Year 11 Chapter 1 PDF

1. BASIC CALCULATIONS, PERCENTAGES AND RATES WHAT’S THE SCORE? Chapter problem Ben bought fish and chips for $11.25 and paid with a $50 note. The cashier gave him a $20 and a $5 note and a few coins for his change. Could the change be wrong? 1.01 Target practice 1.02 Playing darts 1.03 Percentages,...

1. BASIC CALCULATIONS, PERCENTAGES AND RATES WHAT’S THE SCORE? Chapter problem Ben bought fish and chips for $11.25 and paid with a $50 note. The cashier gave him a $20 and a $5 note and a few coins for his change. Could the change be wrong? 1.01 Target practice 1.02 Playing darts 1.03 Percentages, fractions and decimals 1.04 Order of operations 1.05 How much do I pay? 1.06 Estimating costs 1.07 Rounding numbers 1.08 Practical multiples 1.09 After the point 1.10 How much do I get? Keyword activity Solution to the chapter problem Test yourself WHAT WILL WE DO IN THIS CHAPTER? Calculate with numbers Convert between percentages and fractions/decimals Find a fraction or decimal of a quantity Add, subtract, multiply and divide using order of operations, with and without a calculator Estimate and check the reasonableness of answers Understand place value after the decimal point in decimals Round numbers, including to decimal places Solve practical problems involving numbers HOW ARE WE EVER GOING TO USE THIS? Keeping score in sporting events Shopping At work Budgeting: for yourself or for a party Home renovations iStock.com/courtneyk 1.01 Target practice Double le ub Do ub le 3 7 5 Dou 10 b le Dou b le 2 9 Do Do u 25 Dou ble Do u Double = 12 + 8 6 4 le 8 Marc’s score = 2 × 6 + 8 ub b le 1 b le The 2 red crosses on the diagram show where Marc’s throws landed. One throw landed in double 6 and the other in 8. Do Alan coaches a junior football team. He designed a ball-throwing competition to help his players improve the accuracy of their passes. Each player throws a football at the target two times each turn. The first player to reach 100 points is the winner. = 20 Exercise 1.01 Target practice 1 Without using a calculator, work out the scores for each turn. b ble Dou ble Dou ble ble Dou ble ble ble Do le ub Double D o le ub Double D o ub le Dou Dou 6 le 4 ub 8 Do le Dou Dou ub ble 5 ble Dou Double Do ble Do u 1 7 25 Dou 6 10 ble 4 ble 8 Dou 5 ble 1 7 25 Double D ou ble 9 2 3 ble u Do Dou 10 le D 6 ub d 4 Dou Double D ou ble 9 2 3 le b ou 8 ble c 6 5 Dou 4 ble 8 1 7 25 ble 5 10 Dou 1 ble 7 25 Dou 10 Double D ou ble 9 2 3 ble u Do Dou Double D ou ble 9 2 3 ble u Do Do a le ub Double D o 2 Add each set of scores as quickly as you can. a A 7 and a 9 b Double 5 and 8 c Double 6 and double 4 d Double 9 and double 5 e Bullseye and double 10 f A 3 and double 7 3 Jason’s score from 2 throws was 13. Suggest 5 different combinations of points that can make a score of 13. 4 NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 4 How many different ways can you get a score of 7 from 2 throws? List the ways. 5 To win the game, a player’s score must add to exactly 100. David’s score is 77. a How many more points does David need to score to win? b What strategy can David use to win on his next turn? 6 What is the smallest number of throws required to make a total of exactly 100? How is it achieved? 7 Courtney’s score is 81. She decided to aim for a 9 and a 10. With her first throw, Courtney missed the 9 and her ball landed on 3. How can she still make 100 on her second throw? 8 When Alan designed the positions of the numbers on the target, he didn’t put any big numbers next to each other. He put small numbers on each side of the big numbers. Why do you think Alan arranged the numbers in this way? PRACTICAL ACTIVITY DESIGN A PRACTICE COMPETITION What you have to do 1 Design a practice competition to develop the skills of children learning to play a sporting game that you enjoy. 2 Determine the rules and how you will score the game. 3 Use the practice competition with some younger children in your area. 4 How successful was the competition in developing the children’s skills? Shutterstock.com/SpeedKingz 5 Did the scoring help to improve the children’s number skills? ISBN 9780170443906 1. What’s the score? 5 1.02 Playing darts 20 5 1 18 6 11 13 14 4 9 12 Triple points 16 15 8 10 Double points 7 19 17 3 2 There are 20 numbers on a dartboard and each player throws 3 darts on their turn. If a dart lands in the outside ring of a number, the dart scores double points. It scores triple points if it lands in the inside ring. The bullseye is worth 50 points. The green ring around the bullseye is worth 25 points. Exercise 1.02 Playing darts 1 Calculate the score for each player. a b 20 1 18 12 5 20 1 19 3 17 18 19 3 17 10 15 2 16 7 8 16 15 8 10 6 11 6 11 13 14 13 14 4 9 4 9 12 5 7 2 2 What is the highest score possible with 3 darts? 3 How can you score a total of 37 using 3 darts? Suggest 2 possible ways. 6 NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 4 In the dart game ‘201’, players start with 201 points and they keep subtracting the points they score with 3 darts. The winner is the first to reach 0, but they must finish with a double. If a player’s score on one turn is more than their remaining points, no points are subtracted for that turn. Use the rules of the game to answer the rest of the questions in this exercise. With her first set of 3 darts, Madeleine scored a 7, a double 5 and a triple 11. How many points did Madeleine have left after this? 5 After the second round, Tim had 152 points left and Rebecca had 147. Who is winning the game: Tim or Rebecca? 6 Fiona had 126 points remaining before her turn. After her turn, she had 103 points. One of her darts landed on double 1 and another on 5. Where could her third dart have landed? 7 Divya’s score was 86. She threw 2 double 20s and a triple 19. What was Divya’s score after her turn? 8 Before he had his turn Santo had 136 points remaining. His first dart landed on 20, his second landed on triple 4 and his third missed the board. How many points did Santo have left after his turn? 9 Andre needs to score 34 points, finishing with a double, to win a game of 201. How could he win using: a one dart? b c 2 darts? 3 darts? 10 What is the largest number of points a player can have and still win on their next turn? Explain your answer. 11 Renata and Samantha are playing 201, writing their progress scores on a board. Turn Renata Samantha 201 201 a In what number turn did Samantha score a total of 30 points? 1 162 175 b Who threw 20 three times in the same turn? In which turn did she do it? 2 102 143 3 63 85 c What was Samantha’s biggest score in one turn? 4 39 55 5 24 29 d In which turn did Samantha throw a double 3, triple 2 and single 20? 6 0 e Renata finished when she threw a double 7. Explain how it was possible for her to finish with a double 7. f Samantha can tie the game if she can score a total of 29, finishing with a double. Explain why Samantha will need more than one throw to score 29. g What strategy do you suggest Samantha should adopt to try to tie the game? ISBN 9780170443906 1. What’s the score? 7 1.03 Percentages, fractions and decimals A percentage is a fraction whose denominator is 100. Converting a fraction or decimal to a percentage To convert a percentage to a fraction, write the percentage with a denominator of 100 and simplify if needed To convert a percentage into a decimal, divide it by 100 EXAMPLE 1 Convert each percentage to a simplified fraction. a 55% b 1 37 % 2 c 130% Solution 55 100 11 = 20 Write the percentage as a fraction over 100 and simplify. Alternatively, enter 55 a 100 on the calculator and press =. 55% = b Write 130 as a fraction over 100 and simplify. 130% = c Write 37 12 as a fraction over 100 and simplify. 37 12 % = a b c 130 100 13 = 10 3 =1 10 1 ÷ 100 to get 0.375, then use 2 the calculator key that converts a decimal into a fraction. Or enter 37 8 NELSON SENIOR MATHS 11. Essential Mathematics = 37 12 100 37 12 × 2 100 × 2 75 = 200 3 = 8 ISBN 9780170443906 EXAMPLE 2 Convert each percentage to a decimal. a 8% b c 43.6% 18 12 % Solution 8 100 = 8 ÷ 100 = 0.08 a Write the percentage as a division by 100 and simplify. Alternatively, enter 8 ÷ 100 on the calculator and press =. 8% = b Divide 43.6 by 100. 43.6% = c Divide 18 12 by 100. 18 12 % = To mentally divide a number by 100, move the decimal point 2 places to the left. 43.6 100 = 43.6 ÷ 100 = 0.436 18 12 100 = 18.5 ÷ 100 = 0.185 Converting a fraction or decimal to a percentage To convert a fraction or a decimal to a percentage, multiply it by 100 1 1 = 25% because × 100% = 25% 4 4 0.2 = 20% because 0.2 × 100% = 20% EXAMPLE 3 Convert each fraction to a percentage. 11 7 b a 20 8 Solution a Multiply the fraction by 100. 11 11 = × 100% 20 20 = 55% b Multiply the fraction by 100. 7 7 = × 100% 8 8 = 87 12 % ISBN 9780170443906 1. What’s the score? 9 EXAMPLE 4 Convert each decimal to a percentage. a b 0.65 0.267 Solution a Multiply the decimal by 100. To mentally multiply a number by 100, move the decimal point 2 places to the right. b 0.65 = 0.65 × 100% = 65% 0.267 = 0.267 × 100% = 26.7% Multiply the decimal by 100. EXAMPLE 5 Find: 1 of 45 a 5 b 7 of $32 8 c 1 of 1 year 3 d 4 of 2 kg (in grams) 5 Solution a b c d 10 To find a fraction of an amount, we multiply. Check that the answer is reasonable. 1 is a small fraction, so 9 sounds like a 5 1 reasonable answer for of 45. 5 7 is a big fraction so $28 sounds like a 8 7 reasonable answer for of $32. 8 1 Convert 1 year to months first. is less 3 1 than , so the answer should be under 6 2 months, which it is. 4 is more 5 1 than so the answer should be between 2 1000 and 2000 g, which it is. Convert 2 kg to grams first. 1 × 45 = 9 5 7 × $32 = $28 8 1 year = 12 months 1 1 of 1 year = × 12 months 3 3 = 4 months 2 kg = 2 × 1000 g = 2000 g 4 4 of 2kg = × 2000g 5 5 = 1600 g NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 Exercise 1.03 Percentages, fractions and decimals 1 Convert each percentage to a simplified fraction. Example 1 a 60% b 75% c 31% d 8% e 30% f 85% g 99% h 3% i 160% j 135% k 25% l 250% 2 Which decimal is equal to 62 12 %? Select the correct answer A, B, C or D. A B 62.12 62.5 Example C 0.625 D 0.0625 2 3 Convert each percentage to a decimal. a 18% b 82% c 2% d 50% e 120% f 51.1% g 79% h i 16.3% j 4% k 18.7% l 12 12 % 5 41 % 4 Convert each fraction to a percentage. Example 17 7 13 a b c 100 10 50 8 2 5 g h i 5 3 4 590 4 40 m n o 200 9 12 5 Convert each decimal to a percentage. 11 d 20 2 j 1 5 12 p 15 e k q 5 8 27 40 1 1 6 f l r 24 25 1 16 7 11 3 Example a 0.38 b 0.55 c 0.96 d 0.625 e 0.08 f 0.054 g 0.6 h 0.003 i 1.9 j 0.405 k 1.26 l 0.114 4 6 Copy and complete this table. Fraction Decimal a 0.65 b 0.6 Percentage c 20% d 84% e 1 2 f 1 8 36% g h ISBN 9780170443906 5 8 This table continues next page. 1. What’s the score? 11 0.73 i 1 3 j k Example 66 32 % 7 Find: 5 a 3 × 40 5 b 1 × 28 4 c 1 × 24 6 d 2 × 15 3 e 7 × 60 10 f 5 × 16 8 g 3 of 1 km (in metres) 4 h 1 1 day(in (inhours) hours) of 1day 3 i 2 of 1 L (in mL) 5 j 1 of 1 t (in kg) 8 k 5 of 1 year (in months) 6 l 7 of 1 hour (in min) 12 1.04 Order of operations What is the answer to 200 ÷ 20 × 2? Is it 20 or 50? Order of operations BIDMAS is an easy way to remember the order of operations in a mixed calculation. B I Brackets Indices D M A S ÷ × + − To solve this problem, we have special rules for the order in which we do calculations. Brackets ( ) first Indices (powers) next Then any Dividing ÷ or Multiplying ×, working from left to right Finally, any Adding + or Subtracting −, working from left to right So 200 ÷ 20 × 2 = 20, because we do × and ÷ at the same time and we work from left to right. 12 NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 EXAMPLE 6 Evaluate each mixed expression. a 60 − 2 × 52 b (12 + 5 × 3) ÷ 9 + 4 c 11 + (5 + 11) ÷ (12 ÷ 3) Order of operations Most calculators ‘know’ the order of operations. If you press the = key only at the end of the calculation, your calculator’s answer will be correct. Check the answers in this example using a calculator. Solution 60 − 2 × 52 = 60 − 2 × 25 The 5 is squared, so do indices first. a Next comes multiply. = 60 − 50 Subtraction comes last. = 10 Do × before the + inside the brackets first. b (12 + 5 × 3) ÷ 9 + 4 = (12 + 15) ÷ 9 + 4 Then do the ÷. = 27 ÷ 9 + 4 The + 4 comes last. =3+4 =7 11 + (5 + 11) ÷ (12 ÷ 3) = 11 + 16 ÷ 4 Do both sets of brackets first. c Then do the ÷. = 11 + 4 The + outside the brackets comes last. = 15 Exercise 1.04 Order of operations 1 Is each statement true or false? Write the correct value for any false statements. a 4 + 2 × 3 = 18 b 12 − 3 × 4 = 0 2 c 20 ÷ 2 + 2 = 12 d 6 + 18 ÷ 3 = 12 e 2 × 4 = 64 f 48 ÷ 4 × 3 = 4 g 20 − 5 + 8 = 7 h 5 × (20 − 3 × 4) = 40 i 2 × 53 = 250 Example 6 2 Find the value of each expression. Check your answers using a calculator. a 18 − 3 × 5 b 24 ÷ (5 + 3) c 8 × 3 − 10 ÷ 5 2 f 36 ÷ 12 ÷ 3 i 4 × (7 − 2) ÷ (32 + 1) d (2 + 10) × (12 − 9) e 3 × (1 + 4) g 30 ÷ 5 × 2 h 40 × 2 ÷ 8   j 42 + 52 − 3 × 9 k 10 + 5 × 6 l 300 − 20 × 8 n 10 + 42 o 5 × (12 − 3 × 2) m 120 ÷ 4 × 5 3 Copy each statement and insert brackets to make the statement true. a 4 + 7 × 5 = 55 b 60 ÷ 5 + 7 = 5 c 3 × 22 = 36 d 6 + 8 × 9 − 5 = 56 e 3 × 4 + 5 × 2 = 34 f 28 − 4 × 5 × 2 = 16 ISBN 9780170443906 1. What’s the score? 13 4 When Siobhan used her calculator to evaluate 3 + 6 × 5, she got the wrong answer. She pressed the following calculator keys: 3 + 6 = × 5 = Explain why Siobhan’s answer was wrong. 5 In the game of snooker the points scored are determined by the colours of the balls sunk in the correct order. This table shows the value of each colour. Colour Red Yellow Green Brown Blue Pink Black Points 1 2 3 4 5 6 7 In snooker, Brad sank 4 red balls, 2 brown balls, one pink and 2 blacks. In the same game, he lost 15 points for foul shots. a What does the expression 4 × 1 + 2 × 4 + 6 + 2 × 7 represent? b Determine Brad’s score for the game. 6 Mr Healy, the school principal, has a parent complaining about the marking of his daughter’s maths exam. He claims his daughter’s correct answer was marked wrong. This is the question: 48 − 8 × 3 The daughter’s answer was 120. a Why was the daughter’s answer wrong? b How could Mr Healy explain why the daughter’s answer is wrong? c Put brackets in 48 − 8 × 3 to make the daughter’s answer correct. 1.05 How much do I pay? We use numbers in every aspect of our lives: from earning and spending money, through to organising schedules, preparing food, assisting with leisure activities and scoring sporting events. EXAMPLE 7 Muspha’s rent is $42 768 annually. How much is his rent per month? Solution 14 Annually means ‘per year’. There are 12 months in a year. Divide the annual amount by 12 to calculate the monthly amount. Monthly rent = 42 768 ÷ 12 Write your answer. Each month Muspha pays $3564. NELSON SENIOR MATHS 11. Essential Mathematics = 3564 ISBN 9780170443906 EXAMPLE 8 Gillian bought 3 books online and paid with her debit card. She paid $6.40 for postage and $7.95 for each book. Before Gillian bought the book the balance of her debit card was $160. Calculate the balance after her online purchase. Solution Calculate the cost of the postage and the books. Remember to do × before +. Cost = $6.40 + 3 × $7.95 = $30.25 The books and postage cost $30.25. Subtract the amount that Gillian spent from the balance of her debit card. $160 − $30.25 = $129.75 Write your answer. The balance of Gillian’s debit card was $129.75. Exercise 1.05 How much do I pay? Example 1 Finn’s annual business expenses total $82 440. Calculate his average monthly expenses. 7 2 Gazi pays $3250 per month for rent. Calculate the annual rent that Gazi pays. 3 Li pays $12 per hour for parking. How much does he pay when he parks for 7 hours? 4 Aisling’s business spends $1840 per fortnight on electricity. a How much does it spend per week on electricity? b Calculate the amount the business spends annually on electricity. Remember! A fortnight is 2 weeks long and there are 26 fortnights in a year. 5 Rob ordered 2 burgers and some fries for his lunch. The burgers cost $4.80 each and the fries cost $3.20. Rob paid with a $20 note. How much change should he get? Example 8 6 Voula spends 45 minutes at the gym 6 days per week. How many hours does she spend at the gym each week? 7 Joel plays golf 3 times per week. On average his golf games take 3 hours and 15 minutes. a How many hours does Joel play golf per week? b  During holidays Joel plays golf every day. Calculate the number of hours he played golf on his 5-week holiday. 8 Reah hates TV ads. During a one-hour period she counted there were twelve 30-second ads and twenty-four 15-second ads. How many minutes of ads were included in the one-hour period? 9 The Rockets and the Rascals are opposing teams in a rugby league match. Tries are worth 4 points and goals 2 points. The Rockets scored 5 tries and 3 goals and the Rascals scored 6 tries and no goals. Which team won and by how many points? ISBN 9780170443906 1. What’s the score? 15 10 The Boomers and the Snakes are two opposing teams in an AFL match. In AFL, teams score 6 points for a goal and 1 point for a behind. At three-quarter time the Boomers had scored 18 goals and 2 behinds while the Snakes had scored 17 goals and 8 behinds. Who is winning? 11 The BBQ Specialists are catering for Mala’s garden party. They are supplying catering and bar service for 42 people. How much will Mala have to pay? The BBQ specialists Catering Steak and salad with bread rolls: $18.75 per person Bar service Fruit juices and soft drink: $6.50 per person PRACTICAL ACTIVITY THE YEAR 11 PARTY Budget catering Price per person Finger food:..........................................$5.60 BBQ steak:......................................... $9.20 BBQ sausages:................................... $2.10 BBQ seafood:..................................... $8.20 Salad:.................................................... $4.35 Bread rolls:..............................................90c Desserts:............................................. $3.95 Coffee/tea:..............................................$1.10 Cheese platters and biscuits:...... $2.60 Fruit platters:..................................... $2.40 Non-alcoholic drinks:..........................$7.40 Toby is responsible for organising the catering for the Year 11 end-of-year party. The organising committee has allocated $4000 to spend on the food and non-alcoholic drinks for the 180 people expected to attend. Your group’s task is to help Toby decide on an interesting menu for food and drink. Remember to stay within your budget. 16 NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 1.06 Estimating costs WS It’s difficult to mentally calculate the total cost of the items in your supermarket trolley, but careful shoppers can estimate using approximation techniques. Homework Estimating and rounding EXAMPLE 9 Patrick is purchasing the supermarket items shown below: 5 cans of dog food, Caesar salad, blueberries, teabags, strawberries and macadamia nuts. The price of each item is shown. $2.98 $6.99 $4.98 $13.98 Scout Kozakiewicz $4.5 9 $5.50 Patrick has $54 in his wallet. Estimate whether he has enough to pay for his shopping. Solution Mentally round each price up to the nearest dollar. Dog food: 5 × $3 = $15 Caesar salad: $6 Blueberries: $5 Teabags: $7 Strawberries: $5 Macadamia nuts: $14 Estimated price = 15 + 6 + 5 + 7 + 5 + 14 = 52 The items cost approximately $52. When we estimated the costs, each estimate was more than the true cost. The true cost of the items is less than $52. Write the answer to the question. Patrick has enough money to pay for his shopping. ISBN 9780170443906 1. What’s the score? 17 Exercise 1.06 Estimating costs Answer all questions without using a calculator. For Questions 1 to 5, select the correct answer A, B, C or D. 1 Which is the best estimate of the value of $9.99 × 8? A $72 B $80 C $90 D $100 2 What is the best approximate value of $25.99 + $4.10 + $15.05 + $3.90? A $45 B $49 C $55 D $60 3 Gavin ordered 19.5 m of materials at a cost of $29.90 per metre. Estimate the cost of the materials. A $60 B $90 C $600 D $900 4 Seth is buying small chocolates that cost 49c each. Approximately how much will 50 cost? A $9 B $25 C $99 D $2500 5 Davinia buys bags of nuts and bolts to use in her furniture business. The bags contain between 48 and 51 nuts and bolts. She has 11 bags in her supply cupboard. Approximately how many nuts and bolts does she have? A 50 B 550 C 1100 D 2500 iStock.com/amixstudio 6 Christina cuts 18.9 m long lengths of wool for the warp in her loom. She requires 82 lengths. 18 a Approximately how many metres of wool will she need for the 82 warp threads? b The wool Christina uses comes in 400-metre long balls. How many balls of wool will Christina need? c Use a calculator to check your answer to part b. NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 7 When Bryce bought 2 pizzas at $8.95 each and a salad for $5.80, he paid with a $50 note. Approximately how much change should he get? 8 Sebastian has only $60 remaining on his debit card until payday. He has these items in his supermarket basket: 2 packets of sausages @ $5.99 per packet 1 packet of potatoes $4.99 1 carton of milk $3.95 1 carton of yoghurt $6.95 1 packet of breakfast cereal $5.95 a Estimate the cost of the items in Sebastian’s supermarket basket. b Will he have sufficient funds in his account to pay for his shopping? c Does he have enough money to add a $9.80 carton of soft drink to his shopping? Example 9 9 Paige is knitting a scarf. It takes her 29 seconds to complete 1 row. a Explain why we can use the calculation 80 ÷ 2 = 40 to calculate the number of minutes Paige takes to complete 80 rows of knitting. b Paige needs to knit 50 more rows to finish the scarf. Approximately how long will it take her? INVESTIGATION MY SHOPPING DOCKET To complete this investigation you will need samples of supermarket shopping dockets. What you have to do Estimate the total cost of the items on a docket. Compare your estimate to the total on the docket. Was it more or less than the actual total? What is the difference between your estimate and the true cost? Use this formula to calculate how much your estimate is different from the true cost as a percentage. difference × 100% Percentage error = true value Repeat the process for a different shopping docket. Does your percentage error become smaller with practice? Chapter problem You’ve covered the skills required to solve the chapter problem. Can you solve it now? ISBN 9780170443906 1. What’s the score? 19 1.07 Rounding numbers Rounding decimals Calculators often show a lot of decimal places. When Imran used his calculator to divide 5.7 by 1.3, it showed the answer as 4.384 615 385. Imran wants the answer with only 1 decimal place, so he has to round his answer. Imran has to decide whether 4.384 615 385 is closer to 4.3 or 4.4. He will check using just the first 2 decimal places: 4.38. On the number line, he can see that 4.38 is closer to 4.4. 4.3 4.35 4.38 4.4 Remember that 4.30 is the same as 4.3 and 4.40 is the same as 4.4. This is because the 8 in 4.38 is more than 5, which is the halfway mark. Imran will round to 4.4. Rounding numbers When the digit following the decimal place you want to round to is: 4 or smaller − just round down and leave all the extra digits off. 5 or bigger − round up and add 1 to the digit in the decimal place you want to round to, and leave all the extra digits off. This table shows the results when some decimals are correctly rounded to 1 decimal place. Original decimal Rounded to one decimal place 6.802 956 6.8 6.830 5 6.8 6.849 99 6.8 6.853 6.9 6.893 6.9 EXAMPLE 10 Round 18.3721 to one decimal place. Solution Decide whether 18.3721 is closer to 18.3 or 18.4. The figure following the 1st decimal place is 7. Round the 3 up to 4. 20 NELSON SENIOR MATHS 11. Essential Mathematics 18.3721 ≈ 18.4 ISBN 9780170443906 Rounding up or down When we have to round an answer to the nearest whole number, to one decimal place or to the nearest 5 cents, we have to decide whether to round up or round down. Imagine you’ve calculated that you need 2.1 cans of paint to cover the walls in your living room. You’ll have to buy 3 cans, even though 2.1 is closer to 2. Two cans won’t be enough and the shop wouldn’t sell you 0.1 of a can! In this case, we must round UP. In Australia, supermarket bills are rounded to the nearest 5 cents because that is the smallest unit of cash. EXAMPLE 11 Jad bought several items in a supermarket and the bill came to $23.58. How much will he pay? Solution Bills are rounded to the nearest 5c. We have to decide whether $23.58 is closer to $23.55 or $23.60. It is 2c from $23.60 and 3c from $23.55. Jad will pay $23.60 for the items. Exercise 1.07 Rounding numbers 1 Round each number correct to 1 decimal place. Example a 16.1256 b 29.7681 c 14.6472 d 13.2836 e 104.554 f 195.219 10 2 Write 124.727 56 correct to 2 decimal places. 3 Darryn correctly rounded a number to 16.3. Suggest 2 possible values the original number could have been. 4 In surfboard-riding competitions, scores are rounded to 1 decimal place. This is how scores are calculated. Step 1: Each of 5 judges awards a result from 0.0 to 10.0 for the wave. Step 2: The lowest and highest scores are removed. Step 3: The average of the 3 remaining scores is determined by adding them together, then dividing by 3. The average is rounded to 1 decimal place. Step 4: Each competitor’s best 2 scores are added to obtain their final competition score. Here are the scores that the judges gave Sally for her first wave. Use calculations to show that Sally’s score for her first wave is 7.7. 7.5 ISBN 9780170443906 7.3 8.1 7.9 7.6 1. What’s the score? 21 5 Mia and Elissa are competing against each other in a surfboard-riding competition. This table shows the points judges awarded them on 4 waves. Mia’s scores Elissa’s scores Wave 1 Wave 2 Wave 3 Wave 4 Wave 1 Wave 2 Wave 3 Wave 4 4.7 7.2 8.5 7.4 6.9 7.8 5.2 8.2 5.1 7.1 8.5 7.8 7.0 7.9 5.1 8.1 4.9 7.5 8.4 7.6 7.2 7.8 5.3 8.4 5.0 7.3 8.2 7.9 6.9 8.0 5.2 8.0 4.9 7.5 8.6 7.5 7.1 7.8 5.0 8.9 Calculate the score awarded to each surfer for each wave. b Calculate each surfer’s final score for the competition. c Who won the event? Shutterstock.com/Alain Lauga a 6 Express the total of 2 m, 148 cm and 384 mm in metres, correct to 1 decimal place. 7 Ben’s car travels 8.9 km on 1 L of petrol. The car’s fuel tank holds 60.5 L of petrol. How far can the car travel on one full tank of petrol? Express your answer in kilometres, correct to 1 decimal place. 8 Pauline buys knitting wool in hanks from a wool mill. Two of the hanks contained 1560 m each, and the other 3 hanks contained 1875 m each. a Calculate the total length of wool in Pauline’s 5 hanks. b How many kilometres of wool are in the 5 hanks? Express your answer correct to 1 decimal place. 9 Thanh bought 79.3 L of petrol at 155.8c/L 22 a Explain why the value of 80 × 1.5 will give the approximate cost of the petrol in dollars. b Approximately how much will the petrol cost? c Use a calculator to determine the cost of the petrol, correct to the nearest 5 cents. d How different is your approximation to the actual cost? NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 Alamy Stock Photo/redsnapper 10 The gas company replaces the LPG (liquid petroleum gas) in the bottles at Jaye’s house each month. Today’s delivery docket showed a delivery of 151.65 L of LPG priced at $1.35 per litre. a Jaye wanted to estimate the total cost. She calculated that the cost will be between 150 × $1 and 150 × $2. Will the cost be closer to $150 or $300? Give reasons for your answer. b Calculate the value of 151.65 × $1.35 and express the answer correct to 2 decimal places. c Jaye allows $2200 in her annual budget for gas. Is this too much or not enough? Justify your answer. 11 Simone paid $48.62 for 3 metres of fabric. Calculate the amount she paid per metre, correct to the nearest cent. 12 Round each item’s price to the nearest 5 cents. $2.93 Shutterstock.com/Kotema $3.54 11 c Shutterstock.com/Melinda Fawver $2.87 Example b Shutterstock.com/Luminis a 13 Tri bought a bag of fruit priced at $5.78 and he paid with a $10 note. How much change should he receive? ISBN 9780170443906 1. What’s the score? 23 14 At lunchtime, Roberto bought a can of drink marked at $1.38 and a bread roll priced at $1.18 from the local shop. He was charged $2.60 for his purchases: $1.40 for the drink and $1.20 for the roll. Was Roberto charged the correct amount? Give a reason for your answer. 15 Dallas calculated that she requires 18.2 bags of mulch for her garden. How many bags should she buy? 16 Zoe paid $4.25 cash for a bottle of milk. What are the smallest and largest prices that could have been marked on the bottle? 17 Daniel is a landscape gardener. He calculated that he needs 4.5 bags of concrete to complete laying some paving bricks. How many bags will he have to buy? 18 Jess sells boxes of eggs from her hens. Each box contains 12 eggs. How many boxes can she fill with 80 eggs? 19 Calculate each amount, correct to the nearest whole number. a 2 of 700 mL 3 b 3 of 85 m 4 c 5 of 775 g 8 d 0.3 of 165 cm e 0.7 of 25 min f 0.9 of 365 days 20 Calculate each amount, correct to one decimal place. a 2 of 500 mL 3 b 3 of 825 g 4 c 5 of 710 kg 8 d 0.8 of 28 minutes e 0.9 of 19.5 km f 0.75 of 51 seconds PRACTICAL ACTIVITY ACCURACY IN MEASURING LENGTHS In this activity you are going to investigate how accurately each member of your group can measure lengths. Each group will need a tape measure that measures in metres, centimetres and millimetres. What you have to do 1 Choose 3 suitable distances in your school environment; for example, the length of the school verandah, or the distance from your classroom doorway to the nearest tree. 2 Each member of the group measures the lengths as accurately as possible. 3 Compare your group’s measurements. At what level of accuracy (for example, answers in metres correct to 1 decimal place) are the measurements the same? 24 NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 1.08 Practical multiples Nuts and bolts can be sold in packets of 20 or 100, fertiliser in 40 kg bags and copy paper in lots of 500. Many manufacturers package items in multiple quantities for use by tradespeople, and in most cases it’s not possible to buy part of a packet. EXAMPLE 12 Anna starts a dog-walking service and wants to have some advertising pamphlets printed. The printer handles advertising pamphlets only in multiples of 250. Anna wants 600 pamphlets. How many will she have to order? Solution The multiples of 250 mean ‘lots of 250’, that is 1 × 250, 2 × 250, 3 × 250, 4 × 250, 5 × 250, etc. The first 5 multiples are 250, 500, 750, 1000, 1250. Anna can’t order 600 pamphlets so she must choose the next multiple of 250 that is higher. Anna will have to order 750 pamphlets if she requires 600 pamphlets. EXAMPLE 13 Timber is available only in lengths that are multiples of 300 mm. Is it possible to buy a 1 m length of timber? Why, or why not? b Peter needs 2 lengths of timber each 1300 mm long. What is the best way for him to buy the timber? Shutterstock.com/Uber Images a ISBN 9780170443906 1. What’s the score? 25 Solution Determine whether 1 m is a multiple of 300 mm by dividing it by 300 mm. If the answer is a whole number, then it is a multiple of 300 mm. First convert 1 m to mm. a 1 m = 1000 mm 1 m ÷ 300 mm = 1000 mm ÷ 300 mm = 3.333... The answer is not a whole number, so 1 m is not a multiple of 300 mm. It is not possible to buy a 1 m length of timber. Find the total length. b Total length = 2 × 1300 mm = 2600 mm Divide by 300 mm to check if it is a multiple of 300. It isn’t so round up to 9 and buy 9 × 300 mm. 2600 ÷ 300 = 8.666... (not a multiple of 300). Length required = 9 × 300 mm = 2700 mm Cut 2 × 1300 mm from 2700 mm length (100 mm left over). Exercise 1.08 Practical multiples Example 12 Example 13 26 1 Business envelopes are sold in boxes of 200. a At the end of each month, Xanthe posts accounts to all the company’s customers. She ordered 8 boxes of envelopes for the letters. How many envelopes are contained in the boxes Jane ordered? b Mike requires 600 envelopes to post statements to customers. How many boxes of envelopes will he need? c What are the first four multiples of 200? d Tash needs 840 envelopes. How many boxes does she need to order? 2 Timber is available only in multiples of 300 mm. a Explain why you can’t buy a piece of timber 1400 mm long. b If you need a piece of timber 1400 mm long, what length do you need to buy? c Oliver needs 2 pieces of timber, each 1150 mm long. What length of timber should he buy? d A building plan requires a piece of timber 1.04 m long. What length of timber should be bought? NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 3 At the wholesaler, electrical wire is available only in multiples of 100 m. Kelly calculated that for her next job she will need 3 × 74 m lengths of wire and 2 × 180 m lengths. How much wire does she need to buy for the job? 4 Rose likes to make patchwork quilts. At the shop, fabric is available only in lengths that are multiples of 20 cm. Rose requires 15 cm of a pink fabric and 22 cm of a white fabric. Both fabrics cost $22 per metre. What length of each fabric does Rose need to buy? b How much will the fabric that she needs cost? Shutterstock.com/Strakovskaya a 5 Ashok is making a fence from treated pine. The wood is available in lengths that are multiples of 0.3 m from 2.1 m to 3 m. He needs 17 posts each 1.35 m long, and 8 top rails each 2.3 m long. What lengths of timber should he order from the timber yard? 1.09 After the point While 1.5 metres means 1 metre and 50 centimetres, 1.5 hours doesn’t mean 1 hour and 5 minutes, and 1.5 years doesn’t mean 1 year and 5 months. Fortunately, scientific calculators have a ‘degrees, minutes, seconds’ key ( or DMS ) that converts 1.5 hours into 1 h 30 min, but in other situations we have to think for ourselves! EXAMPLE 14 What does the.45 represent in the following 2 situations? a $16.45 b 16.45 metres Solution a For money, the 2 digits after the decimal point represent cents. In $16.45, the.45 represents 45 cents. b For metres, the 2 digits after the decimal point represent centimetres. In 16.45 m, the.45 represents 45 cm. ISBN 9780170443906 1. What’s the score? 27 EXAMPLE 15 Write 2.7 hours in hours and minutes. Solution Change 2.7 hours to hours and minutes using your calculator by pressing = or 2.7 2ndF DMS. 2.7 2.7 hours = 2 h 42 min OR first change the decimal part, 0.7 hours, into minutes by multiplying it by 60 minutes (1 hour). 0.7 hours = 0.7 × 60 Write the answer. 2.7 hours = 2 h 42 min = 42 minutes EXAMPLE 16 Express 1.6 months in months and days, assuming that an ‘average month’ is 30 days. Solution Change 0.6 months into days. Multiply 0.6 by 30. 0.6 month = 0.6 × 30 Write the answer. 1.6 months ≈ 1 month and 18 days. = 18 days Exercise 1.09 After the point Example 14 1 What does the.24 represent in each amount? a b $7.24 7.24 m 2 What does the.9 represent in each measurement? a Example 15 3 Use the minutes. a b $18.9 2.5 h or DMS 18.9 m key on your calculator to express the following times in hours and b 3.8 h c 1.4 h d 2.9 h 4 Follow these steps to convert 4.5 years into years and months. 28 a How many months are there in 1 year? b Multiply the number of months in a year by 0.5. c 4.5 years is 4 years and the number of months you calculated in part b. NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 5 Convert 2.25 years into years and months. 6 a b How many months are equivalent to 0.75 years? Write 5.75 years in years and months. 7 Express each time in months and days. Assume that there are 30 days in a month and write your answers correct to the nearest day. a 6.5 months b c 8.9 months 3.24 months d Example 16 5.3 months 8 A cricket match is measured in ‘overs’. An over consists of 6 balls. a Express 18 balls in ‘overs’. b During a test match the commentator said ‘There are 18.5 overs remaining until the end of play’. If the commentator was using mathematics correctly, how many overs and balls should be remaining? c The commentator meant that there were 18 overs and 5 balls remaining. Why do you think cricket commentators use decimal points in this non-standard way? 1.10 How much do I get? Everyday life is full of occasions when we use calculations. We earn, spend and invest money, pay tax, cook food, buy and sell property. On every occasion, there are calculations involved. EXAMPLE 17 Tom’s taxable income last year was $42 500. This table shows the Australian income tax rates. Use the table to calculate Tom’s income tax for $42 500. Taxable income Tax on this income 0 – $18 200 Nil $18 201 – $37 000 19c for each $1 over $18 200 $37 001 – $90 000 $3572 plus 32.5c for each $1 over $37 000 $90 001 – $180 000 $20 797 plus 37c for each $1 over $90 000 $180 001 and over $54 097 plus 45c for each $1 over $180 000 © Australian Taxation Office for the Commonwealth of Australia Solution $42 500 belongs to the $37 001 – $90 000 row. Use the rule for this row. Write ‘32.5c in the dollar’ as 32.5% = 0.325. Tax = $3572 + 0.325 × ($42 500 – $37 000) Write the answer. Tom will pay $5359.50 in income tax. ISBN 9780170443906 = $5359.50 1. What’s the score? 29 Exercise 1.10 How much do I get? 1 Karen earns $726 per week, but her employer is changing her payroll system to pay wages fortnightly. Calculate Karen’s fortnightly pay. Remember there are 2 weeks in 1 fortnight. 2 After tax and other deductions, Damian’s fortnightly pay is $1425. Damian works 9 days per fortnight. Calculate his net pay per day. 3 Jemma is making mascarpone cheese. Combining 700 ml of milk, 300 ml cream and 1 pinch of starter culture will produce 500 g of the cheese. Jemma needs to make 750 g of mascarpone for dessert. How much milk, cream and starter culture will she need? 4 Tyson has a 4-year apprenticeship and is employed for 5 days per week. During his first year he earns $347.80 per week. If he completes all his trade educational requirements, his wage in the second year will increase by $108 per week. a What will Tyson’s weekly wage be in his second year if he completes all his trade educational requirements? b How much will he earn per year as a second-year apprentice? Remember there are 52 weeks in a year. 5 Alyssa earns $20.21/hour as a receptionist. The timesheet shows the hours she worked last week. Normal hours Monday 7 Tuesday 8 Wednesday 7 Thursday 8 Saturday Overtime 3 4 a How many hours did Allysa work at normal time? b Calculate her pay for the normal hours she worked. c When Allysa works overtime, she is paid 1.5 times as much per hour. Calculate her pay per hour when she works overtime. 30 d Calculate Allysa’s pay for the overtime hours. e Calculate her total pay for the hours she worked last week. NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 6 Anand is a 4th year apprentice in a mining company. He is paid $728.70 per week. Before he receives his pay, Anand’s employer deducts $73.43 for income tax and $14.57 for Medicare. a Calculate Anand’s annual (yearly) pay without deductions. b How much should Anand receive per week for his take-home pay? c Anand receives cheap accommodation and food from the mining company. 1 It costs only of his take-home pay. How much does Anand pay per week for his 4 accommodation and food? d How much of his weekly pay is left after he pays for tax, Medicare, accommodation and food? Shutterstock.com/SeventyFour e Anand saves 0.7 of the money he has left into a regular savings account. How much does he have left to spend? 7 The ingredient list for making a Polish vegetable salad to serve 8 people includes: 300 g carrots 200 g parsnip 120 g celeriac 440 g potatoes 4 eggs 200 g apples 260 g pickled cucumbers Aleksandra wants to make salad for 2 people. How much of each ingredient does she require? 8 Use the tax table on page 29 to calculate the income tax payable on each taxable income. a $55 600 b $99 850 c Example 17 $200 000 9 The taxable income of the CEO of NelsonNetBank was $2.5 million. Use the tax table on page 29 to calculate her income tax. ISBN 9780170443906 1. What’s the score? 31 10 Gareth operates a mobile coffee van. The table shows the sales he made yesterday. Price Number sold Flat white $4.50 20 Cappuccino $5 63 Mocha $5.50 12 Long black $4 28 Late $4.75 14 a Calculate the total number of coffees Gareth sold yesterday. b What was the total sales figure for these coffees? c This total sales figure includes the goods and services tax (GST). Gareth is required to send the GST to the government. Calculate the GST by dividing the value of Gareth’s total sales by 11. Express the answer correct to the nearest dollar. INVESTIGATION GST AT THE SCHOOL CANTEEN GST is a 10% tax on goods and services. It is not charged on essential foods, but it is included in a lot of the foods typically served in a canteen. Fruit and bread don’t have a GST, but cakes, pies, packets of chips and ice cream are subject to GST. Visit the school canteen and obtain a list of 5 items the canteen sells that include GST. Record the selling price of each item. In a typical week, how many of each of the items are sold? Calculate the total amount the canteen receives from selling these items in a typical week. Determine the amount of GST included in the week’s sales of these items by dividing the total value of the sales by 11. (110% ÷ 11% = 10% GST) Concentrate on one of the items. How much GST is included in the price of the item? What is the GST-free price? How would the price of the item change if the rate of GST was increased by half? 32 NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 KEYWORD ACTIVITY CALCULATIONS FIND-A-WORD Calculations find-a-word Use each clue to find 11 keywords. Then copy the puzzle grid below (or print a copy from NelsonNet) and find the same 11 keywords in the grid. 1 Multiply by 2. 2 Multiply by 3. 3 The centre of a dartboard. 4 Number of months in a year. 5 The initials we use to remember the ‘order of operations’ rules. 6 Shopping bills are rounded to the nearest _____ cents. X P O D I V I D E Y T D M I C T N P D L S G T R A U B I D M A S W W B N E G A F I V E J E U R W Z C F R C J A L L O A K T H X N E I V L D M C U R H C Q S E S H Y N K G I V O B 7 Another word for powers, beginning with I. R E S D J R E F P O Y 8 For order of operations, we calculate what’s inside these first. B Y R H D A R T S L R K E L P O X F N S R E D O U B L E M F O Z U 9 A percentage is a fraction with this number as denominator. 10 A game played on a circular board numbered with points values. 11 To convert a percentage to a decimal, you _______ by 100. SOLUTION TO THE CHAPTER PROBLEM Problem Ben bought fish and chips for $11.25 and paid with a $50 note. The cashier gave him a $20 and a $5 note and a few coins for his change. Could the change be wrong? Solution The fish and chips cost approximately $10. Ben’s change should have been approximately $50 − $10 = $40. Ben’s change included a $20 and a $5 note plus a few coins, so the change was probably between $25 and $30. Yes, he has probably been short-changed (by about $10). ISBN 9780170443906 1. What’s the score? 33 1. Qz Exercise 1.01 Exercise 1.02 Exercise 1.03 What’s the score? 1 Add each set of target scores without using a calculator. a A 5 and a 4 b Double 5 and a 3 c Double 3 and double 8 d A 5 and a triple 6 2 Jasper is playing darts. Calculate his score when his 3 darts landed on: a Exercise Exercise 1.04 a 1.04 Exercise 1.05 Exercise 1.05 Exercise 1.06 34 b double 9, 6 and triple 20 13 to a percentage 40 1.05 to a percentage b 18.65% to a decimal d 35% to a fraction 2 4 Find of $45.90. 3 5 Evaluate each expression without using a calculator, then use a calculator to check your answer. a Exercise 5, 7 and double 11 3 Convert: c 1.03 TEST YOURSELF 8+5×2 b 20 ÷ 10 + 3 × 4 c (15 + 9) ÷ (8 − 2) 6 Use a calculator to evaluate each expression, correct to 1 decimal place. a 17.35 + 182.96 − 25.47 b 16.39 + 12.5 ÷ 3.6 c (12.3)2 7 The balance of Ivan’s debit card was $368. He bought a bulk packet of screws and wingnuts for $24.80 plus $9.20 postage from an online hardware store. What was the balance of his debit card after he paid for the purchase and delivery? 8 Isabella ordered the food for her dance group. She ordered 4 chicken wraps and 6 small sushi. The chicken wraps cost $5.50 each and the sushi cost $2.90 each. She paid with a $50 note. How much change should Isabella get? 9 Estimate the value of each purchase. a 6 items valued at $4.99 each b 3 metres of fabric at $8.25 per metre c 95 chocolate frogs at 99c each NELSON SENIOR MATHS 11. Essential Mathematics ISBN 9780170443906 10 Jemima bought 8 balls of wool priced at $11.99. a Estimate how much the wool costs. b The shop charged her $71.94. Is the bill right? 11 a Round 15.32 up to the nearest whole number. b Exercise 1.06 Exercise 1.07 Round 16.75 correct to one decimal place. 12 Round each measurement correct to the nearest whole number. a b 27.9 kg 135.2 m c Exercise 4.5 mm 2 as a decimal, correct to 3 decimal places. 7 14 The buttons that Mai wants to use on a jacket she’s designing are sold in multiples of 3. Mai needs 13 buttons for the jacket. How many buttons will she have to buy? 13 Express 15 Timber is only available in lengths that are multiples of 300 mm. Grigor needs a piece of wood 1930 mm long. What length piece of wood does he need to buy? 16 Convert 3.2 hours into hours and minutes. 1.07 Exercise 1.07 Exercise 1.08 Exercise 1.08 Exercise 1.09 17 Write 8.25 years in years and months. Exercise 18 Toni earns $1265 per fortnight. 1.10 a Calculate her weekly pay. b How much does she earn annually? 19 Ella is planning to make boxes of shortbread as Christmas gifts. The ingredients to make 12 biscuits are: 100 g butter 60 g caster sugar 170 g flour 1 teaspoon of baking power Exercise 1.10 Ella wants to make 84 biscuits. How much of each ingredient will she need? ISBN 9780170443906 1. What’s the score? 35