Percentages: of an amount, increase, decrease, reverse and simple interest
Finding a percentage of an amount, expressing 1 quantity as a percentage of another, percentage change, working backward to an original amount, and simple interest
About three lessons of 45 to 60 minutes
The number that is secretly a fraction out of 100
A shop sign says '20% off'. A phone battery says '15% remaining'. A bank account says '3% interest per year'. Percentage just means 'out of 100', so 20% is really 20/100 (or 0.2), and once that connection is automatic, EVERY percentage problem, no matter how it is phrased, becomes an ordinary multiplication or division.
This unit builds the whole family of percentage skills from that 1 idea: finding a percentage of an amount, expressing 1 amount as a percentage of another, calculating the new value after a percentage change, working backward from a changed value to the original, and simple interest, which is just a percentage increase repeated once per year.
- A £40 jacket with 25% off25% of £40 is £10, so the sale price is £30
- A £200 phone whose price rises 10%the new price is 110% of the original, £220
- £500 saved at 4% simple interest for 3 yearsinterest = £500 x 0.04 x 3 = £60
- A sale price of £75, after a 25% reductionworking backward: the original price was £75 / 0.75 = £100
What students will be able to do
Students will find a percentage of an amount (including over 100%), express 1 quantity as a percentage of another, calculate percentage increase and decrease, work backward from a percentage change to find the original amount, and calculate simple interest.
- I can find a percentage of an amount by converting the percentage to a fraction or decimal and multiplying.
- I can express 1 quantity as a percentage of another by dividing and multiplying by 100.
- I can find the new value after a percentage increase or decrease.
- I can work backward from a value after a percentage change to find the original amount.
- I can calculate simple interest using I = PRT/100 and find the total amount.
Standards this unit teaches
- KS3 Maths: Number; Ratio, proportion and rates of changeUK National Curriculum (England)Number; Ratio, proportion and rates of change
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "define percentage as 'number of parts per hundred', interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare 2 quantities using percentages, and work with percentages greater than 100%" (Number strand); "solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics" (Ratio, proportion and rates of change strand).
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Fraction-decimal equivalence worksheeta percentage is just a fraction with a denominator fixed at 100, so fraction/decimal fluency transfers directly
- Multiplying decimals in the glossaryevery percentage calculation here is a decimal multiplication underneath
- Fraction-decimal equivalence teaching unitpercentages are fractions out of 100, so this fraction/decimal fluency is the direct foundation
Words to teach and display
- Percentage
- a number expressed as 'parts per hundred', written with a % sign
- Percentage increase
- when a value grows by a given percentage of its original amount
- Percentage decrease
- when a value shrinks by a given percentage of its original amount
- Original value (reverse percentage)
- the amount before a percentage change was applied, found by working backward from the changed value
- Simple interest
- interest calculated only on the original principal every period, never on interest already earned (unlike compound interest)
- Principal
- the original amount of money invested or borrowed, before any interest is added
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Percentage of an amount, and expressing a quantity as a percentage
Concrete'Find 30% of 150' means 'find 30/100 of 150', which is exactly the same calculation as finding a fraction of an amount: multiply. Going the other way, 'what percentage is 45 out of 150?' divides first (45/150), then multiplies by 100 to convert that fraction back into a percentage.
Percentages over 100% work exactly the same way: 120% of an amount is simply MORE than the whole amount, for example when a recipe needs '120% of the usual batch size' to make extra.
Find 30% of 150. Then find what percentage 45 is of 150.
- 30% of 150 = (30/100) x 150 = 0.3 x 150 = 45.
- What percentage is 45 of 150? = (45/150) x 100 = 0.3 x 100 = 30%.
Answer: 30% of 150 is 45. 45 out of 150 is 30%.
- Find 15% of 80.
- Why do '30% of 150 = 45' and '45 out of 150 = 30%' use exactly the same 2 numbers, just rearranged?
2. Percentage increase and decrease
PictorialAfter a percentage increase, the new value is the ORIGINAL 100% plus the extra percentage: a 20% increase gives 120% of the original. After a percentage decrease, the new value is 100% minus the percentage removed: a 20% decrease gives 80% of the original.
This means every percentage change can be done in a single multiplication, without a separate 'find the change, then add or subtract it' step: multiply by 1.2 for a 20% increase, or by 0.8 for a 20% decrease.
A jacket costs £80. Its price increases by 15%. A different item costs £60 and decreases by 20%. Find both new prices.
- Increase: 15% increase means the new price is 100% + 15% = 115% of the original. £80 x 1.15 = £92.
- Decrease: 20% decrease means the new price is 100% - 20% = 80% of the original. £60 x 0.8 = £48.
Answer: The jacket's new price is £92. The other item's new price is £48.
- Why does a 15% increase multiply the original by 1.15, not by 0.15?
- A £50 item decreases by 10%. Find the new price.
3. Reverse percentages and simple interest
AbstractA reverse percentage problem gives the AFTER value and asks for the BEFORE (original) value. Because the after value is a percentage multiple of the original (like 115% or 80%), working backward means DIVIDING by that multiplier, never subtracting the percentage from the after value directly.
Simple interest is a percentage increase applied to the SAME original principal every period (never to a growing balance, that would be compound interest instead). The formula I = PRT/100 gives the interest for principal P, rate R% per year, over T years; the total amount is the principal plus all the interest earned.
After a 20% increase, a bike's price is £96. Find the original price. Then find the total amount when £200 is invested at 5% simple interest for 3 years.
- Reverse percentage: a 20% increase means £96 is 120% of the original. Original = £96 / 1.2 = £80.
- Check: 20% of £80 is £16, and £80 + £16 = £96. Correct.
- Simple interest: I = PRT/100 = (200 x 5 x 3)/100 = £30.
- Total amount = principal + interest = £200 + £30 = £230.
Answer: The bike's original price was £80. The total amount after 3 years is £230.
- Why is the original price found by DIVIDING £96 by 1.2, rather than subtracting 20% of £96?
- Find the simple interest on £300 at 4% per year for 2 years.
Common misconceptions and how to address them
MisconceptionTo increase a value by 20%, just add 20 to the number.
Why it happens: The word 'percentage' gets dropped mentally and 20% is treated as if it meant 'the number 20'.
How to address it: Always say the full sentence out loud: 'increase by 20% of the ORIGINAL amount', then calculate that 20% first as its own step before adding it on.
MisconceptionDecreasing a value by 50% and then increasing the result by 50% brings it back to the original value.
Why it happens: Percentage changes feel like they should cancel out symmetrically, the way +5 and -5 do for addition.
How to address it: Work a concrete example: £100 decreased by 50% is £50, then £50 increased by 50% is £75, not £100, because the SECOND 50% is calculated on the smaller, already-reduced amount.
MisconceptionTo reverse a percentage change, just subtract the percentage from the final value (e.g. a value of £96 after a 20% increase means the original was £96 minus 20% of £96).
Why it happens: It looks like the natural 'undo' of the increase step, mirroring how subtraction undoes addition.
How to address it: The 20% that was added was 20% of the ORIGINAL (unknown) value, not 20% of the final value, so subtracting 20% of £96 uses the wrong base and gives the wrong answer. Dividing by 1.2 is the only way that correctly undoes multiplying by 1.2.
MisconceptionSimple interest compounds, growing on top of itself each year like a typical bank account.
Why it happens: The everyday word 'interest' is usually experienced as compound interest (bank savings accounts), so students assume all interest works that way.
How to address it: Point at the number line figure: SIMPLE interest adds the exact same amount every single year, because it is always calculated on the original principal, never on the growing balance. Compound interest (not covered in this unit) would add a growing amount each year instead.
Guided practice (with answers)
1. Find 40% of 60.
Answer: 24, because 40% of 60 = 0.4 x 60 = 24.
2. 24 out of 30 questions were correct. What percentage is that?
Answer: 80%, because (24/30) x 100 = 0.8 x 100 = 80.
3. A £40 item increases in price by 25%. Find the new price.
Answer: £50, because a 25% increase means the new price is 125% of the original: £40 x 1.25 = £50.
4. A £90 item decreases in price by 10%. Find the new price.
Answer: £81, because a 10% decrease means the new price is 90% of the original: £90 x 0.9 = £81.
5. After a 10% decrease, an item costs £72. Find the original price.
Answer: £80, because £72 is 90% of the original, so original = £72 / 0.9 = £80.
6. Find the simple interest on £400 at 3% per year for 4 years.
Answer: £48, because I = PRT/100 = (400 x 3 x 4)/100 = 4800/100 = 48.
Independent practice worksheets
Practise every percentage skill with computed, never-wrong answer keys.
Differentiation
- Keep a printed 'common percentages as fractions' reference card visible (10% = 1/10, 25% = 1/4, 50% = 1/2, 75% = 3/4) until the fraction/decimal conversion is automatic.
- For percentage change, always write the multiplier (e.g. 1.2 or 0.8) down as its own explicit step before multiplying, rather than doing it mentally.
- For reverse percentages, write 'final value = multiplier x original' as a sentence first, then rearrange it, rather than guessing which operation to use.
- Use only whole-number, easily-divisible percentages (10%, 20%, 25%, 50%) until the multiplier method is secure.
- Investigate why 2 successive percentage changes (e.g. +10% then -10%) never exactly cancel out, using algebra: original x 1.1 x 0.9 = original x 0.99.
- Compare simple interest to compound interest on the same principal and rate over 5 years, and explain why compound interest grows faster.
- Solve a multi-step real-world problem combining a percentage discount AND a percentage tax added afterward.
- Find the percentage change (as a single percentage) equivalent to 2 successive percentage changes applied one after another.
Assessment: exit ticket
A three-question exit ticket sampling percentage of an amount, percentage change, and a reverse percentage or simple interest question.
1. Find 35% of 200.
Answer: 70, because 35% of 200 = 0.35 x 200 = 70.
2. A £120 item increases in price by 15%. Find the new price.
Answer: £138, because a 15% increase means the new price is 115% of the original: £120 x 1.15 = £138.
3. Find the simple interest on £600 at 2% per year for 5 years.
Answer: £60, because I = PRT/100 = (600 x 2 x 5)/100 = 6000/100 = 60.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 percentage of an amount plus expressing as a percentage (section 1), Lesson 2 percentage increase/decrease (section 2), Lesson 3 reverse percentages plus simple interest (section 3) plus the exit ticket.
- Every amount in this unit and its matching worksheets is constructed backward from the chosen percentage (the amount is always a whole multiple of 100/gcd(percent, 100)), so every percentage result is a genuine, exact whole number, never a rounded approximation, matching this site's 'never wrong' answer-key policy.
- The multiplier method (multiply by 1 + rate for an increase, 1 - rate for a decrease) is used consistently across increase, decrease and reverse-percentage questions, since it is both faster and more reliable at GCSE than the 'find the change, then add or subtract it' 2-step method.
- Percentages greater than 100% are deliberately included in section 1 (a curriculum requirement), since students who have only ever seen percentages between 0% and 100% often believe a percentage cannot exceed 100%.