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Stage 7 Chapter Percentages

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Objectives Solve percentage problems involving increasing and decreasing by using a multiplier

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**Calculating percentages using fractions**

Remember, a percentage is a fraction out of 100. Find 15% of 90 15% of 90, means “15 hundredths of 90” or 3 9 15 100 × 90 = 15 × 90 100 20 This slide demonstrates how to find a percentage using a fractional operator. 2 = 27 2 = 13 1 2

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**Calculating percentages using decimals**

What is 4% of 9? We can also calculate percentages using an equivalent decimal operator. 4% of 9 = 0.04 × 9 = 4 × 9 ÷ 100 This slide demonstrates how to find a percentage using a decimal operator. Each step should be calculated mentally. = 36 ÷ 100 = 0.36

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**Complete the activity Calculating percentages**

Ask pupils to calculate the given percentages of the central amount using an appropriate written or calculator method. Make the activity more challenging by revealing the proportion and asking what percentage of the central amount it represents.

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

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**Percentage increase Method 2**

If we don’t need to know the actual value of the increase we can find the result in a single calculation. We can represent the original amount as 100% like this: 100% 20% 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. When we add on 20%, we have 120% of the original amount. Finding 120% of the original amount is equivalent to finding 20% and adding it on.

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Percentage increase So, to increase £ by 20% we need to find 120% of £ 120% of £ = 1.2 × £ = £ 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.

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**Percentage increase Here are some more examples using this method:**

Increase £50 by 60%. Increase £86 by 17.5%. 160% × £50 = 1.6 × £50 117.5% × £86 = 1.175 × £86 = £80 = £101.05 Increase £24 by 35% Increase £300 by 2.5%. Talk through each example as it appears. Explain the significance of finding a 17.5% increase in relation to VAT. 135% × £24 = 1.35 × £24 102.5% × £300 = 1.025 × £300 = £32.40 = £307.50

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

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**Percentage decrease Method 2**

We can use this method to find the result of a percentage decrease in a single calculation. We can represent the original amount as 100% like this: 70% 100% 30% 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. When we subtract 30% we have 70% of the original amount. Finding 70% of the original amount is equivalent to finding 30% and subtracting it.

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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.

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**Percentage decrease Here are some more examples using this method:**

Decrease £65 by 20%. Decrease £320 by 3.5%. 80% × £65 = 0.8 × £65 96.5% × £320 = 0.965 × £320 = £52 = £308.80 Decrease £56 by 34% Decrease £1570 by 95%. Talk through each example as it appears. Pupils should be able to confidently subtract any given amount from 100 in their heads. 66% × £56 = 0.66 × £56 5% × £1570 = 0.05 × £1570 = £36.96 = £78.50

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**Complete the activity Percentage increase and decrease**

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