Rational Numbers 11II. Learn and Remember

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$11II. Rational Numbers Learn and Remember 1. A rational number can be expressed in the form of J!..., where q p and q are integers and q "* 0. 2. All integers and fractions are rational numbers.$

1 11II. Rational Numbers Learn and Remember 1. A rational number can be expressed in the form of J!..., where q p and q are integers and q "* All integers and fractions are rational numbers.. To convert a rational number to an equivalent rational number either multiply and divide both numerator and denominator by a non-zero integer.. A rational number is positive if numerator and denominator are either both positive or negative integers whereas a rational number is negative if either the numerator or the denominator is a negative integer. 5. A rational number is said to be in standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than l. 6. There are infinite rational numbers between two rational numbers. 7. Two rational numbers can be added and subtracted by taking the L.C.M. of their denominator and then converting the rational numbers to their equivalent forms having the L.C.M. as denominator.. J!... + (- p) = 0, - P is said to be additive inverse of J!... q q q q 9. Multiplicative inverse means reciprocal of rational t.e.,. a x - 1 = 1,- 1. IS Saltd to be mu1 tip. 1 teative tve iinverse 0f a, a a where a is rational 10. To multiply two rational numbers, ti 1 b Product of their numerators Product 0f ra IOna num ers = Product of their denominators

10 MATHEMATiCS-VII 11 11. To divide one rational number by the other non-zero rational We multiply the rational number by the reciprocal of the other rational TEXTBOOK QUESTIONS SOLVED Exercise 9.

2 10 MATHEMATiCS-VII To divide one rational number by the other non-zero rational We multiply the rational number by the reciprocal of the other rational TEXTBOOK QUESTIONS SOLVED Exercise 9.1 (Page No. 12-1) Q1. List five rational numbers between: - 1 and 0 (ii) - 2 and and - "2and ". - 1 and 0 Let us write - 1 and 0 as rational numbers with denominator We have - 1 = - and 0 = - = 0, So -<-<-<-<-<-<0, or - 1<- < - <- <- < - < The five rational numbers between - 1 and 0 would be ''2''6 (ii) - 2 and - 1 Let us write - 2 and - 1 as rational numbers with denominator We have, - 2= -6- and - 1= So, -6-<-6-<-6-<6<6<6< or - 2< -- <- <- < - <- < Thus, the five rational numbers between - 2 and -1would be ''2''6 Q and Let us write and - 2 into rational numbers with same denominators So, -=-and-= We have, or < - <-- <- <-- <-- < ll <-<--<-<-<--< Thus, the five rational numbers between - and - 5 would be (iu) -.!. and ~ ' 5 ' 15 ' 5 ' 5. Let us write - ~ and ~ into rational numbers with same denominators. So, We have, or = - and - = <-<-<0<-<-<-< <-<-<0<-<-<-< Thus, the fiverational numbers between ""2 and " would be ' 6,0, 6'" Write four more rational numbers in each of the following patterns:

12 MATHEMATiCS-VII 1 - -6-9 -12 5' 10' 15' 20, -1 2 ------- 6 ' - 12' - 1' - 2 '. -2 2 6 ' _' - 6' - 9'. - - 6-9 -12 We have, 5' 10' 15' 20-1 -2 - (ii) ' ' 12'.

3 12 MATHEMATiCS-VII ' 10' 15' 20, ' - 12' - 1' - 2 ' ' _' - 6' - 9' We have, 5' 10' 15' (ii) ' ' 12'. - -x1-6 -x2-9 5 = -W-' 10 = ~, 15 Thus, this pattern follow as: - x x =~'20=5x _ - x x =--w- =25' 5 =~ =0' - - x x =5x7 =5'5 =~ =0 Therefore, the next four rational numbers of this pattern would be, ' 0 ' 5 ' (ii) We have, ' ' lxl -2 -lx2 - -lx = xl' = x 2 ' 12= x Thus, this pattern follows as: -1 -lx lx5 - = x =16' = x5 =20' -1 -lx lx7-7 = x6 =2 ' = x7 =2s Therefore, the next four rational numbers ofthis pattern would be, ' 20' 2 ' 2' Q. (iu) We have, ' - 12' - 1' lx1 2 1x2 (=W-' -12 = - 6x 2 ' 1x 1x -- =--, -- =-- -W -6x -2-6x Thus, this pattern follows as: 1 1x5 5 1 lx6 6 - =-- =--,- =-- = x x x x - 6 =-6x 7 =- 2' - 6 =-6x =- Therefore, the next four rational numbers of this pattern would be, We have, Thus, this pattern ' - 6' - 2' ' -' -6' x1 2 2x1 -=--,-=-- xl - -xl 2x2 6 2x -=--,-= x2-9 -x follows as: 2 2x 2 2x =--=--,- =-- = x x x x =-- =--, - =-- =-- - -x x7-21 Therefore, the next four rational would be, ' - 15' - 1' Give four rational numbers equivalent to: (ii) _ numbers ofthis pattern "9.

4 1 MATHEMATICS-VII Q x x -6 Now - = -- = - - = -- = - '7 7x 2 1' 7 7x 21 5 (ii) =- " x x =-W-=2s'7=~=5 Therefore, four equivalent rational ' 21 ' 2' 5. numbers are 5 5x x 15 Now - =-- =-, - =-- =- '- -x x x x =--=--, - =-- =- _ - x x 5-15 Therefore, four equivalent rational ' - 9' - 12' x2 x 12 Now - = -- =- - =-- =- '9 9 x 2 1' 9 9 x 27 x 16 x 5 20 "9= 9 x = 6' "9= 9 x 5 = 5 Therefore, four equivalent rational ' 27' 6' 5 numbers are numbers.are Draw the number line and represent the following rational numbers on it: " (ii) -"-7 7 " ~ Q5. (ii) Draw a number line and represent number: 15 the positive rational < I I I' 'I I I I ) Draw a number line and represent number: the negative rational ( I 111+1'''', '.'1'1111 I I > Draw a number line and represent number: -7 (,11 I, I I I,,, > Draw a number line and represent number: < I I III 11 ''!' I) the negative rational the positive rational 7 The points, P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by p, Q, Rand S.

16 MATHEMATICS-VII RATIONAl NUMBERS 17 ( vu..) -an d -9-9 -7 21 and 9" USRT APQB (I I I I I I I I I I I I I) - - -2-1 0 1 2 Each part which is between the two numbers is divided into parts.

5 16 MATHEMATICS-VII RATIONAl NUMBERS 17 ( vu..) -an d and 9" USRT APQB (I I I I I I I I I I I I I) Each part which is between the two numbers is divided into parts So A =- P =- Q =- and B =-, ' ' Similarly T - - R - - S = - and U = - '-'-' Thus, the rational numbers represented P, Q, Rand S are i, - " ' and respectively. Q6. Which of the following pairs represent the same rational number? and- - (ii) 20 and _ 25 and (v) - (w) 5 and 20 and- (vi) " and "" 'j;- -7 Thus, 21 'j; 9" not represent the pairs of same rational (ii) 20 and _ =5 I (v) (vi) (vii) Thus, 20 = _ 25 represent the pairs of same rational and Thus, - and - represent the pairs of same rational "5 and = Thus, 5 and 20 represent the pairs of same rational -2 - and =5 Th us, - an d represent 15 thee nai pairs 0f same rationa. I and h. f. I "'j; 9 not represent t e pairs 0 same rationa Because these are positive and negative rational 5 -and numbers.

6 1 MATHEMATics-VII Q7. ~ 5 h. f. I "9 :;:."9 not represent t e pairs 0 same rationa Because these are positive and negative rational numbers. Rewrite the following rational numbers in the simplest form: Sol - = -- = (ii) - = -- = = 72+ = =-- = (ii) Q. Fill in the boxes with the correct symbol out of >, < and = " (ii) "" (v) - "" (vi) -11 II (vii) r=:l 2 7 L2.J " Since, positive number is greater than negative - 0 (ii) On converting both rational numbers into same denominator and comparing, ~ -X7D x5 5x7 7x5 ~ -2 5 r:::.i - 25 L2.J Ss (H.C.F. of and 6 is 2.) (H.C.F. of25 and 5 is 5.) (H.C.F. of and 72 is.) (H.C.F. of and 10 is 2.) (:. Negative smaller number is greater than negative bigger ) - r:::.i Thus, "5 L2J 7' On converting both rational numbers with same denominator and comparing, -7x2 0 1x(-I) ~ ---- x2-16x(-1) -7 '-l 1 Thus, ~ -16' ~~ G -1~ On converting both rational numbers with same denominator and comparing, -X0-7x5 ~ x ~ ~ x5-2 r:::j ~ 20 (...Negative smaller number is greater than negative bigger ) - G -7 Thus -- > - ' (v) r:::.i Thus, L2J (vi) ~ -11 Oil Thus'il -1 G 11 r:::j -7 (vii) 0 ~ 6 (... 0 is greater than every negative ) Q9. Which is greater in each of the following: "' 2 (ii) 6'" - 2 ' - 19

150 MATHEMATICS-VII Sol.. -1 1 2 (u) ' " - 7,- 5. 2 5 ' 2 into same denominators, 2x2 x2 ="6 and 5x 15 2x =6 r=:l 15 "6 L2.J 6 or ~GJ~ 2 5. 2 Thus, "2 IS greater than.

7 150 MATHEMATICS-VII Sol (u) ' " - 7, ' 2 into same denominators, 2x2 x2 ="6 and 5x 15 2x =6 r=:l 15 "6 L2.J 6 or ~GJ~ Thus, "2 IS greater than. - (ii) 6' into same denominators, -x2- -and--=- 6 x2 6 G - Now - > - or - ' Thus, 6 is greater than. - 2 ""' - into same denominators, G - -x = -9 and ~ = 2x(-) x x(-) -9 r=:l - - r=:l L~ - or - L2.J Thus, - is greater than ' Positive number is always greater than negative 1-1 So, " is greater than "". RATIONAl NUMBERS (u) - 7,- 5 into same denominators, x x 7 -- = -- = and-- = -- = x x "5 and "5-115 G -1 2 G Now -- > -- or -- > --, Thus, - * is greater than - ~. QI0. Write the following rational numbers in ascending order: "5'"5' 5 (ii) "'""9' '7 '2"-'"" ' 5' 5 (ii) rational numbers are in ascending order ' 9'" <5< ed into same denominators "9' 9' 9 rational numbers are in ascending order Thus - < - <- ' <-< ' 2 ' - rational numbers are in ascending order <-<-. 7

152 MATHEMATiCS-VII Exercise 9.2 (Page No. 190) Q1. Find the sum: 5 (-11) " + - 5-11 + 9 (vii) -2i + i 5 (-11) 5-11 -6 - " + = -- = "" = 2, ( u..) - 5 +- 5 L.C.M. of and 5 is 15.

8 152 MATHEMATiCS-VII Exercise 9.2 (Page No. 190) Q1. Find the sum: 5 (-11) " (vii) -2i + i 5 (-11) " + = -- = "" = 2, ( u..) L.C.M. of and 5 is So - = - and - = - ' Thus = = -- = - = 2- ' L.C.M. of 10and 15is So, 10 = 0 and 15 = Thus, = = 0 = L.C.M. of 11and 9 is (ii) " + "5 - (- 2) (v) So, _ 11 = 99 and "9 = 99 (vi) Thus, "9 = = 99 = 99 - r- 2) (u) L.C.M. of19 and 57 is (- 2) - 2 So -=-and--=- ' (- 2) (- 2)+ (- 2) - 26 Thus = = = _ ' (vi) + 0 = (vii) = L.C.M. of and 5 is ' Thus = = (-5)+69 ' = 15 = Q2. Find: ~ _ L. _ L.C.M. of2 and 6 is So - = - and - = - ' Thus l-.- _ 17 = 21 _ = 21- = -1 ' (-6) (ii) 6-21 L.C.M. of6 and 21 is ' (") n (- 6) (-7) 15 1 (v) Thus ~ _ (-6) = ~ _ (-1) = 5-~-1), =6=6

MATHEMATICS-VII 15 ~: - (~;) L.C.M. of1and 15is 195. -6-90 -7-91, 1 195 15 195 Thus -6 _ (-7) = -90 _ (-91) = -90-(-91), 1 15 195 195 195-90+ 91 1-7 = 195 = 195-11 L.C.M. of and 11is.

9 MATHEMATICS-VII 15 ~: - (~;) L.C.M. of1and 15is , Thus -6 _ (-7) = -90 _ (-91) = -90-(-91), = 195 = L.C.M. of and 11is , 11 Thus 1 (v) _- ~ = _- 56 = _- -_5_6 '11 L.C.M. of9 and 1is =-- = , Thus, = = --9- Q. Find the product: -7 1 = -9- = (-7) C') -6 9 "2 x (ii) 1~ x r- 9) Ul -x (-2) x "5 (v) - x - (vi) - X-. 9 (-7) 9x(-7) " x = 2x = -- =- 7". 155 x(-9) (ii) - x(-9)= = - = (- 6)x 9-5 (m) 5 x 11 = 5x11 = ss. ~ (-2) _ x(-2) _ -6 (w) 7 x 5-7x x2 6 (v) 11 x "5 = 11x5 = 55 (vi) ~ x () = x() = (- 5)x Q. Find the value of: r- ) +! (..) (-) vu (ii) " (v) (-)+ - =(-)x - =(-2)x= (- ) 1 (- )x1 - (ii) = 5 x 2"= 5x2 = (-) 1 (-)xl - (m) 5 +(-)= - x (-) = 5x(-) = -15 ~ 15 (vi) =-! + ~ = (- 1) x ± = (- 1)x1 = =-!. 2x 6-5" + (-) ;: + (;:) -2 1 (-2) 7 (-2)x7-1 1 (v) 1 + -;;=1 x "1 = 1x1 = 1 = vi) =2+(-2)=(-7)x(E..)= (-7)x1 = -91 = 19. ( x(- 2) (vii) ~ + (-) = ~ x (-65) = x() = -15 =-~ x 00

Description Reflect and Review Teasers Answers Recall basics of fractions. Review

$Description Reflect and Review Teasers Answers Recall basics of fractions. Review$ 1. Revision Recall basics of fractions. are equivalent fractions of. Since Write five equivalent fractions of. 2. Rational Numbers A number is defined as a number which can be represented in the form of,