Percentages (AQA GCSE Maths): Revision Notes
Percentages - GCSE revision guide
Understanding percentages
When we talk about percentages, we're referring to parts of 100. The term "per cent" literally means "out of 100", which is why understanding this concept is crucial for converting between percentages, decimals, and fractions. Once you grasp this fundamental idea, you'll be ready to tackle different types of percentage problems that commonly appear in GCSE mathematics.
Fundamental Concept: All percentage problems stem from the basic understanding that "per cent" means "out of 100". This is why 50% = 50/100 = 0.5, and why this conversion is the foundation of all percentage calculations.
Three fundamental percentage question types
Type 1: Finding x% of y
This is often the most straightforward type of percentage question. To solve these problems, you need to convert the percentage into either a decimal or a fraction, then multiply it by the given value.
Worked Example: Finding 15% of £46
Step 1: Convert percentage to decimal 15% = 15 ÷ 100 = 0.15
Step 2: Multiply by the given amount 0.15 × £46 = £6.90
Therefore, 15% of £46 = £6.90
The key steps are:
- Convert the percentage to a decimal by dividing by 100
- Multiply this decimal by the given amount
- Calculate your final answer
Type 2: Finding the new amount after a percentage change
When dealing with percentage increases or decreases, the most efficient method involves finding what we call a "multiplier". This multiplier represents the decimal equivalent of the percentage change.
Multiplier Rules to Remember:
- For increases: multiplier > 1 (e.g., 5% increase → multiplier = 1.05)
- For decreases: multiplier < 1 (e.g., 26% decrease → multiplier = 0.74)
For increases, your multiplier will be greater than 1. For example, a 5% increase gives you a multiplier of 1.05 (which is 1 + 0.05). For decreases, your multiplier will be less than 1. A 26% decrease gives you a multiplier of 0.74 (which is 1 - 0.26).
Once you've identified your multiplier, simply multiply the original value by this number to get your answer. This method is particularly useful for VAT calculations and similar real-world problems.
Type 3: Expressing x as a percentage of y
This type of question asks you to compare two values and express their relationship as a percentage. The method is straightforward: divide the first value by the second value, then multiply by 100.
The formula can be written as:
Worked Example: Expressing 209 as a percentage of 400
Step 1: Divide the first value by the second 209 ÷ 400 = 0.5225
Step 2: Multiply by 100 to get percentage 0.5225 × 100 = 52.25%
Therefore, 209 is 52.25% of 400
Advanced percentage techniques
Finding percentage change
When you need to calculate how much something has changed as a percentage, you use the percentage change formula. This involves finding the difference between the original and new values, then expressing this change as a percentage of the original amount.
The formula for percentage change is:
This type of question often appears in real-world contexts, such as calculating profit margins, price changes, or statistical variations. The key is to identify what has changed and what the original value was, then apply the formula systematically.
Finding original values (reverse percentages)
This is considered one of the most challenging types of percentage problems, but it becomes manageable once you understand the systematic approach. The key insight is that you need to work backwards from the given information.
Critical Method for Reverse Percentages: This is often the most difficult type of percentage problem for students. The systematic three-step approach is essential for success.
The method involves three clear steps:
- Write the given amount as a percentage of the original value
- Divide this amount by the percentage to find what 1% of the original value equals
- Multiply by 100 to find the complete original value (which represents 100%)
Worked Example: Finding Original House Value
A house increases in value by 10.5% to £132,600. Find the original value.
Step 1: Identify what percentage the new value represents £132,600 represents 110.5% of the original value (100% + 10.5%)
Step 2: Find 1% of the original value 1% = £132,600 ÷ 110.5 = £1,200
Step 3: Find 100% (the original value) Original value = £1,200 × 100 = £120,000
Real-world applications
Simple interest calculations
Simple interest problems involve calculating interest that remains constant over time. Unlike compound interest, simple interest is calculated only on the original amount (principal) and remains the same each year.
To calculate simple interest:
- Work out the interest for one year using the percentage rate
- Multiply by the number of years to find the total interest earned
Worked Example: Simple Interest Calculation
Someone invests £380 at 3% simple interest per year.
Step 1: Calculate annual interest 3% of £380 = 0.03 × £380 = £11.40 per year
Step 2: For multiple years, multiply by number of years For 5 years: £11.40 × 5 = £57.00 total interest
Complex percentage scenarios
Some percentage problems require you to combine multiple techniques or think through multi-step processes. These might involve calculating percentages of percentages, or working with changing populations and statistics.
Worked Example: Complex Gym Membership Problem
In a gym, 80% of members are male and 35% of male members are over 40. What percentage of total members are male and under 40?
Step 1: Find percentage of male members under 40 100% - 35% = 65% of male members are under 40
Step 2: Find what this represents of total membership 65% of 80% = 0.65 × 0.80 = 0.52 = 52%
Therefore, 52% of total members are male and under 40
Key Points to Remember:
- Percentage means "out of 100" - this fundamental understanding helps with all conversions and calculations
- Use multipliers for increases and decreases - increases have multipliers greater than 1, decreases have multipliers less than 1
- For reverse percentages, work systematically - identify what percentage the given value represents, find 1%, then multiply by 100
- Percentage change formula is - always divide by the original value, not the new value
- Practice identifying question types - recognising whether you need to find a percentage, apply a change, or work backwards will guide your method choice