Fractions, decimals & percentages

Adding, Subtracting & Multiplying Fractions

Year 3 · Year 4 · Year 5 · Year 6

✓By the end of this lesson students will be able to add and subtract fractions with the same denominator.
✓By the end of this lesson students will be able to add and subtract fractions with different denominators by finding a common multiple.
✓By the end of this lesson students will be able to multiply proper fractions by whole numbers.
✓By the end of this lesson students will be able to convert between mixed numbers and improper fractions.
✓By the end of this lesson students will be able to add and subtract mixed numbers.

Key concepts

What is a Fraction?

A fraction represents part of a whole. The numerator (the top number) tells you how many parts you have, and the denominator (the bottom number) tells you how many equal parts the whole is divided into.

Equivalent Fractions

Equivalent fractions are fractions that look different but have the same value. You can find equivalent fractions by multiplying or dividing both the numerator and the denominator by the same number. For example, 1/2 is equivalent to 2/4.

Common Denominator

When adding or subtracting fractions, the parts must be the same size. This means they must have the same denominator. This is called a common denominator. You find a common denominator by finding a common multiple of the original denominators.

Proper Fraction

A proper fraction is a fraction where the numerator is smaller than the denominator, meaning its value is less than 1 whole. For example, 3/4 is a proper fraction.

Improper Fraction

An improper fraction is a fraction where the numerator is equal to or larger than the denominator, meaning its value is 1 whole or more. For example, 7/4 is an improper fraction.

Mixed Number

A mixed number combines a whole number and a proper fraction. For example, 1 3/4 is a mixed number.

Adding Fractions

To add fractions, first make sure they have a common denominator. Then, add the numerators and keep the denominator the same. Simplify your answer if possible, and convert to a mixed number if it is an improper fraction.

a/b + c/b = (a+c)/b

Subtracting Fractions

To subtract fractions, first make sure they have a common denominator. Then, subtract the numerators and keep the denominator the same. Simplify your answer if possible, and convert to a mixed number if it is an improper fraction.

a/b - c/b = (a-c)/b

Multiplying a Fraction by a Whole Number

To multiply a fraction by a whole number, multiply the numerator by the whole number. The denominator stays the same. Simplify your answer if possible, and convert to a mixed number if it is an improper fraction.

(a/b) × c = (a × c)/b

Key facts to remember

1A fraction has a numerator (top number) and a denominator (bottom number).
2To find equivalent fractions, multiply or divide both the numerator and denominator by the same number.
3You must have a common denominator to add or subtract fractions.
4An improper fraction has a numerator equal to or larger than its denominator.
5A mixed number has a whole number part and a proper fraction part.
6To multiply a fraction by a whole number, only multiply the numerator by the whole number.
7Always simplify your final answer to its simplest form.
8Convert improper fractions to mixed numbers in your final answer, unless specified otherwise.

Worked examples

Example 1

Calculate 1/3 + 1/6.

I1. Find a common denominator for 3 and 6. The smallest common multiple is 6.

II2. Convert 1/3 to an equivalent fraction with a denominator of 6. To get from 3 to 6, we multiply by 2. So, multiply the numerator by 2 as well: 1 × 2 = 2. Therefore, 1/3 = 2/6.

III3. Now the problem is 2/6 + 1/6.

IV4. Add the numerators: 2 + 1 = 3. Keep the denominator the same. So, we have 3/6.

V5. Simplify the answer. Both 3 and 6 can be divided by 3. 3 ÷ 3 = 1, 6 ÷ 3 = 2. So, 3/6 = 1/2.

Answer

1/2

Example 2

Calculate 3 1/4 - 1 1/2.

I1. Convert the mixed numbers to improper fractions.

II 3 1/4 = (3 × 4 + 1)/4 = 13/4.

III 1 1/2 = (1 × 2 + 1)/2 = 3/2.

IV2. Find a common denominator for 4 and 2. The smallest common multiple is 4.

V3. Convert 3/2 to an equivalent fraction with a denominator of 4. To get from 2 to 4, we multiply by 2. So, multiply the numerator by 2 as well: 3 × 2 = 6. Therefore, 3/2 = 6/4.

VI4. Now the problem is 13/4 - 6/4.

VII5. Subtract the numerators: 13 - 6 = 7. Keep the denominator the same. So, we have 7/4.

VIII6. Convert the improper fraction back to a mixed number. How many times does 4 go into 7? Once, with a remainder of 3. So, 7/4 = 1 3/4.

Answer

1 3/4

Converting to improper fractions often makes subtraction of mixed numbers simpler.

Example 3

Calculate 3/5 × 4.

I1. Multiply the numerator by the whole number: 3 × 4 = 12.

II2. Keep the denominator the same. So, we have 12/5.

III3. Convert the improper fraction to a mixed number. How many times does 5 go into 12? Twice, with a remainder of 2. So, 12/5 = 2 2/5.

Answer

2 2/5

Common mistakes

✗Adding or subtracting the denominators when adding or subtracting fractions.
✗Not finding a common denominator before adding or subtracting fractions.
✗Incorrectly converting between mixed numbers and improper fractions.
✗Forgetting to simplify fractions to their simplest form.
✗When multiplying a fraction by a whole number, multiplying both the numerator and the denominator by the whole number.

Exam tips

★Always show your working step-by-step to gain full marks, even for simple calculations.
★Check if your answer can be simplified by dividing the numerator and denominator by a common factor.
★If your answer is an improper fraction, convert it to a mixed number unless the question specifically asks for an improper fraction.
★For addition and subtraction, ensure you find the lowest common multiple for your common denominator to make calculations easier.

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