UK K12 GCSE/A-Level: Year 6 KS2 Mathematics - Fractions Decimals and Percentages, Equivalence

Learning objectives

By the end of this topic, students will be able to: - Understand the concept of equivalence between fractions, decimals, and percentages. - Identify and create equivalent fractions, decimals, and percentages. - Apply the concept of equivalence to solve problems involving real-world contexts. - Compare and order fractions, decimals, and percentages. - Use equivalent fractions, decimals, and percentages to simplify calculations.

Core concepts

A fraction is a way of showing a part of a whole. It consists of a numerator (the number on top) and a denominator (the number on the bottom). For example, the fraction 1/2 represents one half of a whole.

Decimals are a way of showing fractions with a denominator of 10 or a power of 10. They are written using a decimal point, with the digits after the point representing the numerator. For example, the decimal 0.5 represents one half of a whole.

Percentages are a way of showing a part of a whole as a fraction of 100. They are written using the symbol %, with the number representing the numerator. For example, the percentage 50% represents one half of a whole.

Fractions, decimals, and percentages are equivalent if they represent the same value. For example, the fraction 1/2, the decimal 0.5, and the percentage 50% are all equivalent.

Worked examples

Example 1: Equivalent fractions

Simplify the fraction 6/8 to its simplest form.

To simplify the fraction, we need to find the greatest common divisor (GCD) of 6 and 8. The GCD is 2, so we can divide both the numerator and the denominator by 2 to get:

6 ÷ 2 = 3 8 ÷ 2 = 4

So, the simplified fraction is 3/4.

Example 2: Equivalent decimals

Convert the decimal 0.25 to a fraction.

To convert the decimal to a fraction, we need to find the place value of the digits after the decimal point. In this case, the 2 is in the hundredths place, so we can write the decimal as a fraction with a denominator of 100:

0.25 = 25/100

We can simplify the fraction by dividing both the numerator and the denominator by 25:

25 ÷ 25 = 1 100 ÷ 25 = 4

So, the simplified fraction is 1/4.

Example 3: Equivalent percentages

Convert the percentage 25% to a decimal.

To convert the percentage to a decimal, we need to divide the numerator by 100:

25 ÷ 100 = 0.25

Common misconceptions

• Students may think that equivalent fractions, decimals, and percentages are always the same number. For example, they may think that 1/2 and 0.5 are always equal, but this is not true if the numerator and denominator are different.
• Students may struggle to convert between fractions, decimals, and percentages. For example, they may find it difficult to convert a decimal to a fraction or a percentage.
• Students may not understand the concept of equivalence and may think that fractions, decimals, and percentages are always different.

Exam tips

• Make sure to read the question carefully and understand what is being asked.
• Use equivalent fractions, decimals, and percentages to simplify calculations and make problems easier to solve.
• Check your answers to make sure they are reasonable and make sense in the context of the problem.

MCQs with explanations

Question 1: Equivalent fractions [F]

Which of the following fractions is equivalent to 2/4?

A) 1/2 B) 1/3 C) 3/6 D) 4/8

Correct answer: A) 1/2 Why the distractors fail: B) 1/3 is not equivalent to 2/4 because the numerator and denominator are different. C) 3/6 is equivalent to 1/2, but it is not the simplest form. D) 4/8 is equivalent to 1/2, but it is not the simplest form.

Question 2: Equivalent decimals [H]

Which of the following decimals is equivalent to 0.375?

A) 0.25 B) 0.5 C) 0.75 D) 1.25

Correct answer: C) 0.75 Why the distractors fail: A) 0.25 is one quarter of a whole, not three quarters. B) 0.5 is one half of a whole, not three quarters. D) 1.25 is one and a quarter of a whole, not three quarters.

Question 3: Equivalent percentages [F]

Which of the following percentages is equivalent to 25%?

A) 10% B) 20% C) 50% D) 75%

Correct answer: A) 10% Why the distractors fail: B) 20% is one fifth of a whole, not one quarter. C) 50% is one half of a whole, not one quarter. D) 75% is three quarters of a whole, not one quarter.

Question 4: Equivalent fractions [H]

Which of the following fractions is equivalent to 3/8?

A) 1/4 B) 1/2 C) 3/6 D) 6/8

Correct answer: D) 6/8 Why the distractors fail: A) 1/4 is one eighth of a whole, not three eighths. B) 1/2 is one half of a whole, not three eighths. C) 3/6 is one half of a whole, not three eighths.

Question 5: Equivalent decimals [F]

Which of the following decimals is equivalent to 0.1?

A) 0.01 B) 0.5 C) 1.0 D) 10.0

Correct answer: A) 0.01 Why the distractors fail: B) 0.5 is one half of a whole, not one tenth. C) 1.0 is one whole, not one tenth. D) 10.0 is ten wholes, not one tenth.

Short-answer questions

1. Simplify the fraction 6/8 to its simplest form.

2. Convert the decimal 0.25 to a fraction.

3. Convert the percentage 25% to a decimal.

4. Which of the following fractions is equivalent to 2/4?

A) 1/2 B) 1/3 C) 3/6 D) 4/8

1. Which of the following decimals is equivalent to 0.375?

A) 0.25 B) 0.5 C) 0.75 D) 1.25