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Maths · Fractions, ratio and percentages

Percentages

Multipliers turn every percentage question into one multiplication - increases, decreases, percentage change, finding the original price and interest that grows year after year.

7 key terms
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Last Lesson and Before

Answer each one, then check.

1

What is 10% of 80?

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8
2

Write 0.35 as a percentage.

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35%
3

Write \(\frac{3}{4}\) as a percentage.

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75%
4

Work out \(1.2 \times 50\).

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60

Learning Objectives

1Find a percentage of an amount, with and without a calculator.
2Increase and decrease by a percentage using a multiplier.
3Write one quantity as a percentage of another and find a percentage change.
4Find the original amount after a percentage change (reverse percentages).
5Work out simple and compound interest and depreciation.

Percentages of Amounts

Per cent means "out of 100".

Without a calculator
Build from 10%, 5% and 1%: 35% of £180 = 30% + 5% = £54 + £9 = £63.
With a calculator
Use the decimal: 35% of 180 = \(0.35 \times 180 = 63\).
As a fraction of the whole
15 out of 60 is \(\frac{15}{60} \times 100 = 25\%\).

Multipliers

A multiplier does a percentage change in one step.

Increase by 15%
100% + 15% = 115%, so multiply by 1.15.
Decrease by 15%
100% − 15% = 85%, so multiply by 0.85.
More examples
Increase by 3%: \(\times 1.03\). Decrease by 40%: \(\times 0.6\). Increase by 100%: \(\times 2\).
Why use them
They work for every amount and every percentage, and they make repeated changes easy.

Percentage Increase with a Multiplier

Increase £240 by 15%.

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1 The multiplier \(100\% + 15\% = 115\% = 1.15\)
2 Multiply \(240 \times 1.15 = 276\)

Answer£276

Percentage Change

The price of a jacket rises from £60 to £75. Find the percentage increase.

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1 Find the change \(75 - 60 = 15\)
2 Divide by the ORIGINAL amount \(\dfrac{15}{60} = 0.25\)
3 Convert to a percentage \(0.25 \times 100 = 25\%\)

Answer25% increase

Reverse Percentages

When you know the amount AFTER a percentage change, divide by the multiplier to get back to the original.

The trap
After a 20% increase to £90, the original is NOT £90 minus 20% of £90.
The method
£90 is 120% of the original, so the original is \(90 \div 1.2 = £75\).
Check
\(75 \times 1.2 = 90\).

A Reverse Percentage

In a sale, prices are reduced by 30%. A coat costs £56 in the sale. What was its original price?

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1 The sale price is 70% of the original multiplier \(0.7\)
2 Divide by the multiplier \(56 \div 0.7 = 80\)
3 Check \(80 \times 0.7 = 56\)

Answer£80

Simple and Compound Interest

Interest is money added to savings (or a loan) each year.

Simple interest
The same amount each year, worked out on the original sum: 3% of £2000 is £60 a year.
Compound interest
Interest is added to the total, so next year's interest is on a bigger amount.
The formula
Amount \(= \text{original} \times \text{multiplier}^{\text{years}}\).
Depreciation
The same idea for losing value: a car losing 12% a year is \(\times 0.88\) each year.

Compound Interest

£2000 is invested at 3% compound interest per year. How much is it worth after 4 years?

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1 The multiplier for one year \(1.03\)
2 Four years: multiply by 1.03 four times \(2000 \times 1.03^4\)
3 Work it out \(2251.017\ldots\)

Answer£2251.02

Simple or Compound?

Simple interest on £2000 at 3%

Year 1: £60
Year 2: £60
Year 3: £60
Year 4: £60
Total after 4 years: £2240

Compound interest on £2000 at 3%

Year 1: £60
Year 2: £61.80
Year 3: £63.65
Year 4: £65.56
Total after 4 years: £2251.02

The Best Deal

(a) A shop increases a £50 price by 20%, then reduces the new price by 20%. What is the final price? (b) Bank A pays 4% simple interest. Bank B pays 3.8% compound interest. Which gives more on £1000 after 5 years? (c) A phone costs £252 after a 16% discount. What did it cost before?

1. Write each multiplier.

2. Multiply, or divide for a reverse percentage.

3. Check your answer makes sense.

A good answer shows: (a) \(50 \times 1.2 \times 0.8 = £48\) - not £50. (b) A: \(1000 + 5 \times 40 = £1200\). B: \(1000 \times 1.038^5 = £1205.03\). Bank B. (c) \(252 \div 0.84 = £300\).

Can I...?

1Find a percentage of an amount without a calculator.
2Write a percentage as a multiplier.
3Increase or decrease by a percentage.
4Write one amount as a percentage of another.
5Work out a percentage change.
6Find the original amount (reverse percentage).
7Work out compound interest.
8Work out depreciation.

Summary & Exam Focus

Increase by \(p\%\): multiply by \(1 + \frac{p}{100}\). Decrease: multiply by \(1 - \frac{p}{100}\).
Percentage change = change \(\div\) original \(\times\) 100.
Reverse percentage: divide by the multiplier.
Compound interest: original \(\times\) multiplier to the power of the number of years.

Exam focus

After a 15% decrease, the price of a TV is £391. What was the price before the decrease? (3 marks) (3 marks)

In a reverse percentage question, the amount you are given is NOT 100%. Divide by the multiplier - never take the percentage off the new amount.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Percentage: A number out of 100.
Multiplier: The number you multiply by to make a percentage change, e.g. 1.15 for a 15% increase.
Percentage change: \(\dfrac{\text{change}}{\text{original}} \times 100\).
Reverse percentage: Finding the original amount from the amount after a change.
Simple interest: Interest worked out on the original amount only.
Compound interest: Interest added to the total, so later interest is on a bigger amount.
Depreciation: A loss in value over time.

Practice questions

Have a go at each one before you open its answer.

Question 1 Non-calculator 2 marks

Work out 35% of £180.
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Model answer

10% = £18, so 30% = £54. 5% = £9. 35% = £54 + £9 = £63.

Mark scheme
- A correct method, e.g. 10% = 18 and 5% = 9 — M1
- £63 — A1
Question 2 Non-calculator 2 marks

The price of a jacket rises from £60 to £75. Work out the percentage increase.
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Model answer

Increase = £15. \(\dfrac{15}{60} \times 100 = 25\%\).

Mark scheme
- \(\frac{15}{60}\) — M1
- 25% — A1
Question 3 Calculator 3 marks

After a 15% decrease, the price of a TV is £391. What was the price before the decrease?
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Model answer

£391 is 85% of the original. \(391 \div 0.85 = £460\).

Mark scheme
- 85% or 0.85 used — P1
- \(391 \div 0.85\) — P1
- £460 — A1
Question 4 Calculator 3 marks

Priya invests £5000 for 3 years at 2.5% per year compound interest. How much will the investment be worth at the end of 3 years?
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Model answer

\(5000 \times 1.025^3 = 5384.453\ldots\), so £5384.45.

Mark scheme
- 1.025 used as a multiplier — M1
- \(5000 \times 1.025^3\) — M1
- £5384.45 — A1

Quick check

What multiplier increases an amount by 7%?
1. A0.07
2. B1.7
3. C1.07
4. D0.93
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C: 1.07

100% + 7% = 107% = 1.07.
A price falls from £80 to £60. What is the percentage decrease?
1. A25%
2. B20%
3. C33.3%
4. D75%
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A: 25%

The change is £20, and \(\frac{20}{80} = 25\%\). Always divide by the original.
After a 10% increase a bike costs £330. What was the original price?
1. A£297
2. B£300
3. C£320
4. D£363
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B: £300

£330 is 110% of the original: \(330 \div 1.1 = £300\).

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