Page 1 FRACTIONS AND DECIMALS 29 29 29 29 29 2.1 INTRODUCTION Y ou have learnt fractions and decimals in earlier classes. The study of fractions included proper, improper and mixed fractions as well as their addition and subtraction. We also studied comparison of fractions, equivalent fractions, representation of fractions on the number line and ordering of fractions. Our study of decimals included, their comparison, their representation on the number line and their addition and subtraction. W e shall now learn multiplication and division of fractions as well as of decimals. 2.2 HOW WELL HAVE YOU LEARNT ABOUT FRACTIONS? A proper fraction is a fraction that represents a part of a whole. Is 7 4 a proper fraction? Which is bigger, the numerator or the denominator? An improper fraction is a combination of whole and a proper fraction. Is 7 4 an improper fraction? Which is bigger here, the numerator or the denominator? The improper fraction 7 4 can be written as 3 1 4 . This is a mixed fraction. Can you write five examples each of proper, improper and mixed fractions? EXAMPLE 1 Write five equivalent fractions of 3 5 . SOLUTION One of the equivalent fractions of 3 5 is 3 3 2 6 5 5 2 10 × = = × . Find the other four. Chapter 2 Fractions and Decimals 2020-21 © NCERT not to be republished Page 2 FRACTIONS AND DECIMALS 29 29 29 29 29 2.1 INTRODUCTION Y ou have learnt fractions and decimals in earlier classes. The study of fractions included proper, improper and mixed fractions as well as their addition and subtraction. We also studied comparison of fractions, equivalent fractions, representation of fractions on the number line and ordering of fractions. Our study of decimals included, their comparison, their representation on the number line and their addition and subtraction. W e shall now learn multiplication and division of fractions as well as of decimals. 2.2 HOW WELL HAVE YOU LEARNT ABOUT FRACTIONS? A proper fraction is a fraction that represents a part of a whole. Is 7 4 a proper fraction? Which is bigger, the numerator or the denominator? An improper fraction is a combination of whole and a proper fraction. Is 7 4 an improper fraction? Which is bigger here, the numerator or the denominator? The improper fraction 7 4 can be written as 3 1 4 . This is a mixed fraction. Can you write five examples each of proper, improper and mixed fractions? EXAMPLE 1 Write five equivalent fractions of 3 5 . SOLUTION One of the equivalent fractions of 3 5 is 3 3 2 6 5 5 2 10 × = = × . Find the other four. Chapter 2 Fractions and Decimals 2020-21 © NCERT not to be republished MATHEMATICS 30 30 30 30 30 EXAMPLE 2 Ramesh solved 2 7 part of an exercise while Seema solved 4 5 of it. Who solved lesser part? SOLUTION In order to find who solved lesser part of the exercise, let us compare 2 7 and 4 5 . Converting them to like fractions we have, 2 10 7 35 = , 4 28 5 35 = . Since10 < 28 , so 10 28 35 35 < . Thus, 2 4 < 7 5 . Ramesh solved lesser part than Seema. EXAMPLE 3 Sameera purchased 1 3 2 kg apples and 3 4 4 kg oranges. What is the total weight of fruits purchased by her? SOLUTION The total weight of the fruits 1 3 3 4 kg 2 4 ? ? = + ? ? ? ? = 7 19 14 19 kg kg 2 4 4 4 ? ? ? ? + = + ? ? ? ? ? ? ? ? = 33 1 kg 8 kg 4 4 = EXAMPLE 4 Suman studies for 2 5 3 hours daily . She devotes 4 2 5 hours of her time for Science and Mathematics. How much time does she devote for other subjects? SOLUTION Total time of Suman’s study = 2 5 3 h = 17 3 h Time devoted by her for Science and Mathematics = 4 2 5 = 14 5 h 2020-21 © NCERT not to be republished Page 3 FRACTIONS AND DECIMALS 29 29 29 29 29 2.1 INTRODUCTION Y ou have learnt fractions and decimals in earlier classes. The study of fractions included proper, improper and mixed fractions as well as their addition and subtraction. We also studied comparison of fractions, equivalent fractions, representation of fractions on the number line and ordering of fractions. Our study of decimals included, their comparison, their representation on the number line and their addition and subtraction. W e shall now learn multiplication and division of fractions as well as of decimals. 2.2 HOW WELL HAVE YOU LEARNT ABOUT FRACTIONS? A proper fraction is a fraction that represents a part of a whole. Is 7 4 a proper fraction? Which is bigger, the numerator or the denominator? An improper fraction is a combination of whole and a proper fraction. Is 7 4 an improper fraction? Which is bigger here, the numerator or the denominator? The improper fraction 7 4 can be written as 3 1 4 . This is a mixed fraction. Can you write five examples each of proper, improper and mixed fractions? EXAMPLE 1 Write five equivalent fractions of 3 5 . SOLUTION One of the equivalent fractions of 3 5 is 3 3 2 6 5 5 2 10 × = = × . Find the other four. Chapter 2 Fractions and Decimals 2020-21 © NCERT not to be republished MATHEMATICS 30 30 30 30 30 EXAMPLE 2 Ramesh solved 2 7 part of an exercise while Seema solved 4 5 of it. Who solved lesser part? SOLUTION In order to find who solved lesser part of the exercise, let us compare 2 7 and 4 5 . Converting them to like fractions we have, 2 10 7 35 = , 4 28 5 35 = . Since10 < 28 , so 10 28 35 35 < . Thus, 2 4 < 7 5 . Ramesh solved lesser part than Seema. EXAMPLE 3 Sameera purchased 1 3 2 kg apples and 3 4 4 kg oranges. What is the total weight of fruits purchased by her? SOLUTION The total weight of the fruits 1 3 3 4 kg 2 4 ? ? = + ? ? ? ? = 7 19 14 19 kg kg 2 4 4 4 ? ? ? ? + = + ? ? ? ? ? ? ? ? = 33 1 kg 8 kg 4 4 = EXAMPLE 4 Suman studies for 2 5 3 hours daily . She devotes 4 2 5 hours of her time for Science and Mathematics. How much time does she devote for other subjects? SOLUTION Total time of Suman’s study = 2 5 3 h = 17 3 h Time devoted by her for Science and Mathematics = 4 2 5 = 14 5 h 2020-21 © NCERT not to be republished FRACTIONS AND DECIMALS 31 31 31 31 31 Thus, time devoted by her for other subjects = 17 14 3 5 ? ? - ? ? ? ? h = 17 × 5 14 × 3 – h 15 15 ? ? ? ? ? ? = 85 – 42 h 15 ? ? ? ? ? ? = 43 15 h = 13 2 15 h EXERCISE 2.1 1. Solve: (i) 3 2 5 - (ii) 7 4 8 + (iii) 3 2 5 7 + (iv) 9 4 11 15 - (v) 7 2 3 10 5 2 + + (vi) 2 1 2 3 3 2 + (vii) 1 5 8 3 2 8 - 2. Arrange the following in descending order: (i) 2 9 2 3 8 21 , , (ii) 1 5 3 7 7 10 , , . 3. In a “magic square”, the sum of the numbers in each row, in each column and along the diagonals is the same. Is this a magic square? 4 11 3 11 8 11 9 11 5 11 1 11 2 11 7 11 6 11 4. A rectangular sheet of paper is 1 12 2 cm long and 2 10 3 cm wide. Find its perimeter. 5. Find the perimeters of (i) ? ABE (ii) the rectangle BCDE in this figure. Whose perimeter is greater? 6. Salil wants to put a picture in a frame. The picture is 7 3 5 cm wide. T o fit in the frame the picture cannot be more than 7 3 10 cm wide. How much should the picture be trimmed? (Along the first row 4 9 2 15 11 11 11 11 + + = ). 5 cm 2 3 2 cm 4 3 3 cm 5 7 cm 6 2020-21 © NCERT not to be republished Page 4 FRACTIONS AND DECIMALS 29 29 29 29 29 2.1 INTRODUCTION Y ou have learnt fractions and decimals in earlier classes. The study of fractions included proper, improper and mixed fractions as well as their addition and subtraction. We also studied comparison of fractions, equivalent fractions, representation of fractions on the number line and ordering of fractions. Our study of decimals included, their comparison, their representation on the number line and their addition and subtraction. W e shall now learn multiplication and division of fractions as well as of decimals. 2.2 HOW WELL HAVE YOU LEARNT ABOUT FRACTIONS? A proper fraction is a fraction that represents a part of a whole. Is 7 4 a proper fraction? Which is bigger, the numerator or the denominator? An improper fraction is a combination of whole and a proper fraction. Is 7 4 an improper fraction? Which is bigger here, the numerator or the denominator? The improper fraction 7 4 can be written as 3 1 4 . This is a mixed fraction. Can you write five examples each of proper, improper and mixed fractions? EXAMPLE 1 Write five equivalent fractions of 3 5 . SOLUTION One of the equivalent fractions of 3 5 is 3 3 2 6 5 5 2 10 × = = × . Find the other four. Chapter 2 Fractions and Decimals 2020-21 © NCERT not to be republished MATHEMATICS 30 30 30 30 30 EXAMPLE 2 Ramesh solved 2 7 part of an exercise while Seema solved 4 5 of it. Who solved lesser part? SOLUTION In order to find who solved lesser part of the exercise, let us compare 2 7 and 4 5 . Converting them to like fractions we have, 2 10 7 35 = , 4 28 5 35 = . Since10 < 28 , so 10 28 35 35 < . Thus, 2 4 < 7 5 . Ramesh solved lesser part than Seema. EXAMPLE 3 Sameera purchased 1 3 2 kg apples and 3 4 4 kg oranges. What is the total weight of fruits purchased by her? SOLUTION The total weight of the fruits 1 3 3 4 kg 2 4 ? ? = + ? ? ? ? = 7 19 14 19 kg kg 2 4 4 4 ? ? ? ? + = + ? ? ? ? ? ? ? ? = 33 1 kg 8 kg 4 4 = EXAMPLE 4 Suman studies for 2 5 3 hours daily . She devotes 4 2 5 hours of her time for Science and Mathematics. How much time does she devote for other subjects? SOLUTION Total time of Suman’s study = 2 5 3 h = 17 3 h Time devoted by her for Science and Mathematics = 4 2 5 = 14 5 h 2020-21 © NCERT not to be republished FRACTIONS AND DECIMALS 31 31 31 31 31 Thus, time devoted by her for other subjects = 17 14 3 5 ? ? - ? ? ? ? h = 17 × 5 14 × 3 – h 15 15 ? ? ? ? ? ? = 85 – 42 h 15 ? ? ? ? ? ? = 43 15 h = 13 2 15 h EXERCISE 2.1 1. Solve: (i) 3 2 5 - (ii) 7 4 8 + (iii) 3 2 5 7 + (iv) 9 4 11 15 - (v) 7 2 3 10 5 2 + + (vi) 2 1 2 3 3 2 + (vii) 1 5 8 3 2 8 - 2. Arrange the following in descending order: (i) 2 9 2 3 8 21 , , (ii) 1 5 3 7 7 10 , , . 3. In a “magic square”, the sum of the numbers in each row, in each column and along the diagonals is the same. Is this a magic square? 4 11 3 11 8 11 9 11 5 11 1 11 2 11 7 11 6 11 4. A rectangular sheet of paper is 1 12 2 cm long and 2 10 3 cm wide. Find its perimeter. 5. Find the perimeters of (i) ? ABE (ii) the rectangle BCDE in this figure. Whose perimeter is greater? 6. Salil wants to put a picture in a frame. The picture is 7 3 5 cm wide. T o fit in the frame the picture cannot be more than 7 3 10 cm wide. How much should the picture be trimmed? (Along the first row 4 9 2 15 11 11 11 11 + + = ). 5 cm 2 3 2 cm 4 3 3 cm 5 7 cm 6 2020-21 © NCERT not to be republished MATHEMATICS 32 32 32 32 32 7. Ritu ate 3 5 part of an apple and the remaining apple was eaten by her brother Somu. How much part of the apple did Somu eat? Who had the larger share? By how much? 8. Michael finished colouring a picture in 7 12 hour . V aibhav finished colouring the same picture in 3 4 hour. Who worked longer? By what fraction was it longer? 2.3 MULTIPLICATION OF FRACTIONS Y ou know how to find the area of a rectangle. It is equal to length × breadth. If the length and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its area would be 7 × 4 = 28 cm 2 . What will be the area of the rectangle if its length and breadth are 7 1 2 cm and 3 1 2 cm respectively? Y ou will say it will be 7 1 2 × 3 1 2 = 15 2 × 7 2 cm 2 . The numbers 15 2 and 7 2 are fractions. T o calculate the area of the given rectangle, we need to know how to multiply fractions. W e shall learn that now . 2.3.1 Multiplication of a Fraction by a Whole Number Observe the pictures at the left (Fig 2.1). Each shaded part is 1 4 part of a circle. How much will the two shaded parts represent together? They will represent 1 1 4 4 + = 1 2× 4 . Combining the two shaded parts, we get Fig 2.2 . What part of a circle does the shaded part in Fig 2.2 represent? It represents 2 4 part of a circle . Fig 2.1 Fig 2.2 2020-21 © NCERT not to be republished Page 5 FRACTIONS AND DECIMALS 29 29 29 29 29 2.1 INTRODUCTION Y ou have learnt fractions and decimals in earlier classes. The study of fractions included proper, improper and mixed fractions as well as their addition and subtraction. We also studied comparison of fractions, equivalent fractions, representation of fractions on the number line and ordering of fractions. Our study of decimals included, their comparison, their representation on the number line and their addition and subtraction. W e shall now learn multiplication and division of fractions as well as of decimals. 2.2 HOW WELL HAVE YOU LEARNT ABOUT FRACTIONS? A proper fraction is a fraction that represents a part of a whole. Is 7 4 a proper fraction? Which is bigger, the numerator or the denominator? An improper fraction is a combination of whole and a proper fraction. Is 7 4 an improper fraction? Which is bigger here, the numerator or the denominator? The improper fraction 7 4 can be written as 3 1 4 . This is a mixed fraction. Can you write five examples each of proper, improper and mixed fractions? EXAMPLE 1 Write five equivalent fractions of 3 5 . SOLUTION One of the equivalent fractions of 3 5 is 3 3 2 6 5 5 2 10 × = = × . Find the other four. Chapter 2 Fractions and Decimals 2020-21 © NCERT not to be republished MATHEMATICS 30 30 30 30 30 EXAMPLE 2 Ramesh solved 2 7 part of an exercise while Seema solved 4 5 of it. Who solved lesser part? SOLUTION In order to find who solved lesser part of the exercise, let us compare 2 7 and 4 5 . Converting them to like fractions we have, 2 10 7 35 = , 4 28 5 35 = . Since10 < 28 , so 10 28 35 35 < . Thus, 2 4 < 7 5 . Ramesh solved lesser part than Seema. EXAMPLE 3 Sameera purchased 1 3 2 kg apples and 3 4 4 kg oranges. What is the total weight of fruits purchased by her? SOLUTION The total weight of the fruits 1 3 3 4 kg 2 4 ? ? = + ? ? ? ? = 7 19 14 19 kg kg 2 4 4 4 ? ? ? ? + = + ? ? ? ? ? ? ? ? = 33 1 kg 8 kg 4 4 = EXAMPLE 4 Suman studies for 2 5 3 hours daily . She devotes 4 2 5 hours of her time for Science and Mathematics. How much time does she devote for other subjects? SOLUTION Total time of Suman’s study = 2 5 3 h = 17 3 h Time devoted by her for Science and Mathematics = 4 2 5 = 14 5 h 2020-21 © NCERT not to be republished FRACTIONS AND DECIMALS 31 31 31 31 31 Thus, time devoted by her for other subjects = 17 14 3 5 ? ? - ? ? ? ? h = 17 × 5 14 × 3 – h 15 15 ? ? ? ? ? ? = 85 – 42 h 15 ? ? ? ? ? ? = 43 15 h = 13 2 15 h EXERCISE 2.1 1. Solve: (i) 3 2 5 - (ii) 7 4 8 + (iii) 3 2 5 7 + (iv) 9 4 11 15 - (v) 7 2 3 10 5 2 + + (vi) 2 1 2 3 3 2 + (vii) 1 5 8 3 2 8 - 2. Arrange the following in descending order: (i) 2 9 2 3 8 21 , , (ii) 1 5 3 7 7 10 , , . 3. In a “magic square”, the sum of the numbers in each row, in each column and along the diagonals is the same. Is this a magic square? 4 11 3 11 8 11 9 11 5 11 1 11 2 11 7 11 6 11 4. A rectangular sheet of paper is 1 12 2 cm long and 2 10 3 cm wide. Find its perimeter. 5. Find the perimeters of (i) ? ABE (ii) the rectangle BCDE in this figure. Whose perimeter is greater? 6. Salil wants to put a picture in a frame. The picture is 7 3 5 cm wide. T o fit in the frame the picture cannot be more than 7 3 10 cm wide. How much should the picture be trimmed? (Along the first row 4 9 2 15 11 11 11 11 + + = ). 5 cm 2 3 2 cm 4 3 3 cm 5 7 cm 6 2020-21 © NCERT not to be republished MATHEMATICS 32 32 32 32 32 7. Ritu ate 3 5 part of an apple and the remaining apple was eaten by her brother Somu. How much part of the apple did Somu eat? Who had the larger share? By how much? 8. Michael finished colouring a picture in 7 12 hour . V aibhav finished colouring the same picture in 3 4 hour. Who worked longer? By what fraction was it longer? 2.3 MULTIPLICATION OF FRACTIONS Y ou know how to find the area of a rectangle. It is equal to length × breadth. If the length and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its area would be 7 × 4 = 28 cm 2 . What will be the area of the rectangle if its length and breadth are 7 1 2 cm and 3 1 2 cm respectively? Y ou will say it will be 7 1 2 × 3 1 2 = 15 2 × 7 2 cm 2 . The numbers 15 2 and 7 2 are fractions. T o calculate the area of the given rectangle, we need to know how to multiply fractions. W e shall learn that now . 2.3.1 Multiplication of a Fraction by a Whole Number Observe the pictures at the left (Fig 2.1). Each shaded part is 1 4 part of a circle. How much will the two shaded parts represent together? They will represent 1 1 4 4 + = 1 2× 4 . Combining the two shaded parts, we get Fig 2.2 . What part of a circle does the shaded part in Fig 2.2 represent? It represents 2 4 part of a circle . Fig 2.1 Fig 2.2 2020-21 © NCERT not to be republished FRACTIONS AND DECIMALS 33 33 33 33 33 The shaded portions in Fig 2.1 taken together are the same as the shaded portion in Fig 2.2, i.e., we get Fig 2.3. Fig 2.3 or 1 2× 4 = 2 4 . Can you now tell what this picture will represent? (Fig 2.4) Fig 2.4 And this? (Fig 2.5) Fig 2.5 Let us now find 1 3× 2 . We have 1 3× 2 = 1 1 1 3 2 2 2 2 + + = We also have 1 1 1 1+1+1 3×1 3 + + = = = 2 2 2 2 2 2 So 1 3× 2 = 3×1 2 = 3 2 Similarly 2 ×5 3 = 2×5 3 = ? Can you tell 2 3× 7 = ? 3 4 × ? 5 = The fractions that we considered till now, i.e., 1 2 2 3 , , , 2 3 7 5 and 3 5 were proper fractions. = = = 2020-21 © NCERT not to be republishedRead More

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