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ID : pk-8-Rational-Numbers [1] Grade 8 Rational Numbers For more such worksheets visit www.edugain.com Answer t he quest ions (1) 2 What is the multiplicative inverse of ? 23 (2) If p = q (3) (4) pxm , then what is the relation between m and n? qxn Write the rational number that are equal to its negative. Consider the f ollowing statement a +( b c + d e )=( f a b + c d )+ e . What property is f being described here? Choose correct answer(s) f rom given choice (5) If 8 = -4 -32 , then X = ___________ X a. 12 b. 16 c. -16 d. None of these (6) How many rational numbers are there between any two non equal rational numbers ? a. 30000 b. Inf inite c. 0 d. 433 (7) Which of the statements below is f alse? a. All negative numbers can be represented as rational numbers b. T here is no f raction that cannot be represented as a rational number c. T here is no integer that cannot be d. A decimal number cannot be a rational represented as a rational number number (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [2] (8) What is the largest rational number that can be f ormed using 2 of the numbers below in the p f orm of ? q 47, 163, 340, 127, 202, 48 202 a. b. 47 48 47 c. d. 202 (9) p If 340 47 = q m , then which of the f ollowing is always true: n q a. 340 = p m b. n c. p x n = m x q p n = m q d. p x m = n x q (10) If the sum of both the horizontal and the vertical rows are the same, f ind the missing rational number. 4 31 2 6 31 31 2 31 a. 5 b. 31 c. 4 31 (11) -3 3 31 d. 2 31 is _________________ 0 a. neither positive nor negative rational number b. positive rational number c. negative rational number d. either positive or negative rational number Fill in t he blanks (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [3] (12) What rational number results f rom the f ollowing: A) 4 + 5 × 9 - 7 + 322 ÷ 91 = B) 2 + 5 × 9 - 2 + 164 ÷ 81 = (13) T he rational number that are equal to their reciprocals are & . (14) Fill in the blank to make the two rational numbers equivalent A) 10 = B) 70 16 C) 17 41 (15) 108 = 252 28 D) = 18 T he average of the middle two rational numbers if -3 15 , -4 30 , -4 15 , -4 are arranged in 2 . © 2016 Edugain (www.edugain.com). All Rights Reserved (C) 2016 Edugain (www.Edugain.com) 288 31 492 ascending order is = Many more such worksheets can be generated at www.edugain.com Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [4] Answers (1) 23 2 Step 1 If you look at the question caref ully, you will notice that you have 2 . 23 Step 2 Multiplicative inverse: T he Multiplicative Inverse of a number is its reciprocal. T he product of a number and the reciprocal of a number is 1. Multiplicative inverse of a number n is represented as 1 . n So, the multiplicative inverse of 2 is 23 23 . 2 Step 3 T heref ore the multiplicative inverse of 2 23 (C) 2016 Edugain (www.Edugain.com) is 23 . 2 Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [5] (2) m= n Step 1 If you look at the question caref ully, you will notice that you have the relation p q = pxm qxn . Step 2 Now p = q pxm qxn ⇒p × (q × n) = q × (p × m) ⇒ (p × q) × n = (q × p) × m ⇒ n = m ⇒ n q×p = m ⇒ n p×q p×q p×q =1 m ⇒n=m or ⇒m= n Step 3 T heref ore the relation between m and n is m = n . (3) 0 Step 1 Rational numbers: A rational number is any number that can be expressed as the quotient or f raction p/q of two integers, p and q, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number. Rational number zero(0) is the only rational number that is equal to its negative. Step 2 T heref ore zero(0) is the only rational number that is equal to its negative. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [6] (4) Associative property Step 1 If you look at the question caref ully, you will notice that the f ollowing statement is given a c +( b d + e )=( f a b + c )+ d e . f Step 2 According to the statement sum of a +( b e + c d + e ) is equal to the sum of ( f a b + c ) d . f Which is equal to the associative property. Associative property is states that if you are adding or multiplying it does not matter where you put the parenthesis.In other words, associative property states that you can add or multiply regardless of how the numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. For example : 2 + (3 + 5) = (2 + 3) + 5 10 = 10 Step 3 T heref ore the property of above statement is associative property. (5) b. 16 Step 1 According to the question, we have to f ind the value of X in the given equation: 8 = -32 -4 X Step 2 Now, 8 = -32 -4 X ⇒ 8 × X = -4 × -32 ⇒ 8 X = 128 ⇒X= 128 8 ⇒ X = 16 Step 3 T heref ore the value of X is 16 . (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [7] (6) b. Inf inite Step 1 Rational Number: A rational number is a number that can be expressed as a f raction. A rational number is said to have numerator and denominator . Step 2 For example: 3 or 1.5 , 2 5 or 2.5 2 rational numbers between these two rational numbers are 1.51, 1.52, 1.53, 1.54, 1.52, 1.6 and so on upto inf inite. Step 3 T heref ore, Infinite rational numbers are there between any two non equal rational numbers. (7) d. A decimal number cannot be a rational number Step 1 A rational number is a number which can be written in the f orm p where p and q both are q integers and q is not equal to zero. Step 2 All integers can be written in the f orm of p/1 (where p is any integer). Hence, all integers are rational numbers. T heref ore, the statement "T here is no integer that cannot be represented as a rational number" is true. Step 3 All f ractions are rational numbers as they are represented in p f orm where p and q are q integers. T heref ore, the statement "T here is no fraction that cannot be represented as a rational number" is true. Step 4 All integers (-ve or +ve) can be written in the f orm p/1 where p is the integer. T heref ore, the statement "All negative numbers can be represented as rational numbers" is true. Step 5 A decimal number can be written in p/q f orm, if digits af ter decimal are either terminating or repeating. T heref ore, the statement "A decimal number cannot be a rational number" is false. Step 6 T he f alse statement is "A decimal number cannot be a rational number" and hence, the correct answer is d. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [8] (8) 340 d. 47 Step 1 If you look at the question caref ully, you will notice that you have to f ind out the value of largest rational number that can be f ormed using 2 of the numbers below in the f orm of p q 47, 163, 340, 127, 202, 48. Step 2 Rational number: A rational number is a number that can be written as a ratio. T hat means it can be written as a f raction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. For largest rational number numerator (p) is the largest number and denominator (q) is the smallest number. Now, p = 340 and q = 47 Hence, the largest rational number is 340 . 47 Step 3 T heref ore, the largest rational number that can be f ormed using 2 of the numbers is 340 47 . (9) c. p x n = m x q Step 1 If you look at the question caref ully, you will notice that you have the relation p = q m . n Step 2 You have p = q m n By cross multiply ⇒pxn= mxq Step 3 T heref ore p x n = m x q is always true. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [9] (10) 4 c. 31 Step 1 Let the missing rational number be x. Step 2 2 Sum of the numbers in the horizontal row = + 31 8 = 6 +x 31 +x 31 Step 3 Sum of the numbers in the vertical row = 4 + 31 6 31 + 2 31 12 = 31 Step 4 It is given that the sum of the horizontal row is same as the sum of the vertical row. T hat is, 2 31 ⇒ 6 + +x= 31 8 31 +x= 31 ⇒x= + 6 31 + 2 31 12 31 12 31 ⇒x= 4 - 8 31 4 31 Step 5 T heref ore, the missing rational number is 4 . 31 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [10] (11) a. neither positive nor negative rational number Step 1 Rational numbers: A rational number is any number that can be expressed as the quotient or f raction p/q of two integers, p and q, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number. -3 is not a rational number because denominator of the rational numbers can not be 0 zero(0). Step 2 T heref ore -3 is neither positive nor negative rational number . 0 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [11] (12) A) 592 4 + 5 × 9 - 7 + 322 ÷ 91 = 13 Step 1 In order to solve this expression, we will have to f ollow the correct precedence of operators i.e. Division, Multiplication, Addition and Subtraction (in that order). Step 2 Now, 4 + 5 × 9 - 7 + 322 ÷ 91 can be solved as: 4 +5×9- 7 + 322 91 4 +5×9 - 7 + 46 13 = 4 + 45 - 7 + 46 13 = 49 - 7 + 46 13 = 42 + 46 13 = (42 × 13) + 46 13 = 546 + 46 13 = 592 13 Step 3 T heref ore, the result of 4 + 5 × 9 - 7 + 46 ÷ 91 is 592 . 13 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [12] B) 3809 2 + 5 × 9 - 2 + 164 ÷ 81 = 81 Step 1 In order to solve this expression, we will have to f ollow the correct precedence of operators i.e. Division, Multiplication, Addition and Subtraction (in that order). Step 2 Now, 2 + 5 × 9 - 2 + 164 ÷ 81 can be solved as: 2+5×9- 2+ 164 81 = 2 + 45 - 2 + 164 81 = 47 - 2 + 164 81 164 = 45 + 81 = (45 × 81) + 164 81 = 3645 + 164 81 = 3809 81 Step 3 T heref ore, the result of 2 + 5 × 9 - 2 + 164 ÷ 81 is 3809 . 81 (13) 1 -1 Step 1 Rational numbers: A rational number is any number that can be expressed as the quotient or f raction p/q of two integers, p and q, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number. Rational numbers 1 and -1 are the only numbers which are equal to their reciprocals. Step 2 T heref ore rational numbers 1 and -1 are equal to their reciprocals. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [13] (14) A) 112 T o make the two rational numbers equivalent, f irst of all divide the given greatest numerator/denominator by the smallest numerator/denominator and multiply the numerator/denominator which is in the f ront of blank by the result. Now the number which make the rational numbers 10 and 70 equivalent is 16 112. B) 12 T o make the two rational numbers equivalent, f irst of all divide the given greatest numerator/denominator by the smallest numerator/denominator and divide the numerator/denominator which is in the f ront of blank by the result. Now the number which make the rational numbers and 108 equivalent is 252 28 12. C) 204 T o make the two rational numbers equivalent, f irst of all divide the given greatest numerator/denominator by the smallest numerator/denominator and multiply the numerator/denominator which is in the f ront of blank by the result. Now the number which make the rational numbers 17 and 41 equivalent is 492 204. D) 496 T o make the two rational numbers equivalent, f irst of all divide the given greatest numerator/denominator by the smallest numerator/denominator and multiply the numerator/denominator which is in the f ront of blank by the result. Now the number which make the rational numbers 18 and 288 equivalent is 31 496. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : pk-8-Rational-Numbers [14] (15) -7 30 Step 1 T o compare f ractions, f irst we need to make sure that all denominators are same, so we can just compare the numerators of f ractions. Step 2 T he LCM of the denominators 15, 30, 15 and 2 = 30 Step 3 Now, divide the LCM by the denominators and multiply the result with the numerator and denominator as f ollowing: -3 × 2 -4 × 1 , 15 × 2 -6 or , 30 × 1 -4 , 30 , 30 -4 × 2 15 × 2 -8 , , -4 × 15 2 × 15 -60 30 30 Step 4 Let’s arrange the given numbers in ascending order, we get: -60 , 30 -8 , 30 -6 30 , -4 30 Step 5 -8 Now, the average of the middle two rational numbers = + 30 -6 30 2 = -14 30 = × 1 2 -7 30 Step 6 T hus, the average of the middle two rational numbers is -7 . 30 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited