Upcoming SlideShare
×

# Calculate percentage change

4,990
-1

Published on

Calculate percentage change

Published in: Education, Technology
1 Like
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total Views
4,990
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
91
0
Likes
1
Embeds 0
No embeds

No notes for slide
• Explain that we can easily write a whole number percentage as a fraction over 100. However, if there is a common factor between the numerator and the denominator, we must cancel the fraction down to its lowest terms. If the percentage is not a whole number, as in the last example, we must find an equivalent fraction with a whole number numerator and denominator. We can then cancel if necessary.
• Explain that to convert a percentage to a decimal we simply divide the percentage by 100. This can be done mentally.
• If the denominator of the fraction is a factor of 100 we can make an equivalent fraction over a hundred. Remind pupils that as long as we multiply the numerator and the denominator of a fraction by the same number it will have the same value. Talk through both examples. For the second example, we write the fraction as an improper fraction first. An alternative would be to recognise that 1 = 100% and then convert 7 / 25 to 28 / 100 . Adding 100% and 28% gives 128%.
• Explain that we can also multiply fractions by 100% to convert them to percentages. Stress that 3 / 8 is equal to 3 / 8 × 100% because multiplying by 100% is the same as multiplying by 1. However, 3 / 8 is not equal to 3 / 8 × 100. Stress to pupils the importance of writing the % sign at every stage of the calculation. Talk through the example and tell pupils that this method will work with any fraction regardless of the denominator.
• Explain again that 100% means the same as ‘1 whole’ or ‘all of it’. So multiplying by 100% is the same as multiplying by 1. The amount remains unchanged.
• Encourage pupils to to check the answer given by the calculator by estimating the given fraction as a fraction of 100. For example, if the fraction is less than 1 / 2 we would expect the corresponding percentage to be less than 50%.
• Talk through the example on the board. Start by asking pupils to tell you what fraction of the sweets are orange and then go through each step on the board.
• Stress that when we write one amount as a percentage of another we must use the same units.
• This slide demonstrates how to find a percentage using a fractional operator.
• This slide demonstrates how to find a percentage using a decimal operator. Each step should be calculated mentally.
• Discuss suitable values for each estimate. Explain that it is not always necessary to round to the nearest 10% because other percentages, such as 25% or 75% might be easier to work out.
• Explain that by converting the percentage to a decimal we can also work out percentages on a calculator. Discuss how we could estimate the answer before using a calculator. For example, we can estimate the answer by working out 40% of £65. 10% of £65 is £6.50 so 40% is 4 × £6.50 = £26.
• Ask pupils to estimate 57% of £80. For example, 60% of £80 = 6 × £8 = £48. Then ask pupils how they think we could use 57% as a fraction to work out 57% of £80 using a calculator. Reveal the correct sequence of key presses on the board. This method requires more key presses (it is less efficient). However, many pupils find it easier to remember because we can think of ‘per cent’ as ‘÷ 100’ and ‘of 80’ as ‘× 80’.
• Ask pupils if they know what 37.5% is as a fraction ( 3 / 8 ). Ask pupils to estimate what 37.5% of £59 is. Discuss the unitary method for calculating percentages. Discuss how we could write 22.125 to the nearest penny.
• Explain that to calculate a percentage increase or decrease we start by working out the actual increase or decrease by finding the difference between the new amount and the old amount. This is then written as a fraction of the original amount and converted to a percentage by multiplying it by 100%. Compare this method to writing one number as a percentage of another (see slides 15 and 16 for examples).
• Stress that the actual increase must be found as a fraction of the original amount and not the new amount. Ask pupils if they can convert 0.7 / 3.5 to a percentage mentally before revealing the answer.
• Explain that percentage profit and percentage loss work in the same way as percentage increase and decrease as the next two examples show.
• Ask pupils if the number of shares bought makes any difference to the dealer’s percentage loss. Establish that it makes a difference to her actual loss but not to her percentage loss. This is because the percentage is a proportion of the amount lost.
• Stress that this method requires two calculations. One to find the actual increase and one to add this amount to the original value. This method is most useful when we need to know the actual value of the increase. In most cases, however, we only need to know the end result. Although this method is reliable when done correctly, there is more room for error because it is easy to forget to do the second part of the calculation. Completing the calculation in one step as shown in Method 2 is more efficient.
• Remind pupils that 100% of the original amount is like one times the original amount: it remains unchanged. 100% means ‘all of it’. When we add on 20% we are adding on 20% to the original amount, 100%, to make 120% of the original amount. Ask pupils why might it be better to find 120% than to find 20% and add it on. Establish that it’s quicker because we can do the calculation in one step. Discuss how 120% can be written as a decimal.
• Talk through each example as it appears. Explain the significance of finding a 17.5% increase in relation to VAT.
• Stress that, as with the similar method demonstrated for percentage increases, this method requires two calculations. One to find the reduction and one to subtract this amount from the original price.
• Remind pupils again that 100% of the original amount is like one times the original amount, it remains unchanged. 100% is ‘all of it’. Explain that when we subtract 30% we are subtracting 30% from the original amount, 100%, to make 70% of the original amount. Ask pupils why might it be better to find 70% than to find 30% and take it away. Again, establish that it’s quicker. We can do the calculation in one step. Discuss how 70% can be written as a decimal. Conclude that to decrease an amount by 30% we multiply it by 0.7. Give more verbal examples as necessary.
• Talk through each example as it appears. Pupils should be able to confidently subtract any given amount from 100 in their heads.
• In this example, establish that we must have multiplied the original amount by 0.85 (100% – 15%) to get the new price. To get back to the original amount we can use inverse operations and divide the new amount by 0.85 to get back to the original amount. Explain this process using equations with p for the unknown original price.
• Explain how we can use inverse operations to find the original price before the discount using a diagram.
• Some pupils may find it easier to remember a unitary method as shown by this example.
• It is very common to assume that when two percentage changes are combined, the total percentage change can be found by adding. Discuss why this is incorrect.
• Pupils may wish to discuss this problem by giving the original investment a value, £100 say. If this increases by 10% Jenna will have £110. When £110 is reduced by 10% Jenna will have £99. This is equivalent to a 1% loss overall.
• We can see that the actual increase gets larger each time. In the first year £25 is added on. In the second year it’s £26.25, in the third year it’s £27.56 and in the fourth year it’s £28.94. This is an example of exponential growth. Ask pupils to verify that £500 × 1.05 4 = £607.75 using the x y key on their calculators.
• Again ensure that pupils can use the x y key on their calculators to complete these calculations.
• Ask pupils to calculate each answer using the x y key on their calculators.
• Explain that a percentage depreciation is equivalent to a percentage decrease.
• ### Calculate percentage change

1. 1. Download Free GCSE Maths Past Papers and video lessons <ul><li>Why you should download past exam papers </li></ul><ul><li>Get used to exam style questions.it ’s a very well known fact that practicing exam questions puts you more at ease with the way they are formatted. </li></ul><ul><li>Improve your confidence.The only way to improve your confidence is by learning to answer exam questions. </li></ul><ul><li>Identify similar questions & practice them until you can do them easily.after practicing few papers, you will notice some topics tend to show up more often than others.  This is a perfect opportunity for you to take note and revise them really well. </li></ul><ul><li>Assess yourself and see which topic you need to revise.Once you have completed a paper, see how many marks you are getting on average.  This will give you an indication of what grade you might end up with. </li></ul><ul><li>Make revision notes from the questions.If you found a question slightly tricky, find the solution and make notes for revision. </li></ul>Download
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
42. 42. Download Free GCSE Maths Past Papers and video lessons <ul><li>Why you should download past exam papers </li></ul><ul><li>Get used to exam style questions.it ’s a very well known fact that practicing exam questions puts you more at ease with the way they are formatted. </li></ul><ul><li>Improve your confidence.The only way to improve your confidence is by learning to answer exam questions. </li></ul><ul><li>Identify similar questions & practice them until you can do them easily.after practicing few papers, you will notice some topics tend to show up more often than others.  This is a perfect opportunity for you to take note and revise them really well. </li></ul><ul><li>Assess yourself and see which topic you need to revise.Once you have completed a paper, see how many marks you are getting on average.  This will give you an indication of what grade you might end up with. </li></ul><ul><li>Make revision notes from the questions.If you found a question slightly tricky, find the solution and make notes for revision. </li></ul>Download
43. 43. Download Free GCSE Maths Past Papers and video lessons <ul><li>Why you should download past exam papers </li></ul><ul><li>Get used to exam style questions.it ’s a very well known fact that practicing exam questions puts you more at ease with the way they are formatted. </li></ul><ul><li>Improve your confidence.The only way to improve your confidence is by learning to answer exam questions. </li></ul><ul><li>Identify similar questions & practice them until you can do them easily.after practicing few papers, you will notice some topics tend to show up more often than others.  This is a perfect opportunity for you to take note and revise them really well. </li></ul><ul><li>Assess yourself and see which topic you need to revise.Once you have completed a paper, see how many marks you are getting on average.  This will give you an indication of what grade you might end up with. </li></ul><ul><li>Make revision notes from the questions.If you found a question slightly tricky, find the solution and make notes for revision. </li></ul>Download
1. #### A particular slide catching your eye?

Clipping is a handy way to collect important slides you want to go back to later.