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, , ,  , p, , where p and q are, q, integers and q ¹ 0, is called a rational number., , • A number that can be expressed in the form, , p, is said, q, to be in the lowest form or simplest form if p and q have no common, factor other than 1 and q ¹ 0., , • Lowest form of a rational number – A rational number, , Addition, subtraction, multiplication and division of rational, numbers are done in the same way as we do for fractions., • Rational numbers are closed under the operations of addition,, subtraction and multiplication., • The operations of addition and multiplication for rational numbers, are, (i) commutative,, , (ii) associative, , • The rational number 0 is the additive identity for rational numbers., • The rational number 1 is the multiplicative identity for rational, numbers., , 12/04/18

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, • The additive inverse of the rational number, , a, –a, is, and vice-versa., b, b, , • The reciprocal or multiplicative inverse of the rational number, is, , a, b, , a c, c, if × = 1., b d, d, , • Distributivity of rational numbers – For all rational numbers a,, b and c, a (b + c) = ab + ac, a (b – c) = ab – ac, • Rational numbers can be represented on a number line., • Between any two given rational numbers there are infinitely many, rational numbers. The idea of mean helps us to find rational, numbers between two given rational numbers., , , In examples 1 to 3, there are four options out of which one is correct., Choose the correct answer., Example 1 : Which of the following is not true?, (a), , 2 5 5 2, + = +, 3 4 4 3, , (b), , 2 5 5 2, − = −, 3 4 4 3, , (c), , 2 5 5 2, × = ×, 3 4 4 3, , (d), , 2 5 2 4, ÷ = ×, 3 4 3 5, , Solution, , : The correct answer is (b)., , Example 2, , : Multiplicative inverse of, , Solution, , (a) 1, (b) –1, : The correct answer is (d)., , 0, is, 1, , (c) 0, , (d) not defined, , −3, , Example 3, , 1, , : Three rational numbers lying between 4 and, are, 2, (a) −, , 1, 3, , 0,, 2, 4, , (b), , –1 1 3, , ,, 4 4 4, ,  , 12/04/18

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, –1, 1, ,0,, 4, 4, , (c), Solution, , (d), , –5, 1, ,0,, 4, 4, , : The correct answer is (c)., , In examples 4 and 5, fill in the blanks to make the statements true., Example 4, , : The product of a non-zero rational number and its, reciprocal is ________., , Solution, , : 1, , Example 5, , : If x =, , Solution, , :, , 1, 6, y, and y =, then xy − = _______., 3, 7, x, , −16, 7, , In examples 6 and 7, state whether the given statements are true or, false., Example 6, , : Every rational number has a reciprocal., , Solution, , : False, , Example 7, , :, , Solution, , : True, , Example 8, , : Find, , Solution, , :, , −5, −4, is, larger, than, 4 ., 5, , 4 14 2, ×, ÷ ., 7, 3, 3, , 4 14 2, 4  14 3 , ×, ÷ = × × , 7,  3 2, 7 3 3, , =, , 4, ×7 = 4, 7, 2, , Example 9, , : Using appropriate properties, find 3 ×, , Solution, , :, , −5, 7 2, −2, + + ×, ., 7, 3 3, 7, , 2  −5  7 2  −2 , ×,  + + ×, , 3  7  3 3  7 , , =, , −5 2 2 2 7, × − × +, 7 3 7 3 3, ,  , , 12/04/18

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,  −5 2  2 7, − × +, = ,  7 7 3 3, , = −, , 2 7 5, + =, 3 3 3, , Example 10 : Let O, P and Z represent the numbers 0, 3 and -5, respectively on the number line. Points Q, R and S are, between O and P such that OQ = QR = RS = SP., What are the rational numbers represented by the points, Q, R and S. Next choose a point T between Z and O so, that ZT = TO. Which rational number does T represent?, Z, , Solution, , :, , –5, , O, –4, , –3, , –2, , –1, , 0, , P, 1, , 2, , 3, , As OQ = QR = RS = SP, and OQ + QR + RS + SP = OP, therefore Q, R and S divide OP into four equal parts., So, R is the mid-point of OP, i.e., , R=, , 0+3 3, =, 2, 2, , Q is the mid-point of OR, i.e., , Q=, , 1, 2, , S=, , 1 3,  9,  + 3 =, 2 2,  4, , T=, , 0 + (–5) –5, =, 2, 2, , and S is the mid-point of RP, i.e., , 3 3, , 0 +  =, 2 4, , , 3, 3, 9, therefore, Q = , R = and S=, 4, 2, 4, , Also Z T = TO, So,, , T is the mid-point of OZ, i.e., ,   , 1. Explain the first step in solving an addition equation with fractions, having like denominators., 2. Explain the first step in solving an addition equation with fractions, having unlike denominators., ,  , 12/04/18

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, 4, ha. He wants to divide it, 5, equally among his one son and two daughters. Find the, area of each one’s share., (ha means hectare; 1 hectare = 10,000 m2), , Example 11 : A farmer has a field of area 49, , Solution, , : 49, , 4, 249, ha =, ha, 5, 5, , Each share =, , 1 249, 83, 3, ×, ha =, ha = 16 ha, 3, 5, 5, 5, , 2, 4, Example 12 : Let a, b, c be the three rational numbers where a = , b =, 3, 5, , and c = −, , Solution, , 5, 6, , Verify:, (i) a + (b + c) = (a + b) +c (Associative property of addition), (ii) a × (b × c) = (a × b) × c (Associative property of, multiplication), : (i) L.H.S, = a + (b +c), =, , 2  4  −5  , +, +, , 3  5  6  , , =, , 2  24 − 25 , +, 3  30 , , =, , 2  −1 , +, , 3  30 , , 20 − 1 19, =, 30, 30, R.H.S. of (i) = (a + b) + c, , =, ,  2 4   −5 , =  + +, , 3 5  6 ,  10 + 12   −5 , = , +, ,  15   6 , , =, So,, , 22 5 44 − 25 19, − =, =, 15 6, 30, 30, , 2  4  −5    2 4   −5 , +, +,  = + +, , 3  5  2    3 5   6 ,  , , 12/04/18

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, Hence verified., (ii) L.H.S = a × (b × c), =, , R.H.S., , 2  4  −5  , ×, ×, , 3  5  6  , =, , 2  −20  2  −2 , ×,  = ×, , 3  30  3  3 , , =, , 2 × ( −2) −4, =, 3×3, 9, , = (a × b) × c,  2 4   −5 , =  × ×, , 3 5  6 , , =, , 2 × 4 −5, ×, 3×5 6, , =, , 8  −5 , ×, , 15  6 , , =, , 8 × ( −5) − 40 − 4, =, =, 15 × 6, 90, 9, , 2  4  −5    2 4   −5 , ×, ×, ×, ×,  =, , 3  5  6    3 5   6 , , So,, , Example 13 : Solve the following questions and write your observations., , Solution, , (i), , 5, +0=?, 3, , (ii), , −2, +0=?, 5, , (iii), , 3, +0=?, 7, , (iv), , 2, ×1=?, 3, , (v), , −6, ×1=?, 7, , (vi), , 9, ×1=?, 8, , (ii), , −2, −2, +0=, 5, 5, , (iii), , 3, 3, +0=, 7, 7, , : (i), , 5, 5, +0=, 3, 3, , Rational Numbers, Integers, Whole Numbers, ,  , 12/04/18

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, (iv), , 2, 2, ×1=, 3, 3, , (v), , −6, −6, ×1 =, 7, 7, , (vi), , 9, 9, ×1=, 8, 8, , Observation, From (i) to (iii), we observe that: (i) When we add 0 to a rational number we, get the same rational number., From (iv) to (vi), we observe that: (ii) When we multiply a rational number, by 1 we get the same rational number., (iii) Therefore, 0 is the additive identity of rational numbers and 1 is the, ‘multiplicative identity’ of rational numbers., Example 14 : Write any 5 rational numbers between, Solution, , :, , 7, −5, and ., 6, 8, , −5 −5 × 4 −20, =, =, 6, 6×4, 24, 7 7 × 3 21, =, =, 8 8 × 3 24, , and, , Thus, rational numbers, between, , −19 −18 −17, 20, ,, ,, ,....., 24 24 24, 24, , lie, , −5, 7, and ., 6, 8, , Example 15 : Identify the rational number which is different, from the other three :, Solution, , :, , 2 −4 1 1, ,, , , . Explain your reasoning., 3 5 2 3, , −4, 5 is the rational number which is different from the, , other three, as it lies on the left side of zero while others, lie on the right side of zero on the number line., Example 16 : Problem Solving Strategies, Problem, , : The product of two rational numbers is –7. If one of the, number is –10, find the other., , Solution, , : Understand and explore, • What information is given in the question?, One of the two rational numbers, Product of two rational numbers, • What are you finding?, The other rational number,  , , 12/04/18

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, Plan a strategy, • Let the unknown rational number be x. Form an, equation with the conditions given. Then solve the, equation., Solve, Let the other rational number be x, –10 × x = –7, x=, , –7, 7, ,x=, –10, 10, , Check, –10 ×, , 7, = –7. Hence, the result is correct., 10, ,   , Some other easier ways to find the answer., Is the product greater than both the rational numb or less than both the, rational numbers?, , , Focus on Graphic Organisers, You can use an information frame to organize information about a, mathematical concept or property, such as the commutative property of, addition., , WORDS, The order in which you add two, numbers does not change the sum, EXAMPLE, 3+5=5+3, , COMMUTATIVE PROPERTY, OF ADDITION, , ALGEBRA, a+b=b+a, , VOCABULARY HELP, The word commute means travel, to move, , Make an information frame for the distributive property.,  , 12/04/18

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, , , , , In questions 1 to 25, there are four options out of which one is correct., Choose the correct answer., p, 1. A number which can be expressed as, where p and q are integers, q, and q ≠ 0 is, (a) natural number., (b) whole number., (c) integer., (d) rational number., , p, is said to be a rational number if, q, and q are integers., and q are integers and q ≠ 0, and q are integers and p ≠ 0, and q are integers and p ≠ 0 also q ≠ 0., , 2. A number of the form, (a), (b), (c), (d), , p, p, p, p, , 3 ( −5) −19, +, =, shows that, 8, 7, 56, (a) rational numbers are closed under addition., , 3. The numerical expression, , (b) rational numbers are not closed under addition., (c) rational numbers are closed under multiplication., (d) addition of rational numbers is not commutative., 4. Which of the following is not true?, (a) rational numbers are closed under addition., (b) rational numbers are closed under subtraction., (c) rational numbers are closed under multiplication., (d) rational numbers are closed under division., 3 1 1  −3 , + = +,  is an example to show that, 8 7 7  8 , (a) addition of rational numbers is commutative., , 5. −, , (b) rational numbers are closed under addition., (c) addition of rational number is associative., (d) rational numbers are distributive under addition., 6. Which of the following expressions shows that rational numbers are, associative under multiplication., (a), , 2  −6 3, 2 −6  3, ×, ×  =  ×, ×, 3  7 5, 3 7  5,  , , 12/04/18

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, , (b), , 2  −6 3, 2 3 −6 , ×, ×  = × ×, , 3  7 5, 3 5 7 , , (c), , 2  −6 3, 3 2  −6, ×, ×  =  × ×, 3  7 5, 5 3 7, ,  2 −6  3,  −6 2  3, = , × ×, (d)  ×, ×, 3 7  5,  7 3 5, , 7. Zero (0) is, (a) the identity, (b) the identity, (c) the identity, (d) the identity, 8. One (1) is, (a) the identity, (b) the identity, (c) the identity, (d) the identity, , for addition of rational numbers., for subtraction of rational numbers., for multiplication of rational numbers., for division of rational numbers., for addition of rational numbers., for subtraction of rational numbers., for multiplication of rational numbers., for division of rational numbers., , 9. The additive inverse of, , −7, is, 19, , −7, 7, 19, (b), (c), (d), 19, 19, 7, 10. Multiplicative inverse of a negative rational number is, , (a), , (a), (b), (c), (d), 11. If x, (a), (b), (c), (d), , a positive rational number., a negative rational number., 0, 1, + 0 = 0 + x = x, which is rational number, then 0 is called, identity for addition of rational numbers., additive inverse of x., multiplicative inverse of x., reciprocal of x., , 12. To get the product 1, we should multiply, (a), , −19, 7, , 8, 21, , (b), , −8, 21, , (c), , 21, 8, , 8, by, 21, , (d), , −21, 8, ,  , 12/04/18

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, 13. – (–x) is same as, (a) – x, , (b) x, , 14. The multiplicative inverse of −1, , (c), , 1, x, , −1, x, , (d), , 7, −8, , 1, is, 7, , 8, −8, 7, (b), (c), 7, 7, 8, 15. If x be any rational number then x + 0 is equal to, , (a), , (a) x, (b) 0, 16. The reciprocal of 1 is, (a) 1, (b) –1, 17. The reciprocal of –1 is, (a) 1, (b) –1, 18. The reciprocal of 0 is, (a) 1, (b) –1, , (d), , (c) – x, , (d) Not defined, , (c) 0, , (d) Not defined, , (c) 0, , (d) Not defined, , (c) 0, , (d) Not defined, , 19. The reciprocal of any rational number, , p, , where p and q are integers, q, , and q ≠ 0, is, (a), , p, q, , (b) 1, , (c) 0, , (d), , q, p, , 20. If y be the reciprocal of rational number x, then the reciprocal of y, will be, x, y, (a) x, (b) y, (c), (d), y, x, −3  −7 , ×, 21. The reciprocal of,  is, 8  13 , 104, −104, 21, −21, (b), (c), (d), 21, 21, 104, 104, 22. Which of the following is an example of distributive property of, multiplication over addition for rational numbers., , (a), , (a) −, , 1  2  − 4    1 2   1  − 4 , × +, + − ×, = − ×, , 4  3  7    4 3   4  7 , , (b) −, , 1 2  − 4   1 2   − 4 , × +, ×, −,  =, , 4  3  7    4 3   7 , ,  , , 12/04/18

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, , (c) −, , 1 2  − 4   2  1  − 4, × +,   = + − ×, 4 3  7   3  4  7, , (d) −, , 1  2  − 4    2  − 4  1, × +,   =  +,  −, 4  3  7    3  7  4, , 23. Between two given rational numbers, we can find, (a) one and only one rational number., (b) only two rational numbers., (c) only ten rational numbers., (d) infinitely many rational numbers., Plan a strategy, • Some problems contain a lot of information. Read the entire, problem carefully to be sure you understand all the facts., You may need to read it over several times, perhaps aloud so, that you can hear yourself and understand it well., • Then decide which information is most important (prioritise)., Is there any information that is absolutely necessary to solve, the problem? This information is most important., • Finally, put the information in order (sequence). Use, comparison words like before, after, longer, shorter, and so on, to help you. Write down the sequence before you try to solve, the problem., Read the problem given below, and then answer the questions, that follow, • Five friends are standing in line for the opening of a show., They are in line according to their arrival. Shreya arrived 3, minutes after Sachin. Roy took his place in line at 9:01 P.M., He was 1 minute behind Reena and 7 minutes ahead of Shreya., The first person arrived at 9:00 P.M. Babu showed up 6, minutes after the first person. List the time of each person’s, arrival., (a) Whose arrival information helped you determine each, person’s arrival time?, (b) Can you determine the order without the time?, (c) List the friends’ order of arrival from the earliest to the, last., ,  , 12/04/18

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, 24., , x +y, is a rational number., 2, (a) Between x and y, , (b) Less than x and y both., (c) Greater than x and y both., (d) Less than x but greater than y., 25. Which of the following statements is always true?, (a), , x −y, is a rational number between x and y., 2, , (b), , x +y, is a rational number between x and y., 2, , (c), , x ×y, is a rational number between x and y., 2, , (d), , x ÷y, is a rational number between x and y., 2, , In questions 26 to 47, fill in the blanks to make the statements true., 26. The equivalent of, , 5, , whose numerator is 45 is ___________., 7, , 27. The equivalent rational number of, , 7, , whose denominator is 45 is, 9, , ___________., 15, 35, and, , the greater number is __________., 20, 40, 29. The reciprocal of a positive rational number is ___________., , 28. Between the numbers, , 30. The reciprocal of a negative rational number is ___________., 31. Zero has ___________ reciprocal., 32. The numbers ___________ and ___________ are their own reciprocal., 33. If y be the reciprocal of x, then the reciprocal of y2 in terms of x will, be ___________., 34. The reciprocal of, , 2  –4, ×,  is ___________., 5  9 , , 35. (213 × 657)–1 = 213–1 × ___________., 36. The negative of 1 is ___________., ,  , , 12/04/18

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, , Writing Strategy:, Write a Convincing Argument, Your ability to write a convincing, argument proves that you have, understanding of the concept. An, effective argument should include the, following four parts:, (1) A goal, (2) A response to the goal, ,   any, , , , , r, ,  2 and 210 . Fo a l l y, su, , 0, hat u, are 1, r,, Comp m b e r s , w r n u m b e, u, e, s, n, t, a, two, g r e a number, e, h, t, r, t?, g i v e s he greate, ponen, x, e, t, e, using e or as th exception., as, one, the b, least, t, a, Give, , (3) Evidence to support the response, (4) A summary statement, Step 1 : Identify the goal, For any two numbers, explain whether using the greater number as the, base will generally result in a greater number or using it as the exponent., Find one exception., Step 2 : Provide a response to the goal, Using the greater number as the exponent usually gives the greater, number., Step 3 : Provide evidence to support your response, , 2, 0 and r,, 1, r, e, be, mb, he nu eater num w i l l, t, r, o, r, F, t, the g, o n e n ber., Using t h e e x p, r num, s, 1 0 , a n a greate, t i, resul, 100, 2, 10 =, 1024, 0, 21 =, 1024, 100 < 0, 21, 2, 10 <, , rs, umbe r, n, e, h, for t, reate, ption sing the g ent, e, c, x, E, on, U, e exp, d 3., 2 an r, 3, as th a greater, e, numb t result in, o, n, will, er., numb, 32 = 9, 23 = 8, 9>8, 3, 32 > 2, , Step 4 : Summarise your argument, Generally, for any two numbers, using the greater number as the exponent, instead of as the base will result in a greater number., ,  , 12/04/18

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, , 37. For rational numbers, , e, a c e , a c, ×  +  = _________ +, ,, and, we have, f, b d f , b d, , ________., −5, is ________ than –3., 7, 39. There are ________ rational numbers between any two rational, numbers., , 38., , 40. The rational numbers, , 1, −1, and, are on the ________ sides of zero on, 3, 3, , the number line., 41. The negative of a negative rational number is always a ________ rational, number., 42. Rational numbers can be added or multiplied in any __________., 43. The reciprocal of, , −5, is ________., 7, , 44. The multiplicative inverse of, , 4, is _________., 3, , 45. The rational number 10.11 in the from, 46., , p, is _________., q, , 1 2 3 , 1 2 , ×  +  =  ×  + _________., 5 7 8 , 5 7 , , 47. The two rational numbers lying between –2 and –5 with denominator, as 1 are _________ and _________., In each of the following, state whether the statements are true (T) or, false (F)., 48. If, , x, is a rational number, then y is always a whole number., y, , 49. If, , p, is a rational number, then p cannot be equal to zero., q, , 50. If, , r, is a rational number, then s cannot be equal to zero., s, , 51., , 5, 2, lies between, and 1., 6, 3,  , , 12/04/18

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, 52., , 5, 1, lies between and 1., 10, 2, , 53., , −7, lies between –3 and –4., 2, , 54., , 9, lies between 1 and 2., 6, , 55. If a ≠ 0, the multiplicative inverse of, 56. The multiplicative inverse of, 57. The additive inverse of, , 58. If, , a, b, is ., b, a, , −3, 5, is ., 5, 3, , 1, is –2., 2, , x, c, x c, is the additive inverse of, , then, + = 0., y, d, y d, , 59. For every rational number x, x + 1 = x., 60. If, , x, x c, c, is the additive inverse of, , then − = 0, y, y d, d, , 61. The reciprocal of a non-zero rational number, number, , q, is the rational, p, , q, ., p, , 62. If x + y = 0, then –y is known as the negative of x, where x and y are, rational numbers., 63. The negative of the negative of any rational number is the number, itself., 64. The negative of 0 does not exist., 65. The negative of 1 is 1 itself., 66. For all rational numbers x and y, x – y = y – x., 67. For all rational numbers x and y, x × y = y × x., ,  , 12/04/18

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, 68. For every rational number x, x × 0 = x., 69. For every rational numbers x, y and z, x + (y × z) = (x + y) × (x + z)., 70. For all rational numbers a, b and c, a (b + c) = ab + bc., 71. 1 is the only number which is its own reciprocal., 72. –1 is not the reciprocal of any rational number., 73. For any rational number x, x + (–1) = –x., 74. For rational numbers x and y, if x < y then x – y is a positive rational, number., 75. If x and y are negative rational numbers, then so is x + y., 76. Between any two rational numbers there are exactly ten rational, numbers., 77. Rational numbers are closed under addition and multiplication but, not under subtraction., 78. Subtraction of rational number is commutative., 79. −, , 3, is smaller than –2., 4, , 80. 0 is a rational number., 81. All positive rational numbers lie between 0 and 1000., 82. The population of India in 2004 - 05 is a rational number., 83. There are countless rational numbers between, 84. The reciprocal of x –1 is, , 5, 8, and ., 6, 9, , 1, ., x, , 85. The rational number, , 57, lies to the left of zero on the number line., 23, , 86. The rational number, , 7, lies to the right of zero on the number line., −4, , −8, lies neither to the right nor to the left of, −3, zero on the number line., , 87. The rational number, ,  , , 12/04/18

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, , 88. The rational numbers, , 1, and –1 are on the opposite sides of zero on, 2, , the number line., 89. Every fraction is a rational number., 90. Every integer is a rational number., 91. The rational numbers can be represented on the number line., 92. The negative of a negative rational number is a positive rational, number., 93. If x and y are two rational numbers such that x > y, then x – y is, always a positive rational number., 94. 0 is the smallest rational number., 95. Every whole number is an integer., 96. Every whole number is a rational number., 97. 0 is whole number but it is not a rational number., 98. The rational numbers, , 1, 5, and − are on the opposite sides of 0 on, 2, 2, , the number line., 99. Rational numbers can be added (or multiplied) in any order, −4 −6 −6 −4, ×, =, ×, 5 5, 5 5, , 100. Solve the following: Select the rational numbers from the list which, are also the integers., 9 8 7 6 9 8 7 6 5 4 3 3 1 0 –1 –2 −3 −4 −5 −6, , , , , , , , , , , , , , , ,, ,, ,, ,, ,, ,, 4 4 4 4 3 3 3 3 2 2 1 2 1 1 1 1 2 2 2 2, , 101. Select those which can be written as a rational number with, denominator 4 in their lowest form:, 7 64 36 − 16 5 140, ,, ,, ,, ,, ,, 8 16 − 12 17 − 4 28, ,  , 12/04/18

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, 102. Using suitable rearrangement and find the sum:, (a), , 4  − 4  3  −13 , +, + +, , 7  9  7  9 , , (b) −5 +, , 7 3, 5 −4, + + ( −3) +, +, 10 7, 14, 5, , 103. Verify – (– x) = x for, (i) x =, , 3, 5, , (ii) x =, , −7, 9, , (iii) x =, , 13, −15, , 104. Give one example each to show that the rational numbers are closed, under addition, subtraction and multiplication. Are rational numbers, closed under division? Give two examples in support of your answer., 105. Verify the property x + y = y + x of rational numbers by taking, (a) x =, , 1, 1, , y=, 2, 2, , (b) x =, , −2, −5, , y=, 3, 6, , −3, 20, −2, −9, , y=, , y=, (d) x =, 7, 21, 5, 10, 106. Simplify each of the following by using suitable property. Also name, the property., , (c) x =, , 1 1  1, , (a)  ×  +  × 6 , 2 4  2, , ,  1 2  1 2 , −, ×, (b)  ×,  5 15   5 5 , , (c), , −3  3  −5  , × +, , 5  7  6 , , 107. Tell which property allows you to compute, 1 5 7 , 1 5  7, ×  ×  as  ×  ×, 5 6 9, 5 6  9, , 108. Verify the property x × y = y × z of rational numbers by using, 1, (a) x = 7 and y =, 2, , (b) x =, , 9, 2, and y =, 4, 3, , −5, 14, −3, −4, and y =, (d) x =, and y =, 7, 15, 8, 9, 109. Verify the property x × (y × z) = (x × y) × z of rational numbers by, using, , (c) x =, , (a) x = 1, y =, , −1, 1, and z =, 2, 4, , (b) x =, , 2, −3, 1, , y=, and z =, 3, 7, 2, ,  , , 12/04/18

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, (c) x =, , −2, −5, 1, , y=, and z =, 7, 6, 4, , (d) x = 0, y =, , 1, 2, , and What is the name of this property?, 110. Verify the property x × (y + z) = x × y + x × z of rational numbers by, taking., (a) x =, , −1, 3, 1, , y= , z=, 2, 4, 4, , (b) x =, , −1, 2, 3, , y= , z=, 2, 3, 4, , (c) x =, , −2, −4, −7, , y=, , z=, 3, 6, 9, , −1, 2, −3, , y=, , z=, 5, 15, 10, 111. Use the distributivity of multiplication of rational numbers over, addition to simplify, , (d) x =, , (a), , 3  35 10 , ×, +, 5  24 1 , , 2  7 21 , ×, −, 7 16 4 , 112. Simplify, , (c), , (b), , −5  8 16 , ×, +, 4  5 15 , , (d), , 3 8, , ×  − 40 , 4 9, , , (a), , 32 23 22, +, ×, 5 11 15, , (b), , 3 28 14, ×, ÷, 7 15 5, , (c), , 3 −2 −5, +, ×, 7 21 6, , (d), , 7 1, 1, +, −, 8 16 12, , 113. Identify the rational number that does not belong with the other, three. Explain your reasoning, −5 −1 −4 −7, ,, ,, ,, 11 2 9 3, , 114. The cost of, , 19, 171, metres of wire is Rs., . Find the cost of one metre, 4, 2, , of the wire., 115. A train travels, , 1445, 17, km in, hours. Find the speed of the train in, 2, 2, , km/h., ,  , 12/04/18

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, 116. If 16 shirts of equal size can be made out of 24m of cloth, how much, cloth is needed for making one shirt?, 117., , 7, of all the money in Hamid’s bank account is Rs. 77,000. How, 11, much money does Hamid have in his bank account?, 1, 1, m long rope is cut into equal pieces measuring 7 m each., 3, 3, How many such small pieces are these?, , 118. A 117, , 119., , 120., , 1, 1, of the class students are above average,, are average and rest, 6, 4, are below average. If there are 48 students in all, how many students, are below average in the class?, 2, 1, of total number of students of a school come by car while, of, 5, 4, students come by bus to school. All the other students walk to school, 1, walk on their own and the rest are escorted by their, 3, parents. If 224 students come to school walking on their own, how, many students study in that school?, , of which, , 121. Huma, Hubna and Seema received a total of Rs. 2,016 as monthly, allowance from their mother such that Seema gets, gets and Hubna gets 1, , 1, of what Huma, 2, , 2, times Seema’s share. How much money do, 3, , the three sisters get individually?, 122. A mother and her two daughters got a room constructed for, Rs. 62,000. The elder daughter contributes, , 3, of her mother’s, 8, , 1, of her, 2, mother’s share. How much do the three contribute individually?, , contribution while the younger daughter contributes, , 123. Tell which property allows you to compare, 2 3 5, 2 5  3, ×  ×  and  ×  ×, 3 4 7, 3 7  4, ,  , , 12/04/18

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, 124. Name the property used in each of the following., (i) −, (ii) −, , 7 −3 −3 −7, ×, =, ×, 11 5, 5 11, 2  3 −1   −2 3   −2 −1 , ×, +, =, ×, +, ×, 3  4 2   3 4   3, 2 , , (iii), , 1  4  − 4   1 4   − 4 , +, +, +, +,  =, 3  9  3    3 9   3 , , (iv), , −2, −2, 2, +0 =0+, =−, 7, 7, 7, , (v), , 3, 3 3, ×1 = 1 × =, 8, 8 8, , 125. Find the multiplicative inverse of, (i) −1, , 1, 8, , 126. Arrange the numbers, , (ii) 3, , 1 13 5, ,, , in the descending order., 4 16 8, , 127. The product of two rational numbers is, be, , 7, , find the other., 9, , 128. By what numbers should we multiply, be, , 1, 3, , −14, . If one of the numbers, 27, , −15, so that the product may, 20, , −5, ?, 7, , 129. By what number should we multiply, , −8, so that the product may, 13, , be 24?, 130. The product of two rational numbers is –7. If one of the number, is –5, find the other?, 131. Can you find a rational number whose multiplicative inverse is –1?, 132. Find five rational numbers between 0 and 1., 133. Find two rational numbers whose absolute value is, , 1, ., 5, ,  , 12/04/18

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, 134. From a rope 40 metres long, pieces of equal size are cut. If the length, of one piece is, , 10, metre, find the number of such pieces., 3, , 1, metres long rope is cut into 12 equal pieces. What is the length, 2, of each piece?, , 135. 5, , 136. Write the following rational numbers in the descending order., 8 −9 −3, 2, ,, ,, , 0,, 7 8 2, 5, , 137., , Find (i), , 0÷, , 2, 3, , (ii), , 1 −5 −21, ×, ×, 3 7, 10, , 138. On a winter day the temperature at a place in Himachal Pradesh, was –16°C. Convert it in degree Fahrenheit (0F) by using the formula., C F – 32, =, 5, 9, , 139. Find the sum of additive inverse and multiplicative inverse of 7., 1, 140. Find the product of additive inverse and multiplicative inverse of – ., 3, , 141. The diagram shows the wingspans of different species of birds. Use, the diagram to answer the question given below:, , (a) How much longer is the wingspan of an Albatross than the, wingspan of a Sea gull?, (b) How much longer is the wingspan of a Golden eagle than the, wingspan of a Blue jay?, ,  , , 12/04/18

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, 142. Shalini has to cut out circles of diameter 1, strip of dimensions 8, , 1, cm from an aluminium, 4, , 3, 1, cm by 1 cm. How many full circles can, 4, 4, , Shalini cut? Also calculate the wastage of the aluminium strip., , 1, cup of sugar. Another recipe for the, 2, same fruit salad requires 2 tablespoons of sugar. If 1 tablespoon is, , 143. One fruit salad recipe requires, , equivalent to, , 1, cup, how much more sugar does the first recipe, 16, , require?, 144. Four friends had a competition to see how far could they hop on one, foot. The table given shows the distance covered by each., Name, , Distance covered (km), , Seema, , 1, 25, , Nancy, , 1, 32, , Megha, , 1, 40, , Soni, , 1, 20, , (a) How farther did Soni hop than Nancy?, (b) What is the total distance covered by Seema and Megha?, (c) Who walked farther, Nancy or Megha?, 145. The table given below shows the distances, in kilometres, between, four villages of a state. To find the distance between two villages,, ,  , 12/04/18

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, locate the square where the row for one village and the column for, the other village intersect., , (a) Compare the distance between Himgaon and Rawalpur to, Sonapur and Ramgarh?, (b) If you drove from Himgaon to Sonapur and then from Sonapur, to Rawalpur, how far would you drive?, 146. The table shows the portion of some common materials that are, recycled., Material, , Recycled, , Paper, , 5, 11, , Aluminium cans, , 5, 8, , Glass, , 2, 5, , Scrap, , 3, 4, , (a) Is the rational number expressing the amount of paper recycled, 1, 1, more than, or less than ?, 2, 2,  , , 12/04/18

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, 1, ?, 2, (c) Is the quantity of aluminium cans recycled more (or less) than, , (b) Which items have a recycled amount less than, , half of the quantity of aluminium cans?, (d) Arrange the rate of recycling the materials from the greatest to, the smallest., 147. The overall width in cm of several wide-screen televisions are 97.28 cm,, 98, , 4, 1, cm, 98, cm and 97.94 cm. Express these numbers as rational, 9, 25, , numbers in the form, , p, and arrange the widths in ascending order., q, , 2, 148. Roller Coaster at an amusement park is m high. If a new roller, 3, 3, coaster is built that is, times the height of the existing coaster,, 5, what will be the height of the new roller coaster?, , 149. Here is a table which gives the information about the total rainfall, for several months compared to the average monthly rains of a town., p, Write each decimal in the form of rational number ., q, Month, , Above/Below, normal (in cm), , May, , 2.6924, , June, , 0.6096, , July, , – 6.9088, , August, , – 8.636, , 150. The average life expectancies of males for several states are shown in, the table. Express each decimal in the form, , p, and arrange the, q, , states from the least to the greatest male life expectancy., State-wise data are included below; more indicators can be found in, the “FACTFILE” section on the homepage for each state., ,  , 12/04/18

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, State, , Male, , Andhra Pradesh, Assam, Bihar, Gujarat, Haryana, Himachal Pradesh, Karnataka, Kerala, Madhya Pradesh, Maharashtra, Orissa, Punjab, Rajasthan, Tamil Nadu, Uttar Pradesh, West Bengal, , 61.6, 57.1, 60.7, 61.9, 64.1, 65.1, 62.4, 70.6, 56.5, 64.5, 57.6, 66.9, 59.8, 63.7, 58.9, 62.8, , India, , 60.8, , p, form, q, , Lowest terms, , Source: Registrar General of India (2003) SRS Based Abridged Lefe Tables. SRS Analytical, Studies, Report No. 3 of 2003, New Delhi: Registrar General of India. The data are, for the 1995-99 period; states subsequently divided are therefore included in their, pre-partition states (Chhatisgarh in MP, Uttaranchal in UP and Jharkhand in, Bihar), , 7, 1, cm long has a hem of 3 cm. How long will the, 8, 8, skirt be if the hem is let down?, , 151. A skirt that is 35, , 152. Manavi and Kuber each receives an equal allowance. The table shows, the fraction of their allowance each deposits into his/her saving, account and the fraction each spends at the mall. If allowance of, each is Rs. 1260 find the amount left with each., Where money goes, , Fraction of allowance, Manavi, , Kuber, , Saving Account, , 1, 2, , 1, 3, , Spend at mall, , 1, 4, , 3, 5, , Left over, , ?, , ?,  , , 12/04/18

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,    , 70, , – 21, , 25 24, , 1. Given below is a magic square. Place the numbers 95 , –133 , 95 , 38, in the appropriate squares so that sum in each row, column and, diagonal is equal., , 32, 38, −18, − 57, 22, 38, 1, 19, , 18, 38, , 4, 38, , ?, , ?, , ?, , ?, , −16, − 38, , 45, 57, , − 14, −38, 104, 152, − 20, − 95, 60, 114, , Hint: (Rewrite each rational number in its lowest term.), 2. Solve the given crossword filling up the given boxes. Clues are, given below for across as well as downward filling. Also, for across, and down clues, clue number is written at the corner of the boxes., Answers of clues have to be filled in their respective boxes., Down 1:, , 2, 5, and, are _______ numbers., 3, 4, , Down 2:, , The _______ inverse of, , Down 3:, , The addition and multiplication of whole number integers, and rational numbers is _________., , Down 4:, , Since, , Down 5:, , The number line extends _______ on both the sides., , Down 6:, , The _______ of two integers may not lead to the formation, of another integer., , Down 7:, , The multiplication of a number by its reciprocal, gives_______., , Down 8:, , Rational numbers can be represented on a _______ line., , a, –a, is, ., b, b, , 1, doesn’t exist hence 0 has no ________., 0, ,  , 12/04/18

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, Across 1: There are _______ rational numbers between two integers., Across 2: The multiplication of rational numbers is commutative and, ______., Across 3: The addition and ______ of two rational numbers lead to, the formation of another rational number., Across 4: All the positive integers excluding 0 are known as _______, numbers., Across 5: For any rational a ; a ÷ 0 is _______., Across 6: Reciprocal is also known as the multiplicative, _____________., ,  , , 12/04/18

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, 3. Break the Code, Solve this riddle by reducing each rational number to its lowest, term. The magnitude of the numerator of rational number so obtained, gives you the letter you have to encircle in the word following it. Use, the encircled letters to fill in the blanks given below:, S.No., , Rational Number, , (1), , 12, 30, , SPIN, , (2), , 24, –36, , TYPE, , (3), , 39, 52, , WITH, , (4), , – 48, 144, , HOST, , (5), , 27, 90, , SHARP, , (6), , –34, –170, , GAIN, , (7), , 76, 95, , PROOF, , (8), , 46, 92, , RAIN, , (9), , 29, 116, , AWAY, , (10), , 14, –42, , SWEET, , _____ _____, 1, 2, , _____, 3, , _____, 4, , _____, 5, , Lowest Term, , _____, 6, , _____, 7, , Word, , _____, 8, , ____, 9, , ____, 10, ,  , 12/04/18

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UNIT-1, , ÷, , æ2 1ö, ¸ç - ÷, è5 2ø, , 1, × (– 2 ), , ÷3, 5, , 1, × (–12 ), , 2, 5, , 1 2, 1, ÷ (– ÷ (, 3 5, 2, , × –1, 3, , ), , ×(–2), , –2, , ), , ), , ×, , 4, 1, 6×, 3, 3, , ×, , 3 1, + ÷(–2), 4 12, , ), , 1, 3, , Its, reciprocal, , + Its additive, identity, , × Its multiplicative, inverse, , × Its multiplicative, inverse, , (, , (, × Its additive, inverse, , ÷–3, 4, , (, , (, , (–25 +203, , ÷, , (, , ÷ –1 1, 3, , 2, , ONE, , RATIONAL NUMBERS, , 31, , 12/04/18

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MATHEMATICS, Rough Work, , 32, , EXEMPLAR PROBLEMS, , 12/04/18