Comparing and Working with Percentages Revision Notes | GCSE AQA Higher Maths

1. Why Percentages Make Comparisons Fair

Imagine two shops both reduce their prices. Shop A cuts £10 off a £40 jacket; Shop B cuts £10 off a £200 coat. The cash saving is identical — but which deal is actually better? Raw numbers hide the answer. Percentages reveal it.

Converting quantities to percentages puts them on the same scale, so you can compare fairly. This is the core idea behind both spec requirements in this lesson: comparing two quantities using percentages, and working confidently with percentages that go above 100%.

2. Comparing Two Quantities Using Percentages

To compare two quantities as a percentage, you express one quantity as a fraction of the other, then multiply by 100. The key decision is: which quantity is your reference (the denominator)?

Back to the jacket example: Shop A's discount is $\dfrac{10}{40} \times 100 = 25\%$. Shop B's discount is $\dfrac{10}{200} \times 100 = 5\%$. Shop A is clearly the better deal — percentages made that visible instantly.

Worked example: Comparing Two Test Scores as Percentages

Priya scores 36 out of 45 in a maths test. Mohammed scores 52 out of 80 in a different maths test. Who performed better? [3 marks]

1. $\dfrac{36}{45} \times 100$ — Divide her score by the total marks available, then multiply by 100. [M1]
2. $\dfrac{36}{45} = 0.8$, so $0.8 \times 100 = 80\%$ — 36 ÷ 45 = 0.8 exactly. Priya scored 80%.
3. $\dfrac{52}{80} \times 100$ — Same method — divide by his total, multiply by 100. [M1]
4. $\dfrac{52}{80} = 0.65$, so $0.65 \times 100 = 65\%$ — 52 ÷ 80 = 0.65. Mohammed scored 65%.
5. $80\% > 65\%$, so Priya performed better. — State the comparison explicitly — examiners want a clear conclusion. [A1]

Answer: Priya performed better: 80% compared to Mohammed's 65%.

Notice that the method works for any two quantities — prices, distances, populations, or scores — as long as you are clear about which value is the reference. The reference goes on the bottom of the fraction.

3. Percentages Greater Than 100%

Here is where many students hit a mental block: can a percentage be more than 100? The answer is yes — and it comes up regularly in GCSE Higher exam questions.

A percentage greater than 100% means the quantity you are measuring is larger than the reference quantity. If a town's population grows from 4,000 to 10,000, the new population is $\dfrac{10{,}000}{4{,}000} \times 100 = 250\%$ of the original. The population is two and a half times what it was — perfectly sensible, and greater than 100%.

The calculation method is identical to before — you are still dividing one quantity by a reference and multiplying by 100. The only difference is that the result lands above 100.

Worked example: Calculating a Percentage Greater Than 100%

A company's revenue was £80,000 last year. This year it is £124,000. Express this year's revenue as a percentage of last year's revenue. [2 marks]

1. Part = £124,000 (this year). Reference = £80,000 (last year). — The reference is last year's revenue — that is what we are comparing against. [M1]
2. $\dfrac{124{,}000}{80{,}000} \times 100$ — Part over reference, multiplied by 100.
3. $\dfrac{124{,}000}{80{,}000} = \dfrac{124}{80} = 1.55$ — Divide numerator and denominator by 1,000 first to keep numbers manageable.
4. $1.55 \times 100 = 155\%$ — The result exceeds 100% because this year's revenue is greater than last year's. [A1]

Answer: This year's revenue is 155% of last year's revenue.

Notice that 155% means the revenue is 1.55 times the original — it has grown by 55% on top of the original 100%. That connection between the percentage and the multiplier is worth keeping in mind.

4. Connecting the Two Ideas: A Comparison Problem

Exam questions often combine both skills — you compare two quantities and one of the percentages turns out to be above 100%. This is where the method needs to feel automatic.

Worked example: Comparing Two Quantities Where One Percentage Exceeds 100%

Aisha's electricity bill was £95 in January. In February it was £114. Ben's bill was £60 in January and £66 in February. Who had the larger percentage increase in their bill? [4 marks]

1. $114 - 95 = £19$ increase — Calculate the change first: new minus original.
2. $\dfrac{19}{95} \times 100 = 0.2 \times 100 = 20\%$ — $19 \div 95 = 0.2$ exactly. Aisha's bill rose by 20%. [M1, A1]
3. $66 - 60 = £6$ increase — Same method for Ben.
4. $\dfrac{6}{60} \times 100 = 0.1 \times 100 = 10\%$ — $6 \div 60 = 0.1$. Ben's bill rose by 10%. [M1, A1]
5. $20\% > 10\%$, so Aisha had the larger percentage increase. — Even though Aisha's cash increase (£19) was larger than Ben's (£6), the question asks for percentage — always answer what is asked.

Answer: Aisha had the larger percentage increase: 20% compared to Ben's 10%.

Now consider a variation: what if Ben's February bill had been £126 instead of £66? His percentage increase would be $\dfrac{126}{60} \times 100 = 210\%$ of the original — a percentage greater than 100%, meaning his bill is 2.1 times the original (a 110% increase). Both skills appear in the same question.

5. Deliberate Practice

These questions cover both spec requirements. Work through each one before checking the solution — the method is the same every time, but the context changes.

6. Exam Technique and Common Mistakes

The method for percentage comparison is consistent, but there are predictable places where marks get dropped. The following guidance targets the errors that appear most often in mark schemes.

Bringing both spec requirements together: comparing quantities using percentages and working with percentages above 100% are the same skill applied in different contexts. The formula never changes — only the size of the result does.