Fractions and Decimals

Welcome dear students! Today we are going to learn about Fractions and Decimals from Class 7 Maths.

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. What will be the area of the rectangle if its length and breadth are 7 1/2 cm and 3 1/2 cm respectively? You will say it will be 7 1/2 × 3 1/2 = 15/2 × 7/2 cm². The numbers 15/2 and 7/2 are fractions. To calculate the area of the given rectangle, we need to know how to multiply fractions. We shall learn that now.

Let us begin with section 2.1.1, Multiplication of a Fraction by a Whole Number. Observe the pictures at the left, Figure 2.1. Each shaded part is 1/4 part of a circle. How much will the two shaded parts represent together? They will represent 1/4 + 1/4 = 2/4. Combining the two shaded parts, we get Figure 2.2. What part of a circle does the shaded part in Figure 2.2 represent? It represents 2/4 part of a circle. The shaded portions in Figure 2.1 taken together are the same as the shaded portion in Figure 2.2, i.e., we get Figure 2.3, or 2 × 1/4 = 2/4. Can you now tell what this picture will represent? Figure 2.4 shows three shaded quarters, representing 3 × 1/4 = 3/4. And Figure 2.5 shows four shaded quarters, representing 4 × 1/4 = 4/4, which is 1 whole.

Let us now find 3 × 1/2. We have 3 × 1/2 = 1/2 + 1/2 + 1/2 = 3/2. We also have 1/2 + 1/2 + 1/2 = 1+1+1 over 2 = 3/2. So 3 × 1/2 = 3×1 over 2 = 3/2. Similarly, 2 × 5/3 = 2×5 over 3 = 10/3. Can you tell 2 × 3/7? It is 2×3 over 7 = 6/7. 4 × 2/5? It is 4×2 over 5 = 8/5. The fractions that we considered till now, i.e., 1/2, 2/3, 2/7 and 3/5 were proper fractions. For improper fractions also we have, 2 × 5/3 = 2×5 over 3 = 10/3. Try, 3 × 8/7 = 24/7. 4 × 5/7 = 20/7. 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.

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Let us solve Exercise 2.1 together. Question 1. Find: (a) 2 × 3/7. Multiply 2 by 3 to get 6. Keep denominator 7. Answer is 6/7. (b) 9 × 6/7. Multiply 9 by 6 to get 54. Keep denominator 7. Answer is 54/7. As a mixed fraction, 7 goes into 54 seven times with remainder 5, so 7 5/7. (c) 3 × 1/8. Multiply 3 by 1 to get 3. Keep denominator 8. Answer is 3/8. (d) 13 × 6/11. Multiply 13 by 6 to get 78. Keep denominator 11. Answer is 78/11. Convert to mixed: 11 goes into 78 seven times with remainder 1, so 7 1/11.

Question 2. Represent pictorially: 2 × 2/5 = 4/5. Imagine two identical rectangles, each divided into 5 equal vertical strips. Shade 2 strips in the first, and 2 strips in the second. Count all shaded strips across both. You will have 4 shaded strips out of 5 total parts in one whole, representing 4/5.

Question 3. Multiply and reduce to lowest form and convert into a mixed fraction: (i) 3 × 7/5 = 21/5, which is 4 1/5. (ii) 4 × 1/3 = 4/3, which is 1 1/3. (iii) 6 × 2/7 = 12/7, which is 1 5/7. (iv) 2 × 5/9 = 10/9, which is 1 1/9. (v) 4 × 2/3 = 8/3, which is 2 2/3. (vi) 5 × 6/2 = 30/2, which simplifies to 15. (vii) 4 × 11/7 = 44/7, which is 6 2/7. (viii) 4 × 20/5 = 80/5, which simplifies to 16. (ix) 1 × 13/3 = 13/3, which is 4 1/3. (x) 3 × 15/5 = 45/5, which simplifies to 9.

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Question 4. Shade: (i) 1/2 of the circles in box (a). Look at your textbook figure, count the total circles, divide by 2, and shade exactly half of them. (ii) 2/3 of the triangles in box (b). Count the total triangles, divide by 3, multiply by 2, and shade that many triangles. (iii) 3/5 of the squares in box (c). Count the total squares, divide by 5, multiply by 3, and shade that many squares.

Question 5. Find: (a) 1/2 of (i) 24 equals 1/2 × 24 = 12. (ii) 46 equals 1/2 × 46 = 23. (b) 2/3 of (i) 18 equals 2/3 × 18 = 12. (ii) 27 equals 2/3 × 27 = 18. (c) 3/4 of (i) 16 equals 3/4 × 16 = 12. (ii) 36 equals 3/4 × 36 = 27. (d) 4/5 of (i) 20 equals 4/5 × 20 = 16. (ii) 35 equals 4/5 × 35 = 28.

Question 6. Multiply and express as a mixed fraction: (a) 3 × 5 1/5. First convert 5 1/5 to improper fraction: 5 × 5 + 1 = 26, so 26/5. Multiply 3 × 26/5 = 78/5. Convert to mixed: 15 3/5. (b) 5 × 6 3/4. Convert 6 3/4 to 27/4. Multiply 5 × 27/4 = 135/4. Convert to mixed: 33 3/4. (c) 7 × 2 1/4. Convert to 9/4. Multiply 7 × 9/4 = 63/4. Convert to mixed: 15 3/4. (d) 4 × 6 1/3. Convert to 19/3. Multiply 4 × 19/3 = 76/3. Convert to mixed: 25 1/3. (e) 3 1/4 × 6. Convert 3 1/4 to 13/4. Multiply 13/4 × 6 = 78/4 = 39/2. Convert to mixed: 19 1/2. (f) 3 2/5 × 8. Convert to 17/5. Multiply 17/5 × 8 = 136/5. Convert to mixed: 27 1/5.

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Question 7. Find: (a) 1/2 of (i) 2 3/4. Convert to 11/4. 1/2 × 11/4 = 11/8, which is 1 3/8. (ii) 4 2/9. Convert to 38/9. 1/2 × 38/9 = 19/9, which is 2 1/9. (b) 5/8 of (i) 3 5/6. Convert to 23/6. 5/8 × 23/6 = 115/48, which is 2 19/48. (ii) 9 2/3. Convert to 29/3. 5/8 × 29/3 = 145/24, which is 6 1/24.

Question 8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed 2/5 of the water. Pratap consumed the remaining water. (i) How much water did Vidya drink? Calculate 2/5 of 5. 2/5 × 5 = 10/5 = 2 litres. (ii) What fraction of the total quantity of water did Pratap drink? Total is 5 litres. Vidya drank 2 litres. Remaining is 5 - 2 = 3 litres. Fraction is 3/5.

Now let us explore Fraction as an operator 'of'. Observe Figure 2.6. The two squares are exactly similar. Each shaded portion represents 1/2 of 1. So, both the shaded portions together will represent 1/2 of 2. Combine the 2 shaded 1/2 parts. It represents 1. So, we say 1/2 of 2 is 1. We can also get it as 1/2 × 2 = 1. Thus, 1/2 of 2 = 1/2 × 2 = 1. Look at Figure 2.7. Each shaded portion represents 1/2 of 1. The three shaded portions represent 1/2 of 3. Combine the 3 shaded parts. It represents 1 1/2 i.e., 3/2. So, 1/2 of 3 is 3/2. Also, 1/2 × 3 = 3/2. Thus, 1/2 of 3 = 1/2 × 3 = 3/2. So we see that 'of' represents multiplication. Farida has 20 marbles. Reshma has 1/5 of the number of marbles what Farida has. How many marbles Reshma has? As 'of' indicates multiplication, Reshma has 1/5 × 20 = 4 marbles. Similarly, 1/2 of 16 is 1/2 × 16 = 16/2 = 8. Can you tell, what is (i) 1/2 of 10? It is 5. (ii) 1/4 of 16? It is 4. (iii) 2/5 of 25? It is 10.

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TRY THESE. Find: (i) 3/5 × 2/7. Multiply numerators 3 × 2 = 6. Multiply denominators 5 × 7 = 35. Answer is 6/35. (ii) 4/9 × 6. Multiply 4 × 6 = 24. Denominator 9. Simplify 24/9 by dividing by 3. Answer is 8/3.

EXAMPLE 1. In a class of 40 students, 1/5 of the total number of students like to study English, 2/5 of the total number like to study Mathematics and the remaining students like to study Science. (i) How many students like to study English? Number of students who like English equals 1/5 of 40 = 1/5 × 40 = 8. (ii) How many students like to study Mathematics? Number equals 2/5 of 40 = 2/5 × 40 = 16. (iii) What fraction of the total number of students like to study Science? Number of students who like English and Mathematics equals 8 + 16 = 24. Number who like Science equals 40 - 24 = 16. Required fraction is 16/40, which simplifies to 2/5.

Now section 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. Each part will be 9/4 of the strip. She took one part and divided it in two equal parts by folding the part once. What will one of the pieces represent? It will represent 1/2 of 9/4 or 1/2 × 9/4. To find this, we first learn 1/2 × 1/3. How do we find 1/3 of a whole? We divide the whole in three equal parts. Each represents 1/3. Take one part and shade it as shown in Figure 2.8. How will you find 1/2 of this shaded part? Divide this one-third shaded part into two equal parts. Each represents 1/2 of 1/3 i.e., 1/2 × 1/3, shown in Figure 2.9. Take out 1 part and name it A. A represents 1/2 × 1/3. What fraction is A of the whole? Divide each of the remaining 1/3 parts also in two equal parts. Now there are six equal parts. A is one of these. So A is 1/6 of the whole. Thus, 1/2 × 1/3 = 1/6. The whole was divided in 6 = 2 × 3 parts and 1 = 1 × 1 part was taken out. Thus, 1/2 × 1/3 = 1×1 over 2×3. Similarly, 1/3 × 1/2 = 1/6. Hence 1/2 × 1/3 = 1/3 × 1/2 = 1/6. Find 1/3 × 1/4 and 1/4 × 1/3. Both equal 1/12. Find 1/2 × 1/5 and 1/5 × 1/2. Both equal 1/10. Fill in these boxes: (i) 1/2 × 1/7 = 1×1 over 2×7 = 1/14. (ii) 1/5 × 1/7 = 1/35. (iii) 1/7 × 1/2 = 1/14. (iv) 1/7 × 1/5 = 1/35.

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EXAMPLE 2. Sushant reads 1/3 part of a book in 1 hour. How much part of the book will he read in 2 1/5 hours? Part read in 1 hour = 1/3. Part read in 2 1/5 hours = 2 1/5 × 1/3. Convert 2 1/5 to 11/5. So 11/5 × 1/3 = 11×1 over 5×3 = 11/15.

Let us now find 1/2 × 5/3. We know 5/3 = 1/3 × 5. So 1/2 × 5/3 = 1/2 × 1/3 × 5 = 5/6. Also, 5/6 = 1×5 over 2×3. Thus, 1/2 × 5/3 = 5/6. This is shown in Figure 2.10. Each of the five equal shapes are parts of five similar circles. To obtain one shape, divide a circle in three equal parts, then divide each part in two equal parts. One part represents 1/2 × 1/3 = 1/6. Total of such parts is 5 × 1/6 = 5/6. Similarly, 3/5 × 1/7 = 3×1 over 5×7 = 3/35. We can find 2/3 × 7/5 as 2×7 over 3×5 = 14/15. So, we multiply two fractions as Product of Numerators over Product of Denominators.

Value of the Products. Product of two whole numbers is bigger. What happens with fractions? Consider product of two proper fractions: 2/3 × 4/5 = 8/15. 8/15 is less than 2/3 and less than 4/5. Product is less than each of the fractions. 1/2 × 5/7 = 5/14, less than both. 3/8 × 5/7 = 15/56, less than both. 2/9 × 4/7 = 8/63, less than both. You will find that when two proper fractions are multiplied, the product is less than each. Now multiply two improper fractions: 7/3 × 5/2 = 35/6. 35/6 is greater than 7/3 and greater than 5/2. Product is greater than each. 6/5 × 8/3 = 48/15, greater than both. 9/7 × 2/3 = 18/21 = 6/7. 3/8 × 7/4 = 21/32. Product of two improper fractions is greater than each. Let us multiply a proper and an improper fraction: 2/3 × 7/5 = 14/15. Here, 14/15 is less than 7/5 and greater than 2/3. The product is between them. Check for 6/5 × 2/8 = 12/40 = 3/10. 3/10 is less than 6/5 and greater than 2/8. 8/3 × 4/5 = 32/15. 32/15 is less than 8/3 and greater than 4/5.

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TRY THESE. Find: 1/3 × 4/5 = 4/15. 2/3 × 1/5 = 2/15. Find: 8/3 × 4/7 = 32/21 = 1 11/21. 3/4 × 2/3 = 6/12 = 1/2.

Let us solve Exercise 2.2. Question 1. Find: (i) 1/4 of (a) 1/4 = 1/4 × 1/4 = 1/16. (b) 3/5 = 1/4 × 3/5 = 3/20. (c) 4/3 = 1/4 × 4/3 = 4/12 = 1/3. (ii) 1/7 of (a) 2/9 = 1/7 × 2/9 = 2/63. (b) 6/5 = 1/7 × 6/5 = 6/35. (c) 3/10 = 1/7 × 3/10 = 3/70.

Question 2. Multiply and reduce to lowest form: (i) 2/3 × 2 2/3. Convert 2 2/3 to 8/3. 2/3 × 8/3 = 16/9 = 1 7/9. (ii) 2/7 × 7/9 = 14/63 = 2/9. (iii) 3/8 × 6/4 = 18/32 = 9/16. (iv) 9/5 × 3/5 = 27/25 = 1 2/25. (v) 1/3 × 15/8 = 15/24 = 5/8. (vi) 11/2 × 3/10 = 33/20 = 1 13/20. (vii) 4/5 × 12/7 = 48/35 = 1 13/35.

Question 3. Multiply fractions: (i) 2/5 × 5 1/4. Convert 5 1/4 to 21/4. 2/5 × 21/4 = 42/20 = 21/10 = 2 1/10. (ii) 6 2/5 × 7/9. Convert 6 2/5 to 32/5. 32/5 × 7/9 = 224/45 = 4 44/45. (iii) 3/2 × 5 1/3. Convert 5 1/3 to 16/3. 3/2 × 16/3 = 48/6 = 8. (iv) 5/6 × 2 3/7. Convert 2 3/7 to 17/7. 5/6 × 17/7 = 85/42 = 2 1/42. (v) 3 2/5 × 4/7. Convert 3 2/5 to 17/5. 17/5 × 4/7 = 68/35 = 1 33/35. (vi) 2 3/5 × 3. Convert 2 3/5 to 13/5. 13/5 × 3 = 39/5 = 7 4/5. (vii) 3 4/7 × 3/5. Convert 3 4/7 to 25/7. 25/7 × 3/5 = 75/35 = 15/7 = 2 1/7.

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Question 4. Which is greater: (i) 2/7 of 3/4 or 3/5 of 5/8. 2/7 × 3/4 = 6/28 = 3/14. 3/5 × 5/8 = 15/40 = 3/8. Compare 3/14 and 3/8. Cross multiply: 3 × 8 = 24, 3 × 14 = 42. 24 is greater, so 3/8 is greater. (ii) 1/2 of 6/7 or 2/3 of 3/7. 1/2 × 6/7 = 6/14 = 3/7. 2/3 × 3/7 = 6/21 = 2/7. 3/7 is greater.

Question 5. Saili plants 4 saplings, in a row. Distance between adjacent is 3/4 m. Distance between first and last: There are 3 intervals. 3 × 3/4 = 9/4 = 2 1/4 m.

Question 6. Lipika reads 1 3/4 hours everyday. Reads in 6 days. Total hours: 1 3/4 = 7/4. 7/4 × 6 = 42/4 = 21/2 = 10 1/2 hours.

Question 7. Car runs 16 km using 1 litre. Distance in 2 3/4 litres: 2 3/4 = 11/4. 16 × 11/4 = 176/4 = 44 km.

Question 8. (a) (i) Provide number in box such that 2/3 × box/10 = 10/30. The box must be 5. (ii) Simplest form of 10/30 is 1/3. (b) (i) Provide number in box such that 3/5 × box/15 = 24/75. The box must be 8. (ii) Simplest form of 24/75 is 8/25.

Now section 2.2, Division of Fractions. John has a paper strip of length 6 cm. He cuts into 2 cm strips. He gets 6 ÷ 2 = 3 strips. John cuts another 6 cm strip into 3/2 cm strips. He gets 6 ÷ 3/2 strips. A strip of 15/2 cm cut into 3/2 cm gives 15/2 ÷ 3/2 pieces. We need to divide whole by fraction or fraction by fraction.

Section 2.2.1, Division of Whole Number by a Fraction. Find 1 ÷ 1/2. Divide whole into halves. Number of halves is 1 ÷ 1/2. Figure 2.11 shows two half parts. So 1 ÷ 1/2 = 2. Also, 1 × 2/1 = 2. Thus, 1 ÷ 1/2 = 1 × 2/1. Similarly, 3 ÷ 1/4 = number of 1/4 parts in 3 wholes. Figure 2.12 shows 12 parts. So 12. Also, 3 × 4/1 = 12. Thus, 3 ÷ 1/4 = 3 × 4/1 = 12. Find similarly 3 ÷ 1/2 = 6.

Reciprocal of a fraction. The number 2/1 is obtained by interchanging numerator and denominator of 1/2. Similarly, 3/1 by inverting 1/3. Observe products: 1/7 × 7 = 1. 5/4 × 4/5 = 1. 9 × 1/9 = 1. 2/7 × 7/2 = 1. 2/3 × 3/2 = 6/6 = 1. 5/9 × 9/5 = 1. Non-zero numbers whose product with each other is 1 are called reciprocals. Reciprocal of 1/9 is 9. Reciprocal of 2/7 is 7/2. Reciprocal of 2/3 is 3/2. THINK, DISCUSS AND WRITE. (i) Will reciprocal of proper fraction be proper? No, it will be improper. (ii) Will reciprocal of improper be improper? No, it will be proper.

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Thus, 1 ÷ 1/2 = 1 × reciprocal of 1/2. 3 ÷ 1/4 = 3 × reciprocal of 1/4. 3 ÷ 1/2 = 3 × 2/1 = 6. 2 ÷ 3/4 = 2 × 4/3 = 8/3. 5 ÷ 2/9 = 5 × 9/2 = 45/2. Thus, to divide whole number by fraction, multiply by reciprocal.

TRY THESE. Find: (i) 7 ÷ 2/5 = 7 × 5/2 = 35/2 = 17 1/2. (ii) 6 ÷ 4/7 = 6 × 7/4 = 42/4 = 21/2 = 10 1/2. (iii) 2 ÷ 8/9 = 2 × 9/8 = 18/8 = 9/4 = 2 1/4. While dividing whole by mixed fraction: 4 ÷ 2 2/5. Convert 2 2/5 to 12/5. 4 ÷ 12/5 = 4 × 5/12 = 20/12 = 5/3 = 1 2/3. 5 ÷ 3 1/3. Convert to 10/3. 5 ÷ 10/3 = 5 × 3/10 = 15/10 = 3/2 = 1 1/2.

Section 2.2.2, Division of a Fraction by a Whole Number. 3/4 ÷ 3 = 3/4 × 1/3 = 3/12 = 1/4. So, 2/3 ÷ 7 = 2/3 × 1/7 = 2/21. What is 5/7 ÷ 6? 5/7 × 1/6 = 5/42. 2/7 ÷ 8 = 2/7 × 1/8 = 2/56 = 1/28. While dividing mixed fractions by whole numbers: 2 2/3 ÷ 5. Convert to 8/3. 8/3 ÷ 5 = 8/3 × 1/5 = 8/15. 2 4/5 ÷ 3. Convert to 14/5. 14/5 ÷ 3 = 14/5 × 1/3 = 14/15. 3 2/5 ÷ 2. Convert to 17/5. 17/5 ÷ 2 = 17/5 × 1/2 = 17/10 = 1 7/10.

Section 2.2.3, Division of a Fraction by Another Fraction. 1/3 ÷ 6/5 = 1/3 × reciprocal of 6/5 = 1/3 × 5/6 = 5/18. Similarly, 8/5 ÷ 2/3 = 8/5 × reciprocal of 2/3 = 8/5 × 3/2 = 24/10 = 12/5 = 2 2/5. 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3.

TRY THESE. Find: (i) 3/5 ÷ 1/2 = 3/5 × 2/1 = 6/5 = 1 1/5. (ii) 1/2 ÷ 3/5 = 1/2 × 5/3 = 5/6. (iii) 2 1/2 ÷ 3/5. Convert 2 1/2 to 5/2. 5/2 × 5/3 = 25/6 = 4 1/6. (iv) 1 9/5 ÷ 6/2. Convert 1 9/5 to 14/5. 6/2 is 3. 14/5 ÷ 3 = 14/15. TRY THESE. Find: (i) 6 ÷ 1 5/3. Convert 1 5/3 to 8/3. 6 ÷ 8/3 = 6 × 3/8 = 18/8 = 9/4 = 2 1/4. (ii) 7 ÷ 2 4/7. Convert 2 4/7 to 18/7. 7 ÷ 18/7 = 7 × 7/18 = 49/18 = 2 13/18.

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Now Exercise 2.3. Question 1. Find: (i) 3/4 ÷ 12 = 3/4 × 1/12 = 3/48 = 1/16. (ii) 5/6 ÷ 14 = 5/6 × 1/14 = 5/84. (iii) 7/8 ÷ 3 = 7/8 × 1/3 = 7/24. (iv) 8/3 ÷ 4 = 8/3 × 1/4 = 8/12 = 2/3. (v) 1/2 ÷ 3 2/3. Convert 3 2/3 to 11/3. 1/2 × 3/11 = 3/22.

Question 2. Find the reciprocal and classify. (i) 3/7 reciprocal is 7/3, improper. (ii) 5/8 reciprocal is 8/5, improper. (iii) 9/7 reciprocal is 7/9, proper. (iv) 6/5 reciprocal is 5/6, proper. (v) 12/7 reciprocal is 7/12, proper. (vi) 1/8 reciprocal is 8, whole number. (vii) 1/11 reciprocal is 11, whole number.

Question 3. Find: (i) 7/2 ÷ 3 = 7/2 × 1/3 = 7/6. (ii) 4/5 ÷ 9 = 4/5 × 1/9 = 4/45. (iii) 6/7 ÷ 13 = 6/7 × 1/13 = 6/91. (iv) 4 1/3 ÷ 3. Convert to 13/3. 13/3 × 1/3 = 13/9. (v) 3 1/2 ÷ 4. Convert to 7/2. 7/2 × 1/4 = 7/8. (vi) 4 3/7 ÷ 7. Convert to 31/7. 31/7 × 1/7 = 31/49.

Question 4. Find: (i) 2 1/5 ÷ 2. Convert to 11/5. 11/5 × 1/2 = 11/10. (ii) 4 2/9 ÷ 3. Convert to 38/9. 38/9 × 1/3 = 38/27. (iii) 3 8/7 ÷ 7. Convert to 29/7. 29/7 × 1/7 = 29/49. (iv) 1 3/2 ÷ 3/5. Convert to 5/2. 5/2 × 5/3 = 25/6. (v) 3 1/2 ÷ 8/3. Convert to 7/2. 7/2 × 3/8 = 21/16. (vi) 2 1/1 ÷ 5/2. Convert to 3. 3 × 2/5 = 6/5. (vii) 1 2/3 ÷ 1 5/3. Convert to 5/3 ÷ 8/3. 5/3 × 3/8 = 5/8. (viii) 1 1/2 ÷ 1 5/5. Convert to 3/2 ÷ 2. 3/2 × 1/2 = 3/4.

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Now we move to section 2.3 MULTIPLICATION OF DECIMAL NUMBERS. Reshma purchased 1.5 kg vegetable at the rate of ₹ 8.50 per kg. To find the cost, we calculate 8.50 × 1.50. Both 8.5 and 1.5 are decimal numbers. Let us learn how to multiply them. First we find 0.1 × 0.1. Since 0.1 = 1/10, we have 1/10 × 1/10 = 1/100 = 0.01. Let us describe the pictorial representation in Figure 2.13. It shows a square divided into 10 equal vertical strips. One strip is shaded, representing 1/10. Figure 2.14 takes that shaded strip and divides it into 10 equal horizontal parts. One small part is dotted. This dotted square represents one part out of 100, or 0.01. Hence, 0.1 × 0.1 = 0.01. Notice 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, which is 1 + 1.

Let us now find 0.2 × 0.3. We have 0.2 × 0.3 = 2/10 × 3/10. Figure 2.15 shows a square divided into 10 columns. Three columns are shaded. Each shaded column is divided into 10 rows, and two rows in each are dotted. There are 6 dotted squares out of 100, representing 0.06. Thus, 0.2 × 0.3 = 0.06. Observe that 2 × 3 = 6 and the product has two decimal places. To multiply decimals, first multiply them as whole numbers ignoring the decimal point. Then count the total digits to the right of the decimal in both factors. Place the decimal point in the product by counting that many places from the right. Let us find 1.2 × 2.5. Multiply 12 and 25 to get 300. Both have 1 decimal place, so count 1 + 1 = 2 digits from the right in 300. We get 3.00 or 3. Try these: Find 1.5 × 1.6. Multiply 15 and 16 to get 240. Two decimal places give 2.40. Find 2.4 × 4.2. Multiply 24 and 42 to get 1008. Two decimal places give 10.08. While multiplying 2.5 and 1.25, first multiply 25 and 125 to get 3125. Count 1 + 2 = 3 decimal places. Thus, 2.5 × 1.25 = 3.125. Find 2.7 × 1.35. Multiply 27 and 135 to get 3645. Three decimal places give 3.645.

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Try these: Find 2.7 × 4. Multiply 27 and 4 to get 108. One decimal place gives 10.8. Find 1.8 × 1.2. Multiply 18 and 12 to get 216. Two decimal places give 2.16. Find 2.3 × 4.35. Multiply 23 and 435 to get 10005. Three decimal places give 10.005. Arrange the products in descending order: 10.8, 10.005, 2.16.

Example 3: The side of an equilateral triangle is 3.5 cm. Perimeter = 3 × 3.5 cm = 10.5 cm. Example 4: Length of a rectangle is 7.1 cm and breadth is 2.5 cm. Area = 7.1 × 2.5 cm² = 17.75 cm².

Section 2.3.1 covers Multiplication of Decimal Numbers by 10, 100 and 1000. When a decimal is multiplied by 10, 100, or 1000, the digits stay the same, but the decimal point shifts to the right by as many places as there are zeros over one. For example, 1.76 × 10 = 17.6, × 100 = 176.0, × 1000 = 1760.0. 2.35 × 10 = 23.5, × 100 = 235, × 1000 = 2350. 12.356 × 10 = 123.56, × 100 = 1235.6, × 1000 = 12356. 0.5 × 10 = 5, × 100 = 50, × 1000 = 500. Try these: 0.3 × 10 = 3. 1.2 × 100 = 120. 56.3 × 1000 = 56300. Now, 2.97 × 10 = 29.7. 2.97 × 100 = 297. 2.97 × 1000 = 2970. To help Reshma, 8.50 × 1.5 = 12.75.

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Exercise 2.4. Question one: 0.2 × 6 = 1.2. 8 × 4.6 = 36.8. 2.71 × 5 = 13.55. 20.1 × 4 = 80.4. 0.05 × 7 = 0.35. 211.02 × 4 = 844.08. 2 × 0.86 = 1.72. Question two: Area = 5.7 × 3 = 17.1 cm². Question three: 1.3 × 10 = 13. 36.8 × 10 = 368. 153.7 × 10 = 1537. 168.07 × 10 = 1680.7. 31.1 × 100 = 3110. 156.1 × 100 = 15610. 3.62 × 100 = 362. 43.07 × 100 = 4307. 0.5 × 10 = 5. 0.08 × 10 = 0.8. 0.9 × 100 = 90. 0.03 × 1000 = 30. Question four: 55.3 × 10 = 553 km. Question five: 2.5 × 0.3 = 0.75. 0.1 × 51.7 = 5.17. 0.2 × 316.8 = 63.36. 1.3 × 3.1 = 4.03. 0.5 × 0.05 = 0.025. 11.2 × 0.15 = 1.68. 1.07 × 0.02 = 0.0214. 10.05 × 1.05 = 10.5525. 101.01 × 0.01 = 1.0101. 100.01 × 1.1 = 110.011.

Now section 2.4 DIVISION OF DECIMAL NUMBERS. Savita needs 1.9 cm pieces from a 9.5 cm strip. She calculates 9.5 ÷ 1.9. Section 2.4.1 covers Division by 10, 100 and 1000. 31.5 ÷ 10 = 3.15. ÷ 100 = 0.315. ÷ 1000 = 0.0315. The decimal point shifts left by as many places as there are zeros over one. 231.5 ÷ 10 = 23.15. ÷ 100 = 2.315. ÷ 1000 = 0.2315. 1.5 ÷ 10 = 0.15. ÷ 100 = 0.015. ÷ 1000 = 0.0015. 29.36 ÷ 10 = 2.936. ÷ 100 = 0.2936. ÷ 1000 = 0.02936. Try these: 235.4 ÷ 10 = 23.54. ÷ 100 = 2.354. ÷ 1000 = 0.2354.

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Section 2.4.2 covers Division of a Decimal Number by a Whole Number. 6.4 ÷ 2 = 3.2. 19.5 ÷ 5 = 3.9. Try these: 35.7 ÷ 3 = 11.9. 25.5 ÷ 3 = 8.5. Try these: 43.15 ÷ 5 = 8.63. 82.44 ÷ 6 = 13.74. 12.96 ÷ 4 = 3.24. Example 5: Average of 4.2, 3.8 and 7.6 is (4.2 + 3.8 + 7.6) ÷ 3 = 15.6 ÷ 3 = 5.2.

Section 2.4.3 covers Division of a Decimal Number by another Decimal Number. 25.5 ÷ 0.5 = 51. 22.5 ÷ 1.5 = 15. Try these: 20.3 ÷ 0.7 = 29. 15.2 ÷ 0.8 = 19. Try these: 3.96 ÷ 0.4 = 9.9. 2.31 ÷ 0.3 = 7.7. Try these: 15.5 ÷ 5 = 3.1. 126.35 ÷ 7 = 18.05. 33.725 ÷ 0.25 = 134.9. 27 ÷ 0.03 = 900. Example 6: Number of sides = 12.5 ÷ 2.5 = 5. Example 7: Average distance = 89.1 ÷ 2.2 = 40.5 km.

Exercise 2.5. Question one: 0.4 ÷ 2 = 0.2. 0.35 ÷ 5 = 0.07. 2.48 ÷ 4 = 0.62. 65.4 ÷ 6 = 10.9. 651.2 ÷ 4 = 162.8. 14.49 ÷ 7 = 2.07. 3.96 ÷ 4 = 0.99. 0.80 ÷ 5 = 0.16. Question two: 4.8 ÷ 10 = 0.48. 52.5 ÷ 10 = 5.25. 0.7 ÷ 10 = 0.07. 33.1 ÷ 10 = 3.31. 272.23 ÷ 10 = 27.223. 0.56 ÷ 10 = 0.056. 3.97 ÷ 10 = 0.397. Question three: 2.7 ÷ 100 = 0.027. 0.3 ÷ 100 = 0.003. 0.78 ÷ 100 = 0.0078. 432.6 ÷ 100 = 4.326. 23.6 ÷ 100 = 0.236. 98.53 ÷ 100 = 0.9853. Question four: 7.9 ÷ 1000 = 0.0079. 26.3 ÷ 1000 = 0.0263. 38.53 ÷ 1000 = 0.03853. 128.9 ÷ 1000 = 0.1289. 0.5 ÷ 1000 = 0.0005.

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Question five: 7 ÷ 3.5 = 2. 36 ÷ 0.2 = 180. 3.25 ÷ 0.5 = 6.5. 30.94 ÷ 0.7 = 44.2. 0.5 ÷ 0.25 = 2. 7.75 ÷ 0.25 = 31. 76.5 ÷ 0.15 = 510. 37.8 ÷ 1.4 = 27. 2.73 ÷ 1.3 = 2.1. Question six: Distance per litre = 43.2 ÷ 2.4 = 18 km.

Let us review what we have discussed. We multiply fractions by multiplying numerators and denominators separately. A fraction acts as an operator 'of'. The product of two proper fractions is less than each. The product of a proper and improper fraction lies between them. The product of two improper fractions is greater than both. Reciprocals are found by inverting fractions. We divide fractions by multiplying by the reciprocal. For decimals, we multiply as whole numbers, count decimal places, and place the point. Multiplying by 10, 100, 1000 shifts the decimal right. Dividing by them shifts it left. Dividing decimals involves shifting the decimal to make the divisor whole, then dividing normally.

Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]