Fractions, Decimals and PercentagesAQA GCSE Maths: Revision notes

Terminating decimals are decimals that end (like 0.5, 0.75, 0.125). These can always be converted to exact fractions.

Converting decimals to fractions:

1. Write the decimal as a fraction with a power of 10 as the denominator (e.g., 0.3 = 3/10)
2. Simplify by dividing numerator and denominator by their highest common factor (HCF)

Example: 0.24 = 24/100 = 6/25 (dividing by HCF of 4)

Converting fractions to decimals:

• Divide the numerator by the denominator using long division
• If the division terminates, you have a terminating decimal
• Example: 3/8 = 0.375

Key insight: A fraction in lowest terms converts to a terminating decimal only if the denominator has only factors of 2 and 5. For example, 1/8 = 0.125 (8 = 2³), but 1/3 will not terminate.

Recurring decimals are decimals where one or more digits repeat infinitely (e.g., 0.333... = 0.3̇, 0.142857142857... = 0.1̇4̇2̇8̇5̇7̇).

Recognising recurring decimals:

• A fraction in lowest terms produces a recurring decimal if the denominator has prime factors other than 2 and 5
• Common examples: 1/3 = 0.333..., 1/6 = 0.1666..., 1/7 = 0.142857...

Converting recurring decimals to fractions:

For a single repeating digit (e.g., 0.7̇):

1. Let x = 0.777...
2. Multiply by 10: 10x = 7.777...
3. Subtract: 10x − x = 7.777... − 0.777...
4. 9x = 7, so x = 7/9

For multiple repeating digits (e.g., 0.3̇5̇ where both 3 and 5 repeat):

1. Let x = 0.353535...
2. Multiply by 100 (since 2 digits repeat): 100x = 35.353535...
3. Subtract: 100x − x = 35
4. 99x = 35, so x = 35/99

For mixed recurring (e.g., 0.1̇6̇ where 1 is fixed and 6 repeats):

1. Let x = 0.1666...
2. Multiply by 10: 10x = 1.666...
3. Multiply by 100: 100x = 16.666...
4. Subtract: 100x − 10x = 16.666... − 1.666...
5. 90x = 15, so x = 15/90 = 1/6

Rule: Multiply by 10ⁿ where n is the number of repeating digits, then subtract the original equation.

A percentage is a number of parts per hundred. The symbol % literally means 'out of 100'.

Converting between percentages and fractions/decimals:

• Percentage to decimal: Divide by 100 (e.g., 35% = 0.35)
• Percentage to fraction: Write as a fraction out of 100, then simplify (e.g., 35% = 35/100 = 7/20)
• Decimal to percentage: Multiply by 100 (e.g., 0.45 = 45%)
• Fraction to percentage: Convert to decimal then multiply by 100, or find equivalent fraction with denominator 100 (e.g., 3/5 = 60%)

Percentage as an operator: When a percentage 'operates on' a quantity, it means you multiply the quantity by the percentage (as a decimal or fraction). This is used when finding a percentage of an amount.

Finding a percentage of an amount:

• Method 1: Convert percentage to decimal and multiply. Example: 15% of 80 = 0.15 × 80 = 12
• Method 2: Convert percentage to fraction and multiply. Example: 15% of 80 = (15/100) × 80 = 12

Fractions as operators: Similarly, a fraction operates on a quantity through multiplication. Example: 3/4 of 120 = (3/4) × 120 = 90

In ratio problems: Fractions can represent parts of a ratio. If a ratio is 3:5 (total 8 parts), then 3/8 of the total is the first quantity and 5/8 is the second.

Percentage change measures how much a quantity has changed relative to its original value.

Percentage increase/decrease formula:

Percentage change = (Change / Original) × 100%

Where Change = Final value − Original value (positive for increase, negative for decrease).

Using multipliers (percentage change method): Rather than calculating the change separately, you can multiply the original by a multiplier.

• For a percentage increase of x%: multiply by (1 + x/100)
• For a percentage decrease of x%: multiply by (1 − x/100)

Example: A price of £80 increases by 15%. New price = £80 × 1.15 = £92 Example: A price of £80 decreases by 20%. New price = £80 × 0.80 = £64

Successive percentage changes: When multiple percentage changes occur in sequence, multiply the multipliers together. Example: A value of £100 increases by 10%, then decreases by 5%. Final value = £100 × 1.10 × 0.95 = £104.50

Finding the original value (reverse percentages): If you know the final value and the percentage change, you can find the original.

Rearranged formula: Original = Final / Multiplier

Example: A price increased by 20% to become £180. Original = £180 / 1.20 = £150 Example: A price decreased by 15% to become £85. Original = £85 / 0.85 = £100

Step-by-step reverse percentage:

1. Identify the percentage change and calculate the multiplier
2. Divide the final value by the multiplier
3. Check your answer by applying the original percentage change forward

Simple interest is interest calculated on the original amount only. The interest stays the same each period.

Simple interest formula:

Interest = (Principal × Rate × Time) / 100

Or: Final amount = Principal + Interest

Where:

• Principal = the original amount
• Rate = the annual interest rate as a percentage
• Time = the number of years

Example: £500 invested at 4% simple interest for 3 years. Interest = (500 × 4 × 3) / 100 = £60 Final amount = £500 + £60 = £560

Compound interest is interest calculated on the principal and previously earned interest. The amount grows exponentially.

Compound interest formula:

Final amount = Principal × (1 + Rate/100)ⁿ

Where n = number of compounding periods (usually years)

Example: £500 invested at 4% compound interest for 3 years. Final amount = £500 × (1.04)³ = £500 × 1.124864 = £562.43 Interest earned = £562.43 − £500 = £62.43

Comparison:

Compound interest with different compounding periods: If interest compounds more than once per year (monthly, quarterly, etc.), adjust the formula:

• Divide the annual rate by the number of periods per year
• Multiply the time by the number of periods per year

Example: £1000 at 6% compounded quarterly for 2 years. Final amount = £1000 × (1 + 6/400)⁸ = £1000 × (1.015)⁸ = £1126.16