1}{4}=?$ | $\frac{1}{9}+\frac{1}{4}=?$ |
| $\frac{1}{5}+\frac{1}{5}=\frac{10}{25}$ | — | — |
Follow-up Question:
Can you find a rule that explains the pattern in the table?
👀 show answer
Missing values:
$\frac{1}{9}+\frac{1}{3}=\frac{4}{9}$
$\frac{1}{7}+\frac{1}{4}=\frac{11}{28}$
$\frac{1}{9}+\frac{1}{4}=\frac{13}{36}$
Rule of the pattern:
When adding fractions of the form $\frac{1}{a}+\frac{1}{b}$, the result can be written as
$\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}$
The numerator becomes the sum of the denominators, and the denominator becomes their product.