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7 Decimals and fractions CHAPTER 7.1 Arithmetic of decimals In a decimal number the decimal point separates the whole number part from the part that is less than 1 When adding or subtracting decimals line up the decimal points first. To multiply by a decimal ignore the decimal point and do the multiplication with whole numbers. Then decide on the position of the decimal point. Example 1 a Multiply 5.12 by 4.6 Solution 1 51112 a 1 1 1 1 21146 3101712 210141810 213151512 Estimate 5 5 25 b b Multiply 3.4 by 0.2 Ignore the decimal points and do the multiplication with whole numbers. Round 5.12 to 5 and round 4.6 to 5 An estimate for the answer is 5 5 5.12 4.6 23.552 This means that the decimal point will go between the 3 and the 5 as 23.552 is close to 25 5.12 4.6 23.552 The number of decimal places in the answer is 3 which is the same as the total number of decimal places in the question. This rule is another way of finding the position of the decimal point in the answer. 314 212 618 3.4 0.2 0.68 Do the multiplication with whole numbers. The total number of decimal places in the question is 2 so there must be 2 decimal places in the answer. To divide a number by a decimal multiply both the number and the decimal by a power of 10 (10, 100, 1000 …) to make the decimal a whole number. It is much easier to divide by a whole number than by a decimal. Example 2 Divide 20 by 0.4 Solution 2 10 200 20 0.4 4 50 0 0 4 2 So 20 0.4 50 92 10 To make 0.4 a whole number multiply it by 10 So multiply both 20 and 0.4 by 10 Then divide in the usual way. 7.1 Arithmetic of decimals CHAPTER 7 Example 3 Divide 58.2 by 0.03 Solution 3 1191410 2 81 21 0 35 100 5820 58.2 0.03 3 100 To make 0.03 a whole number multiply it by 100 So multiply both 58.2 and 0.03 by 100 Then divide in the usual way. So 58.2 0.03 1940 Exercise 7A 1 Write each of the following sets of numbers in order of size. Start with the smallest number each time. a 0.373 0.37 0.73 0.333 0.733 b 15.8 15.38 15.3 15.833 15.803 c 0.045 0.05 0.0545 0.055 0.0454 d 6.067 6.006 6.07 6.06 6.077 6.076 e 8.092 8.9 8.02 8.09 8.2 8.29 8.92 2 Work out a 8.2 9.7 e 13.1 5.69 i 4 6.2 8.77 b 5.67 0.94 f 87.34 45.9 j 23 15.6 3.45 c 12.45 3.49 g 345.06 24.8 d 76.29 64.67 h 15.2 8.953 3 Work out a 8.57 3.21 e 8.6 3.42 i 8.72 6.04 b 19.31 7.16 f 14.6 4.31 j 7.34 3.286 c 56.43 12.56 g 9 3.4 d 67.65 45.8 h 17 5.43 4 Work out a 9.62 10 e 8.2 100 i 0.21 1000 b 67.231 100 f 0.9 100 j 6.08 100 c 0.83 10 g 8.41 100 k 0.0134 10 d 0.0065 1000 h 43.2 1000 l 56.1 100 5 Work out a 6.34 0.4 e 2.16 0.3 b 4.21 0.3 f 0.54 0.8 c 0.02 0.4 g 0.723 0.06 d 0.08 0.3 h 3.15 0.8 6 Work out a 3.1 4.2 e 8.6 2.4 b 0.36 1.4 f 9.2 0.15 c 3.6 2.3 g 0.064 0.73 d 7.4 0.53 h 0.095 3.4 7 Work out the cost of 0.6 kg of carrots at 25p per kilogram. Give your answer in pounds. 8 Work out the cost of 1.6 m of material at £4.20 per metre. 9 Work out a 456 100 e 0.9 10 i 0.054 10 b 72.3 10 f 67.2 100 j 2.31 100 c 0.76 10 g 7 1000 k 45 1000 d 53 100 h 4 100 l 6.01 100 10 Work out a 12 0.2 e 4.2 0.3 i 6.12 0.003 b 5 0.2 f 0.72 0.03 j 0.035 0.7 c 26 0.4 g 0.145 0.5 k 0.048 0.6 d 9 0.04 h 19.2 0.03 l 0.00828 0.09 93 Decimals and fractions CHAPTER 7 11 Five people share £130.65 equally. Work out how much each person will get. 12 A bottle of lemonade holds 1.5 litres. A glass will hold 0.3 litres. How many glasses can be filled from the bottle of lemonade? 7.2 Manipulation of decimals 8.4 Using a calculator 42 0.2 84 420 so multiplying the numerator by 10 without altering the denominator multiplies the 0.2 answer by 10 8.4 4.2 so multiplying the denominator by 10 without altering the numerator divides the 2 answer by 10. 0.84 4.2 so dividing the numerator by 10 without altering the denominator divides the 0.2 answer by 10. 8.4 420 so dividing the denominator by 10 without altering the numerator multiplies the 0.02 answer by 10. Similar results are obtained by using other powers of 10 8400 For example 42 000 (the numerator 8.4 has been multiplied by 1000 without altering the 0.2 denominator, so the answer has been multiplied by 1000). Sometimes the numerator and denominator are both changed. 8.4 84 For example starting with the result 42 it is possible to write down the value of 0.2 0.002 The numerator 8.4 has been multiplied by 10 and the denominator has been divided by 100 so the 84 answer has been multiplied by 1000, that is 42 000 0.002 Example 4 16.3 16.3 Given that 6.52 work out the value of 2.5 25 Solution 4 16.3 16.3 25 2.5 10 94 16.3 16.3 Starting with multiply the denominator by 10 to get 2.5 25 16.3 6.52 10 25 Multiplying the denominator by 10 without altering the numerator divides the answer by 10 16.3 0.652 25 To get the answer divide 6.52 by 10 7.2 Manipulation of decimals CHAPTER 7 Example 5 3.46 25.5 Given that 25.95 find the value of each of the following. 3.4 34.6 2.55 a 0.34 2.595 0.34 b 25.5 3.46 10 25.5 10 34.6 2.55 3.4 10 0.34 3.46 25.5 Starting with multiply 3.46 by 10 and divide 3.4 25.5 by 10 (so the value of the numerator is not altered) 34.6 2.55 and divide the denominator by 10 to get 3.4 34.6 2.55 25.95 10 0.34 Dividing the denominator by 10 without altering the numerator multiplies the answer by 10 34.6 2.55 259.5 0.34 To get the answer multiply 25.95 by 10 3.46 25.5 25.95 3.4 3.46 25.5 Multiply both sides of 25.95 by 3.4 3.4 25.95 3.4 3.46 25.5 Divide both sides of 3.46 25.5 25.95 3.4 by 25.5 Solution 5 a b 25.95 10 3.4 10 2.595 0.34 25.5 25.5 25.95 3.4 Starting with divide 25.95 by 10 and divide 3.4 25.5 by 10 (so the value of the numerator is divided by 100 and 2.595 0.34 the denominator is not altered) to get 25.5 2.595 0.34 3.46 100 25.5 Dividing the numerator by 100 without altering the denominator divides the answer by 100 2.595 0.34 0.0346 25.5 To get the answer divide 3.46 by 100 Exercise 7B 1 Given that 6.4 2.8 17.92 work out a 64 28 b 640 2.8 c 0.64 28 d 0.64 0.028 c 0.183 1.25 d 0.183 12.5 c 0.132 55 d 0.0132 550 c 304 4.75 d 3.04 0.475 2 Given that 18.3 1.25 14.64 work out a 183 1.25 b 1.83 1.25 3 Given that 13.2 5.5 72.6 work out a 132 5.5 b 1.32 0.55 4 Given that 30.4 4.75 6.4 work out a 30.4 47.5 b 3.04 4.75 95 Decimals and fractions CHAPTER 7 23.2 5.1 5 Given that 34.8 work out 3.4 23.2 51 232 51 a b 3.4 3.4 23.2 5.1 c 34 232 51 d 34 17.2 4.5 6 Given that 32.25 work out 2.4 172 45 17.2 4.5 a b 2.4 240 17.2 45 c 240 1.72 0.45 d 0.24 7 Given that 23 56 1288 work out a 0.23 560 b 1288 5.6 c 12.88 0.23 d 1288 (23 28) 8 Given that 52 32 1664 work out a 0.52 0.32 b 1664 5.2 c 16.64 0.32 d 166.4 0.64 9 Given that 884 34 26 work out a 8.84 340 b 884 2.6 c 8.84 260 d 884 (3.4 2.6) 10 Given that 1512 36 42 work out a 15.12 3.6 b 1.512 0.036 c 15.12 420 d 1.512 0.84 144 28 11 Given that 96 work out 42 14.4 28 1.44 2.8 a b 0.42 420 14.4 2.8 c 9.6 4.2 9.6 d 0.028 84 45 12 Given that 108 work out 35 8.4 4.5 0.84 4.5 b a 350 0.035 8.4 0.45 c 10.8 10.8 0.35 d 840 1872 13 Given that 1300 work out 1.22 1872 18.72 a b 2 12 1.22 187.2 c 2 0.12 936 d 2 120 7.3 Conversion between decimals and fractions A terminating decimal is a decimal which ends. For example 0.56, 0.0004 and 4.57 are terminating decimals. All terminating decimals can be converted to fractions using place value. 0.7 170 6 0.76 170 100 7 0 6 100 100 7 6 100 96 6 0.06 100 7.3 Conversion between decimals and fractions Example 7 Write 0.024 as a fraction. Give your fraction in its simplest form. Write 3.7 as a fraction. Solution 7 . 2 4 2 4 1000 3 The heading of the last column with a figure in it is thousandths, so the denominator is 1000 2 4 0.024 1000 1 2 6 3 500 250 125 tent hs 0 unit s hund redt hs thou sand ths 0 tent hs unit s Solution 6 . 3.7 3170 hund redt hs thou sand ths Example 6 CHAPTER 7 7 The 3 is the whole number part, the .7 is 170 Example 9 All fractions can be changed into decimals. Example 8 Write the following fractions as decimals. 2 3 b a 190 100 Solution 8 a 190 0.9 b 2 3 100 0.23 Write the following fractions as decimals. b 1215 a 25 Solution 9 Method 1 (non-calculator using equivalent fractions) 4 4 b 1215 a 25 140 0.4 100 0.44 Method 2 (calculator) a 25 means 2 5 Using a calculator b 1 1 25 means 11 25 Using a calculator 2 5 0.4 11 25 0.44 2 5 1 1 25 0.4 0.44 Short division is suitable for changing 25 to a decimal because the denominator is small. 2 5 means 2 5 2.0 is the same as 2 so divide 2.0 by 5 2 0. 4 2 .0 52 2 5 0.4 5 does not divide into 2 so put down a zero and carry. 5 divides into 20 four times. Not all fractions can be written as terminating decimals. For example 13 1 3 0.333 33 … which is a recurring decimal. A recurring decimal is a decimal in which one or more figures repeat. 0.11111111 … , 0.565 656 56 … and 9.762 333 33 … are also recurring decimals. To show that a figure recurs put a dot above the figure. . . So 0.333 33 … is written as 0.3 and 13 0.3 Sometimes more than one figure recurs. 3 11 0.272 727 … 3 11 A dot is placed above each recurring figure. .. So 131 0.27 97 Decimals and fractions CHAPTER 7 Example 10 Write the following fractions as decimals. b 1232 c 57 a 79 Solution 10 a 79 means 7 9 Work out 7 9 on a calculator. Using a calculator b . 7 9 0.777 777 … 0.7 The 7 recurs so put a dot above the 7 1 3 22 Work out 13 22 on a calculator. means 13 22 Using a calculator c .. 13 22 0.590 909 0 … 0.59 0 The 90 recurs so put a dot above each of these figures. Do not put a dot above the 5 as it does not recur. 5 7 Work out 5 7 on a calculator. means 5 7 Using a calculator . . 5 7 0.714 285 714 … 0.714 285 A group of six figures recurs. There isn’t enough room to see all the figures recurring but you can see that the same pattern of figures is starting again. When more than two figures recur just two dots are used, one above the first figure in the recurring group and one above the last figure in the group. Fractions written in their simplest form with denominators 2, 4, 5, 8, 10, 16, 20, … will convert to terminating decimals. Fractions written in their simplest form with denominators 3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, … will convert to recurring decimals. In general if a fraction written in its simplest form has a denominator with a prime factor other than 2 or 5, it will convert to a recurring decimal. Example 11 Work out whether the following fractions will convert to terminating or recurring decimals. b 3270 a 380 Solution 11 a 380 145 15 3 5 Write the fraction in its simplest form. Write the denominator as the product of its prime factors. 3 is a prime factor as well as 5 will convert to a recurring decimal. 8 30 b 7 20 Just consider the fraction part. 20 2 2 5 Write the denominator as the product of its prime factors. The only prime factors are 2 and 5 7 will convert to a terminating decimal. 20 98 7.4 Converting recurring decimals to fractions CHAPTER 7 The fractions and decimals in the table are some that are used frequently and should be learnt. Fraction Decimal 1 100 1 10 1 5 1 4 1 3 1 2 2 3 3 4 0.01 0.1 0.2 0.25 . 0.3 0.5 . 0.6 0.75 Exercise 7C 1 Write each of the decimals as a fraction in its simplest form. a 0.7 b 0.14 c 0.093 d 0.006 e 0.2 f 2.5 g 25.08 h 2.84 2 Write the following fractions as decimals. 3 7 3 b c a 190 100 100 e 8 1000 3 Write the following as equivalent fractions and then as decimals. 9 b 570 c 285 d a 45 10 100 100 500 1000 e 3 20 4 Write down the following fractions as decimals. b 14 c 23 a 12 e 3 4 e 1 1 16 d d 56 1 1000 1 5 100 5 Use short division to change these fractions to decimals. b 38 c 16 a 35 6 Use a calculator to change these fractions to decimals. 2 3 b 490 c d 58 a 312 125 7 By writing the denominator in terms of its prime factors, decide whether the following fractions will convert to recurring or terminating decimals. 6 b 1372 c 485 d 1432 e f 3670 a 490 125 8 Use a calculator to change these fractions to decimals. b 89 c 151 d 172 a 56 e 2 7 7.4 Converting recurring decimals to fractions All recurring decimals can be converted to fractions. To convert a recurring decimal to a fraction: ● introduce a letter, usually x ● form an equation by putting x equal to the recurring decimal ● multiply both sides of the equation by 10 if 1 digit recurs, by 100 if 2 digits recur, by 1000 if 3 digits recur and so on ● subtract the original equation from the new equation ● rearrange to find x as a fraction. 99 Decimals and fractions CHAPTER 7 Example 11 . Convert the recurring decimal 0.2 to a fraction. Solution 11 Let x 0.2222 … 10x 2.222 … 10x 0.222 … 9x 2 x 29 . 0.2 29 Put x equal to the recurring decimal. Multiply both sides of the equation by 10 as 1 digit recurs. Subtract the equations. Divide both sides by 9 Example 12 . . Convert the recurring decimal 0.2371 to a fraction. Solution 12 Let x 0.237 137 1 … 1000x 237.1371 … 1000x 0.2371 … 999x 236.9 236.9 x 999 2369 9990 . . 2369 0.2371 9990 Put x equal to the recurring decimal. Care is needed here, the 2 does not recur. Multiply both sides of the equation by 1000 ( 103) as 3 digits recur. Subtract the equations. Divide both sides by 999 Multiply both the numerator and denominator by 10 to change the decimal in the numerator to an integer. Example 13 .. Convert the recurring decimal 3.08 6 to a fraction. Solution 13 .. 3.08 6 3 0.086 86… Let x 0.086 86 … 100x 8.686 …. 1000x 0.086 86… 99x 8.6 8.6 x 99 86 990 43 495 .. 4 3 3.08 6 3 495 100 Consider the decimal part 0.086 86 … Put x equal to the recurring decimal. Care is needed here. The 0 does not recur. Multiply both sides of the equation by 100( 102) as 2 digits recur. Subtract the equations. Divide both sides by 99 Multiply both the numerator and denominator by 10 to change the decimal in the numerator to an integer. Simplify the fraction. Include the whole number part in the answer. 7.5 Rounding to significant figures CHAPTER 7 Example 14 . Given that 0.2 29 . Express the recurring decimal 0.32 as a fraction. Solution 14 . . 0.32 0.3 0.02 . 0.3 0.2 10 130 29 110 To use the information given in the question split up the decimal 0.32222 … 0.3 0.02222 … Rewrite the recurring part of the decimal using the given result. Change all decimals to fractions. 130 920 2970 920 Write fractions with a common denominator so they can be added. 2990 . 0.32 2990 Exercise 7D Convert each recurring decimal to a fraction. Give each fraction in its simplest form. .. 1 0.777 77 … 2 0.343 434 … 3 0.915 915 … 4 0.18 . . . .. .. 5 0.317 6 0.05 7 0.326 8 0.701 . .. .. . 9 0.23 10 6.83 11 2.106 12 7.352 .. .. 13 Given that 161 0.54 write the recurring decimal 0.554 as a fraction. .. .. 14 Given that 353 0.15 write the recurring decimal 0.215 as a fraction. . . 15 Given that 16 0.16 write the recurring decimal 0.4016 as a fraction. 7.5 Rounding to significant figures A number rounded to one significant figure has only one figure that is not zero. 5937 rounded to one significant figure is 6000 0.006 183 rounded to one significant figure is 0.006 A number rounded to two significant figures is more accurate than a number rounded to one significant figure. To round 5937 to two significant figures look at the third figure (3). As this is less than 5, do not change the previous figure (9) and write zeros in the tens column and the units column. So 5937 rounded to two significant figures is 5900 To round 0.006 183 to two significant figures, look at the third figure (8) after the zeros at the beginning. As this is more than 5, increase the previous figure (1) by 1. Remember to include the zeros at the beginning in your answer. So 0.006 183 rounded to two significant figures is 0.006 2 To round whole numbers greater than one to three significant figures, look at the fourth figure. 101 Decimals and fractions CHAPTER 7 To round decimals to three significant figures, look at the fourth figure after the zeros at the beginning. 5937 rounded to three significant figures is 5940 0.006 183 rounded to three significant figures is 0.006 18 Example 15 Round a 3462 to one significant figure c 0.3469 to three significant figures Solution 15 a 3462 3462 rounds to 3000 to one significant figure. b 7.38 7.38 rounds to 7.4 to two significant figures. c 0.3469 0.3469 rounds to 0.347 to three significant figures. d 0.0201 0.0201 rounds to 0.020 to two significant figures. b 7.38 to two significant figures d 0.0201 to two significant figures The second figure is 4. As this is less than 5, the 3 stays as it is and a zero goes in all the other places. The third figure is 8 As this is more than 5, increase the 3 by 1 The fourth figure after the zero at the beginning is 9 As this is more than 5, increase the 6 by 1 The third figure after the zeros at the beginning is 1 As this is less than 5, the zero before the 1 stays as it is. The zero at the end is needed as it is the second significant figure. Example 16 6.73 4.5 Use your calculator to work out the value of 12.03 9.73 Give your answer correct to two significant figures. Solution 16 6.73 4.5 11.23 Use a calculator to work out the value of the numerator. 12.03 9.73 2.3 Use a calculator to work out the value of the denominator. 11.23 2.3 4.882 608 696 … The line in a fraction means divide. Now use a calculator to work out 11.23 2.3 Write down all the figures shown on your calculator. 4.882 608 696 4.9 correct to two significant figures To give the answer to two significant figures, look at the third figure (8). As this is more than 5, increase the figure before it by 1 Exercise 7E 102 1 Round these numbers to one significant figure a 8234 b 76 420 e 0.381 f 0.004 56 c g 453 0.109 d h 72 532.4 Chapter summary CHAPTER 7 2 Round these numbers to two significant figures a 4263 b 8719 c e 798 f 0.005 62 g 685 703 d h 3.84 0.4032 3 Round these numbers to three significant figures a 8736 b 56.24 c e 0.030 56 f 87.98 g 27.839 6 735 412 d h 0.786 21 907.189 d h l 82.14 (2) 0.002 345 1 (2) 0.000 481 6 (3) 4 Round these to the number of significant figures given in the brackets a 6712 (1) b 8614 (3) c 6926 (2) e 876.3 (3) f 12.52 (3) g 0.0426 (1) i 7.6024 (3) j 8.795 (2) k 508 342 (3) 5 Use your calculator to work out the value of the following. Give each answer correct to three significant figures. a 5421 23 b 423 871 d 3250 720 0.32 e 9.6 13.21 9.1 c 0.0562 0.041 f 27.31 8.96 4.56 9.8 Chapter summary You should now know that: in a decimal number the decimal point separates the whole number part from the part that is less than one to multiply by a decimal ignore the decimal point and do the multiplication with whole numbers. Then decide on the position of the decimal point to divide by a decimal write the division as a fraction then multiply numerator and denominator by a power of 10 to find an equivalent fraction with an integer as the denominator if a number in the numerator of an expression is multiplied by a power of 10 (or a number in the denominator is divided by a power of 10) then the value of the expression is multiplied by the same power of 10 if a number in the numerator of an expression is divided by a power of 10 (or a number in the denominator is multiplied by a power of 10) then the value of the expression is divided by the same power of 10 terminating decimals can be converted to fractions by using place value fractions can be converted to decimals by using equivalent fractions or division some fractions convert to recurring decimals when the denominator of a fraction written in its simplest form has prime factors containing only 2s andor 5s then the fraction will convert to a terminating decimal; otherwise the fraction will convert to a recurring decimal every recurring decimal can be converted to a fraction numbers can be rounded to significant figures. 103 Decimals and fractions CHAPTER 7 Chapter 7 review questions 1 Work out a 5.6 10 d 0.0062 100 b 76.2 100 e 0.87 1000 c 9 100 2 Work out a 0.2 0.3 d 0.4 0.08 b 1.2 0.6 e 6.1 4.2 c 0.37 0.5 f 0.32 5.6 3 Work out a 6.25 0.5 d 56 0.2 b 75.6 0.3 e 46.2 0.03 c 47.7 0.09 f 0.84 0.004 4 0.3 0.06 0.058 0.26 a Write these four decimals in order of size. Start with the smallest decimal. b Write 0.3 as a fraction. c Work out 0.3 0.26 d Work out 0.058 100 5 a Work out 41.3 100 b Work out 0.4 0.6 (1388 January 2002) c Work out 5.2 1.37 (1388 March 2003) 6 Change 78 to a decimal. 7 Karen says that 1234 can be converted into a terminating decimal. Lucy says that the fraction converts to a recurring decimal. Who is correct? You must give a reason for your answer. 8 By writing the denominator as the product of its prime factors, decide whether the following fractions will convert to a recurring decimal or a terminating decimal. b 3575 c 725 d 1996 a 196 9 a Write 0.35 as a fraction. Give your answer in its simplest form. b Write 38 as a decimal. 10 Use your calculator to write each fraction as a decimal. b 1116 c 191 a 2430 11 1.54 450 693 Use this result to write down the answer to a 1.54 45 b 1.54 4.5 c 0.154 0.45 12 Using the information that 97 123 11 931 write down the value of a 9.7 12.3 b 0.97 123 000 c 11.931 9.7 (1387 June 2002) d 7 90 (1387 May 2002) (1387 June 2003) 13 a Express 49 as a recurring decimal. .. b Convert the recurring decimal 0.136 to a fraction in its simplest form. 14 a Change 131 to a decimal. .. b Prove that the recurring decimal 0.39 1333 104 (1387 June 2005) Chapter 7 review questions CHAPTER 7 . 15 Express the recurring decimal 2.06 as a fraction. Write your answer in its simplest form. (1388 March 2005) 16 Change to a single fraction .. a the recurring decimal 0.13 .. b the recurring decimal 0.513 (1388 March 2002) 17 a is an integer such that 1 a 9 b is an integer such that 1 b 9 .. ab Prove that 0.0ab 990 .. 18 a Express 0.27as a fraction in its simplest form. b x is an integer such that 1 x 9 .. x Prove that 0.0x 99 .. 19 a Convert the recurring decimal 0.36 to a fraction. .. b Convert the recurring decimal 2.136 to a mixed number. Give your answer in its simplest form. .. 20 The recurring decimal 0.72 can be written as the fraction 181 .. Write the recurring decimal 0.572 as a fraction. (1388 January 2003) (1388 March 2004) (1387 November 2005) 21 Round the following to the number of significant figures given in the brackets a 3546 (1) b 3546 (2) c 0.005 62 (1) d 23.76 (2) e 2.4387 (3) f 696 213 (2) 105