Fractions and Decimals PDF

20 MATHEMATICS Chapter 2 Fractions and Decimals 2.1 MULTIPLICATION OF FRACTIONS You know how to find the area of a rectangle. It is equal to length ×...

20 MATHEMATICS Chapter 2 Fractions and Decimals 2.1 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 cm and 2 1 1 1 15 7 15 3 cm respectively? You will say it will be 7 × 3 = × cm2. The numbers 2 2 2 2 2 2 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. 2.1.1 Multiplication of a Fraction by a Whole Number 1 Observe the pictures at the left (Fig 2.1). Each shaded part is 4 part of a circle. How much will the two shaded parts represent together? 1 1 1 They will represent+ = 2×. 4 4 4 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 2024-25  FRACTIONS AND DECIMALS 21 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 1 2 or 2× =. 4 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 1 1 1 3 We have 3× = + + = 2 2 2 2 2 1 1 1 1+1+1 3×1 3 We also have + + = = = 2 2 2 2 2 2 1 3×1 3 So 3× = = 2 2 2 2 2 ×5 Similarly ×5 = =? 3 3 2 3 Can you tell 3× =? 4× =? 7 5 1 2 2 3 3 The fractions that we considered till now, i.e., , , ,and were proper fractions. 2 3 7 5 5 2024-25 22 MATHEMATICS For improper fractions also we have, 5 2 × 5 10 2× = = 3 3 3 8 7 Try, = ? 4× = ? 3× 7 5 Thus, to multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same. TRY THESE 2 9 1 13 1. Find: (a) ×3 (b) ×6 (c) 3 × (d) ×6 7 7 8 11 If the product is an improper fraction express it as a mixed fraction. 2 4 2. Represent pictorially : 2× = 5 5 To multiply a mixed fraction to a whole number, first convert the TRY T HESE mixed fraction to an improper fraction and then multiply. 3 5 57 1 Find: (i) 5 × 2 19 7 Therefore, 3× 2 = 3× = =8. 7 7 7 7 4 (ii) 1 × 6 2 22 9 Similarly, 2× 4 = 2× =? 5 5 Fraction as an operator ‘of ’ Observe these figures (Fig 2.6) The two squares are exactly similar. 1 Each shaded portion represents of 1. 2 1 So, both the shaded portions together will represent of 2. 2 1 Combine the 2 shaded parts. It represents 1. 2 1 1 So, we say of 2 is 1. We can also get it as × 2 = 1. 2 2 1 1 Thus, of 2 = × 2 = 1 Fig 2.6 2 2 2024-25  FRACTIONS AND DECIMALS 23 Also, look at these similar squares (Fig 2.7). 1 Each shaded portion represents of 1. 2 1 So, the three shaded portions represent of 3. 2 Combine the 3 shaded parts. 1 3 It represents 1 i.e.,. 2 2 1 3 1 3 Fig 2.7 So, of 3 is. Also, × 3 =. 2 2 2 2 1 1 3 Thus, of 3 = × 3 =. 2 2 2 So we see that ‘of’ represents multiplication. 1 Farida has 20 marbles. Reshma hasth of the number of marbles what 5 Farida has. How many marbles Reshma has? As, ‘of ’ indicates multiplication, 1 so, Reshma has × 20 = 4 marbles. 5 1 1 16 Similarly, we have of 16 is ×16 = = 8. 2 2 2 TRY THESE 1 1 2 Can you tell, what is (i) of 10?, (ii) of 16?, (iii) of 25? 2 4 5 1 EXAMPLE 1 In a class of 40 students of the total number of studetns like to study 5 2 English, of the total number like to study Mathematics and the remaining 5 students like to study Science. (i) How many students like to study English? (ii) How many students like to study Mathematics? (iii) What fraction of the total number of students like to study Science? S OLUTION Total number of students in the class = 40. 1 (i) Of these of the total number of students like to study English. 5 2024-25 24 MATHEMATICS 1 1 Thus, the number of students who like to study English = of 40 = × 40 = 8. 5 5 (ii) Try yourself. (iii) The number of students who like English and Mathematics = 8 + 16 = 24. Thus, the number of students who like Science = 40 – 24 = 16. Thus, the required fraction is 16. 40 E XERCISE 2.1 1. Which of the drawings (a) to (d) show : 1 1 2 1 (i) 2 × (ii) 2 × (iii) 3 × (iv) 3 × 5 2 3 4 (a) (b) (c) (d) 2. Some pictures (a) to (c) are given below. Tell which of them show: 1 3 1 2 3 1 (i) 3 × = (ii) 2 × = (iii) 3 × =2 5 5 3 3 4 4 = (a) (b) = (c) 3. Multiply and reduce to lowest form and convert into a mixed fraction: 3 1 6 2 2 (i) 7 × (ii) 4 × (iii) 2 × (iv) 5 × (v) ×4 5 3 7 9 3 5 4 4 1 3 (vi) ×6 (vii) 11× (viii) 20 × (ix) 13 × (x) 15 × 2 7 5 3 5 2024-25  FRACTIONS AND DECIMALS 25 1 2 4. Shade: (i) of the circles in box (a) (ii) of the triangles in box (b) 2 3 3 (iii) of the squares in box (c). 5 (a) (b) (c) 5. Find: 1 2 (a) of (i) 24 (ii) 46 (b) of (i) 18 (ii) 27 2 3 3 4 (c) of (i) 16 (ii) 36 (d) of (i) 20 (ii) 35 4 5 6. Multiply and express as a mixed fraction : 1 3 1 (a) 3× 5 (b) 5 × 6 (c) 7× 2 5 4 4 1 1 2 (d) 4 × 6 (e) 3 ×6 (f) 3 ×8 3 4 5 1 3 2 5 5 2 7. Find: (a) of (i) 2 (ii) 4 (b) of (i) 3 (ii) 9 2 4 9 8 6 3 8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that 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? 2.1.2 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 2024-25  26 MATHEMATICS 1 9 folding the part once. What will one of the pieces represent? It will represent of or 2 4 1 9 ×. 2 4 1 9 Let us now see how to find the product of two fractions like ×. 2 4 1 1 To do this we first learn to find the products like ×. 2 3 1 (a) How do we find of a whole? We divide the whole in three equal parts. Each of 3 1 Fig 2.8 the three parts represents of the whole. Take one part of these three parts, and 3 shade it as shown in Fig 2.8. A 1 1 (b) How will you find of this shaded part? Divide this one-third ( ) shaded part into 2 3 1 1 1 1 two equal parts. Each of these two parts represents of i.e., × (Fig 2.9). 2 3 2 3 Fig 2.9 1 1 Take out 1 part of these two and name it ‘A’. ‘A’ represents ×. 2 3 1 (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. 1 1 1 1 So, ‘A’ is of the whole. Thus, × =. 6 2 3 6 1 How did we decide that ‘A’ was of the whole? The whole was divided in 6 = 2 × 3 6 parts and 1 = 1 × 1 part was taken out of it. 1 1 1 1×1 Thus, × = = 2 3 6 2×3 1 1 1×1 or × = 2 3 2×3 2024-25  FRACTIONS AND DECIMALS 27 1 1 The value of × can be found in a similar way. Divide the whole into two equal 3 2 parts and then divide one of these parts in three equal parts. Take one of these parts. This will represent 1 × 1 i.e., 1. 3 2 6 1 1 1 1×1 Therefore × = = as discussed earlier. 3 2 6 3× 2 1 1 1 1 1 Hence × = × = 2 3 3 2 6 1 1 1 1 1 1 1 1 Find × and × ; × and × and check whether you get 3 4 4 3 2 5 5 2 1 1 1 1 1 1 1 1 × = × ; × = × 3 4 4 3 2 5 5 2 TRY THESE Fill in these boxes: 1 1 1×1 1 1 (i) × = = (ii) × = = 2 7 2×7 5 7 1 1 1 1 (iii) × = = (iv) × = = 7 2 7 5 1 E XAMPLE 2 Sushant reads part of a book in 1 hour. How much part of the book 3 1 will he read in 2 hours? 5 1 SOLUTION The part of the book read by Sushant in 1 hour =. 3 1 1 1 So, the part of the book read by him in 2 hours = 2 × 5 5 3 11 1 11× 1 11 = × = = 5 3 5× 3 15 1 5 5 1 Let us now find ×. We know that = ×5. 2 3 3 3 1 5 1 1 1 5 So, × = × ×5= 5= 2 3 2 3 6 6 2024-25  28 MATHEMATICS 5 1× 5 1 5 1× 5 5 Also, =. Thus, × = =. 6 2×3 2 3 2×3 6 This is also shown by the figures drawn below. Each of these five equal shapes (Fig 2.10) are parts of five similar circles. Take one such shape. To obtain this shape we first divide a circle in three equal parts. Further divide each of these three parts in two equal parts. One part out of it is the shape we considered. What will it represent? 1 1 1 1 5 It will represent × =. The total of such parts would be 5 × =. 2 3 6 6 6 Fig 2.10 TRY T HESE 3 1 3 ×1 3 Similarly × = =. 5 7 5× 7 35 1 4 2 1 Find: × ; × 2 7 2 7 2 × 7 14 3 5 3 5 We can thus find × as × = =. 3 5 3 5 3 × 5 15 Product of Numerators So, we find that we multiply two fractions as. Product of Denominators Value of the Products TRY THESE You have seen that the product of two whole numbers is bigger than each of the two whole numbers. For example, 3 × 4 = 12 and 12 > 4, 12 > 3. What 8 4 3 2 happens to the value of the product when we multiply two fractions? Find: × ; ×. Let us first consider the product of two proper fractions. 3 7 4 3 We have, 2 4 8 8 2 8 4 × = < , < Product is less than each of the fractions 3 5 15 15 3 15 5 1 2 × = --------- --------,-------- -------------------------------------- 5 7 3
× = --------,-------- -------------------------------------- 5 8 2 4 8 × = --------,-------- --------------------------------------
9 45 2024-25  FRACTIONS AND DECIMALS 29 You will find that when two proper fractions are multiplied, the product is less than each of the fractions. Or, we say the value of the product of two proper fractions is smaller than each of the two fractions. Check this by constructing five more examples. Let us now multiply two improper fractions. 7 5 35 35 7 35 5 × = > , > Product is greater than each of the fractions 3 2 6 6 3 6 2 6
24 × = --------,-------- ---------------------------------------- 5 3 15 9 7 63 × = --------,-------- ---------------------------------------- 2 8 3 8 24 × = --------,-------- ---------------------------------------- 7 14 We find that the product of two improper fractions is greater than each of the two fractions. Or, the value of the product of two improper fractions is more than each of the two fractions. Construct five more examples for yourself and verify the above statement. 2 7 Let us now multiply a proper and an improper fraction, say and. 3 5 2 7 14 14 7 14 2 We have × =. Here, < and > 3 5 15 15 5 15 3 The product obtained is less than the improper fraction and greater than the proper fraction involved in the multiplication. 6 2 8 4 Check it for × , ×. 5 8 3 5 E XERCISE 2.2 1. Find: 1 1 3 4 (i) of (a) (b) (c) 4 4 5 3 1 2 6 3 (ii) of (a) (b) (c) 7 9 5 10 2024-25 30 MATHEMATICS 2. Multiply and reduce to lowest form (if possible) : 2 2 2 7 3 6 9 3 (i) ×2 (ii) × (iii) × (iv) × 3 3 7 9 8 4 5 5 1 15 11 3 4 12 (v) × (vi) × (vii) × 3 8 2 10 5 7 3. Multiply the following fractions: 2 1 2 7 3 1 5 3 (i) ×5 (ii) 6 × (iii) ×5 (iv) ×2 5 4 5 9 2 3 6 7 2 4 3 4 3 (v) 3 × (vi) 2 × 3 (vii) 3 × 5 7 5 7 5 4. Which is greater: 2 3 3 5 1 6 2 3 (i) of or of (ii) of or of 7 4 5 8 2 7 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 _____. 2.2 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. 2024-25  FRACTIONS AND DECIMALS 31 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. 2 15 3 A paper strip of length cm can be cut into smaller strips of length cm each to give 2 2 15 3 ÷ pieces. 2 2 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. 2.2.1 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. 1 1 The number of such half ( ) parts would be 1÷. Observe the figure (Fig 2.11). How 2 2 many half parts do you see? There are two half parts. 1 1 1 2 1 2 So, 1÷ = 2. Also, 1× = 1 × 2 = 2. Thus, 1 ÷ = 1 × 2 2 2 1 2 1 1 1 Similarly, 3 ÷ = number of parts obtained when each of the 3 whole, are divided Fig 2.11 4 4 1 into equal parts = 12 (From Fig 2.12) 4 1 1 1 1 1 1 4 4 4 4 4 4 1 1 1 1 1 1 4 4 4 4 4 4 Fig 2.12 4 1 4 Observe also that, 3× = 3 × 4 = 12. Thus, 3 ÷ = 3 × = 12. 1 4 1 1 2 Find in a similar way, 3 ÷ and 3 ×. 2 1 2024-25 32 MATHEMATICS Reciprocal of a fraction 2 The number can be obtained by interchanging the numerator and denominator of 1 1 1 3 1 or by inverting. Similarly, is obtained by inverting. 2 2 1 3 Let us first see about the inverting of such numbers. Observe these products and fill in the blanks : 1 5 4 7× = 1 × = --------- 7 4 5 1 2 × 9 = ------ × ------- = 1 9 7 2 3 2×3 6 5 × = = =1 ------ × = 1 3 2 3× 2 6 9 Multiply five more such pairs. The non-zero numbers whose product with each other is 1, are called the 5 9 9 5 reciprocals of each other. So reciprocal of is and the reciprocal of is. What 9 5 5 9 1 2 is the receiprocal of ? ? 9 7 2 3 You will see that the reciprocal of is obtained by inverting it. You get. 3 2 THINK, DISCUSS AND WRITE (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 1 2 1 1÷ = 1× = 1× reciprocal of. 2 1 2 1 4 1 3÷ = 3× = 3× reciprocal of. 4 1 4 1 3÷ = ------ = ----------------------. 2 3 3 4 So, 2 ÷ = 2 × reciprocal of = 2 ×. 4 4 3 2 5÷ = 5× ------------------- = 5 × ------------- 9 2024-25  FRACTIONS AND DECIMALS 33 Thus, to divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction. TRY THESE 2 4 8 Find : (i) 7 ÷ (ii) 6 ÷ (iii) 2 ÷ 5 7 9 l While dividing a whole number by a mixed fraction, first convert the mixed fraction into improper fraction and then solve it. 2 12 1 10 TRY THESE Thus, 4 ÷ 2 = 4÷ = ? Also, 5 ÷ 3 =3÷ =? 5 5 3 3 1 Find: (i) 6 ÷ 5 2.2.2 Division of a Fraction by a Whole Number 3 3 4 l What will be ÷ 3? (ii) 7 ÷ 2 4 7 3 3 3 3 1 3 1 Based on our earlier observations we have: ÷3= ÷ = × = = 4 4 1 4 3 12 4 2 2 1 5 2 So, ÷7 = × = ? What is ÷ 6 , ÷8? 3 3 7 7 7 l While dividing mixed fractions by whole numbers, convert the mixed fractions into improper fractions. That is, 2 8 2 3 2 ÷ 5 = ÷ 5 = ------ ; 4 ÷ 3 = ------ = ------; 2 ÷ 2 = ------ = ------ 3 3 5 5 2.2.3 Division of a Fraction by Another Fraction 1 6 We can now find ÷. 3 5 1 6 1 6 1 5 5 ÷ = × reciprocal of   . 3 5 3 5 3 6 18 8 2 8 2 1 3 Similarly, ÷ = × reciprocal of =? and, ÷ = ? 5 3 5 3 2 4 TRY THESE 3 1 1 3 1 3 1 9 Find: (i) ÷ (ii) ÷ (iii) 2 ÷ (iv) 5 ÷ 5 2 2 5 2 5 6 2 2024-25 34 MATHEMATICS E XERCISE 2.3 1. Find: 3 5 7 8 (i) 12 ÷ (ii) 14 ÷ (iii) 8 ÷ (iv) 4 ÷ 4 6 3 3 1 4 (v) 3 ÷ 2 (vi) 5 ÷ 3 3 7 2. Find the reciprocal of each of the following fractions. Classify the reciprocals as proper fractions, improper fractions and whole numbers. 3 5 9 6 (i) (ii) (iii) (iv) 7 8 7 5 12 1 1 (v) (vi) (vii) 7 8 11 3. Find: 7 4 6 1 (i) ÷2 (ii) ÷5 (iii) ÷7 (iv) 4 ÷ 3 3 9 13 3 1 3 (v) 3 ÷ 4 (vi) 4 ÷ 7 2 7 4. Find: 2 1 4 2 3 8 1 3 1 8 (i) ÷ (ii) ÷ (iii) ÷ (iv) 2 ÷ (v) 3 ÷ 5 2 9 3 7 7 3 5 2 3 2 1 1 2 1 1 (vi) ÷1 (vii) 3 ÷ 1 (viii) 2 ÷ 1 5 2 5 3 5 5 2.3 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 deci- mals. Let us now learn the multiplication of two decimal numbers. First we find 0.1 × 0.1. 1 1 1 Now, 0.1 =. So, 0.1 × 0.1 = × = 10 10 10 1×1 1 = = 0.01. 10 ×10 100 Let us see it’s pictorial representation (Fig 2.13) 1 The fraction represents 1 part out of 10 equal 10 Fig 2.13 parts. 2024-25  FRACTIONS AND DECIMALS 35 1 The shaded part in the picture represents. 10 We know that, 1 1 1 1 1 × means of. So, divide this th part into 10 equal parts and take one 10 10 10 10 10 part out of it. Thus, we have, (Fig 2.14). Fig 2.14 1 th The dotted square is one part out of 10 of the part. That is, it represents 10 1 1 × 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. 2 3 We have, 0.2 × 0.3 = × 10 10 1 1 As we did for × , let us divide the square into 10 equal 10 10 3 parts and take three parts out of it, to get. Again divide each Fig 2.15 10 2024-25 36 MATHEMATICS 2 3 of these three equal parts into 10 equal parts and take two from each. We get ×. 10 10 2 3 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. Thus, 0.2 × 0.3 = 0.06. Observe that 2 × 3 = 6 and the number of digits to the right of the decimal point in 0.06 is 2 (= 1 + 1). Check whether this applies to 0.1 × 0.1 also. Find 0.2 × 0.4 by applying these observations. While finding 0.1 × 0.1 and 0.2 × 0.3, you might have noticed that first we multiplied them as whole numbers ignoring the decimal point. In 0.1 × 0.1, we found 01 × 01 or 1 × 1. Similarly in 0.2 × 0.3 we found 02 × 03 or 2 × 3. Then, we counted the number of digits starting from the rightmost digit and moved towards left. We then put the decimal point there. The number of digits to be counted is obtained by adding the number of digits to the right of the decimal point in the decimal numbers that are being multiplied. Let us now find 1.2 × 2.5. Multiply 12 and 25. We get 300. Both, in 1.2 and 2.5, there is 1 digit to the right of the decimal point. So, count 1 + 1 = 2 digits from the rightmost digit (i.e., 0) in 300 and move towards left. We get 3.00 or 3. Find in a similar way 1.5 × 1.6, 2.4 × 4.2. While multiplying 2.5 and 1.25, you will first multiply 25 and 125. For placing the decimal in the product obtained, you will count 1 + 2 = 3 (Why?) digits starting from the rightmost digit. Thus, 2.5 × 1.25 = 3.225 Find 2.7 × 1.35. TRY THESE 1. Find: (i) 2.7 × 4 (ii) 1.8 × 1.2 (iii) 2.3 × 4.35 2. Arrange the products obtained in (1) in descending order. E XAMPLE 3 The side of an equilateral triangle is 3.5 cm. Find its perimeter. S OLUTION All the sides of an equilateral triangle are equal. 2024-25  FRACTIONS AND DECIMALS 37 So, length of each side = 3.5 cm Thus, perimeter = 3 × 3.5 cm = 10.5 cm E XAMPLE 4 The length of a rectangle is 7.1 cm and its breadth is 2.5 cm. What is the area of the rectangle? S OLUTION Length of the rectangle = 7.1 cm Breadth of the rectangle = 2.5 cm Therefore, area of the rectangle = 7.1 × 2.5 cm2 = 17.75 cm2 2.3.1 Multiplication of Decimal Numbers by 10, 100 and 1000 23 235 Reshma observed that 2.3 = whereas 2.35 =. Thus, she found that depending 10 100 on the position of the decimal point the decimal number can be converted to a fraction with denominator 10 or 100. She wondered what would happen if a decimal number is multiplied by 10 or 100 or 1000. Let us see if we can find a pattern of multiplying numbers by 10 or 100 or 1000. Have a look at the table given below and fill in the blanks: 176 1.76 × 10 = × 10 = 17.6 2.35 ×10 =___ 12.356 × 10 =___ 100 176 1.76 × 100 = × 100 = 176 or 176.0 2.35 ×100 = ___ 12.356 × 100 =___ 100 176 1.76 × 1000 = × 1000 = 1760 or 2.35 ×1000 = ___ 12.356 × 1000 = ___ 100 1760.0 5 0.5 × 10 = × 10 = 5 ; 0.5 × 100 = ___ ; 0.5 × 1000 = ___ 10 Observe the shift of the decimal point of the products in the table. Here the numbers are multiplied by 10,100 and 1000. In 1.76 × 10 = 17.6, the digits are same i.e., 1, 7 and 6. Do you observe this in other products also? Observe 1.76 and 17.6. To which side has the decimal point shifted, right or left? The decimal point has shifted to the right by one place. Note that 10 has one zero over 1. In 1.76×100 = 176.0, observe 1.76 and 176.0. To which side and by how many digits has the decimal point shifted? The decimal point has shifted to the right by two places. Note that 100 has two zeros over one. Do you observe similar shifting of decimal point in other products also? 2024-25  38 MATHEMATICS So we say, when a decimal number is multiplied by 10, 100 or 1000, the digits in the product are same as in the decimal number but the decimal point in the product is shifted to the right by as, many of places as there are zeros over one. Based on these observations we can now say TRY T HESE 0.07 × 10 = 0.7, 0.07 × 100 = 7 and 0.07 × 1000 = 70. Find: (i) 0.3 × 10 Can you now tell 2.97 × 10 = ? 2.97 × 100 = ? 2.97 × 1000 = ? (ii) 1.2 × 100 Can you now help Reshma to find the total amount i.e., ` 8.50 × (iii) 56.3 × 1000 150, that she has to pay? E XERCISE 2.4 1. Find: (i) 0.2 × 6 (ii) 8 × 4.6 (iii) 2.71 × 5 (iv) 20.1 × 4 (v) 0.05 × 7 (vi) 211.02 × 4 (vii) 2 × 0.86 2. Find the area of rectangle whose length is 5.7cm and breadth is 3 cm. 3. Find: (i) 1.3 × 10 (ii) 36.8 × 10 (iii) 153.7 × 10 (iv) 168.07 × 10 (v) 31.1 × 100 (vi) 156.1 × 100 (vii) 3.62 × 100 (viii) 43.07 × 100 (ix) 0.5 × 10 (x) 0.08 × 10 (xi) 0.9 × 100 (xii) 0.03 × 1000 4. A two-wheeler covers a distance of 55.3 km in one litre of petrol. How much distance will it cover in 10 litres of petrol? 5. Find: (i) 2.5 × 0.3 (ii) 0.1 × 51.7 (iii) 0.2 × 316.8 (iv) 1.3 × 3.1 (v) 0.5 × 0.05 (vi) 11.2 × 0.15 (vii) 1.07 × 0.02 (viii) 10.05 × 1.05 (ix) 101.01 × 0.01 (x) 100.01 × 1.1 2.4 DIVISION OF DECIMAL NUMBERS Savita was preparing a design to decorate her classroom. She needed a few coloured strips of paper of length 1.9 cm each. She had a strip of coloured paper of length 9.5 cm. How many pieces of the required length will she get out of this strip? She thought it would 9.5 be cm. Is she correct? 1.9 Both 9.5 and 1.9 are decimal numbers. So we need to know the division of decimal numbers too! 2.4.1 Division by 10, 100 and 1000 Let us find the division of a decimal number by 10, 100 and 1000. Consider 31.5 ÷ 10. 2024-25  FRACTIONS AND DECIMALS 39 315 1 315 31.5 ÷ 10 = × = = 3.15 10 10 100 315 1 315 Similarly, 31.5 ÷ 100 = × = = 0.315 10 100 1000 Let us see if we can find a pattern for dividing numbers by 10, 100 or 1000. This may help us in dividing numbers by 10, 100 or 1000 in a shorter way. 31.5 ÷ 10 = 3.15 231.5 ÷ 10 =___ 1.5 ÷ 10 =___ 29.36 ÷ 10 =___ 31.5 ÷ 100 = 0.315 231.5 ÷ 10 =___ 1.5 ÷ 100 =___ 29.36 ÷ 100 =___ 31.5 ÷ 1000 = 0.0315 231.5 ÷ 1000 =___ 1.5 ÷ 1000 =___ 29.36 ÷1000 =___ 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 TRY T HESE quotient. To which side and by how many digits? The decimal Find: (i) 235.4 ÷ 10 point has shifted to the left by one place. Note that 10 has one zero over 1. (ii) 235.4 ÷100 Consider now 31.5 ÷ 100 = 0.315. In 31.5 and 0.315 the (iii) 235.4 ÷ 1000 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 2.4.2 Division of a Decimal Number by a Whole Number Let us find 6.4. Remember we also write it as 6.4 ÷ 2. TRY THESE 2 (i) 35.7 ÷ 3 = ?; 64 64 1 (ii) 25.5 ÷ 3 = ? So, 6.4 ÷ 2 = ÷2 = × as learnt in fractions.. 10 10 2 64 × 1 1 × 64 1 64 1 32 = = = × = × 32 = = 3. 2 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. TRY THESE 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 (i) 43.15 ÷ 5 = ?; would be one digit to its right. You will get 3.9. (ii) 82.44 ÷ 6 = ? 2024-25  40 MATHEMATICS TRY T HESE 1296 1296 1 1 1296 1 Now, 12.96 ÷ 4 = ÷4 = × = × = × 324 = 3.24 Find: (i) 15.5 ÷ 5 100 100 4 100 4 100 (ii) 126.35 ÷ 7 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. EXAMPLE 5 Find the average of 4.2, 3.8 and 7.6. 4.2 + 3.8 + 7.6 S OLUTION The average of 4.2, 3.8 and 7.6 is = = 5.2. 3 2.4.3 Division of a Decimal Number by another Decimal Number 25.5 Let us find i.e., 25.5 ÷ 0.5. 0.5 255 5 255 10 We have 25.5 ÷ 0.5 = ÷ = × = 51. Thus, 25.5 ÷ 0.5 = 51 10 10 10 5 25.5 What do you observe? For , we find that there is one digit to the right of the 0.5 decimal in 0.5. This could be converted to whole number by dividing by 10. Accordingly 25.5 was also converted to a fraction by dividing by 10. Or, we say the decimal point was shifted by one place to the right in 0.5 to make it 5. So, there was a shift of one decimal point to the right in 25.5 also to make it 255. 22.5 225 Thus, 22.5 ÷ 1.5 = 1.5 = 15 = 15 TRY THESE 20.3 15.2 7.75 42.8 5.6 Find and in a similar way. Find: (i) (ii) (iii) 0. 7 0. 8 0.25 0.02 1.4 Let us now find 20.55 ÷ 1.5. 3.96 2.31 We can write it is as 205.5 ÷ 15, as discussed above. We get 13.7. Find ,. 0.4 0.3 2024-25  FRACTIONS AND DECIMALS 41 33.725 3372.5 Consider now,. We can write it as (How?) and we get the quotient 0.25 25 27 as 134.9. How will you find ? We know that 27 can be written as 27.00. 0.03 27 27.00 2700 So, = = = 900 0.03 0.03 3 E XAMPLE 6 Each side of a regular polygon is 2.5 cm in length. The perimeter of the polygon is 12.5cm. How many sides does the polygon have? SOLUTION The perimeter of a regular polygon is the sum of the lengths of all its equal sides = 12.5 cm. 12.5 125 Length of each side = 2.5 cm. Thus, the number of sides = = =5 2.5 25 The polygon has 5 sides. E XAMPLE 7 A car covers a distance of 89.1 km in 2.2 hours. What is the average distance covered by it in 1 hour? SOLUTION Distance covered by the car = 89.1 km. Time required to cover this distance = 2.2 hours. 89.1 891 So distance covered by it in 1 hour = = = 40.5 km. 2.2 22 E XERCISE 2.5 1. Find: (i) 0.4 ÷ 2 (ii) 0.35 ÷ 5 (iii) 2.48 ÷ 4 (iv) 65.4 ÷ 6 (v) 651.2 ÷ 4 (vi) 14.49 ÷ 7 (vii) 3.96 ÷ 4 (viii) 0.80 ÷ 5 2. Find: (i) 4.8 ÷ 10 (ii) 52.5 ÷ 10 (iii) 0.7 ÷ 10 (iv) 33.1 ÷ 10 (v) 272.23 ÷ 10 (vi) 0.56 ÷ 10 (vii) 3.97 ÷10 3. Find: (i) 2.7 ÷ 100 (ii) 0.3 ÷ 100 (iii) 0.78 ÷ 100 (iv) 432.6 ÷ 100 (v) 23.6 ÷100 (vi) 98.53 ÷ 100 4. Find: (i) 7.9 ÷ 1000 (ii) 26.3 ÷ 1000 (iii) 38.53 ÷ 1000 (iv) 128.9 ÷ 1000 (v) 0.5 ÷ 1000 2024-25 42 MATHEMATICS 5. Find: (i) 7 ÷ 3.5 (ii) 36 ÷ 0.2 (iii) 3.25 ÷ 0.5 (iv) 30.94 ÷ 0.7 (v) 0.5 ÷ 0.25 (vi) 7.75 ÷ 0.25 (vii) 76.5 ÷ 0.15 (viii) 37.8 ÷ 1.4 (ix) 2.73 ÷ 1.3 6. A vehicle covers a distance of 43.2 km in 2.4 litres of petrol. How much distance will it cover in one litre of petrol? W HAT HAVE WE DISCUSSED? 1. We have learnt how to multiply fractions. Two fractions are multiplied by multiplying their numerators and denominators seperately and writing the product as product of numerators 2 5 2 × 5 10. For example, × = =. product of denominators 3 7 3× 7 21 1 1 2. A fraction acts as an operator ‘of ’. For example, of 2 is × 2 = 1. 2 2 3. (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. 4. A reciprocal of a fraction is obtained by inverting it upside down. 5. 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 5 10 For example, 2 ÷ = 2 × = 5 3 3 (b) While dividing a fraction by a whole number we multiply the fraction by the reciprocal of the whole number. 2 2 1 2 For example, ÷7= × = 3 3 7 21 (c) While dividing one fraction by another fraction, we multuiply the first fraction by 2 5 2 7 14 the reciprocal of the other. So, ÷ = × =. 3 7 3 5 15 6. 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 2024-25  FRACTIONS AND DECIMALS 43 7. 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 8. 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. 2024-25