MATHEMATICS AND STATISTICS. Apply numeric reasoning in solving problems Internally assessed credits Factors, multiples and primes The set of whole numbers is infinite (continues without end). 0,, 2,,, 5, 6, 7, 8, 9, 0,, 2 The following ideas should be familiar.. The multiples of are, 6, 9, 2, 5, [multiply by, 2,,, 5, ] 2. The factors of 2 are,, 7, 2 (factor pairs: 2 = 2, 7 = 2) [,, 7, 2 divide into 2 exactly, with no remainder]. is a prime number since it has exactly two factors, and. 2 is a composite number as it has more than two factors.. The lowest common multiple (LCM) of 6 and 8 is 2. [the first multiple of 8 (8, 6, 2, ) that is also a multiple of 6, is 2] 5. The highest common factor (HCF) of 5 and 6 is 9. [5 = 9 5, 6 = 9 ; and 5 have no common factor] 6. The prime factorisation of 60 is 2 2 5 or 2 2 5. A factor tree is shown (prime factors circled) 60 2 0 2 5 5 [60 = 2 0 = 2 2 5 = 2 2 5 (prime factor tree)] Exercise A: Factors, multiples and primes. a. List all the factors of the following numbers: i. 0 ii. 2 iii. 7 iv. 25 b. What is the highest common factor of 0 and 2? 2. a. List the first four multiples of the following numbers: i. 9 ii. 2 iii. 2 iv. 25 b. What is the lowest common multiple of 9 and 2?. a. Which of the following numbers are primes? 7 9 2 2 25 27 29 b. List the next two prime numbers after 29.. Find whole numbers between 0 and 60 inclusive which are: a. multiples of 9 b. factors of 20 c. prime numbers d. composite numbers with exactly four factors Ans. p. 5
2 Achievement Standard 9026 (Mathematics and Statistics.) 5. A red light flashes every 2 seconds. A blue light flashes every 0 seconds. At 0 a.m. exactly, Mia notices that both lights flashed together. She decided to wait until both lights flashed together again. How long did she have to wait? e. Use the quicker strategy to find the LCM of i. 8 and 60 ii. 2 and 2 8. a. Draw a factor tree and use it to express the following numbers as a product of their prime factors. i. 0 6. Jenny takes her dog to the vet on the first of the month every 6 months. Louis takes his cat to the vet on the first of the month every months. Zane takes his hamster to the vet on the first of the month every 9 months. One month, all three took their animals to the vet on the same day. How many months will pass before this happens again? ii. 70 7. Two strategies are shown for finding the LCM of 5 and 20. a. List the multiples of 5 (the smaller number), until you reach a number which is a multiple of 20 (the bigger number). iii. 65 b. List the multiples of 20 (the bigger number), until you reach a number which is a multiple of 5 (the smaller number). c. What is the LCM of 5 and 20? d. Explain which strategy is quicker, and why. b. The volume of a cuboid is: Volume = length width height. A cuboid has sides whose lengths are whole numbers greater than. Find the dimensions (length, width and height) of the cuboid if it has volume i. 0 mm ii. 70 cm iii. 65 m
Apply numeric reasoning in solving problems 9. a. Complete the table of factor pairs of 0, e.g. 0, 2 70 0 2 70 20 b. Complete the table of factor pairs of 68 b. Use the strategy in a. to work out the HCF of i. 8 and 57 ii. 9 and 65 68 28 7 8 c. A class has a number of children that is between 20 and 0. Each child in the class must pay a fee for art materials and the cost of the bus for a class trip. Both fees are a whole number of dollars and all children pay. The total art fee collected is $0. The total bus cost collected is $68. i. How many children are in the class? iii. 69 and. The prime factorisation of a = 2 2 5. The prime factorisation of b = 2 5. a. i. If a is doubled it becomes 2a. What is the prime factorisation of 2a? ii. If b is tripled, it becomes b. What is the prime factorisation of b? ii. How much is the art fee per child? iii. What is the LCM of a and b? iii. How much does each child pay for the bus trip? b. i. Is 5 a common factor of a and b? 0. a. i. One factor pair multiplying to 5 is 5. Find the other factor pair. ii. Is 2 2 a common factor of a and b? iii. Is a common factor of a and b? ii. One factor pair of is. Find the other factor pair. iv. What is the HCF of a and b? Explain your strategy. iii. Explain how you can use these results to find the HCF of and 5.
Achievement Standard 9026 (Mathematics and Statistics.) The integers The integers are the whole numbers and their opposites (shown using a negative sign). The integers are shown on the number line below. 5 2 0 2 5 6 Exercise B: The integers. Find the value of a. + 6 b. 72 8 c. 5 + 5 d. 2 e. 5 + 9 f. 8 2 Ans. p. 5 Each number on the number line is greater than numbers to its left and less than numbers to its right, e.g. 2 is greater than 5 ( 2 > 5) and 5 is less than zero ( 5 < 0). Calculating with integers The number line illustrates addition and subtraction of integers: when adding a positive integer move right (to increase); when subtracting a positive integer move left (to decrease). + 7 = 5 = 9 2 = 2 0 9 8 7 6 5 2 0 2 5 x When adding or subtracting integers: Two like signs in a row mean addition (+), e.g. = + = 7 Two unlike signs in a row mean subtraction ( ), e.g. 7 + 0 = 7 0 = When multiplying or dividing two integers: If the signs are the same the answer is positive, e.g. 2 = 6 2 = 6 If the signs are opposite, the answer is negative, e.g. 2 = 8 2 = In word problems, the correct operation needs to be identified, so that the calculation can be written down and worked out. Mila s bank account is overdrawn by $65. She deposits $2 and pays for $76 of groceries. Write down some calculations that can be used to find her bank account balance after these transactions. A. Mila s balance after deposit = 65 + 2 [ overdrawn is a negative balance] = $7 Mila s balance after groceries = 7 76 = $7 Mila now has a balance of $7 g. 7 + 6 h. 25 + i. 8 8 j. 2 6 k. 2 5 l. 72 6 m. 8 6 n. ( 6) 2 o. 2 p. 2. A lift runs from the basement, six floors below ground level, to the top of the building, 20 floors above ground level. Jerome s office is on the th floor. He gets in the lift and goes down 7 floors. a. How many more floors below Jerome is the basement? b. How many floors up will Jerome need to go to get to Beth s office on the 0th floor?. It is recommended to customers that a packet of frozen peas be stored at 8 C. The supermarket refrigerator stores its frozen goods at 5 C. How much colder is this than a domestic refrigerator?. The temperature is recorded as C at 5 a.m. By 8 a.m. the temperature has gone up by C. a. What is the temperature at 8 a.m.? Over the next six hours the temperature goes up steadily by C every two hours. b. What is the temperature at 2 p.m.? 5. Kim doubles an integer, then adds 2. The result is 2. What was the integer?
Apply numeric reasoning in solving problems 5 Order of operations When simplifying expressions with several operations, follow the order BEDMAS: B Brackets first E Exponents (powers) next DM Division/multiplication next (left to right) AS Addition/subtraction last (left to right). 8 5 = 8 5 [ before ] = 8 + 5 = 2 2. (8 ) 5 = 5 5 [brackets first] = 25 Exercise C: Order of operations. Use the correct order of operations to find the values of the following expressions. a. 6 7 + 8 b. 5 + 5 c. 7 ( ) d. 2 + 8 2 Ans. p. 52 A calculator follows the correct order of operations for integers. Take care to include hidden brackets in calculations whenever they occur in quotient expressions. e. 2 + 8 9 To simplify 5 using a calculator, the 8 6 numerator and denominator should be bracketed. Press f. 5 7 g. ( (5 )) to get the answer 0.5 In word problems, identify the operations then write down the required calculation using the correct order of the operations. Then calculate your answer. Q. Jack buys a cabbage and kg of carrots at $2 per kilogram. If he pays with a $20 note and gets $9 change, how much was the cabbage? A. $9 change from $20 means that the vegetables cost 20 9 = $ The carrots cost 2 = $8, so Cost of cabbage = 8 [subtracting off the cost of the carrots from total spent] = $ Scientific calculators follow the correct order of operations. Note that some calculators allow you to enter fractions as they are written, e.g. 2, so that brackets 2 are not required. h. (2 + ) ( ) i. 22 ( + 5) 6 2 j. (5 2) 2 9
6 Achievement Standard 9026 (Mathematics and Statistics.) k. ( 5) 2 2 2 l. 2 2 9 0 5. A carton holding 8 packets of gum weighs 756 g altogether. Later in the week there are packets of gum left in the carton, and the total weight is 6 g. a. What is the weight of a packet of gum? m. 2 2 + b. Find the weight of the carton. n. (( 2 6) 2 ) 2 o. 2 + + ( 2) 2 2. A company makes a loss of 2 million dollars in its first year of trading. In the following two years it makes a profit of $625 000 per year. What is the average profit or loss per year made by the company over the three years? 6. Identify any errors between one line and the next in the following working. Circle the errors, then work out the correct answer. (5 9) 2 2( 2 5) 2 = (5 7) 2 2(6 5) 2 = 8 2 2() 2 = 6 2 2 = 6 = 60. Libby receives pay of $7 which she deposits in her cheque account. She then writes out cheques for $558 and $5. If her account is now $78 overdrawn what was her cheque account balance before she received her pay?. A competition mark is worked out by doubling the sum of the middle four scores, then adding the average of the top and bottom scores. Martha scores: 7 7 6 8 5 9 What is her competition mark? 7. On an internet stationery website, rulers cost 85 cents each and pencils can be bought separately, or at a cost of $.20 cents for a packet of twelve pencils. Postage costs $2. A customer uses the website to order some rulers and pencils. Explain what is being calculated using the following calculation. 85 6 + 20 2 + (20 2) 6 + 200 00
Apply numeric reasoning in solving problems 7 Powers of numbers Repeated multiplication can be written using powers. Squares and cubes of numbers Squaring a number means multiplying the number by itself. It is useful to be familiar with the squares of the natural numbers. These are:,, 9, 6, 25, 6, 9, 6, 8, 00, [ 2 = = ; 2 2 = 2 2 = ; 2 = = 9; 2 = = 6; 5 2 = 5 5 = 25; 6 2 = 6 6 = 6; 7 2 = 7 7 = 9; 8 2 = 8 8 = 6; 9 2 = 9 9 = 8, etc.] Exercise D: Powers of numbers. Evaluate the following powers. a. 5 2 b. 26 2 c. 9 2 d. 5 e. f. 7 g. 6 h. 0 2. Find the value of the following: a. ( 5) 2 b. ( ) c. 6 Ans. p. 52 These numbers are sometimes called perfect squares. The first few perfect cubes should be familiar too:, 8, 27, 6, [ = = ; 2 = 2 2 2 = 8; = = 27; = = 6, etc.] Calculators have special keys for squaring numbers and cubing numbers. Alternatively the general power key can be used Higher powers of numbers s of higher powers follow.. 5 5 5 5 5 5 = 5 6 6 factors Using a calculator, press: to get a value of 5 625 2. = ( ) 5 5 factors [note the use of brackets when negative numbers are being raised to a power] Using a calculator, press: Note: Some calculators have to get 2 instead of d. 2 2 e. 7 2 ( 7) 2 f. ( 2) ( 2 ). A square has side length 2 cm. a. Find its area. A second square has a side length of 2 cm. b. How many times bigger is the area of the second square compared with the area of the first?. Find the next number in each of the following sequences: a., 9, 25, 9, b., 8, 27, 6, 5. The volume of a hemisphere is 2 πr. Find the volume of a hemisphere when the approximation π = is used and r = 5 cm.
8 Achievement Standard 9026 (Mathematics and Statistics.) Ans. p. 52 Zero and negative integer powers of numbers By using the laws of indices, definitions can be found for integer powers. Zero power x 0 x = x [since x n x m = x n + m ] but x = x, therefore: x 0 =. 5 0 = 2. ( 5) 0 =. 2 0 = Negative integer powers x n x n = x 0 which is [since x n x m = x n + m ] Dividing both sides by x n gives the following relationship: x n = x n Inverting this gives the rule: = xn n x. 6 2 = 6 2 2. 5 = 5 = 6 = 25 Exercise E: Zero and negative integer powers of numbers. Evaluate the following without a calculator. Leave answers as fractions. a. 2 b. c. 6 d. 2 7 e. 0 0 f. ( 5) 2 2. Evaluate the following without a calculator. 2 a. b. c. d. 2 2 5 e. 2 2. Use a calculator to evaluate a. (0.8) 2. 8 = 8 2 [separating into fractions] 2 = 8 2 [since = x n ] x n = 8 6 = 28 To raise a fraction to a negative power, use the following rule: n n a b = b a b. (0.0) 2 c. (0.25) d. 2 2 e. (2 2 ). Find x if a. x = 0.00. 2 = 2 2. 2 = 2 b. x 2 = 0.25 = 2 = 6 c. x = 2 Calculators can be used to evaluate numbers raised to zero and negative integer powers. d. = 8 x e. x 2 = 9 6
ANSWERS Exercise A: Factors multiples and primes (page ). a. i., 2,, 5, 6, 0, 5, 0 b. 6 ii., 2,,, 6, 8, 2, 2 iii., 7 iv, 5, 25 2. a. i. 9, 8, 27, 6 b. 6 ii. 2, 2, 6, 8 iii. 2, 8, 26, 68 iv. 25, 250, 75, 500. a. 7, 9, 2, 29 b., 7. a. 5, 5 b. 0, 8, 60 c.,, 7, 5, 59 d. 6, 5, 57 5. 60 seconds (or minute) 6. 6 months (or years) 7. a. 5, 0, 5, 60 b. 20, 0, 60 c. 60 d. Multiples of bigger number is quicker as step size is larger. e. i. 80 ii. 96 8. a. i. 2 5 ii. 2 5 7 iii. 5 b. i. 2 mm mm 5 mm ii. 2 cm 5 cm 7 cm iii. m 5 m m 9. a. 0 2 70 5 5 28 7 20 0 b. 68 2 8 56 2 6 28 7 2 8 2 2 c. i. 28 ii. $5 iii. $6 0. a. i. 7 ii. 2 7 iii. 7 is a common factor, and 2 and have no factors in common so 7 is the HCF. b. i. 9 ii. iii.. a. i. 2 5 ii. 2 5 iii. 2 5 b. i. Yes ii. Yes iii. No iv. 2 2 5 = 500 Multiply smaller powers of common prime factors. Exercise B: The integers (page ). a. 8 b. 6 c. 0 d. 6 e. f. 20 g. h. 9 i. 0 j. 2
52 Answers ANSWERS k. 60 l. 2 m n. 6 o. 6 p. 27 2. a. floors below b.. 7 C colder. a. 27 C b. 8 C 5. 7 Exercise C: Order of operations (page 5). a. 5 b. 5 c. 2 d. 2 e. f. 6 g. 27 h. 6 i. 9 j. 0 k. 9 l. m. 2 n. 969 o. 2. Loss of $250 000 per year. $99. 6 5. a. 5 g b. 76 g 6. (5 9 ) 2 2( 2 5) 2 = (5 7) 2 2(6 5) 2 = 8 2 2() 2 = 6 2 2 = 6 = 60 Correct working is (5 9 ) 2 2( 2 5) 2 = (5 7) 2 2(6 5) 2 = 8 2 2() 2 = 6 2 = 2 76 2 = 2 2 = 2 5 7. Numerator is cost (in cents) of 6 rulers plus two packets of pencils plus 6 individual pencils plus postage. Dividing by 00 gives the total cost in dollars (which is $5.0). Exercise D: Powers of numbers (page 7). a. 2 025 b. 676 c. 6 d. 2 e. 6 f. 2 0 g. h 0 000 2. a. 25 b. 6 c. 296 d. 2 e. 98 f. 0. a. cm 2 b. times. a. 8 b. 25 5. 6 750 cm Exercise E: Zero and negative integer power of numbers (page 8). a. d. 6 28 b. c. e. f. 2. a. 6 b. 27 c. d.. a. 2 2 e. 8 26 25 6 9 25 or.5625 b. 625 6 c. 52 d. 2 e. 2 or 2.57 ( d.p.) 7. a. 0 b. 2 c. 0.5 or 2 d. e. Exercise F: Roots of numbers and fractional powers (page 0). a. 7 b. 2 c. 8 d. e. 7 f. 2 2. a. 7 b. c. d. 5 e. 2. a. i. ii. iii. iv. 6, 8, 9, 50, 69 9, 2, 6 00, 07, 2
INDEX BEDMAS 5 benchmark (numbers or fractions) composite compound interest consecutive 0 cube of a number 7 cube root 9 decimal 26 decimal place 29 denominator (of fraction) direct proportion 2 equivalent fraction estimating 29 exchange rate 9 factor factor tree fraction fraction part (of a decimal) 26 Goods and Services Tax (GST) GST-exclusive GST-inclusive percentage (%) 5 percentage increase/decrease 7 perfect square 7 power of a number 7 prime prime factorisation principal rate 9 ratio 6 rational number 9 root of a number 9 rounding numbers 29 significant figure 29 simple interest simplest form of fraction simplified fraction square of a number 7 square root 9 standard form 2 surd 9 whole number part (of a decimal) 26 whole numbers highest common factor (HCF) integer inverse proportion 2 irrational number 9 lowest common multiple (LCM) multiple number line numerator of fraction