Exercise 1.1

Q.1 Using appropriate properties find.
(i) −23×35+52−35×16
(ii) 25×(−37)−16×32+114×25
Sol. (i) Given, −23×35+52−35×16
= −23×35−35×16+52 (Using Commutative property)
= 35(−23−16)+52 (Using Distributive property)
= 35(−4−16)+52
= 35(−56)+52
= 35×−56+52
= −36+52
= −3+156
= 126
= 2

(ii) Given, 25×(−37)−16×32+114×25
= 25×(−37)+114×25−16×32 (Using Commutative property)
= 25(−37+114)−16×32 (Using Distributive property)
= 25(−6+114)−16×32
= 25(−514)−16×32
= 25×−514−14
= −17−14
= −4−728
= −1128

Q.2 Write the additive inverse of each of the following.
(i)28
(ii) −59
(iii)−6−5
(iv)2−9 (v) 19−6
Sol. (i) 28
We know, 28+(−28)=28−28=0
Hence, the additive inverse of 28 is (−28).

(ii) −59
We know, −59+59=−5+59=0
Hence, the additive inverse of −59 is 59.

(iii) −6−5
We know, −6−5=65
Now, 65+(−65)=6−65=0
Hence, the additive inverse of −6−5is −65.

(iv) 2−9
We know, 2−9+29=−2+29=0
Hence, the additive inverse of 2−9is 29.

(v) 19−6
We know, 19−6+196=−19+196=0
Hence, the additive inverse of 19−6is 196.

Q.3 Verify that – (-x) = x for.
(i) x=1115
(ii) x=−1317

Sol. (i) x=1115
The additive inverse of x=1115 is −x=−1115
Thus, 1115+(−1115)=0
Now, the additive inverse of −1115 is 1115
Thus, −(−1115)=1115
Hence, proved that−(−x)=x.

(ii) x=−1317
The additive inverse of x=−1317is −x=1317
Thus, −1317+137=0
The additive inverse of 1317is −1317
Hence, proved that −(−x)=x.

Q.4 Find the multiplicative inverse of the following.
(i) -13
(ii)−1319
(iii) 15
(iv) −58×−37
(v) −1×−25
(vi)1
Sol. The multiplicative inverse is defined as the reciprocal of the given number.
(i) -13
Hence, the multiplicative inverse of -13 is equal to −113

(ii) −1319
Hence, the multiplicative inverse of −1319is equal to 19−13

(iii) 15
Hence, the multiplicative inverse of 15 is equal to 51 or 5.

(iv) −58×−37
We know that, −58×−37 =(−5)×(−3)8×7=1556
Hence, multiplicative inverse of 1556 is 5615

(v) −1×−25
We know that, −1×−25=25
Hence, multiplicative inverse of 25 is 52

(vi)1
We know that, -1 is equal to 1−1 = -1
Hence, multiplicative inverse of -1 is -1.

Q.5 Name the property under multiplication used in each of the following.
(i) −45×1=1×−45=−45
(ii) −1317×−27=−27×−137
(iii) −1929×29−19=1

Sol. (i) −45×1=1×−45=−45
We know that, 1 is the multiplicative identity for rational numbers.
Hence, the property of multiplicative identity is used here.

(ii) −1317×−27=−27×−137
When rational numbers are swapped between one operators and still their result does not change, then we say that the numbers follow the commutative property for that operation.
Hence, commutative property is used here.

(iii) −1929×29−19=1
The reciprocal of is 29−19
Thus, multiplicative inverse property is used here.

Q.6 Multiply 613by the reciprocal of −716.
Sol. We know that, the reciprocal of −716 is 16−7
So,613×16−7=6×1613×(−7)
=96−91

Q.7 Tell what property allows you to compute 13×(6×43) as (13×6)×43
Sol. When rational numbers are rearranged between one or more same operations and still their result does not change then we say that they follow the associative property for that operation.
Thus, given equation follows the associative property.

Q.8 Is  the multiplicative inverse of −118 ? Why or why not?
Sol. We can write, −118=−78
Now, multiplying both numbers we get, 89×−78=−79≠1
The result is not equal to 1.
Hence, −118 is not the multiplicative inverse of 89.

Q.9 Is 0.3 the multiplicative inverse of 13? Why or why not?
Sol. We know that, 0.3=310
The multiplicative inverse of 310 is 103 
Again, we know that 103=313
Hence, 313 is the multiplicative inverse of 0.3

Q.10 Write.
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
Sol. (i) Zero (0) is the rational number that does not have a reciprocal.
(ii) 1 and – 1 are the rational numbers that are equal to their reciprocals.
(iii) Zero (0) is the rational number that is equal to its negative.

Q.11 Fill in the blanks.
(i) Zero has __________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals.
(iii) The reciprocal of – 5 is _____________.
(iv) Reciprocal of 1/x, where x ≠ 0is ______________.
(v) The product of two rational numbers is always a _____________.
(vi) The reciprocal of a positive rational number is ____________.
Sol. (i) No.
(ii) 1 and – 1.
(iii) -1/5.
(iv) x.
(v) Rational Number.
(vi) Positive.

Exercise 1.2

Q.1 Represent these numbers on the number line.
(i)74
(ii) −56
Sol. (i) 74

(ii) −56

Q.2 Represent −211,−511,−911 on the number line.
Sol.

Q.3 Write five rational numbers which are smaller than 2.
Sol. There can be infinite rational numbers smaller than 2.
The random five rational numbers smaller than 2 are:1,12,13,0,−12.

Q.4 Find ten rational numbers between −25and 12.
Sol. The five rational numbers between −25and 12are −310,−210,−110,0,110

Q.5 Find five rational numbers between
(i) 23and45
(ii) −32and53
(iii) 14and12
Sol. (i) 23and45
The given numbers can be written as 2×153×15=3045 and 4×95×9=3645
Hence, five rational numbers between 23and45are 3145,3245,3345,3445,3545

(ii) −32and53
The given numbers can be written as −3×32×3=−96 and 5×23×2=106
Hence, five rational numbers between −32and53are−86,−76,−1,−56,−46

(iii) 14and12
The given numbers can be written as 1×84×8=832and 1×162×16=1632
Hence, five rational numbers between 14and12are 932,1032,1132,1232,1332

Q.6 Write five rational numbers greater than – 2
Sol. There can be infinite rational numbers greater than -2.
The random five rational numbers greater than -2 are: -1, 0, 1, ½ and 2.

Q.7 Find ten rational numbers between35and34.
Sol. The given numbers can be written as 3×205×20=60100 and 3×254×25=75100
Hence, ten rational numbers between 35and34are61100,62100,63100,64100,65100,66100,67100,68100,69100,70100