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Number skills 1 There are many factors that affect the environment of our planet. One of these is world population. The data supplied in the table give the estimated or projected world Year World population population for the middle 1950 2 555 078 074 of the year. In which 1960 3 039 332 401 ten-year period did (or 1970 3 707 610 112 will) the world population increase the most? 1980 4 456 705 217 In which ten-year period 1990 5 283 755 345 is the percentage increase 2000 6 080 141 683 the largest? 2010 6 823 634 553 This chapter refreshes your skills in working with 2020 7 518 010 600 numbers expressed as 2030 8 140 344 240 fractions, decimals, 2040 8 668 391 454 percentages or in index form and applying those 2050 9 104 205 830 skills to real-life situations.
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2 Maths Quest 9 for VictoriaOrder of operations Anton has calculated the answer to 5 + 6 × 4 as 44, while Marco insists that the answer is 29. Who is correct? In mathematics, it is important to ensure that everybody obtains the same result from a calcu- lation; so the order in which mathematical oper- ations are worked is important. The order of operations requires that: 1 all brackets are evaluated ﬁrst, beginning with the innermost brackets 2 then, all multiplication and division are evaluated, working from left to right 3 and ﬁnally, any addition and subtraction, working from left to right. To obtain the correct answer to the calculation 5 + 6 × 4, we must complete the operations of + and × in the correct order. That is, × ﬁrst then +. 5+6×4 = 5 + 24 = 29 WORKED Example 1 Evaluate each of the following without using a calculator. a 4 + 12 − 5 + 6 b 4 + 12 − (5 + 6) c 6 + 21 ÷ 7 d [4 × (5 + 8)] ÷ 2 THINK WRITE a 1 Write the calculation. a 4 + 12 − 5 + 6 2 Perform the addition and subtraction = 16 − 5 + 6 from left to right. = 11 + 6 3 Write the answer. = 17 b 1 Write the calculation. b 4 + 12 − (5 + 6) 2 Evaluate the bracket ﬁrst. = 4 + 12 − 11 3 Perform the addition and subtraction =5 from left to right and write the answer. c 1 Write the calculation. c 6 + 21 ÷ 7 2 Perform the division. =6+3 3 Perform the addition. =9 d 1 Write the calculation. d [4 × (5 + 8)] ÷ 2 2 Remove the brackets by working the = [4 × 13] ÷ 2 innermost bracket ﬁrst. = 52 ÷ 2 3 Divide and write the answer. = 26 In other examples you will need to read the question carefully to interpret the correct order of operations and the correct way to write the calculation.
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Chapter 1 Number skills 3 WORKED Example 2 Mum bought 2 packets of Easter eggs to hide in the garden for her 4 children to ﬁnd. Each packet contained 20 eggs. While she was hiding them, the dog ate 4 eggs, Dad ate 3, and 1 was squashed. If all the other eggs were found, and each child found the same number of eggs, how many eggs did each child have? THINK WRITE 1 Write a mathematical sentence showing what happened. Find the total number of eggs and subtract the number that were eaten or squashed. Then divide by the number of children looking for eggs. [2 × 20 − (4 + 3 + 1)] ÷ 4 2 Use order of operations to solve the = (2 × 20 − 8) ÷ 4 problem. = [40 − 8] ÷ 4 = 32 ÷ 4 =8 3 Write the answer in a sentence. Each child found 8 eggs. remember remember Evaluate in the following order. 1. Brackets ﬁrst, beginning with the innermost pair, then working through to the outermost pair. 2. Multiplication and division in order from left to right. 3. Addition and subtraction in order from left to right. 1A Order of operations MathWORKED 1 Evaluate each of the following without using a calculator. cadExample 1 a 3 + 12 − 5 + 6 b 7 + 5 − 11 + 2 − 3 c 10 − 2 − 3 + 4 Order of d 18 − 11 + 4 + 12 − 14 e 25 + 5 − 10 + 2 − 10 f 32 − 8 + 6 − 7 − 5 operations g 10 × 6 × 4 × 2 h 18 × 4 × 3 × 0 i 80 ÷ 4 ÷ 5 j 25 ÷ 5 × 6 k 8×2÷4×3 l 72 ÷ 2 ÷ 6 × 3 Math cad m 16 + 2 × 5 n 80 ÷ 2 + 28 o 12 − 14 × 0 Ascending and p (4 + 6) × 8 q (35 − 11) ÷ 6 r (7 + 2 − 3) × 8 descending s 12 ÷ (9 − 3) t 75 ÷ (12 + 13) order
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4 Maths Quest 9 for Victoria d 2 multiple choice hca a What is 12 × (4 + 2) ÷ 8 equal to?Mat Adding A 10 B 12 C 9 D 8 E 1 whole numbers DIY b What is 36 ÷ 3 ÷ 4 + 2 equal to? d A 2 B 72 C 50 D 5 E 10 hca c What is 8 × 5 + 3 × (8 − 5) is equal to?Mat Subtracting A 49 B 192 C 59 D 129 E 339 whole numbers DIY 3 Evaluate each of the following. d a 8 × 9 − 10 × 6 b 14 × (3 + 2) ÷ 7 c 72 ÷ (2 + 7 × 1) hca d 80 ÷ 5 − 60 ÷ 6 e 35 × (8 + 4 − 6 × 2) f (13 − 3) × 2 + 4 × 6Mat Multiplying g (17 − 12) ÷ 5 × 2 h (14 + 7 − 8) × 6 i [14 + (2 × 6 − 3)] × 4 whole j [(2 + 1) × 7 − 3 × 5] − 6 ÷ 3 k {[(3 + 9) ÷ 12] + 4 × 4} − 17 numbers DIY l {40 − [(8 + 2) × 3 − 5]} ÷ 5 m 16 ÷ 4 + 24 ÷ 6 + 5 × 5 − 19 hca d n 108 ÷ 4 × (4 − 4) × 4 o {11 + (4 + 3) × 2 + 5 × 6 + (8 – 2) × 5} × 4Mat p [16 × 3 ÷ 2 + 40 ÷ 4 × 2 − 3 × 11 + 14] ÷ 5 + (6 × 2 + 4) × 2 − (7 × 5 + 2) Dividing whole WORKED 4 Takiko has brought 3 packs of nut biscuits to share with the 20 members of her class. If numbers Example each pack contains 12 nut biscuits and 3 girls and 5 boys are allergic to nuts or don’t eat DIY 2 biscuits, so don’t have any; how many nut buscuits will each of the other class members receive? 5 The Wimbletons wanted to buy a tennis racquet for each of their 3 children. The normal price of a racquet is $100 but the shop is offering a special deal. If two racquets are bought at the same time, the price is reduced by $25 for each one. If the Wimbletons buy one at the normal price and two on the special deal, how much do they pay altogether? Write an equation to show how you could have found the answer.
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Chapter 1 Number skills 5Integers Integers include positive whole numbers, negative whole numbers and zero. They can be represented on the number line. –5 –4 –3 –2 –1 0 1 2 3 4 5 The rules for using integers are: Rule 1 When adding integers with the same sign, keep the sign and add; −3 + −2 = −5. Rule 2 When adding integers with different signs, ﬁnd the difference and use the sign of the number further from zero; –3 + 4 = 1. Rule 3 When subtracting integers, add the opposite; 5 − −7 = 12. Rule 4 When multiplying integers, the following rules are obeyed. (a) Positive × Positive = Positive 5 × 8 = 40 (b) Positive × Negative = Negative 5 × −8 = −40 (c) Negative × Positive = Negative −5 × 8 = −40 (d) Negative × Negative = Positive −5 × −8 = 40 Rule 5 When dividing integers, use the same rules as for multiplication. (a) 16 ÷ 2 =8 (b) 16 ÷ −2 = −8 (c) −16 ÷ 2 = −8 (d) −16 ÷ −8 = 2 WORKED Example 3 Calculate each of the following without the use of a calculator and using the correct order of operations. a −15 × −5 ÷ 3 b 7 + −5 − −8 c 4 − 60 ÷ (−4 − 6) THINK WRITE a 1 Write the calculation. a −15 × −5 ÷ 3 2 Multiplication and division are the only = 75 ÷ 3 operations; so work from left to right. = 25 b 1 Write the calculation. b 7 + −5 − −8 2 Addition and subtraction are the only = 2 − −8 operations; so work from left to right. =2+8 = 10 c 1 Write the calculation. c 4 − 60 ÷ (−4 − 6) 2 Work the brackets. = 4 − 60 ÷ −10 3 Perform the division. = 4 − −6 4 Perform the subtraction. =4+6 = 10
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6 Maths Quest 9 for Victoria WORKED Example 4 Insert operation signs to make this equation true. 5 K 3 K 4 K 1 = −2 (Trial and error is a suitable method.) THINK WRITE 1 The answer (−2) is less than the ﬁrst 5 − 3 − 4 − 1 = −3 ≠ −2 number in the question; so try subtraction. 2 The result of the ﬁrst try (−3) is a little too 5 − 3 − 4 + 1 = −1 ≠ −2 small; so change the last sign to +. 3 The result of the second try (−1) is too big; 5−3−4×1 so try multiplying the last digit, which is 1, =5−3−4 remembering to use the order of operations. = −2 remember remember 1. When adding integers with the same sign, keep the sign and add. 2. When adding integers with different signs, ﬁnd the difference and use the sign of the number further from zero. 3. When subtracting integers, add the opposite; for example 5 – –7 = 12. 4. When multiplying and dividing integers, like signs give positive answers, unlike signs give negative answers. 5. When using order of operations, evaluate brackets before multiplication and division, then evaluate addition and subtraction. 1B Integers 1.1 WORKED 1 Calculate each of the following without the use of a calculator and using the correct HEET Example order of operations.SkillS 3a a −7 + 12 b −14 + 7 c −18 − 8 d 25 − 24 − 2 e −2 − 3 − 6 f −7 − 11 + 5 g 14 − 15 + 11 h 13 − 19 − 6 + 9 i 10 × 2 ÷ 5 j 6 × −3 × −2 k −4 × −3 × −5 l 64 ÷ −16 ÷ 4 m −12 × 4 ÷ 16 n −120 ÷ −10 × 2 o 36 ÷ −6 × −5 p −6 × −1 × −10 ÷ 4 HEET 1.2 WORKED 2 Calculate each of the following without the use of a calculator and using the correctSkillS Example 3b order of operations. a 8 + −7 + −3 b 15 − 18 + −8 c 6 + −7 + −10 d 6 − −7 d e −5 − −2 f 7 − −2 − 7 g 4 + −8 − −5 h −7 − −13 hca i −9 + −9 − −9 j 4 + −6 + −2 − −1 k −3 − 6 − −10 + −5Mat Order of 3 multiple choice operations with integers a −7 − 8 + 2 − 3 is equal to: reads A −2 B −16 C −14 D −20 E 0 L Sp he b −12 × −8 ÷ −4 × 2 is equal to: etEXCE Arithmetic A 12 B −12 C 48 D −48 E 316 timer c 9 + −5 − −4 + 2 − −1 is equal to: A 9 B 1 C 23 D 11 E −1
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Chapter 1 Number skills 7WORKED 4 Calculate each of the following without the use of a calculator and using the correctExample order of operations. 3c a −3 + 3 × 3 b −9 − 2 × 6 c 15 ÷ 5 − 5 d 7×0−5 e 6 × (0 − 6) f −14 × 2 − 2 × 10 g 2 − 6 × 3 h 8 + 2 × −5 i 3×8−5×7 j 12 × −3 − 4 k 0 × 3 × −6 + 6 l −90 ÷ −5 − 26 m 5 × (−3 + 5) + 7 n 128 ÷ −16 + 3 × −5 o (3 + 7) ÷ −2 + −4 p −60 ÷ −4 × 3 − 43 q 28 ÷ −2 × (2 − 5) r 56 ÷ 7 + 70 ÷ −10 s 94 ÷ 2 + 3 × −3 t 14 − 4 × (5 + −6) 5 multiple choice a What does 5 × −4 − 10 × −6 equal? A −40 B 40 C 80 D −80 E 180 b What does 5 × (−4 − 10) × −6 equal? A 420 B −420 C 180 D −180 E −40 c (–2 – –4) × (8 × 5 − 4) is equal to: A 244 B −216 C 16 D −48 E 72 d –64 ÷ 8 – –8 is equal to: A 4 B −4 C 0 D −16 E 16 e The correct operation signs to make 2 K −5 K −2 K −5 = −3 a true statement are: A ×, −, − B ×, +, + C ×, ÷, + D −, +, × E −, ×, +WORKED 6 Insert operation signs to make these equations true:Example 4 a 5 K 6 = 11 b 7 K −4 = −28 c −18 K −2 = 9 d −7 K −3 = −4 e 3K4K5=2 f 7 K 2 K 3 = 17 g −5 K −4 K 10 = 2 h 6 K 3K 3 = 0 i 8 K 5 K 2 = −2 j 2 K 3 K 5 K 4 = 26 k 16 K 8 K 8 = −6 l 12 K 18 K 2 = 21 m 12 K 18 K −2 = 21 n −8 K 4 K −2 = 0 o 10 K 3 K 4 K 2 = 0 p 5 K 2 K −3 K −3 = 2 GAME 7 Thanh stands on a cliff top 68 m above sea level and drops a stone into the water. time It stops on the bottom 27 m below sea level. How far has the stone fallen? Number skills — 001 8 The temperature range in Melbourne on 29 April was 7°C. If the maximum temperature was 15°C, what was the minimum temperature?
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8 Maths Quest 9 for Victoria Golf scores In golf, par is the number of strokes considered necessary to complete a hole in expert play. A birdie is a score of one stroke under par and a bogey is one stroke over par. An eagle is a score of 2 strokes under par while 3 strokes under par is called an albatross. A double bogey is 2 strokes over par and a triple bogey is 3 strokes over par. 1 Use integers to represent: a par b a birdie c a bogey d an eagle e an albatross f a double bogey g a triple bogey. 2 Which score for a hole would be the most difﬁcult to achieve? 3 Leon and Dion have ﬁnished a round of 18 holes with the following information shown on their scorecards. Leon Dion pars 4 pars 6 birdies 3 birdies 2 bogeys 6 bogeys 4 eagles 1 eagles 0 double bogeys 2 double bogeys 2 albatrosses 0 albatrosses 1 triple bogeys 2 triple bogeys 3 Final score Final score What integer represents each person’s ﬁnal score as a number of strokes over, under or at par? 4 Who wins this round of golf? 5 Two professional golfers achieve overall ﬁnal scores for 18 holes of −8 and −6. a What does this mean? b Who achieved a better score for this round of golf? c How many strokes did each player make for the 18 holes if the course is considered to be a par 71 course?
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Chapter 1 Number skills 9Estimation and rounding Rounding to a given number of decimal places Ms Shopper’s bill at the supermarket comes to $94.68 and she pays $94.70. Mr Shopper’s bill is $83.72 and he pays $83.70. The bills have been rounded to the nearest 5 cents because the 5-cent is the smallest coin used. Ms Shopper’s bill has been rounded up because 68 cents is closer to 70 cents than to 65 cents. Mr Shopper’s bill has been rounded down because 72 cents is closer to 70 cents than to 75 cents. Measuring distances is another one of the many practical situations where it is necessary to round an answer to a given number of decimal places. For example, the distance between two towns is given to the nearest kilometre. It is not practical or useful to the average motorist that the distance between Melbourne and Sydney by a certain route is 1024.352 km. We give the distance simply as 1024 km. The accuracy of measurement is limited by what is practical and by the accuracy of the instrument being used to take the measurement. For example, with your ruler it would not be possible to measure anything more accurately than to the nearest millimetre. The measurement 5.6713 cm ≈ 5.7 cm because 5.6713 is closer to 5.7 than it is to 5.6. The rounded answer, 5.7, is the closest approximation to the exact answer. To round an answer to a given number of decimal places, consider only the ﬁrst digit after the required number of decimal places. If that digit is 0, 1, 2, 3 or 4, then leave it and all following digits out of the answer. If that digit is 5, 6, 7, 8 or 9, then the last digit to be written is increased by 1 and all else is left out. Many calculators are able to round off using the FIX function. On some scientiﬁc calculators, you need to press MODE ﬁrst. On a TI graphics calculator, press MODE , arrow down to the second row, then arrow across to highlight the number that corresponds to the required number of decimal places. Press ENTER to set this rounding condition. To undo this operation, press MODE , arrow down to highlight FLOAT and press ENTER . Any rounded answer is not an exact answer but a close approximation. WORKED Example 5 Round 15.439 657 to: a 1 decimal place b 3 decimal places. THINK WRITE a 1 Write the number. a 15.439 657 2 Look at the second decimal place to determine whether to ≈ 15.4 leave it or to round it up. The digit is 3; so rewrite the number without all digits after the ﬁrst decimal place. b 1 Write the number. b 15.439 657 2 Look at the fourth decimal place to determine whether to ≈ 15.440 leave it or to round it up. The digit is 6; so increase the third decimal place by 1. Note: Adding 1 to 9 gives 10, thus 439 becomes 440 and the zero must be included. Note: The more decimal places, the closer the approximation is to the exact answer.
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10 Maths Quest 9 for Victoria There were 70 000 people at the Melbourne Cricket Ground for Australia’s one-day match against the West Indies. Rounding to a given number of signiﬁcant ﬁgures Although the caption describes a crowd of 70 000, in reality there may have been 70 246 people. The number has been rounded to 1 signiﬁcant ﬁgure because the rest of the number, 246, has no impact on our image of the size of the crowd. When using very large or very small numbers, rounding to a given number of signiﬁcant ﬁgures is often used. To round to 1 signiﬁcant ﬁgure means having only 1 non-zero digit beginning from the left with the other digits being zeros. The number 367 rounded to 1 signiﬁcant ﬁgure is 400 because 367 is closer to 400 than to 300. To write 452 correct to 2 signiﬁcant ﬁgures, we need to consider whether 452 is closer to 450 or 460. It is closer to 450, and 4 and 5 are the 2 signiﬁcant ﬁgures. The method of deciding whether to leave or round up is the same as rounding to a number of decimal places. WORKED Example 6Round 347 629 to: a 1 signiﬁcant ﬁgure b 3 signiﬁcant ﬁgures.THINK WRITEa 1 Write the number. a 347 629 2 Look at the second signiﬁcant ﬁgure to determine whether to ≈ 300 000 leave it or to round it up. The digit is 4, so rewrite the number, replacing all digits after the ﬁrst signiﬁcant ﬁgure with zeros.b 1 Write the number. b 347 629 2 Look at the fourth signiﬁcant ﬁgure to determine whether to it ≈ 348 000 leave or to round it up. The digit is 6 so write the answer by adding 1 to the third digit and replace all other digits with zeros.
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Chapter 1 Number skills 11 Note: The more signiﬁcant ﬁgures taken, the closer the approximation is to the exact answer. When the ﬁrst non-zero signiﬁcant ﬁgure appears after the decimal point, any zeros before that ﬁgure are not signiﬁcant. WORKED Example 7Round 0.004 502 6 to 3 signiﬁcant ﬁgures.THINK WRITE1 Write the number. 0.004 502 62 The ﬁrst signiﬁcant ﬁgure is the 4. ≈ 0.004 50 Round to 3 signiﬁcant ﬁgures beginning with the 4. The last zero must be included in the answer because it is one of the signiﬁcant ﬁgures. Estimation Rounding is also used when making an estimation or mental approximation of an answer. Estimation is a method of checking the reasonableness of an answer or a calcu- lator computation. We can estimate an answer by rounding the numbers in the question to simple numbers that can be calculated mentally. WORKED Example 8Estimate answers to the following without calculating the exact answer.a 31 × 58 b 46 679 + 2351 × 65THINK WRITEa 1 Write the calculation. a 31 × 58 2 Round each number to 1 signiﬁcant ﬁgure. ≈ 30 × 60 3 Perform the mental calculation. = 1800b 1 Write the calculation. b 46 679 + 2351 × 65 2 Round each number to 1 signiﬁcant ﬁgure. ≈ 50 000 + 2000 × 70 3 Perform the mental calculation. = 50 000 + 140 000 = 190 000 remember remember 1. When rounding to a given number of decimal places, count only those places after the decimal point. 2. When rounding to a given number of signiﬁcant ﬁgures, begin counting from the ﬁrst non-zero digit. 3. A quick mental estimation can be used to check the accuracy of calculations. 4. Rounding is often used to convey a concept of size rather than an exact number.
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12 Maths Quest 9 for Victoria 1C Estimation and rounding WORKED 1 Round the following to: i 1 decimal place ii 2 decimal places iii 3 decimal places. Example 5 a 5.893 27 b 67.805 629 c 712.137 84 d 81.053 72 e 504.896 352 2 Round the following to 0 decimal places. (To 0 decimal places means to the nearest whole number.) a 25.68 b 317.19 c 1027.8 d 19.53 reads 3 Round the following to 1 decimal place. L Sp he a 3047.2735 b 24.7392 c 8.2615 d 19.9804 etEXCE Rounding WORKED 4 Round the following to: i 1 ii 2 iii 3 iv 4 signiﬁcant ﬁgures. and Example 6 a 574 248 b 430 968 c 28 615 d 1 067 328 e 458 610 significant figures DIY 5 Round the numbers in question 2 to 2 signiﬁcant ﬁgures. hca d WORKED 6 Round the following correct to 3 signiﬁcant ﬁgures. ExampleMat 7 a 0.085 246 b 0.000 580 4 c 0.000 008 067 3 Rounding d 0.006 765 73 e 0.000 026 973 f 0.000 352 1 WORKED 7 Estimate answers to the following without calculating the exact answer. d Example hca a 183 ÷ 58 b 78 × 11 c 632 + 169 d 1010 ÷ 98 8Mat Estimation e 17 × 19 f 476 ÷ 8 + 52 g (51 + 68) × 12 h 68 + 19 × 9 i 5 × (78 − 59) j 42 × 8 + 18 × 5 k 176 ÷ 18 + 689 ÷ 7 l 397 m 473 × 248 n 657 − 239 ÷ 49 o 12 345 + 549 × 146 8 multiple choice a The number 49.954 correct to 1 decimal place is: A 49.9 B 49.0 C 50 D 50.0 E 50.1 b The number 3 056 084 correct to 3 signiﬁcant ﬁgures is: A 3 050 000 B 3 056 000 C 3 057 000 D 306 E 3 060 000 c The number 0.008 065 3 correct to 3 signiﬁcant ﬁgures is: A 0.008 B 0.008 065 C 0.008 06 D 0.008 07 E 0.806 d A number rounded to 2 decimal places is 6.83. The original number could have been: A 6.835 B 6.831 C 6.8372 D 6.85 E 6.8 SHE ET 1.1 9 Each of the 178 students who attend the Year 9 Social has to pay $55. If the cost of Work hiring the band is $1000, estimate how much money would be available to pay for the supper and the security people. QUEST S MAT H GE 1 In 1832, a young runner named Mensen Ehrnot reportedly ran nearly EN 8950 km over a 59-day period. On each of those days he ran 16 hours CH L and rested for 8 hours. Estimate how many kilometres he ran, on AL average, per hour. 2 In the hundred consecutive whole numbers from 1 to 100, how many times does each of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 occur?
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Chapter 1 Number skills 13 1 1 Evaluate 9 − 13 − 14. 2 Evaluate 8 − 8 ÷ 4. 3 Evaluate (13 + 5 × 7) ÷ 12. 4 Evaluate −25 + −10 − −50. 5 Evaluate −84 ÷ 12 × 3. 6 Evaluate −18 + (−9 + 11) × 14. 7 Insert signs to make the following equation true. 5 K 21 K 7 K 5 = 20 8 Round 1.746 582 to 4 decimal places. 9 Round 0.006 059 9 to 4 signiﬁcant ﬁgures. 10 Give an estimate for 78 + 43 + 55 − 86.Decimal numbers Decimal numbers are so much a part of everyday life that we need to be able to use them, put them in order and convert them to simple fractions and percentages. When using either your graphics calculator or a scientiﬁc calculator, enter the calcu- lation as written and the calculator will perform the calculation using the correct order of operations. There are, however, many things that we need to be able to do ourselves with decimals without the aid of a calculator. Ordering decimal numbers Ascending order means from lowest to highest and descending order means from highest to lowest. This is done by ﬁrst writing each number with the same number of decimal places, adding zeros where necessary. We then look at the left-most digit. The greater this digit, the greater the decimal number. If the left-most digits are the same, we move to the next digit, and so on. WORKED Example 9Write the following decimal numbers in ascending order:0.66, 0.606, 0.6.THINK WRITE 1 Write the numbers. 0.66, 0.606, 0.6 2 Write all numbers with the largest 0.660, 0.606, 0.600 number of decimal places, in this case 3, then compare. 3 Write the original numbers in 0.6, 0.606, 0.66 ascending order after looking at the second and third decimal places.
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14 Maths Quest 9 for Victoria Finite or terminating decimal numbers Finite decimal numbers have a ﬁxed or ﬁnite number of decimal places and can be written as a fraction with a denominator that is a multiple of 10. If the decimal number has 1 decimal place, the denominator of the fraction is 10; if there are 2 decimal places, the denominator is 100; if there are 3 decimal places, the denominator is 1000 and so on. In each case the numerator is the decimal number without the decimal point. These fractions are simpliﬁed where possible. WORKED Example 10Convert each of the following to fractions in simplest form:a 0.65 b 1.2 c 0.6275.THINK WRITEa 1 Write the decimal number. a 0.65 13 2 There are 2 decimal places, so write as a fraction 65 = ------------ - with a denominator of 100 and simplify by 100 20 cancelling. (You may use a calculator to simplify.) 13 3 Write the answer. = ----- - 20b 1 Write the decimal number. b 1.2 6 2 There is 1 decimal place, so write as a fraction with a 12 = ------- - denominator of 10 and simplify by cancelling. (You 10 5 may use a calculator to simplify.) 1 3 Write the answer as a mixed number. = 1 -- - 5c 1 Write the decimal number. c 0.6275 251 2 There are 4 decimal places, so write the fraction with 6275 = --------------------- - a denominator of 10 000 and simplify. (You may use 10 000 400 a calculator to simplify.) 251 3 Write the answer. = -------- - 400 Converting decimal numbers to percentages To convert a decimal number to a percentage, we multiply the decimal number by 100 and include the % sign. WORKED Example 11 Convert 0.357 to a percentage. THINK WRITE 1 Write the decimal number. 0.357 2 Multiply the decimal number by 100 by moving the = (0.357 × 100)% decimal point 2 places to the right. Remember to include = 35.7% the percentage sign.
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Chapter 1 Number skills 15 remember remember 1. To order decimal numbers, write each with the same number of decimal places and compare. 2. To write ﬁnite decimal numbers as fractions, make the denominator an appropriate multiple of 10 and simplify where possible. The number of zeros in the denominator should be the same as the number of digits after the decimal point. 3. To convert a decimal number to a percentage, multiply by 100 and include the percentage sign. 1D Decimal numbers 1 Calculate each of the following. Math 1.3 a 6.56 + 3.214 b 4.87 − 2.493 c 5.6 × 7.04 HEET cad SkillS d 5.75 ÷ 0.25 e (4.5 + 2.1) × 3.5 f (8.6 − 4.4) ÷ 7 Operations g 4.8 − 2.16 ÷ 0.18 h 3.2 × (6.4 + 0.78) i 7.2 ÷ 0.12 × 6 with decimal j 7.2 ÷ (0.12 × 6) k 5.8 × (3.1 ÷ 0.4) l 6.2 + 3.5 × 2 numbers 2 Calculate each of the following, rounding your answers to 2 decimal places. HEET 1.4 SkillS a 6.46 × 2.356 b 8.12 × 5.4 ÷ 9.6 c 8 ÷ 0.35 + 2.1 d (6.509 + 4.804) ÷ 0.341 e 3.2 × 4.057 − 13.91 ÷ 2.43WORKED 3 Write each of the following sets of decimal numbers in ascending order. 1.5Example a 0.66, 0.4, 0.71 b 2.3, 0.23, 23 c 0.7, 1.32, 1.04 HEET SkillS 9 d 1.02, 1.1, 1.22 e 0.5, 0.56, 0.06 f 0.323, 0.4, 0.35 4 Write each of the following sets of decimal numbers in descending order. a 0.24, 0.204, 0.2004 b 0.062, 0.081, 0.11 c 0.7, 0.77, 0.707 d 0.082, 0.09, 0.0802 e 1.2304, 1.23, 1.204 f 0.359, 0.39, 0.3592 5 multiple choice a The expression 6.43 × 2.356 ÷ (2.1 − 0.365) correct to 2 decimal places is equal to: A 0.36 B 6.85 C 87.31 D 8.73 E 6.84 b The false statement is: A 0.67 < 0.7 B 0.506 < 0.51 C 0.735 > 0.73 D 0.203 < 1.3 E 0.085 > 0.85 c The expression −0.9 + 6.5 × 0.004 − 1.2 ÷ 0.6 is equal to: A −1.074 B −2.874 C −2.64 D −20.874 E −0.84 d A good estimate for 5.2 × 0.2 + 1.18 ÷ 0.012 is: A 101 B 11 C 110 D 99.373 E 1010WORKEDExample 6 Convert each of the following to fractions in simplest form. XCE L Spread HEET 1.6 sheet E a 0.9 b 0.6 c 0.16 d 0.27 e 0.78 SkillS 10 Converting f 0.15 g 0.08 h 1.5 i 2.84 j 0.125 decimals to k 0.484 l 0.963 m 0.775 n 0.0625 o 0.8875 fractionsWORKED 7 Convert each of the following to percentages. L Spread 1.7Example XCE HEET a 0.72 b 0.31 c 0.89 d 0.57 e 0.9 sheet E SkillS 11 f 0.06 g 0.782 h 0.6175 i 0.0094 j 1.35 Converting k 1.602 l 11 m 2.3 n 5.75 o 2.485 decimals to percentages
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16 Maths Quest 9 for Victoria 8 multiple choice a In simplest form and as a fraction 0.3125 is equal to: 3125 5 A --------------- 10 000 - B 31.25 C ----- 16 - D 31 1 -- 4 - E 13 ----- 40 - b As a percentage 0.0875 is equal to: 7 A 0.875% B 8.75% C 87.5% D ----- % 80 - E 875% c The number 0.656 25 is equal to: A 13----- 20 - B 53 ----- 80 - C 11----- 16 - D 21 ----- 32 - E 5 -- 8 - 9 Francis is paid $11.50 an hour for babysitting. If he works for 7 hours over the weekend, how much does he earn altogether? 10 Yvette babysits for 5 hours after school each Friday. She is paid $10 an hour. a How much does she earn each week? b If she banks $3.25 of the money each week, how much does she have left to spend? QUEST SM ATH GE 1 Allison, Bhiba, Chris and Dinesh ordered one box of apples to share EN equally between them. However, no one was present when the box was 1 CH L delivered. Allison arrived and took -- of the apples. Later, Bhiba came - 4 AL 1 and took 3-- of the apples left in the box. Then Chris came and did the - same. Finally Dinesh arrived and took his rightful share of the remaining apples. If 9 apples remained in the box, how many apples were in the box originally? 2 Mitchell has mown 0.6 of the lawn. He still has 50 m2 of lawn to mow. What is the total area of the lawn? 3 A train 0.5 km long is travelling at a speed of 80 km/h. How long will it take the train to go completely through a tunnel which is 1.5 km long?
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17 Chapter 1 Number skillsWhat type of creature is a KATYDID creature KATYDID and where are its ears? where are Answer the decimal questions to find the puzzle’s code. = 3 – as a decimal = 8.6 = 2.8 + 3.6 4 – 4.9 = = = 0.3 + 0.4 = 10% as a decimal = 0.5 × 8.4 = = = 5 23 = 93% as a decimal = – 2 = –– 50 as a decimal = = = = 5 × 0.3 = 12.7 = 8.34 – 9.87 – 6.54 = 7 = 1.2 – 0.8 = –– 20 as a decimal = 6.3 ÷ 0.63 = = = = 22% as a decimal = 0.67 = 60% as a decimal + 0.53 = = = 12 ÷ 0.5 = 1.1 × 0.8 = 0.2 × 20 = = = = 4 – as a decimal = 1.64 ÷ 0.4 = 0.87 5 + 1.33 = = = 51% as a decimal = 1.6 = 5.26 + 1.87 – 0.95 = = = 4.5 = 6 × 0.8 = 17 –– as a decimal 4 × 1.2 = = = 2.374 = 7.63 = 3 – as a decimal + 3.926 – 3.23 8 = 5.4 2.5 4.0 3.7 1.8 0.46 2.2 0.375 4.8 0.22 24 0.4 0.8 6.3 6.4 4.4 4.1 0.93 0.75 7.13 0.88 0.6 0.1 0.51 4.2 1.5 0.65 10 0.7 1.2 4.25 2.83 0.35
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18 Maths Quest 9 for VictoriaFractions There are many essential skills that you will need with fractions. You can review them in the exercise below and by the matching SkillSHEET. You should be able to simplify fractions and convert between improper fractions and mixed numbers. You should also be able to use your calculator efﬁciently. Graphics Calculator tip! Obtaining an a fraction expressed as answer As with any calculation involving fractions, if you wish to have an answer expressed as a fraction then each calculation needs to end by pressing MATH , selecting 1: Frac and pressing ENTER . M For example, to simplify 28 on your graphics calcu- ----- 44 - lator, enter 28 ÷ 44 then press MATH choose option 1: Frac, then press ENTER . This can be seen in the M screen at right. Note: The graphics calculator gives all answers as improper fractions and will not give answers as mixed numbers. It is important that we know how to perform calculations using fractions both with and without a calculator. 28 Without a calculator, we would simplify ----- 44 - by dividing both the numerator and the denominator by the highest common factor (HCF) of both. The HCF of 28 and 44 is 4. 7 28 28 ----- = --------- - 11 - 44 44 7 = ----- - 11 WORKED Example 12 Evaluate the following. a 3 + 5 -- 4 - -- 6 - b 3 -- 4 - × 5 -- 6 - c 21 ÷ -- 4 - 3 -- 5 - THINK WRITE a 1 Write the fraction calculation. a 3 -- 4 - + 5 -- 6 - 2 Write both fractions with the same = 9 ----- 12 - + 10 ----- 12 - denominator by using equivalent fractions. 3 Add the fractions and simplify the answer = 19 ----- 12 - by writing it as a mixed number. = 1 ----- 12 7 - 31 b 1 Write the fraction calculation and cancel b ---- - × 5 ---- - 4 62 where applicable. 2 Multiply numerators and multiply = 5 -- 8 - denominators.
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Chapter 1 Number skills 19THINK WRITEc 1 Write the fraction calculation. c 21 ÷ -- 4 - 3 -- 5 - 2 Change the mixed number to an improper fraction. = 9 -- 4 - ÷ 3 -- 5 - 93 3 Times and tip, (change the division sign to a multiplication = ---- - × 5 ---- - 4 31 sign and tip the second fraction) and cancel. 4 Multiply numerators and multiply denominators; then = 15 ----- 4 - simplify the answer by writing the fraction as a mixed number. = 33 -- 4 - Graphics Calculator tip! Fraction calculations To perform the calculations in worked example 12 on a graphics calculator, the following steps need to be followed: (a) Enter 3 ÷ 4 + 5 ÷ 6, press MATH , choose 1: Frac then press ENTER . The result is given M as 19 . The graphics calculator gives all answers as ----- 12 - improper fractions. (b) Enter 3 ÷ 4 × 5 ÷ 6, press MATH , choose 1: Frac then press ENTER . M (c) Enter (2 + 1 ÷ 4) ÷ (3 ÷ 5), press MATH , choose 1: Frac then press ENTER . M WORKED Example 13 3Find -- 7 - of 98.THINK WRITE 31 Write the calculation. -- of 98 7 - 98 142 Change the ‘of’ to ×, write the whole number over 1 and = 3 ---- - × ---------- - 71 1 cancel if applicable.3 Multiply numerators and multiply denominators. = 42 Writing fractions with the same denominator allows us to compare the size of fractions. WORKED Example 14Write the fractions 2 , 8 , -- -- - - 3 9 5 -- 6 - in ascending order.THINK WRITE 2 8 51 Write the fractions. -- , -- , -- - - - 3 9 62 Write all fractions as equivalent fractions by ﬁnding the = 12 16 15 ----- , ----- , ----- - 18 18 18 - - lowest common denominator, in this case 18.3 Rewrite the original fractions in the correct order. = 2, 5, -- -- - - 3 6 8 -- 9 -
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20 Maths Quest 9 for Victoria Another way of writing fractions in order is to convert each fraction to a decimal number before comparing them. Converting fractions to decimal numbers To convert a fraction to a decimal number, divide the numerator by the denominator. WORKED Example 15 7 Convert -- 8 - to a decimal number. THINK WRITE 7 1 Write the fraction. -- 8 - 0.875 2 Divide the numerator by the denominator. 8 ) 7.000 3 Write the fraction and the equivalent decimal number. 7 -- 8 - = 0.875 Converting fractions to percentages To convert a fraction to a percentage, multiply the fraction by 100 and include the % sign. WORKED Example 16 23 Convert ----- 40 - to a percentage. THINK WRITE 23 1 Write the fraction. ----- 40 - 100 5 2 Multiply by 100, include the percentage sign and = ( -------- × 23 2 ---------- 1 - )% 40 cancel if applicable. 3 Multiply the numerators and multiply the denominators. = 115 -------- % 2 - 4 Simplify by writing as a mixed number. = 57 1 % -- 2 - remember remember 1. To write fractions in simplest form, divide the numerator and the denominator by the highest common factor (HCF) of both. 2. To change improper fractions to mixed numbers, divide the numerator by the denominator and express the remainder as a fraction in simplest form. 3. To change a mixed number into an improper fraction, multiply the whole number by the denominator, add the numerator and write the result over the denominator. 4. To add or subtract fractions, form equivalent fractions with the same denominator, then add or subtract the numerators. 5. To multiply fractions, cancel if possible, then multiply the numerators, multiply the denominators and simplify if appropriate. 6. To divide fractions, times and tip, then simplify if possible. 7. To add, subtract, multiply or divide mixed numbers, change the mixed numbers to improper fractions ﬁrst. (When subtracting, an alternative method is to make the second fraction into a whole number after writing the fractions with the same denominator.) 8. To write fractions in order, express them as equivalent fractions and compare. 9. To ﬁnd a fraction of an amount, multiply the fraction by the amount. 10. To convert a fraction to a decimal number, divide the numerator by the denominator. 11. To convert a fraction to a percentage, multiply the fraction by 100 and include the % sign.
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