Rational Numbers - Solutions

 CBSE Class –VII Mathematics

NCERT Solutions
Chapter 9 Rational Numbers (Ex. 9.1)

Question 1. List five rational numbers between:

(i) −1 and 0

(ii) −2 and −1

(iii) −45 and −23

(iv) −12 and 23

Answer: (i) −1 and 0

Let us write −1 and 0 as rational numbers with denominator 6.

⇒ −1=−66 and 0 = 06

∴ −66<−56<−46<−36<−26<−16<0

⇒ −1<−56<−23<−12<−13<−16<0

Therefore, five rational numbers between −1 and 0 would be

−56,−23,−12,−13,−16

(ii) −2 and −1

Let us write −2 and −1 as rational numbers with denominator 6.

⇒ −2=−126 and −1=−66

∴ −126<−116<−106<−96<−86<−76<−66

⇒ −2<−116<−53<−32<−43<−76<−1

Therefore, five rational numbers between −2 and −1 would be

−116,−53,−32,−43,−76

(iii) −45 and −23

Let us write −45 and −23 as rational numbers with the same denominators.

⇒ −45=−3645 and −23=−3045

∴ −3645<−3545<−3445<−3345<−3245<−3145<−3045

⇒ −45<−79<−3445<−1115<−3245<−3145<−23

Therefore, five rational numbers between −45 and −23 would be

−79,−3445,−1115,−3245,−3145,−23

(iv) −12 and 23

Let us write −12 and 23 as rational numbers with the same denominators.

⇒ −12=−36 and 23=46

∴ −36<−26<−16<0<16<26<36<46

⇒ −12<−13<−16<0<16<13<12<23

Therefore, five rational numbers between −12 and 23 would be −13,−16,0,16,13.

Question 2. Write four more rational numbers in each of the following patterns:

(i) −35,−610,−915,−1220,.........

(ii) −14,−28,−312,..........

(iii) −16,2−12,3−18,4−24,.........

(iv) −23,2−3,4−6,6−9,..........

Answer: (i) −35,−610,−915,−1220,.........

⇒ −3×15×1,−3×25×2,−3×35×3,−3×45×4,.........

Therefore, the next four rational numbers of this pattern would be

−3×55×5,−3×65×6,−3×75×7,−3×85×8 = −1525,−1830,−2135,−2440

(ii) −14,−28,−312,..........

⇒ −1×14×1,−1×24×2,−1×34×3,..........

Therefore, the next four rational numbers of this pattern would be

−1×44×4,−1×54×5,−1×64×6,−1×74×7 = −416,−520,−624,−728

(iii) −16,2−12,3−18,4−24,.........

⇒ −1×16×1,1×2−6×2,1×3−6×3,1×4−6×4,.........

Therefore, the next four rational numbers of this pattern would be

1×5−6×5,1×6−6×6,1×7−6×7,1×8−6×8 = 5−30,6−36,7−42,8−48

(iv) −23,2−3,4−6,6−9,..........

⇒ −2×13×1,2×1−3×1,2×2−3×2,2×3−3×3,..........

Therefore, the next four rational numbers of this pattern would be

2×4−3×4,2×5−3×5,2×6−3×6,2×7−3×7 = 8−12,10−15,12−18,14−21

Question 3. Give four rational numbers equivalent to:

(i) −27

(ii) 5−3

(iii) 49

Answer: (i) −27

−2×27×2=−414,−2×37×3=−621,−2×47×4=−828,−2×57×5=−1035

Therefore, four equivalent rational numbers are −414,−621,−828,−1035.

(ii) 5−3

5×2−3×2=10−6,5×3−3×3=15−9,5×4−3×4=20−12,5×5−3×5=25−15

Therefore, four equivalent rational numbers are 10−6,15−9,20−12,25−15.

(iii) 49

4×29×2=818,4×39×3=1227,4×49×4=1636,4×59×5=2045

Therefore, four equivalent rational numbers are 818,1227,1636,2045.

Question 4. Draw the number line and represent the following rational numbers on it:

(i) 34

(ii) −58

(iii) −74

(iv) 78

Answer: (i) 34

(ii) −58

(iii) −74

(iv) 78

Question 5. 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, R and S.

Answer: Each part which is between the two numbers is divided into 3 parts.

Therefore, A = 63, P = 73, Q = 83 and B = 93

Similarly T = −33, R = −43, S = −53 and U = −63

Thus, the rational numbers represented P, Q, R and S are 73,83,−43 and −53 respectively.

Question 6. Which of the following pairs represent the same rational numbers:

(i) −721 and 39

(ii) −1620 and 20−25

(iii) −2−3 and 23

(iv) −35 and −1220

(v) 8−5 and −2415

(vi) 13 and −19

(vii) −5−9 and 5−9

Answer: (i) −721 and 39

⇒ −721 = −13 and 39 = 13 [Converting into lowest term]

∵ −13≠13

∴ −721≠39

(ii) −1620 and 20−25

⇒ −1620 = −45 and 20−25 = 4−5=−45

[Converting into lowest term]

∵ −45 = −45

∴ −1620 = 20−25

(iii) −2−3 and 23

⇒ −2−3 = 23 and 23 = 23 [Converting into lowest term]

∵ 23 = 23

∴ −2−3 = 23

(iv) −35 and −1220

⇒ −35 = −35 and −1220 = −35 [Converting into lowest term]

∵ −35 = −35

∴ −35 = −1220

(v) 8−5 and −2415

⇒ 8−5 = −85 and −2415 = −85 [Converting into lowest term]

∵ −85 = −85

∴ 8−5 = −2415

(vi) 13 and −19

⇒ 13 = 13 and −19 = −19 [Converting into lowest term]

∵ 13≠−19

∴ 13≠−19

(vii) −5−9 and 5−9

⇒ −5−9 = 59 and 5−9 = 59 [Converting into lowest term]

∵ 59≠5−9

∴ −5−9≠5−9

Question 7. Rewrite the following rational numbers in the simplest form:

(i) −86

(ii) 2545

(iii) −4472

(iv) −810

Answer: (i) −86 = −8÷26÷2 = −43 [H.C.F. of 8 and 6 is 2]

(ii)2545 = 25÷545÷5 = 59

[H.C.F. of 25 and 45 is 5]

(iii)−4472 = −44÷472÷4 = −1118 [H.C.F. of 44 and 72 is 4]

(iv) −810 = −8÷210÷2 = −45 [H.C.F. of 8 and 10 is 2]

Question 8. Fill in the boxes with the correct symbol out of <, > and =:

(i) −57 23

(ii) −45 −57

(iii) −78 14−16

(iv) −85 −74

(v) 1−3 −14

(vi) 5−11 −511

(vii) 0 −76

Answer: (i) −57 < 23 Since, the positive number if greater than negative number.

(ii) −4×75×7 −5×57×5 ⇒ −2835 < −2535 ⇒ −45 < −57

(iii) −7×28×2 14×(−1)−16×(−1) ⇒ −1416 = −1416 ⇒−78 = 14−16

(iv) −8×45×4 −7×54×5 ⇒ −3220 > −3520 ⇒ −85 > −74

(v) 1−3 −14 ⇒1−3 < −14

(vi) 5−11 −511 ⇒5−11 = −511

(vii) 0 > −76 Since, 0 is greater than every negative number.

Question 9. Which is greater in each of the following:

(i) 23,52

(ii) −56,−43

(iii) −34,2−3

(iv) −14,14

(v) −327,−345

Answer: (i) 2×23×2=46 and 5×32×3=156

Since 46 < 156

Therefore 23 < 52

(ii) −5×16×1=−56 and −4×23×2=−86

Since −56 > −86 Therefore −56 > −43

(iii) −3×34×3=−912 and 2×(−4)−3×(−4)=−812

Since −912 < −812

Therefore −34 < 2−3

(iv) −14 < 14 Since positive number is always greater than negative number.

(v) −327=−237=−23×57×5=−11535 and −345=−195=−19×75×7=−13335

Since −11535 > −13335

Therefore−327 > −345

Question 10. Write the following rational numbers in ascending order:

(i) −35,−25,−15

(ii) 13,−29,−43

(iii) −37,−32,−34

Answer: (i) −35,−25,−15⇒ −35<−25<−15

(ii) 13,−29,−43 ⇒39,−29,−129 [Converting into same denominator]

Now −129<−29<39 ⇒ −43<−29<13

(iii) −37,−32,−34

⇒