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TABLE OF CONTENTS 1 RATIONAL NUMBERS 1 1.1 PROPERTIES OF RATIONAL NUMBERS 1 1.2 REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE 5 1.3 DECIMAL REPRESENTATION OF RATIONAL NUMBERS 9 2 LINEAR EQUATIONS IN ONE VARIABLE 13 2.1 LINEAR EQUATIONS IN ONE VARIABLE 13 2.2 LINEAR EQUATIONS IN ONE VARIABLE–APPLICATIONS 16 2.3 SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE SIDES 19 2.4 SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE SIDES –APPLICATIONS 21 2.5 REDUCING EQUATIONS TO SIMPLER FORM ( LINEAR) 23 3 CONSTRUCTION OF QUADRILATERALS 26 3.1 QUADRILATERALS AND THEIR PROPERTIES 26 3.2 CONSTRUCTION : LENGTHS OF FOUR SIDES AND ONE DIAGONAL ARE GIVEN 28 3.3 CONSTRUCTION : LENGTHS OF THREE SIDES AND TWO DIAGONALS ARE GIVEN 29 3.4 CONSTRUCTION: LENGTHS OF TWO ADJACENT SIDES AND THREE ANGLES ARE GIVEN 30 3.5 CONSTRUCTION : LENGTHS OF THREE SIDES AND TWO INCLUDED ANGLES ARE GIVEN 31 3.6 CONSTRUCTION OF SPECIAL TYPES OF QUADRILATERALS 32 4 EXPONENTS AND POWERS 33 4.1 POWERS WITH NEGATIVE EXPONENTS 33 4.2 APPLICATION OF EXPONENTS TO EXPRESS NUMBERS IN STANDARD FORM 39 5 COMPARING QUANTITIES USING PROPORTION 42 5.1 COMPOUND RATIO 42 5.2 FINDING DISCOUNTS 45 5.3 COMPOUND INTEREST 48

6 SQUARE ROOTS AND CUBE ROOTS 52 6.1 PROPERTIES OF SQUARE NUMBERS 52 6.2 PYTHAGOREAN TRIPLETS FINDING SQUARE ROOTS BY PRIME FACTORIZATION METHOD 55 6.3 FINDING SQUARE ROOT BY DIVISION METHOD 57 6.4 CUBIC NUMBERS 60 6.5 CUBE ROOTS 62 7 FREQUENCY DISTRIBUTION TABLES AND GRAPHS 64 7.1 BASIC MEASURES OF CENTRAL TENDENCY 64 7.2 ORGANISATION OF GROUPED DATA 69 7.3 GRAPHICAL REPRESENTATION OF DATA 72 8 EXPLORING GEOMETRICAL FIGURES 77 8.1 CONGRUENCY OF SHAPES 77 8.2 SYMMETRY 84 9 AREA OF PLANE FIGURES 89 9.1 AREA OF TRAPEZIUM 89 9.2 AREA OF CIRCLE 96 10 DIRECT AND INVERSE PROPORTIONS 100 10.1 DIRECT PROPORTION 100 10.2 INVERSE PROPORTION 104 10.3 COMPOUND PROPORTION 107 109 PROJECT BASED QUESTIONS

CHAPTER 1 RATIONAL NUMBERS EXERCISE 1.1 PROPERTIES OF RATIONAL NUMBERS 1.1.1 Key Concepts i. Rational Numbers (Q): Numbers which can be expressed in the form p , where ‘p ’ and ‘q ’ q are both integers and q 0 and p, q are relatively prime. e.g.: 3 , − 3 , 5, 0 etc. 5 7 ii. Positive rational number: A rational number is positive if its numerator and denominator are either both positive or both negative. e.g.: 5 , −3 etc. 3 −2 iii. Negative rational number: A rational number is negative if its numerator and denominator are of opposite signs. e.g.: − 5 , 3 etc. 7 −11 p iv. Standard form of a rational number: A rational number q is said to be in standard form if ‘p’ and ‘q’ are integers having no common factors other than 1. e.g., 33 can be expressed as 3 in its standard form. 44 4 a c a c v. Closure property: If b and d are any two rational numbers then b + d is also a rational number. vi. Commutative property: If a and c are any two rational numbers then a + c = c + a . b d b d d b The change in the order of two numbers does not alter their sum. vii. Associative property: If a , c and e are any three rational numbers, then f bd a + c + e = a + c + e . The change in the order of three numbers does not alter their sum. b d f b d f viii. When the number ‘0’ is added to any rational number, the sum is the same rational number. e.g.: 1 + 0 = 1. So, 0 is called the additive identity. ix. The additive identity for integers and whole numbers is also ‘0’. x. When we multiply any rational number by ‘1’, we get the same rational number as product. So, 1 is called the multiplicative identity. xi. Any two numbers whose sum is ‘0’ are called additive inverses of each other. For any rational number 'a', if a + (-a) = 0 and (-a) + a = 0, then 'a' and '(-a)' are called the additive inverses of each other. xii. Any two numbers whose product is ‘1’ are called multiplicative inverses of each other. e.g.: 3 × 1 = 1. So, 3 and 1 are multiplicative inverses of each other. 3 3 a ,c e are any three rational numbers, then xiii. Distributive property: If bd and f and . EXERCISE 1.1. PROPERTIES OF RATIONAL NUMBERS 1

1.1.2 Additional Questions . Objective Questions 1. [AS1] The negative rational number among the following is (A) 4 (B) 7 (C) −8 3 3 (D) 0 2. [AS1] −8 + 4 = −4 is an example for closure property with respect to . (A) Addition (B) Subtraction (C) Multiplication (D) Division 3. [AS1] 8 + (7 + 4) = (8 + 7) + 4 shows that the set of whole numbers satisﬁes the property with respect to addition. (A) Closure (B) Commutative (C) Identity (D) Associative 4. [AS1] The multiplicative inverse of 7 is . 8 (B) 8 7 (A) −7 (D)None of these 8 −8 (C) 7 5. [AS1] 11×(−7) = (−7)×11 is an example for the property of integers with respect to multiplication. (A) Closure (B) Commutative (C) Identity (D) Associative EXERCISE 1.1. PROPERTIES OF RATIONAL NUMBERS 2

6. [AS1] 13 × −18 = . 6 91 (A) −5 (B) 3 546 7 (C) −5 (D) −3 97 7 7. [AS1] If a rational number is multiplied by its additive identity then the result is . (A) 0 (B) The number itself (C)1 (D)None of these 8. [AS1] If a is the reciprocal of b then the reciprocal of ab is . (A) a (B) b (C) 1 (D) a Very Short Answer Type Questions b 9 [AS1] Answer the following questions in one sentence. (ii) The sum of two rational numbers is the additive identity. If one of them is −9 , then ﬁnd the second 14 rational number. 10 [AS3] Fill in the blanks. Identify and write the numbers in the given collection under the appropriate category: 0, 1, 22, −2, −5, 1 , 0.5, 4 1 22 (A number can be written in more than one group.) (i) Natural numbers: (ii) Whole numbers: (iii) Integers: (iv) Rational numbers: (v) Every integer is also a . EXERCISE 1.1. PROPERTIES OF RATIONAL NUMBERS 3

Short Answer Type Questions 11 [AS2] Verify whether the set of integers satisﬁes closure property with respect to addition. 12(i) [AS2] Verify whether the natural numbers 9 and 17 satisfy the commutative property with respect to addition. (ii) [AS2] Verify whether the set of rational numbers satisﬁes commutative property with respect to addition, multiplication and subtraction by taking an example. 13(i) [AS2] Verify whether the set of whole numbers satisﬁes associative property with respect to multiplication by taking an example. (ii) [AS2] Verify whether the set of rational numbers satisﬁes associative property with respect to addition by taking an example. 14(i) [AS3] Write the additive identity and additive inverse of the number 8 in the set of integers. (ii) [AS3] Find the additive identity, additive inverse, multiplicative identity and multiplicative inverse of rational number 5 . 3 15(i) [AS3] Write the properties under addition for the set of integers. (ii) [AS3] Write the properties of whole numbers with respect to addition. 16(i) [AS3] Write the properties satisﬁed by natural numbers with respect to multiplication. (ii) [AS3] Write the properties of integers with respect to multiplication. Long Answer Type Questions 17 [AS2] Verify the properties of rational numbers with respect to subtraction and check whether it is a Group or not with respect to subtraction. 18 [AS2] Verify the properties of rational numbers with respect to division and check whether it is a Group or not with respect to division. 19 [AS2] Verify the properties of rational numbers with respect to multiplication. EXERCISE 1.1. PROPERTIES OF RATIONAL NUMBERS 4

EXERCISE 1.2 REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE 1.2.1 Key Concepts i. Rational numbers can be represented on a number line. ii. There are countless rational numbers in between any two given rational numbers. iii. The idea of mean helps us to ﬁnd rational numbers between two given rational numbers. 1.2.2 Additional Questions Objective Questions on the number line is . (B) 7 1. [AS5] The rational number that lies to the left of 3 (A) 9 (D)None of these 4 (C) 5 4 2. [AS5] The rational number that lies to the right of on the number line is . (A) −23 (B) −32 9 9 (C) −43 (D) 0 9 3. [AS1] The rational number between 3 and 7 is . 49 55 (B) 7 (A) 72 12 (C) 12 (D) 72 7 55 EXERCISE 1.2. REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE 5

4. [AS5] The rational number 3 lies between and on the number line. (A) 20 and 16 (B) 20 and 24 77 77 (C) −12 and 9 (D)None of these 77 −3 and −7 on the number line is . 4 5. [AS5] The rational number that lies between 4 (A) 3 (B) 2 (C) 1 (D) −1 6. [AS5] The value of P represented on the number line l is . (A) 4 (B) 14 6 6 (C) 4 (D)1 4 3 6 7. [AS5] P represents 3 on the number line given. The rational number represented by Q is . 5 (A) 4 (B) 1 4 5 5 (C) 4 (D)1 4 3 6 EXERCISE 1.2. REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE 6

8. [AS5] The values of P and Q on the number line are −11 and −1 . R is a number equidistant from 16 8 P and Q on the number line. The value of R is . (A) −7 16 (B) −13 16 (C) −3 4 (D) −23 Very Short Answer Type Questions 9 [AS5] Fill in the blanks. Observe the following number line and ﬁll in blanks given. (i) The rational number represented by A is . (ii) The rational number 9 is represented by the letter . 5 (iii) The rational number represented by G is . (iv) The rational number 2 1 is represented by the letter . 5 (v) The letter F represents the rational number . 10 [AS5] Answer the following questions in one sentence. 7 on the number (i) What will be the position of the result when we add 3 to the rational number 8 4 line? (ii) What will be the position of the result when we subtract 2 from the rational number 5 on the 3 6 number line? Short Answer Type Questions 11(i) [AS5] Identify the rational numbers P and Q on the number line given. EXERCISE 1.2. REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE 7

(ii) [AS5] Identify the rational numbers A, B and C represented on the number line given. Long Answer Type Questions 12 [AS1] Find ﬁve rational numbers between −2 and 4 . 35 13 [AS5] Represent the rational numbers −2 and 3 on a number line. Find two rational numbers 34 between −2 and 3 and also represent them on the number line. 3 4 14 [AS5] Represent the rational numbers −1 , 4 , −2 , 5 on a number line and write them in 2334 the ascending order of their magnitude. EXERCISE 1.2. REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE 8

EXERCISE 1.3 DECIMAL REPRESENTATION OF RATIONAL NUMBERS 1.3.1 Key Concepts i. Rational numbers are in the form of p , where q 0; p and q are integers. q ii. We can convert a rational number into a decimal by division. iii. Terminating decimals: The decimal numbers with only a ﬁnite number of digits in the decimal part. iv. Non–terminating recurring decimals: In the division process, sometimes, the division never comes to an end. The remainder keeps repeating after a certain number of steps. In these cases, a digit or a set of digits in the quotient repeats in the same order. Such decimals are called non-terminating recurring decimals. 1.3.2 Additional Questions Objective Questions 1. [AS1] Of the following, a terminating decimal is . (A) 7 (B) 8 4 9 (C) −3 (D) 1 7 14 (A) 9 (B) 67 17 45 (C) 45 (D)None of these 67 3. [AS3] The rational number which is the multiplicative inverse of −4 is . 7 (A) 4 (B) −4 77 (C) −7 (D) 7 4 4 EXERCISE 1.3. DECIMAL REPRESENTATION OF RATIONAL NUMBERS 9

4. [AS4] 96.75 expressed as a rational number is . (A) 9657 (B) 387 100 4 (C) 9675 (D) 9675 10 1000 5. [AS1] The decimal representation of the rational number 21 is . 16 (A) 1312.5 (B) 131.25 (C) 13.125 (D) 1.3125 (A) 225 (B) 6.5 841 (D) 0.65 (B) 7 29 (C) 1 (D) 841 225 (A) −6.5 (C) 65.0 EXERCISE 1.3. DECIMAL REPRESENTATION OF RATIONAL NUMBERS 10

8. [AS2] If the multiplicative inverse of a rational number is –3.45, then the rational number is . (A) 20 (B) −69 69 20 (C) −69 (D)None of these 2 9. [AS4] 0.125 expressed as a rational number is . (B) 5 (A) 25 . 2 4 (D) 1 (C) 1 80 8 10. [AS4] 36 expressed as a decimal is (B) 0.3675 75 (A) 0.48 (C) 36.75 (D) 0.7536 Very Short Answer Type Questions 11 [AS1] Answer the following questions in one sentence. (i) Express the value of −4 + 3 × 14 as a decimal. 74 5 (ii) Calculate −7 ÷ 35 + 17 and write the answer as a decimal. 16 32 4 (iii) Find a rational number between 3 and 4 and represent it as a decimal. 43 12 [AS1] Answer the following questions in one sentence. (i) If 'p' is the additive inverse of19 then ﬁnd the value of 'p' upto 2 places of decimals. 4 (ii) If 'x' is the inverse element of 13 w.r.t. subtraction in Q then ﬁnd the value of 'x' upto 2 decimals. 5 (iii) If 7.45 is the rational number in between 27 and P then ﬁnd the value of P. 4 13 [AS4] Answer the following questions in one sentence. (i) Express 69 as a decimal. 12 (ii) Express 369.45 as a rational number. EXERCISE 1.3. DECIMAL REPRESENTATION OF RATIONAL NUMBERS 11

(iii) Express 125.375 as a rational number. Short Answer Type Questions 14(i) [AS1] The cost of 72 metres of rope is Rs.123. Find its cost per metre. 34 (ii) [AS1] What should be subtracted from −3 so as to get 5 6? 4 15(i) [AS4] Express the decimal 23.75 as a rational number. (ii) [AS4] Express the following decimals as rational numbers. a) 0.55 b) 0.378 16 (i) [AS4] Express 0.46 as a rational number. (ii) [AS4] Express 0.375 in the form of rational number. 17 [AS1] Find the area of a rectangular flower bed whose length is 28 5 m and breadth is 15 1 m. 8 2 18 [AS1] If 2 of a number exceeds 1 of the same number by 28, ﬁnd the number. 34 19 [AS4] Amit earns Rs.16000 per month. He spends 1 of his income on food, 3 of the remaining 4 10 on house rent and 5 of the remainder on the education of children. How much money is still left 21 with him? EXERCISE 1.3. DECIMAL REPRESENTATION OF RATIONAL NUMBERS 12

CHAPTER 2 LINEAR EQUATIONS IN ONE VARIABLE EXERCISE 2.1 LINEAR EQUATIONS IN ONE VARIABLE 2.1.1 Key Concepts i. Equation: A statement of equality which contains one or more unknown quantities or variables is called an equation. e.g.: 3x + 7 = 12, x +2 1 = 5 ii. Degree ‘1 ’ equations are called linear equations. iii. Linear equation: An equation in which the highest index of the variable present is ‘1’ is called a linear equation. e.g.: 3x + 7y − 8z − 2 = 7 iv. If a linear equation has only one variable, then it is called a linear equation in one variable or a simple equation. The equation of the form ax + b = 0 or ax = b, where a, b are constants, is called a linear equation in one variable. v. Rules for solving linear equations in one variable: a. Rule 1: Same quantity (number) can be added to both sides of an equation. b. Rule 2: Same quantity can be subtracted from both sides of an equation. c. Rule 3: Both sides of an equation can be multiplied by the same non–zero number. d. Rule 4: Both sides of an equation may be divided by the same non–zero number. 2.1.2 Additional Questions Objective Questions 1. [AS3] The degree of a linear equation is . (A) 0 (B) 1 (C) 2 (D) 3 2. [AS3] A linear equation among the following, is . (A) 5x − 12 = 0 (B) 6x − 12 = 6x − 5 (C)3x2 − 5x + 7 (D)None of these EXERCISE 2.1. LINEAR EQUATIONS IN ONE VARIABLE 13

3. [AS3] A linear equation in one variable among the following is . (A) 3p − 12 = 4q − 12 (B) 5p 2 − 7p + 2 = 0 (C)(4p)2 − 2 = 0 (D)7p − 8 = 0 4. [AS1] The value of p in 3p − 8 = 12 is . (A) 20 (B) 17 (C) 20 (D) 23 3 5. [AS1] The value of x in 3x − 6 = 14 − 2x is . (A) 4 (B) 20 (C) 20 (D) 10 3 6. [AS4] The cost of a pen and a pencil together is Rs.19, when expressed as a linear equation is . (A) x + y = 19 (B) x − y = 19 (C) x × y = 19 (D) x ÷ y = 19 Very Short Answer Type Questions 7 [AS1] Answer the following questions in one sentence. Solve the equation: 3x + 2 = 11 8 [AS2] State true or false. [] (i) 4x + 6y – 7 = 0 is a linear equation in one variable. [] [] (ii) 3x2 – 2x + 4 = 0 is not a linear equation. [] (iii) 7x + 4 = 2x + 5 is not a linear equation in one variable. (iv) 4x = 13 is a linear equation in one variable. [AS2] Answer the following questions in one sentence. (v) 3x – 5 = 4x – 1 is a linear equation in . EXERCISE 2.1. LINEAR EQUATIONS IN ONE VARIABLE 14

9 [AS2] State true or false. [ ] (i) xy + xyz + zx is a simple equation. [ ] [ ] (ii) 2x2 – 3x + 5 = 0 is not a simple equation. [ ] (iii) 2x + 3y = 15 is a simple equation in one variable. (iv) 3x + 13 = 12 + 5x is a simple equation. [AS2] Answer the following questions in one sentence. (v) Write general form of a simple equation. 10 [AS3] Answer the following questions in one sentence. (i) 5 less than 7 times a number is the same as 4 more than 3 times the same number. Express this as a linear equation. (ii) 9 more than 3 times a number is the same as 12 less than 2 times the same number. 4 3 Express this as a linear equation. Short Answer Type Questions 11 [AS1] Solve: 4x − 7 = − 3x + 14 12 [AS3] A number is such that it is as much greater than 84 as it is less than 108. Express this in the form of a linear equation and solve it. Long Answer Type Questions 13 [AS1] Solve the linear equation: 4x + 4 = 7 1 3 2 14 [AS3] The sum of the digits of a 2-digit number is 8. When 18 is added to the number the digits get reversed. Express this in the form of a linear equation in one variable and ﬁnd the number. EXERCISE 2.1. LINEAR EQUATIONS IN ONE VARIABLE 15

EXERCISE 2.2 LINEAR EQUATIONS IN ONE VARIABLE–APPLICATIONS 2.2.1 Key Concepts i. The value of the variable, which when substituted in the given equation, makes L.H.S. = R.H.S., is called a solution or root of the given equation. 2.2.2 Additional Questions Objective Questions 1. [AS3] Two angles differ by 12◦ when expressed as an equation using variables x and y is . (A) x − y = 12 (B) x − y − 13 =0 (C) x = 12y (D)y = 12x 2. [AS3] \"The sum of two numbers is 56 and their difference is 12\" in the equation form is . (A) x + y = 56 x − y = 12 (B) x + y = 12 x − y = 56 (C) x + 56 = y x − y = 12 (D) x = 56 + y y = 12 − x 3. [AS4] The length of a rectangular plot is three times its breadth. If its breadth is 8 m then the length of the plot is . (A) 512 m (B) 11 m (C)24 m (D)None of these EXERCISE 2.2. LINEAR EQUATIONS IN ONE VARIABLE–APPLICATIONS 16

4. [AS3] If Ramu earns Rs. x a day and Rahim earns Rs. 7 more than twice the earnings of Ramu then the earning of Rahim is Rs. . (A) Rs. 7x − 2 (B) Rs. 14x (C) Rs. 7x + 2 (D) Rs. 2x + 7 5. [AS4] The present age of Sita’s mother is 5 less than six times that of Sita’s present age. Present age of Sita's mother is years. (A) 6x − 5 (B) 5x + 6 (C)5x − 6 (D)6x + 5 6. [AS3] The present age of Rajesh is 2 years less than thrice that of the 12 year old boy Ranga. The present age of Rajesh is . (A) 36 years (B) 30 years (C)34 years (D)42 years 7. [AS4] \"The age of Suma is 5 more than 3 times the age of her son\", represented as a linear equation is . (A) 3x + 5 (B) 5x + 3 (C) 15 x (D)None of these Very Short Answer Type Questions 8 [AS1] Answer the following questions in one sentence. (i) The digit in the units place of a number is three times the digit in its tens place. The number formed by reversing the digits is 36 more than the given number. Find the number. (ii) Find the solution of 3x − 7 = 3 − 2x. 9 [AS4] Answer the following questions in one sentence. (i) After 12 years Rajesh shall be 3 times as old as he was 4 years ago. Express this as a linear equation. (ii) Express x + 11 = 1 as a verbal statement. 5 15 10 [AS5] Answer the following questions in one sentence. Represent as a linear equation: \"The age of Jaya is 12 years more than four times the age of Sona\". EXERCISE 2.2. LINEAR EQUATIONS IN ONE VARIABLE–APPLICATIONS 17

Short Answer Type Questions 11(i) [AS1] Two complementary angles differ by 14 ◦. Find the angles. (ii) [AS1] The sum of two numbers is 46 and their difference is 18. Find the numbers. 12(i) [AS1] A sum of Rs. 400 is in the denominations of Rs. 5 and Rs. 10. If the total number of notes is 57, ﬁnd the number of notes of each denomination. (ii) [AS1] A sum of Rs. 870 is in the denominations of Rs. 10 and Rs. 20. If the total number of notes is75, ﬁnd the number of notes of each denomination. 13(i) [AS5] Represent \"A number whose double is 45 greater than its half\" as a linear equation and solve it. (ii) [AS5] Represent \"Four ﬁfth of a number is more than three fourth of the number by 4” in the form of a linear equation in one variable using the variable 'x'. Hence ﬁnd the number. 14 [AS4] The present age of Ramu’s father is three times that of Ramu. After ﬁve years, the sum of their ages would be 70 years. Find their present ages. 15 [AS5] An altitude of a triangle is ﬁve –thirds the length of its corresponding base. If the altitude was increased by 4 cm and the base was decreased by 2 cm, the area of the triangle would remain the same. Represent these conditions in the form of linear equations and solve them. 16 [AS5] 50 kg of an alloy of lead and tin contains 60% lead. How much lead must be melted into it to make an alloy that contains 75% lead? Represent as a linear equation and solve it. EXERCISE 2.2. LINEAR EQUATIONS IN ONE VARIABLE–APPLICATIONS 18

EXERCISE 2.3 SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE SIDES 2.3.1 Key Concepts i. Equations that have variables on both sides can be simpliﬁed and solved using the rules for solving linear equations. 2.3.2 Additional Questions Objective Questions 1. [AS1] If 2x − 7 = x + 3 then x = . (A) 21 (B) 4 (D) 10 (C) 21 2 2. [AS1] If 3x − 5 = x − 12 then x = . (A) 17 (B) −7 2 (C) 7 (D) –17 2 3. [AS1] If 7x = 9x then x = . 5 (A) 21 (B) 21 9 (D)None of these (C) 0 4. [AS1] If 3p − 17 = 2p + 20 then p = . (A) 37 (C) −3 (B) 3 (D) −37 EXERCISE 2.3. SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE. . . 19

5. [AS1] If 5m −3= 2m +7 then m = . 4 4 (A) 40 (B) 120 (D) 10 (C) 40 3 6. [AS5] 2x + 3 = 4x − 5 represented with x as subject is . (A) x = 4 (B) 2x = 8 (C)−2x = 8 (D)None of these Very Short Answer Type Questions 7 [AS1] Answer the following questions in one sentence. (i) Solve : 8 − 3x = 5x − 12 22 (ii) Find the value of x : 7x − 4 = 3x + 20 Short Answer Type Questions 8(i) [AS1] Solve: 5x = 3x + 12 (ii) [AS1] Solve: 2x + 3 = x + 10 Long Answer Type Questions 9 [AS1] Solve : 5 (2z − 3) = 2 7 3z − 2 EXERCISE 2.3. SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE. . . 20

EXERCISE 2.4 SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE SIDES –APPLICATIONS 2.4.1 Key Concepts i. Statements involving unknown quantities can be expressed in the form of equations. ii. The unknown quantities are represented by variables. iii. The equations can be solved to ﬁnd the solutions (values of the variables). 2.4.2 Additional Questions Objective Questions . 1. [AS1] On solving 18x = −13x + 62, we get (A) x = 1 (B) x = 2 (C) x = 3 (D) x = 12.4 2. [AS2] The pair of equations y = 0 and y = −5 has solution(s). (A) Two (B) One (C)Inﬁnitely many (D) No 3. [AS1] The value of 'x' in 8x = 6x + 10 is . (A) 4 (B) 5 (C) 2 (D) 8 4. [AS1] The value of 'x' in 7 − x = 2x − 7 is . (B) 5 5 (D) 10 (A) 6 . (C) 8 5. [AS1] The value of 'x' in: x + 7 = 2x + 2 is 3 3 (A) x = 10 (B) x = 15 (C) x = 7 (D) x = 5 EXERCISE 2.4. SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE. . . 21

Very Short Answer Type Questions 6 [AS5] Answer the following questions in one sentence. Represent the condition \"Three fourth of a number added to 10 gives seven –eighth of the same number subtracted from 35” as a linear equation. Short Answer Type Questions 7(i) [AS4] Radha is 40 years old and Sindhu is 4 years old. In how many years will Radha be twice as old as Sindhu? (ii) [AS4] How many kilos of tea worth Rs. 72 per kg should be mixed with 10 kg of tea worth Rs. 90 per kg to produce a mixture which will cost Rs.78 per kg? Long Answer Type Questions 8 [AS1] A number is twice another number. When the larger is subtracted from 50, the result is 2 more than when the smaller is subtracted from 40. Find the numbers. 9 [AS1] The units digit of a two digit number is 3. The number is seven times the sum of its digits. Find the number. 10 [AS1] The ﬁgure given is a square. Find the value of x and also the length of the side of the square. EXERCISE 2.4. SOLVING EQUATION THAT HAS VARIABLES ON BOTH THE. . . 22

EXERCISE 2.5 REDUCING EQUATIONS TO SIMPLER FORM ( LINEAR) 2.5.1 Key Concepts i. Method of cross multiplication: Cross multiplication method is used to solve a linear equation in which the L.H.S. or R.H.S. or both are in fractional form. a. Multiply the numerator of the L.H.S. by the denominator of the R.H.S. b. Multiply the numerator of the R.H.S. by the denominator of the L.H.S. c. Equate these two expressions. d. Solve for the unknown quantity. 2.5.2 Additional Questions Objective Questions 1. [AS4] Two trains run with speeds x kmph and y kmph. If they run in opposite directions on parallel tracks, then the speed to be considered is . (A) (x + y) kmph (B) (x − y) kmph (C)(xy) kmph (D) x km ph y 2. [AS4] If the speed of a boat in still water is x kmph and the speed of the stream is y kmph, then the speed of the boat upstream is . (A) (x + y) kmph (B) (x − y) kmph (C) xy kmph (D) x kmph y 3. [AS2] The numerator of a fraction is 3 less than its denominator. Then the fraction is . (A) x (B) x + 3 x−3 x (C) x(x + 3) (D) x − 3 x−3 x EXERCISE 2.5. REDUCING EQUATIONS TO SIMPLER FORM ( LINEAR) 23

4. [AS2] The speed of a car is x kmph. If the speed of another car is 10 kmph more than twice the speed of the first car, then the speed of the second car is km ph. (A) 10x + 2 (B) 2x + 10 (C) x + 20 (D)2x − 10 5. [AS2] The digit in the tens place of a number is 3 more than twice the digit in its units place. Then the digit in tens place of the number is . (A) 2x + 3 (B) 2x −3 (C)3x + 2 (D)3x − 2 6. [AS2] If the units digit of a number is three more than twice the digit in its tens place, then the maximum value of the digit in tens place is . (A) 1 (B) 2 (C) 3 (D) 4 7. [AS4] If the age of Rajesh is twice the age of Swetha and the sum of their ages is 99 years, then the age of Swetha is . (A) 33 years (B) 99 years (C)66 years (D)22 years Very Short Answer Type Questions 8 [AS3] Answer the following questions in one sentence. The sum of three consecutive integers is 51. Express this as a linear equation. 9 [AS5] Answer the following questions in one sentence. A and B can complete a work in 13 days and 15 days respectively. Represent the equation representing the work done by them together in one day. 10 [AS1] The denominator of a fraction is 3 more than its numerator. If 1 is added to both the numerator and the denominator, the fraction becomes 1 . Find the fraction. 2 11 [AS1] What number when increased by 7% of itself gives 1605? 12 [AS4] A and B can do a piece of work in 6 days. B alone can do the work in 18 days. In how many days can A alone do the same work? EXERCISE 2.5. REDUCING EQUATIONS TO SIMPLER FORM ( LINEAR) 24

13 [AS4] The present ages of A and B are in the ratio 5 : 4. Ten years hence, the ages of A and B will be in the ratio 7 : 6. Find the present ages of A and B. 14 [AS4] A man covers a distance of 25 km in 4 hours partly by walking and partly by running. If he walks at 5 kmph and runs at 7.5 kmph, ﬁnd the distance covered by him while walking. EXERCISE 2.5. REDUCING EQUATIONS TO SIMPLER FORM ( LINEAR) 25

CHAPTER 3 CONSTRUCTION OF QUADRILATERALS EXERCISE 3.1 QUADRILATERALS AND THEIR PROPERTIES 3.1.1 Key Concepts i. A quadrilateral is a closed ﬁgure having four sides, four angles and two diagonals. ii. Sum of the interior angles of a quadrilateral is 360°. iii. To construct a quadrilateral uniquely, we should have the data of at least ﬁve elements. iv. The following data about the ﬁve elements is enough to construct a quadrilateral: a. 4 sides and 1 diagonal b. 3 sides and both diagonals c. 4 sides and 1 angle d. 3 sides and 2 included angles e. 3 angles and 2 included sides v. It is possible to construct a quadrilateral with less than 5 parts, but some other relations between them are to be given. vi. In all cases, it is important to draw a rough diagram and label it as per the measurements given. 3.1.2 Additional Questions Objective Questions 1. [AS3] A simple closed ﬁgure formed by four line segments is a . (A) Triangle (B) Quadrilateral (C) Pentagon (D) Rhombus 2. [AS3] The number of independent measurements required to construct a quadrilateral is . (A) 3 (B) 4 (C) 5 (D) 6 EXERCISE 3.1. QUADRILATERALS AND THEIR PROPERTIES 26

3. [AS3] In a parallelogram, the are equal. (A) Diagonals (B) Adjacent sides (C) Opposite sides (D) None of these 4. [AS3] In a rhombus, the bisect perpendicularly. (A) Diagonals (B) Adjacent sides (C) Opposite sides (D) None of these 5. [AS3] In a trapezium, the are parallel. (A) Adjacent sides (B) Diagonals (C) A pair of opposite sides (D) None of these Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) How does a trapezium differ from a parallelogram? (ii) A quadrilateral has all its four angles of the same measure. What is the measure of each angle? (iii) The sum of two angles of a quadrilateral is 180°. What is the sum of the other two angles? (iv) Two adjacent sides of a parallelogram are 5 m and 8 m respectively. Find its perimeter. (v) One of the diagonals of a rhombus is equal to one of its sides. Find the angles of the rhombus. Short Answer Type Questions 7(i) [AS3] State the properties of a parallelogram. (ii) [AS3] State the properties of a square. Long Answer Type Questions 8 [AS3] Write the steps to construct a rhombus with diagonals 8 cm and 12 cm. 9 [AS5] Construct a quadrilateral ABCD in which AB = 4 cm, BC = 3.5 cm, CD = 5 cm, AD = 5.5 cm and ∠B = 75°. 10 [AS5] Construct a quadrilateral MNOP where MN = NO = 3.4 cm, MP = OP = 5.2 cm and ∠MNO =120 ◦ . EXERCISE 3.1. QUADRILATERALS AND THEIR PROPERTIES 27

EXERCISE 3.2 CONSTRUCTION : LENGTHS OF FOUR SIDES AND ONE DIAGONAL ARE GIVEN 3.2.1 Key Concepts i. To construct a quadrilateral, first draw a rough diagram and label it as per the measurements given. ii. When the lengths of four sides and a diagonal are given, begin the construction with any one side (two vertices). iii. Then, the third vertex is got by the intersection of another side and the diagonal. iv. Then, from the ﬁrst and the third vertices, we can get the fourth vertex. 3.2.2 Additional Questions Long Answer Type Questions 1 [AS5] Construct a quadrilateral ABCD such that AB = 4 cm, BC = 3.5 cm, CD = 4.2 cm, DA = 3 cm and AC = 5 cm. 2 [AS5] Construct a quadrilateral PQRS in which PQ = 4.5 cm, QR = 4 cm, RS = 6.5 cm, SP = 3cm and QS = 6.5 cm. 3 [AS5] Construct a rhombus MNOP in which MO = 5.8 cm and PN = 7 cm. 4 [AS5] Construct a rhombus ABCD whose side AB = 4 cm and one of the diagonals AC = 6 cm. 5 [AS5] Check whether a parallelogram ABCD can be constructed with the measures AB = 4 cm and BC = 3 cm. Construct it if it is possible. 6 [AS5] Construct a parallelogram PQRS in which PQ = 4.5 cm, QR = 3.5 cm and PR = 5.4 cm. EXERCISE 3.2. CONSTRUCTION : LENGTHS OF FOUR SIDES AND ONE DI. . . 28

EXERCISE 3.3 CONSTRUCTION : LENGTHS OF THREE SIDES AND TWO 3.3.1 Key Concepts DIAGONALS ARE GIVEN i. To construct a quadrilateral, first draw a rough diagram and label it as per the measurements given. ii. When the lengths of three sides and two diagonals are given, begin the construction with the middle side. iii. Then, the third vertex is got by the intersection of the second side and one diagonal. iv. Then, the fourth vertex is got by the intersection of the third side and the second diagonal. 3.3.2 Additional Questions Long Answer Type Questions 1 [AS5] Construct a quadrilateral ABCD in which BC = 4.2 cm, CA = 5.8 cm, AD = 4.7 cm, CD = 5.3 cm and BD = 6.7 cm. 2 [AS5] Check if a quadrilateral ABCD can be constructed with the given measures. If yes, construct it. AD = 5 cm, ∠ADC = 60◦, ∠BCD = 75◦, CD = 7 cm and CB = 4.5 cm EXERCISE 3.3. CONSTRUCTION : LENGTHS OF THREE SIDES AND TWO D. . . 29

EXERCISE 3.4 CONSTRUCTION: LENGTHS OF TWO ADJACENT SIDES AND THREE ANGLES ARE GIVEN 3.4.1 Key Concepts i. To construct a quadrilateral, first draw a rough diagram and label it as per the measurements given. ii. When the lengths of two adjacent sides and the measurements of three angles are given, begin the construction with the included side (whose two adjacent angles are known). iii. Then, construct two rays for both the angles and mark the third vertex with the measurement of the second known side on the respective ray. iv. From this vertex, construct a ray with the measurement of the third angle. v. This ray intersects the other ray to give the fourth vertex. 3.4.2 Additional Questions Long Answer Type Questions 2 [AS5] Construct a parallelogram ABCD whose base is 5 cm, side is 6 cm and base angles are 60◦ and 120◦ respectively. 3 [AS5] Construct a rectangle of length 4.5 cm and diagonal 6 cm. 4 [AS5] Construct a parallelogram ABCD, in which AB = 5.4 cm, BC = 3.8 cm and the exterior angle at B is 70 ◦. EXERCISE 3.4. CONSTRUCTION: LENGTHS OF TWO ADJACENT SIDES AND. . . 30

EXERCISE 3.5 CONSTRUCTION : LENGTHS OF THREE SIDES AND TWO INCLUDED ANGLES ARE GIVEN 3.5.1 Key Concepts i. To construct a quadrilateral, first draw a rough diagram and label it as per the measurements given. ii. When the lengths of three sides and the measurements of two included angles are given,begin the construction with the middle side. iii. Then, construct two rays with the measurements of the two included angles. iv. Now cut off the lengths of the other two sides on the respective rays to get the third and fourth vertices. 3.5.2 Additional Questions Long Answer Type Questions 1 [AS5] Construct a quadrilateral ABCD in which BC = 6 cm, CD = 6 cm, DA = 4 cm, ∠C = 80° and ∠D = 50°. 2 [AS5] Construct a trapezium ABCD in which AB is parallel to DC, AB = 8 cm, BC = 4 cm, CD =3.5 cm and ∠B = 40◦ . EXERCISE 3.5. CONSTRUCTION : LENGTHS OF THREE SIDES AND TWO I. . . 31

EXERCISE 3.6 CONSTRUCTION OF SPECIAL TYPES OF QUADRILATERALS 3.6.1 Key Concepts i. To construct a quadrilateral uniquely, we should have the measures of at least ﬁve elements. ii. It is possible to construct a quadrilateral with less than ﬁve measurements, but some other relations between them are to be given. iii. If the quadrilateral is a square, we only need one measurement (side) to construct it. 3.6.2 Additional Questions Long Answer Type Questions 1 [AS5] Construct a rhombus ABCD in which two diagonals are 4.6 cm and 3.4 cm respectively. Measure each side of the rhombus. 2 [AS5] Construct a square of side 6 cm. EXERCISE 3.6. CONSTRUCTION OF SPECIAL TYPES OF QUADRILATERALS 32

CHAPTER 4 EXPONENTS AND POWERS EXERCISE 4.1 POWERS WITH NEGATIVE EXPONENTS 4.1.1 Key Concepts i. Very large numbers are easy to read when written in exponential form. e.g.: 100000 = 10 5 . ii. When a number is multiplied by itself for many number of times, then we can write it in exponential form. e.g.: 5 × 5 × 5 × 5 = 54, a × a × a . . . .(m times) = am . Here ‘a’ is called the base and ‘m’is called the exponent. iii. (−a)n = an when n is even and − an when ‘n’ is odd. iv. Let a be any rational number and ‘n’ be a positive integer. Then a n = a × a × a × .....n times = an b b b b b bn v. Laws of exponents: Let a, b be any two rational numbers and m, n be any two integers. Then a. a m× a n = a m+n b b b m÷ an a m−n b. a = b c. b am n b mn d. a b = b a a × c n= b n× c n b d d −n bn e. a = a b f. a 0=1 b 4.1.2 Additional Questions Objective Questions 1. [AS3] 3 × 3 × 3 × 2 × 2 × 2 × 2 expressed in exponential form is . (A) 35 × 25 (B) 34 × 24 (C)33 × 24 (D)34 × 23 EXERCISE 4.1. POWERS WITH NEGATIVE EXPONENTS 33

2. [AS3] Expressing 196 as the product of prime factors in exponential form gives . (A) 23 × 72 (B) 22 × 72 (C)22 × 73 (D)23 × 73 3. [AS1] 2 2 × 32 × 5 = . (A) 90 (B) 900 (C) 360 (D) 180 4. [AS1] The value of 6 −3 is . (A) 216 (B) 1 216 (C) 36 (D) 1 36 5. [AS3] Expressing 3.0245 × 10−8 in usual form gives . (A) 0.00030245 (B) 0.000030245 (C) 0.000000030245 (D) 0.000000000030245 6. [AS1] The value of (−3)2×2−1 is . (A) –3 (B) –9 (C) –27 (D) –54 (A) 9 (B) − 9 (C) 1 (D) − 1 9 9 (A) −5 −4 (B) 5 −4 2 2 5 4 (D) −5 4 3 2 (C) EXERCISE 4.1. POWERS WITH NEGATIVE EXPONENTS 34

(A) 3 (B) 6 (C)− 6 (D)− 3 14. [AS3] The equivalent of 4− 4 with the base 2 is . (A) 6 (B) 2−8 2 (C) 28 (D) 2−6 15. [AS4] The size of a plant cell is 0.00001275 metres. Its exponentialform is m. (A) 1.275 × 10−6 (B) 1.275 × 10−5 (C)1.275 × 106 (D)1.275 × 107 EXERCISE 4.1. POWERS WITH NEGATIVE EXPONENTS 35

16. [AS4] The charge of an electron is 0.0000000000000000016 coulombs. Its exponential form is coulombs. (A) 1.6 × 10−18 (B) 1.6 × 1018 (C)1.6 × 1017 (D)1.6 × 10−17 17. [AS4] The diameter of the Sun is 1.4 × 109metres. Its usual form is . (A) 1400000 metres (B) 14000000000 metres (C)1400000000 metres (D)140000000000 metres Very Short Answer Type Questions 18 [AS1] Answer the following questions in one sentence. (i) Find the value of the following: −2 . (27) 3 (ii) The value of (−2)−5 is (iii) Find the value of −5 −1 3 . −2 (iv) Find the value of: (512) 9 19 [AS2] Answer the following questions in one sentence. (i) Can (32)3 be written as 323? Support your answer. (ii) By what number should we divide (−15)−1 so that the quotient may be equal to (−15)−1 ? (iii) Is 35 = 38 ? Justify your answer. 33 (iv) Is 2−3 can be written as 1 ? Give support to your answer. (−2)3 (v) Is 70 can be written as 7 ? Give support to your answer. 20 [AS3] Answer the following questions in one sentence. (i) Express the number 0.000000625 in exponential form. EXERCISE 4.1. POWERS WITH NEGATIVE EXPONENTS 36

(ii) Express the following in exponential form: √5 35 (iii) Express 0.0000000523 in exponential form. (iv) Express the following in exponential form: 3 1252 21 [AS4] Answer the following questions in one sentence. (i) The mass of a molecule of hydrogen gas is about 3.34 × 10−21 tons. Write this as a decimal. (ii) A helium atom has a diameter of 2.2 × 10−8cm. Write this as a decimal. (iii) Pluto is at a distance of 5913000000 m from the Sun. Write this in exponential form. Short Answer Type Questions 22 [AS1] Simplify : 1 −3 1 −3 3 −3 ×× 344 23 (i) [AS1] Find the value of m for which 5m ÷ 5−3 = 55. (ii) [AS1] Find m so that (−3)m+1 × (−3)5 = (−3)7 . 24(i) [AS2] By what number should 5 −1be multiplied so that the product may be equal to 53 ? (ii) [AS2] By what number should (−64)−1 be divided so that the quotient may be 4−1 ? 25(i) [AS3] Express(−4)4× (−4)−10 in exponential form with a positive exponent. (ii) [AS3] Express 43 × 3 3 as a power of a rational number with a negative exponent. −4 26(i) [AS4] An electron’s mass is approximately 9.1093826 × 10−31 kilograms. What is this mass in grams. (ii) [AS4] The cells of bacteria double itself in every hour. Find the number of cells after 8 hours, if initially there was 1 cell. Express the answer in powers. Long Answer Type Questions 27 [AS1] Find the values of the following expressions: (i) 14 15 1 6 × × 222 × 215 EXERCISE 4.1. POWERS WITH NEGATIVE EXPONENTS 37

(ii) 5 −7 8 −5 × 85 61 (iii) (27)5 ÷ (27) 5 (iv) 6−1 − 8−1 −1 + 2−1 − 3−1 −1 (v) −2 −4 1 −4 × 38 28 [AS1] Simplify: (i) 1 −2 1 −2 1 −2 3 −1−1 4 3 (ii)6−1 + + + 2 2 29 [AS1] Evaluate: (i) 1 −1 1 −1−1 3 4 − (ii) 3−5 × 10−5 × 125 5−7 × 6−5 30 [AS2] (i) By what number should 1 −1 be multiplied so that the product may be equal to −4 −1 ? 27 (ii) By what number should (−15)−2 be divided so that the quotient may be equal to (−5)−2 ? 31 [AS2] By what number should be (−8)−2 be multiplied so that the product may be equal to (−6)−2? 32 [AS2] Show that (i) −1 −3 −1 −5 = −1 −3+5 (ii) 4 −23 7 23 ÷ 7 = 4 44 4 33 [AS3] Express as a power of a rational number with a positive exponent. (i) 4 −2 4 (ii) 3 3 ÷ 38 × 32 5 7 7 7 34 [AS4] In a stack there are 5 books each of thickness 20 mm and 5 paper sheets each of thickness 0.016 mm. What is the total thickness of the stack? EXERCISE 4.1. POWERS WITH NEGATIVE EXPONENTS 38

EXERCISE 4.2 APPLICATION OF EXPONENTS TO EXPRESS NUMBERS IN STANDARD FORM 4.2.1 Key Concepts i. Number in standard form: A number written as m × 10n is said to be in standard form if ‘m’ is a decimal number (where,1 ≤ m ≤ 9) and ‘n’ is a positive or negative integer. 5 e.g.: 150000 = 1.5 × 10 ii. Very small numbers can be expressed in standard form using negative exponents. e.g.: 0.000000625 = 6.25 × 10 −7 iii. If ‘a’ is a non–zero rational number and ‘n’ is a positive integer, then i. an = a × a × a . . . .n times ii. a−n = 1 an iii. a0 = 1 4.2.2 Additional Questions Objective Questions 1. [AS3] The number 235.324 in the standard form is . (A) 2.35324 × 10 5 (B) 2.35324 × 102 (C)2.35324 × 10−2 (D)2.35324 × 10−5 2. [AS3] The general form of the number 7.6489 × 10−10 is . (A) 0.0000074689 (B) 0.00000000076489 (C) 0.000000074689 (D) 0.00000074689 3. [AS3] 0.000000000000003024586 in standard form is . (A) 3.024586 × 10 −10 (B) 3.024586 × 10−12 (C)3.024586 × 10−15 (D)3.024586 × 10−18 4. [AS3] 9.020504603 × 1015 in general form is . (A) 9020504603000000000 (B) 902050460300000 (C) 902050460300000000 (D) 9020504603000000 EXERCISE 4.2. APPLICATION OF EXPONENTS TO EXPRESS NUMBERS IN . . . 39

5. [AS3] 3425.861 × 10−5 in general form is . (A) 0.03425861 (B) 0.0003425861 (C) 0.0000003425861 (D) 3.425861 Very Short Answer Type Questions 6 [AS3] Answer the following questions in one sentence. (i) Express 0.000000061 in the standard form. (ii) Express 15360000000000 in the standard form. (iii) Express 1.596 × 10−6 in the usual form. Short Answer Type Questions 7(i) [AS2] Observe the procedure of solving the problem given and rectify the errors. Factorize 432: 432 = 2 × 216 = 2 × 2 × 18 =2×2×2×9 =2×2×2×3×3 (ii) [AS2] Observe the following, identify and rectify the errors. Factorize 560: 560 = 2 × 230 = 2 × 2 × 115 = 2 × 2 × 5 × 23 8(i) [AS3] Express the following numbers in the usual form: a) 4.37 × 105 b) 3.65 × 10 7 EXERCISE 4.2. APPLICATION OF EXPONENTS TO EXPRESS NUMBERS IN . . . 40

(ii) [AS3] Express the following numbers in the usual form: a) 5.8 × 10 7 b) 32.5 × 10 −4 c) 3.71529 × 10 7 Long Answer Type Questions 9 [AS3] Write the following numbers in the standard form. (i) 0.0000456 (ii) 0.0000000125 (iii) 0.0000000085 (iv) 35400000000 (v) 0.00000675 ×105 10 [AS4] The distance between the sun and the earth is (1.496 × 1011) m and the distance between the earth and the moon is (3.84 × 10 8 ) m. During a solar eclipse, if the moon comes in between the earth and the sun, what is the distance between the moon and the sun? EXERCISE 4.2. APPLICATION OF EXPONENTS TO EXPRESS NUMBERS IN . . . 41

CHAPTER 5 COMPARING QUANTITIES USING PROPORTION EXERCISE 5.1 COMPOUND RATIO 5.1.1 Key Concepts i. Compound ratio of the given two simple ratios is expressed like a simple ratio as the product of antecedents to the product of consequents. ii. If a : b and c : d are any two ratios, then their compound ratio is: a × c = ac i.e., ac : bd. b d bd iii. A percentage compares numbers to 100. The word percent means per every hundred. 5.1.2 Additional Questions Objective Questions 1. [AS1] The ratio 6 : 8 in lowest terms is . (A) 4 : 3 (B) 3 : 4 (C)6 : 1 (D)1 : 8 2. [AS1] The equivalent ratio of 5 : 6 is . (A) 15 : 12 (B) 10 : 18 (C)6 : 5 (D)15 : 18 3. [AS1] The compound ratio of 2 : 3 and 4 : 5 is . (A) 6 : 8 (B) 8 : 8 (C)8 : 15 (D)6 : 15 EXERCISE 5.1. COMPOUND RATIO 42

4. [AS1] The compound ratio of 4 : 5 and 3 : p is 3 : 10. Then the value of p is . (A) 8 (B) 10 (C) 6 (D) 12 5. [AS1] If an amount of Rs. 300 is divided in the ratio 3 : 7, then the smaller portion is . (A) Rs. 100 (B) Rs. 90 (C)Rs. 210 (D)Rs. 30 Short Answer Type Questions 6(i) [AS1] Find the compound ratio of 6 : 10 and 20 : 15 and express it in lowest terms. (ii) [AS1] The compound ratio of 4 : 7 and 3 : 5 is the same as the compound ratio of 3 : 7 and 2 : p. Find the value of p. 7(i) [AS1] The compound ratio of 8 : 9 and 7 : p is 14 : 27. Find the value of p. (ii) [AS1] The compound ratio of q : 17 and 6 : 5 is 36 : 51. Find the value of q. 8(i) [AS5] Observe the picture given and ﬁnd the ratio of shaded parts to unshaded parts. EXERCISE 5.1. COMPOUND RATIO 43

(ii) [AS5] Observe the picture given and ﬁnd the ratio of shaded parts to the unshaded parts. Long Answer Type Questions 9 [AS4] Anitha scored 320 marks out of 400 marks and Rita scored 300 marks out of 360 marks. Whose performance is better? 10 [AS4] The population of a town in 2011 is 56, 40, 000 out of which 5, 40, 000 are children below 18 years and 21, 25, 000 are women. Find the ratio of men to women in that village and also ﬁnd the ratio of children to the total population. 11 [AS4] In a gram panchayat elections the winning candidate secured 560 votes majority over the opponent losing candidate. If there are 7680 people in the village of which 120 members did not vote, ﬁnd the ratio of the votes secured by winning candidate to those secured by the lost candidate. EXERCISE 5.1. COMPOUND RATIO 44

EXERCISE 5.2 FINDING DISCOUNTS 5.2.1 Key Concepts i. Discount is a decrease percent of market price. ii. Price reduction is called rebate or discount. It is calculated on marked price or list price. iii. Proﬁt or loss is always calculated on cost price. iv. VAT is charged by the shopkeeper from the customer and given to the state government. It is charged on selling price of the item and is added to the value of the bill. v. VAT will be on the selling price of an item and will be included in the bill. vi. Proﬁt is an example of increase percent of cost price and loss is an example of decrease percent of cost price. vii. VAT is an increase percent on selling price. 5.2.2 Additional Questions Objective Questions 1. [AS1] The marked price of an article is Rs. 1060 and the shop keeper sold it for Rs. 945. Then the discount given is . (A) Rs. 100 (B) Rs. 150 (C)Rs. 115 (D)Rs. 125 2. [AS1] If S P = Rs. 725 and CP = Rs. 560 then proﬁt is . (A) Rs. 165 (B) Rs. 240 (C)Rs. 75 (D)None of these 3. [AS1] If the marked price of an article is Rs. 250 and its selling price is Rs. 190, then the discount is . (A) Rs. 60 (B) Rs. 440 (C)Rs. 50 (D)Rs. 500 EXERCISE 5.2. FINDING DISCOUNTS 45

4. [AS1] If the proﬁt is Rs. 60 on an article that costs Rs. 600, then the proﬁt percentage is . (A) 11 % (B) 10% (C) 9% (D) 15% 5. [AS1] On a total bill of Rs. 8500 a discount of 18% was given. Discount is . (A) Rs. 1200 (B) Rs. 1000 (C) Rs.1500 (D) Rs. 1530 6. [AS1] A dealer lists his articles at 20% above cost price and allows a discount of 10%. His gain percent is . (A) 10% (B) 8% (C) 9% (D) 8 1 % 4 7. [AS2] The percentage above the cost price that an article must be the marked so as to gain 33% after allowing a discount of 5 % is . (A) 35% (B) 38% (C) 40% (D) 42% 8. [AS3] The formula to ﬁnd discount % is . (A) M.P × 100% discount (B) discount × 100% M.P (C) Pr o f it × 100% C.P (D) 100 × M.P discount 9. [AS4] At a clearance sale, all goods are on sale at 30% discount. If I buy a skirt marked at Rs. 500, the amount I need to pay is . (A) Rs.300 (B) Rs.450 (C) Rs.250 (D) Rs.350 Very Short Answer Type Questions 10 [AS1] Answer the following questions in one sentence. (i) M.P = Rs.1300, Discount = 20%. Find the discount. (ii) M.P = Rs. 900, Discount = Rs. 150. Find the SP . EXERCISE 5.2. FINDING DISCOUNTS 46

11 [AS4] Answer the following questions in one sentence. A pair of shoes marked at Rs. 800. A shop gives 25% discount. Find the discount. 12 [AS4] Answer the following questions in one sentence. (i) What should be the basic price of a watch, whose price including 10% VAT is Rs. 800? (ii) What would be the sale price of a dress marked at Rs. 160 if a discount of 20% is given on it? (iii) What would be the buying price of a shirt at Rs.50, when 5% of ST (sales tax) is added on the purchase? 13(i) [AS2] Five times a number is a 400% increase in the number. If we take one ﬁfth of the number what would be the decrease in percent? (ii) [AS2] By what percent is Rs. 3000 less than Rs. 3600? Is it the same as the percent by which Rs. 3600 is more than Rs. 3000? 14(i) [AS4] A packet of spices is marked 30% above the cost price. If the shopkeeper allows a discountof 20% on the marked price, what is his gain or loss percent? (ii) [AS4] The selling price of a toy car is Rs. 540. If the proﬁt made by shopkeeper is 20%, what is the cost price of this toy? 15(i) [AS4] An article is sold for Rs. 748 at a loss of 6 1 %. Find its cost price. 2 (ii) [AS4] A dress with marked price Rs. 360 is sold at a discount of 10%. If the shopkeeper still makes a proﬁt of 20%, what is the cost price of the dress? 16(i) [AS4] The price of a computer is Rs. 18, 000. ST(service tax) is charged on it at the rate of 12%. Find the amount which the customer pays. (ii) [AS4] Mr. Sharma purchased some electronic items for Rs. 972 including 8% VAT. Find their price before VAT was added. 17 [AS2] Show that two successive discounts of 20% and 10% is equal to a single discount of 28%. 18 [AS4] The list price of a frock is Rs. 220. A discount of 20% is announced on sales. What is the amount of discount on its sale price? 19 [AS4] A washing machine with marked price Rs. 8500 is available at successive discounts of 10% and 5%. What is its selling price? EXERCISE 5.2. FINDING DISCOUNTS 47

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