Integers

(a) By observing the above-given data, we can represent the temperature of cities as follows:

(b) From the above table, it is clear that:

Hottest place temperature = 22^oC

Coolest place temperature = -8^oC

Therefore,

Temperature diference = Hottest place temperature – Coolest place temperature

= 22 – (-8)

= 30^oC

Hence,

Temperature between the hottest and coolest place is 30^oC

(c) From the above table, it is clear that:

Temperature of Lahaul Spiti = -8^oC

Temperature at Srinagar = -2^oC

Therefore,

Temperature difference of both cities = Temperature of Lahulspiti - Temperature of Srinagar

= -8 – (-2)

= -8 + 2

= - 6^oC

Hence,

The temperature difference between Lahaul Spiti and Srinagar is -6^oC.

(d) From the above table, it is clear that:

Temperature of Srinagar = -2^oC

Also,

Temperature of Shimla = 5^oC

Therefore,

Temperature of Srinagar and Shimla taken together = -2 + 5

= 3^oC

Since,

3^oC < 5^oC

3^oC < Temperature of Shimla

It is clear that temperature of Srinagar and Shimla taken together is less than the temperature of Shimla.

It is given in the question that,

Scores of Jack in 5 successive rounds = 25, -5, -10, 15 and 10

Therefore,

The total score of Jack at the and will be the sum of scores of 5 successive rounds

Therefore,

Total score of Jack = 25 + (-5) + (-10) + 15 + 10

= 25 – 5 – 10 + 15 + 10

= 25 – 15 + 15 + 10

= 25 + 10

= 35

It is given in the question that,

Temperature of Srinagar on Monday = -5^oC

∵ The temperature of Srinagar on Tuesday = Temperature of Srinagar on Monday – 2^oC

= - 5 – 2

= - 7^oC

The temperature of Srinagar on Wednesday = Temperature of Srinagar on Tuesday + 4^oC

= - 7 + 4

= - 3^oC

Therefore,

The temperature of Srinagar on Tuesday and the temperature of Srinagar on Wednesday was – 7^oC and -3^oC respectively.

It is given in the question that,

The height of plane = 5000 m

Also,

Depth of submarine = - 1200 m

Therefore,

The distance between plane and submarine = 5000 – (-1200)

= 5000 + 1200

= 6200 m

It is given in the question that,

The amount deposited in bank account = Rs 2000

Also,

Amount withdrawn from bank account = Rs 1642

Therefore,

Balance left in account = Amount deposited in bank account + Amount withdrawn from the bank account

= 2000 + (-1642)

= 2000 – 1642

= 358

Therefore, money left in Mohan’s account after withdrawal is Rs 358.

We have,

Distance towards the east is represented by positive whereas distance towards the west is represented by a negative integer

It is given in the question that,

Distance traveled in the east direction = 20 km

Also,

Distance traveled in west direction = 30 km

Therefore,

Distance travelled from A = 20 + (-30)

= - 10 km

As the distance is negative, so we will represent the distance travelled by Rita from A towards the west direction.

(i) Sum of the rows of the box,
5 - 1 - 4 = 0
-5 - 2 + 7 = 0
0 + 3 - 3 = 0
Sum of the columns,
5 - 5 + 0 = 0
-5 - 2 + 7 = 0
0 + 3 - 3 = 0
Sum of the diagonals,
5 - 2 - 3 = 0
0 - 2 - 4 = -6
As one of the diagonals does not equal to 0, it is not a magic square.
(ii) Sum of the rows,
1 - 10 + 0 = -9
-4 - 3 - 2 = -9
-6 + 4 - 7 = -9
Sum of the columns,
1 - 4 - 6 = -9
-10 - 3 + 4 = -9
0 - 2 - 7 = -9
Sum of the columns,
1 - 3 - 7 = -9
0 - 3 - 6 = 9
As the sum of rows, columns and diagonals are equal. This is a magic square.

(i) We have given in the question that,

a = 21

b = 18

We have to prove that:

a – (-b) = a + b

Now,

a – (-b) = 21 – (-18)

= 21 + 18

= 39

Also,

a + b = 21 + 18

= 39

Therefore,

a – (-b) = a + b

Hence, verified

(ii) We have given in the question that,

a = 118

b = 125

We have to prove that:

a – (-b) = a + b

Now,

a – (-b) = 118 – (-125)

= 118 + 125

= 243

Also,

a + b = 118 + 125

= 243

Therefore,

a – (-b) = a + b

Hence, verified

(iii) We have given in the question that,

a = 75

b = 84

We have to prove that:

a – (-b) = a + b

Now,

a – (-b) = 75 – (-84)

= 75 + 84

= 159

Also,

a + b = 75 + 84

= 159

Therefore,

a – (-b) = a + b

Hence, verified

(iv) We have given in the question that,

a = 28

b = 11

We have to prove that:

a – (-b) = a + b

Now,

a – (-b) = 28 – (-11)

= 28 + 11

= 39

Also,

a + b = 28 + 11

= 39

Therefore,

a – (-b) = a + b

Hence, verified

(a)(-8) + (-4) ….. (-8) - (-4)

We have,

(-8) + (-4)

= -8 – 4

= - 12

Also,

(-8) – (-4)

= - 8 + 4

= - 4

Hence,

-12 < - 4

⇒ (-8) + (-4) < (-8) - (-4)

(b)(-3) + 7 – (19) ….. 15 – 8 + (-9)

We have,

(-3) + 7 – (19)

= - 3 + 7 – 19

= - 22 + 7

= - 15

Also,

15 – 8 + (-9)

= 15 – 8 – 9

= 15-17

= - 2

Hence,

- 15 < - 2

⇒ (-3) + 7 – (19) < 15 – 8 + (-9)

(c)23 – 41 + 11 ….. 23 – 41 – 11

We have,

23 - 41 + 11

= -18 + 11

= - 7

Also,

23 – 41 – 11

= 23 – 52

= - 29

Hence,

- 7 > - 29

23 – 41 + 11 > 23 – 41 – 11

(d) 39 + (-24) – (15) ….. 36 + (-52) – (-36)

We have,

39 + (-24) – (15)

= 39 – 24 - 15

= 39 – 39

= 0

Also,

36 + (-52) – (- 36)

= 36 – 52 + 36

= 72 – 52

= 20

Hence,

0 < 20

39 + (-24) – (15) < 36 + (-52) – (-36)

(e)– 231 + 79 + 51 ….. – 399 + 159 + 81

We have,

- 231 + 79 + 51

= - 231 + 130

= - 101

Also,

- 399 + 159 + 81

= - 399 + 240

= - 159

Hence,

- 101 > - 159

– 231 + 79 + 51 > – 399 + 159 + 81

(i) Let us assume, down moved steps be represented by positive whereas up moved steps be represented by negative integers

It is given that,

Initially the monkey will be at step = 1

After the 1^st jump monkey’s step = 1 + 3 = 4

After the 2^nd jump monkey’s step = 4 + (-2) = 2

After the 3^rd jump monkey’s step = 2 + 3 = 5

After the 4^th jump monkey’s step = 5 + (-2) = 3

After the 5^th jump monkey’s step = 3 + 3 = 6

After the 6^th jump monkey’s step = 6 + (-2) = 4

After the 7^th jump monkey’s step = 4 + 3 = 7

After the 8^th jump monkey’s step = 7 + (-2) = 5

After the 9^th jump monkey’s step = 5 + 3 = 8

After the 10^th jump monkey’s step = 8 + (-2) = 6

After the 11^th jump monkey’s step = 6 + 3 = 9

Hence,

From above results it is clear that the monkey will be at water level i.e.,9 after 11 jumps

(ii) From above, we have

Initial step of monkey = 9

Therefore,

After the 1^st jump monkey’s step = 9 + (-4) = 5

After the 2^nd jump monkey’s step = 5 + 2 = 7

After the 3^rd jump monkey’s step = 7 + (-4) = 3

After the 4^th jump monkey’s step = 3 + 2 = 5

After the 5^th jump monkey’s step = 5 + (-4) = 1

Hence,

It is clear from the above results is that after 5 jumps the monkey will reach back at the top

(iii) If we will represent the down steps by negative and up steps by positive than the moves are are as follows:

We have,

Moves in (i):

- 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 = 8

Also,

Moves in part (ii):

4 – 2 + 4 – 2 + 4 = 8

Therefore,

Moves in part (ii) shows those steps who moves up.

(a) It is given in the question that,

Sum = - 7

Therefore,

Pair of integers having sum – 7 is as follows:

- 8 + (1) = - 7

(b) It is given in the question that,

Difference = - 10

Therefore,

Pair of integers having the difference – 10 is as follows:

- 12 - (-2)

= - 12 + 2

= - 10

(c) It is given in the question that,

Sum = 0

Therefore,

Pair of integers having sum 0 is as follows:

4 + (-4) = 0

It is given in the question that,

Difference = 8

Therefore,

Pair of negative integers having difference 8 is as follows:

- 2 - (-10)

= - 2 + 10

= 8

It is given in the question that,

Sum = - 5

Therefore,

Negative integer and positive integer having sum – 5 is as follows:

- 8 + (3) = - 5

It is given in the question that,

Difference = - 3

Therefore,

Negative integer and positive integer having difference – 3 is as follows:

- 2 - (1)

= - 2 – 1

= - 3

It is given in the question that,

Scores of Team A = -40, 10, 0

Therefore,

Total score of Team A = - 40 + 10 + 0

= - 30

Also,

Scores of Team B = 10, 0, -40

Therefore,

Total score of Team B = 10 + 0 + (-40)

= 10 – 40

= - 30

Hence,

It is clear from the above results that the scores of both teams are equal.

Yes, we can also add integers in any order.

As in the above case scores of both, the teams are equal in numerical terms but different in order.

(i) We have,

(-5) + (-8) = (-8) + (…..)

By the commutative property a+b=b+a

We have,

(-5) + (-8) = (-8) + (-5)

(ii) We have,

- 53 + (…..) = - 53

As we know 0 is the number which is added to a number to give the same number.

So,

- 53 + (0) = - 53

(iii) We have,

17 + (…..) = 0

answer should be zero, and we know if we add the negative of the number itself it will give zero,

So,

17 + (-17) = 0

(iv) We have,

[13 + (-12)] + (…..) = 13 + [(-12) + (-7)]

We have,

[13 + (-12)] + (-7) = 13 + [(-12) + (-7)]

(v) We have,

(-4) + [15 + (-3)] = [(-4) + 15] + (…..)

We have,

(-4) + [15 + (-3)] = [(-4) + 15] + (-3)

(a) We have,

3 × (-1) = - 3

(b) We have,

(-1) × 225 = - 225

(c) We have,

(-21) × (-30) = -21 × -30

= 630

(d) We have,

(-316) × (-1) = - 316 × - 1

= 316

(e) We have,

(-15) × 0 × (-18) = - 15 × 0 × -18

= 0 × 270

= 0

(f) We have,

(-12) × (-11) × 10 = 12 × -11 × 10

= 12 × - 110

= - 1320

(g) We have,

9 × (-3) × (-6) = 9 × -3 × -6

= - 27 × - 6

= 162

(h) We have,

(-18) × (-5) × (-4) = - 18 × - 5 × - 4

= 90 × - 4

= - 360

(i) We have,

(-1) × (-2) × (-3) × 4 = -1 × -2 × -3 × 4

= 2 × - 12

= - 24

(j) We have,

(-3) × (-6) × (-2) × (-1) = -3 × -6 × -2 × -1

= 18 × 2

= 36

(a) We have,

L.H.S = 18 × [7 + (- 3)]

= 18 × [7 – 3]

= 18 × [4]

= 72

Also, we have:

R.H.S = [18 × 7] + [18 × (-3)]

= 126 + (-54)

= 126 - 54


= 72

Therefore,

L.H.S = R.H.S

Hence,

18 × [7 + (- 3)] = [18 × 7] + [18 × (-3)]

_(b)We have,

L.H.S = (-21) × [(-4) + (-6)]

= (-21) × [- 4 – 6]

= (-21) × (-10)

= 210

Also, we have:

R.H.S = [(-21) × (-4) + [(-21) × (-6)]

= 84 + 126

= 210

Also, we have:

R.H.S = [(-21) × (-4)] + [(-21) × (-6)]

= [21 x 4] + [21 x 6]

= 84 + 126

Therefore,

L.H.S = R.H.S

Hence,

[(-21) × (-4) + [(-21) × (-6)] = [(-21) × (-4)] + [(-21) × (-6)]

If we look the question then there will be three possible cases for this question:

Case 1: When a is any positive integer

So,

(-1) × a = - a

Case 2: When a is any negative integer

So,

(-1) × (-a) = a

Also,

Case 3: When a is 0

So,

(-1) × 0 = 0

Therefore,

From the above results, the possible answers are a, 0 and – a

(a) We have,

22 × (-1) = -22

(b)

We have,

-37 × (-1) = 37

(c) We have,

0 × (-1) = 0

In this question, we have

- 1 × 5 = - 5

- 1 × 4 = - 4 = - 5 + 1

- 1 × 3 = - 3 = - 4 + 1

- 1 × 2 = 2 = - 3 + 1

- 1 × 1 = - 1 = - 2 + 1

- 1 × 0 = 0 = - 1 + 1

Therefore,

- 1 × (-1) = 0 + 1 = 1

(a) We have,

26 × (-48) + (-48) × (-36)

We know that,

(b × a = a × b)

Also,

(a × b + a × c) = a (b + c)

Therefore,

= (-48) × 26 + (-48) × (-36)

= (-48) [26 + (-36)]

= (-48) [26 – 36]

= (-48) [-10]

= 480

(b) We have,

8 × 53 × (-125)

We know that,

(b × a = a × b)

Also,

a × (b × c) = (a × b) × c

Therefore,

= 8 × [53 × (-125)]

= 8 × [(-125) × 53]

= [8 × (-125)] × 53

= [-1000] × 53

= [-1000] × 53

= - 53000

(c) We have,

15 × (-25) × (-4) × (-10)

Therefore,

= 15 × [(-25) × (-4)] × (-10)

= 15 × (100) × (-10)

= 15 × (-1000)

= -15000

(d) We have,

(-41) × 102

We know that,

a × (b + c) = (a × b) + (a × c)

Therefore,

= (-41) × (100 + 2)

= (-41) × 100 + (-41) × 2

= - 4100 – 82

= - 4182

(e) We have,

625 × (-35) + (-625) × 65

We know that,

(a × b + a × c) = a (b + c)

Therefore,

= 625 × [(-35) + (-65)]

= 625 × [-100]

= - 62500

(f) We have,

7 × (50 - 2)

We know that,

a × (b – c) = (a × b)- (a × c)

Therefore,

= (7 × 50) – (7 × 2)

= 350 – 14

= 336

(g) We have,

(-17) × (-29)

We know that,

a × (b + c) = (a × b)+ (a × c)

Therefore,

= (-17) × [-30 + 1]

= [(-17) × (-30)] + [(-17) × 1]

= [510] + [-17]

= 493

(h) We have,

(-57) × (-19) + 57 = 57 × 19 + 57

We know that,

(a × b) + (a × c) = a × (b + c)

Therefore,

= 57 × 19 + 57 × 1

= 57 [19 + 1]

= 57 × 20

= 1140

It is given in the question that,

Initial temperature = 40^oC

Also,

Change in temperature per hour = - 5^oC

Therefore,

Change in temperature after 10 hours = (-5) × 10

= - 50^oC

Hence,

Final temperature = 40^oC + (-50^oC)

= - 10^oC

(i) It is given in the question that,

Marks given to students for 1 correct answer = 5

Therefore,

Marks given to students for 4 correct answers = 5 × 4

= 20

Also,

Marks given to students for 1 wrong answer = - 2

Therefore,

Marks given to students for 6 wrong answers = - 2 × 6

= - 12

Hence,

Marks obtained by Mohan = 20 – 12

= 8

(ii) It is given in the question that,

Marks given to students for 1 correct answer = 5

Therefore,

Marks given to students for 5 correct answers = 5 × 5

= 25

Also,

Marks are given to students for 1 wrong answer = - 2

Therefore,

Marks given to students for 5 wrong answers = - 2 × 5

= - 10

Hence,

Marks obtained by Reshma = 25 – 10

= 15

(iii) It is given in the question that,

Marks given to students for 1 correct answer = 5

Therefore,

Marks given to students for 2 correct answers = 5 × 2

= 10

Also,

Marks given to students for 1 wrong answer = - 2

Therefore,

Marks given to students for 5 wrong answers = - 2 × 5

= - 10

Hence,

Marks obtained by Heena = 10 – 10

= 0

(a) In the question profit is denoted by a positive integer whereas loss is denoted by a negative integer

It is given in the question that:

Profit earned while selling 1 bag of white cement = Rs 8

Therefore,

Profit earned while selling 3000 bags of white cement = 8 × 3000

= Rs 24000

Also,

Loss during selling 1 bag of grey cement = Rs – 5

Therefore,

Loss during 5000 bags of grey cement = - 5 × 5000

= - 25000

Hence,

Total proft or loss earned by the company = Profit + Loss

= 24000 + (-25000)

= - 1000

Hence,

A loss of Rs 1000 is being incurred by the company

(b) It is given in the question that,

Loss during selling 1 bag of grey cement = Rs – 5

Therefore,

Loss during 6400 bags of grey cement = - 5 × 6400

= - 32000

Let the number of bags of white cement to be sold be x

It is also given that,

Profit earned while selling 1 bag of white cement = Rs 8

Therefore,

Profit earned while selling x bags of white cement = x × 8

= 8x

For having a condition of no profit and no loss, we have:

Profit + Loss = 0

8x + (-32000) = 0

8x = 32000

x =

= 4000

Therefore, 4000 bags of white cement must be sold

(a) We have,

(-3) × (…..) = 27

So,

(-3) × (-9) = 27

(b) We have,

5 × (…..) = - 35

So,

5 × (-7) = - 35

(c) We have,

(…..) × (-8) = - 56

So,

7 × (-8) = - 56

(d) We have,

(…..) × (-12) = 132

So,

(-11) × (-12) = 132

(a) We have,

(-30) ÷ 10

=

= - 3

(b) We have,

50 ÷ (-5)

=

= - 10

(c) We have,

(-36) ÷ (-9)

=

= 4

(d) We have,

(-49) ÷ 49

=

= - 1

(e) We have,

13 ÷ [(-2) + 1]

= 13 ÷[-2 + 1]

= 13 ÷ (-1)

=

= - 13

(f) We have,

0 ÷ (-12)

=

= 0

(g) We have,

(-31) ÷ [(-30) + (-1)]

= (-31) ÷ [-30 – 1]

= (-31) ÷ (-31)

=

= 1

(h) We have,

[(-36) ÷ 12] ÷ 3

= ÷ 3

= (-3) ÷ 3

=

= - 1

(i) We have,

[(-6) + 5)] ÷ [(-2) + 1]

= [-6 + 5] ÷ [-2 + 1]

= (-1) ÷ (-1)

=

= 1

(a) It is given in the question that,

a = 12

b = - 4

And,

c = 2

We have to verify: a ÷ (b + c) (a ÷ b) + (a ÷ c)

Now,

a ÷ (b + c)

= 12 ÷ (-4 + 2)

= 12 ÷ (-2)

=

= - 6

Also,

(a ÷ b) + (a ÷ c)

= [12 ÷ (- 4)] + [12 ÷ 2]

= +

= - 3 + 6

= 3

Therefore,

- 6 3

Hence,

a ÷ (b + c) (a ÷ b) + (a ÷ c)

Verified

(b) It is given in the question that,

a = (- 10)

b = 1

And,

c = 1

We have to verify: a ÷ (b + c) (a ÷ b) + (a ÷ c)

Now,

a ÷ (b + c)

= (-10) ÷ (1 + 1)

= (-10) ÷ (2)

=

= - 5

Also,

(a ÷ b) + (a ÷ c)

= [(-10) ÷ 1] + [(-10) ÷ 1]

= +

= - 10 + (- 10)

= - 10 - 10

= - 20

Therefore,

- 5 - 20

Hence,

a ÷ (b + c) (a ÷ b) + (a ÷ c)

Verified

(a) We have,

369 ÷ ……. = 369

So,

369 ÷ 1 = 369

(b) We have,

(-75) ÷ …… = - 1

So,

(-75) ÷ 75 = - 1

(c) We have,

(-206) ÷ …… = 1

So,

(-206) ÷ (-206) = 1

(d) We have,

- 87 ÷ ….. = 87

So,

- 87 ÷ (-1) = 87

(e) We have,

…… ÷ 1 = - 87

So,

(-87) ÷ 1 = - 87

(f) We have,

…… ÷ 48 = - 1

So,

(-48) ÷ 48 = - 1

(g) We have,

20 ÷ …… = - 2

So,

20 ÷ (-10) = - 2

(h) We have,

…… ÷ (4) = - 3

So,

(-12) ÷ (4) = - 3

_{Five such pairs are as follows:}

_{(a) (3, - 1)}

_{This is because,}

_{3 ÷ (-1) = - 3}

_{(b) (-3, 1)}

_{This is because,}

_{(-3) ÷ 1 = - 3}

_{(c) (9, - 3)}

_{This is because,}

_{9 ÷ (-3) = - 3}

_{(d) (-9, 3)}

_{This is because,}

_{(-9) ÷ 3 = - 3}

_{(e) (12, -4)}

_{This is because,}

_{12 ÷ (-4) = - 3}

Method 1:

Given: Temperature at 12 noon = 10^oC

Change in temperature per hour = - 2^oC

To Find: Time at which temperature is 8ºC below zero = -8ºC.

Temperature Difference = -8 - 10 = - 18ºC.

-2ºC change is in 1 hour
1ºC change will be in 1/2 hour.
18ºC change will be in 1/2 × 18 hours.
18ºC change will be in 9 hours.
So, the time at which the temperature is -8ºC = 12 noon + 9 hours = 9 p.m

Now, temperature at 12 midnight = 12 noon + 12 hours
Therefore, we need to find a change in temperature in 12 hours.

In 1-hour change is -2ºC, in 12 hours change will be 12 × -2ºC = -24ºC.

So, the temperature at midnight = 10ºC - 24ºC = - 14ºC


Method 2:
Given: Temperature at 12 noon = 10^oC

Change in temperature per hour = - 2^oC

Therefore,

Temperature at 1:00 PM = 10^oC + (-2^oC) = 8^oC

Temperature at 2:00 PM = 8^oC + (-2^oC) = 6^oC

Temperature at 3:00 PM = 6^oC + (-2^oC) = 4^oC

Temperature at 4:00 PM = 4^oC + (-2^oC) = 2^oC

Temperature at 5:00 PM = 2^oC + (-2^oC) = 0^oC

Temperature at 6:00 PM = 0^oC + (-2^oC) = - 2^oC

Temperature at 7:00 PM = - 2^oC + (-2^oC) = - 4^oC

Temperature at 8:00 PM = - 4^oC + (-2^oC) = - 6^oC

Temperature at 9:00 PM = - 6^oC + (-2^oC) = - 8^oC

Hence,

The temperature will be 8^oC below zero at 9:00 PM

Now,

Change in temperature in 12 hours = - 2^oC × 12

= - 24^oC

Also,

At midnight the temperature will be = 10 + (-24)

= - 14^oC

Hence,

The temperature at midnight will be 14^oC below zero.

(i) It is given in the question that,

Marks given for every correct answer = +3

And,

Marks given for every wrong answer = - 2

Also, it is given that:

Marks obtained by Radhika = 20

Correct answer = 12

Therefore,

Marks obtained for correct answers = 12 × 3

= 36

Hence,

Marks obtained for incorrect answers = Total marks – Marks obtained for correct answers

= 20 – 36

= - 16

As, marks obtained for 1 wrong answer = - 2

Hence,

Number of incorrect answers =

= 8

(ii) It is given in the question that,

Marks given for every correct answer = +3

And,

Marks given for every wrong answer = - 2

Also, it is given that:

Marks obtained by Mohini = - 5

Correct answer = 7

Therefore,

Marks obtained for correct answers = 7 × 3

= 21

Hence,

Marks obtained for incorrect answers = Total marks – Marks obtained for correct answers

= -5 – 21

= - 26

As, marks obtained for 1 wrong answer = - 2

Hence,

Number of incorrect answers =

= 13

In the above question the distance descended is denoted by a negative integer

It is given in the question that,

Initial height = 10 m

Also,

Final depth = - 350 m

Therefore,

Total distance to be descended by the elevator = (-350) – (10)

= - 360 m

Also,

Time taken by the elevator to descend – 6m = 1 minute

Hence,

Time taken by the elevator to descend – 360 m =

= 60 minutes

= 1 hour