Integers Exercise 1.1 |

**1)1. Evaluate: **

**(i) 15 +(- 8) **

15 + (-8) = 7

**(ii) (-16)+9**

(-16)+ 9 = -7

**(iii) (-7) + (-23) **

(-7) + (-23) = -30

**(Iv) (-32) + 47 **

(-32) + 47 = 15

**(v) 53 +(-26) **

53 + (-26) = 27

**(vi) (-48) + (-36)**

(-48) + (-36) = -84

**2) Find the sum of: **

** (i) 153 and -302 **

153 + (-302) = -149

**(ii) 1005 and -277**

1005 + (-277) = 728

**(iii) -2035 and 297 **

(-2035) + 297 = -1738

**(iv) -489 and -324 **

(-489) + (-324) = -813

**(v) -1000 and 438 **

(-1000) + 438 = -562

**(vi) -238 and 500**

(-238) + 500 = 262

**3) Find the additive inverse of:**

**Solution:**

**(i)-83**

Additive inverse of -83 = -(-83) = 83

**(ii)256**

Additive inverse of 256 = -(256) = -256

**(iii)0**

Additive inverse of 0 = -(0) = 0

**(iv)-2001**

Additive inverse of 2001 = -(-2001) = 2001

**4) Subtract:**

**Solution:**

**(i) 28 from -42**

-42 – 28 = (-42) + (-28) = -70

**(ii) -36 from 42**

42 -(-36) = 42 + 36 = 78

**(iii) -37 from -53**

-53 – (-37) = (-53) – (-37) = -16

**(iv) -66 from -34**

-34 – (-66) = -34 + 66 = 32

**(v) 318 from 0**

0 – 318 = -318

**(vi) -153 from -240**

(-240) – (-153) = -87

**(vii) -64 from 0**

0 – (-64) = 0 + 64 = 64

**(viii) -56 from 144**

144 – (-56) = 144 + 56 = 200

**5) Subtract the sum of -1032 and 878 from -34.**

**Solution:**

Sum of -1032 and 878 = -1032 + 878 = -154

Subtracting the sum from -34, we get

-34 – (-154)

= (-34)+ 154

= 120

**6) Subtract -134 from the sum of 38 and -87.**

**Solution:**

First, we will calculate the sum of 38 and -87.

38 + (-87) = -49

Now, subtracting -134 from the sum, we get:

-49 – (-134)

=(-49) + 134

= 85

**7) Fill in the blanks:**

**(i) {(-13) + 27}4- (-41) = (-13) + {27 +(……..)}**

**(ii) (-26) + {(-49) + (- 83)1 = {(-26)+ (-49)1+ (…..) **

**(iii) 53 + (-37) = (-37) + (……….) **

**(iv) (-68) + (-76) = (……….) +(- 68)**

**(v) (-72) + (……..) = -72 **

**(vi) -(-83) =………**

**(vii) (-60) – (………)= -59 **

**(viii) (-31) +(……….) = -40**

**Solution:**

(i) -41 ( Associative property)

(ii) -83 ( Associative property)

(iii) 53 (v Commutative property)

(iv) -76 ( Commutative property)

(v) 0( Additive identity)

(vi) 83 ( Additive inverse)

(vii) (-60) – (-59) = -1

(viii) (-40) – (-31) = -9

**8)Simplify: (-13 – (-27)) + (-25 – (-40)). **(-13 – (-27)) + (-25 – (-40))

= (-13 + 27) + (-25 + 40)

=14 + 15

= 29

**9) Find 36 – (-64) and (-64) -36. Are they equal?**

36-(-64)= 36+64= 100

Now, (-64) – 36 = (-64) + (-36) = -100

Here, 100 is not equal -100

hus, they are not equal.

**10) Ifa =-8,b=-7,c=6,verify that(a+b)+c=a+(b+c).**

(a+b)+c=(-8+(-7))+6=-15+6=-9

a +(b+c)= – 8 +(-7+6)=-13+(-1)=-9

Hence, (a + b) + c = a + (b + c) [i.e., Property of Associativity]

**11) If a = -9 and b = -6, show that (a – b) Not Equal To (b – a).**

Here, (a – b) = -9 – (-6) = -3

Similarly, (b – a) = -6 – (-9) = 3

:. (a-b) not equal to (b-a)

**12) The sum of two integers is -16. If one of them is 53. find the other.**

Let the other integer be a. Then, we have:

53+a=-16

=>a = -16 – 53 = -69

The other integer is -69.

**13) The sum of two integers is 65. If one of them is -31, find the other.**

Let the other integer be a.

Then, -31 + a= 65

=>a= 65 – (-31) = 96

The other integer is 96.

**14) The difference of an integer a and (-6) is 4. Find the value of a.**

**
** We have:

a – (-6) = 4

=>a=4+(-6)=-2

:. a = -2

**15)Write a pair of integers whose sum gives **

**(i) zero; **

**(ii) a negative integer; **

**(iii) an integer smaller than both the integers; **

**(iv) an integer greater than both the integers; **

**(v) an integer smaller than only one of the integers.**

(i) Consider the integers 8 and -8. Then, we have: 8 + (-8) = 0

(ii) Consider the integers 2 and (-9). Then, we have: 2 + (-9)= -7, which is a negative integer.

(iii) Consider the integers -4 and -5. Then, we have: (-4) + (-5) = -9, which is smaller than -4 and -5.

(iv) Consider the integers 2 and 6. Then, we have: 2 + 6 = 8, which is greater than both 2 and 6.

(v) Consider the integers 7 and -4. Then, we have: 7 + (-4) = 3, which is smaller than 7 only.

**16)For each of the following statements, write (T) for true and (F) for false:**

**(i) The smallest integer is zero. **

**(ii) -10 is greater than -7. **

**(iii) Zero is larger than every negative integer.**

**(iv) The sum of two negative integers is a negative integer. **

**(v) The sum of a negative integer and a positive integer is always a positive integer.**

(i) F (false). -3. -90 and -100 are also integers. We cannot determine the smallest integer. since they are infinite.

(ii) F (false). -10 is less than -7.

(iii) T (true). Al negative integers are less than zero.

(iv) T (true).

(v) F (false). Example: -9 + 2 = -7

**EXERCISE – 1B**

**1)Multiply:
(i) 16 by 9**

**(ii) 18 by -6 **

**(iii) 36 by -11 **

**(iv) -28 by 14 **

**(v) -53 by 18 **

**(vi) -35 by 0 **

**(vii) 0 by -23 **

**(viii) -16 by -12 **

**(ix) -105 by -8 **

**(x) -36 by -50 **

**(xi) -28 by -1 **

**(xii) 25 by -11**

(i) 16 x 9 = 144

(ii) 18 x (-6) = -(18×6) = -108

(iii) 36 x (-11) = – (36×11) = -396

(iv) (-28) x14 = -(28×14) = -392

(v) (-53) X 18 = -(53×18) = -954

(vi) (-35) x 0 = 0

(vii) 0 x (-23) = 0

(viii) (-16) x (-12) = 192

(ix) (-105) x (-8) = 840

(x) (-36) x (-50) = 1800

(xi) (-28) x (-1) = 28

(xii) 25 X (-11) = (25×11) = -275

**2)Find each of the following products: **

**(i) 3x4x(-5) **

**(ii) 2 x (-5)x (-6) (iii) (-5)x (-8)x (-3) **

**(iv) (-6)x 6 x (-10) **

**(v) 7x (-8)x3 **

**(vi) (-7)x (-3)x 4**

(i) 3 x 4 x (-5) = (12) x (-5) = -60

(ii) 2 x(-5)x(-6)=(-10)x(-6)=60

(iii) (-5) x (-8) x (-3) = (-5) x (24) = -120

(iv) (-6)x6 x(-10)=6 x(60)=360

(v) 7 x (-8) x 3 = 21 x (-8) = -168

(vi) (-7) x (-3) x 4 = 21 x 4 = 84

**3)Find each of the following products: **

**(i) (- 4)x (-5)x(- 8)x(-10) **

**(ii) (-6)x (- 5)x (-7) x (-2) x (-3) **

**(iii) (-60)x (-10)x (-5) x (-1) **

**(iv) (-30)x (-20)x (-5) **

**(v) (-3)x (-3)x (-3)x … 6 times **

**(vi) (-5)x (-5) x (-5) x … 5 times **

**(vii) (-1)x (-1)x (-1)x … 200 times **

**(viii) (-1)x (-1)x (-1)x …171 times**

(i) Since the number of negative integers in the product is even. the product will be positive. (4) x (5) x (8) x (10) = 1600

(ii) Since the number of negative integers in the product is odd. the product will be negative. -(6) x (5) x (7) x (2) x (3) = -1260

(iii) Since the number of negative integers in the product is even, the product will be positive. (60) x (10) x (5) x (1) = 3000

(iv) Since the number of negative integers in the product is odd. the product will be negative. -(30) x (20) x (5) = -3000

(v) Since the number of negative integers in the product is even. the product will be positive. (-3)6 = 729

(vi) Since the number of negative integers in the product is odd. the product will be negative. (-5)5 = -3125

(vii) Since the number of negative integers in the product is even the product will be positive. (-1)200= 1

(viii) Since the number of negative integers in the product is odd. the product will be negative. (-1)171 = -1

**4)What will be the sign of the product, if we multiply 90 negative integers and 9 positive integers?**

Multiplying 90 negative integers will yield a positive sign as the number of integers is even.

Multiplying any two or more positive integers always gives a positive integer.

The product of both(the above two cases) the positive and negative integers is also positive.

Therefore, the final product will have a positive sign.

**5) What will be the sign of the product, if we multiply 103 negative integers and 65 positive Integers?**

Multiplying 103 negative integers will yield a negative integer, whereas 65 positive integers will give a positive integer. The product of a negative integer and a positive integer is a negative integer.

**6) Simplify: **

**(i) (-8) x 9 + (- 8) x 7 **

**(ii) 9 x (-13)+ 9 x (-7) **

**(iii) 20 x (-16) + 20 x14 **

**(iv) (-16)x (-15) + (-16) x (-5) **

**(v) (-11)x (-15)+ (-11) x (-25) **

**(vi) 10 x (-12)+5 x (-12) **

**(vii) (-16)x (- 8) + (- 4) x (-8) **

**(viii) (-26) x 72 + (-26) x 28**

(i) (-8) x (9 + 7) [using the distributive law]

= (-8) x 16 = -128

(ii) 9 x (-13 + (-7)) [using the distributive law]

=9 x(-20)=-180

(iii) 20 x (-16 + 14) [using the distributive law]

= 20 x (-2) = -40

(iv) (-16) x (-15 + (-5)) [using the distributive law]

= (-16) x (-20) = 320

(v) (-11) x (-15 +(-25)) [using the distributive law]

= (-11) x (-40) = 440

(vi) (-12) x (10 + 5) [using the distributive law]

= (-12) x 15 = -180

(vii) (-16 + (-4)) x (-8) [using the distributive law]

= (-20) x (-8) = 160

(viii) (-26) x (72 + 28) [using the distributive law]

= (-26) x100 = -2600

**7)Fill In the blanks: **

**(i) (-6)x(……) = 6 **

**(ii) (-18)x (……) = (-18) **

**(iii) (-8) x (-9) = (-9) x (……)**

**(iv) 7 x (-3) = (-3)x (……) **

**(v) I(-5) x x (- 6) = (……)x {3 x (- 6)} **

**(vi) (-5)x(……)=0**

(i) (-6) x (x) = 6

x = 6 / – 6

x = -1

Thus, x = (-1)

(ii) 1 [Multiplicative identity]

(iii) (-8) [Commutative law]

(iv) 7 [Commutative law]

(v) (-5) [Associative law]

(vi) 0 [Property of zero]

**8)In a class test containing 10 questions, 5 marks are awarded for every correct answer and (-2) marks are awarded for every incorrect answer and 0 for each question not attempted. **

**(i) Ravi gets 4 correct and 6 incorrect answers. What is his score? Oil **

**(ii) Reenu gets 5 correct and 5 Incorrect answers. What Is he score? (iii) Heenagets 2 correct and 5 incorrect answers. What is her score?**

**Solution:
** We have 5 marks for correct answer and (-2) marks for an incorrect answer.

Now, we have the following:

(i) Ravi’s score = 4 x 5 + 6 x (-2)

= 20 + (-12) =8

(ii) Reenu’s score = 5 x 5 + 5 x (-2)

= 25 – 10 = 15

(iii) lieena’s score = 2 x 5+ 5 x (-2)

= 10 – 10 = 0

**9)Which of the following statements are true and which are false? **

**(i) The product of a posttive and a negative integer is negative. **

**(ii) The product of two negative integers is a negative integer. **

**(iii) The product of three negative integers is a negative Integer **

**(iv) Every integer when multiplied with -1 gives its multiplicative inverse.**

**(v) Multiplication on integers is commutative. **

**(vi) Multiplication on integers is associative. **

**(vii) Every nonzero integer has a multiplicative inverse as an integer.**

(i) True.

(ii) False. Since the number of negative signs is even, the product will be a positive integer.

(iii) True. The number of negative signs is odd.

(iv) False. a x (-1) = -a, which is not the multiplicative inverse of a.

(v) True. axb=bxa

(vi) True. (axb)xa=ax(bxa)

(vii) False. Every non-zero integer a has a multiplicative inverse 1a, which is not an integer.

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