Understanding fractions
🎯 In this topic you will
- Represent proper and improper fractions as division.
- Use proper and improper fractions as operators.
🧠 Key Words
- denominator
- improper fraction
- mixed number
- numerator
- operator
- proper fraction
Show Definitions
- denominator: The bottom number in a fraction that shows how many equal parts the whole is divided into.
- improper fraction: A fraction in which the numerator is greater than or equal to the denominator.
- mixed number: A number made up of a whole number and a proper fraction written together.
- numerator: The top number in a fraction that shows how many parts are being considered.
- operator: A fraction used to act on a quantity, meaning it tells you to multiply the quantity by that fraction.
- proper fraction: A fraction in which the numerator is smaller than the denominator.
Understanding Improper Fractions
An improper fraction represents a number greater than one whole. For example, the fraction $\frac{7}{4}$ shows more than a single whole.
Converting to a Mixed Number
For example, the improper fraction $\frac{7}{4}$ is equal to $1\frac{3}{4}$, which is written as a mixed number. In this unit, we will work with proper fractions, improper fractions and mixed numbers.
❓ EXERCISES
1. Represent these divisions as fractions.
a. $5$ divided by $6$
b. $6$ divided by $5$
c. $10$ divided by $4$
d. $4$ divided by $10$
👀 Show answer
a. $\dfrac{5}{6}$
b. $\dfrac{6}{5}$
c. $\dfrac{10}{4}$
d. $\dfrac{4}{10}$
2. Ahmed, Carlos, Ludvik, Oliver and Rajiv share $3$ cakes between them. What fraction of a cake does each person get?
👀 Show answer
There are $5$ people sharing $3$ cakes, so each gets $\dfrac{3}{5}$ of a cake.
3. Which word can you use to complete this sentence?
When you find $\dfrac{6}{5}$ of a number you are using a fraction as an ________.
👀 Show answer
operator
4. Calculate.
a. $\dfrac{3}{4}$ of $\$16$
b. $\dfrac{5}{4}$ of $\$12$
c. $\dfrac{5}{2}$ of $4$ metres
👀 Show answer
a. $\dfrac{3}{4}\times 16 = 12$
b. $\dfrac{5}{4}\times 12 = 15$
c. $\dfrac{5}{2}\times 4 = 10$ metres
5. Halima swims $\dfrac{1}{2}$ of $500$ metres and Bella swims $\dfrac{3}{10}$ of $800$ metres. Who swims further? Explain how you know.
👀 Show answer
Halima: $\dfrac{1}{2}\times 500 = 250$ metres
Bella: $\dfrac{3}{10}\times 800 = 240$ metres
Halima swims further.
6. Leroy and Wayne each have $90$ bricks. Leroy uses $\dfrac{3}{5}$ of his bricks to build a wall. Wayne uses $\dfrac{5}{6}$ of his bricks to build a wall. How many bricks do they have left altogether? Show your working. Discuss your answer with your partner. Do you agree?
👀 Show answer
Leroy uses $\dfrac{3}{5}\times 90 = 54$, so he has $36$ left.
Wayne uses $\dfrac{5}{6}\times 90 = 75$, so he has $15$ left.
Total left $= 36 + 15 = 51$ bricks.
7. Copy and complete this table to show fractions of $24$.
| Fraction | $\dfrac{1}{4}$ | $\dfrac{3}{4}$ | $\dfrac{5}{4}$ | $\dfrac{7}{4}$ | $\dfrac{9}{4}$ | $\dfrac{11}{4}$ |
|---|---|---|---|---|---|---|
| Amount | $6$ | $18$ | $30$ | $42$ | $54$ | $66$ |
👀 Show answer
Multiply $24$ by each fraction.
8. Calculate.
a. $\dfrac{9}{4}$ of $16$
b. $\dfrac{7}{5}$ of $35$
c. $\dfrac{8}{7}$ of $14$
d. $\dfrac{4}{3}$ of $15$
👀 Show answer
a. $36$
b. $49$
c. $16$
d. $20$
🧠 Think like a Mathematician
Task: Imagine that you roll two $1$–$6$ dice and use them to make an improper fraction.
$\dfrac{5}{3}$ is an improper fraction.
Your challenge: Write all the different improper fractions you could make.
Follow-up Questions:
Show Answers
- 1: There are $15$ improper fractions (all cases where the numerator is greater than the denominator using numbers $1$ to $6$).
- 2: The greatest improper fraction is $\dfrac{6}{1} = 6$.
- 3: A complete list is made by systematically comparing every possible numerator and denominator from $1$ to $6$ and selecting only those where the numerator is larger.