4 Percentages Chapter notes


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1 4 Percentages Chapter notes GCSE Specification concepts and skills Find a percentage of a quantity (N o): 4. Use percentages to solve problems (N m): 4., 4.2, 4., 4.4 Use percentages in reallife situations: VAT, value of profit or loss, simple interest, income tax calculations, find prices after a percentage increase or decrease, compound interest, depreciation, percentage profit and loss (N m): 4., 4.2, 4., 4.4 Use decimals to find quantities (N o): 4.2 Understand the multiplicative nature of percentages as operators (N o): 4.2 Use a multiplier to increase or decrease by a percentage in any scenario where percentages are used (N o): 4.2 Calculate and represent repeated proportional change using a multiplier raised to a power (N m, N o): 4. Express a given number as a percentage of another number (N o): 4.4 Support materials Support sheet 4. You can convert between fractions, percentages and decimals and order them Prior key knowledge, skills and concepts Students should already be able to: convert a fraction into a decimal find equivalent fractions find a fraction of a quantity multiply and divide fractions and integers write one quantity as a fraction of another quantity. Main teaching and learning Resources Calculators Resource sheet 4. [4.] Elicit from pupils which method(s) they have used previously to work out the percentage of a quantity and make a judgement about which method(s) to support when using worked examples in class. Complete Example with them using the chosen method(s). Remind the students of possible contexts in which percentages occur, for example in pay rises, income tax, VAT, hire purchase deposits and discounts. Ensure they are confident in using their calculator to work out percentages. Complete Example 2. [4.2] Discuss with students how to tackle problems where there is a need to add or subtract a percentage of a quantity and, if appropriate, explain the advantage of using decimal multipliers to solve such problems. Model the solution of problems like this using Examples and 4. [4.] [H] Remind students of the difference between simple interest and compound interest. [H] Use Example to demonstrate how to work out compound interest. You could compare the interest gained in this example with the simple interest you would get with the same amount of capital over the same time period. [H] Use Example 6 to show pupils how a similar process works for questions involving depreciation. Discuss other contexts in which a compound percentage reduction might occur, e.g. decrease in population. [4.4] Remind students how to work out one quantity as a percentage of another and discuss possible contexts where this would be used, e.g. marks in a test, nutrition information on food labels, percentage profit or loss. Use Examples 8 and 9 to model possible examination questions. 0
2 4 Percentages Chapter notes Exam tips Be careful not to read of as off. Check whether you just have to work out a percentage or whether the question requires you to then add or subtract your answer from a quantity. Be careful with decimal places when converting decimals to percentages. For example, 0.8 is equivalent to 80% and 0.08 is equivalent to 8%. Take care when changing a fraction into a percentage when the denominator is neither a factor nor a multiple of 0. Remember the link between working out the fraction as a decimal and converting it to a percentage. Do lots of practise with calculator methods. Mistakes in calculator use are a common cause of dropped marks. Do not try to use a buildup method for percentages when you are allowed a calculator. Although this is a valid method, it is easy to make errors in the arithmetic. You should use an appropriate method depending on the type of question. Remember to use the multiplier 0.8, not 0., to decrease a quantity by %. [H] When working out percentage profit or loss (or increase or decrease), remember not to use the final amount in this calculation. You should use the original amount. [H] Make sure you understand the difference between simple interest and compound interest. Always check your answers to help eliminate errors. Enrichment Students could investigate the tax system and work out how much tax is paid by employees earning pa, pa and pa. Uptodate tax rates can be found at Students could research the interest rates being offered by banks and building societies. Choose an amount to invest and, for each rate, work out how much interest they would get after one year. More able students could be given a reverse percentage problem (e.g. If the sale price of a shirt is 2 after a % discount, what is the normal price?). What is 60 as a percentage of 40? (0%) Discuss situations that could lead to percentages greater than 0%. Plenary Ask students to write 48% as a fraction and as a decimal and explain their answers. Display a variety of questions, e.g. % of 60, 2% of 6, and for each one ask whether it is best tackled by a mental method or by using a calculator. Discuss how to find VAT using a mental method and using a calculator method. Display pictures of several items for sale, each with a price. Ask students to work out the price of each after a % reduction or after VAT at % is added. Ask students to work out simple questions involving percentage profit and loss. Peter bought a bicycle for 60 and then sold it to a friend for 7. Work out his percentage profit. (2%) Ask students how they would use a calculator to work out the result of increasing an amount by three successive increases of 4%, by two successive decreases of %, etc. Students could play or make up their own dominoes game like the one described in Resource sheet 4..
3 4 Percentages Support material Questions in this exercise are targeted at the grades indicated. 4. You can convert between fractions, percentages and decimals and order them (N l) Key Points You can put a list with fractions, decimals and percentages in order of size by changing them to the same type of number. Example Write as a decimal: a 6% b % a 6% = 6 0 b % = = = 0.0 Write the percentage as a fraction. So 6% = 0.6 So % = 0.0 Exercise A G Write the following percentages as fractions in their simplest form. a 40% b 7% c % d 90% e % f 80% g 84% h 7% G 2 64% of the spectators at a football match were male. Write down the fraction of the spectators that were male. Give your fraction in its simplest form. Example 2 Write as a percentage: Method Without using a calculator: = 6 Write as an equivalent fraction with denominator 6 = tenths written as a decimal is 0.6 So = 60% 0.6 is equivalent to 60% Method 2 Using a calculator: = Use a calculator to divide the numerator by the denominator. = 0.6 Remember 0.60 and 0.6 are equivalent decimals. So = 0.6 2
4 4 Percentages Support material Exercise B G Write the following fractions as decimals: a b 8 0 c 7 0 d 6 00 G 2 Write the following fractions as decimals. Do not use a calculator. a 4 b c 0 d 2 2 G Write the following fractions as decimals. You may use a calculator. a 7 Example b c 7 8 d 6 Write the following numbers in order of size. Start with the smallest number. 0.4 % 8 8 = 0.7 Change 8 into a decimal is already a decimal. % = 0. Change % into a decimal Write the decimals in order of size. % Write each number in its original form. Exercise C G a Write 2% as a decimal. b Write as a decimal. 4 c Which is bigger, 2% or? 4 G 2 a Write 74% as a decimal. b Write 7 as a decimal. 7 c Which is bigger, 74% or? G Write each of the following lists in order of size, starting with the smallest number. a 2 48% 0.4 b 0.7 F 4 Write each of the following lists in order of size, starting with the smallest number. a 0.2 b 2 % 68% 0.6 % %
5 4 Percentages Skills worksheet Questions in this worksheet are targeted at the grades indicated. F Safiyah got 7 out of in a spelling test. What is 7 out of as a percentage? Remember: Write one amount as a percentage of another, form a fraction, then multiply by 0. F 2 In a maths test, Laura scored 4 out of 60 and in a science test she scored 2 out of 0. In which test did she do best? Hint: Write both scores as percentages. E There are 260 students in Year 7. On Friday % of these students were absent. How many students were absent on Friday? D 4 Last year a garage sold 270 cars. % of these were new cars. Work out how many new cars the garage sold last year. D Tamsin bought a flat costing 000. When she sold the flat she made a profit of 7%. Work out how much profit Tamsin made. D 6 Work out the sale price of each item, below. Remember: Write one amount as a percentage of another, form a fraction, then multiply by D 7 Adam s monthly salary is 860. His salary is increased by 4%. Work out Adam s new monthly salary. Remember: To increase a quantity, the multiplier will be greater than. For example, to add on 0% to a quantity, use. as the multiplier. 4
6 4 Percentages Skills worksheet D 8 A stereo costs 64 plus VAT at %. Calculate the total cost of the stereo. D 9 A train ticket costs 4. The price is increased by 6.%. Work out the new price of the train ticket. D A football stadium has a capacity of At a match one weekend, 8900 people were present. Work out the percentage of the stadium that was occupied. C In a sale, prices are reduced by 2%. A TV set normally costs 42. Calculate its price in the sale. Remember: To decrease a quantity, the multiplier will be less than. For example, to subtract 40% from a quantity, use 0.6 as the multiplier. Questions 2, and 4 are Higher tier only. C 2 In a sale, the price of an MP player is reduced from 68 to.04. Work out the percentage reduction in the price of the MP player. reduction Remember: Percentage reduction = original amount 0. C 000 is invested in a bank for 2 years. Compound interest is paid at a rate of 4% per annum. How much money is in the bank after 2 years? C 4 Alan bought a car for 600. Each year the total cost of the car depreciated by %. Work out the value of the car 2 years after he bought it.
7 4 Percentages Chapter test Questions in this test are targeted at the grades indicated. F Write the following percentages as decimals. a 7% b 8% c 90% () F 2 Write 8% as a fraction in its simplest form. () G a What is 7% of 0? (2) E b What is % of 4.80? (2) E AO2 *4 In a test, Wendy scored 24 out of 60, Jenni scored 4% and Kavish scored of the marks. 2 Which of the three scored the highest in this test? You must show all your working. (4) D Carl works for hours each week in a shop. From the beginning of March, Carl s hours were reduced by %. Work out how many hours Carl works each week in March. Give your answer to the nearest hour. () D AO2/ *6 Three shops were selling the same type of washing machine. Sam s Store Was 0 Now 0% off Eli s Emporium 0 deposit plus 7 monthly payments of 2.40 Monty s Machines 8 plus % VAT Helen wants to buy the cheapest washing machine. From which shop should she buy the washing machine? You must show how you decided on your answer. (6) Questions 7, 8 and 9 are Higher tier only. C 7 Neil bought a watch for 60 and sold it a week later for 2. Work out his percentage loss. Write your answer correct to 2 significant figures. (2) C AO2/ 8 Here are the details of how income tax is calculated for the tax year : The first 647 is free of income tax. The next 0 92 is charged at %. Any further income up to is taxed at 40%. John has a taxable income of Calculate how much tax John will pay. (4) C AO2/ 9 Stephen invested 000 for three years at % compound interest. His sister Millie invested 000. The value of Millie s investment increased by 9.2% over the three years. Who made the better investment? You must show all your calculations. () (Total = 0 marks) 6
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