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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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Teacher resources

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Student handouts

The same files the students see, to print or hand out.

Last Lesson and Before

Answer each one, then check.

  1. 1

    What is 10% of 80?

    Show answerHide answer

    8

  2. 2

    Write 0.35 as a percentage.

    Show answerHide answer

    35%

  3. 3

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

    Show answerHide answer

    75%

  4. 4

    Work out \(1.2 \times 50\).

    Show answerHide answer

    60

Learning Objectives

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

Show the solutionHide the solution
  1. 1 The multiplier \(100\% + 15\% = 115\% = 1.15\)
  2. 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.

Show the solutionHide the solution
  1. 1 Find the change \(75 - 60 = 15\)
  2. 2 Divide by the ORIGINAL amount \(\dfrac{15}{60} = 0.25\)
  3. 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?

Show the solutionHide the solution
  1. 1 The sale price is 70% of the original multiplier \(0.7\)
  2. 2 Divide by the multiplier \(56 \div 0.7 = 80\)
  3. 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?

Show the solutionHide the solution
  1. 1 The multiplier for one year \(1.03\)
  2. 2 Four years: multiply by 1.03 four times \(2000 \times 1.03^4\)
  3. 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...?

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

Questions and answers

7 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Non-calculator 2 marks Easier

Work out 35% of £180.

Mark scheme — 2 marks available

  • A correct method, e.g. 10% = 18 and 5% = 9 — M1
  • £63 — A1

Model answer

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

2. Exam question Non-calculator 2 marks Easier

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

Mark scheme — 2 marks available

  • \(\frac{15}{60}\) — M1
  • 25% — A1

Model answer

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

3. Exam question Calculator 3 marks Easier

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

Mark scheme — 3 marks available

  • 85% or 0.85 used — P1
  • \(391 \div 0.85\) — P1
  • £460 — A1

Model answer

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

4. Exam question Calculator 3 marks Easier

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?

Mark scheme — 3 marks available

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

Model answer

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

5. Multiple choice 1 mark Easier

What multiplier increases an amount by 7%?

  1. A 0.07
  2. B 1.7
  3. C 1.07 Correct
  4. D 0.93

Why: 100% + 7% = 107% = 1.07.

6. Multiple choice 1 mark Core

A price falls from £80 to £60. What is the percentage decrease?

  1. A 25% Correct
  2. B 20%
  3. C 33.3%
  4. D 75%

Why: The change is £20, and \(\frac{20}{80} = 25\%\). Always divide by the original.

7. Multiple choice 1 mark Stretch

After a 10% increase a bike costs £330. What was the original price?

  1. A £297
  2. B £300 Correct
  3. C £320
  4. D £363

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