Fractions as operators
🎯 In this topic you will
- Describe a unit fraction as a fraction with a numerator of 1.
- Use a unit fraction as an operator to find a fraction of a quantity by dividing (for example, find one-fifth by dividing by 5, and one-sixth by dividing by 6).
🧠 Key Words
- operator
- unit fraction
Show Definitions
- operator: A symbol or instruction that tells you what calculation to do to a number or quantity (for example, “take one-fifth of” means “divide by 5”).
- unit fraction: A fraction with a numerator of 1 (such as 1/5 or 1/6) that represents one equal part of a whole.
🍳 Cooking for Fewer People
When you are cooking you may need to cook for a smaller number of people than the recipe suggests.
🧩 Halving a Recipe
If you want to halve a recipe, you must work out all of the amounts using fractions.
🥠 Making Half the Biscuits
To use this recipe to make eight gingerbread biscuits you would need to halve all of the ingredients.
How to Work it Out: $\dfrac{1}{2}\text{ of }350 = 175$
So you need 175 g of plain flour.
❓ EXERCISES
$1.$ What is $\dfrac{1}{3}$ of $\$12$?
👀 Show answer
$2.$ Copy and complete the following.
a. $24 \div 3$ is equivalent to $\square$ of $24$
b. $16 \div 8$ is equivalent to $\square$ of $16$
👀 Show answer
a. Dividing by $3$ is the same as finding $\dfrac{1}{3}$, so $24 \div 3$ is equivalent to $\dfrac{1}{3}$ of $24$.
b. Dividing by $8$ is the same as finding $\dfrac{1}{8}$, so $16 \div 8$ is equivalent to $\dfrac{1}{8}$ of $16$.
$3.$
a. What is one-tenth of $30$?
b. What is $\dfrac{1}{5}$ of $45$?
c. What is one-quarter of $40$?
👀 Show answer
a. One-tenth of $30$ is $30 \div 10 = 3$.
b. $\dfrac{1}{5}$ of $45$ is $45 \div 5 = 9$.
c. One-quarter of $40$ is $\dfrac{1}{4}$ of $40$, so $40 \div 4 = 10$.
$4.$ Copy and complete these diagrams to find fractions of amounts of money.
👀 Show answer
For $\$24$: $\dfrac{1}{2}=\$12$, $\dfrac{1}{3}=\$8$, $\dfrac{1}{4}=\$6$, $\dfrac{1}{6}=\$4$, $\dfrac{1}{8}=\$3$.
For $\$32$: $\dfrac{1}{2}=\$16$, $\dfrac{1}{4}=\$8$, $\dfrac{1}{8}=\$4$, $\dfrac{3}{4}=\$24$.
$5.$ Ajay says, ‘To find a tenth of a number I divide by $10$, and to find a fifth of a number I divide by $5$’. Is he correct?
Explain your reasoning to your partner, then write down your thoughts.
👀 Show answer
$6.$ Which would you choose: $\dfrac{1}{3}$ of $\$15$ or $\dfrac{1}{4}$ of $\$24$?
Check your answer with your partner.
Explain how you worked out your answer.
👀 Show answer
$7.$ Here are some numbers.
$10\ \ \ 20\ \ \ 30\ \ \ 40\ \ \ 50\ \ \ 60\ \ \ 70\ \ \ 80$
Write one of these numbers in each box to make the fraction sentences correct.
You can use each number once only.
$\dfrac{1}{2}$ of $\square$ $=$ $\square$ $\dfrac{1}{4}$ of $\square$ $=$ $\square$ $\dfrac{1}{5}$ of $\square$ $=$ $\square$
👀 Show answer
🧠 Think like a Mathematician
□ represents a whole number in each calculation. Investigate the largest value of □ in this set of calculations.
Question: What is the largest possible value of □ that makes all of the following calculations true?
Equipment: Pencil and paper (a calculator is optional).
Method:
- Work out the value of □ for each calculation.
- Check each answer is a whole number and that it satisfies the original calculation.
- Compare all the values you found and identify the largest.
- Write a clear explanation of the steps you used (including how you “undo” operations).
Calculations:
Follow-up Questions:
👀 show answer
1) Values of □:
- $\frac{1}{4}$ of $40$ = 10
- $\frac{1}{4}$ of □ = $4$ ⟹ □ = 16
- $\frac{1}{3}$ of □ = $9$ ⟹ □ = 27
- $\frac{1}{3}$ of $21$ = 7
- $64$ ÷ $8$ = 8
- $\frac{1}{5}$ of $45$ = 9
- $24$ ÷ $4$ = 6
2) Largest value of □:27
3) Explanation (how it was worked out): “Of” means multiply by the fraction (which is the same as dividing by the denominator). So $\frac{1}{4}$ of $40$ is $40 ÷ 4 = 10$, and $\frac{1}{3}$ of $21$ is $21 ÷ 3 = 7$. When □ is the number being fractioned, you undo the fraction by multiplying: if $\frac{1}{4}$ of □ is $4$, then □ is $4 × 4 = 16$; if $\frac{1}{3}$ of □ is $9$, then □ is $9 × 3 = 27$. For division statements, compute directly: $64 ÷ 8 = 8$ and $24 ÷ 4 = 6$. Comparing all results, the largest is $27$.