Rational Numbers

9 RATIONAL NUMBER CONTENTS • • • • For example, 3 = Definition 7 11 5 , , are rational numbers but not 9 13 7 natural numbers. but, Equality of Rational numbers Comparison of Rational numbers Representation of Rational numbers on Number line • (ii) Zero is a rational 0 0    0 = = = ... 1 2   number because (iii) Every integer is a rational number, but every rational number may not be an integer. Rational numbers between two given rational numbers • • 3 5 9 ,5= ,9= and so on. 1 1 1 −2 −5 0 3 5 , , , , etc. are all 1 1 1 1 1 3 −5 rationals, but rationals like , , etc. are 2 2 not integers. For example, Operation on Rational numbers Multiplication of Rational numbers on a number line (iv) Rational numbers can be positive and negative. 2 −7 8 −9 , , , etc. are positive rational 3 − 8 11 − 3 numbers Eg : DEFINITION p , q −2 7 −8 11 , , , etc. are negative rational 3 8 11 − 20 numbers. p is called numerator (Nr) and q is called denominator (Dr) (v) Every positive rational number is greater than zero. Results (vi) Every negative rational number is less than zero. Since every number is divisible by 1, we can say that : (vii) Every positive rational number is greater than every negative rational number. (i) Every natural number is a rational number, but every rational number need not be a natural number. (viii) Every negative rational number is smaller than every positive rational number. A rational number is a number of the form where p, q are integers and q ≠ 0. & EXAMPLES Ex.1 Write down the numerator of each of the following. Also find which is positive and negative rational number. (i) −1 −2 (ii) −3 5 (iii) Write Sol. Multiplying both the numerator 2 denominator of by –28, we have 5 7 −5 −1 in which numerator is –1. It is −2 a positive rational number. (iii) Numerator is 7 and negative rational number. Ex.2 −8 3 (ii) 5 −2 (iii) 5 4 Denominator are 3, –2, 4 respectively. Sol. Equivalent Rational Numbers Q Rational no. can be written with different Nr and Dr. Eg : −5 −5 × 2 −10 = = 7 7×2 14 ∴ −5 −10 is same as 7 14 −5 −5 × 3 −15 = = 7 7×3 21 −5 −15 is same as 7 21 −5 −5 × −1 5 = = 7 7 × −1 − 7 −5 5 is same as 7 −7 Such rational number that are equal to each other are said to be equivalent to each other. Ex.3 Sol. Lowest Form of a Rational Number A rational number is said to be in lowest form if the numerator and the denominator have no common factor other than 1. Write the denominator of each of the following rational numbers : (i) EXAMPLES −5 Express as a rational number with 6 numerator –15. By multiplying both the numerator and −5 denominator of by 3, 6 −5 (−5) × 3 −15 we get, = = 6 6×3 18 Therefore, the required rational number −15 is . 18 and 2 × (–28) −56 = . 5 × ( −28) − 140 Sol. (i) We have (ii) Numerator is –3 and negative rational number. 2 in an equivalent form so that the 5 numerator is equal to –56. Ex.4 EXAMPLE Write the following rational numbers in the lowest form : −36 −64 (i) (ii) 180 256 Sol. (i) Here, HCF of 36 and 180 is 36, therefore, we divide the numerator and denominator of −36 by 36, we have 180 −36 ÷ 36 −1 = 180 ÷ 36 5 −36 −1 So, the lowest form of is 180 5 (ii) Here, HCF of 64 and 256 is 64. Dividing the numerator and denominator of −64 by 64, we have 256 −64 ÷ 64 −1 = 256 ÷ 64 4 −64 −1 So, the lowest form of is . 256 4 Standard Form of a Rational Number p A rational number is said to be in its standard q form if (i) its denominator ‘q’ is positive (ii) the numerator and denominator have no common factor other than 1. 2 −1 5 For example : , , , etc. 3 2 7 Ex.5 EXAMPLES Ex.6 Sol. EQUALITY OF RATIONAL NUMBERS Write the rational number 5 −7 with a positive denominator. 5 5 × (−1) −5 = = − 7 (−7) × (−1) 7 Ex.10 Are the rational numbers which is the required answer. Ex.7 Express the rational number 14 − 21 Sol. in standard form. Sol. Ex.8 Sol. Ex.9 14 . − 21 1. Its denominator is negative. Multiply both the numerator and denominator by – 1 to change it to positive, i.e., 14 14 × (–1) −14 = = − 21 (−21) × (−1) 21 The given rational number is 2. The greatest common divisor of 14 and 21 is 7. Dividing both numerator and denominator by 7, we have −14 ( −14) ÷ 7 −2 = = 21 21 ÷ 7 3 Which is the required answer. −144 Express in the standard form : . − 504 To change in the lowest form, divide both numerator and denominator by g.c.d. of 144 and 504 i.e., 72 144 ÷ 72 2 = , which is the required answer. 504 ÷ 72 7 Find x such that : −1 8 −48 (i) = (ii) = 2. 5 x x −1 8 = , by cross multiplication, 5 x we get (–1) × x = 5 × 8 ⇒ −x = 40 Multiplying both sides by (–1), we get (–x) × (–1) = 40 × (–1) x = –40. −48 −48 2 (ii) Given that = 2 or = , by cross x x 1 multiplication, we get −48 –48 × 1 = 2 × x or x= = −24. 2 Sol. (i) Given that, Method-1 : If two or more rational numbers have the same standard form, we say that the given rational numbers are equal. 8 −50 and equal? − 12 75 We first express these given rational numbers in the standard form. The first rational number is 8 . − 12 (i) Multiplying both the numerator and denominator by –1. We have, 8 8 × (−1) −8 = = − 12 (−12) × ( −1) 12 (ii) Dividing both the numerator and denominator by the greatest common divisor of 8 and 12, which is 4. We have, 8 (−8) ÷ 4  − 2  = =  − 12 12 ÷ 4  3  Again, the second rational number is −50 75 (i) The denominator is positive. (ii) Dividing both numerator and denominator by the greatest common divisor of 50 and 75, which is 25. We have, − 50 (−50) ÷ 25  − 2  = =  75 75 ÷ 25  3  Clearly, both the rational numbers have the same standard form. Therefore, 8 −50 = − 12 75 Method-2 : In this method, to test the equality of a c two rational numbers, say and , we use cross b d multiplication in the following way : a c = b d Then a × d = b × c If a × d = b × c, we say that the two rational a c numbers and are equal. b d Ex.11 Check the equality of the rational numbers −7 3 and . 21 −9 Sol. To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order. The given rational numbers are For example, −7 3 and . 21 −9 −9 −5 and , 13 3 By cross multiplication, we get we first compare (–7) × (–9) = 21 × 3 i.e., 9 5 and . 13 3 63 = 63. Clearly, both sides are same. Thus, we can say that −7 3 = . 21 − 9 Comparing fraction. We compare two unequal fractions, each is written as another equal fraction so that both have the same denominators. Then the fraction with greater numerator is greater. 7 5 and , find the L.C.M. 6 8 of 6 and 8 (it is 24) and 7 7 × 4 28 = = 6 6 × 4 24 28 15 > 24 24 ⇒ 7 5 > 6 8 −9 −5 > . 13 3 Note : Every positive rational number is greater than negative rational number. Comparing integers. There are two methods for comparing the given integers. (i) Represent the two integers by the points (say A and B) on the number line. The point to the right of the other represents the greater. (ii) We write the integers in ascending order of infinity many integers. 5 5 × 3 15 = = 8 8 × 3 24 As (Q 9 × 3 < 13 × 5 ⇒ 27 < 65) and conclude that COMPARISON OF RATIONAL NUMBERS Example : To compare 9 5 < 13 3 …–5,–4,–3,–2,–1,0,+1,+2,+3,+4,+5,… (as 28 > 15) Quicker method of comparison of a c a c and is that > b d b d Clearly, every positive integer (+1, +2, +3, +4 etc.) is greater than every negative integer (…–4, –3, –2, –1) and 0. Every non-negative integer i.e., whole number (0, +1, +2, +3, +….) is greater than every negative integer (…–3, –2, –1,…) Let x and y be integers if ad > bc. and if x > y ⇒ –x < – y. 7 5 > as (7 × 8 > 6 × 5) 6 8 e.g. 8 > 5 ⇒ –8 < –5. Therefore, we have EXAMPLES Ex.12 Which of the two rational numbers −7 −7 × 4 −28 = = 10 10 × 4 40 5 and − 12 −4 is greater ? 9 Sol. −5 −5 × 5 −25 = = 8 8×5 40 The given rational numbers are or 5 −4 and − 12 9 −5 −4 and . 12 9 Comparing the numerators of these numbers, we find that Now, we write the rational numbers −5 and 12 −4 in the form of common denominator. 9 −5 ( −5) × 9 −45 = = 12 12 × 9 108 −4 denominator of ) 9 −4 (−4) × 12 −48 = = 9 9 × 12 108 −5 by the denominator of ) 12 (multiplying or −4 −5 < 9 12 −4 5 < 9 − 12 Ex.13 Arrange the rational numbers −7 5 3 , , in 10 − 8 − 4 ascending order. Sol. Therefore, the ascending order is : −30 −28 −25 < < 40 40 40 or −3 −7 −5 < < 4 10 8 or 3 −7 5 < < . − 4 10 − 8 REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE Since, –48 < –45, we have or –30 < –28 < –25. (multiplying by the Again, −48 −45 < 108 108 −3 −3 × 10 −30 = = 4 4 × 10 40 The given rational numbers are : −7 5 3 , , 10 − 8 − 4 The given rational numbers with positive denominators are −7 −5 −3 , , 10 8 4 We now convert these numbers so that they have common denominator. For this, we take the LCM of 10, 8 and 4. The LCM is 40. We know that the natural numbers, whole numbers and integers can be represented on a number line. For representing an integer on a number line, we draw a line and choose a point O on it to represent ‘0’. We can represent this point ‘O’ by any other alphabet also. Then we mark points on the number line at equal distances on both sides of O. Let A, B, C, D be the points on the right hand side and A', B', C', D' be the points on the left of O as shown in the figure. –4 –3 –2 –1 0 1 2 3 4 D' C' B' A' O A B C D The points on the left side of O, i.e., A', B', C', D', etc. represent negative integers –1, –2, –3, –4 whereas, points on the right side of O, i.e., A, B, C, D represent positive integers 1, 2, 3, 4 etc. Clearly, the points A and A' representing the integers 1 and –1 respectively are on opposite sides of O, but at equal distance from O. Same is true for B and B' ; C and C' and other points on the number line. −1 −4 and 7 7 Q Dr is 7 ∴ we divide OA in 7 equal parts by points B, C, D, E, F, G. So, these points represent (1) Natural Numbers Eg : A B C D 1 2 3 4----------- −1 −2 −3 −4 −5 −6 −7 , , , , , , respectively. 7 7 7 7 7 7 7 −1 −4 ∴ by B and by E. 7 7 −7 −5 −3 −2 −1 7 G 7 E 7 7 7 (2) Whole Numbers A B C D 0 1 2 3 4--------- (3) Integers –1 −6 F −4 A D' C' B' A' O A B - - - - --- –4 –3 –2 –1 0 1 7 2--- ∴ If we move on number line from right to left we are getting smaller numbers. EXAMPLES Ex.14 Represent 13 13 and − on number line. 3 3 Sol. 1 13 −4 =− 3 3 P' E' D' C' B' A' O A B –5 –4 –3 –2 –1 OA = distance of 1 from 0 C 1 13 4 = 3 3 P D E l 0 +1 +2 +3 +4 +5 Draw a line l and mark zero on it OD′ = distance of –4 from 0 D'A = distance between –4 and 1. etc. 13 1 1 =4 =4+ 3 3 3 and − 13 1  = − 4 +  3 3  Therefore, from O mark OA, AB, BC, CD and DE to the right of O such that (4) Rational Numbers (a) If Nr < Dr : We divide line segment OA (i.e. distance between 0 & 1) in equal parts as denominator (Dr). 1 3 0 c O 2 3 3 3 d 1 A Q Dr is 3, so we divide OA in three equal parts by points c and d. ∴ c= 7 (b) If N > Dr Q negative numbers are less than positive numbers 1 2 Eg : and = 3 3 B O r Negative numbers are in left side of zero (0) & positive numbers are in right side. Also D C 1 2 3 , d = and A = = 1 3 3 3 OA = AB = BC = CD = DE = 1 unit. Clearly, Point A,B,C,D,E represents the Rational numbers 1, 2, 3, 4, 5 respectively. Since we have to consider 4 complete units and a part of the fifth unit, therefore divide the fifth unit DE into 3 equal parts. Take 1 part out of these 3 parts. Then point P is the 13 representation of number on the number 3 line. Similarly, take 4 full unit lengths to the left of 0 and divide the fifth unit D'E' into 3 equal parts. Take 1 part out of these three equal parts. Thus, P' represents the rational 13 number − . 3 Ex.15 Represent the rational number 7 on the 4 RATIONAL NUMBERS BETWEEN TWO GIVEN RATIONAL NUMBERS number line. If x and y are any two rational numbers such that x+y x < y, then x < < y or that the rational 2 x+y number lies between x and y. 2 Let N is the point that represents the integer 7 on the number line and is on the right hand side of the point O. Divide the segment ON 7 into four (Denominator of ) equal parts 4 (with the help of a ruler). Let A, B, C be the points of division as shown in the figure. Sol. Note : There are countless rational numbers between any two rational numbers between any two distinct rational numbers, there is always a rational number. 7 0 O A B C N EXAMPLES Then OA = AB = BC = CN. Therefore, the point A represents the rational 7 −7 number . Similarly, can be represented 4 4 on the number line on the left hand side of ‘O’. Ex.17 Write five 1 1 and . 2 3 Sol. Ex.16 Draw the number line and represent the following rational numbers on it. (i) 3 8 (ii) − 5 3 3 on number line, we 8 first draw a number line and mark the point O on it representing ‘0’ (zero), we find the point P on the number line representing the positive integer 3 as shown in figure. 3 O A B C D E F G P Now divide the segment OP into 8 equal parts. Let A and B be points of division so that OA = AB = BC = CD = DE = EF = FG = GP. 1 By construction, OA is of OP. Therefore, 8 3 A represents the rational number . 8 (ii) –5 Q –5/3 B A 3 O numbers between To find five rational numbers between 1 and 2 1 take higher common multiple of 2 and 3 3 1 1 12 12 that is 36 then = × = and 3 3 12 36 1 1 18 18 So five rational numbers = × = 2 2 18 36 1 1 13 14 15 16 between and are , , , 2 3 36 36 36 36 17 and . 36 Sol. (i) In order to represent 0 rational Ex.18 Write three more numbers in the following pattern : −2 −4 −6 −8 , , , ,...... 5 10 15 20 Sol. −4 −2 × 2 −6 −2 × 3 −8 −2 × 4 = , = , = 10 5 × 2 15 5 × 3 20 5 × 4 Thus, we observe a pattern in these numbers. The other numbers would be −2 × 5 −10 −2 × 6 −12 −2 × 7 −14 = , = , = . 5×5 25 5 × 6 30 5 × 7 35 Ex.19 Name the rational numbers denoted by A, B, C, D. A D B (i) C –4 –3 –2 –1 0 Sol. Ex.23 Give two rational numbers equivalent to 1 2 3 4 Sol. −8 9 (ii) 2 5 (iii) 11 −6 (iv) −3 7 (i) Rational numbers equivalent to : −8 −16 −24 are , etc. 9 18 27 1 1 A → –3 , B → –1, C → 2 , D → –2. 2 3 (ii) Rational numbers equivalent to : −3 +7 Ex.20 Represent , on number line. 2 4 Sol. We can write, 2 4 6 are , etc. 11 22 33 −3  1 7  3 = −1 + , = 1 +  2  2 4  4 (iii) Rational numbers equivalent to : 5 10 15 are , etc. −6 − 12 − 18 − 3 1 = −1 2 2 –2 –1 7 3 =1 4 4 0 1 Ex.24 Fill in the blank boxes with the correct symbol out of >, < or = 7 5 −8 −5 −3 7 −1 −9 , , , , , , , . 8 − 4 3 − 7 8 − 4 − 22 + 4 (i) 0 Negative rational numbers (iii) 0 = 5 −8 −3 7 −9 , , , , . −4 3 8 −4 4 Positive rational numbers = Ex.22 Reduce Sol. −3 −6 −9 are , etc. 7 14 21 2 Ex.21 Identify the positive and negative rational numbers : Sol. (iv) Rational numbers equivalent to : Sol. 7 −5 −1 , , . 8 − 7 − 22 −2 2 64 , , in standard form. − 5 16 − 128 −2 −2 × −1 2 = = − 5 − 5 × −1 5 2 2÷2 1 = = 16 16 ÷ 2 8 64 64 ÷ (−64) −1 = = . − 128 ( −128) ÷ (−64) 2 (Q H.C.F. of 64 and 128 = 64) (i) 0 + < (iii) 0 < 1 7 8 9 + −4 7 (ii) 2 (ii) 2 −4 7 > 1 7 Ex.25 Write two rational numbers between Sol. (Q H.C.F. of 2 and 16 = 2) 8 9 Two rational numbers between 1 1 and . 4 5 1 1 and 4 5 5 4 , 20 20 50 40 Means two rational numbers between , . 200 200 Means two rational numbers between i.e., 41 42 49 , ,........ (any two) 200 200 200 Ex.26 Write the following rational numbers in ascending and descending order : −6 1 0 −3 , , , . 5 5 5 −5 Sol. Ascending order : Ex.29 Find ten rational −3 8 and . 11 11 –3 < –2 < –1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 This gives us 10 integers between –3 and 8. −6 0 1 −3 , , , . 5 5 5 −5 −2 −1 1 2 3 4 5 6 7 , ,0, , , , , , , are 11 11 11 11 11 11 11 11 11 the ten desired rational numbers. So, −3 1 0 −6 Descending order : , , , . −5 5 5 5 Sol. Given rational numbers are between We know that Sol. Ex.27 Find a rational number between numbers −2 1 and . 3 4 OPERATION ON RATIONAL NUMBERS There are four basic operations on rational numbers : −2 1 and . 3 4 −2 1 and 3 4 can be obtained by the following method : (i) Addition (ii) Subtraction (iii) Multiplication (iv) Division. Then a rational number between (i) Addition : If two rational numbers are to be added, we first express each one of them as rational number with positive denominator. (Addition of two rational numbers) ÷ 2 −5 −2 1  −8 + 3 + ÷2= ÷2  ÷2= 4 12  3  12  = There are two possibilities : −5 1 −5 × 1 −5 × = = 12 2 12 × 2 24 (1) Either they have same denominators, or (2) They have different denominators. So, the rational number between Rational Numbers with Same Denominator −2 1 −5 and is . 3 4 24 Ex.28 Find a rational number between Sol. Let us add −7 −6 and . 6 7 8 −6 and . 5 5 Represent the numbers on the number line. (Addition of two rational numbers) ÷ 2 −2 −1 0 1 2 3 4 5 6 7 8 9 10 =1 =2 5 5 5 5 5 5 5 5 5 5 5 5 5  − 7  − 6   − 49 + (−36)  +  ÷ 2 =  ÷2  42    6  7  [LCM of 6 and 7 is 42] −85 1 −85 − 85  − 49 − 36  = × = . ÷2= ÷2= 42 42 2 84  42  Therefore, a rational number between and −6 −85 is . 7 84 −7 6 Here, the distance between two consecutive points 1 −6 8 is . For , move 6 steps to the left of and 5 5 5 2 we reach at . 5 So, 8  − 6  8 + ( −6) 2 + = = 5  5  5 5 Converting to the same denominators, we have EXAMPLES Ex.30 Add : Sol. −5 −17 and . 9 9 Given rational numbers are −8 (−8) × 3 −24 = = 5 5×3 15 −5 −17 and . 9 9 −4 (−4) × 5 −20 = = 3 3× 5 15 And Adding these two numbers, we have ( −5) ( −17) (−5) + ( −17) −22 + = = 9 9 9 9 So, Which is the required answer. −23 5 Ex.31 Add : and 28 − 28 Sol. 5 We first express as a rational number − 28 with positive denominator. We have, Now, 5 × (−1) −5 = ( −28) × (−1) 28 = ( −24) + (−20) −44 = 15 15 Which is the required answer. Ex.33 Add : Sol. − 23  − 5  − 23 5 + − = 28  28  28 28 8 , 3. 10 We have, 3 which can be written as Multiplying both the 3 denominator of by 10, 1 [since (+) × (–) = –] = − 8  − 4  − 24  − 20  + + =  5  3  15  15  −23 − 5 −28 = = −1 28 28 we get Addition of Rational numbers with Different Denominators Therefore, Sol. 8 4 and . −5 −3 Clearly, they have different denominators. Here, first we express the given rational numbers into standard forms. The given rational numbers are i.e., 8 8 × (−1) −8 = = − 5 (−5) × (−1) 5 And, 4 4 × (−1) −4 = = − 3 ( −3) × (−1) 3 Now, 8 4 ( −8) (−4) + = + −5 −3 5 3 and 8 8 3 8 30 +3= + = + 10 10 1 10 10 = 8 4 and . −5 −3 numerator 3 3 × 10 30 = = 1 1 × 10 10 In this case, we convert the given rational numbers to a common denominator and then add. Ex.32 Add 3 . 1 8 + 30 38 19 = = 10 10 5 Which is the required answer. Ex.34 Simplify : Sol. 4 3 −2 −11 + + + . 3 5 5 3 4 3 − 2 − 11  4 − 11   3 − 2  + + + = + + +  3 5 5 3 3  5 5  3 = 4 − 11 3 − 2 −7 1 −7 × 5 1 × 3 + = + = + 3 5 3 5 3× 5 5× 3 (changing them to same denominator) = −35 3 −32 + = . 15 15 15 Ex.35 Add Sol. 7 −5 and . 9 9 (ii) Subtraction : If we add the additive inverse of a rational number and other rational number then this is called subtraction of two rational numbers. So the subtraction is inverse process of addition and the term add the negative of use for subtraction. 7  − 5  7 + (−5) 7 − 5 2 + = = . = 9  9  9 9 9 In case, if denominator of the rational number is negative, first we make it (denominator) Positive and then add. Ex.38 Find value of 6 4 Ex.36 Add : and . −5 5 Sol. EXAMPLES 6 4 −6 4 −6 + 4 −2 + = + = = . −5 5 5 5 5 5 Sol. 2 4 − . 3 5 2 4 + additive inverse of . 3 5 −8 −5 Ex.37 Find the sum of and . 5 3 = 2  − 4  2×5  − 4×3 + + =  3  5  3× 5  5× 3  LCM of 5 and 3 is 15. = −8 −8 × 3 −24 = = 5 5× 3 15 10 −12 10 + (−12) + = 15 15 15 = −2 . 5 Sol. ⇒ So, −5 −5 × 5 −25 = = 3 3× 5 15 Ex.39 Find value of − 8  − 5  − 24  − 25  − 24 − 25 + + = = 5  3  15  15  15 Sol. − 8  − 5  − 49 + = 5  3  15 2  −5 2  −5 −  = + additive inverse of   7  3  7  3  Note : Addition of rational numbers is closure (the sum is also rational) commutative (a + b = b + a) and associative(a + (b + c)) = ((a + b) + c) = 2 5 + 7 3 = 2× 3 5× 7 + 7 × 3 3× 7 = 6 35 6 + 35 41 + = = . 21 21 21 21 Additive inverse The negative of a rational number is called additive inverse of the given number. Rational number Additive inverse x y −x y −3 7 3 7 13 17 −13 17 Note : Zero is the only rational no. which is its even negative or inverse. Ex.40 Subtract Sol. 2  −5 − . 7  3  −5 −3 from . 8 7 The given rational numbers are Therefore, −3  −5 −3 5 − + = 7  8  7 8 = −24 35 −24+ 35 11 + = = 56 56 56 56 −5 −3 and . 8 7 Ex.41 Simplify : Sol. 1 −2 −2 + − 6 5 15 Ex.44 The sum of two rational numbers is We have, one of them is 1 −2 −2 1 2 2 + − = − + 6 5 15 6 5 15 Sol.   −2 2  =  Since −   15  15   = 5 − 12 + 4 9 − 12 −3 −1 = = = = 30 30 30 10 The number will be obtained by subtracting −5 5 from . 8 9 So, = = −3 −9 −3 9 − = + 5 10 5 10 = −3 × 2 + 9 × 1 10 = −6 + 9 3 = 10 10 3 . 10 a and c are two rational numbers, then b a ac ×c = . b b 85 . 72 27 11 −5 ? 33 Eg : Find product of −5 and 9. 7 −5 −5 9 −5 × 9 −45 ×9 = × = = 7 7 1 7 ×1 7 We have, difference of the given number and −5 the required number = 33 Therefore, = (a) Let Ex.43 What number should be subtracted from Given number = The other number (iii) Multiplication : 40 + 45 85 = 72 72 Therefore, required number is Sol. −3 5 −9 10 Therefore, required number is 5  − 5  5 5 5×8 + 5× 9 − = + = 9  8  9 8 72 so as to get Given, sum of the numbers = = Sum of the numbers – One of the numbers −5 Ex.42 What number should be added to so that 8 5 the sum is ? 9 Sol. −9 . Find the other. 10 One of the numbers = 1× 5 − 2 × 6 + 2 × 2 [LCM of 6, 5, 15 is 30] 30 −3 . If 5 27 11 27 −5 27 5 − = + 11 33 11 33 i.e., a c a × c ac × = = b d b × d bd e.g., 1 −3 1 × ( −3) −3 × = = . 2 4 2× 4 8 (i) On multiplying two rational numbers, we get result as a rational number. 27 × 3 + 5 × 1 81 + 5 86 = = 33 33 33 Therefore, required number is (b) When we multiply two rational numbers : e.g., 86 . 33 −3 5 −3 × 5 −15 × = = 4 7 4×7 28 (closure property) (ii) (iii) 1 1 1 1 × = × (commutative i.e., on 3 4 4 3 changing the order the result remains same) 1 1 1 1 1 1 ×  ×  =  ×  × (associative) 3  4 5 3 4 5 (iv) If 0 is multiplied to any rational number, the result is always zero. (v) Reciprocal of a (+ve) rational number is (+ve) and reciprocal of (–ve) rational number is (–ve). To divide one rational number by other rational numbers we multiply the rational number by the reciprocal of the other a c i.e., ÷ b d = e.g., 4 0× 4 0 (a) 0 × = = =0 5 5 5 −2 −2 × 0 0 (b) ×0 = = = 0. 3 3 3 a c a d ad × reciprocal of = × = b d b c bc For example, 5 −3 5 ÷ = × reciprocal of −6 2 −6  − 3    2  5 +2 × −6 −3 5× 2 10 5 = = = . − 6 × (−3) 18 9 = (iv) Division : (a) Let 5 −3 ÷ −6 +2 a b be a rational number then its reciprocal will be (vi) Zero divided by any rational number is always equal to zero. b . a (i) The product of a rational number with its reciprocal is always 1. 2 −3 =0; 0÷ =0 5 8 Note : For example, −3 −8 (b) × =1 8 3 5 7 (a) × = 1 7 5 (ii) Zero has no reciprocal as reciprocal of 0 1 0 (which is not defined)   is 1 0   (iii) The reciprocal of a rational number is called the multiplicative inverse of rational number. (iv) 1 and –1 are the only rational numbers which are their own reciprocal. Reciprocal of 1 = For example, 0 ÷ 1 = 1. 1 Reciprocal of –1 = 1 = –1. −1 (i) When a rational number (except zero) is divided by another rational number (except 0) the quotient is always a rational number. (closed under division) a c  ad  ÷ =  b d  bc  number. i.e., is also rational (ii) Division of any rational number by itself gives the quotient 1. For example, 4 4 ÷ = 1. 5 5 (iii) When a rational number is divided by 1, the quotient is a rational number itself. For example, 3 3 ÷1 = . 4 4 EXAMPLES (i) Ex.45 Find the sum : (i) 6  − 11  +  4  4  (ii) −4 +0 3 −4 (−3) (v) + . 19 57 (i) 6  − 11  6 + (−11) 6 − 11 −5 + = = .  = 4  4  4 4 4 7 3 (ii) + 3 7 (iii) 1 4 (iv) − 3 + 2 3 5 (iii) Sol. 7 3 + 3 7 5 13 − 24 36 −7 −6 −  13  25  Sol. (i) = (ii) 15 − 26 −11 = 72 72 4  −5 4 5 − + = 63  21  63 21 (Q LCM of 21 and 63 = 63) = (iii) 4 ×1 + 5 × 3 4 + 15 19 = = . 63 63 63 −7 −6 −7 6 − + = 13  25  13 25 Q LCM of 13 and 25 = 325) = (iv) −7 × 25 + 6 × 13 −175 + 78 −97 = = 325 325 325 −5 8 −5 × 11 − 8 × 8 − = 8 11 88 (Q LCM of 8 and 11 = 88) = (v) − 3 = −55 − 64 −119 = 88 88 1 22 3 −3=− − (Q LCM of 7 & 1 = 7) 7 7 1 −22 − 3 × 7 −22 − 21 −43 = = . 7 7 7 Ex.47 Find the product : Ex.46 Find : −5 8 − 8 11 (Q LCM of 24 and 36 = 72) 7 3 49 9 49 + 9 58 + = + = = . 3 7 21 21 21 21 Alternative Method : 7 3 7 × (7) + 3 × (3) + = 3 7 21 (Q LCM of 3 and 7 = 21) 49 + 9 58 = = . 21 21 −4 −4 0 ( −4) + 0 −4 (iii) +0 = + = = . 3 3 3 3 3 1 4 (iv) − 3 + 2 3 5 −10 14 −10 × 5 + 14 × 3 = + = 3 5 15 (Q LCM of 3 and 5 = 15) −50 + 42 −8 = = . 15 15 −4 (−3) (v) + 19 57 −4 × 3 + (−3) × 1 −12 + (−3) = = 57 57 (Q LCM of 19 and 57 = 57) −12 − 3 −15 = = . 57 57 (iv) 5 13 5 × (3) − 13 × 2 − = 24 36 72 (Q LCM of 3 and 7 = 21) So, 4  −5 −  63  21  1 (v) − 3 − 3 7 7 7 × 7 49 = = 3 3 × 7 21 3 3× 3 9 = = 7 7 × 3 21 (ii) (i) 7  −5 ×  2  4  (ii) 4 −5 × −5 4 (iv) (iii) Sol. (i) 4  −2 ×  7  5  = 7  − 5  7 × (−5) − 35 × = = 2  4  2× 4 8 7 7 (−9) 7 × (−9) −63 × (−9) = × = = 10 10 1 10 × 1 10 4 × (−5) 4 −5 −20 (iii) × = = =1 ( −5) × 4 −5 4 − 20 (ii) (iv) 6 1 −6 −6 1 (v)  ×  ÷1 =  ÷ =  9   9  1 9 1 7 × ( −9) 10 −2 −6 So,  .  ÷1 = 9 3   Ex.49 Multiply :  −8  −5 (i)   by    25   16  4  − 2  4  − 2  4 × (−2) −8 × =  = ×  = 7  5  7  5  7×5 35 Ex.48 Find the value of : 2 (i) (−6) ÷ 3 −4 (ii) ÷2 5 (iv) −2 −2 ÷ 8 8  9   22  (ii)   by    − 11   − 27   −8  −5 Sol. (i) Multiplication of   by   25    16  = 3  −4 (iii) ÷  13  65  2 −6 3 −6 × 3 −18 = × = = = −9 3 1 2 1× 2 2 2 So, (−6) ÷ = −9. 3 −4 −4 1 −4 × 1 (ii) ÷2= × = 5 5 2 5× 2 −2 × 1 −2 −2 = = = 5 5 5 −4 −2 So, ÷2= . 5 5 3  − 4  3 65 3 × 65 ÷ = = × 13  65  13 − 4 13 × ( −4) 15 −15 = = −4 4 So, (iv) 3  − 4  − 15 ÷ . = 13  65  4 −2 −2 −2 8 (−2) × 8 1 × 1 ÷ = × = = =1 8 8 8 − 2 8 × (−2) 1 × 1 So, −2 −2 ÷ = 1. 8 8 −8 −5 −8 × −5 40 × = = 25 16 25 × 16 400 Dividing both the numerator and denominator by the greatest common divisor of 40 and 400 which is 40. −6 (v)   ÷1  9  Sol. (i) (−6) ÷ (iii) −6 × 1 −2 × 1 −2 = = . 9 ×1 3 ×1 3 = 40 ÷ 40 1 = 400 ÷ 40 10  9   22  (ii) Multiplication of   by    − 11   − 27  = 9 22 × − 11 − 27 = 9 × 22 198 = − 11 × −27 297 = 198 ÷ 99 2 = 297 ÷ 99 3  −2 9  2 −6 Ex.50 Simplify :  × + × .  3 5  3 7  Sol. −2 9 2 −6 We have,  × + ×   3 5  3 7  = − 18  − 12  −18 × 7 + (−12 × 5) +  = 15  21  105 = −126 + (−60) −126 − 60 −186 −62 = = = 105 105 105 35 MULTIPLICATION OF RATIONAL NUMBERS ON A NUMBER LINE The product of two rational numbers on the number line can be calculated in the following way. −2 by 3 on a number line, it 7 −2 means 3 jumps of to the left from zero. Now 7 −6 −2 −6 we reach at . Thus we find ×3= , i.e., 7 7 7 −2 −2 3 −2 × 3 −6 ×3= × = = . 7 7 1 7 ×1 7 When we multiply −8 −7 − 6 7 7 7 −5 −4 −3 −2 −1 7 7 7 7 7 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 7 This result reconfirms that the product of two rational numbers is rational number whose numerator is the product of the numerators of the given rational numbers and the denominator is the product of the denominators of the given numbers. ∴ Multiplication is closure (product is rational), commutative (ab = ba) and associative (a(bc) = (ab)c) for rational number.