Maths Workbook_7_P_1.pdf 1 18-10-2019 17:36:07 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________

TABLE OF CONTENTS 1 INTEGERS 1 1.1 INTRODUCTION 1 1.2 OPERATIONS OF INTEGERS – ADDITION 4 1.3 OPERATIONS OF INTEGERS – SUBTRACTION 7 1.4 MULTIPLICATION OF INTEGERS 9 1.5 DIVISION OF INTEGERS 12 1.6 PROPERTIES OF INTEGERS 14 1.7 PRACTICAL PROBLEMS USING NEGATIVE NUMBERS 19 2 FRACTIONS, DECIMALS AND RATIONAL NUMBERS 21 2.1 PROPER AND IMPROPER FRACTIONS 21 2.2 MULTIPLICATION OF FRACTIONS 27 2.3 MULTIPLICATION OF A FRACTION BY A FRACTION 31 2.4 DIVISION OF FRACTIONS 34 2.5 DECIMAL NUMBERS OR FRACTIONAL DECIMALS 39 2.6 MULTIPLICATION OF DECIMAL NUMBERS 42 2.7 INTRODUCTION TO RATIONAL NUMBERS – POSITIVE FRACTIONAL NUMBERS 46 3 SIMPLE EQUATIONS 49 3.1 INTRODUCTION 49 3.2 EQUATIONS – SOLVING THE EQUATION 51 3.3 USAGE OF ALGEBRAIC EQUATIONS IN SOLVING DAY TO DAY PROBLEMS 53 4 LINES AND ANGLES 55 4.1 INTRODUCTION 55 4.2 COMPLEMENTARY ANGLES 61 4.3 SUPPLEMENTARY ANGLES 63 4.4 ADJACENT ANGLES 65

4.5 LINEAR PAIR 68 4.6 VERTICALLY OPPOSITE ANGLES 73 4.7 TRANSVERSAL AND PARALLEL LINES 78 5 TRIANGLE AND ITS PROPERTIES 85 5.1 CLASSIFICATION OF TRIANGLES 85 5.2 ALTITUDES OF TRIANGLES 90 5.3 PROPERTIES OF TRIANGLES 93 5.4 EXTERIOR ANGLE OF A TRIANGLE 95 6 RATIO – APPLICATIONS 98 6.1 RATIO 98 6.2 PROPORTION 101 6.3 UNITARY METHOD 103 6.4 PERCENTAGES 106 6.5 PROFIT AND LOSS 110 6.6 SIMPLE INTEREST 114 7 DATA HANDLING 117 7.1 ORGANISING DATA 117 7.2 MODE 120 7.3 MEDIAN 123 7.4 PRESENTATION OF DATA 125 PROJECT BASED QUESTIONS 132 ADDITIONAL AS BASED PRACTICE QUESTIONS 135

CHAPTER 1 INTEGERS EXERCISE 1.1 INTRODUCTION 1.1.1 Key Concepts i. Integers: • All counting numbers and their negatives including zero are known as integers. • The set of integers can be represented by Z or I = . . . . . . − 4, −3, −2, −1, 0, 1, 2, 3, 4.... ii. Positive Integers: The set I+ = 1, 2, 3, 4 . . . is the set of all positive integers. Clearly, positive integers and natural numbers are the same. iii. Negative Integers: The set I− = {−1, −2, −3....} is the set of all negative integers. ‘0’ is neither positive nor negative. iv. Non-negative Integers: The set 0, 1, 2, 3 . . . is the set of all non-negative integers. 1.1.2 Additional Questions Objective Questions . 1. [AS1] The value of the integer P represented on the number line given is (A) 5 (B) –5 (C) 6 (D) –6 2. [AS1] The greatest among the following integers −7, 9, 0, −13 is . (A) −7 (B) 0 (C) −13 (D) 9 EXERCISE 1.1. INTRODUCTION 1

3. [AS1] One of the integers between − 3 and 4 is . (A) − 4 (B) 7 (C) 1 (D)− 4 4. [AS1] The smallest among the following integers is 6, −2, 3, −8 is . (A) − 8 (B) − 2 (C) 3 (D) 6 5. [AS1] The integers 8, − 6, 0, 3, − 7 in ascending order are . (A) 8, 3, 0, −6, −7 (B) −7, −6, 0, 3, 8 (C)3, 8, 0, −7, −6 (D)−6, −7, 0, 3, 8 Very Short Answer Type Questions 6. [AS1] Identify the integers in between the integers in Column A and match them with the integers in Column B. Column A Column B i. −5 and − 7 a. −14, −6, 0, 2 ii. 1 and 3 b. 0 iii. − 1 and 1 c. − 6 iv. 20 and − 15 d. − 18, − 14, − 6, 0 and 2 v. −100 and 20 e. 2 7 [AS2] Answer the following questions in one sentence. (i) Some integers are marked on the number line. a) Which is the smallest? b) Which is the biggest? EXERCISE 1.1. INTRODUCTION 2

(ii) To which side of the number 0 does the integer –6 lie on the number line? (iii) To which side of the number 0 does the integer 4 lie on the number line? (iv) To which side of the number 0 does the integer -5 lie on the number line? (v) To which side of the number 0 does the integer 3 lie on the number line? 8 [AS2] Fill in the blanks. than the integers to the left of zero. (i) Integers to the right of zero are (ii) If an integer a > b, then a is on the side of b. (iii) Zero is than every positive integer. (iv) Two integers which are on the left o f the integer − on the number line are . (v) (− 9) lies to the of (− 12) on the number line. Short Answer Type Questions 9(i) [AS5] Find the integers which are to the immediate right of the integers marked on the number line. (ii) [AS5] Find the integers which lie to the immediate left of the integers marked on the number line. Long Answer Type Questions 10 [AS1] Write the integers between the pairs of integers given. Also choose the biggest and the smallest integers from them. (iv) – 6, – 9 (v) – 5, – 3 (i) –7, 2 (ii) – 6, 4 (iii) – 3, 2 11 [AS5] Fill in the missing integers on the number line given. EXERCISE 1.1. INTRODUCTION 3

EXERCISE 1.2 OPERATIONS OF INTEGERS – ADDITION 1.2.1 Key Concepts i. On a number line when a positive integer is to be added, you move to the right side of the number; and if a negative integer is to be added you move to the left side of the number. e.g: 7 + (−8) = −1 1.2.2 Additional Questions Objective Questions . 1. [AS1] 7 + (8) + (3) = (A) 18 (B) 12 (D)− 12 (C) −18 (B) –10 2. [AS1] (− 3) + 5 + (− 2) = . (D) 4 (A) 10 (C) 0 (B) − 15 (D) 9 3. [AS1] 5 + (−3) + 7 = . (A) 15 (C)− 9 EXERCISE 1.2. OPERATIONS OF INTEGERS –ADDITION 4

4. [AS1] Representing the addition given on the number line as an equation gives . (A) 3 + 5 = 8 (B) −3 + 11 = 8 (C)−3 + 6 + 5 = 8 (D) None of these 5. [AS1] 22 + (−13) + 25 + (−19) = . (A) 15 (B) −15 (C) 79 (D)− 41 Very Short Answer Type Questions [] 6 [AS3] State true or false. [] (i) –223 > 222 (ii) (–731) + 221 = 510 [AS3] Answer the following questions in one sentence. (iii) When you add a positive integer, in which direction do you move on the number line? (iv) When you add a negative integer, in which direction do you move on the number line? (v) The initial temperature of a freezer was −15◦C. If it was later increased by 3◦ C, what is the resultant temperature of the freezer? Short Answer Type Questions 7(i) [AS1] Solve the following: a) 6 + (– 4) b) 4 + (– 2) EXERCISE 1.2. OPERATIONS OF INTEGERS –ADDITION 5

(ii) [AS1] Solve the following: a) 15 + (– 9) b) 40 + (– 10) c) 9 + (– 6) 8(i) [AS5] Represent the following additions on a number line. a) 5 + 4 b) 6 + (– 3) (ii) [AS5] Simplify using a number line: 12 + (−3) + 7 EXERCISE 1.2. OPERATIONS OF INTEGERS –ADDITION 6

EXERCISE 1.3 OPERATIONS OF INTEGERS – SUBTRACTION 1.3.1 Key Concepts i. On the number line if you subtract a positive integer you move to the left side and if you subtract a negative integer you move to the right side. e.g: (−7) − (−4) = −3 1.3.2 Additional Questions . Objective Questions (B) −5 1. [AS1] 20 − (13) − (−2) = (D) −9 (A) 5 (C) 9 2. [AS1] (−19) − (−11) − (13) = . (A) −21 (C) 17 (B) 21 (D) −17 3. [AS1] −12 expressed as the sum of two integers is . (A) −12 = 10 − 22 (B) −12 = (− 5) + (− 7) (C)−12 = −3 − 9 (D)−12 = 5 + (−7) EXERCISE 1.3. OPERATIONS OF INTEGERS –SUBTRACTION 7

4. [AS1] 12 expressed as the difference of two negative integers is . (A) 12 = 9 + 3 (B) 12 = 21 –9 (C) 12 = 9 –( –3) (D) 12 = –9 –( –21) 5. [AS1] 21 + (−13) − (−9) + 14 = . (A) 31 (B) −31 (C) 57 (D) −57 Short Answer Type Questions 6(i) [AS3] Express –5 as the difference of a negative integer and a whole number. (ii) [AS3] Express – 5 as the difference of two whole numbers. 7(i) [AS5] Represent the following subtraction on the number line: 8 – (–7) (ii) [AS5] Represent the following subtraction on the number line: 16 – 12 Long Answer Type Questions (iii) 25 – 22 (iv) 60 + 30 (v) 120 – 60 8 [AS1] Solve the following: (i) 17 – (– 12) (ii) 13 – (– 6) EXERCISE 1.3. OPERATIONS OF INTEGERS –SUBTRACTION 8

EXERCISE 1.4 MULTIPLICATION OF INTEGERS 1.4.1 Key Concepts i. The product of any two positive integers or any two negative integers is always a positive integer. ii. The product of a positive integer and a negative integer is always a negative integer i.e., two integers with opposite signs always give a negative product. iii. The product of an even number of negative integers is always a positive integer. iv. The product of an odd number of negative integers is always a negative integer. e. g: 5 × −2 = −10 1.4.2 Additional Questions Objective Questions 1. [AS1] 3 × (−5) = . (A) 15 (B) − 15 (C) 8 (D)− 8 2. [AS1] −8 × (−9 + 14) = . (B) –40 (A) 40 (D) –184 (C) 184 (B) 1 (D)− 2 3. [AS1] − 17 × = 34. (A) −1 (C) 2 EXERCISE 1.4. MULTIPLICATION OF INTEGERS 9

4. [AS1] 6 × (− 3) × (− 7) × 2 = . (A) 252 (B) − 252 (C) 32 (D)− 32 5. [AS1] × (−9) × (−6) × 7 = −1134. (A) 8 (C) –3 (B) 3 (D) –8 Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) Find the product of (− 25) and (− 30). (ii) Find the product of 3, (− 2) and (− 6). (iii) (− 3) × 7 × 1 = ________. (iv) −(− 321) × (− 42) = ________. (v) − 1[3(− 3 × 5)] = ________. Short Answer Type Questions 7(i) [AS1] A bonus of Rs. 3 is given on every Rs. 100 deposited in a bank. What is the bonus given on a sum of Rs. 1500? (ii) [AS1] A bicycle factory manufactures 35 cycles a day. In the month of May 2015, the factory works for 25 days. If each cycle is sold for Rs. 1850, find the total sales in the month. 8(i) [AS1] Vinod earns Rs. 1500 a day. How much will he earn in a month of 30 days ? (ii) [AS1] If there is a fall in temperature by 3◦C each day for a week at a certain place, what is the change in temperature over the week? 9(i) [AS5] Represent 3 × (− 2) on the number line. (ii) [AS5] Represent 4 × (− 3) on the number line. EXERCISE 1.4. MULTIPLICATION OF INTEGERS 10

Long Answer Type Questions 10 [AS1] Fill the blanks with an integer to make it a true statement. (i) (− 5) × = 35 (ii) 2 × 3 × = − 30 (iii) × (− 7) = − 49 (iv) × (− 4) = 24 (v) (− 3) × = − 27 11 [AS4] In a quiz every right answer carries 5 marks and every wrong answer carries (–2) marks. If a total of 10 questions are attempted by a student, of which 6 are correct, find his total score. EXERCISE 1.4. MULTIPLICATION OF INTEGERS 11

EXERCISE 1.5 1.5.1 Key Concepts DIVISION OF INTEGERS For any integer a, (B) 30 i. a ÷ 0 is not deﬁned or meaningless. (D)− 9 ii. 0 ÷ a = 0 ( f or 0) (B) 3 iii. a × 1 = a (D)− 40 (B) 27 1.5.2 Additional Questions (D)− 9 (B) 0 Objective Questions (D) None of these 1. [AS1] 27 ÷ (− 3) = . (A) − 30 (C) 9 2. [AS1] (− 30) ÷ −(− 10) = . (A) − 3 (C) 40 3. [AS1] (− 729) ÷ (− 27) = . (A) − 27 (C) 9 4. [AS1] 343 ÷ (–343) = . (A) 1 (C) –1 EXERCISE 1.5. DIVISION OF INTEGERS 12

5. [AS1] (− 5625) ÷ (− 25) = . (A) 225 (C) 25 (B) − 225 (D)− 25 Short Answer Type Questions 6(i) [AS1] Simplify: (– 50) ÷ (– 5) (ii) [AS1] Divide and write the quotient: (125) ÷ (– 25) Long Answer Type Questions 7 [AS1] Simplify: (i) (– 81) ÷ 9 (ii) 81 ÷ – 9 (iii) Is the result in (i) the same as that in (ii)? EXERCISE 1.5. DIVISION OF INTEGERS 13

EXERCISE 1.6 PROPERTIES OF INTEGERS 1.6.1 Key Concepts For any three integers a, b and c, i. a + b is also an integer: Closure property w.r.t addition ii. a − b is also an integer: Closure property w.r.t subtraction. iii. a.b is also an integer: Closure property w.r.t multiplication. iv. a + b = b + a: Commutative property w.r.t addition v. a + (b + c) = (a + b) + c: Associative property w.r.t addition vi. a(b.c) = (a.b)c: Associative property w.r.t multiplication vii. a + 0 = 0 + a = a: Identity property w.r.t addition viii. a.1 = 1.a = a: Identity property w.r.t multiplication ix. a(b + c) = ab + ac: Distributive property 1.6.2 Additional Questions Objective Questions 1. [AS1] − 25 × 19 = − 475 represents the property with respect to multiplication. (A) Closure (B) Commutative (C) Associative (D) Identity 2. [AS1] −12 × 13 = 13 × −12 represents the property with respect to multiplication. (A) Closure (B) Commutative (C) Associative (D) Identity 3. [AS1] 3 × (−5 + 6) = + 3 × 6 (Distributive property) (A) 3 × –5 (B) 3 × 1 (C)3 × 11 (D)3 × –11 EXERCISE 1.6. PROPERTIES OF INTEGERS 14

4. [AS1] The additive identity in the set of the integers is . (A) 1 (B) –1 (C) 0 (D)None of these 5. [AS1] 7 + [(− 4) + (− 6)] = [7 + (− 4)] + (− 6) represents the property with respect to addition. (A) Closure (B) Commutative (C) Identity (D) Associative Very Short Answer Type Questions property of integers under addition. 6 [AS1] Fill in the blanks. (i) 6 + 15 = 21, an integer. This represents the (ii) (− 8) × 5 = − 40 represents the closure property of integers under . (iii) When two negative integers are multiplied, the product is a positive integer. This verifies the property of integers under multiplication. (iv) To show that the set of integers is closed under addition, integer + integer = . 7 [AS1] Fill in the blanks. . (i) [15 + (−8)] + 23 = 15 + [(−8) + 23] represents the (ii) shows the property under addition. (iii) a × (b × c) = (a × b) × c represents the under multiplication. (iv) [(− 5) + (− 4)] + (− 1) = (− 5)+ . EXERCISE 1.6. PROPERTIES OF INTEGERS 15

8 [AS2] Choose the correct answer. (i) 2130 × (− 5) = (− 5) × . (A) 2130 (B) − 2130 (C)5 (D)None of these (ii) 86 + (− 92) = + 86 (A) 86 (B) − 86 (C) 92 (D)− 92 (iii) (− 6) × 15 = 15 × (− 6) is the property of integers under multiplication. (A) Closure (B) Commutative (C) Associative (D)None of these (iv) (−9) × (−1) = shows that multiplication of integers is commutative. (A) (–9) × (–1) (B) (–9) + (–1) (C)1 × (–9) (D)(–1) × (–9) Short Answer Type Questions 9 [AS2] Fill in the blanks. (i) _________________ is an example to show that subtraction of integers is closed. (ii) An example for associative property under multiplication of integers: _____________________________ EXERCISE 1.6. PROPERTIES OF INTEGERS 16

[AS4] Choose the correct answer. (iii) An example to show that addition of integers is commutative: (A) (− 3) + (− 2) = (− 5) (B) (− 3) + (− 2) = (− 2) + (− 3) (C)−3 × 2 = −2 × 3 (D)(− 3) − 2 = − 5 10(i) [AS2] Take any 3 pairs of integers and check whether they are closed under addition and multiplication. (ii) [AS2] Using the pairs of integers given verify whether the set of integers is closed with respect to addition and multiplication. a) (12, − 3) b) (− 8, − 6) 11(i) [AS2] Verify the following: a) 5 + [(−10) + (−15)] = [5 + (−10)] + (−15)] b) [(−2) + (−3)] + (−5) = (−2) + [(−3) + (−5)] (ii) [AS2] Use the integers given and verify whether the set of integers satisfies associative property w.r.t addition and multiplication: 7, − 11, 19 12 [AS2] Verify the following: (i) (−30) + 5 = 5 + (−30) (ii) (– 16) + (– 15) = (– 15) + (– 16) (iii) (– 25) + (– 5) = (– 5) + (– 25) (iv) (9) + (– 5) = (– 5) + 9 (v) (–7) × (3) = 3 × (–7) EXERCISE 1.6. PROPERTIES OF INTEGERS 17

13 [AS2] Verify the following: 18 (i) 2 × [(−3) × (−4)] = [2 × (−3)] × (−4) (ii) [(−7) × (5)] × 2 = (−7) [5 × 2] (iii) [(−2) × 6] × (−4) = −2 × [6 × (−4)] Long Answer Type Questions 14 [AS1] Simplify using a suitable property: (i) [7 × (50)] + [7 × 20] (ii) [6 × (−4)] × 3 (iii) (25) + [(−150) + (32)] (iv) (−20) × [(−125) + (27)] (v) [81 × (−15)] + [81 × (−30)] 15 [AS2] Verify the following: (i) (– 7) × [(– 3) + (– 4)] = [(– 7) × (– 3)] + [(– 7) × (– 4)] (ii) (− 4) × [(− 5) + 6] = [(4) × (− 5)] + [(− 4) × 6] (iii) 15 × [3 + (− 2)] = [15 × 3] + [15 × (− 2)] (iv) 9 × [(−4) + (−6)] = [9 × (−4)] + [9 × (−6)] (v) 25 × [10 + (−15)] = (25 × 10) + [25 × (−15)] EXERCISE 1.6. PROPERTIES OF INTEGERS

EXERCISE 1.7 PRACTICAL PROBLEMS USING NEGATIVE NUMBERS 1.7.1 Key Concepts i. Negative numbers are useful for solving a wide variety of practical problems. e. g: A monkey sits on a tomb that is 25 feet above the ground. He swings up 10 feet, climbs up 6 feet more, then jumps down 13 feet. How far off the ground is the monkey now? Ans. (25 + 10 + 6 − 13) feet = 28 feet 1.7.2 Additional Questions Objective Questions 1. [AS1] A submarine was situated 450 feet below sea level. If it descends 300 feet, its new position is . (A) 750 feet above sea level (B) 150 feet below sea level (C) 750 feet below sea level (D) 150 feet above sea level 2. [AS1] The temperature of a city is − 4◦C. It decreases by 5 ◦C in the evening and it again increases by 2◦C . Then the final temperature of the city is . (A) − 7◦C (B) − 10◦C (C) 7◦C (D) −11◦C 3. [AS1] In a quiz, team A scored − 40, 10, 0 and team B scored 10, 0, − 40 in three successive rounds. The team which scored more is . (A) Team A (B) Team B (C) Both the teams scored the same (D) None of these 4. [AS1] An elevator descends into a mine shaft at the rate of 5 metres per minute. If the descent starts from 10 m above the ground level, the time it will take to reach –300 m is . (A) 60 minutes (B) 70 minutes (C) 78 minutes (D) 62 minutes EXERCISE 1.7. PRACTICAL PROBLEMS USING NEGATIVE NUMBERS 19

5. [AS1] A jar contains 1000 ml of fruit juice. If 120 ml of juice is distributed among 3 people and 90 ml of juice is distributed among 5 people, then the quantity of juice left over in the jar is . (A) 150 ml (B) 160 ml (C)190 ml (D)200 ml Very Short Answer Type Questions 6 [AS1] Choose the correct answer. On a particular day, the temperature of Delhi at 10 a.m was 13◦C but by the mid-night, it fell down by 6◦C . The temperature of Delhi at mid-night is . (A) 19◦C (B) 7◦C (C) 8◦C (D) 9◦C 7 [AS1] Fill in the blanks. (i) The population of a village in 2012 is 1200. It increases by 50 every year. Then the village population in 2016 is . (ii) The temperature of a room is 25◦C, which decreases by 2 ◦C every one hour. The temperature of the room after 10 hours is . Short Answer Type Questions 8 [AS1] At sunrise, the outside temperature was 1◦C below zero. By lunch time, the temperature rose by 17◦C and then fell by 4◦C at night. What was the temperature at the end of the day? 9 [AS1] A submarine hovers at 240 metres below sea level. If it descends 160 metres and then ascends 390 metres, what is its new position? Long Answer Type Questions 10 [AS1] A rice mill earns a profit of Rs. 9 per bag of grade I rice bag sold and loss of Rs. 4 per bag of grade II rice bag sold. a) The mill sells 6000 bags of grade I rice and 5000 bags of grade II rice in a month. What is its proﬁt or loss? b) How many grade I bags must be sold to have neither profit nor loss, if the number of grade II bags sold is 5000? 11 [AS1] A shopkeeper earns a profit of Rs. 30 on one pair of shoes and loses Rs. 20 on a pair of chappals in a clearance sale. If he sells 20 pairs of each, find the total profit or loss he gets. EXERCISE 1.7. PRACTICAL PROBLEMS USING NEGATIVE NUMBERS 20

CHAPTER 2 FRACTIONS, DECIMALS AND RATIONAL NUMBERS EXERCISE 2.1 PROPER AND IMPROPER FRACTIONS 2.1.1 Key Concepts i. A proper fraction is a fraction that represents a part of a whole i.e., a fraction in which the numerator is less than its denominator is called a proper fraction. Example: 1 , 1 , 2 , 5 , 8 ... 2 3 3 6 13 ii. An improper fraction is a fraction that represents a whole or more than a whole i.e., a fraction in which the numerator is more than or equal to its denominator is called an improper fraction. Example: 5 , 4 , 8 , 11 ... 3 3 7 5 iii. Fractions can be represented pictorially. Example: iv. Like fractions can be compared by comparing their numerators. v. Unlike fractions can be compared by converting them into like fractions. vi. An equivalent fraction of a given fraction can be obtained by multiplying its numerator and denominator by the same number. Example: Equivalent fractions for 3 are 5 3 = 3 × 2 = 6 5 5 2 10 3 = 3 × 3 = 9 5 5 3 15 3 = 3 × 4 = 12 and so on. 5 5 4 20 EXERCISE 2.1. PROPER AND IMPROPER FRACTIONS 21

2.1.2 Additional Questions Objective Questions 1. [AS1] An example of a proper fraction is . (A) 4 (B) 5 3 2 (C) 1 (D) 3 1 7 7 2. [AS1] An example of an improper fraction is . (A) 3 (B) 7 4 4 (C) 2 (D) 2 3 5 3. [AS1] An example of a mixed fraction is . (A) 2 (B) 5 3 3 (C) 1 (D) 7 2 2 9 Very Short Answer Type Questions 6 [AS2] Answer the following questions in one sentence. (i) Classify the following fractions as proper or improper fractions. a) 5 3 b) 12 5 EXERCISE 2.1. PROPER AND IMPROPER FRACTIONS 22

c) 2 3 d) 20 51 e) 18 25 (ii) Identify and circle the proper fractions from the following. 4 2 5 3 8 6 10 3 5 4 2 15 7 9 (iii) Write three proper fractions. (iv) Write three improper fractions. (v) Identify and circle the improper fractions among the given fractions. 1 2 1 3 7 5 1 2 2 3 4 5 4 5 7 [AS5] Fill in the blanks. (i) The fraction represented by the shaded part in the figure is . (ii) The fraction represented by the shaded part in the figure is . (iii) 23 The fraction represented by the shaded part in the figure is . [AS5] Answer the following questions in one sentence. (iv) Represent the following fraction as the shaded part of a figure. 4 3 EXERCISE 2.1. PROPER AND IMPROPER FRACTIONS

(v) Represent 1 in the following ﬁgure. 2 Short Answer Type Questions 8 [AS1] Write any three equivalent fractions for each of the following. (i) 2 3 (ii) 4 7 9(i) [AS1] Which is greater? a) 3 (or) 4 7 9 b) 5 (or) 3 8 5 (ii) [AS1] Write the given fractions in a) Ascending order: 1 , 2 , 3 , 5 2 3 4 6 b) Descending order: 2 , 5 , 7 , 1 3 6 9 3 EXERCISE 2.1. PROPER AND IMPROPER FRACTIONS 24

10 [AS5] Shade the given figure to represent an equivalent fraction for the fraction of the shaded part in the given figures. (i) (ii) EXERCISE 2.1. PROPER AND IMPROPER FRACTIONS 25

Long Answer Type Questions 11 [AS1] (i) Ravi ate 3 part of a cake. How much cake is left? 4 (ii) A fruit seller sells 2½ kg fruits in the morning and 5¼ kg of fruits in the evening. How many kilograms of fruits were sold on that day? EXERCISE 2.1. PROPER AND IMPROPER FRACTIONS 26

EXERCISE 2.2 MULTIPLICATION OF FRACTIONS 2.2.1 Key Concepts i. To multiply a fraction by a whole number, we multiply the numerator and the whole number and write the product as the new numerator, keeping the denominator the same. e.g: 5 × 2 = 5 × 2 = 10 ; 4 × 8= 4×8 = 32 7 7 7 9 9 9 ii. In mathematical computation, ‘of’ means multiplication. 2.2.2 Additional Questions Objective Questions 1. [AS1] 3 × 4 = . 7 (A) 7 7 (B) 4 21 (C) 12 21 (D)1 5 7 2. [AS1] Seven times of the fraction 3 = . 14 (A) 1 1 2 (B) 3 98 (C) 2 3 (D) 98 3 EXERCISE 2.2. MULTIPLICATION OF FRACTIONS 27

3. [AS1] Three times a fraction is 7. Then the fraction is . (A) 21 (B) 2 1 3 (C) 3 1 3 (D) 3 7 4. [AS1] 9 × 8 = . 2 (A) 72 16 (B) 9 16 (C) 36 (D) None of these 5. [AS1] −34 × 2 = . 7 (A) 51 72 (B) 5 1 3 (C) 3 2 (D) None of these Short Answer Type Questions 6(i) [AS1] Find the products: a) 3 × 2 7 b) 2 × 3 4 (ii) [AS1] Find the products: a) 4× 3 b) 6× 4 5 7 EXERCISE 2.2. MULTIPLICATION OF FRACTIONS 28

7(i) [AS5] Multiply and show the product pictorially: a) 2 × 4 3 b) 3 × 1 3 (ii) [AS5] a) Write the following additions as multiplications. b) EXERCISE 2.2. MULTIPLICATION OF FRACTIONS 29

Long Answer Type Questions 8 [AS1] Complete the table with the products obtained by multiplying the numbers by the given fractions. × 2 4 1 3 6 3 5 4 8 5 5 7 9 (i) [AS1] Ravi got 90 marks in a class test. Ramu got 1 of Ravi’s marks. How many marks 3 did Ramu get? (ii) [AS1] The monthly income of a person is Rs. 15000. He spent 1 of it for children’s education, 5 1 for family expenditure and saved the remaining amount. Find his total savings in a month. 5 a) How much is spent on his children's education? b) How much is spent for the family? c) What is his monthly savings? EXERCISE 2.2. MULTIPLICATION OF FRACTIONS 30

EXERCISE 2.3 MULTIPLICATION OF A FRACTION BY A FRACTION 2.3.1 Key Concepts i. Product of two fractions = Product o f Numerators Product o f Denominator s e.g: 5 × 4 = 20 3 7 21 ii. The product of two proper fractions is less than each of the fractions. e.g: 2 × 5 = 10 3 7 21 10 < 2 and 10 < 5 21 3 21 7 iii. The product of a proper fraction and an improper fraction is less than the improper fraction and greater than the proper fraction. e.g: 3 × 7 = 21 ; Here, 3 < 21 and 7 > 21 . 4 5 20 4 20 5 20 iv. The product of two improper fractions is greater than each of the fractions. e.g: 7 × 3 = 21 ; Here,1201 > 7 and 21 > 3 . 5 2 10 5 10 2 2.3.2 Additional Questions Objective Questions 1. [AS1] 3 × 7 = . 4 6 (A) 7 (B) 11 8 10 (C) 8 (D) 7 7 5 2. [AS1] 3 × 7 × 27 = . 7 9 5 (A) 21 (B) 1 4 5 63 (C) 5 (D) 45 9 7 3. [AS1] The product of73 and a fraction is68 . Then the fraction is . (A) 24 (B) 11 42 13 (C) 56 (D)None of these 18 EXERCISE 2.3. MULTIPLICATION OF A FRACTION BY A FRACTION 31

4. [AS1] 2 × 7 × 3 = . 9 4 5 (A) 12 (B) 23 18 29 (C) 17 (D) 7 41 30 5. [AS1] −3 × 2 = . 7 4 (A) −1 (B) −3 35 14 (C) −1 (D) −6 12 12 Very Short Answer Type Questions 6 [AS1] Choose the correct answer. (i) 4 × 5 = . 15 8 (A) 2 (B) 1 5 6 (C) 5 (D) 6 6 (ii) 6 × 13 = . 13 15 (A) 13 (B) 5 6 13 (C) 13 (D) 2 5 5 (iii) 1 × 3 = . 3 4 (A) 3 (B) 1 4 3 (C) 1 (D) 3 4 (iv) 1 1 × 2 3 = . 4 4 (A) 5 (B) 11 4 4 (C) 55 (D) 16 16 55 EXERCISE 2.3. MULTIPLICATION OF A FRACTION BY A FRACTION 32

(v) 2 2 × 5 = . 5 12 (A) 12 (B) 5 5 12 (C) 1 (D) 1 12 Short Answer Type Questions 7(i) [AS1] A bike consumes 251 litres of petrol in 1 hour. How many litres of petrol is needed to run the bike for 2 1 hours? 2 (ii) [AS1] Evaluate: 1 × 6 × 30 3 5 24 8(i) [AS1] Usha solves 1 of the problems of a chapter in an hour. If she solves problems for 5 the chapter does she complete? 3 1 hours, what part 2 of (ii) [AS1] A bag of potatoes weighs 4 1 kg. What will be the weight of 212 such bags? 2 9(i) [AS5] Find 1× 1 using diagrams. 4 3 (ii) [AS5] Find 3 of 4 using figures. 4 5 Long Answer Type Questions 10 [AS1] Find the product of the fractions 5 and 2 and show that the product is greater or less than 6 7 each of the given fractions. 11 [AS1] 5 part of a water tank can be ﬁlled in 1 hour. What part of it can be ﬁlled in 2 hours 16 40 minutes? 12 [AS1] 7 part of the total distance was covered by a car using 3 1 litres of petrol. How much 24 8 money will a person spend on petrol to cover the total distance, if petrol costs Rs 70 per litre? EXERCISE 2.3. MULTIPLICATION OF A FRACTION BY A FRACTION 33

EXERCISE 2.4 DIVISION OF FRACTIONS 2.4.1 Key Concepts i. To divide a whole number by a fraction, multiply the whole number by the reciprocal of the given fraction. ii. To divide a fraction by a whole number, multiply the fraction by reciprocal of the given whole number. For example: 3 ÷ 2 = 3 × 7 7 ÷ 9 = 7 × 1 7 2 5 5 9 iii. To divide one fraction by another, multiply the ﬁrst fraction by the reciprocal of second fraction. For example: 3 ÷ 5 = 3 × 8 5 8 5 5 2.4.2 Additional Questions Objective Questions 1. [AS1] 3 ÷ 4 = . 4 3 (A) 12 (B) 1 7 (C) 7 (D) 9 7 16 2. [AS1] 12 ÷ = 3 5 4 (A) 2 (B) 5 3 (C) 3 1 (D) None of these 5 3. [AS1] ÷ −7 = 7 9 2 (A) 49 (B) −49 18 18 (C) 14 (D) −14 11 11 4. [AS1] 3 ÷ =1 7 (A) 3 (B) 7 7 3 (C) 10 (D) 3 3 10 EXERCISE 2.4. DIVISION OF FRACTIONS 34

5. [AS1] 9 ÷ 1 = . 7 (A) 7 (B) 7 16 9 (C) 9 (D) 16 7 7 Very Short Answer Type Questions 6. [AS1] Match the fractions in Column A with their reciprocals in Column B. Column A Column B i. 2 a. 9 5 8 ii. 8 b. 5 9 2 iii. 5 1 c. 7 3 18 iv. 2 4 d. 5 7 17 v. 3 2 e. 3 5 16 7 [AS1] Fill in the blanks. (i) The multiplicative inverse of a fraction is also known as its . (ii) The multiplicative inverse of 4 1 is . 3 (iii) The multiplicative inverse of a number is 73. Then the number is . (iv) The multiplicative inverse of 0 . (v) The multiplicative inverse of –1 is . 8 [AS1] Fill in the blanks. (i) 1 ÷ 4 = 2 (ii) 5 ÷ 8 = 6 (iii) 84 ÷ 12 = 35 (iv) 105 ÷ 35 = 24 (v) 10 ÷ 3 = 3 EXERCISE 2.4. DIVISION OF FRACTIONS 35

9 [AS1] Choose the correct answer. (i) 2 ÷ 4 = . 1 5 (A) 2 × 5 4 (B) 2 × 4 5 (C) 1 × 4 2 5 (D) 1 × 5 2 4 (ii) 1 ÷ 2 = . 3 5 (A) 1 × 2 3 5 (B) 3 × 2 5 (C) 1 × 5 3 2 (D) 3 × 5 2 (iii) 5 ÷ reciprocal of 3 = . 8 4 (A) 5 ÷ 3 8 4 (B) 5 × 3 8 4 (C) 5 ÷ 3 4 8 (D) 5 × 4 8 3 EXERCISE 2.4. DIVISION OF FRACTIONS 36

(iv) Reciprocal of 1 ÷ 3 = . 8 7 (A) 1 × 7 8 3 (B) 1 × 3 8 7 (C) 8 ÷ 3 7 (D) 8 × 3 7 (v) 4 ÷ 7 = . 7 4 (A) 4 × 7 7 4 (B) 4 × 4 7 7 (C) 7 × 7 4 4 (D) 7 ÷ 7 4 4 Short Answer Type Questions 10(i) [AS1] Find: a) 3 ÷ 1 4 b) 5 ÷ 1 3 c) 7 ÷ 2 3 d) 6 ÷ 3 5 (ii) [AS1] Evaluate: 20 ÷ 1 5 11 [AS1] What is the reciprocal of 231? 12 [AS2] What fraction is the reciprocal of a proper fraction? Explain with an example. 13 [AS2] Find the sum of 3 and the reciprocal of 7 and multiply it by the reciprocal of 4 5 . 4 3 7 Long Answer Type Questions 14 [AS1] The area of a rectangle is 64 m2 . If its length is 8 m, ﬁnd its breadth. 15 EXERCISE 2.4. DIVISION OF FRACTIONS 37

15 [AS1] Divide the product of 2 1 and 3 1 by 5 1 . 3 7 2 16 [AS1] The product of two fractions is 24 . One of the fractions is 76. Find the other. 42 17 [AS1] Saritha cuts a cake and eats 1 of it. The remaining cake is divided equally among her four 2 friends. What fraction of the cake does each of her friends get? 18 [AS2] The sum of two fractions 9 and −2 multiplied by the reciprocal of another fraction is 23 . 13 3 39 Find the fraction. EXERCISE 2.4. DIVISION OF FRACTIONS 38

EXERCISE 2.5 DECIMAL NUMBERS OR FRACTIONAL DECIMALS 2.5.1 Key Concepts i. When we multiply a decimal number by 10, 100, 1000, . . . ., we move the decimal point in the number to the right by as many places as the number of zeros in the multiplier. e.g.: 1.125 × 10 = 11.25 1.125 × 100 = 112.5 1.125 × 1000 = 1125 1.125 × 10000 = 11250 2.5.2 Additional Questions Objective Questions 1. [AS1] 6 × 1000 + 3 × 100 + 2 × 10 + 7 + 5 + 8 = . 10 100 (A) 632.758 (B) 6327.58 (C) 63275.8 (D) 63.2758 2. [AS1] 25 paise = Rs. . (A) 25 (C) 0.25 (B) 2.5 (D) 0.025 3. [AS1] 700 m = km. (B) 0.07 (A) 0.7 kg. (D) 70 (C) 7 (B) 3 4. [AS1] 750 g = 4 (A) 75 (C) 7.5 (D) 0.075 EXERCISE 2.5. DECIMAL NUMBERS OR FRACTIONAL DECIMALS 39

5. [AS1] 3 × 10 4 + 5 × 103 + 6 × 102 + 7 × 10−1 + 8 × 10−2 . (A) 3567.8 (B) 356.78 (C) 3567.08 (D) 35600.78 Very Short Answer Type Questions 6 [AS1] Fill in the blanks. (i) 5 paise = Rs. . (ii) 125 paise = Rs. . (iii) 35 mm = cm. (iv) 10 cm = mm. (v) 5 m = cm. 7 [AS1] Fill in the blanks. (i) The expanded form of 52.63 is . (ii) The expanded form of 412.054 is . (iii) 3 × 1000 + 1 × 100 + 0 + 1 + 1 + 5× 1 is . 10 1000 (iv) 4× 1000 + 2 × 100 +0 × 10 + 1×1 + 0 × 1 + 1 × 1 + 5 × 1 is ( ). 10 100 1000 40 [AS1] Answer the following questions in one sentence. (v) Write the expanded form of each of the following numbers. a) 20.507 b) 562.043 c) 3120.502 d) 140.138 e) 264.75 EXERCISE 2.5. DECIMAL NUMBERS OR FRACTIONAL DECIMALS

Short Answer Type Questions 8(i) [AS1] Which is bigger? a) 0.5 or 0.05 b) 1.47 or 1.51 (ii) [AS1] Write the following decimals in ascending order. 0.156, 0.0156, 0.165, 0.0165 Long Answer Type Questions 9 [AS1] Radha spent Rs. 15.25 to buy vegetables and Rs. 20.45 to buy fruits. If she had Rs. 50 before spending, how much amount is left with her? EXERCISE 2.5. DECIMAL NUMBERS OR FRACTIONAL DECIMALS 41

EXERCISE 2.6 MULTIPLICATION OF DECIMAL NUMBERS 2.6.1 Key Concepts i. Steps to multiply two decimal numbers. a. Multiply them as whole numbers. b. Count the number of digits in the decimal place of both the numbers and add them. c. Place the decimal point in the product by counting the number of digits from its right-most place. e.g.,: 6.25 × 3.14 625 × 314 = 196250 Sum of the number of digits in decimal places = 2 + 2 = 4 Product = 19.6250 ii. When we divide a decimal number by 10, 100, 1000, . . . ., we move the decimal point in the number to the left by as many places as the number of zeros in the multiplier. a. 435.873 ÷ 10 = 43.5873 b. 4551.3 ÷ 100 = 45.513 c. 8374.2 ÷ 1000 = 8.3742 d. 24.82 ÷ 1000 = 0.02482 iii. To divide a decimal number by a whole number a. Divide them as whole numbers b. Place the decimal point in the quotient as in the decimal number. e.g.: 86.5 ÷ 5 Step a: 865 ÷ 5 = 173 Step b: 17.3 iv. To divide a decimal number by another, a. Shift the decimal point to the right by equal number of places in both to convert the denominator to a whole number. b. Divide them as you would divide a decimal number by a whole number. e.g.: 6.25 ÷ 2.5 6.25 = 62.5 = 2.5 2.5 25 EXERCISE 2.6. MULTIPLICATION OF DECIMAL NUMBERS 42

2.6.2 Additional Questions Objective Questions . 1. [AS1] 2.34 × 1.5 = (A) 3.84 (B) 3.39 (C) 3.51 (D) 2.49 2. [AS1] 701.3457 × 1000 = . (B) 701345.7 (A) 70134.57 (D) 7013457.0 (C) 7013.457 (B) 0.2 (D) 200 3. [AS1] 7.56 ÷ 3.78 = . (A) 2 (C) 20 4. [AS1] 648.3275 ÷ 100 = . (A) 0.6483275 (B) 6483.275 (C) 64.83275 (D) 6.483275 5. [AS1] 57.3456 × 100 = . (B) 5.73456 (A) 0.573454 (D) 573.456 (C) 5734.56 Very Short Answer Type Questions 6 [AS1] Fill in the blanks. (i) 153.2 × 1000 = . (ii) 0.153 × 100 = . (iii) 0.068 × 10000 = . (iv) 6.84 × 1000 = . (v) 0.3204 × 1000 = . EXERCISE 2.6. MULTIPLICATION OF DECIMAL NUMBERS 43

7 [AS1] Fill in the blanks. (i) 56.18 ÷ 2 = . (ii) 680.04 ÷ 4 = . (iii) 85.255 ÷ 5 = . (iv) 105.50 ÷ 5 = (v) 0.225 ÷ 15 = . . Short Answer Type Questions 8(i) [AS1] Multiply the following: a) 8.148 × 200 b) 0.2 × 0.02 × 0.002 × 0.0002 (ii) [AS1] If 412 × 12 = 4944, ﬁnd the value of 4.12 × 0.12. 9(i) [AS1] Ravi purchases 13.75 kg of sugar at Rs. 29.45 per kg. What amount should he pay the shopkeeper? (ii) [AS1] Sumathi purchases 7.50 kg onion, 2.5 kg tomato and 1.25 kg brinjal from a vegetable shop. If the cost per kilogram of onion, tomato and brinjal are Rs. 12.75, Rs. 23.50 and Rs. 14.50 respectively, ﬁnd the total amount Sumathi paid the shopkeeper. 10(i) [AS1] Rahul practises 2.03 hours of tennis each day for 5 days. How many hours of tennis did he practise in total? (ii) [AS1] Fill in the blanks with an appropriate number. a) 0.512 × = 512 b) 0.065 × = 6.5 c) 1.058 × = 10.58 11(i) [AS1] Divide 49.08 by 0.006. (ii) [AS1] Divide 8 by 0.0004. EXERCISE 2.6. MULTIPLICATION OF DECIMAL NUMBERS 44

Long Answer Type Questions 12 [AS1] Divide: 8.12 by 2.5 13 [AS4] Vamsi has to pay Rs. 50 a month for 500 text messages. For each message over 500, he has to pay an extra Rs. 0.55. How much does Vamsi owe if he sent 768 text messages in a month? 14 [AS4] Each of the six buckets of equal capacity holds 120.06 litres of water. How many litres of water can be stored in 12 such buckets? 15 [AS4] A green club is planting trees. Each tree they plant costs them Rs. 25.50. If they have Rs. 510, how many trees can they plant? EXERCISE 2.6. MULTIPLICATION OF DECIMAL NUMBERS 45

EXERCISE 2.7 INTRODUCTION TO RATIONAL NUMBERS – POSITIVE FRACTIONAL NUMBERS 2.7.1 Key Concepts i. The numbers in the form p where p, q are integers such that 'q' is not equal to zero are called q rational numbers. ii. The set of rational numbers is represented by Q. iii. Q includes all integers, positive fractional numbers and negative fractional numbers. iv. All rational numbers can be represented on a number line. 2.7.2 Additional Questions Objective Questions 1. [AS1] One of the rational numbers between 3 and 4 is . (A) 10 (B) 10 2 3 (C) 10 (D) 10 4 5 2. [AS1] The rational number 19 lies between . (B) 19, 20 9 (D)2, 3 (A) 1, 2 (C)9, 10 3. [AS1] Among the following rational numbers, 7 , 3 , 4 and 8 , the greatest one is . 9 4 5 9 (A) 4 (B) 8 5 9 (C) 7 (D) 3 9 4 4. [AS1] The smallest among 6 , 7 , 3 and 8 is . 9 4 4 9 (A) 6 (B) 7 9 4 (C) 3 (D) 8 4 9 EXERCISE 2.7. INTRODUCTION TO RATIONAL NUMBERS –POSITIVE FRAC. . . 46

5. [AS1] There are rational numbers between any two integers. (A) 0 (B) 1 (C) 2 (D)Inﬁnitely many Very Short Answer Type Questions 6 [AS1] State true or false. (i) Every positive integer is a positive rational number. (ii) Zero is a negative rational number. [ ] [ ] (iii) 3 + 1 is the rational number between 2 and 3. [ ] 2 4 [ ] [ ] (iv) There are no rational numbers between 2 and 3. (v) There are no positive fractions between 2 and 4 . 5 5 7 [AS2] Fill in the blanks. (i) −5 11 9 −16 (ii) −3 4 8 (−7) (iii) 3 11 4 8 (iv) 5 −7 −14 12 (v) −7 5 12 12 EXERCISE 2.7. INTRODUCTION TO RATIONAL NUMBERS –POSITIVE FRAC. . . 47

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