Number & Algebra

Fractions and Percentages: Operations and Conversions

Year 5 · Year 6

✓By the end of this lesson students will be able to add and subtract fractions with the same and related denominators.
✓By the end of this lesson students will be able to multiply and divide fractions by whole numbers.
✓By the end of this lesson students will be able to recognise percentages as fractions out of 100.
✓By the end of this lesson students will be able to convert between common fractions, decimals, and percentages.
✓By the end of this lesson students will be able to calculate a percentage of a whole number.

Key concepts

Adding and Subtracting Fractions with the Same Denominator

When fractions have the same denominator (the bottom number), you can add or subtract the numerators (the top numbers) directly. The denominator stays the same. Always simplify your answer if possible.

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

Adding and Subtracting Fractions with Related Denominators

If the denominators are related (one is a multiple of the other), you need to find an equivalent fraction for one of them so that both fractions have the same denominator. This is usually the larger of the two denominators. Then, add or subtract as usual. Always simplify your answer.

a/b + c/d (where d is a multiple of b) -> convert a/b to an equivalent fraction with denominator d.

Multiplying Fractions by Whole Numbers

To multiply a fraction by a whole number, you multiply the numerator of the fraction by the whole number. The denominator stays the same. Remember to simplify your answer or convert to a mixed numeral if it's an improper fraction.

a/b × c = (a × c)/b

Dividing Fractions by Whole Numbers

To divide a fraction by a whole number, you multiply the denominator of the fraction by the whole number. The numerator stays the same. Remember to simplify your answer.

a/b ÷ c = a/(b × c)

What is a Percentage?

A percentage is a way of expressing a number as a fraction of 100. The word 'percent' means 'per hundred' or 'out of 100'. The symbol for percentage is %. For example, 25% means 25 out of 100, which can be written as the fraction 25/100 or the decimal 0.25.

n% = n/100

Converting between Fractions, Decimals and Percentages

You can convert between these forms:\n* **Fraction to Decimal:** Divide the numerator by the denominator.\n* **Decimal to Percentage:** Multiply the decimal by 100 and add the % symbol.\n* **Percentage to Decimal:** Divide the percentage by 100 (remove the % symbol).\n* **Decimal to Fraction:** Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., then simplify.\n* **Percentage to Fraction:** Write the percentage as a fraction with a denominator of 100, then simplify.

Calculating a Percentage of an Amount

To find a percentage of a whole number or amount, first convert the percentage to a decimal or a fraction (usually a fraction out of 100), then multiply it by the amount.

n% of Amount = (n/100) × Amount

Key facts to remember

1A fraction represents a part of a whole.
2The numerator is the top number (how many parts you have), and the denominator is the bottom number (how many parts make a whole).
3To add or subtract fractions, they must have the same denominator.
4To multiply a fraction by a whole number, multiply the numerator by the whole number.
5To divide a fraction by a whole number, multiply the denominator by the whole number.
6A percentage means 'out of 100'.
7To convert a percentage to a decimal, divide by 100.
8To convert a decimal to a percentage, multiply by 100 and add the '%' symbol.

Worked examples

Example 1

Calculate: 3/4 + 1/8

IIdentify the denominators: 4 and 8. They are related because 8 is a multiple of 4.

IIConvert 3/4 to an equivalent fraction with a denominator of 8. To do this, multiply both the numerator and denominator by 2: (3 × 2) / (4 × 2) = 6/8.

IIINow add the fractions with the same denominator: 6/8 + 1/8.

IVAdd the numerators: 6 + 1 = 7.

VKeep the denominator the same: 7/8.

VICheck if the answer can be simplified. 7 and 8 have no common factors other than 1, so it cannot be simplified.

Answer

7/8

Always look for the lowest common multiple (LCM) of the denominators when adding or subtracting fractions with different denominators.

Example 2

A recipe requires 2/3 cup of flour. If you want to make 4 batches of the recipe, how much flour do you need?

IIdentify the operation needed: multiplying the fraction by the whole number of batches.

IISet up the multiplication: 2/3 × 4.

IIIMultiply the numerator (2) by the whole number (4): 2 × 4 = 8.

IVKeep the denominator (3) the same: 8/3.

VConvert the improper fraction to a mixed numeral. Divide 8 by 3: 8 ÷ 3 = 2 with a remainder of 2.

VIWrite the mixed numeral: 2 and 2/3.

Answer

2 2/3 cups of flour

Remember that multiplying a fraction by a whole number means you are finding 'groups of' that fraction.

Example 3

What is 20% of 60?

IConvert the percentage to a fraction or a decimal. As a fraction: 20% = 20/100.

IISimplify the fraction: 20/100 = 1/5.

IIIMultiply the fraction by the amount: 1/5 × 60.

IVMultiply the numerator by the whole number: (1 × 60) / 5 = 60/5.

VDivide the numerator by the denominator: 60 ÷ 5 = 12.

Answer

You could also convert 20% to a decimal (0.20) and then multiply: 0.20 × 60 = 12.

Common mistakes

✗Adding or subtracting denominators when adding or subtracting fractions (e.g., 1/2 + 1/2 = 2/4 instead of 2/2 = 1).
✗Not finding a common denominator when adding or subtracting fractions with different denominators.
✗Forgetting to simplify fractions to their simplest form.
✗Confusing the rules for multiplying and dividing fractions by whole numbers.
✗Incorrectly converting between fractions, decimals, and percentages (e.g., thinking 0.5 is 5% instead of 50%).

Exam tips

★Always read the question carefully to determine the correct operation (addition, subtraction, multiplication, division).
★Show all your working steps clearly, especially when finding common denominators or simplifying fractions.
★Check if your answer makes sense. For example, if you're finding a percentage of an amount, the answer should usually be smaller than the original amount (unless the percentage is over 100%).
★Practise converting between fractions, decimals, and percentages regularly to build fluency.

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