Calculate percentage change

2. Writing percentages as fractions ‘ Per cent’ means ‘out of 100’. To write a percentage as a fraction we write it over a hundred. For example, 46% = Cancelling: 23 50 180% = Cancelling: 9 5 7.5% = Cancelling: 3 40 Click Here To Download Free Exam papers for GCSE Maths 46 100 46 100 = 23 50 180 100 180 100 = 9 5 = 1 4 5 7.5 100 15 200 = 3 40 = 15 200

3. Writing percentages as decimals We can write percentages as decimals by dividing by 100. For example, 46% = = 46 ÷ 100 = 0.46 7% = = 7 ÷ 100 = 0.07 130% = = 130 ÷ 100 = 1.3 0.2% = = 0.2 ÷ 100 = 0.002 Click Here To Download Free Exam papers for GCSE Maths 46 100 7 100 130 100 0.2 100

4. Writing fractions as percentages To write a fraction as a percentage, we can find an equivalent fraction with a denominator of 100. 85 For example, 100 and 85% 100 128 and 128% Click Here To Download Free Exam papers for GCSE Maths = 17 20 × 5 × 5 = 100 85 1 7 25 = = 32 25 × 4 × 4 = 100 128

5. Writing fractions as percentages To write a fraction as a percentage you can also multiply it by 100%. For example, 25 2 Remember, multiplying by 100% does not change the value of the number because it is equivalent to multiplying by 1. Click Here To Download Free Exam papers for GCSE Maths 3 8 = 3 8 × 100% = 3 × 100% 8 = 75% 2 = 37 1 2 %

6. Writing decimals as percentages Decimals can also be converted to percentages by multiplying them by 100%. For example, 0.08 = 0.08 × 100% = 8% 1.375 = 1.375 × 100% = 137.5% Click Here To Download Free Exam papers for GCSE Maths

7. Using a calculator We can also convert fractions to decimals and percentages using a calculator. For example, 5 ÷ 16 × 100% = 31.25% 4 ÷ 7 × 100% = 57.14% (to 2 d.p.) 13 ÷ 8 × 100% = 162.5% Click Here To Download Free Exam papers for GCSE Maths 5 16 = 4 7 = 5 8 = 1 13 8 =

8. One number as a percentage of another There are 35 sweets in a bag. Four of the sweets are orange flavour. What percentage of sweets are orange flavour? Start by writing the proportion of orange sweets as a fraction. 4 out of 35 = Then convert the fraction to a percentage. 20 7 Click Here To Download Free Exam papers for GCSE Maths 4 35 × 100% = 4 35 4 × 100% 35 = 80% 7 = 11 3 7 %

9. One number as a percentage of another Petra put £32 into a bank account. After one year she received 80p interest. To write 80p out of £32 as a fraction we must use the same units. In pence, Petra gained 80p out of 3200p. We then convert the fraction to a percentage. 5 2 = 2.5% What percentage interest rate did she receive? Click Here To Download Free Exam papers for GCSE Maths 80 3200 = 1 40 1 40 × 100% = 100% 40

10. Calculating percentages using fractions Remember, a percentage is a fraction out of 100. 15% of 90, means “15 hundredths of 90” or 3 20 9 2 Find 15% of 90 Click Here To Download Free Exam papers for GCSE Maths 15 100 × 90 = 15 × 90 100 = 27 2 = 13 1 2

11. Calculating percentages using decimals We can also calculate percentages using an equivalent decimal operator. 4% of 9 = 0.04 × 9 = 4 × 9 ÷ 100 = 36 ÷ 100 = 0.36 Click Here To Download Free Exam papers for GCSE Maths What is 4% of 9?

12. Estimating percentages We can find more difficult percentages using a calculator. It is always sensible when using a calculator to start by making an estimate. For example, estimate the value of: 19% of £82  20% of £80 = £16 27% of 38m  25% of 40m = 10m 73% of 159g  75% of 160g = 120g Click Here To Download Free Exam papers for GCSE Maths

13. Using a calculator One way to work out a percentage using a calculator is by writing the percentage as a decimal. For example, What is 38% of £65? So we key in: The calculator will display the answer as 24.7. We write the answer as £24.70 Click Here To Download Free Exam papers for GCSE Maths 38% = 0.38 0 . 3 8 × 6 5 =

14. Using a calculator We can also work out a percentage using a calculator by converting the percentage to a fraction. For example, What is 57% of £80? 57% = = 57 ÷ 100 So we key in: The calculator will display the answer as 45.6 We write the answer as £45.60 Click Here To Download Free Exam papers for GCSE Maths 57 100 5 7 ÷ 1 0 0 × 8 0 =

15. Using a calculator We can also work out percentages on a calculator by finding 1% first and then multiplying by the required percentage. What is 37.5% of £59? 1% of £59 is £0.59 so, 37.5% of £59 is £0.59 × 37.5. We key in: And get an answer of 22.125 We write the answer as £22.13 (to the nearest penny). Click Here To Download Free Exam papers for GCSE Maths 0 . 5 9 × 3 7 . 5 =

16. Finding a percentage increase or decrease Sometimes, we are given an original value and a new value and we are asked to find the percentage increase or decrease. We can do this using the following formulae: Click Here To Download Free Exam papers for GCSE Maths Percentage increase = actual increase original amount × 100% Percentage decrease = actual decrease original amount × 100%

17. Finding a percentage increase The actual increase = 4.2 kg – 3.5 kg = 0.7 kg = 20% A baby weighs 3.5 kg at birth. After 6 weeks the baby’s weight has increased to 4.2 kg. What is the baby’s percentage increase in weight? Click Here To Download Free Exam papers for GCSE Maths The percentage increase = 0.7 3.5 × 100%

18. Finding a percentage decrease The actual decrease = £25 – £17 = £8 32% Click Here To Download Free Exam papers for GCSE Maths All t-shirts were £25 now only £17! What is the percentage decrease? The percentage decrease = 8 25 × 100% =

19. Finding a percentage profit Her actual profit = 50p – 32p = 18p = 56.25% A shopkeeper buys chocolate bars wholesale at a price of 32p per bar. She then sells the chocolate bar in her shop at 50p each. What is her percentage profit? Click Here To Download Free Exam papers for GCSE Maths Her percentage profit = 18 32 × 100%

20. Finding a percentage loss Her actual loss = £3.68 – £3.22 = 46p = 12.5% A share dealer buys a number of shares at £3.68 each. After a week the price of the shares has dropped to £3.22. What is her percentage loss? Make sure the units are the same. Click Here To Download Free Exam papers for GCSE Maths Her percentage loss = 0.46 3.68 × 100%

21. Percentage increase There are two methods to increase an amount by a given percentage. The value of Frank’s house has gone up by 20% in three years. If the house was worth £150 000 three years ago, how much is it worth now? = 0.2 × £150 000 = £30 000 The amount of the increase = 20% of £150 000 The new value = £150 000 + £30 000 = £180 000 Method 1 We can work out 20% of £150 000 and then add this to the original amount.

22. Percentage increase We can represent the original amount as 100% like this: 100% When we add on 20%, 20% we have 120% of the original amount. Finding 120% of the original amount is equivalent to finding 20% and adding it on. Method 2 If we don’t need to know the actual value of the increase we can find the result in a single calculation. Click Here To Download Free Exam papers for GCSE Maths

23. Percentage increase So, to increase £150 000 by 20% we need to find 120% of £150 000. 120% of £150 000 = 1.2 × £150 000 = £180 000 In general, if you start with a given amount (100%) and you increase it by x %, then you will end up with (100 + x )% of the original amount. To convert (100 + x )% to a decimal multiplier we have to divide (100 + x ) by 100. This is usually done mentally. Click Here To Download Free Exam papers for GCSE Maths

24. Percentage increase Here are some more examples using this method: Increase £50 by 60%. 160% × £50 = 1.6 × £50 = £80 Increase £24 by 35% 135% × £24 = 1.35 × £24 = £32.40 Increase £86 by 17.5%. 117.5% × £86 = 1.175 × £86 = £101.05 Increase £300 by 2.5%. 102.5% × £300 = 1.025 × £300 = £307.50 Click Here To Download Free Exam papers for GCSE Maths

25. Percentage decrease There are two methods to decrease an amount by a given percentage. A CD walkman originally costing £75 is reduced by 30% in a sale. What is the sale price? = 0.3 × £75 = £22.50 The sale price = £75 – £22.50 = £52.50 Method 1 We can work out 30% of £75 and then subtract this from the original amount. 30% of £75 The amount taken off =

26. Percentage decrease 100% When we subtract 30% 30% we have 70% of the original amount. 70% Finding 70% of the original amount is equivalent to finding 30% and subtracting it. We can represent the original amount as 100% like this: Method 2 We can use this method to find the result of a percentage decrease in a single calculation. Click Here To Download Free Exam papers for GCSE Maths

27. Percentage decrease So, to decrease £75 by 30% we need to find 70% of £75. 70% of £75 = 0.7 × £75 = £52.50 In general, if you start with a given amount (100%) and you decrease it by x %, then you will end up with (100 – x )% of the original amount. To convert (100 – x )% to a decimal multiplier we have to divide (100 – x ) by 100. This is usually done mentally. Click Here To Download Free Exam papers for GCSE Maths

28. Percentage decrease Here are some more examples using this method: Decrease £320 by 3.5%. 96.5% × £320 = 0.965 × £320 = £308.80 Decrease £1570 by 95%. 5% × £1570 = 0.05 × £1570 = £78.50 Decrease £65 by 20%. 80% × £65 = 0.8 × £65 = £52 Decrease £56 by 34% 66% × £56 = 0.66 × £56 = £36.96 Click Here To Download Free Exam papers for GCSE Maths

29. Reverse percentages Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount. We can solve this using inverse operations. Let p be the original price of the jeans. p × 0.85 = £25.50 so p = £25.50 ÷ 0.85 = £30 Click Here To Download Free Exam papers for GCSE Maths I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them. What is the original price of the jeans?

30. Reverse percentages Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount. I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them. We can show this using a diagram: Price after discount. What is the original price of the jeans? Price before discount. × 0.85% ÷ 0.85%

31. Reverse percentages We can also use a unitary method to solve these type of percentage problems. For example, Christopher’s monthly salary after a 5% pay rise is £1312.50. What was his original salary? The new salary represents 105% of the original salary. 105% of the original salary = £1312.50 1% of the original salary = £1312.50 ÷ 105 100% of the original salary = £1312.50 ÷ 105 × 100 = £1250 This method has more steps involved but may be easier to remember. Click Here To Download Free Exam papers for GCSE Maths

32. Compound percentages A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? When a percentage change is followed by another percentage change do not add the percentages together to find the total percentage change. The second percentage change is found on a new amount and not on the original amount. It is not 30%! Click Here To Download Free Exam papers for GCSE Maths

33. Compound percentages To find a 10% decrease we multiply by 90% or 0.9. A 20% discount followed by a 10% discount is equivalent to multiplying the original price by 0.8 and then by 0.9. To find a 20% decrease we multiply by 80% or 0.8. original price × 0.8 × 0.9 = original price × 0.72 A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? Click Here To Download Free Exam papers for GCSE Maths

34. Compound percentages This is equivalent to a 28% discount. The sale price is 72% of the original price. A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? A 20% discount followed by a 10% discount A 28% discount

35. Compound percentages After a 20% discount it costs 0.8 × £100 = £80 Suppose the original price of the jacket is £100. After an other 10% discount it costs 0.9 × £80 = £72 £72 is 72% of £100. 72% of £100 is equivalent to a 28% discount altogether. A jacket is reduced by 20% in a sale. Two weeks later the shop reduces the price by a further 10%. What is the total percentage discount? Click Here To Download Free Exam papers for GCSE Maths

36. Compound percentages Jenna invests in some shares. After one week the value goes up by 10%. The following week they go down by 10%. Has Jenna made a loss, a gain or is she back to her original investment? To find a 10% increase we multiply by 110% or 1.1. To find a 10% decrease we multiply by 90% or 0.9. original amount × 1.1 × 0.9 = original amount × 0.99 Fiona has 99% of her original investment and has therefore made a 1% loss.

37. Compound interest Jack puts £500 into a savings account with an annual compound interest rate of 6%. How much will he have in the account at the end of 4 years if he doesn’t add or withdraw any money? At the end of each year interest is added to the total amount in the account. This means that each year 5% of an ever larger amount is added to the account. To increase the amount in the account by 5% we need to multiply it by 105% or 1.05. We can do this for each year that the money is in the account.

38. Compound interest At the end of year 1 Jack has £500 × 1.05 = £525 At the end of year 2 Jack has £525 × 1.05 = £551.25 At the end of year 3 Jack has £ 551.25 × 1.05 = £578.81 At the end of year 4 Jack has £578.81 × 1.05 = £607.75 (These amounts are written to the nearest penny.) We can write this in a single calculation as £500 × 1.05 × 1.05 × 1.05 × 1.05 = £607.75 Or using index notation as £500 × 1.05 4 = £607.75 Click Here To Download Free Exam papers for GCSE Maths

39. Compound interest How much would Jack have after 10 years? After 10 years the investment would be worth £500 × 1.05 10 = £814.45 (to the nearest 1p) How long would it take for the money to double? £500 × 1.05 14 = £989.97 (to the nearest 1p) £500 × 1.05 15 = £1039.46 (to the nearest 1p) Using trial and improvement, It would take 15 years for the money to double. Click Here To Download Free Exam papers for GCSE Maths

40. Repeated percentage change We can use powers to help solve many problems involving repeated percentage increase and decrease. For example, The population of a village increases by 2% each year. If the current population is 2345, what will it be in 5 years? To increase the population by 2% we multiply it by 1.02. After 5 years the population will be 2345 × 1.02 5 = 2589 (to the nearest whole) What will the population be after 10 years? After 5 years the population will be 2345 × 1.02 10 = 2859 (to the nearest whole)

41. Repeated percentage change The car costs £24 000 in 2005. How much will it be worth in 2013? To decrease the value by 15% we multiply it by 0.85. After 8 years the value of the car will be £24 000 × 0.85 8 = £6540 (to the nearest pound) The value of a new car depreciates at a rate of 15% a year. There are 8 years between 2005 and 2013. Click Here To Download Free Exam papers for GCSE Maths