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1 unit 8 RATIONAL NUMBERSRATIONAL NUMBERSRATIONAL NUMBERSRATIONAL NUMBERSRATIONAL NUMBERS(A)Main Concepts and Results A number that can be expressed in the form pq, where p and q areintegers and 0q , is called a rational number. All integers and fractions are rational numbers. If the numerator and denominator of a rational number are multipliedor divided by a non-zero integer, we get a rational number which issaid to be equivalent to the given rational number. Rational numbers are classified as positive, zero or negative rationalnumbers. When the numerator and denominator both are positiveintegers or both are negative integers, it is a positive rational either the numerator or the denominator is a negative integer,it is a negative rational number.

2 The number 0 is neither a positive nor a negative rational number. There are unlimited number of rational numbers between two rationalnumbers. A rational number is said to be in the standard form, if itsdenominator is a positive integer and the numerator anddenominator have no common factor other than 1. Two rational numbers with the same denominator can be added byadding their numerators, keeping with the same denominator. Two rational numbers with different denominators are added by firsttaking the LCM of the two denominators and then converting boththe rational numbers to their equivalent forms having the LCM asthe denominator and adding them as NUMBERS 233 unit 8 While subtracting two rational numbers, we add the additive inverseof the rational number to be subtracted to the other rational number.

3 Product of rational numbers = Product of numeratorsProduct of denominators The reciprocal of a non-zero rational number pq is qp. To divide one rational number by the other non-zero rational number,we multiply the first rational number by the reciprocal of the other.(B)Solved ExamplesIn Examples 1 to 4, there are four options, out of which one is the correct 1:Which of the following rational numbers is equivalentto23?(a) 32(b)49(c)46(d)94 Solution:Correct answer is (c).Example 2:Which of the following rational numbers is in standardform?(a)2030(b)104(c)12(d)1 3 Solution:Correct answer is (c).Example 3:The sum of 32 and 12 is(a) 1(b) 2(c)4(d)3 Solution:Correct answer is (a).Example 4:The value of 4 1 33 is(a) 2(b) 3(c)2(d) 1 Solution:Correct answer is (d).

4 15-04-2018 MATHEMATICS234 EXEMPLAR PROBLEMSIn Examples 5 and 6, fill in the blanks to make the statements 5:There are _____ number of rational numbers betweentwo rational :UnlimitedExample 6:The rational number _____ is neither positive :0 (Zero).In Examples 7 to 9, state whether the statements are True or 7:In any rational number pq, denominator is always a non-zero need to be actively involved as you work through each lesson in yourtextbook. To begin with, find the lesson s objective given as main conceptsand Strategy: Read a Lesson for UnderstandingLesson Features15-04-2018 RATIONAL NUMBERS 235 unit 8 Example 8: To reduce the rational number to its standard form, wedivide its numerator and denominator by their HCF.

5 9: All rational numbers are integers . 10:List three rational numbers between 45 and :We convert the rational numbers 45 and 56into rationalnumbers with the same 655 6= 2430=; 5 5 56 6 5= 2530= Divided by Quotient Divided into Added to Plus Sum More than Subtracted from Minus Difference Less than Decreased by Multiplied by Times Product Groups ofSolve Choose an OperationTo decide whether to add, subtract, multiply, or divide to solve a problem,you need to determine the action taking place in the EXEMPLAR PROBLEMSSo,2430 24 430 4= and 2530 25 430 4= 96120= 100120=orHere,96120<97120<98120<99120<10 0120.

6 So, the required numbersare 97,1209899and120120 Alternate solution A rational number between45andis561494 52605 6 ==+ another rational number1974 4921205 60 ==+ one more rational number + 738815-04-2018 RATIONAL NUMBERS 237 unit 81993349 52120 4060 6 ===+ Therefore, three rational numbers between 45and56 are49 9733,and60 12040 Note: There can be many set of AND SUBTRACTING WITH LIKE DENOMINATORSW ordsNumbersFormulaTo add or subtractrational numbers withthe same denominator,add or subtract thenumerators and keepthe same ( 4)555 + += 33,or55 =++=ab a bdddExample11:Which of the following pairs represent equivalent rationalnumbers?

7 (i) 728and1248(ii) 2 16and 324 Solution:(i)728and1248 Now, first rational number is 712 and it is already in thestandard form because there is no common factor in7 and 12 other than , 712 is in its standard form(a)Now, Consider 284828 = 2 2 748 = 2 2 2 2 3 HCF = 2 2 = 415-04-2018 MATHEMATICS238 EXEMPLAR PROBLEMSNow, to reduce the rational numbers to its standard form,we divide the numerator and denominator by their we take HCF of 28 and 48:Now, == 28 28 474848 4 12 (b)From (a) and (b), we can say that the rational numbers712 and 2848 are equivalent.(ii) 2 16and 324 First we multiply the numerator and denominator of 2 3by ( 1), we get 2 ( 2) ( 1) 2 3 ( 3) ( 1) 3 == (a)Now it is in its standard , Consider 1624 HCF of 16 and 24 is 2 2 2 = 816 = 2 2 2 224 = 2 2 2 3 HCF = 2 2 2 = 8So, 16 16 8 22424 83 == (b)From (a) and (b)

8 , we can say that the rational numbers 2 16and 324are not numbers or putting numbers togetherTaking away or finding out how far apart two numbers areCombining groupsSplitting things into equal groups or finding how manyequal groups you can makeActionOperationAdditionSubtractionMu ltiplicationDivision15-04-2018 RATIONAL NUMBERS 239 unit 8 Example 12:Write four more rational numbers to complete thepattern: 1 2 3,,369, _____, _____, _____, :By observing the above pattern, we find thatdenominator is multiple of 3. So we will increase thispattern in this way. 2 1 2 3 1 3,63 293 3 == , 4 1 4123 4 = 1 1 1,3 13 = , 1 4 43 4 12 = Thus, we observe a pattern in these , the other numbers would be 1 5 5 1 6 6 1 7 7,,3 5 153 6183 721 === and 1 8 8=3 824 Example 13:Find the sum of 5 46 and 3 :534764 + = 296+ 314 = 296 + 314 DIVIDING RATIONAL NUMBERS IN FRACTION FORMW ordsNumbersAlgebraTo divide by afraction, multiply bythe reciprocal = =1 4 1 557 5 7 4 28aca d adbd bcbc = =15-04-2018 MATHEMATICS240 EXEMPLAR PROBLEMS= 29 2 31 31212 +.

9 [Since LCM of 6 and 4 is 12]. 29 2 31 312 = 58 9312= 15112=So, the required sum is 14:Find the product of 36 2 and :36 11 41 254747 = Now, product of two rational numbers=Product of numeratorsProduct of denominatorsSo, 36 2547 = 11 4147 = 11 414 7 = 45128 Think and an example of two denominators with no common if 1322516 is positive or negative. how to add 212952+, without first writing them as NUMBERS 241 unit 8 Example 15: Match column I to column II in the following:Column I Column II(i)3 34 4 (a) 1(ii)1 42 3 (b) 23(iii)2( 1)3 (c) 32(iv)3 14 2 (d) 38(v) 5577 (e) 1 Solution:(i) (e), (ii) (d), (iii) (b), (iv) (c), (v) (a)Application on Problem Solving StrategyExample 16 Find the reciprocal of 25 : Understand and Explore the Problem What are you trying to find?

10 The reciprocal of the given number. Plan a Strategy You know the division of rational numbers and the meaningof reciprocal. Apply this knowledge to find the and how you can be sure that a fraction is the sign of a rational number in which the numerator is negativeand the denominator is EXEMPLAR PROBLEMS(C)ExerciseIn each of the following questions 1 to 12, there are four options, outof which, only one is correct. Write the correct rational number is defined as a number that can be expressed inthe form pq, where p and q are integers and(a)q = 0(b)q = 1(c)q 1(d) q 0 Solve Given expression = 2511 55 = 211 55 55= 2 Now find out the reciprocal of 2 The reciprocal of 2 is 12.