MATHEMATICS TEXTBOOK – Grade 5 Name: _________________________ Section: ________Roll No: _______ School: ________________________
Part 1
Contents Part 2 7 Integers 7.1 Introduction to Negative Numbers���������������������������������������������������������������������������� 1 9 Fraction Operations 9.1 Add and Subtract Fractions������������������������������������������������������������������������������������ 28 9.2 M ultiply and Divide Fractions���������������������������������������������������������������������������������� 34 12 Decimal Operations 12.1 A dd and Subtract Decimals����������������������������������������������������������������������������������� 71 12.2 Multiply Decimals���������������������������������������������������������������������������������������������������� 75
Integers7Chapter I Will Learn About • negative numbers. • comparing integers using a number line. • ordering of integers. • rules of addition and subtraction of integers. 7.1 Introduction to Negative Numbers I Think Pooja’s family visited Kuttanad, a region in Kerala. It is well known for rice farming that is carried out around 1 m to 3 m below sea level. She wanted to represent this numerically in her diary. How do you think Pooja can write it? I Recall We have already learnt: • Roman numerals • Indian and international systems of numeration • Representing and comparing numbers on the number line 1
As we move from left to right on the number line, the numbers increase. As we move from right to left on the number line, the numbers decrease. Let us revise this concept by answering the following. 1) Arrange the given numbers in ascending and descending orders. a) 1436, 7001, 5998, 3291 b) 56891, 80149, 32748, 82013 c) 31752, 37251, 35721, 31572 2) Fill in the blanks and complete the sequences given: a) 238, __________, 240, 241, 242, ______________, 244 b) 8997, 8998, ___________, 9000, ______________, 9002 I Remember and Understand We always start counting as 1, 2, 3, 4 and so on. Counting from 1 comes naturally to us. That is why the collection of numbers starting from 1 are called natural numbers. For example, 1, 2, 3, 6, 100, 4536 and so on. The natural numbers along with ‘0’ form the collection of numbers called the whole numbers. Therefore, the smallest whole number is ‘0’ whereas the smallest natural number is ‘1’. On a number line, as we move to the right by one unit, the value increases by 1. Similarly, moving to the left by one unit decreases the value by 1. When we move by one unit to the left of 0, we The numbers +1, +2, + 3 and so denote it as –1, read as negative 1. We write on are usually written as 1, 2, negative numbers with a minus sign to show that 3… These are called positive it is in the opposite direction from 0 to 1. Similarly, numbers. The numbers -1, -2, moving left by 2, 3, 4 and so on, units are denoted -3, …. are called negative as – 2, – 3, – 4 and so on. numbers. 0 is neither positive We extend the number line to the left of 0 and nor negative. write the negative numbers as shown. For each positive number to the right of 0, there is a corresponding negative number to the left of 0. Negative numbers Positive numbers 2
Positive numbers (1, 2, 3, ….), negative numbers (–1, –2, –3, ….) and 0 together form a set of numbers called integers. In other words, the set of whole numbers and negative numbers is called the set of integers. It is denoted by the letter Z. Hence, we write Z = {…, –3, –2, –1, 0, 1, 2, 3, …}. The dots before –3 and after 3 indicate that there are more numbers on either side. Every number to the left of a given number is less than the given number. For example, 5 < 6, 2 < 3, 0 < 1, –1 < 0, –3 < –2 and so on. So, all negative numbers are less than 0 and all positive numbers are greater than 0. In the same way, 6 > 5 whereas –6 < –5. So, the greater a positive integer, the smaller is its corresponding negative integer. Predecessor and successor of a given integer Every integer has an integer that comes before it. This integer is called the predecessor of the given integer. On a number line, the integer just before the given integer (that is, to its left) is called its predecessor. Similarly, every integer has an integer that comes after it. This integer is called the successor of the given integer. On a number line, the integer just after the given integer (that is, to its right) is called its successor. Every number has a predecessor obtained by subtracting 1 from it and a successor obtained by adding 1 to it. So, the smallest negative integer and the largest positive integer cannot be determined. Predecessor Predecessor of -3 of 2 Successor of Successor -3 of 2 Uses of integers The concept of opposites is indicated by the positive and negative numbers. So, we use integers to represent opposites such as the following. a) Moving forward by 5 steps: +5 or 5 Moving backward by 5 steps: –5 b) Profit of ` 10: ` 10 Integers 3
Loss of ` 5: – ` 5 c) Saving of ` 50: ` 50 Expenditure of ` 12: – ` 12 d) Height of 1200 m above sea level: 1200 m Depth of 708 m below sea level: – 708 m e) 10° C above 0° C: +10° C 7° C below 0° C: –7° C Example 1: Represent the following on a number line. a) –4 b) –2 c) 5 d) 3 Solution: Draw a number line with markings at equal intervals. Number them a) according to their distances from the starting point 0. Circle the given number. b) c) d ) Example 2: Write the following using integers. a) A deposit of ` 215 into a savings account. b) A withdrawal of ` 108 from a savings account. c) A mountain is 4326 m above sea level. d) An octopus in water is at a depth of 573 m below sea level. Solution: a) ` 215 b) – ` 108 c) 4326 m d) – 573 m Example 3: Write the predecessors and successors of the following integers. a) 65 b) 308 c) – 4937 d) –2086 4
Solution: Given Predecessor Successor integer (Just before the given (Just after the given a) b) 65 integer) integer) c) 308 64 66 d) -4937 307 309 -2086 -4938 -4936 -2087 -2085 ? Train My Brain Write the predecessor and successor of each of these integers. a) 549 b) - 356 c) -333 d) -286 Integers 5
CHAPTER 9 FOrpaecrtaitoinons9Chapter I Will Learn About • adding and subtracting unlike and mixed fractions. • multiplying fractions by whole numbers and fractions. • dividing whole numbers and fractions by fractions. 9.1 Add and Subtract Fractions I Think Pooja has a round cardboard with some of its portions coloured. She knows that the fractions that represent the coloured portions are unlike fractions. She wanted to find the coloured and uncoloured parts of the cardboard. How do you think Pooja can find that? I Recall We have learnt about the types of fractions. Let us recall them here. 1) A fraction whose numerator is greater than its denominator is called an improper fraction. 2) A fraction whose numerator is less than its denominator is called a proper fraction. 3) A fraction with a combination of a whole number and a proper fraction is called a mixed fraction. 28
I Remember and Understand Addition and Subtraction of Unlike Fractions Let us understand the addition and subtraction of Unlike fractions can be unlike fractions through some examples. added or subtracted by first writing their equivalent like 31 72 fractions and then adding or Example 1: Solve: a) 15 + 10 b) 13 + 39 subtracting the numerators. c) 22 + 7 100 10 31 Solution: a) 15 + 10 [L.C.M. of 15 and 10 is 30.] 3×2 1×3 = 15×2 + 10×3 = 6 + 3 30 30 6+3 9 3 = 30 = 30 = 10 [H.C.F. of 9 and 30 is 3.] 7 2 21 2 21+ 2 23 b) 13 + 39 = 39 + 39 = 39 = 39 [L.C.M. of 13 and 39 is 39.] 22 7 22 70 22 + 70 92 23 c) 100 + 10 = 100 + 100 = 100 =100 = 25 [The L.C.M. of 100 and 10 is 100, and the H.C.F. of 92 and 100 is 4.] Example 2: Solve: a) 8 - 4 b) 17 - 5 c) 14 -– 17 9 11 30 24 25 50 Solution: a) 8 -– 4 = 88 -– 36 [L.C.M. of 9 and 11 is 99.] 9 11 99 99 88- 36 = 52 =99 99 b) 17 - 5 = 68 - 25 [L.C.M. of 24 and 30 is 120.] 30 24 120 120 68 -25 43 = 120 = 120 Fraction Operations 29
c) 14 - 17 = 28 - 17 [L. C. M. of 25 and 50 is 50.] 25 50 50 50 28 - 17 11 = 50 = 50 Addition and Subtraction of Mixed, Improper and Proper Fractions The addition and subtraction of mixed fractions are similar to that of unlike fractions. Let us understand the same through some examples. Example 3: Add: 2 3 + 3 2 5 7 Solved Solve this Steps 23 + 32 12 1 + 15 1 5 7 43 Step 1: Convert all the mixed 2 3 = 2´5+ 3 = 13 ; fractions into improper fractions. 5 5 5 Step 2: Find the L.C.M. and add 32 = 3´7 + 2 = 23 the improper fractions. 7 7 7 23 + 32 = 13 + 23 5 7 5 7 [L.C.M. of 5 and 7 is 35.] Step 3: Find the H.C.F. of the 7´13 + 5´ 23 numerator and the denominator = 35 of the sum. Then reduce the improper fraction to its simplest 91+115 206 form. = 35 = 35 Step 4: Convert the improper The H.C.F. of 206 and 35 is fraction into a mixed fraction. 1. So, we cannot reduce the fraction any further. 206 = 5 31 35 35 Therefore, 23 + 32 5 7 = 5 31 . 35 30
Example 4: Subtract 2 3 from 3 2 57 Steps Solve Solve this 2 3 from 3 2 12 1 from 15 1 Step 1: Convert all the mixed fractions into improper 57 43 fractions. 32 = 3´7 + 2 = 23 ; 7 7 7 2 3 = 2´5 + 3 = 13 55 5 Step 2: Find the L.C.M. 32 – 23 = 23 – 13 and subtract the improper 7 5 7 5 fractions. [L.C.M. of 5 and 7 is 35] Step 3: Find the H.C.F. of the numerator and = 5´23 - 7´13 115 - 91 24 the denominator of the 35 = 35 = 35 difference. Then reduce the proper fraction to its simplest The H.C.F. of 24 and 35 is 1. So, form. we cannot reduce the fraction any further. Step 4: If the difference is an 24 is a proper fraction. So, we improper fraction, convert it 35 into a mixed fraction. cannot convert it into a mixed fraction. Therefore, 32 – 23 = 24 7 5 35 ? Train My Brain Solve the following: a) 13 + 1 b) 41 – 21 c) 25 – 2 1 4 6 4 8 9 3 Fraction Operations 31
9.2 Multiply and Divide Fractions I Think Pooja and each of her 15 friends had a bar of chocolate. Each of them ate 5 of the 12 chocolate. Pooja wants to know how much of the chocolate bar did they eat in all. How do you think Pooja can find this? I Recall Recall that when we find the fraction of a number, we multiply the number by the fraction. After multiplication, we simplify the product to its lowest terms. Similarly, we can multiply a fraction by another fraction too. 34
• Fraction in the simplest terms: A fraction is said to be in its simplest form if its numerator and denominator do not have a common factor other than 1. • Reducing or simplifying fractions: Writing a fraction in such a way that its numerator and denominator have no common factor other than 1 is called reducing or simplifying the fraction to its lowest terms. • Methods used to reduce a fraction: A fraction can be reduced to its lowest terms using: 1) division 2) H.C. F. I Remember and Understand Multiply fractions by whole numbers Let us now learn to find the fraction of a number. Suppose there are 20 shells in a bowl. Vani wants to use 1 of them for an art project. So, she divides the shells into 5 5 (the number in the denominator) equal groups and takes 1 group (the number in the numerator). This gives 5 groups with 4 shells in each group. Therefore, we can say that 1 of 20 is 4. 5 3 Vani’s sister Rani wants to use 10 of the 20 shells. So, she divides the shells into 10 (the number in the denominator) equal groups, and takes 3 groups (the number in the numerator). This gives 2 shells in each group. Hence, Rani takes 6 shells. Therefore, 3 10 of 20 is 6. We write 1 of 20 as 1 × 20 = 20 = 4. 5 5 5 Similarly, 3 of 20 = 3 × 20 = 6. 10 10 Example 10: Find the following: 21 a) 5 of a metre (in cm) b) 10 of a kilogram (in g) Solution: 2 of a metre = 2 ×1m= 2 × 100 cm = 2 × 100 cm = 200 cm a) 5 5 5 5 5 = 40 cm b) 110 of a kilogram = 1 × 1 kg 10 Fraction Operations 35
=110 × 1000 g = 1000 g = 100 g 10 Example 11: Find the following: b) 1 of a day (in hours) 2 4 a) 3 of an hour (in minutes) Solution: a) 2 of an hour = 2 ×1h= 2 × 60 min = 2 × 60 = 120 = 40 min 3 3 3 3 3 b) 1 of a day = 1 × 1 day = 1 × 24 h = 24 h=6h 4 4 4 4 Multiplying a fraction by 2-digit or 3-digit numbers is the same as finding the fraction of a number. Example 12: Find the following: a) 23 of 90 b) 15 of 128 45 32 Solution: a) 23 of 90 = 23 × 90 = 23´90 45 45 45 2070 = 45 = 46 Multiplying the numbers in the numerator and then dividing is tedious. It is especially so when the numbers are large. Therefore, we shall find if any of the numbers in the numerator and the denominator have a common factor. If yes, we take the H.C.F. of the numbers. We then divide the numbers to reduce the fraction to its lowest terms. 23 23 Hence, 45 of 90 = 45 × 90. Here, 45 and 90 have common factors, 3, 5, 9, 15 and 45. The H.C.F. of 45 and 90 is 45. So, divide both 45 and 90 by their H.C.F. 23 × 90 = 23 ´ 90 2 [Divide using the H.C.F. of the numbers.] Therefore, 45 451 = 23 × 2 = 46 b) 15 of 128 = 15 × 128 [H.C.F of 32 and 128 is 32.] 32 32 Divide 32 and 128 by 32, and simplify the multiplication. 15 4 32 × 128 = 15 × 4 = 60 1 36
Multiply fractions by fractions Multiplication of two fractions is simple. If a and c are two fractions where b and d are not equal to zero, b d then a × c = a × c . b d b × d Therefore, product of the fractions = Product of numerators . Product of denominators To multiply mixed numbers, we change them into improper fractions and then proceed. As we know, multiplying the numbers in the numerator and then dividing is tedious. It is especially so when the numbers are large. So, we shall check if any of the numbers in the numerator and the denominator have a common factor. We then reduce the fractions into their lowest terms and then multiply them. Consider an example to understand the concept. 23 15 Example 13: Solve: 45 × 46 Solution: To multiply the given fractions, follow these steps: Step 1: Check if the numerator and denominator have any common factors. Observing the given fractions, we see that, a) (23, 45) and (15, 46) do not have any common factors to be reduced. b) (23, 46) and (15, 45) have common factors. Step 2: Find the H.C.F. of the numerator and the denominator that have common factors. The H.C.F. of 23 and 46 is 23. The H.C.F. of 15 and 45 is 15. Step 3: Reduce the numerator and the denominator that have common factors using their H.C.F. 1 23 ´ 15 1 = 1´1 = 1 45 46 2 3´2 6 3 Therefore, 23 × 15 = 1 . 45 46 6 Fraction Operations 37
Example 14: Solve: a) 2 × 5 7 70 84 45 5 6 b) 35 × 63 c) 54 × 60 11 1´1 1´3 Solution: a) 2 × 5 = 1 × 1 = = 1 5 6 1 3 3 13 b) 1 2 1 2 1´2 =2 = 1 × 9 = 1´9 9 7 70 35 × 63 19 75 84 × 45 = 7 × 5 = 7´5 = 35 = 7 1 c) 54 60 6 5 6´5 30 6 = 16 65 Reciprocal of a fraction To find the reciprocal of a fraction, we A number or a fraction which when multiplied by interchange its numerator and denominator. a given number gives 1 as the product is called • The reciprocal of a number is a fraction. the reciprocal or the multiplicative inverse of 1 the given number. For example, the reciprocal of 20 is 20 . • The reciprocal of a unit fraction is a number. 1 For example, the reciprocal of 7 is 7. • The reciprocal of a proper fraction is an improper fraction. It can be left as it is or converted into a mixed fraction. For example, the reciprocal of 3 is 7 or 2 1 . 7 3 3 • The reciprocal of an improper fraction is a proper fraction. 95 For example, the reciprocal of 5 is 9 . • The reciprocal of a mixed fraction is a proper fraction. For example, the reciprocal of 2 3 is 8. 8 19 Note: 1) The reciprocal of 1 is 1. 2) The reciprocal of 0 does not exist as division by zero is not defined. 38
3) Numbers such as 4, 6, 9 and so on are converted into improper fractions by writing them as 4 , 6 , 9 before finding their reciprocals. 111 4) Fractions are reduced to their lowest terms (if possible) before finding their reciprocals. Let us find the reciprocals of some fractions. Example 15: Find the reciprocals of these fractions. a) 8 b) 4 c) 131 d) 4 17 19 5 Solution: To find the reciprocal of a fraction, we interchange its numerator and denominator. Therefore, the reciprocals of the given fractions are: 17 19 11 5 a) 8 b) 4 c) 3 d) 4 Example 16: Find the multiplicative inverses of these fractions. a) 5 b) 7 5 c) 0 d) 1 e) 33 1 9 3 Solution: To find the multiplicative inverse of a fraction, we interchange its numerator and denominator. The multiplicative inverses of the given fractions are: 19 3 a) 5 b) 68 c) no multiplicative inverse d) 1 e) 100 Note: 0 has no reciprocal or multiplicative inverse because we cannot multiply any number by it to get 1. Zero multiplied by any number is zero. So, 0 is the only number that does not have a multiplicative inverse. ? Train My Brain Solve the following: 14 94 2 14 c) 4 × 3 a) 54 7 b) 7 × 21 15 12 Fraction Operations 39
12Chapter ODpeecrimatailons I Will Learn About • addition and subtraction decimal fractions. • multiplication of decimal fractions with 10, 100 and 1000. • multiplication of decimal numbers by whole and decimal numbers. 12.1 Add and Subtract Decimals I Think Pooja went to an ice cream parlour to purchase some ice creams. She bought strawberry for ` 25.50, vanilla for ` 15.30 and chocolate for ` 32.20. She gave ` 100 to the shopkeeper. She wanted to calculate the total price before the shopkeeper gave the bill. Since the prices were in decimals, she was unable to calculate. Do you know how to find the total cost of the ice creams that Pooja bought? How much change would she get in return? I Recall Addition and subtraction of decimal numbers are similar to that of usual numbers. Let us recall the conversion of unlike decimals to like decimals. 71
Convert the given unlike decimals into like decimals. a) 4.32, 4.031, 4.1, 7.823 b) 0.7, 0.82, 4.513, 0.72 c) 1.82, 7.01, 5.321, 0.8 d) 7.32, 7.310, 7.8, 5.2 I Remember and Understand Addition and subtraction of decimal numbers with the thousandths place is similar to that of decimals with the hundredths place. Write the given decimal numbers such that the digits in their same places are exactly one below the other. Note: T he decimal points of the numbers must be exactly one Before adding below the other. or subtracting decimals, convert Let us see a few examples. the unlike decimals to like Example 1: a) Find the sum of 173.80 and 23.61. decimals. b) Subtract 216.73 from 563.72. Solution: a) b) 12 16 1 5 2/ 6/ 12 1 7 3 .8 0 5 6/ 3/ . 7/ 2/ + 2 3 .6 1 –21 6 . 7 3 1 9 7 .4 1 34 6 . 9 9 Example 2: Solve: a) 294.631 + 306.524 b) 11.904 – 6.207 Solution: a) 1 1 1 b) 11 8 9 14 1/ 1/ . 9/ 0/ 4/ 29 4 . 631 +30 6 . 524 – 6 . 20 7 60 1 . 155 5 . 69 7 ? Train My Brain Solve: a) 347.87 + 67.43 b) 16.53 – 10.73 c) 22.63 – 18.32 72
12.2 Multiply Decimals I Think Pooja bought six different types of toys for ` 236.95 each. She calculated the total cost and paid the amount to the shopkeeper. Do you know how to find the total cost of the toys? I Recall We have already learnt multiplication of numbers. Let us recall the same by answering the following. H TO Th H T O H TO 267 3218 576 ×14 ×34 ×12 I Remember and Understand Multiplication of decimals is similar to multiplication of numbers. Multiply decimals by 1-digit and 2-digit numbers Let us understand multiplication of decimals through a few examples. Decimal Operations 75
Example 10: Solve: a) 276.32 × 6 b) 25.146 × 23 Solution: a) 276.32 × 6 To multiply the given numbers, follow these steps: Step 1: Multiply the numbers as usual without considering the decimal point. T Th Th H T O Step 2: 4 3 11 2 7 6 32 ×6 1 6 5 7 92 Count the number of decimal places in the given number. The number of decimal places in 276.32 is two. Step 3: Count from the right, the number of digits in the product as the number of decimal places in the given number. Then place the decimal point. Therefore, 276.32 × 6 is 1657.92. b) 25.146 × 23 T Th Th H T O 11 1 11 2 5146 ×23 1 + 7 5438 5 0 2920 5 7 8. 3 5 8 Therefore, 25.146 × 23 is 578.358. Multiply decimals by 10,100 and 1000 Example 11: Solve: b) 3.4567 × 100 c) 3.4567 × 1000 a) 3.4567 × 10 Solution: To multiply a decimal number by 10, 100 and 1000, follow these steps: Step 1: Write the decimal number as it is. Step 2: Shift the decimal point to the right by as many digits as the number of zeros in the multiplier. 76
Therefore, a) 3.4567 × 10 = 34.567 (The decimal point is shifted to the right by one digit as the multiplier is 10 which has one zero.) b) 3.4567 × 100 = 345.67 (The decimal point is shifted to the right by two digits as the multiplier is 100 which has two zeros.) c) 3.4567 × 1000 = 3456.7 (The decimal point is shifted to the right by three digits as the multiplier is 1000 which has three zeros.) Multiply a decimal number by another decimal number Multiplication of a decimal number by another decimal number is similar to the multiplication of a decimal number by a number. Let us understand this through an example. Example 12: Solve: 7.12 × 3.7 Solution: Multiply the given numbers as When two decimal numbers are Step1: usual without considering the multiplied, decimal point. a) count the total number of digits 1 after decimal point in both the 7 12 numbers. Say it is ‘n’. × 37 11 b) multiply the two decimal 4 9 84 numbers as usual and place the +2 1 3 6 0 decimal point in the product after 26 3 44 ‘n’ digits from the right. Step 2: Count the number of decimal places in both the multiplicand and the multiplier and add them. The number of decimal places in 7.12 is two. The number of decimal places in 3.7 is one. Total number of decimal places = 2 + 1 = 3 Step 3: Count as many digits in the product from the right as the total number of decimal places. Then place the decimal point. Therefore, 7.12 × 3.7 is 26.344. Sometimes, the number of digits in the product is less than the sum of the number of decimal places in the multiplicand and the multiplier. In such cases, place zeros to the immediate right of the decimal point in the product such that the number of decimal places is equal to the sum of the decimal places in the multiplicand and the multiplier. Decimal Operations 77
? Train My Brain Solve: a) 56.7 × 10 b) 3.08 × 100 c) 8.50 × 1000 78
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