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Understanding fractions

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visibility 47update 4 months agobookmarkshare

🎯 In this topic you will

Learn to represent a fraction as a division of the numerator by the denominator
Use a proper fraction as an operator

🧠 Key Words

denominator
numerator
operator
proper
fraction

Show Definitions

denominator: The number below the line in a fraction that indicates how many equal parts the whole is divided into.
numerator: The number above the line in a fraction that shows how many parts are being considered.
operator: A symbol or function that indicates a mathematical operation, such as addition, subtraction, multiplication, or division.
proper: Refers to a fraction where the numerator is smaller than the denominator, indicating a value less than one.
fraction: A numerical quantity representing a part of a whole, written as one number (numerator) over another (denominator).

Everyday Fraction Use

Fractions are commonly seen in everyday life. For example, if a jumper costs $20, you can use fractions to find out the price during a sale. A common discount is ½ off, meaning half of the original price.

Fractions in Time

Fractions can also help us with time calculations. For example, a rugby game is 80 minutes long. If the game starts at 15:00, you can use fractions to determine the time for a half-time break, which would be at 15:40.

📘 Worked example

1. Calculate $ \dfrac{7}{8} $ of 48.

First calculate $ \dfrac{1}{8} $ of 48.
$ \dfrac{1}{8} $ of 48 = $ \dfrac{48}{8} = 6$

Next calculate $ \dfrac{7}{8} $ of 48.
$ \dfrac{7}{8} $ of 48 = $ 7 \times 6 = 42$

Answer:

1. $ \dfrac{7}{8} $ of 48 = 42

First divide 48 by the denominator, 8, to calculate $ \dfrac{1}{8} $ of 48.
$ 48 \div 8 = 6$

Then multiply the result by the numerator, 7, to calculate $ \dfrac{7}{8} $ of 48.
$ 7 \times 6 = 42$

❓ EXERCISES

1. Marcus divides a cake into five equal pieces. What fraction of the whole cake is each piece?

👀 Show answer

Since the cake is divided into 5 equal pieces, each piece is $ \dfrac{1}{5} $ of the whole cake.

2. Six children share two pizzas equally between them. The diagram shows two ways they can do this. Draw a diagram to show two different ways eight children can share two pizzas equally. How much pizza does each child get?

👀 Show answer

Each pizza can be divided into $6$ pieces to share equally among the six children. For $8$ children, each would get $ \dfrac{1}{8} $ of the pizzas. Since the children are sharing $2$ pizzas, each child will get $ \dfrac{2}{8} = \dfrac{1}{4} $ of the total pizza.

3. Sofia, Marcus, Zara, and Arun share three cakes between them. What fraction of a cake does each child get?

👀 Show answer

Since there are $4$ children sharing $3$ cakes, each child gets $ \dfrac{3}{4} $ of a cake.

4. Calculate:

a. $ \dfrac{2}{3} $ of $15$

b. $ \dfrac{3}{4} $ of $24$

c. $ \dfrac{3}{5} $ of $60$

d. $ \dfrac{6}{7} $ of $84$

👀 Show answer

a. $ \dfrac{2}{3} \times 15 = 10$

b. $ \dfrac{3}{4} \times 24 = 18$

c. $ \dfrac{3}{5} \times 60 = 36$

d. $ \dfrac{6}{7} \times 84 = 72$

5. Arun says, "To find $ \dfrac{3}{10} $ of $20$, I divide by $3$ and multiply by $10$." Arun is not correct. Explain what he has done wrong and correct his statement.

👀 Show answer

Arun should multiply by $ \dfrac{3}{10} $ and then apply it to $20$. The correct process is: $ \dfrac{3}{10} \times 20 = 6$.

6. Zara has $ \dfrac{1}{5} $ of a bottle of milk. There are $100$ ml of milk in her bottle. How much milk was in the bottle when it was full?

👀 Show answer

Since $ \dfrac{1}{5} $ of the bottle equals $100$ ml, the full bottle contains $100 \times 5 = 500$ ml.

7. These four squares are $ \dfrac{1}{4} $ of a whole shape. Draw three different shapes that could be the whole shape.

👀 Show answer

The shape could be a square, rectangle, or another shape with four equal parts. Possible diagrams can include a larger square divided into four smaller squares.

8. One-quarter of a number is $8$. What is the number?

👀 Show answer

If $ \dfrac{1}{4} $ of a number is $8$, then the whole number is $ 8 \times 4 = 32$.

9. $ \dfrac{3}{10} $ of a number is $30$. What is the number?

👀 Show answer

If $ \dfrac{3}{10} $ of a number is $30$, then the number is $ \dfrac{30 \times 10}{3} = 100$.

🧠 Think like a Mathematician

It is Igor’s birthday.

He has 12 cakes.

20 people share the cakes.

Investigate how he could cut the cakes so everyone has an equal share.

You will show you are specialising when you find solutions to the problem.

Show Answers

Answer: To share 12 cakes equally among 20 people, each person will receive $ \dfrac{12}{20} = \dfrac{3}{5} $ of a cake.

📘 What we've learned

We learned that a fraction can be represented as a division of the numerator by the denominator.
We explored how to use a proper fraction as an operator in calculations.

Understanding fractions

🎯 In this topic you will

🧠 Key Words

Everyday Fraction Use

Fractions in Time

❓ EXERCISES

🧠 Think like a Mathematician

📘 What we've learned

Related Past Papers

Related Tutorials

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