Addition and subtraction of fractions
🎯 In this topic you will
- Add and subtract two fractions with the same denominator
- Add and subtract two fractions with denominators that are multiples of each other
🧠 Key Words
- common denominator
- denominator
Show Definitions
- common denominator: A denominator that is the same in two or more fractions, allowing them to be added or subtracted directly.
- denominator: The number below the fraction bar that shows how many equal parts the whole is divided into.
Dividing the Cake
This cake is cut into nine equal pieces.
Sharing Between Friends
Imagine Andri, Ben, Carlos and George each have a piece of cake.
How Much Cake Has Been Eaten?
To find out how much cake has been eaten, we add fractions: $ \tfrac{1}{9} + \tfrac{1}{9} + \tfrac{1}{9} + \tfrac{1}{9} = \tfrac{4}{9} $.
How Much Cake Is Left?
To find out how much cake is left, we subtract the total from one whole: $ 1 - \tfrac{4}{9} = \tfrac{5}{9} $.
What You Will Learn
In this section, you will learn to add and subtract fractions that do not have the same denominator.
❓ EXERCISES
1. Use the fraction wall to calculate.
a. $\dfrac{2}{3} - \dfrac{1}{6}$
b. $\dfrac{1}{6} + \dfrac{1}{3}$
👀 Show answer
a.$\dfrac{2}{3} - \dfrac{1}{6} = \dfrac{1}{2}$
b.$\dfrac{1}{6} + \dfrac{1}{3} = \dfrac{1}{2}$
2. The fractions $\dfrac{9}{10}$ and $\dfrac{1}{5}$ have been shaded on the fraction wall.
Calculate.
a. $\dfrac{9}{10} - \dfrac{1}{5}$
b. $\dfrac{7}{10} - \dfrac{2}{5}$
You can draw diagrams to help you answer the remaining questions in this exercise.
👀 Show answer
a.$\dfrac{9}{10} - \dfrac{1}{5} = \dfrac{7}{10}$
b.$\dfrac{7}{10} - \dfrac{2}{5} = \dfrac{3}{10}$
3. Calculate.
a. $\dfrac{2}{5} + \dfrac{7}{10}$
b. $\dfrac{2}{3} + \dfrac{5}{9}$
c. $\dfrac{11}{12} + \dfrac{3}{4}$
d. $\dfrac{2}{3} + \dfrac{7}{12}$
e. $\dfrac{3}{5} + \dfrac{7}{20}$
f. $\dfrac{3}{4} + \dfrac{3}{8}$
👀 Show answer
a.$\dfrac{2}{5} + \dfrac{7}{10} = 1\dfrac{1}{10}$
b.$\dfrac{2}{3} + \dfrac{5}{9} = 1\dfrac{2}{9}$
c.$\dfrac{11}{12} + \dfrac{3}{4} = 1\dfrac{2}{3}$
d.$\dfrac{2}{3} + \dfrac{7}{12} = 1\dfrac{1}{4}$
e.$\dfrac{3}{5} + \dfrac{7}{20} = \dfrac{19}{20}$
f.$\dfrac{3}{4} + \dfrac{3}{8} = 1\dfrac{1}{8}$
4. Calculate.
a. $\dfrac{5}{6} - \dfrac{1}{3}$
b. $\dfrac{7}{15} - \dfrac{1}{5}$
c. $\dfrac{7}{12} - \dfrac{1}{4}$
d. $\dfrac{7}{8} - \dfrac{3}{4}$
e. $\dfrac{13}{15} - \dfrac{2}{5}$
f. $\dfrac{11}{12} - \dfrac{1}{3}$
Check your answers to questions $3$ and $4$ with your partner.
Compare the methods you used to answer the questions.
👀 Show answer
a.$\dfrac{5}{6} - \dfrac{1}{3} = \dfrac{1}{2}$
b.$\dfrac{7}{15} - \dfrac{1}{5} = \dfrac{4}{15}$
c.$\dfrac{7}{12} - \dfrac{1}{4} = \dfrac{1}{3}$
d.$\dfrac{7}{8} - \dfrac{3}{4} = \dfrac{1}{8}$
e.$\dfrac{13}{15} - \dfrac{2}{5} = \dfrac{7}{15}$
f.$\dfrac{11}{12} - \dfrac{1}{3} = \dfrac{7}{12}$
5. Eva and Mia share a pizza with their dad.
Eva eats $\dfrac{1}{3}$ of the pizza.
Mia eats $\dfrac{1}{6}$ of the pizza.
What fraction of the pizza do they leave for their dad?
👀 Show answer
$\dfrac{1}{3} + \dfrac{1}{6} = \dfrac{1}{2}$ eaten, so they leave $\dfrac{1}{2}$ of the pizza for their dad.
6. Copy the table and write the letters of the calculations in the correct columns.
a. $\dfrac{1}{10} + \dfrac{2}{5}$
b. $\dfrac{4}{5} + \dfrac{3}{10}$
c. $\dfrac{3}{5} + \dfrac{2}{5}$
d. $\dfrac{7}{10} + \dfrac{1}{5}$
| Answer less than $1$ | Answer of $1$ | Answer more than $1$ |
|---|---|---|
👀 Show answer
Answers less than $1$: a, d
Answer of $1$: c
Answer more than $1$: b
7. Write the missing numbers.
a. $\dfrac{\square}{20} + \dfrac{7}{10} = \dfrac{17}{20}$
b. $\dfrac{1}{3} + \dfrac{\square}{12} = \dfrac{9}{12}$
c. $\dfrac{\square}{16} + \dfrac{3}{4} = \dfrac{19}{16}$
👀 Show answer
a.$\dfrac{3}{20}$
b.$\dfrac{5}{12}$
c.$\dfrac{7}{16}$
🧠 Think like a Mathematician
The Ancient Egyptians only used unit fractions, for example $\dfrac{1}{2}$ or $\dfrac{1}{3}$.
Find a way to write $\dfrac{3}{8}$ as the sum of two unit fractions.
Now find $\dfrac{6}{10}$ and $\dfrac{7}{18}$ as the sum of two unit fractions.
Challenge: Try to write $\dfrac{7}{10}$ as the sum of two unit fractions.
Find other examples of a fraction of your choice written as the sum of two unit fractions.
You will show you are specialising when you find solutions to the problem.
Show Answers
- $\dfrac{3}{8}$ can be written as $\dfrac{1}{4} + \dfrac{1}{8}$.
- $\dfrac{6}{10}$ can be written as $\dfrac{1}{2} + \dfrac{1}{5}$ (or other valid pairs).
- $\dfrac{7}{18}$ can be written as $\dfrac{1}{6} + \dfrac{1}{9}$.
- Challenge:$\dfrac{7}{10}$ can be written as $\dfrac{1}{2} + \dfrac{1}{5}$, or $\dfrac{1}{3} + \dfrac{1}{15}$, since more than one solution is possible.
- Other examples (students may find different valid answers):
• $\dfrac{4}{5} = \dfrac{1}{2} + \dfrac{1}{10}$
• $\dfrac{5}{12} = \dfrac{1}{4} + \dfrac{1}{6}$
• $\dfrac{5}{8} = \dfrac{1}{2} + \dfrac{1}{8}$