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More on the Number System Including BODMAS and Rounding Try the following sum: 3+2x5 Try the sum on a non-scientific calculator. What answer does it give? Try the sum on your scientific calculator. What answer does it give? 25 13 Which answer is correct? To avoid confusion, there are rules to follow when a sum involves more than one operation. This method must always be followed even if it is not specified in the question. Scientific calculators are programmed to automatically follow the order of operations. 2 BODMAS • • • • • • B O D M A S BRACKETS () ORDER or POWER OF DIVISION MULTIPLICATION ADDITION eg 23 x + SUBTRACTION - 3 • • • • • • B O D M A S When a sum involves only division and multiplication, the order is not important. Simply work from left to right. When a sum involves only addition and subtraction, the order is not important. Simply work from left to right. • 4x6÷8÷2x5= • 12 – 5 + 9 + 6 – 10 = 4 Examples (BODMAS) • 5x3–2x3+84=? 11 • 5 x (3 – 2) x 3 + 8 4 = ? 17 • (5 x 3 – 2 x 3 + 8) 4 = ? 4.25 • (5 x (3 – 2) x 3 + 8) 4 = ? 5.75 • 9x9+5x5=? 106 • 60 5 + 42 = ? 28 5 Exercises 1. 4 x (12 – 6) + 6 ÷ 3 3. 2 x (20 – 2) 9 2. 36 (5 + 4) + 3 x 6 4. 52 x (8 2) + 3 x 8 Add in brackets to make the following correct: 5. 6. 7. 24 + (7 – 3) x 2 = 32 (9 + 5 )x (9 – 5) = 56 4 x (3 + 2 ) + 32 = 29 6 Rounding off to decimal places Work out 12971 ÷ 23.3 on your calculator. The display shows 556.695279 We can round this off in several ways according to the degree of accuracy we need to work to. Remember, when rounding off the rules are as follows: Always look at the digit to the right of the number of places you are rounding off to. If that digit is 5 or more, round up. If the digit is 4 or less, leave the number as it is. 7 Round off the following to the degree of accuracy specified. a) 37.481 (2 d.p.) b) 113.471 (1 d.p.) c) 30.698 (2 d.p.) d) 27.387 (2 d.p.) e) 28.997 (1 d.p.) f) 39.912 (1 d.p.) g) 456.34567 (3 d.p.) 8 Rounding off to significant figures This is another way of rounding numbers off. NWHC News October 2005 237 956 attend Concert. kdjiejllk kljaldj ijlkdjiowl lijasdjf laudlfj lksajf;lk jalkjdfl ;alsjd jfie hl;hpoijhd lakjdkjf a;lkjd jkf;alj djflajlsdj flasjldfj a;lsdjf ja;lsjd jkd;alkjd kdk;lajsljd kmdls;sdkuriewxl ujuhqsicj lsjhpeoiriujlknwe98osen;l dlj ldfliub 3; kadj kajn’piqa jpsidfpiup joiajspo oijsldjfl lskdjfl /lkdjocsdlrjiuyuhgfgvnbhvt.ljvosy oewjrfl asdf’l josdjnfglsdfo alsdfo uha’sdfjo hsdfn;lajdso jalsdj nawoif lskdjc;o i\js ljfa\shf ‘lksjdfi kdjuef ;oisdlf ;sidf sa;l id lasdjfoiw ‘dpiuf ldjf;osdf jdf’sasijd flaksdjf ashdf;ljsdlif a.lsdjf fjdls;aldh pasjd ;lskdjf jfkdl;saljsd fjkd’a kdjiejllk kljaldj ijlkdjiowl lijasdjf laudlfj lksajf;lk jalkjdfl ;alsjd jfie hl;hpoijhd lakjdkjf a;lkjd jkf;alj djflajlsdj flasjldfj a;lsdjf ja;lsjd jkd;alkjd kdk;lajsljd kmdls;sdkuriewxl ujuhqsicj lsjhpeoiriujlknwe98osen;l dlj ldfliub 3; kadj kajn’piqa jpsidfpiup joiajspo oijsldjfl lskdjfl /lkdjoc 9 The most significant figure in a number is the figure which has the greatest place value in that number, when reading from left to right. 237 → 2 has the greatest place value, therefore is the first significant figure 0.00328 → 3 has the greatest place value and is the first significant figure. The zeros are simply used to indicate place value and are not considered to be significant. Zeros must be used when rounding off to significant figures to preserve place value. Round off 4 500 732 to 2 significant figures (2 s.f) Round off 0.000 364 907 to 1 s.f. 10 Round off the following to the correct number of significant figures. a) 37.48 (3 s.f.) b) 2991 (2 s.f.) c) 0.00892 (2 s.f.) d) 20.374 (3 s.f.) e) 375.21 (2 s.f.) f) 0.000748 (2 s.f.) g) 0.0340078 (3 s.f.) h) 456792 (2 s.f.) i) 2334089 (1 s.f.) 11 Approximating Answers Significant figures can be used to approximate answers. This is a good way of checking that answers are reasonable, especially when using calculators. Numbers are usually rounded off to one significant figure. Approximate 12971 ÷ 23.3 Approximate 89.6 x 10.3 19.7 + 9.8 Approximate 9.672 0.398 12 By rounding each of the numbers to one significant figure, find approximate answers for: a) 39.1 x 18.4 78.1 e) 61.4 x 1.87 49.2 – 28.8 b) 79.2 x 20.9 49.2 c) 42.1 x 2.97 2.017 x 31 d) 218 x 48.1 19.32 13 Sensible Answers You will often arrive at an answer that does not really make sense such as ½ a bus, or ¾ of a person, or 35.46 pence. In the exam you will be expected to look at the problem and decide what is a sensible answer. A college arranges a university trip for 165 students. Each bus can carry 48 passengers. How many buses are needed? Bottles of pop can be bought in packs of six for £2.49. How much does one bottle cost? 14 Using Examples to Show that a Statement is True By means of an example show that the sum of two consecutive numbers is always odd. Show, by means of an example, that the sum of four consecutive numbers in always even. Show, using an example, that the product of two consecutive numbers is always even. Ronnie says that the sum of any two prime numbers is always even. Use an example to show she is incorrect. Harriet says that the product of two even numbers is never divisible by 25. Use an example to show she is wrong. 15 Using statements to calculate similar answers without using a calculator. Given that 227.5 ÷ 35 = 6.5 Find the value of: a) 6.5 x 3.5 b) 227.5 ÷ 350 c) 6.5 x 0.35 Use the calculation 58.5 x 27 = 1579.5 to find: a) 585 x 27 b) 1579.5 ÷ 27 c) 585 x 0.027 16 Given that 487 x 3.53 = 1719.11 Find the value of: a) 487 x 0.0353 b) 48700 x 0.00353 c) 1719.11 487 17