GCSE · Maths · AQA · Spec 8300 · Foundation

Percentage change problems and simple interest

A shop puts a jacket's price up 10%, then takes 10% off in a sale. Is it back where it started? It feels like it should be. It isn't.

The big idea

Every percentage change is one multiplication
0%25%50%75%100%125%100% is always the £80 you started with — drag the bar

the bar: 24/25. Percentage of the original 120%. Change 20% increase. Multiplier 1.20. New amount £96. Back to the original £96 ÷ 1.20 = £80

Percentage of the original120%Change20% increaseMultiplier1.20New amount£96Back to the original£96 ÷ 1.20 = £80

100% is the £80 you started with, and it never moves. Drag to 120%: the multiplier is 120 ÷ 100 = 1.2, so the new amount is £80 × 1.2 = £96. Drag to 85%: 15% less is × 0.85, giving £68. Now watch the last chip — divide any new amount by its multiplier and you land back on £80. Know only the before and after? New ÷ original gives the multiplier: £96 ÷ £80 = 1.2, a 20% increase.

Predict, then check

No calculator yet. Go with your gut first.

A £200 jacket goes up in price by 10%. A month later, the new price goes down by 10%. Where does the price end up?

Spot the mistake

Working backwards to the original

After a 25% increase, a town's population is 15,000. What was the population before the increase?

A student's answer — which line goes wrong?

Simple interest

Work it out once, add it on every year

Maya puts £2,500 in a savings account paying 0.54% simple interest per year. How much is in the account after 4 years?

  1. Simple interest is worked out on the £2,500 Maya put in — and it's the same amount of interest every year.That's the whole idea: calculate it once, then add it on again and again.
  2. missing step
Which line is step 2?

Which calculation?

Read the question before you reach for the calculator

Put each question in the job it's asking you to do. Don't work them out yet — just decide.

Still to sort

Find the new amount (0)

You know the original and the percentage change.

Where the line is: The amount you're given is BEFORE the change. Multiply by the multiplier.

Find the original amount (0)

You know the amount AFTER the change.

Where the line is: The amount you're given is AFTER the change. Divide by the multiplier.

Find the percentage change (0)

You know the amount before AND after.

Where the line is: No percentage is given — you're asked for one. Divide the change by the original.

Simple interest (0)

Money invested or borrowed at a yearly rate, for a number of years.

Where the line is: The interest is worked out once on the starting amount, then added for each year.

8 of 8 still to sort.

Most percentage questions are one of these four jobs. The numbers won't tell you which — the words will.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Find new amounts, work back to originals and calculate simple interest — one clear method at a time.

What you need to know

  • A percentage change is one multiplication: 20% more is × 1.2, and 15% less is × 0.85.
  • To find the original amount, divide the new amount by the multiplier — never take the percentage off the new amount.
  • Percentage change = change ÷ original amount × 100.
  • Simple interest is calculated once on the amount invested, then added for each year.
  • To turn a percentage into a decimal, divide by 100: 0.54% = 0.0054.

The big picture

A percentage change is a multiplication by a decimal multiplier: 20% more is × 1.2 and 15% less is × 0.85. Because it's a multiplication, you undo it by dividing by the same multiplier, which is how you find an original amount. Simple interest is worked out once on the amount invested and then added for every year. Read each question carefully to decide which calculation it needs — and remember that percentage change applies to masses, times and populations as well as money.

Key points

1Keep 100% pinned to the original amount. Every other amount is a percentage of it.
2Multiplier = the new percentage ÷ 100. For an increase, add the change to 100% first; for a decrease, take it away from 100%. The same multiplier can be a decimal or a fraction: 125% = 1.25 = 5/4.
3One multiplication by the multiplier finds the change and adds or subtracts it in a single step — quicker than doing both separately.
4Going backwards is dividing: if £96 is 120% of the original, the original is £96 ÷ 1.2 = £80.
5Up 10% then down 10% leaves you lower than you started (× 1.1 × 0.9 = × 0.99), because the fall is 10% of a bigger number.
6Simple interest earns the same amount every year: one year's interest × the number of years.
7Decide the job first: new amount, original amount, percentage change or simple interest. The words tell you which.
8Percentage change isn't just about money — populations, masses, heights and times all change by percentages in the same way.

Worked example

Problem

A town's population rose from 12,500 to 13,400. Find the percentage increase.

⚠ Watch out

Dividing the change by the NEW amount when finding a percentage change. For a rise from 40 cm to 46 cm, 6 ÷ 46 × 100 ≈ 13% is wrong; the percentage is of the original, so it's 6 ÷ 40 × 100 = 15%.

🧠

Memory hook

100% is the original. Going forwards? Multiply by the multiplier. Going backwards? Divide by it.

✓

Check yourself

Without looking back: a price is £132 after a 10% increase. Can you explain, in one sentence, why the original price was £120 and not £118.80?

Flashcards

(14)
What multiplier increases an amount by 20%?
1.2 — because 100% + 20% = 120%, and 120 ÷ 100 = 1.2.
What multiplier decreases an amount by 15%?
0.85 — because 100% − 15% = 85%, and 85 ÷ 100 = 0.85.
How do you write a percentage as a decimal?
Divide by 100. For example, 0.54% = 0.0054 and 7% = 0.07.
Why is using a multiplier quicker than working out the percentage and then adding it on?
One multiplication does both jobs at once: × 1.2 finds the 20% AND adds it on.
You know the amount AFTER a percentage change. How do you find the original?
Divide the new amount by the multiplier. After a 20% rise to £96: £96 ÷ 1.2 = £80.
Why can't you undo a 25% increase by taking 25% off the new amount?
The 25% was of the original. The new amount is bigger, so 25% of it is too much to take off.
A price goes up 10% and then down 10%. Is it back where it started?
No — it's lower. The fall is 10% of a bigger number. × 1.1 × 0.9 = × 0.99, so it ends at 99%.
How do you find a percentage change?
Change ÷ original amount × 100. Always divide by the amount you started with.
Old amount 50, new amount 60. What does 60 ÷ 50 tell you?
The multiplier: 1.2. So the change was a 20% increase.
What makes interest 'simple' interest?
It's worked out once, on the amount first invested, and the same amount is added every year.
£400 earns 5% simple interest a year. How much interest after 3 years?
One year: £400 × 0.05 = £20. Three years: £20 × 3 = £60.
What's the first question to ask in any percentage problem?
Is the amount I've been given before the change or after it — and what am I asked to find?
Write the multiplier for a 25% increase as a decimal and as a fraction.
125% = 1.25 = 125/100 = 5/4. Multiplying by 1.25 or by 5/4 gives the same answer.
Does percentage change only apply to money?
No. Populations, masses, heights and times change by percentages in exactly the same way.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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