14 PERCENTAGES. Before you start. Objectives

Transcription

1 14 PERCENTAGES One of the most common percentages used to compare schools is the number of students getting five GCSEs grade A* C. Percentages can also be used to compare results across the sexes; traditionally, girls have outdone boys but the gap between the sexes looked as if it was beginning to close in 2009, with 70.5% of girls getting five GCSEs grade A* C, compared to 63.6% of boys. This is the smallest gap since Objectives Before you start In this chapter you will: find a percentage of a quantity find quantities after a percentage increase or decrease express one quantity as a percentage of another find an amount after repeated percentage changes find the original amount after a percentage change. You should know: how to find a fraction of a quantity how to convert between fractions and decimals that percent means out of how to write a percentage as a fraction or a decimal.

2 14.1 Working out a percentage of a quantity 14.1 Working out a percentage of a quantity Objectives You can convert between fractions, decimals and percentages. You can find a percentage of a quantity. Why do this? Percentages are a part of our everyday language. Banks pay interest as a percentage of the money in your bank account. Tax is paid as a percentage of money earned. Get Ready 1. Write these percentages as i fractions in their simplest form ii decimals. a 20% b 12 1_ 1_ % c 60% d % Key Points There are a number of different methods that can be used to work out a percentage of an amount. When using a calculator, first change the percentage to a fraction or decimal then multiply the amount by the fraction or decimal. When not using a calculator, first work out either 10% or 1% of the amount then build up the percentage. Example 1 Colin invests The interest rate is 3% per year. How much interest will Colin receive after 1 year? Method 1 3 3% Convert the percentage to a fraction Multiply the amount by the fraction. Interest will be Watch Out! As the answer is an amount of money, remember to give two decimal places in the answer. Method 2 3% Interest will be Method 3 1% of % of Interest will be Convert the percentage to a decimal. Multiply the amount by the decimal. Work out 1% of the amount. 1 As 1%, divide by. 3% 3 1% so multiply 18.5 by 3. Examiner s Tip Although Method 3 can be used to answer the question, on a calculator paper it is more sensible to use either Method 1 or Method 2 as candidates often make arithmetical errors using the build-up method. 223

3 Chapter 14 Percentages Exercise 14A Questions in this chapter are targeted at the grades indicated. 1 Work out a 30% of 600 b 15% of 40 kg c 5% of d 8% of 62 kg e 20% of 30 f 30% of 150 g 4.2% of 60 km h 17 1_ % of There are 150 shop assistants in a large store. 8% of the shop assistants are male. How many of the shop assistants are male? 3 Danya invests 250. The interest rate is 4% per year. How much interest will she receive after 1 year? 4 A shop has 4600 DVDs. 23% of the DVDs are thrillers. How many of the DVDs in the shop are thrillers? D AO2 5 There are 154 students in Year of these students are girls. 50% of the girls and 10% of the boys attend Spanish lessons. What fraction of these Year 11 students attend Spanish lessons? Give your fraction in its simplest form Finding the new amount after a percentage increase or decrease Objective You can find quantities after a percentage increase or decrease. Why do this? Percentages are often used when a shop has a sale or is offering a discount. A pay rise can be given as a percentage of current earnings. Get Ready 1. Work out 25% of Work out 20% of Write 5 as a fraction of 20. Key Points There are two methods that can be used to increase an amount by a percentage. You can find the percentage of that number and then add this to the starting number. You can use a multiplier. There are two methods that can be used to decrease an amount by a percentage. You can find the percentage of that number and then subtract this from the starting number. You can use a multiplier. 224 multiplier

4 14.2 Finding the new amount after a percentage increase or decrease Example 2 Hugh s salary is a year. His salary is increased by 4%. Work out his new salary. Method 1 4 4% of The increase in his salary is 0. Add the increase to his original salary. Hugh s new salary is Method 2 % 4% 104% 104% Hugh s new salary is His new salary is 104% of is the multiplier. Multiply by (This increases by 4%.) Example 3 a Write down the single number that you can multiply by to increase an amount by 12.5%. b Increase 56 by 12.5%. a % 12.5% 112.5% 112.5% b Use the multiplier worked out in part a. Exercise 14B 1 Write down the single number you can multiply by to work out an increase of a 64% b 3% c 14% d 40% e 13.4% f 12 1_ % g 15% h 2.36% 2 2 a In order to increase an amount by 40%, what single number should you multiply by? b The cost of a theatre ticket is increased by 40% for a special concert. What is the new price if the normal price was 15.40? 3 The table shows the salaries of three workers. Each worker receives a 4.2% salary increase. Work out the new salary of each worker. Helen Tom Sandeep D 225

5 Chapter 14 Percentages D 4 Jenny puts 600 into a bank account. At the end of one year 3.5% interest is added. How much is in her account at the end of 1 year? 5 a Increase 120 by 20%. b Increase 56 kg by 25%. c Increase 2.4 m by 16%. d Increase 1240 by 10.5%. e Increase 126 cm by 2%. Example 4 The value of a car depreciates by 15% each year. The value of a car when new is Work out the value of the car after 1 year. Depreciates means that the value of the car decreases. Method 1 15% of The depreciation in 1 year is 2. Subtract to work out the new value. Value after 1 year Method 2 % 15% 85% 85% The final value is 85% of the original value is the multiplier. Multiply the original amount by Value after 1 year Exercise 14C 1 Write down the single number you can multiply by to work out a decrease of a 7% b 20% c 16% d 27% e 5.6% f 2 1_ 2 % g 7 1_ 4 % h 0.8% D 2 In a sale all prices are reduced by 15%. Work out the sale price of each of the following: a a television set that normally costs 300 b a CD player that normally costs 40 c a computer that normally costs Alan weighs 82 kg before going on a diet. He sets himself a target of losing 5% of his original weight. What is his target weight? 226 depreciates

6 14.3 Working out a percentage increase or decrease 4 A holiday normally costs 850. It is reduced by 12%. How much will the holiday now cost? C 5 Ria buys a car for The value of the car depreciates by 20% each year. Work out the value of the car at the end of: a 1 year b 2 years Working out a percentage increase or decrease Objectives You can express one quantity as a percentage of another. You can find percentage loss or profit. Why do this? To work out how well a business is doing, you might want to work out what percentage profit or loss they have made. Get Ready 1. Work out Work out Write 45 out of 200 as a fraction. Key Points To write one quantity as a percentage of another quantity: write down the first quantity as a fraction of the second quantity convert the fraction to a percentage. Percentage problems sometimes involve percentage profit or percentage loss, where: profit (or increase) percentage profit (or increase) % original amount loss (or decrease) percentage loss (or decrease) % original amount Example 5 a Convert 11 out of 20 to a percentage. b Convert 23 cm out of 4 m to a percentage. a Write the first number as a fraction of the second number % To convert a fraction to a percentage, multiply by. b 4 m cm % Multiply by to convert 4 m into centimetres. Convert the fraction to a percentage. Watch Out! When working with quantities in different units, first make sure that all the units are the same. 227

7 Chapter 14 Percentages Exercise 14D D 1 Write: a 3 as a percentage of 6 b 2 kg as a percentage of 8 kg c 4p as a percentage of 10p d 8 cm as a percentage of 40 cm e 60p as a percentage of 2.40 f 15 mm as a percentage of 6 cm g 36 minutes as a percentage of 1 hour h 50 cm as a percentage of 4 m. 2 Janet scored 36 out of 40 in a German test. Work out her score as a percentage. 3 Jerry took 60 bottles to a bottle bank. 27 of the bottles were green. What percentage of the bottles were green? 4 A 40 g serving of cereal contains 8 g of protein, 24 g of carbohydrates, 4.5 g of fat and 3.5 g of fibre. What percentage of the serving is: a protein b carbohydrates c fat d fibre? Example 6 Karen bought a car for One year later, she sold it for 840. Work out her percentage loss % Her percentage loss is 30%. Subtract the selling price from the original price to find her loss. loss Write down the fraction original price Multiply 360 by to change it to a percentage A02 A03 Example 7 Tony bought a box of 24 oranges for 4. He sold all the oranges for 21p each. Work out his percentage profit p p profit Work out the total amount, in pence, Tony received from selling all the oranges. Subtract the original price from the selling price to find his profit in pence profit original price Write down the fraction % Multiply 104 by to change it to a percentage. 400 Percentage profit 26%. 228

8 14.4 Working out compound interest Exercise 14E 1 Calculate the percentage increase or decrease to the nearest 1%: a 24 to 36 b 12.5 kg to 20 kg c 45 cm to 39.5 cm d 2 minutes to 110 seconds. C * 2 In a sale, the price of a clock is reduced from 32 to Work out the percentage reduction. 3 Rob bought a crate of 40 melons for 30. He sold all the melons for 1.05 each. Work out his percentage profit. 4 David owns three shops selling DVDs. He tells the staff in each of the shops that some of them will receive a bonus. He will give the bonus to the staff who work in the shop that has the biggest percentage increase in the number of DVDs sold from the first half to the second half of the year. DVDs sold Jan Jun DVDs sold Jul Dec Shop A Shop B Shop C Which shop should receive the bonus? You must show how you decided on your answer. 5 Martin goes to a discount centre. He buys 10 trays of drinks. Each tray holds 24 cans of cola and costs Martin sells 150 cans of cola at a fair for 55p each. He sells the rest of the cans for 35p each the next day at a car boot sale. Work out the profit or loss percentage that Martin makes. 6 A badminton club has 44 members. Each member pays 85 per year as a membership fee. The club has to pay a total of 3700 to the sports centre to hire the badminton courts. The sports centre decides to increase the cost of hiring courts by 8.5%. The badminton club will have 46 members next year. Work out the smallest possible percentage rise in club membership fees so that the club can afford to pay the sports centre. Give your answer correct to 1 decimal place. AO2 AO Working out compound interest Objective You can find an amount after repeated percentage changes. Why do this? Percentage changes can happen over a period of time. You may want to work out how much money you will have in two years if you put in your bank account now. Get Ready 1. Write a as a power of 2 b 30% as a decimal c 125% as a decimal 229

9 Chapter 14 Percentages Key Points Banks and building societies pay compound interest. At the end of the first year, interest is paid on the money in an account. This interest is then added to the account. At the end of the second year, interest is paid on the total amount in the account, that is, the original amount of money plus the interest earned in the first year. At the end of each year, interest is paid on the total amount in the account at the start of that year. For example, if 200 is invested in a bank account and interest is paid at a rate of 5% then Year Amount at start of year Amount plus interest To calculate compound interest, find the multiplier: Amount after n years original amount multiplier n Total amount at year end Example is invested for 2 years at 5% per annum compound interest. Work out the total interest earned over the 2 years. Method 1 % 5% 105% 105% Work out the multiplier for an increase of 5%. Multiply the original amount by to find the amount in the account after 2 years. Watch Out! Read this type of question carefully to determine whether you need to work out just the interest or the final amount in the account. The total interest earned over the 2 years is 410. Subtract the original amount to find the interest. Method Work out the interest in the first year. Add the interest to the original amount Work out the interest in the second year. Find the total interest. The total interest earned over the 2 years is compound interest

10 14.4 Working out compound interest Example 9 a Each year the value of a car depreciates by 30%. Find the single number, as a decimal, that the value of the car can be multiplied by to find its value at the end of 4 years. b The value of a house increases by 16% of its value at the beginning of the year. The next year its value decreases by 3% of its value at the start of the second year. Find the single number, as a decimal, that the original value of the house can be multiplied by to find its value at the end of the 2 years. a % 30% 70% 70% is the single number. b % 16% 116% 116% % 3% 97% 97% is the single number. Find the multiplier that represents a decrease of 30%. The depreciation is over 4 years so the single multiplier is 0.7 raised to the power of 4. Find the multiplier for an increase of 16%. Find the multiplier for a decrease of 3%. The value increases and then decreases so find the product of the two multipliers. Example 10 The value of a machine when new is The value of the machine depreciates by 10% each year. Work out its value after 3 years. Method 1 % 10% 90% 90% Work out the multiplier for a decrease of 10%. Multiply the value when new by to find the value after 3 years. The value of the machine after 3 years is Method The value of the machine after 3 years is

11 Chapter 14 Percentages Exercise 14F D C B AO2 1 Work out the multiplier as a single decimal number that represents: a an increase of 20% for 3 years b a decrease of 10% for 4 years c an increase of 4% followed by an increase of 2% d a decrease of 35% followed by a decrease of 20%. 2 0 is invested for 2 years at 5% per annum compound interest. Work out the total amount in the account after 2 years. 3 Mrs Bell buys a house for In the first year, the value of the house increases by 16%. In the second year, the value of the house decreases by 4% of its value at the beginning of that year. a Write down the single number, as a decimal, that the original value of the house can be multiplied by to find its value after 2 years. b Work out the value of the house after the 2 years. 4 Ben says that an increase of 40% followed by an increase of 20% is the same as an increase of 60%. Is Ben correct? You must give a reason for your answer. 5 Jeremy deposits 3000 in a bank account. Compound interest is paid at a rate of 4% per annum. Jeremy wants to leave the money in the account until there is at least 4000 in the account. Calculate the least number of years Jeremy must leave his money in the bank account Calculating reverse percentages Objective You know how to find the original amount given the final amount after a percentage increase or decrease. Why do this? If you found a book marked 60% off in a sale, you can use reverse percentages to work out how much it originally cost. Get Ready 1. Write down the multiplier for: a an increase of 15% b a decrease of 15% c an increase of 4% d a decrease of 4%. Key Points There are two methods that can be used to find the original amount if the final amount after a percentage increase or decrease is known using reverse percentages. A flow diagram can be used to represent a percentage change using multipliers. multiplier original price final amount Drawing a second flow diagram reversing the direction and using the inverse operation shows that to find the original price from the final amount you divide by the multiplier. multiplier original price final amount 232 reverse percentages

12 14.5 Calculating reverse percentages Example 11 In a sale, all prices are reduced by 20%. The sale price of a jacket is Work out the original price of the jacket. Method 1 % 20% 80% Find the multiplier for a decrease of 20%. Original price The original price of the jacket was 42. Divide by the multiplier to find the original price. Method 2 % 20% 80% % So 1% represents 80% of the original price. Divide by 80 to find the value of 1%. So, original price The original price of the jacket was 42. The original price is % so multiply the amount that represents 1% by. (Check: ) Example 12 Method 1 % % 117.5% The price of a new washing machine is 376. This price includes Value Added Tax (VAT) at 17 1_ 2 %. Work out the cost of the washing machine before VAT was added. The original cost was increased by 17.5% so find the multiplier for an increase of 17.5%. Original price Divide by the multiplier to find the amount without VAT. 320 The cost of the washing machine before VAT was added was 320. (Check: ) Method 2 % 17.5% 117.5% 376 represents 117.5% of the original cost % So 1% Divide 376 by to find the value of 1%. So original price The cost of the washing machine before VAT was added was 320. The original cost is % so multiply the amount that represents 1% by. (Check: ) 233

13 Chapter 14 Percentages Exercise 14G B 1 Employees at a firm receive a pay increase of 4%. After the pay increase, Linda earns How much did Linda earn before the pay increase? 2 The price of a new television set is 329. This price includes Value Added Tax (VAT) at 17 1_ 2 %. Work out the cost of the television set before VAT was added. 3 A holiday is advertised at a price of 403. This represents a 35% saving on the brochure price. Work out the brochure price of the holiday. 4 Kunal pays tax at a rate of 22%. After he has paid tax, Kunal receives per week. How much does Kunal earn per week before he pays tax? 5 In one year, the population of an island increased by 3.2% to Work out the population of the island before the increase. 6 Tasha invests some money in a bank account. Interest is paid at a rate of 8% per annum. After 1 year, there is in the account. How much money did Tasha invest? Chapter review There are a number of different methods that can be used to work out a percentage of an amount. When using a calculator, first change the percentage to a fraction or decimal then multiply the amount by the fraction or decimal. When not using a calculator, first work out either 10% or 1% of the amount and build up the percentage. There are two methods that can be used to increase (or decrease) an amount by a percentage. You can find the percentage of that number and then add this to (or subtract this from) the starting number. You can use a multiplier. To write one quantity as a percentage of another quantity, write down the first quantity as a fraction of the second quantity, then change the fraction to a percentage. Percentage problems sometimes involve percentage profit or percentage loss, where: profit (or loss) percentage profit (or loss) original amount % To calculate compound interest, find the multiplier: Amount after n years original amount multiplier n There are two methods that can be used to find the original amount if the final amount after a percentage increase or decrease is known using reverse percentages. A flow diagram can be used to represent a percentage change using multipliers. original price Drawing a second flow diagram reversing the direction and using the inverse operation shows that to find the original price from the final amount you divide by the multiplier. original price multiplier multiplier final amount final amount 234

14 Chapter review Review exercise 1 Work out: a 30% of 800 b 25% of 20 kg c 15% of 70 kg d 17 1_ % of A farmer has a rectangular field. He makes the field 20% longer. If he wants to keep the same area, what would he have to reduce the width by? * 3 A man invests with a guaranteed compound interest rate of 4%. How long will it be before he has doubled his money? 4 The same barbeque set is sold in three different shops. Here are the price labels shown on each barbeque set. Shop A Shop B Shop C (inc. VAT) (inc. VAT) Get 1_ off when you buy 4 Now with 20% discount Plus 15% VAT this barbeque set * Which barbeque is the best buy? 5 Barry has ben asked to compare the pay for four similar jobs advertised in a newspaper. Able Computer Sales Sales Assistant You will spend time in the field, working both from our Manchester headquarters and from home in the North West region. Pay: per annum Compu Systems Sales Agent As a sales agent your pay will be 1800 per month, plus commission of 1% of monthly sales. You can expect to make monthly sales to a minimum value of Which job pays the most? 6 Linda s mark in a Maths test was 36 out of 50. Find 36 out of 50 as a percentage. 7 Jessica s annual income is She pays 1_ of the in rent. 4 She spends 10% of the on clothes. Work out how much of the Jessica has left. Beta IT Support Sales Consultant Full time: 30 hours per week Pay: 15 per hour Tele-sales based in our new offices. Daily hours variable. Digital Hardware Sales Adviser You will be part of a team with a salary of per annum team bonus. Team bonus last year was 20% of salary. D 8 A hotel has 56 guests. 35 of the guests are male. a Work out 35 out of 56 as a percentage. 40% of the 35 male guests wear glasses. b Write the number of male guests who wear glasses as a fraction of the 56 guests. Give your answer in its simplest form. Nov

15 Chapter 14 Percentages D B AO2 9 In April 2004, the population of the European Community was 376 million. In April 2005, the population of the European Community was 451 million. Work out the percentage increase in population. Give your answer correct to 1 decimal place. Nov Bill buys a new lawn mower. The value of the lawn mower depreciates by 20% each year. a Bill says after 5 years the lawn mower will have no value. Bill is wrong. Explain why. Bill wants to work out the value of the lawn mower after 2 years. b By what single decimal number should Bill multiply the value of the lawn mower when new? Nov In a sale normal prices are reduced by 20%. Andrew bought a saddle for his horse in the sale. The sale price of the saddle was 220. Calculate the normal price of the saddle. 12 The value of a car depreciates by 35% each year. At the end of 2007, the value of the car was Work out the value of the car at the end of Exam Question Report 13 Nimer got a pay rise of 5%. His new pay was 1680 per month. Work out his pay per month before he got this pay rise. 86% of students answered this question poorly because they did not find the value of the reduction first. 14 Jim is a plumber. He has to work out the VAT on some equipment. VAT is charged at 17 1_ 2 %. The total cost of the equipment including VAT is Calculate how much the VAT was. Nov 2008 AO2 15 Sophie is offered the following pay deals: A 5% increase this year, followed by a 4% increase next year B 4 1_ 1_ % increase this year, followed by 4 % increase next year 2 2 Which offer should Sophie accept? 236