KS3 · Maths

Finding the original amount after a percentage change

A snack bar with '15% extra free' weighs 230 g. So a normal bar is 230 g minus 15%, which is 195.5 g? Tempting, but wrong. Here's why.

The original is always 100%

The snack bar with 15% extra

the percentage of a normal bar: 23/24. Percentage of a normal bar 115%. Mass 230 g. Which bar is this? The offer bar you're holding

Percentage of a normal bar115%Mass230 gWhich bar is this?The offer bar you're holding

A special-offer snack bar weighs 230 g and has 15% extra. Each cell of the bar is 5% of a normal bar, so the offer bar fills 23 cells.

Watch out: Taking 15% off 230 g gives 195.5 g, and that's wrong. It finds 15% of the offer bar, but the 15% extra was worked out from the normal bar.

Step 1 · Up or down?

Increase or decrease?

Was each new amount made by an increase or a decrease? Sort it, then read what percentage of the original it is.

Still to sort

Increase (0)

The new amount is MORE than 100% of the original.

Where the line is: Words like extra, profit, interest, gains or rise usually mean an increase.

Decrease (0)

The new amount is LESS than 100% of the original.

Where the line is: Words like discount, loss, depreciation, sale or drops usually mean a decrease.

6 of 6 still to sort.

Before any calculation, decide which way the amount moved. That tells you what percentage the new amount is.

Bar models: find one part, then the whole

Problem

(a) A special-offer cereal box holds 600 g, which includes 20% extra free. How much does a normal box hold? (b) A cat lost 30% of its mass and now has a mass of 4.2 kg. What was its mass before? (c) Alex gives 30% of his sweets to his brother. That is 15 sweets. How many sweets did Alex have to start with?

Your turn · One multiplier

Divide by the multiplier

(a) A special-offer bag weighs 336 g and includes 20% extra. What does a normal bag weigh? (b) A cinema ticket bought online with a 6% saving costs £7.05. What is the usual price, c?

(a) 20% extra is an increase, so 336 g is 100% + 20% = 120% of the normal bag's weight, w.
missing step

Which line is step 2?

120% as a multiplier is 1.2, so 1.2 × w = 336120% as a multiplier is 12, so 12 × w = 336120% as a multiplier is 0.8, so 0.8 × w = 336

Check it · Spot the slip

Where does this answer go wrong?

A special-offer box contains 72 apples. It has 20% extra. How many apples are in a normal box?

A student's answer — which line goes wrong?

The normal box is the original, so the normal box is 100%.
The offer box has 20% taken off, so 72 apples is 80% of the normal box.
80% as a multiplier is 0.8.
72 ÷ 0.8 = 90
A normal box contains 90 apples.

Think it through

Which plan would you use?

A bike's price has gone up by 3%. You know the new price, and you want the price before the rise.

Which plan is closest to what you'd do?

3% is 0.3, so I divide the new price by 1.3.The new price is the one I've got, so that's 100%. I'll take 3% of it and subtract.The new price is 103% of the original, so I divide it by 1.03.The new price is 3% of the original, so I divide it by 0.03.

How sure are you?

I'm sureI think soI'm guessing

3% is 0.3, so I divide the new price by 1.3.

This is a common misconception

Close, but the decimal is off. To turn a percentage into a decimal you divide by 100, not 10, so 3% is 0.03, not 0.3. Dividing by 1.3 would undo a 30% rise. The new price is 103% of the original, so divide by 1.03.

The new price is the one I've got, so that's 100%. I'll take 3% of it and subtract.

This is a common misconception

This is the big one to watch. The original price, from before the rise, is the 100%. The new price is 103% of it. If you take 3% of the new price you're using the wrong 100%, so the answer comes out wrong. Divide the new price by 1.03 instead.

The new price is 103% of the original, so I divide it by 1.03.

This matches the evidence

Yes. The original is 100%, the rise takes it to 103%, and 103% as a multiplier is 1.03. Dividing the new price by 1.03 takes you back to 100%. Your answer should come out a little smaller than the new price, because the price went up.

The new price is 3% of the original, so I divide it by 0.03.

This is a common misconception

3% is only the rise. The new price includes the whole of the original as well: 100% + 3% = 103%. Dividing by 0.03 treats the new price as if it were just the increase, so the answer comes out far too big. It's much bigger than the new price, which fails the check straight away.

Work out what percentage the new value is first (here 103%). Then divide by it as a decimal, remembering to divide the percentage by 100: 3% is 0.03.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

In a reverse percentage problem you know the new value and need to find the original value.
The original value is always 100%.
After an increase, the new value is 100% plus the increase: a 20% increase gives 120%.
After a decrease, the new value is 100% minus the decrease: a 30% decrease gives 70%.
To find the original, divide the new value by the multiplier, for example 1.2 for 120% or 0.7 for 70%.

The big picture

A reverse percentage problem gives you the value after a percentage increase or decrease and asks for the original. The original is always 100%. After an increase the new value is more than 100% (a 20% rise gives 120%). After a decrease it is less (a 30% fall leaves 70%). You can find the original with a bar model (find one part, then the whole), with a double number line or ratio table (do the same to both sides), or by dividing the new value by a single multiplier such as 1.2 or 0.7. Then check that the answer points the right way.

Key points

1Decide first: is it an increase or a decrease? Words like extra, profit and interest usually mean an increase. Words like discount, loss, sale and depreciation usually mean a decrease.

2Bar model: split the original (100%) into equal parts, find the value of one part, then multiply up to the whole.

3Double number line or ratio table: the relationship is multiplicative, so multiply both sides by the same number, for example × 100/115.

4Multiplier: divide the percentage by 100 (115% → 1.15, 94% → 0.94, 3% → 0.03), then divide the new value by it.

5Sense check: after an increase the original is smaller than the new value; after a decrease it is larger.

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Worked example

Problem

A TV costs £393.60 online, which is 18% lower than the high-street price. What is the high-street price?

⚠ Watch out

Treating the value you're given as 100% and taking the percentage off it (or adding it on). The given value is the NEW amount. It's the original that is 100%, so divide the new value by its multiplier instead.

🧠

Memory hook

Name it, divide it, check it. Name the percentage the new value is, divide by its multiplier, then check that the original went the right way.

✓

Check yourself

A salary is £34 980 after a 6% pay rise. What was it before the rise? (£34 980 is 106%, so 34 980 ÷ 1.06 = £33 000, which is smaller, as it should be.)

Flashcards

(14)

What is a reverse percentage problem?

You know the value after a percentage change and need to find the original value.

In a reverse percentage problem, which value is 100%?

The original value. It's always the original, never the new value you're given.

After a 20% increase, the new amount is what percentage of the original?

120%: the original 100% plus the 20% increase.

After a 30% decrease, the new amount is what percentage of the original?

70%: 100% minus the 30% decrease.

How do you turn a percentage into a multiplier?

Divide it by 100. For example, 115% → 1.15 and 94% → 0.94.

What is 3% as a decimal?

0.03. Divide by 100, not 10, so it is not 0.3.

Once you have the multiplier, how do you find the original?

Divide the new value by the multiplier.

Name some key words that usually mean a DECREASE.

Discount, loss, depreciation, sale, drops.

Name some key words that usually mean an INCREASE.

Profit, extra, interest, gains, increases.

Bar model: what do you work out before the whole?

The value of one equal part. Then multiply up to all the parts that make 100%.

On a double number line or ratio table, what must you do to both sides?

The same thing: multiply both by the same number, because the relationship is multiplicative.

When is a double number line handier than a bar model?

When the bar can't easily be split into equal parts that show the percentage.

Sense check: how should the original compare with the new value?

After an increase the original is smaller. After a decrease the original is larger.

A profit of 45%: what percentage of the original is the selling price?

145%. Profit is an increase, so add it to 100%. It isn't 55%.

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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Increase or decrease?

Divide by the multiplier

Where does this answer go wrong?

Which plan would you use?

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