Number

Fractions and Percentages: Operations and Changes

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to perform the four basic operations (addition, subtraction, multiplication, division) with fractions.
✓By the end of this lesson students will be able to calculate a percentage of a given amount.
✓By the end of this lesson students will be able to calculate percentage increase and percentage decrease.
✓By the end of this lesson students will be able to solve problems involving fractions and percentages in various contexts.

Key concepts

Adding and Subtracting Fractions

To add or subtract fractions, they must have a common denominator. Find the lowest common multiple (LCM) of the denominators to use as the common denominator. Convert each fraction to an equivalent fraction with this common denominator, then add or subtract the numerators, keeping the denominator the same. Always simplify your final answer to its lowest terms.

a/b + c/d = (ad + bc) / bd (or using LCM of b and d)

Multiplying Fractions

To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify the resulting fraction to its lowest terms. You can often simplify before multiplying by cancelling common factors diagonally or vertically.

(a/b) × (c/d) = (a × c) / (b × d)

Dividing Fractions

To divide by a fraction, you 'keep' the first fraction, 'change' the division sign to a multiplication sign, and 'flip' (find the reciprocal of) the second fraction. Then, multiply the fractions as usual.

(a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)

Percentage of an Amount

To find a percentage of an amount, convert the percentage to a decimal or a fraction (out of 100), and then multiply it by the given amount. For example, to find 25% of an amount, you can calculate 0.25 × amount or (25/100) × amount.

Amount × (Percentage / 100)

Percentage Change

Percentage change measures the extent to which a value has increased or decreased. It is calculated by finding the difference between the new value and the original value (the 'change'), dividing this change by the original value, and then multiplying by 100 to express it as a percentage. If the value increases, it's a percentage increase; if it decreases, it's a percentage decrease.

Percentage Change = ((Change in Amount) / (Original Amount)) × 100%

Key facts to remember

1To add or subtract fractions, you must find a common denominator.
2To multiply fractions, multiply the numerators and multiply the denominators.
3To divide fractions, multiply the first fraction by the reciprocal of the second fraction ('keep, change, flip').
4The word 'of' in maths often means multiplication, especially with fractions and percentages.
5A percentage is a fraction out of 100 (e.g., 75% = 75/100).
6To find a percentage of an amount, convert the percentage to a decimal or fraction and multiply.
7Percentage change is calculated as (Change / Original Amount) × 100%.
8Always simplify fractions to their lowest terms.

Worked examples

Example 1

Calculate (3/4 + 1/6) ÷ 5/8.

IFirst, add the fractions in the bracket: 3/4 + 1/6.

IIFind the lowest common multiple (LCM) of 4 and 6, which is 12.

IIIConvert the fractions: (3/4) = (3×3)/(4×3) = 9/12. (1/6) = (1×2)/(6×2) = 2/12.

IVAdd the converted fractions: 9/12 + 2/12 = 11/12.

VNow, divide the result by 5/8: (11/12) ÷ (5/8).

VITo divide, 'keep, change, flip': (11/12) × (8/5).

VIIMultiply the numerators and denominators: (11 × 8) / (12 × 5) = 88/60.

VIIISimplify the fraction by dividing both numerator and denominator by their highest common factor, which is 4: 88 ÷ 4 = 22, 60 ÷ 4 = 15.

9The simplified fraction is 22/15. This can also be written as a mixed number: 1 7/15.

Answer

1 7/15

Always perform operations in brackets first. Convert mixed numbers to improper fractions before multiplying or dividing.

Example 2

Find 45% of £320.

IConvert the percentage to a decimal: 45% = 45/100 = 0.45.

IIMultiply the decimal by the amount: 0.45 × £320.

IIICalculation: 0.45 × 320 = 144.

Answer

£144

Alternatively, you could find 10% (32), 5% (16), then 40% (4 × 32 = 128), and add 5% (16) to get 128 + 16 = 144.

Example 3

A bicycle originally cost £180. Its price increased to £207. Calculate the percentage increase.

IIdentify the original amount: £180.

IIIdentify the new amount: £207.

IIICalculate the change in amount (increase): New Amount - Original Amount = £207 - £180 = £27.

IVApply the percentage change formula: (Change / Original Amount) × 100%.

V(£27 / £180) × 100%.

VICalculate the fraction: 27/180 = 3/20.

VIIMultiply by 100%: (3/20) × 100% = (3 × 100) / 20 % = 300 / 20 % = 15%.

Answer

15% increase

Remember to always divide by the ORIGINAL amount when calculating percentage change.

Common mistakes

✗Adding or subtracting fractions without first finding a common denominator.
✗Forgetting to 'flip' the second fraction when dividing fractions.
✗Using the new amount instead of the original amount in the denominator when calculating percentage change.
✗Confusing percentage of an amount with percentage increase/decrease.
✗Not converting mixed numbers to improper fractions before multiplying or dividing.

Exam tips

★Always show your full working out for fraction and percentage problems, as method marks are often awarded.
★Read the question carefully to determine if you need to find a percentage of an amount or calculate a percentage change.
★Simplify fractions at each step if possible to keep the numbers smaller and calculations easier.
★Check your answers to ensure they are reasonable in the context of the problem (e.g., a 1000% increase is usually incorrect for everyday scenarios).

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