MATHEMATICS 24 EXAMPLE 2 Ramesh solved part of an exercise while Seema solved of it. Who 75 solved lesser part? SOLUTION In order to find who solved lesser part of the exercise, let us compare 24 and 5. 7428 Converting them to like fractions we have, 210 , = . = 735 535 10 28Since10 < 28 , so < . 35 35 24 Thus, < . 75 Ramesh solved lesser part than Seema. 13 EXAMPLE 3 Sameera purchased 3 kg apples and 4 kg oranges. What is the 24 total weight of fruits purchased by her? ⎛ 13⎞ SOLUTION The total weight of the fruits =⎜3 + 4 ⎟ kg ⎝ 24⎠ ⎛ 719⎞⎛14 19⎞ = ⎜+ ⎟ kg =⎜ +⎟ kg ⎝ 24 ⎠⎝ 44 ⎠ 33 1 = kg = 8 kg 44 24 EXAMPLE 4 Suman studies for 5 hours daily. She devotes 2 hours of her time 35 for Science and Mathematics. How much time does she devote for other subjects? 2 17 SOLUTION Total time of Suman’s study = 5 h = h 334 14 Time devoted by her for Science and Mathematics = 2 5 = 5 h MATHEMATICS 3 7. Ritu ate part of an apple and the remaining apple was eaten by her brother Somu. 5 How much part of the apple did Somu eat? Who had the larger share? By how much? 7 8. Michael finished colouring a picture in hour. Vaibhav finished colouring the same 12 3 picture in hour. Who worked longer? By what fraction was it longer? 4 2.3 MULTIPLICATION OF FRACTIONS You 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 cm2. 1 What will be the area of the rectangle if its length and breadth are 7 2 cm and 1 1 1157 15 3 cm respectively? You will say it will be 7 × 3 = × 2 cm2. The numbers 22222 7 and are fractions. To calculate the area of the given rectangle, we need to know how to 2 multiply fractions. We shall learn that now. 2.3.1 Multiplication of a Fraction by a Whole Number 1 Observe the pictures at the left (Fig 2.1). Each shaded part is part of 4 a circle. How much will the two shaded parts represent together? They 11 1 will represent + = 2× . 444 Fig 2.1 Combining the two shaded parts, we get Fig 2.2 . What part of a circle does the 2 shaded part in Fig 2.2 represent? It represents part of a circle . 4 Fig 2.2 FRACTIONS AND DECIMALS 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 12× 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 1 Let us now find 3× . 2 1 111 3 We have 3× = ++= 2222 2 1 1 1 1+1+1 3×1 3 We also have ++ = = = 2222 22 1 3×1 3 So 3× = = 222 2 2×5 Similarly 3×5 = 3 = ? 23 Can you tell 3×7 =? 4×5 =? 122 3 The fractions that we considered till now, i.e., ,, and were proper fractions. 2375 FRACTIONS AND DECIMALS 12 4. Shade: (i) of the circles in box (a) (ii) of the triangles in box (b) 233 (iii) of the squares in box (c). 5 (a) (b) (c) 5. Find: 12 (a) of (i) 24 (ii) 46 (b) of (i) 18 (ii) 27 23 34 (c) of (i) 16 (ii) 36 (d) of (i) 20 (ii) 35 45 6. Multiply and express as a mixed fraction : 13 1 × 56 (c) × (a) 35 (b) × 72 54 4 11 2 × (d) 46 (e) 3 ×6 (f) 3 ×8 34 5 132 552 7. Find: (a) of (i) 2 (ii) 4 (b) of (i) 3 (ii) 9 8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that 249863 2 contained 5 litres of water. Vidya consumed of the water. Pratap consumed the 5 remaining water. (i) How much water did Vidya drink? (ii) What fraction of the total quantity of water did Pratap drink? 2.3.2 Multiplication of a Fraction by a Fraction Farida had a 9 cm long strip of ribbon. She cut this strip into four equal parts. How did she do it? She folded the strip twice. What fraction of the total length will each part represent? 9 Each part will be of the strip. She took one part and divided it in two equal parts by 4 MATHEMATICS Fig 2.8 A Fig 2.9 19 folding the part once. What will one of the pieces represent? It will represent of 4 or 219 × 4. 219 Let us now see how to find the product of two fractions like × . 24 11To do this we first learn to find the products like × . 23 1 (a) How do we find of a whole? We divide the whole in three equal parts. Each of 3 1 the three parts represents of the whole. Take one part of these three parts, and 3 shade it as shown in Fig 2.8. 11 (b) How will you find of this shaded part? Divide this one-third ( ) shaded part into 23 11 11 two equal parts. Each of these two parts represents of i.e., × (Fig 2.9). 2323 11 Take out 1 part of these two and name it ‘A’. ‘A’ represents × 3. 21 (c) What fraction is ‘A’of the whole? For this, divide each of the remaining parts also 3 in two equal parts. How many such equal parts do you have now? There are six such equal parts. ‘A’is one of these parts. How did we decide that ‘A’was of the whole? The whole was divided in 6 = 2 × 3 So, ‘A’ is 1 6 of the whole. Thus, 1 2 × 1 3 = 1 6 . 1 6 parts and 1 = 1 × 1 part was taken out of it. 111 1×1 Thus, × = = 2362×3 1 1 1×1 or × = 232×3 MATHEMATICS 2. Multiply and reduce to lowest form (if possible) : (i) 2 3 × 2 2 3 (ii) 27 79× (iii) 36× 84 (iv) 93 55 × (v) 115 38 × (vi) 11 3 210 × (vii) 412 57 × 3. Multiply the following fractions: (i) 2 15 5 4 × (ii) 276 59 × (iii) 3 15 2 3 × (iv) 5 32 6 7 × (v) 243 57 × (vi) 32 3 5 × (vii) 433 75 × 4. Which is greater: (i) 2 7 of 3 4 or 3 5 of 5 8 (ii) 1 2 of 6 7 or 2 3 of 3 7 5. Saili plants 4 saplings, in a row, in her garden. The distance between two adjacent 3 saplings is m. Find the distance between the first and the last sapling. 4 3 6. Lipika reads a book for 1 hours everyday. She reads the entire book in 6 days. 4 How many hours in all were required by her to read the book? 3 7. A car runs 16 km using 1 litre of petrol. How much distance will it cover using 2 4 litres of petrol. 2 10 = . 8. (a) (i) Provide the number in the box , such that × 3 30 (ii) The simplest form of the number obtained in is _____. 3 24 = . (b) (i) Provide the number in the box , such that × 5 75 (ii) The simplest form of the number obtained in is _____. 2.4 DIVISION OF FRACTIONS John has a paper strip of length 6 cm. He cuts this strip in smaller strips of length 2 cm each. You know that he would get 6 ÷ 2 =3 strips. FRACTIONS AND DECIMALS 3 John cuts another strip of length 6 cm into smaller strips of length cm each. How 2 3 many strips will he get now? He will get 6 ÷ strips. 215 3 A paper strip of length cm can be cut into smaller strips of length cm each to give 22 15 3 ÷ pieces. 22 So, we are required to divide a whole number by a fraction or a fraction by another fraction. Let us see how to do that. 2.4.1 Division of Whole Number by a Fraction 1 Let us find 1÷. 2 We divide a whole into a number of equal parts such that each part is half of the whole. 11 The number of such half ( ) parts would be 1÷ . Observe the figure (Fig 2.11). How 22 many half parts do you see? There are two half parts. 12 12 So, 1 ÷ = 2. Also, 1× = 1 × 2 = 2. Thus, 1 ÷ = 1 × 21 21 11 Similarly, 3 ÷= number of parts obtained when each of the 3 whole, are divided 44 1 into equal parts = 12 (From Fig 2.12) 4 MATHEMATICS Reciprocal of a fraction 2 The number can be obtained by interchanging the numerator and denominator of 1 113 1 or by inverting . Similarly, is obtained by inverting 3. 2 21 Let us first see about the inverting of such numbers.Observe these products and fill in the blanks : 17 7 × = 1 54 45 × = -------- 1 9 9 × = ----- 2 7 × -------= 1 23 32 × = 23 32 × × = 6 6 = 1 ------5 9 × = 1 Multiply five more such pairs. The non-zero numbers whose product with each other is 1, are called the 59 95 reciprocals of each other. So reciprocal of is and the reciprocal of is . What 9559 12is the receiprocal of ? ? 97 23 You will see that the reciprocal of is obtained by inverting it. You get 2. 3 (i) Will the reciprocal of a proper fraction be again a proper fraction? (ii) Will the reciprocal of an improper fraction be again an improper fraction? Therefore, we can say that 12 1 1 ÷ 2 =1×1 = 1× reciprocal of 2. 14 1 3 ÷ 4 =3×1 = 3× reciprocal of 4 . 1 3 ÷ = ------= ----------------------. 23 34 So, 2 ÷ = 2 × reciprocal of = 2× . 443 2 5 ÷ = 5 × -------------------= 5 × ------------ 9 MATHEMATICS 4. Express in kg: (i) 200 g (ii) 3470 g (iii) 4 kg 8 g 5. Write the following decimal numbers in the expanded form: (i) 20.03 (ii) 2.03 (iii) 200.03 (iv) 2.034 6. Write the place value of 2 in the following decimal numbers: (i) 2.56 (ii) 21.37 (iii) 10.25 (iv) 9.42 (v) 63.352. 7. Dinesh went from place A to place B and from there to place C. A is 7.5 km from B and B is 12.7 km from C.Ayub went from placeAto place D and from there to place C. D is 9.3 km from A and C is 11.8 km from D. Who travelled more and by how much? 8. Shyama bought 5 kg 300 g apples and 3 kg 250 g mangoes. Sarala bought 4 kg 800 g oranges and 4 kg 150 g bananas. Who bought more fruits? 9. How much less is 28 km than 42.6 km? 2.6 MULTIPLICATION OF DECIMAL NUMBERS Reshma purchased 1.5kg vegetable at the rate of ` 8.50 per kg. How much money should she pay? Certainly it would be ` (8.50 × 1.50). Both 8.5 and 1.5 are decimal numbers. So, we have come across a situation where we need to know how to multiply two decimals. Let us now learn the multiplication of two decimal numbers. First we find 0.1 × 0.1. 1 11 1×1 1 Now, 0.1 = . So, 0.1 ×0.1 = × = = = 0.01. 10 10 1010×10100 Let us see it’s pictorial representation (Fig 2.13) 1 The fraction represents 1 part out of 10 equal parts. 10 1 The shaded part in the picture represents 10 . th part into 10 equal parts and take one part We know that, 1 10 1× 10 means 1 10 of 1 10 . So, divide this 1 10 out of it. Fig 2.13 FRACTIONS AND DECIMALS Thus, we have, (Fig 2.14). Fig 2.14 1 The dotted square is one part out of 10 of the th part. That is, it represents 10 11 × or 0.1 × 0.1. 10 10 Can the dotted square be represented in some other way? How many small squares do you find in Fig 2.14? There are 100 small squares. So the dotted square represents one out of 100 or 0.01. Hence, 0.1 × 0.1 = 0.01. Note that 0.1 occurs two times in the product. In 0.1 there is one digit to the right of the decimal point. In 0.01 there are two digits (i.e., 1 + 1) to the right of the decimal point. Let us now find 0.2 × 0.3. 23 We have, 0.2 ×0.3 = × 10 10 11 As we did for , let us divide the square into 10 10 10 3 equal parts and take three parts out of it, to get . Again 10 divide each of these three equal parts into 10 equal parts and 23 take two from each. We get × . 10 10 23 The dotted squares represent × or 0.2 × 0.3. (Fig 2.15) 10 10 Since there are 6 dotted squares out of 100, so they also reprsent 0.06. Fig 2.15 FRACTIONS AND DECIMALS Take 31.5 ÷ 10 = 3.15. In 31.5 and 3.15, the digits are same i.e., 3, 1, and 5 but the decimal point has shifted in the quotient. To which side and by how many digits? The decimal point has shifted to the left by one place. Note that 10 has one zero over 1. Consider now 31.5 ÷ 100 = 0.315. In 31.5 and 0.315 the digits are same, but what about the decimal point in the quotient? It has shifted to the left by two places. Note that 100 has two zeros over1. So we can say that, while dividing a number by 10, 100 or 1000, the digits of the number and the quotient are same but the decimal point in the quotient shifts to the left by as many places as there are zeros over 1. Using this observation let us now quickly find: 2.38 ÷ 10 = 0.238, 2.38 ÷ 100 = 0.0238, 2.38 ÷ 1000 = 0.00238 2.7.2 Division of a Decimal Number by a Whole Number Let us find 6.4 . Remember we also write it as 6.4 ÷ 2. 2 64 641 So, 6.4 ÷ 2 = ÷ 2 = × as learnt in fractions.. 10102 64 ×1 × 1 164 641 32 = ==× × 32 == 32 10 × 2 10 × 2 10 2= 10 10 . Or, let us first divide 64 by 2. We get 32. There is one digit to the right of the decimal point in 6.4. Place the decimal in 32 such that there would be one digit to its right. We get 3.2 again. To find 19.5 ÷ 5, first find 195 ÷5. We get 39. There is one digit to the right of the decimal point in 19.5. Place the decimal point in 39 such that there would be one digit to its right. You will get 3.9. 1296 1296 1 1 1296 1 Now, 12.96 ÷ 4 = ÷4 = ×4= × = ×324 = 3.24 100 100 100 4 100 Or, divide 1296 by 4. You get 324. There are two digits to the right of the decimal in 12.96. Making similar placement of the decimal in 324, you will get 3.24. Note that here and in the next section, we have considered only those divisions in which, ignoring the decimal, the number would be completely divisible by another number to give remainder zero. Like, in 19.5 ÷ 5, the number 195 when divided by 5, leaves remainder zero. However, there are situations in which the number may not be completely divisible by another number, i.e., we may not get remainder zero. For example, 195 ÷ 7. We deal with such situations in later classes. Thus, 40.86 ÷ 6 = 6.81 MATHEMATICS 5. (a) The product of two proper fractions is less than each of the fractions that are multiplied. (b) The product of a proper and an improper fraction is less than the improper fraction and greater than the proper fraction. (c) The product of two imporper fractions is greater than the two fractions. 6. A reciprocal of a fraction is obtained by inverting it upside down. 7. We have seen how to divide two fractions. (a) While dividing a whole number by a fraction, we multiply the whole number with the reciprocal of that fraction. 3 510 For example, 2 ÷=2× = 5 33 (b) While dividing a fraction by a whole number we multiply the fraction by the reciprocal of the whole number. 2 1 2 For example, 2 ÷=7× = 3 37 21 (c) While dividing one fraction by another fraction, we multuiply the first fraction by 25 2714 the reciprocal of the other. So, ÷=× = . 37 3515 8. We also learnt how to multiply two decimal numbers. While multiplying two decimal numbers, first multiply them as whole numbers. Count the number of digits to the right of the decimal point in both the decimal numbers.Add the number of digits counted. Put the decimal point in the product by counting the digits from its rightmost place. The count should be the sum obtained earlier. For example, 0.5 × 0.7 = 0.35 9. To multiply a decimal number by 10, 100 or 1000, we move the decimal point in the number to the right by as many places as there are zeros over 1. Thus 0.53 × 10 = 5.3, 0.53 × 100 = 53, 0.53 × 1000 = 530 10. We have seen how to divide decimal numbers. (a) To divide a decimal number by a whole number, we first divide them as whole numbers. Then place the decimal point in the quotient as in the decimal number. For example, 8.4 ÷ 4 = 2.1 Note that here we consider only those divisions in which the remainder is zero. (b) To divide a decimal number by 10, 100 or 1000, shift the digits in the decimal number to the left by as many places as there are zeros over 1, to get the quotient. So, 23.9 ÷ 10 = 2.39,23.9 ÷ 100 = 0 .239, 23.9 ÷ 1000 = 0.0239 (c) While dividing two decimal numbers, first shift the decimal point to the right by equal number of places in both, to convert the divisor to a whole number. Then divide. Thus, 2.4 ÷ 0.2 = 24 ÷ 2 = 12.