Websink Prep UKGCSE MathsTextbook contentsMaths textbooksYear 1Year 2Year 4Year 5Year 6GCSEBrowse allPast papersFind a tutorBecome a tutor
GCSE Maths · EdexcelChapter 5 · Percentages
Chapter 5 · Higher tier

GCSE Maths Percentages (Higher): worked examples and questions with answers

By the end of this chapter you will be able to

  • Explain a percentage as a number of parts per hundred and change between fractions, decimals and percentages
  • Find a percentage of an amount with and without a calculator
  • Write one quantity as a percentage of another and compare quantities using percentages
  • Increase and decrease an amount by a percentage using a multiplier, including percentages greater than 100%
  • Work out percentage change, profit and loss
  • Solve reverse percentage (original value) problems
  • Calculate simple interest and compound interest
  • Solve repeated percentage change problems: growth and decay, depreciation, and find the rate or the number of years (Higher)

You should already know

  • Fractions, equivalent fractions and fractions of an amount (Chapter 3)
  • Multiplying and dividing by 10 and 100 (Chapter 1)
  • Decimals and rounding (Chapters 1 and 4)

Where this leads

  • Ratio and proportion (Chapter 15)
  • Compound measures (Chapter 16)
  • Pie charts and averages (Chapter 24)
  • Exponential graphs and growth and decay (Chapter 25)

Edexcel GCSE (9–1) Mathematics 1MA1, Higher tier. Content references: R9, N12, R16.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FWork out. Do not use a calculator.
(a)3.5 × 100
(b)45 ÷ 100
(c)260 ÷ 10
(d)0.07 × 100
2
FWrite 3660 in its simplest form.
3
FWrite 0.375 as a fraction in its simplest form.
4
FWork out 38 of 64.
5
FRound 17.4567 to 2 decimal places.
5.1

Percentages, fractions and decimals

Edexcel 1MA1: R9, N12
Grade 1-4

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Percentages, fractions and decimalsComing soon

You meet percentages every day: “30% off” in a shop, “battery 80%” on a phone, “a 5% pay rise”. A percentage tells you how many parts out of every 100. The words per cent come from Latin and mean “for each hundred”. The symbol is %.

Picture a square made of 100 small squares. If 35 of the small squares are shaded, then 35 out of 100 are shaded. We write this as 35%, or as the fraction 35100, or as the decimal 0.35. They are three different ways of writing the same amount.

35 out of 100 squares are shaded: 35% = 35100 = 0.35.

Because a percentage is always out of 100, 1% is just one hundredth: 1% = 1100 = 0.01. Some percentages are used so often that they are worth learning by heart. They sit at the same spot on a number line, whichever way you write them.

0%0025%1/40.2550%1/20.575%3/40.75100%11fractiondecimal
Common percentages, fractions and decimals on the same line.
PercentageFractionDecimal
1%11000.01
10%1100.1
20%150.2
25%140.25
50%120.5
75%340.75
100%1 (the whole)1

Changing between the three forms

Percentage to decimal: divide by 100. The digits move two places to the right in the place-value table, so 45% = 0.45 and 6% = 0.06 (not 0.6).

Decimal to percentage: multiply by 100. So 0.7 = 70% and 0.05 = 5%.

Percentage to fraction: write the percentage over 100, then simplify. 45% = 45100 = 920.

Fraction to percentage: if the denominator goes into 100 (2, 4, 5, 10, 20, 25, 50) make an equivalent fraction with denominator 100: 320 = 15100 = 15%. If it does not (such as 8), divide the top by the bottom to get a decimal, then multiply by 100: 38 = 3 ÷ 8 = 0.375 = 37.5%.

Fraction3/4Decimal0.75Percentage75%top ÷ bottom× 100÷ 100place valueThree ways to write the same amountBlue arrows go left to right. Orange arrows go back again.
Fraction, decimal and percentage: the arrows show how to move between them.

Percentages bigger than 100% and smaller than 1%

100% means the whole amount, so 150% means the whole amount and a half as much again. As a decimal 150% = 1.5, and as a fraction 150% = 150100 = 112. Likewise a percentage under 1%, such as 0.5%, is just a very small part of the whole: 0.5% = 0.005. We use percentages bigger than 100% to describe things that have grown a lot, for example “the price is now 250% of what it was”.

the original: 100%+50%0%100%150%150% of the original is more than the original
150% = 100% (the whole) + 50% (a half as much again) = 1.5.

Key ideas

  • Per cent means out of 100: 1% = 1100 = 0.01.
  • % to decimal: divide by 100. Decimal to %: multiply by 100.
  • % to fraction: put it over 100 and simplify.
  • Fraction to %: make the denominator 100, or divide top by bottom, then multiply by 100.
  • To compare a fraction, a decimal and a percentage, write them all as decimals.
  • Learn: 50% = 12, 25% = 14, 75% = 34, 10% = 110, 20% = 15.
Worked example 5.1.1
Write 35% as a fraction in its simplest form.

Method. Write the percentage as a fraction out of 100, then simplify by dividing the top and the bottom by their highest common factor.

  1. Write the percentage as a fraction out of 100. Per cent means “out of 100”.
    35% = 35100
  2. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 35: 1, 5, 7, 35
    Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
    The highest common factor is 5.
  3. Divide the top and the bottom by 5. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    35 ÷ 5 = 7 and 100 ÷ 5 = 20
    35100 = 720

Check. As a decimal, 35% = 0.35 and 720 = 0.35. ✓

Watch out. Always simplify. 40% = 40/100 is not finished until you reach 2/5.

Worked example 5.1.2
Write 6% as a decimal.

Method. A percentage is a number out of 100, so to change it to a decimal divide by 100 (move every digit two places to the right of the decimal point).

  1. Write the percentage as a number over 100. Per cent means “out of 100”.
    6% = 6100
  2. Divide by 100. Dividing by 100 moves each digit two places to the right in the place-value table (the decimal point moves two places left).
    6 ÷ 100 = 0.06
  3. Check the size. A percentage below 100% must give a decimal below 1, and a small percentage a small decimal.
    6% = 0.06

Check. Go back the other way: 0.06 × 100 = 6, which is 6%. ✓

Watch out. The decimal point moves two places, not one. 5% is 0.05, not 0.5.

Worked example 5.1.3
Write 38 as a percentage.

Method. To change a fraction to a percentage, find an equivalent fraction with denominator 100, or divide the top by the bottom and multiply by 100.

  1. Write the fraction. A fraction already compares a part with a whole.
    38
  2. Divide the top by the bottom. This changes the fraction into a decimal.
    3 ÷ 8 = 0.375
  3. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.375 × 100 = 37.5%

Check. 37.5% = 37.5 ÷ 100 = 0.375, and 3 ÷ 8 = 0.375. ✓

Watch out. Percentages are out of 100. A fraction like 3/8 cannot be written over 100 directly, so divide instead.

Worked example 5.1.4
Write these in order, smallest first:   35,   0.55,   62%

Method. To compare fractions, decimals and percentages, write them all as decimals. Then put the decimals in order.

  1. Change every number into a decimal. Decimals are easy to compare digit by digit, so we use them as a common language.
    35 = 3 ÷ 5 = 0.6
    0.55 is already a decimal: 0.55
    62% = 62 ÷ 100 = 0.62
  2. Put the decimals in order. We want smallest first. Compare the digits from the left, starting with the tenths.
    0.55 , 0.6 , 0.62
  3. Write the original numbers in that order. The answer should use the forms given in the question.
    0.55 ,  35 ,  62%

Check. Read the final list again as decimals: it goes up each time. ✓

Watch out. Do not compare the digits of different forms directly. 3/5 is not smaller than 0.4 just because 3 is smaller than 4.

Worked example 5.1.5
Write 17.5% as a fraction in its simplest form.

Method. Write the percentage as a fraction out of 100, then simplify by dividing the top and the bottom by their highest common factor.

  1. Write the percentage as a fraction out of 100. Per cent means “out of 100”.
    17.5% = 17.5100
  2. Remove the decimal point: multiply the top and the bottom by 10. A fraction should have whole numbers on the top and bottom. Multiplying both by the same number does not change its value.
    17.5100 = 1751000
  3. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 175: 1, 5, 7, 25, 35, 175
    Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
    The highest common factor is 25.
  4. Divide the top and the bottom by 25. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    175 ÷ 25 = 7 and 1000 ÷ 25 = 40
    1751000 = 740

Check. As a decimal, 17.5% = 0.175 and 740 = 0.175. ✓

Watch out. Always simplify. 40% = 40/100 is not finished until you reach 2/5.

Worked example 5.1.6
Write 512 as a percentage. Give your answer correct to 1 decimal place.

Method. To change a fraction to a percentage, find an equivalent fraction with denominator 100, or divide the top by the bottom and multiply by 100.

  1. Write the fraction. A fraction already compares a part with a whole.
    512
  2. Divide the top by the bottom. This changes the fraction into a decimal.
    5 ÷ 12 = 0.416667…
  3. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.416667… × 100 = 41.666667…%
  4. Round the answer. Round to 1 decimal place as the question asks.
    41.666667…% = 41.7% (to 1 d.p.)

Check. 41.7% = 41.7 ÷ 100 = 0.417, and 5 ÷ 12 = 0.416667…. ✓

Watch out. Percentages are out of 100. A fraction like 3/8 cannot be written over 100 directly, so divide instead.

Watch out

Moving the decimal point the wrong way. 6% is 0.06, not 0.6, because 6% is 6 out of 100. And 0.6 is 60%, not 0.6%. Always ask: “is 6% a small part of the whole?” (yes, so the decimal must be small).

Higher tier. Not every fraction gives a neat percentage. 13 = 0.3333… (the 3s go on for ever), so 13 = 33.333…%. We round: 13 ≈ 33.3% to 1 decimal place. For a percentage with a decimal part, such as 12.5%, remove the decimal point before simplifying: 12.5% = 12.5100 = 1251000 = 18.

Exercise 5A  Percentages, fractions and decimals

Non-calculator
1
FIn the hundred square, 37 of the small squares are shaded.
(a)Write the shaded part as a fraction out of 100.
(b)Write the shaded part as a percentage.
(c)Write the shaded part as a decimal.
2
FWrite each percentage as a decimal.
(a)8%
(b)45%
(c)70%
(d)3%
(e)125%
(f)12.5%
3
FWrite each percentage as a fraction in its simplest form.
(a)50%
(b)20%
(c)15%
(d)45%
(e)4%
(f)85%
4
FWrite each decimal as a percentage.
(a)0.27
(b)0.9
(c)0.05
(d)0.375
(e)1.2
(f)2.5
5
FWrite each fraction as a percentage. (Hint: make the denominator 100.)
(a)710
(b)925
(c)320
(d)1750
(e)1120
(f)1325
6
FWrite each fraction as a percentage. You need to divide the top by the bottom.
(a)38
(b)58
(c)740
(d)916
7
FWrite each fraction as a decimal and as a percentage.
(a)35
(b)720
(c)1725
(d)18
8
FWrite these in order, starting with the smallest.
(a)(a) 35,  0.55,  62%
(b)(b) 0.4,  720,  36%
(c)(c) 45%,  920,  0.405,  25
9
RMia says, “5% as a decimal is 0.5.” Is she right? Explain your answer.
10
RWhich is bigger, 0.4 or 4%? Show how you know.
11
PSIn a class of 25 students, 18 walk to school.
(a)What percentage of the class walk to school?
(b)What percentage do not walk to school?
12
PSThree phones show their battery levels as 0.35,  38 and 36%. Write the three battery levels in order, largest first.
13
PSOne shop has “14 off all T-shirts”. Another has “20% off all T-shirts”. Which shop has the bigger discount?
14
RIs 37 bigger or smaller than 40%? Show your working. (Do not use a calculator.)
15
FWrite each fraction as a percentage. Give your answers correct to 1 decimal place.
(a)13
(b)56
(c)29
(d)712
16
FWrite each percentage as a fraction in its simplest form.
(a)0.8%
(b)2.5%
(c)12.5%
(d)0.05%
17
RWrite down a fraction with denominator 40 that is bigger than 30% but smaller than 35%.
5.2

Percentages of an amount

Edexcel 1MA1: R9, N12
Grade 1-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Percentages of an amountComing soon

“Find 35% of £240.” Here 35% works like an instruction: take 35 hundredths of the number. Mathematicians call a fraction or a percentage used in this way an operator, because it operates on a number and changes it. You already know that “35 of 40” means “divide by 5, then multiply by 3”. A percentage of an amount works in exactly the same way, and the word of means multiply.

Without a calculator: build up from 10% and 1%

Some percentages are very easy to find. 10% is one tenth, so divide by 10. 1% is one hundredth, so divide by 100. 50% is a half, and 25% is a quarter. To find any other percentage, split it into pieces that you can do. For example 35% = 10% + 10% + 10% + 5%, and 5% is half of 10%.

24242424242424242424The whole bar is 240 = 100%. Each piece is 10% = 24.0%50%100%
Splitting 240 into ten equal pieces, one for every 10%.

With a calculator: use a decimal

Write the percentage as a decimal (divide by 100) and multiply. So 35% of 240 is 0.35 × 240 = 84. This is quick, but you must be careful to divide by 100 first: 8% is 0.08 (not 0.8). When the answer is money, round to the nearest penny, which means two decimal places.

It is a good habit to estimate first. For example, 48% of £603 is close to 50% of £600, which is £300. If your calculator says £2,894, you know you have made a slip.

Key ideas

  • “Of” means multiply; a percentage can be used as an operator on an amount.
  • 10% = divide by 10.   1% = divide by 100.   50% = divide by 2.   25% = divide by 4.
  • 5% = half of 10%.   2.5% = a quarter of 10%.
  • Build any percentage from these: 35% = 10% + 10% + 10% + 5%.
  • Calculator: decimal × amount, e.g. 17.5% of 64 = 0.175 × 64.
  • Round money to 2 decimal places (the nearest penny).
Worked example 5.2.1
Work out 30% of £240. Do not use a calculator.

Key words. Per cent means “out of 100”. A percentage is a fraction whose denominator is 100, written with the % sign. A multiplier is the number you multiply by to get the new amount in one step.

Method. To find a percentage of an amount without a calculator, find 10% (divide by 10) or 1% (divide by 100) first. Then build up to the percentage you need.

  1. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
    10% of £240 = £240 ÷ 10 = £24
  2. Work out each part. Multiply to build up to the percentage you want.
    30% = 3 × 10% = 3 × £24 = £72
  3. Write the answer. Include the units.
    30% of £240 = £72

Check. Another way: 30% = 0.3 as a decimal, and 0.3 × £240 = £72. ✓

Watch out. 10% means divide by 10 once. Do not divide by 10 and then by 10 again.

Worked example 5.2.2
Work out 7% of 450 g. Do not use a calculator.

Method. To find a percentage of an amount without a calculator, find 10% (divide by 10) or 1% (divide by 100) first. Then build up to the percentage you need.

  1. Split the percentage into easy parts. We can find 10% and 1% very quickly, so we build the percentage out of them.
    7% = 5% + 2%
  2. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
    10% of 450 g = 450 g ÷ 10 = 45 g
  3. Find 1%. 1% is one hundredth, so divide the amount by 100 (each digit moves two places to the right).
    1% of 450 g = 450 g ÷ 100 = 4.5 g
  4. Work out each part. Use the 10% or 1% you found.
    5% = half of 10% = 45 g ÷ 2 = 22.5 g
    2% = 2 × 1% = 2 × 4.5 g = 9 g
  5. Add the parts together. The parts together make up the whole percentage.
    22.5 g + 9 g = 31.5 g

Check. Another way: 7% = 0.07 as a decimal, and 0.07 × 450 g = 31.5 g. ✓

Watch out. 10% means divide by 10 once. Do not divide by 10 and then by 10 again.

Worked example 5.2.3
Work out 17.5% of £360. Do not use a calculator.

Method. To find a percentage of an amount without a calculator, find 10% (divide by 10) or 1% (divide by 100) first. Then build up to the percentage you need.

  1. Split the percentage into easy parts. We can find 10% and 1% very quickly, so we build the percentage out of them.
    17.5% = 10% + 5% + 2.5%
  2. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
    10% of £360 = £360 ÷ 10 = £36
  3. Work out each part. Use the 10% or 1% you found.
    10% = £36
    5% = half of 10% = £36 ÷ 2 = £18
    2.5% = a quarter of 10% = £36 ÷ 4 = £9
  4. Add the parts together. The parts together make up the whole percentage.
    £36 + £18 + £9 = £63

Check. Another way: 17.5% = 0.175 as a decimal, and 0.175 × £360 = £63. ✓

Watch out. 10% means divide by 10 once. Do not divide by 10 and then by 10 again.

Worked example 5.2.4
Use a calculator to work out 23% of 85 kg.

Method. Write the percentage as a decimal, then multiply it by the amount.

  1. Change the percentage into a decimal. Per cent means “out of 100”, so divide by 100.
    23% = 23 ÷ 100 = 0.23
  2. Write the multiplication. To find a percentage of an amount, multiply the amount by the percentage written as a decimal (“of” means multiply).
    23% of 85 kg = 0.23 × 85 kg
  3. Work it out on the calculator. Type the multiplication in carefully.
    0.23 × 85 kg = 19.55 kg

Check. Estimate: 10% of 85 is 8.5, so 23% is 2.3 lots of that: about 19.55. ✓

Watch out. Divide the percentage by 100 first. 8% is 0.08, not 0.8.

Worked example 5.2.5
A savings account pays 6.5% interest on £2,340 in a year. Work out the interest.

Method. Write the percentage as a decimal, then multiply it by the amount.

  1. Change the percentage into a decimal. Per cent means “out of 100”, so divide by 100.
    6.5% = 6.5 ÷ 100 = 0.065
  2. Write the multiplication. To find a percentage of an amount, multiply the amount by the percentage written as a decimal (“of” means multiply).
    6.5% of £2,340 = 0.065 × £2,340
  3. Work it out on the calculator. Type the multiplication in carefully.
    0.065 × £2,340 = £152.10

Check. Estimate: 10% of 2,340 is 234, so 6.5% is 0.65 lots of that: about 152.10. ✓

Watch out. Divide the percentage by 100 first. 8% is 0.08, not 0.8.

Watch out

10% is not “divide by 10 and then by 10 again”. You divide by 10 once. And finding the percentage is often only part of the job: read the question to see whether it wants the percentage itself, or what is left after taking it away.

Exercise 5B  Percentages of an amount (non-calculator)

Non-calculator
1
FWork out each percentage. Do not use a calculator.
(a)10% of 80
(b)20% of 150
(c)30% of 240
(d)50% of 64
(e)25% of 120
(f)5% of 200
2
FWork out each amount of money.
(a)15% of £360
(b)35% of £80
(c)75% of £48
(d)12% of £250
(e)2% of £450
(f)1% of £7,600
3
FWork out each measurement.
(a)45% of 600 g
(b)90% of 30 m
(c)60% of 250 ml
(d)8% of 400 kg
(e)70% of 3 km
(f)15% of 90 minutes
4
PSA coat costs £80. In a sale the price is reduced by 35%.
(a)Work out the reduction.
(b)Work out the sale price.
5
PSA school has 800 students. 45% of them are girls.
(a)How many girls are there?
(b)How many boys are there?
6
PSSam earns £1,600 each month. He saves 15% of it.
(a)How much does Sam save each month?
(b)How much does Sam save in a year?
7
PSA bag has 240 sweets. 25% are red and 5% are green. How many sweets are neither red nor green?
8
PSWhich is bigger, 20% of 350 or 30% of 240? Show your working.
9
RShow that 20% of 60 is equal to 60% of 20.
10
RAli says, “To find 15% of 80 you divide 80 by 15.” Explain why he is wrong, and find the correct answer.
11
FWork out each amount. Do not use a calculator.
(a)2.5% of £1,200
(b)12.5% of 640
(c)7.5% of 920
(d)0.5% of 3,000
12
RWithout using a calculator, explain why 8% of 50 is equal to 50% of 8.

Exercise 5C  Percentages of an amount (calculator)

Calculator allowed
1
FUse a calculator to work out each amount.
(a)12% of 650
(b)35% of 840
(c)7% of 1,250
(d)18% of 2,400
(e)4.5% of 360
(f)62% of 85
2
FWork out each amount of money. Give your answers to the nearest penny.
(a)17.5% of £84.60
(b)23% of £149.99
(c)6.5% of £2,450
(d)3.2% of £19,500
3
PSA restaurant adds a 12.5% service charge to a bill of £86.40. Work out the service charge.
4
PSPriya earns £28,500 a year. She pays income tax of 20% on all the money she earns over £12,570. Work out how much tax she pays.
5
PSA town has 48,500 people. 14.6% of them are aged over 65. How many people is this?
6
PSAn exam is marked out of 180. The pass mark is 65% of the total. Work out the pass mark.
7
RA shop offers 12% cashback on purchases, but the cashback is never more than £25. Amira spends £250. How much cashback does she get?
8
FEstimate 48% of £603 by rounding. Then use a calculator to find the exact amount.
(a)Estimate the amount.
(b)Work out the exact amount.
9
RBen says, “150% of 40 is less than 40, because percentages are parts of 100.” Is he right? Show how you know.
10
PSA survey asks 1,250 people which drink they prefer. 38.4% choose tea. How many people choose tea?
11
PSA charity raises £18,400. It spends 3.5% on costs and the rest goes to the cause. How much goes to the cause?
12
FWork out each amount.
(a)0.35% of 82,000
(b)0.8% of £6,250
13
RShow that 15% of 260 is equal to 26% of 150.
5.3

One quantity as a percentage of another

Edexcel 1MA1: R9, N12
Grade 2-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: One quantity as a percentage of anotherComing soon

Suppose 18 of the 25 students in a class walk to school. What percentage walk? You are being asked to write “18 out of 25” as a number out of 100. This is the reverse of the last section: instead of finding the part, you already know the part and the whole, and you want the percentage.

The method has two steps. Step 1: write the part over the whole as a fraction, partwhole. Step 2: change that fraction to a percentage. If the denominator goes into 100, make an equivalent fraction with denominator 100 (1825 = 72100 = 72%). If it does not, divide the top by the bottom and multiply by 100 (4260: 42 ÷ 60 = 0.7, and 0.7 × 100 = 70%). Both give the same kind of answer, and on a calculator the quick way is always part ÷ whole × 100.

Check the units first

The part and the whole must be measured in the same units before you divide. To write 45 minutes as a percentage of 2 hours, first change 2 hours to 120 minutes. Writing 452 would be a mistake because minutes and hours are different sizes.

Comparing using percentages

Which is the better test result: 18 out of 25, or 31 out of 40? The totals are different, so you cannot just compare 18 and 31. Percentages are all out of 100, so they put the results on the same scale. This is the main reason percentages are useful: they let you make a fair comparison.

Rent £400Food £250Travel £150Saving £120Other £80
A pie chart shows each part as a share of a whole. Dan spends £1,000 a month: rent is 400 out of 1,000, which is 40%.

The percentage can also be more than 100%. If you compare a new amount with an older, smaller one, the part is bigger than the whole you compare with. For example, 612 members this year compared with 480 last year: 612 ÷ 480 × 100 = 127.5%. That means there are 127.5% as many members as last year, so the club has grown.

Key ideas

  • Percentage = part ÷ whole × 100.
  • The part and the whole must be in the same units: change hours to minutes, pounds to pence, and so on.
  • To compare two results with different totals, write both as percentages and compare those.
  • A percentage can be bigger than 100% when the part is bigger than the amount it is compared with.
  • Check: the percentage of the whole should give you back the part.
Worked example 5.3.1
Write 42 as a percentage of 60.

Method. To write one quantity as a percentage of another: write it as a fraction (part over whole), then change the fraction into a percentage.

  1. Write the part over the whole as a fraction. A percentage compares a part with the whole. Both numbers must be in the same units.
    4260
  2. Divide the top by the bottom. This changes the fraction into a decimal.
    42 ÷ 60 = 0.7
  3. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.7 × 100 = 70%

Check. Check: 70% of 60 = 0.7 × 60 = 42, which is the part we started with. ✓

Watch out. Divide by the whole (the amount you are comparing with), not by the part.

Worked example 5.3.2
Write 45 minutes as a percentage of 2 hours.

Method. To write one quantity as a percentage of another: write it as a fraction (part over whole), then change the fraction into a percentage.

  1. Change both amounts to the same units. The part is in minutes and the whole is in hours. 1 hour = 60 minutes, so 2 hours = 120 minutes.
    2 hours = 2 × 60 = 120 minutes
  2. Write the part over the whole as a fraction. A percentage compares a part with the whole. Both numbers must be in the same units.
    45120
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    45 ÷ 120 = 0.375
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.375 × 100 = 37.5%

Check. Check: 37.5% of 120 = 0.375 × 120 = 45, which is the part we started with. ✓

Watch out. Divide by the whole (the amount you are comparing with), not by the part.

Worked example 5.3.3
Zara scored 18 out of 25 in a test. Yusuf scored 31 out of 40 in a different test. Who did better?

Method. Percentages are all out of 100, so they let us compare amounts that have different totals.

  1. Write Zara as a percentage. To compare fairly we need the same total, so we use percentages (out of 100).
    1825 = 72100 = 72%
  2. Write Yusuf as a percentage. Do exactly the same for the second one.
    31 ÷ 40 = 0.775
    0.775 × 100 = 77.5%
  3. Compare the percentages. The bigger percentage is the better result.
    72% and 77.5%, so Yusuf is higher.

Check. Cross-check with decimals: 0.72 and 0.775; Yusuf is bigger. ✓

Watch out. Compare the percentages, not the raw scores. A bigger score can be a smaller percentage.

Worked example 5.3.4
A club had 480 members last year and has 612 members this year. Write this year’s number of members as a percentage of last year’s.

Method. To write one quantity as a percentage of another: write it as a fraction (part over whole), then change the fraction into a percentage.

  1. Write the part over the whole as a fraction. A percentage compares a part with the whole. Both numbers must be in the same units.
    612480
  2. Divide the top by the bottom. This changes the fraction into a decimal.
    612 ÷ 480 = 1.275
  3. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    1.275 × 100 = 127.5%

Check. Check: 127.5% of 480 = 1.275 × 480 = 612, which is the part we started with. ✓

Watch out. Divide by the whole (the amount you are comparing with), not by the part.

Worked example 5.3.5
In a class of 28 students, 11 are left-handed. What percentage of the class is left-handed? Give your answer to 1 decimal place.

Method. To write one quantity as a percentage of another: write it as a fraction (part over whole), then change the fraction into a percentage.

  1. Write the part over the whole as a fraction. A percentage compares a part with the whole. Both numbers must be in the same units.
    1128
  2. Divide the top by the bottom. This changes the fraction into a decimal.
    11 ÷ 28 = 0.392857…
  3. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.392857… × 100 = 39.285714…%
  4. Round the answer. Round to 1 decimal place as the question asks.
    39.285714…% = 39.3% (to 1 d.p.)

Check. Check: 39.3% of 28 is about 11.00, which is very close to 11. ✓

Watch out. Divide by the whole (the amount you are comparing with), not by the part.

Worked example 5.3.6
A is 40% of B. B is 25% of C. Write A as a percentage of C.

Method. Turn each percentage into a decimal multiplier, then multiply the two multipliers.

  1. Write A in terms of B. 40% as a decimal is 0.4. “A is 40% of B” means A = 0.4 × B.
    A = 0.4 × B
  2. Write B in terms of C. 25% as a decimal is 0.25.
    B = 0.25 × C
  3. Put the second line into the first. Replace B with 0.25 × C.
    A = 0.4 × 0.25 × C = 0.1 × C
  4. Change the multiplier into a percentage. Multiply by 100.
    0.1 = 10%

Check. Try C = 200: B is 25% of 200 = 50, and A is 40% of 50 = 20. Then 20 is 10% of 200. ✓

Watch out. Do not add 40% and 25%. The second percentage is of a different amount.

Watch out

Divide by the whole, the amount you are comparing with — and keep the units the same. “45 minutes out of 2 hours” is 45 out of 120, not 45 out of 2. Also, 18 out of 25 and 25 out of 18 are different: the order matters.

Exercise 5D  One quantity as a percentage of another (non-calculator)

Non-calculator
1
FWrite the first number as a percentage of the second. Do not use a calculator.
(a)9 out of 20
(b)12 out of 50
(c)18 out of 25
(d)3 out of 4
(e)30 out of 200
(f)7 out of 10
2
FWrite the first number as a percentage of the second. You will need to divide.
(a)21 out of 60
(b)45 out of 120
(c)9 out of 40
(d)66 out of 120
3
FChange to the same units first, then write the first amount as a percentage of the second.
(a)30 cm out of 2 m
(b)45 minutes out of 3 hours
(c)60p out of £2.40
(d)250 g out of 2 kg
(e)600 m out of 2.4 km
(f)15 seconds out of 1 minute
4
PSA bag has 40 counters. 14 of them are red. What percentage of the counters are red?
5
PSThe pie chart shows how Dan spends his £1,000 monthly pay. Work out the percentage he spends on each.
Rent £400Food £250Travel £150Saving £120Other £80
(a)Rent (£400)
(b)Food (£250)
(c)Travel (£150)
(d)Saving (£120)
(e)Other (£80)
6
PSA netball team played 32 matches and won 24 of them. What percentage did they win?
7
PSRhys scored 36 out of 50 in a maths test and 52 out of 80 in a science test. In which test did he do better? Show your working.
8
RTom says, “8 out of 20 is 8%, because the top number is 8.” Explain his mistake and write 8 out of 20 as a percentage.
9
RAt Shop A, 18 out of 24 customers said they were satisfied. At Shop B, 27 out of 36 said they were satisfied. Is one shop better than the other? Give a reason.
10
PSThe price of a phone rose from £400 to £500. Write the new price as a percentage of the old price.
11
RA is 40% of B. B is 25% of C. Write A as a percentage of C.
12
R(a) Write 45 as a percentage of 36.  (b) Write 36 as a percentage of 45.  (c) Explain why your answers are different.

Exercise 5E  One quantity as a percentage of another (calculator)

Calculator allowed
1
FWrite the first number as a percentage of the second. Give your answers to 1 decimal place where necessary.
(a)17 out of 45
(b)23 out of 36
(c)5 out of 7
(d)129 out of 150
(e)8 out of 13
(f)345 out of 420
2
PSIn a survey of 240 people, 91 said they prefer cats to dogs. What percentage prefer cats? Give your answer to 1 decimal place.
3
PSA bag holds 750 g of flour. A baker uses 285 g. What percentage of the flour does the baker use?
4
PSThe population of a town rises from 48,500 to 52,310. Write the new population as a percentage of the old population. Give your answer to 1 decimal place.
5
PSA runner completes 7.5 km of a 12 km charity route. What percentage of the route has the runner completed?
6
PSAmy scored 68 out of 85 in a quiz. Ben scored 74 out of 95 in a harder quiz. Who got the higher percentage? Show your working.
7
PSElla scored 14 goals from 22 shots. Fatima scored 19 goals from 30 shots. Who has the better shooting record? Show your working.
8
PSLena scored 134 marks out of 200. The pass mark is 65%. Did Lena pass? Explain.
9
PSA company has an income of £1.2 million. Its profit is £84,000. Write the profit as a percentage of the income.
10
PSThe table shows how many students passed a test in three classes. Put the classes in order, highest pass rate first.
ClassPassedStudents
A1320
B2032
C1725
11
FSales last year were £28,000. Sales this year are £35,000. Write this year’s sales as a percentage of last year’s sales.
12
FWrite each as a percentage.
(a)0.35 out of 1.4
(b)7.2 out of 4.5
13
RChen says, “A percentage of something can never be more than 100%.” Is he right? Use an example to explain.
5.4

Percentage increase and decrease

Edexcel 1MA1: R9
Grade 3-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Percentage increase and decreaseComing soon

Prices go up, prices come down. A phone is “15% off”, rent “rises by 4%”, a population “grows by 2%”. In each case the original amount is changed by a percentage of itself. There are two good ways to find the new amount.

Method 1: find the change, then add or subtract

Work out the percentage of the original amount (as in Section 5.2). For an increase, add it to the original. For a decrease, subtract it. This is easy without a calculator because it uses 10% and 1% steps.

Method 2: use a multiplier

The original amount is 100%. An increase of 15% makes the new amount 100% + 15% = 115% of the original, and 115% as a decimal is 1.15. So increasing by 15% is the same as multiplying by 1.15. A decrease of 15% leaves 100% − 15% = 85%, and 85% as a decimal is 0.85. The number you multiply by is called the multiplier. It takes you from the old amount to the new amount in one step, which is quick on a calculator.

Original£80New amount£92× 1.15a 15% increaseone multiplication does it all
Increase £80 by 15%: multiply by 1.15 to get £92.
ChangeNew amount as a percentageMultiplier
Increase by 10%110%1.1
Increase by 5%105%1.05
Increase by 35%135%1.35
Decrease by 10%90%0.9
Decrease by 5%95%0.95
Decrease by 35%65%0.65

Increases of 100% or more

An increase can be bigger than 100%. An increase of 100% means you add on the whole amount again, so the amount doubles (multiplier 2). An increase of 150% gives 100% + 150% = 250% of the original, a multiplier of 2.5. A decrease can never be more than 100%, because you cannot take away more than you have.

the original: 100%+150%0%100%250%250% of the original is more than the original
An increase of 150% makes the new amount 250% of the original (a multiplier of 2.5).

From a multiplier back to a percentage

Look at the multiplier. A multiplier above 1 means an increase: 1.08 is 108%, which is an increase of 8%. A multiplier below 1 means a decrease: 0.6 is 60%, which is a decrease of 40%. And 2.5 is 250%, an increase of 150%.

Key ideas

  • Original amount = 100%.
  • Increase of p%: new amount is (100 + p)% of the original. Multiplier = 1 + p/100.  e.g. +15% → 1.15.
  • Decrease of p%: new amount is (100 − p)% of the original. Multiplier = 1 − p/100.  e.g. −15% → 0.85.
  • New amount = original × multiplier.
  • An increase of 100% doubles the amount; an increase over 100% has a multiplier over 2.
  • Without a calculator: find the percentage, then add it (increase) or subtract it (decrease).
Worked example 5.4.1
Increase £80 by 15%.

Key words. Per cent means “out of 100”. A percentage is a fraction whose denominator is 100, written with the % sign. A multiplier is the number you multiply by to get the new amount in one step.

Method. To increase an amount by a percentage, find the multiplier and multiply. The multiplier for a 15% increase is 1.15.

  1. Find the percentage the new amount will be. An increase of 15% means we add 15% of the original. The original is 100%, so the new amount is 100% + 15%.
    100% + 15% = 115% of the original
  2. Write that percentage as a decimal: this is the multiplier. Divide by 100.
    115% = 115 ÷ 100 = 1.15
  3. Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.
    £80 × 1.15 = £92

Check. Other way: 15% of £80 = 12; £80 + 12 = 92. ✓

Watch out. Do not multiply by 0.15 only: that finds the 15% but not the new amount. The multiplier includes the original 100%.

Worked example 5.4.2
In a sale all prices are reduced by 12%. A bike costs £250. Work out the sale price. Do not use a calculator.

Method. Find 12% of the amount, then subtract it from the original.

  1. Split the percentage into easy parts. We can find 10% and 1% very quickly, so we build the percentage out of them.
    12% = 10% + 2%
  2. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
    10% of £250 = £250 ÷ 10 = £25
  3. Find 1%. 1% is one hundredth, so divide the amount by 100 (each digit moves two places to the right).
    1% of £250 = £250 ÷ 100 = £2.50
  4. Work out each part. Use the 10% or 1% you found.
    10% = £25
    2% = 2 × 1% = 2 × £2.50 = £5
  5. Add the parts together. The parts together make up the whole percentage.
    £25 + £5 = £30
  6. Subtract it from the original amount. We want the amount after the decrease.
    £250 − £30 = £220

Check. Multiplier check: 88% = 0.88, and 0.88 × £250 = £220. ✓

Watch out. Remember the last step. Finding the percentage is not the final answer: add it on or take it off.

Worked example 5.4.3
Decrease 64 kg by 35%.

Method. To decrease an amount by a percentage, find the multiplier and multiply. The multiplier for a 35% decrease is 0.65.

  1. Find the percentage the new amount will be. A decrease of 35% means we subtract 35% of the original. The original is 100%, so the new amount is 100% − 35%.
    100% − 35% = 65% of the original
  2. Write that percentage as a decimal: this is the multiplier. Divide by 100.
    65% = 65 ÷ 100 = 0.65
  3. Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.
    64 kg × 0.65 = 41.6 kg

Check. Other way: 35% of 64 kg = 22.4; 64 kg − 22.4 = 41.6. ✓

Watch out. Do not multiply by 0.35 only: that finds the 35% but not the new amount. The multiplier is what is left after the decrease.

Worked example 5.4.4
Write down the multiplier for a decrease of 7%.

Method. A decrease of 7% means the new amount is 93% of the original. Write 93% as a decimal.

  1. Work out the new percentage. The original is 100%. For a decrease we subtract 7%.
    100% − 7% = 93%
  2. Divide by 100. To change a percentage to a decimal.
    93% = 93 ÷ 100 = 0.93
  3. Check the size of the multiplier. A decrease should give a multiplier below 1.
    0.93 is below 1, so it makes the amount smaller.

Check. Multiplying 100 by 0.93 gives 93, which is a 7% decrease on 100. ✓

Watch out. A decrease multiplier is less than 1; an increase multiplier is more than 1.

Worked example 5.4.5
A TV costs £340 before VAT. VAT of 20% is added. Work out the total price.

Method. To increase an amount by a percentage, find the multiplier and multiply. The multiplier for a 20% increase is 1.2.

  1. Find the percentage the new amount will be. An increase of 20% means we add 20% of the original. The original is 100%, so the new amount is 100% + 20%.
    100% + 20% = 120% of the original
  2. Write that percentage as a decimal: this is the multiplier. Divide by 100.
    120% = 120 ÷ 100 = 1.2
  3. Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.
    £340 × 1.2 = £408

Check. Other way: 20% of £340 = 68; £340 + 68 = 408. ✓

Watch out. Do not multiply by 0.2 only: that finds the 20% but not the new amount. The multiplier includes the original 100%.

Worked example 5.4.6
Increase 40 by 150%.

Method. To increase an amount by a percentage, find the multiplier and multiply. The multiplier for a 150% increase is 2.5.

  1. Find the percentage the new amount will be. An increase of 150% means we add 150% of the original. The original is 100%, so the new amount is 100% + 150%.
    100% + 150% = 250% of the original
  2. Write that percentage as a decimal: this is the multiplier. Divide by 100.
    250% = 250 ÷ 100 = 2.5
  3. Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.
    40 × 2.5 = 100

Check. Other way: 150% of 40 = 60; 40 + 60 = 100. ✓

Watch out. Do not multiply by 1.5 only: that finds the 150% but not the new amount. The multiplier includes the original 100%.

Worked example 5.4.7
A price is multiplied by 2.5. What is the percentage increase?

Method. Compare the multiplier with 1 (which is 100%). More than 1 is an increase; less than 1 is a decrease.

  1. Write the multiplier as a percentage. Multiply by 100.
    2.5 × 100 = 250%
  2. Compare with 100%. The original is 100%. 250% is more than 100%.
    250% − 100% = 150%
  3. Say it in words.
    a 150% increase

Check. 100 × 2.5 = 250, which is 150 more than 100. ✓

Worked example 5.4.8
Decrease £86.50 by 17.5%. Give your answer to the nearest penny.

Method. To decrease an amount by a percentage, find the multiplier and multiply. The multiplier for a 17.5% decrease is 0.825.

  1. Find the percentage the new amount will be. A decrease of 17.5% means we subtract 17.5% of the original. The original is 100%, so the new amount is 100% − 17.5%.
    100% − 17.5% = 82.5% of the original
  2. Write that percentage as a decimal: this is the multiplier. Divide by 100.
    82.5% = 82.5 ÷ 100 = 0.825
  3. Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.
    £86.50 × 0.825 = £71.3625
  4. Round the answer. Round to 2 decimal places.
    £71.3625 = £71.36

Check. Other way: 17.5% of £86.50 = 15.14; £86.50 − 15.14 = 71.36. ✓

Watch out. Do not multiply by 0.175 only: that finds the 17.5% but not the new amount. The multiplier is what is left after the decrease.

Watch out

Finding the percentage is not the end. To increase by 15% you must add the 15% to the original (or multiply by 1.15, not by 0.15). And a decrease multiplier is less than 1: for a 15% decrease use 0.85, not 1.15 or 0.15.

Exercise 5F  Increase and decrease (non-calculator)

Non-calculator
1
FIncrease each amount by the percentage shown. Do not use a calculator.
(a)80 by 10%
(b)£150 by 20%
(c)240 by 5%
(d)£60 by 25%
(e)400 by 30%
(f)£35 by 40%
2
FDecrease each amount by the percentage shown. Do not use a calculator.
(a)£80 by 10%
(b)120 by 25%
(c)£250 by 20%
(d)64 by 50%
(e)£90 by 30%
(f)300 by 15%
3
FWrite down the multiplier for each change.
(a)an increase of 10%
(b)an increase of 4%
(c)a decrease of 20%
(d)a decrease of 3%
(e)an increase of 35%
(f)a decrease of 45%
4
FEach number is a multiplier. Write down the percentage increase or decrease it stands for.
(a)1.2
(b)1.07
(c)0.9
(d)0.65
(e)1.5
(f)2
5
PSA jacket costs £120. In a sale it is reduced by 30%. Work out the sale price.
6
PSA train fare of £48 goes up by 5%. Work out the new fare.
7
PSA plumber charges £60 plus VAT at 20%. Work out the total cost.
8
PSIn 2020 a village had 2,400 people. By 2025 the population had increased by 15%. How many people lived in the village in 2025?
9
RMark says, “To increase a price by 8% you multiply it by 0.08.” Explain why he is wrong. What should he multiply by?
10
RWork out the percentage change in each case.
(a)A price is doubled. What is the percentage increase?
(b)A price is multiplied by 3. What is the percentage increase?
(c)A price is halved. What is the percentage decrease?
11
FWork out each new amount. Do not use a calculator.
(a)Increase £800 by 12.5%
(b)Decrease 560 by 7.5%
(c)Increase £1,200 by 2.5%
(d)Decrease 240 by 17.5%
12
RShow that increasing 60 by 50% gives the same answer as decreasing 150 by 40%.

Exercise 5G  Increase and decrease (calculator)

Calculator allowed
1
FUse a multiplier to work out each new amount.
(a)Increase £85 by 8%
(b)Decrease £250 by 12%
(c)Increase 640 by 3.5%
(d)Decrease 3,400 by 17%
(e)Increase 1,850 by 22%
(f)Decrease 280 by 4.5%
2
FWork out each new amount. Give your answers to the nearest penny.
(a)Increase £249.99 by 15%
(b)Decrease £64.50 by 12.5%
(c)Increase £1,875 by 2.9%
3
FIncrease each amount by the percentage shown. (Remember the multiplier can be more than 2.)
(a)40 by 150%
(b)£80 by 200%
(c)25 by 120%
4
PSOla earns £2,150 each month. Her pay is increased by 3%. Work out her new monthly pay.
5
PSA laptop costs £650 before VAT. VAT is 20%. Work out the total cost.
6
PSChloe earns £11.50 an hour. Her pay is increased by 4%. Work out her new hourly pay.
7
PSA car is worth £14,500 when new. In its first year its value falls by 18%. What is it worth after one year?
8
PSThe population of a city is 125,000. It increases by 2.4%. Work out the new population.
9
PSA landlord increases the monthly rent of £720 by 6.5%. Work out the new rent.
10
RA £50 jumper is reduced by 20%. Liam says, “The sale price is £10, because 20% of 50 is 10.” Explain his mistake and work out the correct sale price.
11
FWrite down the multiplier for each change.
(a)an increase of 0.5%
(b)a decrease of 0.5%
(c)an increase of 12.5%
(d)a decrease of 7.25%
12
RA shop raises all its prices by 10%. Mina says, “So a pair of shoes that cost £60 now costs £66.” Is she right? Show your working.
5.5

Percentage change, profit and loss

Edexcel 1MA1: R9
Grade 4-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Percentage change, profit and lossComing soon

“The price of a phone went from £400 to £500.” That is an increase of £100, but is it a big increase? £100 matters more on a £400 phone than on a £40,000 car. To judge a change fairly we write it as a percentage of the original amount. This is the percentage change.

Old£400New£400+£100The change is compared with the ORIGINAL (old) amount.
A price rises from £400 to £500. The change is £100, which is 25% of the original £400.

The rule is easy to remember:

percentage change = changeoriginal × 100%

where the change is the bigger number take away the smaller number. If the new amount is bigger it is a percentage increase; if it is smaller it is a percentage decrease.

Why divide by the original? Because the percentage tells us what share of the starting amount has been added or lost. The starting amount is the 100%. This is the most common mistake: dividing by the new amount.

Profit and loss

A shop buys goods at a cost price and sells them at a selling price. If the selling price is higher there is a profit; if it is lower there is a loss. The profit or loss is a change, so the same rule works: percentage profit = profitcost price × 100%. The cost price is the original amount.

Percentages bigger than 100%

If a quantity more than doubles, the percentage increase is bigger than 100%. A plant that grows from 12 cm to 30 cm has grown by 18 cm, and 18 ÷ 12 × 100 = 150%. A decrease, however, can never be more than 100%.

Higher tier. Percentage points. If an interest rate goes from 4% to 6%, it has gone up by 2 percentage points (6 − 4 = 2). But as a percentage change it has gone up by 24 × 100% = 50%. Always check whether a question asks for percentage points or a percentage change. Also notice that a rise and the fall that brings you back are different percentages: a price that rises from £50 to £60 (up 20%) must fall by only 16.7% to return to £50, because the fall is measured against the bigger number £60.

Key ideas

  • Percentage change = change ÷ original × 100%.
  • Change = bigger − smaller. Say increase or decrease.
  • Always divide by the original (the starting amount).
  • Profit = selling price − cost price. Percentage profit = profit ÷ cost price × 100%. Loss works the same way.
  • An increase can be over 100%. A decrease cannot.
Worked example 5.5.1
The price of a ticket rises from £40 to £50. Work out the percentage increase.

Key words. Per cent means “out of 100”. A percentage is a fraction whose denominator is 100, written with the % sign. A multiplier is the number you multiply by to get the new amount in one step.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £50 − £40 = £10 (increase)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    1040
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    10 ÷ 40 = 0.25
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.25 × 100 = 25%
  5. Say whether it is an increase or a decrease.
    25% increase

Check. Check: 25% of £40 = £10, and £40 + £10 = £50. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.2
The number of students at a school falls from 250 to 215. Work out the percentage decrease.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    250 − 215 = 35 (decrease)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    35250
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    35 ÷ 250 = 0.14
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.14 × 100 = 14%
  5. Say whether it is an increase or a decrease.
    14% decrease

Check. Check: 14% of 250 = 35, and 250 − 35 = 215. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.3
A shop buys a chair for £60 and sells it for £84. Work out the percentage profit.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £84 − £60 = £24 (profit)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    2460
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    24 ÷ 60 = 0.4
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.4 × 100 = 40%
  5. Say whether it is an increase or a decrease.
    40% profit

Check. Check: 40% of £60 = £24, and £60 + £24 = £84. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.4
A dealer buys a motorbike for £850 and sells it for £680. Work out the percentage loss.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £850 − £680 = £170 (loss)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    170850
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    170 ÷ 850 = 0.2
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.2 × 100 = 20%
  5. Say whether it is an increase or a decrease.
    20% loss

Check. Check: 20% of £850 = £170, and £850 − £170 = £680. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.5
A plant is 12 cm tall. A month later it is 30 cm tall. Work out the percentage increase in its height.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    30 cm − 12 cm = 18 cm (increase)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    1812
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    18 ÷ 12 = 1.5
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    1.5 × 100 = 150%
  5. Say whether it is an increase or a decrease.
    150% increase

Check. Check: 150% of 12 cm = 18 cm, and 12 cm + 18 cm = 30 cm. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.6
A house was worth £245,000. It is now worth £261,500. Work out the percentage increase. Give your answer to 1 decimal place.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £261,500 − £245,000 = £16,500 (increase)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    16,500245,000
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    16,500 ÷ 245,000 = 0.067347…
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.067347… × 100 = 6.734694…%
  5. Round the answer. Round to 1 decimal place as the question asks.
    6.734694…% = 6.7% (to 1 d.p.)
  6. Say whether it is an increase or a decrease.
    6.7% increase

Check. Check: 6.7% of £245,000 is about 16,415.00, which is very close to the actual change of £16,500. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.7
A ticket price falls from £80 to £60. Later it rises from £60 back to £80. Work out the percentage change each time. Give the second answer to 1 decimal place.

Method. Use percentage change = change ÷ original × 100% twice. The original is different each time.

  1. Fall from £80 to £60: the change. Bigger take away smaller.
    £80 − £60 = £20
  2. Divide by the original, £80, and multiply by 100. The original is where the first change started.
    £20 ÷ £80 × 100 = 25%
  3. Rise from £60 to £80: the change. The change is again £20.
    £80 − £60 = £20
  4. Divide by the new original, £60, and multiply by 100. This time the original is £60.
    £20 ÷ £60 × 100 = 33.333…% = 33.3%
  5. Say what happened. The same £20 is a different percentage because the original amount is different.
    A 25% decrease, then a 33.3% increase.

Check. Check: 25% of 80 = 20 and 80 − 20 = 60. 33.3% of 60 = 20 and 60 + 20 = 80. ✓

Watch out. Divide by the original each time. The original changes when the direction changes.

Watch out

Divide by the original amount, not the new amount. A price that goes from £40 to £50 rises by 25% (10 ÷ 40), not 20% (10 ÷ 50).

Exercise 5H  Percentage change, profit and loss (non-calculator)

Non-calculator
1
FWork out the percentage change in each case. Say whether it is an increase or a decrease.
(a)£40 to £50
(b)80 to 60
(c)£200 to £250
(d)50 to 65
(e)400 to 300
(f)£15 to £18
2
FWork out the percentage profit or loss in each case.
(a)Bought for £60, sold for £84
(b)Bought for £500, sold for £450
(c)Bought for £25, sold for £30
(d)Bought for £80, sold for £60
3
PSA gym has 400 members in January and 460 members in June. Work out the percentage increase in the number of members.
4
PSThe price of a bus pass goes up from £80 to £88. Work out the percentage increase in the price.
5
PSA shop buys a pair of trainers for £50 and sells them for £65. Work out the percentage profit.
6
PSThe population of a village falls from 800 to 680. Work out the percentage decrease.
7
PSA phone costs £500. After a year its price falls to £375. Work out the percentage decrease in the price.
8
PSA market trader buys scarves for £8 each and sells them for £14 each. Work out the trader’s percentage profit.
9
PSA sunflower is 12 cm tall in week 1 and 30 cm tall in week 2. Work out the percentage increase in height.
10
PSShop A raises the price of a kettle from £120 to £150. Shop B raises the price of a toaster from £80 to £104. Which shop has the bigger percentage increase?
11
RA price rises from £50 to £60, which is a 20% increase. Lee says, “So it needs to fall by 20% to get back to £50.” Is he right? Show how you know.
12
R(a) The unemployment rate fell from 8% to 6%. By how many percentage points did it fall?  (b) Work out the percentage decrease in the rate.
(a)Change in percentage points
(b)Percentage decrease
13
RA shopkeeper buys a jacket for £200 and marks it up by 40% to get the price on the label. In a sale the jacket is sold for 25% off the label price. Does the shopkeeper make a profit or a loss on the jacket? Work out the percentage.

Exercise 5I  Percentage change (calculator)

Calculator allowed
1
FWork out the percentage change in each case. Give your answers correct to 1 decimal place.
(a)36 to 41
(b)8.5 to 7.2
(c)1,250 to 980
(d)64 to 100
(e)15.5 to 21.7
2
PSA shop buys a lamp for £18.50 and sells it for £27.95. Work out the percentage profit. Give your answer to 1 decimal place.
3
PSNaz buys a car for £12,400 and sells it two years later for £9,650. Work out his percentage loss. Give your answer to 1 decimal place.
4
PSThe rainfall in a town was 84 mm in March and 99 mm in April. Work out the percentage increase. Give your answer to 1 decimal place.
5
PSThe price of a share falls from £4.20 to £3.57. Work out the percentage decrease.
6
PSA town’s population was 24,800 in 2015, 27,280 in 2020 and 29,100 in 2025. In which five-year period was the percentage increase bigger? Show your working.
7
FWrite down the percentage change when a number goes from 80 to 45. Give your answer to the nearest whole number.
8
RSales were 200 in 2023, 300 in 2024 and 450 in 2025. Ava says, “Sales rose by 50% each year, so over the two years they rose by 100%.” Is she right? Show how you know.
5.6

Reverse percentages: finding the original amount

Edexcel 1MA1: R9
Grade 4-7

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Reverse percentages: finding the original amountComing soon

In a sale a coat is reduced by 30% and now costs £63. What did it cost before the sale? This is a reverse percentage (or original value) problem. The amount you are given is the new amount, after the change, and you must work backwards to find the original.

The tempting mistake is to work out 30% of £63 and add it on. That is wrong, because the 30% was taken off the original price, which was bigger than £63, not off £63. Instead use the idea that the original is always 100%.

Bar model (not to scale of the numbers, to scale of the percentages)Original price = 100%New = 70% = £63
A 30% reduction leaves 70% of the original, so £63 is 70% of the original price.

After a 30% reduction the coat costs 100% − 30% = 70% of the original, and we know that 70% is £63.

Method 1: find 1%, then 100%

If 70% is £63, then 1% is £63 ÷ 70 = £0.90, and 100% is 100 × £0.90 = £90. This is the same “unitary” idea you use for ratio: go to one, then scale up.

Method 2: divide by the multiplier

The multiplier for a 30% decrease is 0.7. We know original × 0.7 = £63. To undo a multiplication we divide: original = 63 ÷ 0.7 = £90. For an increase of, say, 20%, the multiplier is 1.2 and the original is the new amount ÷ 1.2.

It is just the same when a percentage of a whole is given. If 15% of the pupils in a school is 21 pupils, then 1% is 21 ÷ 15 = 1.4 pupils, and the whole school (100%) is 140 pupils.

Higher tier. You can also set this up as an equation. Let the original be x. After an increase of 25% the new amount is 1.25x. If the new amount is 75 then 1.25x = 75, so x = 75 ÷ 1.25 = 60. When the division does not come out exactly, keep all the digits on your calculator and round only at the end (to the nearest penny for money).

Key ideas

  • The original amount is 100%.
  • After an increase of p%, the new amount is (100 + p)% of the original. After a decrease of p%, it is (100 − p)%.
  • Method 1: new amount = (100 ± p)%, so divide by (100 ± p) to get 1%, then multiply by 100.
  • Method 2: original = new amount ÷ multiplier.
  • Check by going forwards: original × multiplier should give the new amount.
  • Never find p% of the new amount and add or subtract it.
Worked example 5.6.1
In a sale all prices are reduced by 30%. A coat costs £63 in the sale. Work out its price before the sale.

Key words. Per cent means “out of 100”. A percentage is a fraction whose denominator is 100, written with the % sign. A multiplier is the number you multiply by to get the new amount in one step.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the percentage that the new amount represents. The original amount is 100%. After a decrease of 30% the new amount is 100% − 30% of the original.
    100% − 30% = 70%
  2. Write down what we know. The new amount is a known percentage of the unknown original.
    70% = £63
  3. Find 1%. Divide by 70 to go from 70% down to 1%.
    £63 ÷ 70 = £0.90
  4. Find 100%. The original is 100%, so multiply 1% by 100.
    £0.90 × 100 = £90

Check. Check by going forwards: £90 × 0.7 = £63, which matches the new amount. ✓

Watch out. Do not work out 30% of the new amount and add it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.2
A bill including VAT at 20% is £144. Work out the bill before VAT was added.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the percentage that the new amount represents. The original amount is 100%. After an increase of 20% the new amount is 100% + 20% of the original.
    100% + 20% = 120%
  2. Write down what we know. The new amount is a known percentage of the unknown original.
    120% = £144
  3. Find 1%. Divide by 120 to go from 120% down to 1%.
    £144 ÷ 120 = £1.20
  4. Find 100%. The original is 100%, so multiply 1% by 100.
    £1.20 × 100 = £120

Check. Check by going forwards: £120 × 1.2 = £144, which matches the new amount. ✓

Watch out. Do not work out 20% of the new amount and subtract it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.3
Fatima’s salary was increased by 4%. It is now £27,248. Work out her salary before the increase.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the multiplier. The original is 100%. After an increase of 4% the new amount is 104% of the original, which is 1.04 as a decimal.
    104% = 104 ÷ 100 = 1.04
  2. Write what the multiplier does. original × multiplier = new amount.
    original × 1.04 = £27,248
  3. Go backwards: divide by the multiplier. To undo a multiplication, divide.
    £27,248 ÷ 1.04 = £26,200

Check. Check by going forwards: £26,200 × 1.04 = £27,248, which matches the new amount. ✓

Watch out. Do not work out 4% of the new amount and subtract it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.4
In a school, 21 pupils walk to school. This is 15% of the pupils in the school. How many pupils are in the school?

Method. If you know what a percentage of the whole is, find 1% first, then multiply by 100 to find the whole (100%).

  1. Write down what we know. Part of the whole has been given as a percentage.
    15% of the whole = 21 pupils
  2. Find 1%. Divide by 15. That takes us from 15% to 1%.
    21 pupils ÷ 15 = 1.4 pupils
  3. Find 100%. The whole is 100%, so multiply 1% by 100.
    1.4 pupils × 100 = 140 pupils

Check. 15% of 140 pupils = 0.15 × 140 pupils = 21 pupils. ✓

Watch out. The whole is 100%. Do not stop at 1%.

Worked example 5.6.5
After a 12% price increase, a laptop bag costs £250. Work out the original price. Give your answer to the nearest penny.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the multiplier. The original is 100%. After an increase of 12% the new amount is 112% of the original, which is 1.12 as a decimal.
    112% = 112 ÷ 100 = 1.12
  2. Write what the multiplier does. original × multiplier = new amount.
    original × 1.12 = £250
  3. Go backwards: divide by the multiplier. To undo a multiplication, divide.
    £250 ÷ 1.12 = £223.2143…
  4. Round the answer. Round to 2 decimal places.
    £223.2143… = £223.21

Check. Check by going forwards: £223.21 × 1.12 = £249.9952, which matches the new amount. ✓

Watch out. Do not work out 12% of the new amount and subtract it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.6
After an increase of 25% a number is 75. Use an equation to find the original number.

Method. Call the original number x, write the increase as a multiplier, and solve the equation.

  1. Write the multiplier. An increase of 25% means the new number is 125% of the original, which is 1.25.
    100% + 25% = 125% = 1.25
  2. Form an equation. Original × multiplier = new number.
    1.25x = 75
  3. Solve it: divide both sides by 1.25. Dividing both sides by the same number keeps the equation balanced and leaves x on its own.
    x = 75 ÷ 1.25 = 60
  4. Check. Increase 60 by 25%: 25% of 60 = 15 and 60 + 15 = 75.
    60 × 1.25 = 75

Check. 60 + 25% of 60 = 60 + 15 = 75. ✓

Watch out. Make sure the equation uses the multiplier, 1.25, not 0.25.

Watch out

Do not take 30% of the new price and add it on. £63 + 30% of £63 = £81.90, which is wrong. The 30% was 30% of the original. Going forwards: £90 reduced by 30% is £63. A good habit is to check by applying the percentage change to your answer.

Exercise 5J  Reverse percentages (non-calculator)

Non-calculator
1
FFind the original amount in each case.
(a)After a 10% decrease: £45
(b)After a 20% increase: £60
(c)After a 25% decrease: £30
(d)After a 50% increase: £45
(e)After a 30% decrease: £35
(f)After a 5% increase: 21 kg
2
FWork out the whole amount in each case.
(a)10% is 8
(b)25% is 15
(c)20% is 30
(d)5% is 6
(e)40% is 18
(f)15% is 21
3
FAfter a 25% increase, a number is 50. What was the number before the increase?
4
PSIn a sale, trainers are reduced by 40%. The sale price is £54. What was the price before the sale?
5
PSA garage bill is £180 including VAT at 20%. How much was the bill before VAT?
6
PSThe number of students in a school increased by 5% this year. There are now 1,260 students. How many were there last year?
7
PSAisha’s hourly pay was increased by 5%. She now earns £12.60 an hour. What did she earn an hour before the rise?
8
PSA club has some members. 36 of them are girls, which is 60% of the members.
(a)How many members does the club have?
(b)How many boys are in the club?
9
PSTwo shops sell the same bike. Shop A reduces its price by 20% and sells the bike for £360. Shop B reduces its price by 10% and sells it for £396. Which shop had the higher price before its reduction? Show your working.
10
PSThe price, p pounds, of a camera was increased by 12%. The new price is £84. Form an equation and solve it to find p.
11
FThe 45 students in the school choir are 25% of all the students in the school. How many students are in the school?
12
R20% of a number is 12. Work out the number, and then work out 35% of the number.
13
RThe price of a coat is reduced by 30% in a sale, and it now costs £63. Priya says, “The original price was £63 + 30% of £63 = £81.90.” Explain her mistake and find the original price.

Exercise 5K  Reverse percentages (calculator)

Calculator allowed
1
FFind the original amount in each case. Give money answers to the nearest penny.
(a)After a 12% increase: £250
(b)After a 20% decrease: £399
(c)After a 17.5% increase: £84.60
(d)After a 13% decrease: £13,050
(e)After a 3.5% increase: 12,420
(f)After a 2% decrease: £7.84
2
PSA car loses 18% of its value in its first year. After one year it is worth £10,578. What was it worth when it was new?
3
PSA house price rose by 6% in a year. It is now £236,000. What was its price a year ago, to the nearest pound?
4
PSA dress is reduced by 28% in a sale. The sale price is £37.80. Work out the original price.
5
PSJo’s yearly pay is increased by 3.2% to £28,200. Work out her yearly pay before the rise. Give your answer to the nearest penny.
6
FWork out the original amount in each case.
(a)A population falls by 4.5% to 71,625
(b)A mass falls by 12% to 5.28 kg
7
PSA shop takes 20% off the price of a coat. VAT of 20% is then added to the reduced price, and the customer pays £72. Work out the price of the coat before either change.
8
RA phone costs £96 after a 20% increase. Hamza says the price before the increase was £76.80, because 96 × 0.8 = 76.80. Is he right? Show how you know.
5.7

Simple interest

Edexcel 1MA1: R9
Grade 3-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Simple interestComing soon

When you save money in a bank, the bank pays you for letting it use your money. This payment is called interest. When you borrow money you pay interest to the lender. Interest is usually given as a percentage rate per year, written “3% p.a.” (per annum is Latin for “each year”). The money you first put in or borrow is called the principal.

With simple interest the bank pays the same amount every year. It is always worked out on the original principal, never on the interest that has been added. So simple interest is just “a percentage of an amount” (Section 5.2), repeated for each year.

050010001500010001105021100311504120051250£1,000 at 5% simple interest: total after each year
The total grows by the same £50 every year: simple interest makes a straight-line pattern.

To find simple interest:

  1. Work out the interest for one year: the rate % of the principal.
  2. Multiply by the number of years.
  3. If you want the total, add the interest to the principal.

Short for the first two steps: interest = principal × rate × years ÷ 100 (with the rate written as a number, not as a decimal). If the time is given in months, change it to years first: 6 months = 0.5 years, 18 months = 1.5 years.

Working backwards

If you know the interest you can find the rate or the time. If a savings account earned £300 in 3 years from £2,000, the interest for one year was £300 ÷ 3 = £100, and £100 is 100 ÷ 2,000 × 100 = 5% of the principal. So the rate is 5% p.a.

Key ideas

  • Principal = the money you start with. Rate = percentage per year (p.a.).
  • Simple interest is the same each year: rate% of the original principal.
  • Interest = P × R × T ÷ 100   (R = rate as a number, T = years).
  • Total = principal + interest.
  • Change months to years: 6 months = 0.5 years.
  • Rate = (interest for one year) ÷ principal × 100.
Worked example 5.7.1
Hana puts £1,500 into an account that pays 3% simple interest per year. How much is in the account after 4 years?

Key words. Interest is money paid for saving or borrowing. Simple interest is a fixed percentage of the original amount, paid each year.

Method. Simple interest is the same amount every year. Find one year’s interest, then multiply by the number of years.

  1. Write down what we know. The principal is the money at the start; the rate is the percentage per year; the time is in years.
    Principal = £1,500,  rate = 3% per year,  time = 4 years
  2. Find the interest for ONE year. Simple interest is the same every year: it is 3% of the original amount each time.
    3% of £1,500 = 0.03 × £1,500 = £45
  3. Multiply by the number of years. The same interest is added every year, so multiply by 4.
    £45 × 4 = £180
  4. Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.
    £1,500 + £180 = £1,680

Check. Formula check: I = P × R × T ÷ 100 = 1,500 × 3 × 4 ÷ 100 = 180. ✓

Watch out. Simple interest always uses the original amount, never the amount with interest added.

Worked example 5.7.2
Work out the simple interest on £800 at 2.5% per year for 3 years.

Key words. Interest is money paid for saving or borrowing. Simple interest is a fixed percentage of the original amount, paid each year.

Method. Simple interest is the same amount every year. Find one year’s interest, then multiply by the number of years.

  1. Write down what we know. The principal is the money at the start; the rate is the percentage per year; the time is in years.
    Principal = £800,  rate = 2.5% per year,  time = 3 years
  2. Find the interest for ONE year. Simple interest is the same every year: it is 2.5% of the original amount each time.
    2.5% of £800 = 0.025 × £800 = £20
  3. Multiply by the number of years. The same interest is added every year, so multiply by 3.
    £20 × 3 = £60

Check. Formula check: I = P × R × T ÷ 100 = 800 × 2.5 × 3 ÷ 100 = 60. ✓

Watch out. Simple interest always uses the original amount, never the amount with interest added.

Worked example 5.7.3
Sam borrows £2,400 at 5% simple interest per year. Work out the interest he pays after 6 months.

Key words. Interest is money paid for saving or borrowing. Simple interest is a fixed percentage of the original amount, paid each year.

Method. Simple interest is the same amount every year. Find one year’s interest, then multiply by the number of years.

  1. Change the time to years. The rate is per year, so the time must be in years. 6 months is half a year.
    6 months = 6 ÷ 12 = 0.5 years
  2. Write down what we know. The principal is the money at the start; the rate is the percentage per year; the time is in years.
    Principal = £2,400,  rate = 5% per year,  time = 0.5 years
  3. Find the interest for ONE year. Simple interest is the same every year: it is 5% of the original amount each time.
    5% of £2,400 = 0.05 × £2,400 = £120
  4. Multiply by the number of years. The same interest is added every year, so multiply by 0.5.
    £120 × 0.5 = £60

Check. Formula check: I = P × R × T ÷ 100 = 2,400 × 5 × 0.5 ÷ 100 = 60. ✓

Watch out. Simple interest always uses the original amount, never the amount with interest added.

Worked example 5.7.4
Ravi invests £2,000 and earns £300 simple interest in 3 years. Work out the yearly rate of interest.

Method. Find the interest for one year, then write it as a percentage of the original amount.

  1. Write down what we know. The principal is the money at the start. The interest is the extra money earned.
    Principal = £2,000,  interest = £300,  time = 3 years
  2. Find the interest for one year. The total interest was earned over 3 years, so divide by 3.
    £300 ÷ 3 = £100
  3. Write one year’s interest as a percentage of the original amount. Rate = part over whole, times 100.
    £100 ÷ £2,000 × 100 = 5%

Check. 5% of £2,000 = £100, and 3 × £100 = £300. ✓

Watch out. Divide by the number of years first. The percentage asked for is the yearly rate.

Worked example 5.7.5
How long does it take £2,000 to earn £300 at 5% simple interest per year?

Method. Find the interest for one year, then see how many years give the total interest.

  1. Write down what we know. The principal is the money at the start; the rate is the percentage per year.
    Principal = £2,000,  rate = 5% per year,  total interest wanted = £300
  2. Find the interest for one year. It is 5% of the original amount.
    5% of £2,000 = £100
  3. Divide the total interest by one year’s interest. This tells us how many years are needed.
    £300 ÷ £100 = 3

Check. 3 × £100 = £300. ✓

Worked example 5.7.6
Anna invests some money at 1.5% simple interest per year. After 4 years she has earned £72 interest. Work out how much she invested.

Method. Find the interest for one year, then use it to find what 1.5% of the principal is, and so the principal.

  1. Find the interest for one year. The interest is the same each year, so divide the total interest by the number of years.
    £72 ÷ 4 = £18
  2. Write what you know. 1.5% of the principal is the interest for one year.
    1.5% of the principal = £18
  3. Find 1%. Divide by 1.5.
    £18 ÷ 1.5 = £12
  4. Find 100%, the principal. Multiply by 100.
    £12 × 100 = £1,200

Check. 1.5% of £1,200 = £18, and 4 × £18 = £72. ✓

Watch out. This is a reverse-percentage step: the 1.5% is of the principal.

Watch out

Simple interest is worked out on the original amount every year. Do not add the interest on and then take a percentage of the bigger number: that is compound interest (next section). Also remember to add the interest to the principal when the question asks for the total in the account.

Exercise 5L  Simple interest

Non-calculator
1
FWork out the simple interest earned in each case.
(a)£400 at 3% p.a. for 2 years
(b)£2,000 at 2% p.a. for 5 years
(c)£650 at 10% p.a. for 3 years
(d)£1,200 at 1.5% p.a. for 4 years
(e)£5,000 at 4% p.a. for 3 years
(f)£840 at 5% p.a. for 2 years
2
FWork out the total amount in the account at the end of the time.
(a)£1,500 at 3% p.a. for 4 years
(b)£800 at 2.5% p.a. for 3 years
(c)£3,000 at 4% p.a. for 5 years
3
FZara puts £2,000 in a savings account at 3% simple interest per year. How much is in the account after 2 years?
4
PSMo pays £36 simple interest each year on a loan. The rate of interest is 6% per year. How much did Mo borrow?
5
PSBen borrows £2,400 for 6 months. The loan has simple interest of 5% per year. How much interest does Ben pay?
6
PSDev borrows £600 and pays 8% simple interest per year for 2 years. How much does he pay back in total?
7
PSBank A pays 3% simple interest per year on £2,000 for 4 years. Bank B pays 4% simple interest per year on £1,500 for 5 years. Which bank pays more interest, and how much more?
8
PSNia invests £1,600. She earns £192 simple interest in 3 years. Work out the yearly rate of interest.
9
PSHow many years will it take £2,500 to earn £500 at 4% simple interest per year?
10
RFred says, “With simple interest, the interest you earn in year 2 is bigger than the interest in year 1.” Is he right? Use £1,000 at 5% per year to explain.
11
PSSumi invests some money at 1.5% simple interest per year. After 4 years she has earned £72 interest. How much did she invest?
12
FWork out the simple interest on £5,000 at 4.2% per year for 30 months.
13
RShow that 6% simple interest for 5 years gives the same interest as 5% simple interest for 6 years on the same principal.
5.8

Compound interest, growth and decay

Edexcel 1MA1: R9, R16
Grade 4-9

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Compound interest, growth and decayComing soon

Simple interest pays the same amount every year. Most savings accounts work differently. With compound interest the interest is added to the account each year, and next year’s interest is worked out on the new, bigger amount. So you earn interest on your interest. Look at £1,000 at 5% a year.

Year 1: 5% of £1,000 = £50, so you have £1,050. Year 2: 5% of £1,050 = £52.50, so you have £1,102.50. With simple interest you would have had only £1,100.

YearsSimple interest totalCompound interest total
0£1,000.00£1,000.00
1£1,050.00£1,050.00
2£1,100.00£1,102.50
3£1,150.00£1,157.63
4£1,200.00£1,215.51
5£1,250.00£1,276.28
years1234567891011201260140015401680compound (orange) grows faster than simple (blue)Amount (£)
The orange curve (compound) bends upwards; the blue line (simple) is straight.

The multiplier shortcut

Each year the amount is multiplied by 1.05. After 2 years it has been multiplied by 1.05 twice: 1.05 × 1.05 = 1.052. After n years it is multiplied by 1.05n. So

final amount = starting amount × multipliern

where n is the number of years. On a calculator use the power key (xy or ^). Do not round in the middle of a calculation; round the final money answer to the nearest penny.

Repeated percentage change: growth and decay

The same idea works for anything that changes by a fixed percentage again and again. A population that grows 3% each year, bacteria that increase 20% every hour: use a multiplier above 1 (growth). A car that loses 15% of its value each year, or a medicine that falls by 30% each hour in the blood: use a multiplier below 1 (decay). A fall in value is called depreciation.

years123456300060009000120001500018000Value (£)
A car worth £18,000 loses 15% of its value each year. The fall is fastest at first.

Higher tier. Because the amount is multiplied each time, you can work backwards or solve for the unknown:

Key ideas

  • Compound interest: interest is added each year and then earns interest itself.
  • Final amount = start × multipliern, with n = number of years (or other periods).
  • Growth by p%: multiplier = 1 + p/100.   Decay by p%: multiplier = 1 − p/100.
  • Compound interest earned = final amount − starting amount.
  • Do not round until the end. Money: nearest penny. Population: nearest whole number.
  • Compound interest is always at least as much as simple interest, and more after year 1.
Worked example 5.8.1
Maya invests £2,000 for 3 years at 5% compound interest per year. How much does she have at the end? Work year by year.

Key words. Compound interest is interest paid on the original amount and on the interest already added. Depreciation is a fall in value.

Method. Compound interest means the interest is added each year, so next year’s interest is worked out on the bigger amount. Use the multiplier 1.05 once for each year.

  1. Write down what we know. The starting amount, the percentage change each time, and how many times it happens.
    Start = £2,000,  increase = 5% per year,  number of years = 3
  2. Find the multiplier for one year. Each year the amount grows by 5%, so it becomes 105% of the previous amount. The multiplier is 105% as a decimal.
    105% = 105 ÷ 100 = 1.05
  3. Apply the multiplier once for each year. Each year we multiply the amount we have by 1.05.
    After year 1: £2,000 × 1.05 = £2,100
    After year 2: £2,100 × 1.05 = £2,205
    After year 3: £2,205 × 1.05 = £2,315.25

Check. Estimate: with simple interest the total would be £2,300. Compound interest is a little more because the interest earns interest, and £2,315.25 is a little above £2,300. ✓

Watch out. Use the new amount each year as the starting point for the next year.

Worked example 5.8.2
Work out the total amount when £5,000 is invested for 4 years at 3% compound interest per year.

Key words. Compound interest is interest paid on the original amount and on the interest already added. Depreciation is a fall in value.

Method. Compound interest means the interest is added each year, so next year’s interest is worked out on the bigger amount. Use the multiplier 1.03 once for each year.

  1. Write down what we know. The starting amount, the percentage change each time, and how many times it happens.
    Start = £5,000,  increase = 3% per year,  number of years = 4
  2. Find the multiplier for one year. Each year the amount grows by 3%, so it becomes 103% of the previous amount. The multiplier is 103% as a decimal.
    103% = 103 ÷ 100 = 1.03
  3. Raise the multiplier to the power 4. The same increase happens 4 times, so we multiply by 1.03 4 times. That is the multiplier to the power 4.
    1.034 = 1.125509…
  4. Multiply the starting amount by 1.034. This gives the amount after all the changes in one calculation.
    £5,000 × 1.125509… = £5,627.54405
  5. Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.
    £5,627.54405 = £5,627.54

Check. Estimate: with simple interest the total would be £5,600. Compound interest is a little more because the interest earns interest, and £5,627.54 is a little above £5,600. ✓

Watch out. Do not multiply by 1.03 and then by 4. Use the multiplier to the power 4.

Worked example 5.8.3
Work out the compound interest earned when £4,000 is invested for 2 years at 2.5% per year.

Key words. Compound interest is interest paid on the original amount and on the interest already added. Depreciation is a fall in value.

Method. Compound interest means the interest is added each year, so next year’s interest is worked out on the bigger amount. Use the multiplier 1.025 once for each year.

  1. Write down what we know. The starting amount, the percentage change each time, and how many times it happens.
    Start = £4,000,  increase = 2.5% per year,  number of years = 2
  2. Find the multiplier for one year. Each year the amount grows by 2.5%, so it becomes 102.5% of the previous amount. The multiplier is 102.5% as a decimal.
    102.5% = 102.5 ÷ 100 = 1.025
  3. Raise the multiplier to the power 2. The same increase happens 2 times, so we multiply by 1.025 2 times. That is the multiplier to the power 2.
    1.0252 = 1.050625
  4. Multiply the starting amount by 1.0252. This gives the amount after all the changes in one calculation.
    £4,000 × 1.050625 = £4,202.50
  5. Subtract the original amount to find the interest. The interest is only the extra money, not the total.
    £4,202.50 − £4,000 = £202.50

Check. Estimate: with simple interest the total would be £4,200. Compound interest is a little more because the interest earns interest, and £4,202.50 is a little above £4,200. ✓

Watch out. Do not multiply by 1.025 and then by 2. Use the multiplier to the power 2.

Worked example 5.8.4
Compare the total from £1,000 invested for 3 years at 4% per year with (a) simple interest and (b) compound interest. How much more does compound interest give?

Method. Work out each total, then subtract.

  1. Simple interest: interest for one year. 4% of £1,000.
    4% of £1,000 = £40
  2. Simple interest: total. Three years of £40 added to the £1,000.
    £1,000 + 3 × £40 = £1,120
  3. Compound interest: the multiplier. 104% as a decimal.
    1.04
  4. Compound interest: total. Multiply by 1.04 three times.
    £1,000 × 1.043 = £1,000 × 1.124864 = £1,124.864 = £1,124.86
  5. Find the difference. Subtract the simple total from the compound total.
    £1,124.86 − £1,120 = £4.86

Check. Compound interest in year 2 is 4% of £1,040 = £41.60 (more than £40), and in year 3 it is bigger again. ✓

Watch out. Compound interest only beats simple interest from year 2 onwards.

Worked example 5.8.5
A car is bought for £14,000. It loses 15% of its value every year. Work out its value after 3 years.

Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.

Method. Repeated percentage decrease: multiply by 0.85 once for each year.

  1. Write down what we know. The starting amount, the percentage change each time, and how many times it happens.
    Start = £14,000,  decrease = 15% per year,  number of years = 3
  2. Find the multiplier for one year. Each year the amount falls by 15%, so it becomes 85% of the previous amount. The multiplier is 85% as a decimal.
    85% = 85 ÷ 100 = 0.85
  3. Raise the multiplier to the power 3. The same decrease happens 3 times, so we multiply by 0.85 3 times. That is the multiplier to the power 3.
    0.853 = 0.614125
  4. Multiply the starting amount by 0.853. This gives the amount after all the changes in one calculation.
    £14,000 × 0.614125 = £8,597.75

Check. Estimate: losing 15% of the original each time would leave £7,700. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than £7,700. ✓

Watch out. Do not multiply by 0.85 and then by 3. Use the multiplier to the power 3.

Worked example 5.8.6
The population of a village is 12,500. It grows by 4% every year. Estimate the population after 5 years, to the nearest whole number.

Key words. Compound interest is interest paid on the original amount and on the interest already added. Depreciation is a fall in value.

Method. Compound interest means the interest is added each year, so next year’s interest is worked out on the bigger amount. Use the multiplier 1.04 once for each year.

  1. Write down what we know. The starting amount, the percentage change each time, and how many times it happens.
    Start = 12,500,  increase = 4% per year,  number of years = 5
  2. Find the multiplier for one year. Each year the amount grows by 4%, so it becomes 104% of the previous amount. The multiplier is 104% as a decimal.
    104% = 104 ÷ 100 = 1.04
  3. Raise the multiplier to the power 5. The same increase happens 5 times, so we multiply by 1.04 5 times. That is the multiplier to the power 5.
    1.045 = 1.216653…
  4. Multiply the starting amount by 1.045. This gives the amount after all the changes in one calculation.
    12,500 × 1.216653… = 15,208.16128
  5. Round the answer. Round to the nearest whole number.
    15,208.16128 = 15,208

Check. Estimate: with simple interest the total would be 15,000. Compound interest is a little more because the interest earns interest, and 15,208 is a little above 15,000. ✓

Watch out. Do not multiply by 1.04 and then by 5. Use the multiplier to the power 5.

Worked example 5.8.7
£1,000 is invested at 4% compound interest per year. After how many whole years will it be worth more than £1,500?

Method. Use the multiplier to a power and try more and more years until the target is reached.

  1. Find the multiplier for one year. Write the percentage as a decimal.
    104% = 1.04
  2. Try years one at a time. We cannot solve this directly yet, so we use trial and improvement: work out the amount after 1, 2, 3, … years using the multiplier to a power.
    After 1 year: £1,000 × 1.041 = £1,040
    After 2 years: £1,000 × 1.042 = £1,081.60
    After 3 years: £1,000 × 1.043 = £1,124.864 = £1,124.86
    After 4 years: £1,000 × 1.044 = £1,169.85856 = £1,169.86
    After 5 years: £1,000 × 1.045 = £1,216.6529… = £1,216.65
    After 6 years: £1,000 × 1.046 = £1,265.3190… = £1,265.32
    After 7 years: £1,000 × 1.047 = £1,315.9318… = £1,315.93
    After 8 years: £1,000 × 1.048 = £1,368.5691… = £1,368.57
    After 9 years: £1,000 × 1.049 = £1,423.3118… = £1,423.31
    After 10 years: £1,000 × 1.0410 = £1,480.2443… = £1,480.24
    After 11 years: £1,000 × 1.0411 = £1,539.4541… = £1,539.45
  3. Find the first time the target is passed. We want the amount to be more than £1,500.
    After 11 years the amount is £1,539.45, which is above £1,500. After 10 it was not.

Check. After 10 years the amount is £1,480.24, which has not yet passed £1,500, so 11 is the first. ✓

Watch out. The answer is a whole number of years: say the first year in which the target is reached or passed.

Worked example 5.8.8
£3,000 is invested at a fixed compound rate of interest. After 3 years it is worth £5,184. Work out the yearly percentage rate.

Method. The multiplier to the power n gives the overall change. To find the yearly multiplier, divide the end amount by the start amount and take the n-th root.

  1. Write the equation. Let the yearly multiplier be m. After 3 years the start amount has been multiplied by m3.
    £3,000 × m3 = £5,184
  2. Divide both sides by the start amount. This isolates m3.
    m3 = £5,184 ÷ £3,000 = 1.728
  3. Take the cube root of both sides. The cube root undoes the power 3. On a calculator use the x1/3 or the n-th root key.
    m = cube root of 1.728 = 1.2
  4. Turn the multiplier into a percentage. 1.2 = 120%, and the original is 100%.
    120% − 100% = 20% increase

Check. £3,000 × 1.23 = £3,000 × 1.728 = £5,184. ✓

Watch out. Do not divide the total percentage change by the number of years. The change compounds.

Worked example 5.8.9
£8,000 is invested for 2 years. The interest rate is 3% in the first year and 2.5% in the second year. Work out the total at the end.

Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.

  1. Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).
    increase of 3%: 103% = 1.03
    increase of 2.5%: 102.5% = 1.025
  2. Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.
    1.03 × 1.025 = 1.05575
  3. Multiply the starting amount by the overall multiplier. This gives the amount after both changes.
    £8,000 × 1.05575 = £8,446

Check. Apply the changes one at a time and you reach the same amount. ✓

Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.

Watch out

Do not work out the percentage once and multiply it by the number of years. 5% for 3 years is not 15%: the multiplier is 1.053 = 1.157625, which is a 15.76% increase. Also take care with the power: 1.053 means 1.05 × 1.05 × 1.05, not 1.05 × 3.

Exercise 5M  Compound interest, growth and decay

Calculator allowed
1
FWork out the total amount in each case. Give your answers to the nearest penny.
(a)£1,000 at 5% per year for 2 years
(b)£2,000 at 3% per year for 3 years
(c)£5,000 at 4% per year for 2 years
(d)£800 at 2% per year for 3 years
(e)£1,500 at 6% per year for 2 years
(f)£10,000 at 1.5% per year for 4 years
2
FWork out the compound interest earned in each case.
(a)£4,000 at 2.5% per year for 2 years
(b)£6,000 at 3% per year for 2 years
(c)£2,500 at 4% per year for 3 years
3
PSEve invests £3,000 for 3 years at 4% compound interest per year.
(a)Work out the total amount at the end of 3 years.
(b)Work out the compound interest she earns.
4
PSHiba invests £2,000 for 3 years at 4% per year. Work out how much more interest she gets if the interest is compound interest rather than simple interest.
5
PSA town has a population of 25,000. It grows by 3% each year. Work out the population after 4 years, to the nearest whole number.
6
PSA scientist starts with 500 bacteria. The number of bacteria increases by 20% every hour. Estimate the number after 5 hours, to the nearest whole number.
7
PSA car is worth £14,000 when new. It loses 15% of its value every year.
(a)Work out the value of the car after 3 years.
(b)Work out the value of the car after 5 years.
8
PSA new phone costs £800. It loses 25% of its value each year. Work out its value after 2 years. Work year by year.
9
RJoe says, “£1,000 at 10% compound interest for 2 years gives the same as 20% simple interest, because 10% + 10% = 20%.” Is he right? Work out both totals.
10
RDan wants to work out the value of £1,000 after 3 years at 5% compound interest. He writes: 1,000 × 1.15 = £1,150. Explain his mistake and work out the correct value.
11
FWork out each amount. Give your answers to the nearest penny or nearest whole number as appropriate.
(a)£8,500 at 2.8% compound interest per year for 6 years
(b)A forest of 14,500 trees that decreases by 3.2% each year, after 7 years
(c)A bike worth £640 that loses 18.5% each year, after 4 years
12
PSA bank pays interest of 2% every six months, added to the account each time. Work out the total in the account after 2 years when £10,000 is invested. (Hint: how many six-month periods are there in 2 years?)
13
R£1,000 is invested at 5% compound interest per year. Show that the interest earned in year 2 is £52.50. Explain why it is more than the interest in year 1.

Exercise 5N  Growth and decay: years, rates and starting amounts

Calculator allowed
1
FFind the smallest whole number of years in each case.
(a)£1,000 invested at 4% compound interest per year: first year it is worth more than £1,500
(b)£5,000 invested at 3% compound interest per year: first year it is worth more than £6,000
(c)A £20,000 machine loses 12% of its value each year: first year it is worth less than £10,000
2
FWork out the yearly percentage rate in each case.
(a)£3,000 grows to £5,184 in 3 years
(b)£2,500 grows to £3,025 in 2 years
(c)£16,000 falls to £12,960 in 2 years
3
FWork out the starting amount in each case.
(a)After 3 years at 10% growth per year the amount is £6,655
(b)After 2 years at 5% decrease per year the value is £9,025
(c)After 4 years at 20% growth per year the amount is £10,368
4
PSThe interest rates on an account are 4% in year 1, 3% in year 2 and 2.5% in year 3. Work out the total after 3 years when £5,000 is invested.
5
PSA sample of a radioactive substance has a mass of 80 g. Its mass decreases by 15% every day. Work out its mass after 4 days, correct to 2 decimal places.
6
RA savings account pays 8% compound interest per year. Show that the money more than doubles in 10 years.
7
RThe population of a country grows by 2% each year. Priya says, “It will double in 35 years.” Use a calculator to show that she is nearly right but not quite.
5.9

Percentage problems: choosing the method and comparing

Edexcel 1MA1: R9, N12, R16
Grade 4-8

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Percentage problems: choosing the method and comparingComing soon

Exam questions rarely tell you which percentage method to use. Your job is to read the question and decide. The table below is a checklist: look for the key words and pick the matching method.

The question says…What to do
“Find 35% of £240”Percentage of an amount: 0.35 × 240, or build up from 10% and 1%
“Increase by 8%”, “reduced by 30%”, “add VAT”Multiplier: 1.08, 0.7, 1.2 — new amount = original × multiplier
“What percentage is 18 of 25?”Part ÷ whole × 100
“Work out the percentage increase/decrease/profit/loss”Change ÷ original × 100
“After a 20% rise the price is £84. Find the original”Reverse percentage: divide by the multiplier (1.2)
“Simple interest”Same amount each year: principal × rate × years ÷ 100
“Compound interest”, “grows by 4% each year”Multiplier to the power n

One change after another

Suppose a price of £200 goes up by 10% and then the new price goes down by 10%. Does it come back to £200? No. The 10% rise adds £20, giving £220. The 10% fall is 10% of £220, which is £22, giving £198. The two percentages are of different amounts. In multiplier form: 1.1 × 0.9 = 0.99, so the overall change is a 1% decrease. When there is one change after another, multiply the multipliers; never add or subtract the percentages.

£200× 1.1£220× 0.9£198
Up 10% then down 10% is the same as multiplying by 1.1 × 0.9 = 0.99: a 1% decrease overall.

Comparing offers

To compare two offers, work out the final price (or the saving) for each, using the same starting amount, and then compare. If no starting price is given, use 100 or an easy number: percentages are the same for any starting amount. A fraction and a percentage can be compared too: one third off (33.3%) is a bigger discount than 30% off.

A percentage of a percentage, and fractions with percentages

Take it one step at a time and keep track of what the 100% is each time. If 45% of a school is in Key Stage 3, and one third of those are in Year 7, then Year 7 is one third of 45% = 15% of the whole school.

Higher tier. You can prove general facts with algebra. A rise of x% followed by a fall of x% has overall multiplier (1 + x/100)(1 − x/100) = 1 − x2/10 000, which is always less than 1, so the result is always a loss. For x = 20: 1 − 400/10 000 = 0.96, a 4% decrease.

Key ideas

  • Choose the method from the key words (see the table).
  • For several changes in a row, multiply the multipliers. Do not add the percentages.
  • A rise of p% followed by a fall of p% is always a (small) loss.
  • To compare offers, use the same starting amount (use 100 if none is given).
  • Keep track of what the 100% is at each step.
  • Always read the last line of the question: it tells you what to give as your final answer.
Worked example 5.9.1
A price of £200 is increased by 10%. The new price is then decreased by 10%. Work out the final price.

Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.

  1. Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).
    increase of 10%: 110% = 1.1
    decrease of 10%: 90% = 0.9
  2. Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.
    1.1 × 0.9 = 0.99
  3. Multiply the starting amount by the overall multiplier. This gives the amount after both changes.
    £200 × 0.99 = £198

Check. Apply the changes one at a time and you reach the same amount. ✓

Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.

Worked example 5.9.2
A price is increased by 20% and then the new price is decreased by 20%. Work out the overall percentage change.

Method. Find the overall multiplier by multiplying the separate multipliers, then turn it back into a percentage change.

  1. Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).
    increase of 20%: 120% = 1.2
    decrease of 20%: 80% = 0.8
  2. Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.
    1.2 × 0.8 = 0.96
  3. Turn the overall multiplier into a percentage change. Subtract 1 (the original, 100%) and multiply by 100.
    (0.96 − 1) × 100 = −4%
  4. Say it in words.
    an overall decrease of 4%

Check. Try it with 100: apply each change in turn to 100 and compare the result with 100 times the overall multiplier. ✓

Watch out. Do not add or subtract the percentages. They are applied to different amounts.

Worked example 5.9.3
A TV costs £400. Shop A offers 15% off. Shop B offers £50 off. Which shop gives the lower price? Do not use a calculator.

Method. Work out the price in each shop and compare.

  1. Shop A: find 15% of £400. 10% of 400 = 40 and 5% = 20, so 15% = 40 + 20.
    £40 + £20 = £60
  2. Shop A: subtract. The reduction is taken off the price.
    £400 − £60 = £340
  3. Shop B: subtract £50.
    £400 − £50 = £350
  4. Compare. £340 is less than £350.
    Shop A is cheaper, by £10.

Check. 0.85 × 400 = 340. ✓

Watch out. A bigger-looking discount is not always the bigger saving: work out the actual prices.

Worked example 5.9.4
A tank holds 800 litres. It is 38 full. Then 40% of the water in the tank is used. How many litres are left?

Method. Find the amount in the tank, then find what is used, then subtract.

  1. Find the water in the tank. 38 of 800: divide by 8, then multiply by 3.
    800 ÷ 8 = 100,  100 × 3 = 300 litres
  2. Find 40% of the water. 10% of 300 = 30, so 40% = 4 lots of 30.
    4 × 30 = 120 litres used
  3. Subtract. The water left is what was there take away what was used.
    300 − 120 = 180 litres

Check. 60% of the water is left: 0.6 × 300 = 180. ✓

Watch out. The 40% is of the 300 litres in the tank, not of the full 800.

Worked example 5.9.5
45% of the 600 students in a school are in Key Stage 3. One third of the Key Stage 3 students are in Year 7. What percentage of all the students in the school are in Year 7?

Method. Find the number in Year 7, then write it as a percentage of the whole school.

  1. Find the Key Stage 3 students. 10% of 600 = 60, so 40% = 240 and 5% = 30.
    240 + 30 = 270
  2. Find the Year 7 students. One third of 270.
    270 ÷ 3 = 90
  3. Write 90 as a percentage of 600. Part over whole, times 100.
    90600 = 15100 = 15%

Check. One third of 45% is 45% ÷ 3 = 15%. ✓

Watch out. Compare with the whole school (600), not with Key Stage 3 (270).

Worked example 5.9.6
Show that increasing by x% and then decreasing by x% always gives a decrease overall. Find the overall percentage decrease when x = 20.

Method. Write the two multipliers using x, multiply them, and compare with 1.

  1. Write the two multipliers. Increase by x%: 1 + x/100. Decrease by x%: 1 − x/100.
    (1 + x/100) and (1 − x/100)
  2. Multiply them. This is a difference of two squares: (1 + a)(1 − a) = 1 − a2.
    (1 + x/100)(1 − x/100) = 1 − x2/10 000
  3. Compare with 1. x2/10 000 is always positive, so the multiplier is less than 1.
    So the overall change is always a decrease.
  4. Put in x = 20. 202 = 400.
    1 − 400/10 000 = 1 − 0.04 = 0.96, which is a 4% decrease

Check. Check with 100: +20% gives 120, then −20% of 120 = 24 gives 96, which is a 4% decrease. ✓

Watch out. A rise and a fall of the same percentage do not cancel.

Worked example 5.9.7
A price is increased by 20% and the new price is then decreased by 10%. The final price is £216. Work out the original price.

Method. Combine the two changes into one multiplier, then divide.

  1. Find the multiplier for each change. 20% increase: 120% = 1.2. 10% decrease: 90% = 0.9.
    1.2 and 0.9
  2. Multiply the multipliers. The changes happen one after the other.
    1.2 × 0.9 = 1.08
  3. Divide the final price by the overall multiplier. original × 1.08 = £216, so original = £216 ÷ 1.08.
    £216 ÷ 1.08 = £200

Check. £200 × 1.2 = £240; £240 × 0.9 = £216. ✓

Watch out. Do not reverse the two changes one at a time unless you divide by each multiplier.

Watch out

Percentages do not simply add or cancel. “Up 20% then down 20%” is not back to the start. And never apply the second percentage to the original amount: it applies to the new amount after the first change.

Exercise 5O  Mixed percentage problems (non-calculator)

Non-calculator
1
RSay which method you would use for each problem (you do not need to work out the answer): percentage of an amount, increase or decrease with a multiplier, percentage change, or reverse percentage.
(a)A bike costs £250. It is reduced by 12%. Find the sale price.
(b)A rent rises from £40 to £46. Find the percentage rise.
(c)After a 20% rise the rent is £600. Find the old rent.
(d)There are 120 pupils. 35% of them play football. How many play?
2
FA price is changed twice. Work out the final price in each case.
(a)£200: up 10%, then down 10%
(b)£500: down 20%, then up 20%
(c)£80: up 50%, then down 50%
(d)£1,000: up 10%, then up 10%
3
PSLaptop Shop A takes 12% off a £600 laptop. Shop B takes £70 off the same laptop. Which shop is cheaper, and by how much? Do not use a calculator.
4
PSA jacket costs £120. Shop A has 25% off. Shop B has one third off. Which shop gives the bigger discount? By how much? Do not use a calculator.
5
PSA tank holds 600 litres. It is 25 full. Then 30% of the water in the tank is used. How many litres of water are left in the tank?
6
PSIn a class of 30 students, 40% are boys. Of the boys, 25% wear glasses.
(a)How many boys wear glasses?
(b)What percentage of the whole class is this?
7
RDan has a £20 shirt. It is reduced by 10%. Then the sale price is reduced by a further 10%. Dan says, “So altogether the shirt is 20% off, and costs £16.” Is he right? Show how you know.
8
PSA pair of shoes costs £90. The price is reduced by 20% in a sale. Jess then uses a voucher that takes a further 10% off the sale price. How much does Jess pay?
9
PSAnna earns £30,000 a year and gets a 4% pay rise. Ben earns £28,000 a year and gets an 8% pay rise. Who earns more after the rises, and by how much? Do not use a calculator.
10
PSA test has 40 questions. 30% of the questions are on algebra and 25% are on geometry. The rest are on number. How many questions are on number?
11
RShow that an increase of 20% followed by a decrease of 20% is equivalent to an overall decrease of 4%.
12
RThe price of a ticket is increased by x% and then decreased by x%. Show that the overall multiplier is 1 − x2/10 000.
13
PSA price is increased by 20% and then the new price is decreased by 10%. The final price is £216. Work out the original price.
14
RA is 20% more than B. B is 20% less than C. Is A more than C, less than C, or equal to C? Explain.

Exercise 5P  Mixed percentage problems (calculator)

Calculator allowed
1
PSA holiday for a family costs £1,250. The family pays a 15% deposit, then pays the rest in 5 equal payments.
(a)Work out the deposit.
(b)Work out the amount of each of the 5 payments.
2
PSA boiler costs £480 before VAT. VAT at 20% is added. Gabi then gets a 5% discount on the price including VAT for paying early. How much does Gabi pay?
3
PSRavi invests £4,000 in Account A, which pays 3% simple interest per year for 5 years. Sam invests £4,000 in Account B, which pays 2.5% compound interest per year for 5 years. Who has more money at the end, and how much more?
4
PS360 people are asked how they travel to work. 35% walk, 38 drive, and the rest cycle. How many people cycle?
5
PSA shop buys a lamp for £25. It wants to make a profit of 36% on the cost price. Work out the selling price of the lamp.
6
PSA coat is reduced by 15% in a sale. The sale price is £51. What would the sale price have been if the coat had been reduced by 25% instead?
7
PSIn a country, income tax is 20% on the first £37,700 of taxable income and 40% on all taxable income above £37,700. Work out the tax on a taxable income of £50,000.
8
RA share price rises by 12% one week and falls by 12% the next week. Hannah says, “It is back to where it started.” Is she right? Work out the overall percentage change.
9
PSA paint mix is 40% blue paint and the rest is yellow paint. There are 12 litres of blue paint in the mix.
(a)How many litres of paint are in the mix altogether?
(b)How many litres of yellow paint are there?
10
PSA machine costs £25,000. It loses 20% of its value in the first year and then 12% of its value in each year after that. Work out its value after 4 years.
11
PSThe population of a town is 80,000. It grows by the same percentage each year and reaches 100,000 after 6 years. Work out the yearly percentage growth, correct to 1 decimal place.
12
RThe price of a games console is increased by 25%. The shop wants to bring the price back to the original price. Show that it needs a decrease of 20%.

Chapter summary

SkillMethod
Percentage, fraction, decimal% to decimal: ÷ 100. Decimal to %: × 100. % to fraction: over 100 and simplify. Fraction to %: make the denominator 100, or divide then × 100
Percentage of an amountWithout a calculator: build from 10% (÷ 10) and 1% (÷ 100). With a calculator: decimal × amount
One quantity as a percentage of anotherpart ÷ whole × 100 (same units!)
Comparing using percentagesWrite each as a percentage of its own total, then compare
Increase or decrease by p%New amount = original × multiplier. Multiplier = 1 + p/100 (increase) or 1 − p/100 (decrease)
Percentages over 100%100% + 150% = 250% so multiplier 2.5. An increase of 100% doubles
Percentage change, profit, losschange ÷ original × 100
Reverse percentageoriginal = new amount ÷ multiplier (or find 1%, then 100%)
Simple interestInterest = principal × rate × years ÷ 100. Same amount each year
Compound interest, growth, decayfinal = start × multipliern. Interest = final − start
One change after anotherMultiply the multipliers. Never add the percentages
Higher: years, rate, starting amountYears: try powers. Rate: n-th root of (final ÷ start). Start: divide by multipliern

Can you…?

  • I can write a percentage as a fraction and a decimal, and change a fraction or decimal to a percentage.
  • I can find a percentage of an amount without a calculator (using 10% and 1%) and with a calculator.
  • I can write one number as a percentage of another, changing units first if I need to.
  • I can compare two results using percentages.
  • I can increase or decrease an amount by a percentage using a multiplier, including increases of more than 100%.
  • I can work out a percentage change, percentage profit and percentage loss.
  • I can find the original amount after a percentage increase or decrease (reverse percentages).
  • I can work out simple interest, with the time in years or months.
  • I can work out compound interest and repeated percentage growth or decay using multipliern.
  • I know that percentage changes one after another are multiplied, not added.
  • Higher: I can find the number of years or the yearly rate in a compound problem, and work backwards to a starting amount.

Key terms

Percentage
A number of parts per hundred, written with the % sign. 35% means 35 out of 100.
Per cent
Out of 100.
Multiplier
The number you multiply by to get the new amount in one step. A 15% increase has multiplier 1.15; a 15% decrease has multiplier 0.85.
Percentage change
The change as a percentage of the original amount: change ÷ original × 100.
Original amount
The starting amount, which is 100% in a percentage-change problem.
Profit / loss
Selling price minus cost price: a profit if positive, a loss if negative. Percentage profit or loss is found out of the cost price.
Reverse percentage
A problem where you know the amount after a percentage change and must find the original amount.
Principal
The amount of money first invested or borrowed.
Simple interest
Interest that is the same every year because it is always a percentage of the original amount.
Compound interest
Interest worked out on the original amount plus the interest already added.
Per annum (p.a.)
Each year.
Depreciation
A fall in the value of something (like a car) over time. Usually a repeated percentage decrease.
Growth and decay
A repeated percentage increase (growth) or decrease (decay).
Percentage points
The difference between two percentages. A rate that goes from 4% to 6% rises by 2 percentage points (but by 50% as a percentage change).

Exercise 5Z  Check-up (non-calculator)

Non-calculator
1
Write 45% (a) as a decimal and (b) as a fraction in its simplest form.
(a)As a decimal
(b)As a fraction in its simplest form
2
Work out 15% of £240.
3
Write 18 as a percentage of 24.
4
Increase £60 by 20%.
5
Decrease 80 kg by 35%.
6
The price of a book rises from £25 to £30. Work out the percentage increase.
7
After a 20% increase in price, a scooter costs £84. Work out the price before the increase.
8
Work out the simple interest on £600 invested at 4% per year for 3 years.
9
Write these in order, smallest first:   0.6,   55%,   58
10
£1,000 is invested at 10% compound interest per year. Work out the total after 2 years. Work year by year.

Total 49 marks. Allow about 49 minutes.

Exercise 5X  Exam-style questions

Calculator allowed
1
F(Non-calculator) Write 720 as a percentage.[2 marks]
2
F(Non-calculator) Work out 35% of £80.[2 marks]
3
PSJamie scored 34 out of 40 in Test A and 41 out of 50 in Test B. In which test did Jamie do better? You must show your working.[3 marks]
4
PSA coat costs £90. In a sale the price is reduced by 30%. Work out the sale price.[3 marks]
5
PSA phone costs £420 before VAT. VAT is charged at 20%. Work out the total cost of the phone including VAT.[2 marks]
6
PSA house was worth £240,000 last year. It is now worth £264,000. Work out the percentage increase in the value of the house.[3 marks]
7
PSIn a sale all prices are reduced by 15%. The sale price of a jacket is £68. Work out the price of the jacket before the sale.[3 marks]
8
PSPriya invests £2,500 for 4 years in an account that pays 3.2% simple interest per year. Work out the total amount in the account at the end of 4 years.[3 marks]
9
PSBilal invests £5,000 for 3 years at 2% compound interest per year. Work out the value of his investment at the end of 3 years.[3 marks]
10
PSA car is worth £16,000 when it is new. In the first year its value falls by 20%. In the second year its value falls by 15% of its value at the end of the first year. Work out the value of the car at the end of the second year.[4 marks]
11
RRent is increased by 5% each year for 2 years. Show that the total percentage increase over the 2 years is 10.25%.[3 marks]
12
PSThis year a shop’s sales were £60,000. This is 150% of last year’s sales. Work out last year’s sales.[3 marks]
13
PS£2,000 is invested at 5% compound interest per year. Work out the smallest whole number of years after which the investment will be worth more than £2,500. You must show your working.[4 marks]
14
PSThe value of a van falls by the same percentage each year. After 3 years its value has fallen from £15,000 to £10,935. Work out the yearly percentage decrease.[4 marks]
15
PSThe population of a town increased by 8% in the first year and then decreased by 5% in the second year. At the end of the second year the population was 12,312. Work out the population at the start of the first year.[4 marks]
16
RAn interest rate falls from 4% to 3%. Alex says, “The rate has fallen by 1%.” Explain why Alex’s statement is not complete, and write down two correct statements about the change.[3 marks]
← Chapter 4: Decimals, rounding and estimationChapter 6: Standard form, indices and surds →

Websink Prep UK is an independent site. It is not affiliated with, endorsed by or sponsored by AQA, Pearson Edexcel, OCR, WJEC, Eduqas, CCEA, SQA, GL Assessment, CEM or the Department for Education. All exam board names and trademarks belong to their owners. Past papers are linked from the exam boards' own websites and are not hosted here.

About Websink Prep UK · Find a tutor · Privacy · Terms · Cookie settings