UNIT 4 FRACTIONS AND DECIMALS (A) Main Concepts and Results • A fraction is a number representing a part of a whole. This whole may be a single object or a group of objects. • A fraction whose numerator is less than the denominator is called a proper fraction, otherwise it is called an improper fraction. 5 41 • Numbers of the type 3 ,8 ,2 etc. are called mixed fractions 79 5(numbers). • An improper fraction can be converted into a mixed fraction and vice versa. • Fractions equivalent to a given fraction can be obtained by multiplying or dividing its numerator and denominator by a nonzero number. • A fraction in which there is no common factor, except 1, in its numerator and denominator is called a fraction in the simplest or lowest form. • Fractions with same denominators are called like fractions and if the denominators are different, then they are called unlike fractions. • Fractions can be compared by converting them into like fractions and then arranging them in ascending or descending order. • Addition (or subtraction) of like fractions can be done by adding (or subtracting) their numerators. • Addition (or subtraction) of unlike fractions can be done by converting them into like fractions. • Fractions with denominators 10,100, etc. can be written in a form, using a decimal point, called decimal numbers or decimals. • Place value of the place immediately after the decimal point (i.e. 1 1 tenth place) is , that of next place (i.e. hundredths place) is10 100 and so on. • Fractions can be converted into decimals by writing them in the form with denominators 10,100, and so on. Similarly, decimals can be converted into fractions by removing their decimal points and writing 10,100, etc in the denominators, depending upon the number of decimal places in the decimals. • Decimal numbers can be compared using the idea of place value and then can be arranged in ascending or descending order. • Decimals can be added (or subtracted) by writing them with equal number of decimal places. • Many daily life problems can be solved by converting different units of measurements such as money, length, weight, etc. in the decimal form and then adding (or subtracting) them. (B) Solved Examples In examples 1 and 2, write the correct answer from the given four options: Example 1. Which of the following fractions is the smallest? 11 11 11 11 (A) (B) (C) (D)9 7 10 6 Solution: Answer is (C) Example 2: 0.7625 lies between (A) 0.7 and 0.76 (B) 0.77 and 0.78 (C) 0.76 and 0.761 (D) 0.76 and 0.763 Solution: Answer is (D) Example 3: Solution: Example 4: Solution: Example 5: Solution: Example 6: Solution: Example 7: Solution: Example 8: Solution: Fill in the blanks so that the statement is true: Decimal 8.125 is equal to the fraction ________. 65 1 8125 8 or 8 8 (because 8.125 = 1000 ) Fill in the blanks so that the statement is true: 6.45 – 3.78 = _________. 2.67 State true or false: 2 The fraction 14 is equal to 14.2.52 False [Hint: 14 = 14.2] 10Fill in the blanks using > or < : 8 16 – 4589 88× 2 16 == 45 45 × 2 90 16 16 8 16 Now, < , so, < 90 89 45 89 Another method: 8 × 89 = 712 and 16 × 45 =720 8 16 As 712 < 720, therefore < 45 89 12 Express as a decimal. 2512 12 ×4 = 25 25 × 4 48 == 0.48 10 0 Convert 5809g to kg. 5809 Since 1000g = 1kg, therefore, 5809g = kg1000 = 5.809kg. Example 9: Solution: Example 10: Solution: Example 11: Solution: Example 12: Solution: Round off 87.952 to tenths place. For rounding off to tenths place, we look at the hundredths place. Here the digit is 5. So, the digit at the tenths place (9) will be increased by 1 (i.e., it will become 9 + 1) Hence, rounding off 87.952 to tenths place, we get 88.0 (Note: Do not write it as 88.) 35 5Add the fractions and 816 35 43 5 5 + = + 816 816 43 × 25 86 5 =+ = +8×2 16 16 16 86 + 5 91 11 == = 5 16 16 16 What should be added to 37.28 to obtain 46.8? Here, we want to fill in the box in 37.28 + = 46.8. For this, We will have to find 46.8 – 37.28. We perform this operation as follows by writing the two numbers having equal number of decimal places: (Since 46.8 = 46.80) 46.80 – 37.28 Hence, the required number to be added 9.52 to 37.28 is 9.52. Arrange the following in ascending order. 2.2, 2.023, 2.0226, 22.1, 20.42 We have to arrange them from the smallest to the greatest number. We arrange them as follows (using the idea of place value and comparing their digits at different places); 2.0226, 2.023, 2.2, 20.42, 22.1 (Note:Using the < sign, these numbers can also be written as 2.0226 < 2.023 < 2.2 < 20.42 < 22.1.) Solution: Weight of apples = 2kg 280g = 2280g (Since 1kg = 1000g) Weight of bananas = 3kg 375g = 3375g Weight of grapes = 225g Weight of oranges = 5kg 385g = 5385g Total weight = 2280g + 3375g + 225g + 5385g 2280g + 3375g + 225g + 5385g 11265 11265g Thus, total weight = 11265g = kg1000 = 11.265kg i.e. 11kg 265g Example 14: What is wrong in the following? 7 57 + 5 12 += = = 2 424 + 2 6 7 57 + 5 Solution: Writing += is wrong. It should be as follows: 424 + 2 7 57 10 + =+ (Converting into like fractions)424 4 7 + 10 17 = (Only numerators are added)44 In questions 1 to 20, out of the four options, only one answer is correct. Choose the correct answer. 4 1. The fraction which is not equal to is 5 40 12 16 9 (A) 50 (B) 15 (C) 20 (D) 15 5 2. The two consecutive integers between which the fraction lies are 7(A) 5 and 6 (B) 0 and 1 (C) 5 and 7 (D) 6 and 7 1 3. When is written with denominator as 12, its numerator is4(A)3 (B) 8 (C)24 (D) 12 4. Which of the following is not in the lowest form? 7 15 13 27 (A) (B) (C) (D)5 20 33 28 5 20 = 5. If , then value of p is8 p (A)23 (B) 2 (C)32 (D) 16 6. Which of the following is not equal to the others? 6 12 15 18 (A) (B) (C) (D)8 16 25 24 7. Which of the following fractions is the greatest? 555 5 (A) (B) (C) (D)769 8 8. Which of the following fractions is the smallest? 793 5 (A) (B) (C) (D)888 8 4 15 9. Sum of and is 1717 19 11 19 2 (A) (B) (C) (D)17 17 34 17 5 19 10. On subtracting from , the result is99 24 14 14 14 (A) (B) (C) (D)9 918 0 11. 0.7499 lies between (A) 0.7 and 0.74 (B) 0.75 and 0.79 (C) 0.749 and 0.75 (D) 0.74992 and 0.75 12. 0. 023 lies between (A) 0.2 and 0.3 (B) 0.02 and 0.03 (C) 0.03 and 0.029 (D) 0.026 and 0.024 11 13. can be expressed in the form14. The mixed fraction 5 can be expressed as15. 0.07 + 0.008 is equal to (A) 0.15 (B) 0.015 (C) 0.078 (D) 0.78 16. Which of the following decimals is the greatest? (A) 0.182 (B) 0.0925 (C) 0.29 (D) 0.038 17. Which of the following decimals is the smallest? (A) 0.27 (B) 1.5 (C) 0.082 (D) 0.103 18. 13.572 correct to the tenths place is (A) 10 (B) 13.57 (C) 14.5 (D) 13.6 19. 15.8 – 6.73 is equal to (A) 8.07 (B) 9.07 (C) 9.13 (D) 9.25 20. The decimal 0.238 is equal to the fraction 7(A) 1 7 4 (B) 1 4 7 (C) 4 1 7 (D) 1 11 7 4 733 39 33 39 (A) 7 (B) 7 (C) 4 (D) 4 119 238 119 119 (A) 500 (B) 25 (C) 25 (D) 50 In questions 21 to 44, fill in the blanks to make the statements true: 21. A number representing a part of a _________ is called a fraction. 22. A fraction with denominator greater than the numerator is called a _________ fraction. 23. Fractions with the same denominator are called _________ fractions. 24. 13 is a _________ fraction. 5 1818 7 25. 5 is an ______ fraction. 26. 19 is a ______ fraction. 5 3 27. 8 and 8 are ______ proper fractions. 6 6 28. 11 and 13 are ______ proper fractions. 6 29. The fraction in simplest form is ______.1517 30. The fraction in simplest form is ______.3418 90 31. and are proper, unlike and ______ fractions.1356752 32. 8 is equal to the improper fraction ______.787 33. is equal to the mixed fraction ______.726 34. 9 ++ is equal to the decimal number ______.10 100 35. Decimal 16.25 is equal to the fraction ______. 36. Fraction is equal to the decimal number ______.7 2517 41 67 24 +−37. = ______. 38. = ______.99 14 14 17 1 15 + 39 −39. = ______. 40. = ______.22 44 41. 4.55 + 9.73 = ______. 42. 8.76 – 2.68 = ______. 43. The value of 50 coins of 50 paisa = Rs ______. 44. 3 Hundredths + 3 tenths = ______. In each of the questions 45 to 65, state whether the statement is true or false: 45. Fractions with same numerator are called like fractions. 46. Fraction is in its lowest form. 18 3915 45 47. Fractions and are equivalent fractions.48. The sum of two fractions is always a fraction. 49. The result obtained by subtracting a fraction from another fraction is necessarily a fraction. 50. If a whole or an object is divided into a number of equal parts, then each part represents a fraction. 51. The place value of a digit at the tenths place is 10 times the same digit at the ones place. 52. The place value of a digit at the hundredths place is times the 391171 10same digit at the tenths place. 53. The decimal 3.725 is equal to 3.72 correct to two decimal places. 54. In the decimal form, fraction = 3.125. 55. The decimal 23.2 = 23 56. The fraction represented by the 25 82 5 3 shaded portion in the adjoining figure is . 857. The fraction represented by the 5 unshaded portion in the adjoining figure is . 925 631 8 88 58. += 59. – = 19 19 38 18 15 3 7 11 3 60. += 61. 3.03 + 0.016 =3.019 12 12 2 16 13 >62. 42.28 – 3.19 = 39.09 63. 25 25 64. 19.25 < 19.053 65. 13.730 = 13.73 In each of the questions 66 to 71, fill in the blanks using ‘>’, ‘<’ or ‘=’ : 11 14 8 95 66. ... 16 15 67. ... 15 14 12 32 68. ... 69. 3.25... 3.4 75 200 18 25 70. ...1.3 71. 6.25... 15 4 72. Write the fraction represented by the shaded portion of the adjoining figure: 73. Write the fraction represented by the unshaded portion of the adjoining figure: 74. Ali divided one fruit cake equally among six persons. What part of the cake he gave to each person? 75. Arrange 12.142, 12.124, 12.104, 12.401 and 12.214 in ascending order. 76. Write the largest four digit decimal number less than1using the digits 1, 5, 3 and 8 once. 77. Using the digits 2, 4, 5 and 3 once, write the smallest four digit decimal number. 78. Express as a decimal. 11 202 79. Express 6 as an improper fraction.32 80. Express 3 as a decimal. 581. Express 0.041 as a fraction. 82. Express 6.03 as a mixed fraction. 83. Convert 5201g to kg. 84. Convert 2009 paise to rupees and express the result as a mixed fraction. 85. Convert 1537cm to m and express the result as an improper fraction. 86. Convert 2435m to km and express the result as mixed fraction. 23 1 5 87. Arrange the fractions ,, and in ascending order.3426674 3 88. Arrange the fractions ,, and in descending order.785 43 89. Write as a fraction with denominator 44. 4 5 90. Write as a fraction with numerator 60. 6129 91. Write as a mixed fraction. 892. Round off 20.83 to nearest tenths. 93. Round off 75.195 to nearest hundredths. 94. Round off 27.981 to nearest tenths. 3 2 95. Add the fractions 8 and 3 . 96. Add the fractions 3 and 3 6 . 8 4 1 1 97. Subtract from . 6 2 98. Subtract 1 8 from 100 . 3 9 1 1 99. Subtract 1 from 6 . 4 2 100. Add 1 1 4 and 1 6 2 . 101. Katrina rode her bicycle 1 6 2 km in the morning and 3 8 4 km in the evening. Find the distance travelled by her altogether on that day. 102. A rectangle is divided into certain number of equal parts. If 16 of the 1 parts so formed represent the fraction , find the number of parts4 in which the rectangle has been divided. 9 103. Grip size of a tennis racquet is 11 cm. Express the size as an80improper fraction. 1 104. On an average of the food eaten is turned into organism’s own10body and is available for the nextlevel of consumer in a food chain. What fraction of the food eaten is not available for the next level? 105. Mr. Rajan got a job at the age of 24 years and he got retired from the job at the age of 60 years. What fraction of his age till retirement was he in the job? 106. The food we eat remains in the stomach for a maximum of 4 hours. For what fraction of a day, does it remain there? 107. What should be added to 25.5 to get 50? 108. Alok purchased 1kg 200g potatoes, 250g dhania, 5kg 300g onion, 500g palak and 2kg 600g tomatoes. Find the total weight of his purchases in kilograms. 109. Arrange in ascending order: 0.011, 1.001, 0.101, 0.110 110. Add the following: 20.02 and 2.002 111. It was estimated that because of people switching to Metro trains, about 33000 tonnes of CNG, 3300 tonnes of diesel and 21000 tonnes of petrol was saved by the end of year 2007. Find the fraction of : (i) the quantity of diesel saved to the quantity of petrol saved. (ii) the quantity of diesel saved to the quantity of CNG saved. 112. Energy content of different foods are as follows: Food Energy Content per kg. Wheat Rice Potatoes (Cooked) Milk 3.2 Joules 5.3 Joules 3.7 Joules 3.0 Joules Which food provides the least energy and which provides the maximum? Express the least energy as a fraction of the maximum energy. 1 113. A cup is full of milk. What part of the cup is still to be filled by 3milk to make it full? 13 114. Mary bought 3 m of lace. She used 1 m of lace for her new dress. 24How much lace is left with her? 115. When Sunita weighed herself on Monday, she found that she had 13 gained 1 5kg. Earlier her weight was 46 kg. What was her weight48on Monday? 13 116. Sunil purchased 12 litres of juice on Monday and 14 litres of juice24on Tuesday. How many litres of juice did he purchase together in two days? 31 117. Nazima gave 2 litres out of the 5 litres of juice she purchased to42her friends. How many litres of juice is left with her? 1 118. Roma gave a wooden board of length 150 cm to a carpenter for41 making a shelf. The Carpenter sawed off a piece of 40 cm from it. 5What is the length of the remaining piece? 1 1 119. Nasir travelled 3 km in a bus and then walked 1 km to reach a 2 8town. How much did he travel to reach the town? 3 120. The fish caught by Neetu was of weight 3 kg and the fish caught by41 Narendra was of weight 2 kg. How much more did Neetu’s fish2weigh than that of Narendra? 3 121. Neelam’s father needs 1 m of cloth for the skirt of Neelam’s new 41 dress and m for the scarf. How much cloth must he buy in all? 2122. What is wrong in the following additions? (a) (b) 12 1 8 = 8 6 24 2 11 1 +4 = 4 +2 44 4 3 21 =12 =8 =8 8 63 123. Which one is greater? 1 metre 40 centimetres + 60 centimetres or 2.6 metres. 124. Match the fractions of Column I with the shaded or marked portion of figures of Column II: Column I Column II (i) 6 4 (ii) 6 10 (A) (B) 125. Find the fraction that represents the number of natural numbers to total numbers in the collection 0, 1, 2, 3, 4, 5. What fraction will it be for whole numbers? 126. Write the fraction representing the total number of natural numbers in the collection of numbers –3, – 2, –1, 0, 1, 2, 3. What fraction will it be for whole numbers? What fraction will it be for integers? 7 2 127. Write a pair of fractions whose sum is and difference is . 128. What fraction of a straight angle is a right angle? 129. Put the right card in the right bag. Cards Bags 11 11 3 (i) 7 4 (ii) 4 9 (iii) (iv) 8 8 9