Percentage Basics and Percentage of Amounts Revision Notes | GCSE AQA Higher Maths

1. What Are Percentages Really?

You see percentages everywhere - exam results, shop discounts, phone battery levels. But what does '73%' actually mean? It's not just a number with a funny symbol.

Think of percentage as a universal language for comparing parts to wholes. When your phone shows 73% battery, it means 73 parts out of every 100 parts of total battery capacity are still available. The word 'percent' literally breaks down as 'per cent' - meaning 'for each hundred'.

This 'out of 100' idea makes percentages incredibly useful for comparing different situations. Whether you're comparing test scores out of different totals or discount rates at different shops, percentages provide a common scale. Understanding percentages requires recognising that they are fractions and decimals presented in a different form. 50% equates to $\frac{50}{100} = \frac{1}{2} = 0.5$. This connection unlocks the power of percentage calculations.

Understanding percentages requires recognising that they are simply fractions and decimals presented in a different form. 50% equates to $\frac{50}{100} = \frac{1}{2} = 0.5$. This connection unlocks the power of percentage calculations.

2. The Percentage Toolkit

Converting between percentages, decimals and fractions is your first essential skill. These three forms are different ways of expressing the same relationship, like saying 'half', '0.5' or '50%' - they all mean the same thing.

The multiplicative view is crucial for exam success. When you see 30%, think 'multiply by 0.3'. When you see 150%, think 'multiply by 1.5'. This mental shift from 'percentage arithmetic' to 'multiplication' not only simplifies problems but also strengthens your understanding. Application of percentages in various contexts helps you grasp their real importance.

For finding what percentage one quantity is of another, you need the reverse process. This is where many students get confused, but there's a reliable method that works every time.

3. Working Through Standard Problems

Let's tackle the most common percentage calculations you'll meet in exams. These step-by-step methods will work for any similar problem, so focus on understanding the process rather than memorising specific examples.

Worked example: Finding a Percentage of a Quantity

Find 35% of £240

1. $35\% = \frac{35}{100} = 0.35$ — Divide by 100 to get decimal form
2. $0.35 \times 240$ — Percentage of quantity means multiply
3. $0.35 \times 240 = 84$ — 35 × 240 ÷ 100 = 8400 ÷ 100 = 84
4. $35\% \text{ of } £240 = £84$ — Always include the original units

Answer: £84

Notice how we converted to decimal first - this avoids fraction arithmetic and reduces errors. The key insight is that 'percentage of' always means multiplication.

Worked example: Expressing One Quantity as a Percentage of Another

Express 18 out of 25 as a percentage

1. $\frac{18}{25}$ — Part over whole - 18 is the part, 25 is the whole
2. $18 \div 25 = 0.72$ — Divide to get decimal form
3. $0.72 \times 100 = 72$ — Multiply by 100 to get percentage
4. $\frac{18}{25} = 72\%$ — Don't forget the % symbol

Answer: 72%

The fraction-to-decimal-to-percentage route is reliable and works with any numbers. Some students try shortcuts, but this method never fails.

Worked example: Percentage Increase Calculation

The price of petrol increases from £1.20 to £1.32 per litre. Find the percentage increase.

1. $£1.32 - £1.20 = £0.12$ — New value minus original value
2. $\frac{0.12}{1.20}$ — Increase divided by original amount
3. $0.1$ — 12 ÷ 120 = 0.1
4. $0.1 \times 100 = 10$ — Multiply by 100 for percentage
5. Percentage increase = $10\%$ — Always specify 'increase' or 'decrease'

Answer: 10% increase

4. Practice Your Method

Before we look at exam traps, let's consolidate the core methods. These practice problems cover the essential techniques you need to master.

These problems test the fundamental skills. If you found any tricky, revisit the step-by-step examples above. The methods are consistent - once you've mastered them, you can tackle any percentage problem.

5. Exam Traps and Common Mistakes

Percentage questions are mark-losers for many students, not because the maths is hard, but because of predictable errors. Let's identify these traps so you can avoid them.

The biggest trap is mixing up 'percentage of' with 'what percentage is'. These are completely different calculations, but students often confuse them under exam pressure. 'Find 30% of 50' uses multiplication. 'What percentage is 30 of 50?' uses division.

Another common error is the 'wrong denominator' mistake in percentage changes. When calculating percentage increase or decrease, you must always divide by the original amount, not the new amount. This catches many students out.

Unit errors are also frequent. If the question gives amounts in pounds and pence, or metres and centimetres, you must convert to the same units before calculating. Mixed units lead to answers that are 100 times too big or too small.

Rounding errors can cost marks too. Unless the question specifies rounding, give exact answers. If you must round, do it only at the final step - never round intermediate calculations as errors accumulate.

6. Real-World Applications

Percentages aren't just exam topics - they're essential life skills. From calculating tips and discounts to understanding interest rates and statistics, percentage fluency affects your daily decisions and financial literacy.

In retail, percentage discounts help you compare deals. A 25% discount on a £80 item saves you £20, making the final price £60. But a 30% discount on a £70 item saves £21, making it £49 - the better deal despite the smaller original discount percentage.

Statistics in news and social media often use percentages to make claims seem more dramatic. Understanding percentage changes helps you evaluate these claims critically. A 100% increase sounds huge, but if something goes from 1 to 2 cases, that's still only 1 additional case.

In finance, percentage calculations determine loan costs, investment returns, and tax implications. A credit card charging 18% annual interest means you pay £18 extra for every £100 borrowed over a year. This multiplicative understanding helps you make informed financial decisions.