##### Question 1:

Following number line shows the temperature in degree celsius ($\xb0$C) at different places on a particular day.

Observe this number line and write the temperature of the places marked on it.

##### Answer:

Temperature of Lahulspiti = –8$\xb0$C

Temperature of Srinagar = –2$\xb0$C

Temperature of Shimla = 5$\xb0$C

Temperature of Ooty = 14$\xb0$C

Temperature of Bangalore = 22$\xb0$C

##### Question 2:

Following number line shows the temperature in degree celsius ($\xb0$C) at different places on a particular day.

What is the temperature difference between the hottest and the coldest places among the above?

##### Answer:

Temperature of hottest place = 22$\xb0$C

Temperature of coldest place = –8$\xb0$C

Difference between hottest and coldest place

= 22$\xb0$C – (–8$\xb0$C)

= 22$\xb0$C + 8$\xb0$C

= 30$\xb0$C

##### Question 3:

Following number line shows the temperature in degree celsius ($\xb0$C) at different places on a particular day.

What is the temperature difference between Lahulspiti and Srinagar?

##### Answer:

Temperature of Lahulspiti = –8$\xb0$C

Temperature of Srinagar = –2$\xb0$C

Difference between temperature of

Lahulspiti and Srinagar

= –8$\xb0$C – (–2$\xb0$C)

= –8$\xb0$C + 2$\xb0$C

= –6$\xb0$C

##### Question 4:

Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar?

##### Answer:

Temperature of Srinagar and Shimla

= –2$\xb0$C + 5$\xb0$C = 3$\xb0$C

Yes, temperature of Srinagar and Shimla together is less than the temperature of Shimla but
it is not less than the temperature of Srinagar.

##### Question 5:

In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack’s scores in five successive rounds were 25, –5, –10, 15 and 10, what was his total at the end?

##### Answer:

Total of Jack’s score

= 25 + (–5) + (–10) + 15 + 10

= 25 – 5 – 10 + 15 + 10

= 25 + 15 + 10 – 5 –10

= 50 – 15 = 35

Thus, Jacks's final score was 35.

##### Question 6:

At Srinagar temperature was –5$\xb0$C on Monday and then it dropped by 2$\xb0$C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4$\xb0$C. What was the temperature on this day?

##### Answer:

Temperature on Monday = –5$\xb0$C

Since temperature dropped by 2$\xb0$C on Tuesday

$\therefore $ Temperature on Tuesday = –5$\xb0$C – 2$\xb0$C = –7$\xb0$C.

Now temperature rose by 4$\xb0$C on Wednesday

$\therefore $ Temperature on Wednesday = –7$\xb0$C + 4$\xb0$C = –3$\xb0$C.

##### Question 7:

A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them?

##### Answer:

Vertical distance between submarine and plane

= 5000 m + 1200 m

= 6200 m

$\therefore $ Thus, vertical distance between submarine and plane is 6200 m.

##### Question 8:

Mohan deposits ₹2000 in his bank account and withdraws ₹1642 from it, the next day If
withdrawal of amount from the account is represented by a negative integer, then how will
you represent the amount deposited?

Find the balance in Mohan’s account after the withdrawal.

##### Answer:

Amount deposited by Mohan = ₹2000

Amount withdraws by Mohan = ₹1642

Balance of Mohan’s account = ₹(2000 – 1642) = ₹358

Withdrawal of amount is represented by a negative integer.

$\therefore $ Deposite of amount is represented by a positive integer.

##### Question 9:

Rita goes 20 km towards east from a point A to the point B. From B, she moves 30 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance travelled towards west? By which integer will you represent her final position from A?

##### Answer:

Rita moves firstly towards east which is represented by a positive integer then she moves
towards west which is opposite to east. So, distance travelled towards west is represented
by negative integer.

Final position of Rita from A = 20 km – 30 km = –10 km

$\therefore $ Rita is 10 km towards west.

##### Question 10:

In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square.

##### Answer:

(i) Sum of first row = 5 + (–1) + (–4)

= 5 – 1 – 4 = 4 – 4 = 0

Sum of 2nd row = –5 + (–2) + 7

= –5 – 2 + 7 = – 7 + 7 = 0

Sum of 3rd row = 0 + 3 + (–3)

= 0 + 3 – 3 = 3 – 3 = 0

Sum of 1st column = 5 + (–5) + 0

= 5 – 5 = 0

Sum of 2nd column = (–1) + (–2) + 3

= – 1 – 2 + 3 = – 3 + 3 = 0

Sum of 3rd column = (–4) + 7 + (–3)

= – 4 + 7 – 3 = 3 – 3 = 0

Sum of 1st diagonal = 5 + (–2) + (–3)

= 5 – 2 – 3 = 5 – 5 = 0

Sum of 2nd diagonal = – 4 + (–2) + 0

= – 4 – 2 = –6

$\therefore $ Sum of each row, each column and one diagonal is 0 but sum of 2nd diagonal is
–6.

$\therefore $ (i) square is not a magic square.

(ii) Sum of first row = 1 + (–10) + 0 = 1 – 10 = –9

Sum of 2nd row = (–4) + (–3) + (–2)

= – 4 – 3 – 2 = –9

Sum of 3rd row = (–6) + 4 + (–7)

= – 6 + 4 – 7 = – 2 – 7 = –9

Sum of 1st column = 1 + (–4) + (–6)

= 1 – 4 – 6 = 1 – 10 = –9

Sum of 2nd column = (–10) + (–3) + 4

= –10 – 3 + 4 = – 13 + 4 = –9

Sum of 3rd column = 0 + (–2) + (–7)

= 0 – 2 – 7 = –9

Sum of 1st diagonal = 1 + (–3) + (–7)

= 1 – 3 – 7

= 1 – 10 = –9

Sum of 2nd diagonal = 0 + (–3) + (–6)

= 0 – 3 – 6

= –9

$\therefore $ Sum of each row, each column and each diagonal is –9.

$\therefore $ (ii) square is a magic square.

##### Question 11:

Verify a – (–b) = a + b for the following values of a = 21, b = 18.

##### Answer:

Given that, a = 21, b = 18

Consider a – (–b) = 21 –(–18) = 21 + 18 = 39

and a + b = 21 + 18 = 39

$\therefore $ a – (–b) = a + b

Hence, verified.

##### Question 12:

Verify a – (–b) = a + b for the following values of a = 118, b = 125.

##### Answer:

Given that, a = 118, b = 125

Consider a – (–b) = 118 – (–125) = 118 + 125 = 243

and a + b = 118 + 125 = 243

$\therefore $ a – (–b) = a + b

Hence, verified.

##### Question 13:

Verify a – (–b) = a + b for the following values of a = 75, b = 84.

##### Answer:

Given that, a = 75, b = 84

Consider a – (–b) = 75 – (–84) = 75 + 84 = 159

and a + b = 75 + 84 = 159

$\therefore $ a – (–b) = a + b

Hence, verified.

##### Question 14:

Verify a – (–b) = a + b for the following values of a = 28, b = 11.

##### Answer:

Given that, a = 28, b = 11

Consider a – (–b) = 28 – (–11) = 28 + 11 = 39

and a + b = 28 + 11 = 39

$\therefore $ a – (–b) = a + b

Hence, verified.

##### Question 15:

Use the sign of >, < or=in the blank space to make the statement true:

(–8) + (–4) _________ (–8) – (–4)

##### Answer:

(–8) + (–4) < (–8) – (–4)

##### Question 16:

Use the sign of >, < or=in the blank space to make the statement true:

(–3) + 7 – (19) _________ 15 – 8 + (–9)

##### Answer:

(–3) + 7 –(19) < 15 – 8 + (–9)

##### Question 17:

Use the sign of >, < or=in the blank space to make the statement true:

23 – 41 + 11 _________ 23 – 41 – 11

##### Answer:

23 – 41 + 11 > 23 – 41 – 11

##### Question 18:

Use the sign of >, < or=in the blank space to make the statement true:

39 + (–24) –(15) _________ 36 + (–52) –(–36)

##### Answer:

39 + (–24) –(15) < 36 + (–52) –(–36)

##### Question 19:

Use the sign of >, < or=in the blank space to make the statement true:

– 231 + 79 + 51 _________ – 399 + 159 + 81

##### Answer:

– 231 + 79 + 51 > – 399 + 159 + 81

##### Question 20:

A water tank has steps inside it. A monkey is sitting on the topmost (i.e., the first step). water level is at the ninth step.

He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level?

##### Answer:

Let, number of steps moved down is represented by negative integers and number of steps moved
up is represented by positive integers Total number of steps to be moved up and down = 9

Number of steps moved down in one jump = 3

Number of steps moved up in one jump = 2

According to question,

Number of jumps in which, he will reach the water level

= –9 + (–3 + 2) = –10

Hence, in 11 jumps he will reach the water level.

##### Question 21:

A water tank has steps inside it. A monkey is sitting on the topmost (i.e., the first step). water level is at the ninth step.

After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step?

##### Answer:

Let, number of steps moved down is represented by negative integers and number of steps moved
up is represented by positive integers Total number of steps to be moved up and down = 9

In one jump number of steps moved up = 4

In one jump number of steps moved down = –2

$\therefore $ Distance covered in single jump = 4 – 2 = 2

Now, total number of steps for top step = 9

Hence, he will reach back the top step in 5 jumps.

##### Question 22:

A water tank has steps inside it. A monkey is sitting on the topmost (i.e., the first step). water level is at the ninth step.

If the number of steps moved down is represented by negative integers and the number of steps
moved up by positive integers, represent his moves in

(i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach
the water level?

(ii) He jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps
will he reach back the top step?

by completing the following;

(a) – 3 + 2 –........... = –8

(b) 4 – 2 + ........... = 8.

In (a) the sum (–8) represents going down by eight steps. So, what will the sum 8 in (b)
represent?

##### Answer:

Let, number of steps moved down is represented by negative integers and number of steps moved
up is represented by positive integers Total number of steps to be moved up and down = 9

(a) – 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 = –8

(b) 4 – 2 + 4 – 2 + 4 = 8

Sum 8 in (b) represents going up 8 steps.

##### Question 23:

Write down a pair of integer whose sum is –7.

##### Answer:

One such pair could be

(–3) + (–4) or 2 + (–9)

##### Question 24:

Write down a pair of integer whose difference is –10.

##### Answer:

One such pair could be

5 – 15 or – 13 – (–3)

##### Question 25:

Write down a pair of integer whose sum is 0.

##### Answer:

One such pair could be

(–3) + 3 or 7 + (–7)

##### Question 26:

Write a pair of negative integers whose difference gives 8.

##### Answer:

One such pair could be

(–13) – (–21) or (–6) – (–14)

##### Question 27:

Write a negative integer and a positive integer whose sum is –5.

##### Answer:

One such pair could be

3 + (–8) or (–7) + 2

##### Question 28:

Write a negative integer and a positive integer whose difference is –3.

##### Answer:

One such pair could be

– 2 – (1)

##### Question 29:

In a quiz, team A scored –40, 10, 0 and team B scored 10, 0, –40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?

##### Answer:

Team A scored total scores = – 40 + 10 + 0 = – 30

Team B scored total scores = 10 + 0 + (–40) = –30

$\therefore $ Both teams A and B scored equal.

Yes, we can add integers in any order.

##### Question 30:

Fill in the blanks to make the following statements true:

(–5) + (–8) = (–8) + (________)

##### Answer:

(–5) + (–8) = (–8) + (–5)

##### Question 31:

Fill in the blanks to make the following statements true:

–53 + ________ = –53.

##### Answer:

– 53 + 0 = –53

##### Question 32:

Fill in the blanks to make the following statements true:

17 + ________ = 0

##### Answer:

17 + (–17) = 0

##### Question 33:

Fill in the blanks to make the following statements true:

[13 + (–12)] + (________) = 13 + [(–12) + (–7)]

##### Answer:

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

##### Question 34:

Fill in the blanks to make the following statements true:

(–4) + {15 + (–3)] = [–4 + 15] + ________

##### Answer:

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

##### Question 35:

Write a pair of integers whose sum gives a negative integer.

##### Answer:

If a and b be any integers such that a < b.

then a + (–b) = a – b; is an integer whose sum is a negative integer.

##### Question 36:

Write a pair of integers whose sum gives zero.

##### Answer:

If a is any integer, then

a + (–a) = a – a = 0.

##### Question 37:

Write a pair of integers whose sum gives an integer smaller than both the integers.

##### Answer:

For this both integers a and b should be

negative, i.e. –a + (–b) = –a – b

e.g. (i) –2 + (–3) = –2 – 3 = –5

(ii) –10 + (–8) = –10 – 8 = –18 and so on.

##### Question 38:

Write a pair of integers whose sum gives an integer smaller than only one of the integers.

##### Answer:

For this, one integer should be positive and other should be negative, i.e.

(i) – 5 + 4 = – 5 + 4 = –1; which is smaller than 4.

(ii) 9 + (–6) = 9 – 6 = 3; which is smaller than 9.

##### Question 39:

Write a pair of integers whose sum gives an integer greater than both the integers.

##### Answer:

For this both integers a and b should be positive, i.e.

4 + 5 = 9; which is greater than both integers 18 + 10 = 28; which is greater than both
integers and so on.

##### Question 40:

Write a pair of integers whose difference gives a negative integer.

##### Answer:

For this two integers a and b be such that a < b i.e., a – b

(i) 4 – (+ 5) = 4 – 5 = –1

(ii) (–9) + 5 = –9 + 5 = –4 and so on.

##### Question 41:

Write a pair of integers whose difference gives zero.

##### Answer:

For any integer a,

a – a = 0

(i) 5 – 5 = 5 – 5 = 0

(ii) (–5) – (–5) = –5 + 5 = 0

##### Question 42:

Write a pair of integers whose difference gives an integer smaller than both the integers.

##### Answer:

(i) 10 – (8) = 10 – 8 = 2

(ii) 25 – 16 = 9 and so on.

##### Question 43:

Write a pair of integers whose difference gives an integer greater than only one of the integers.

##### Answer:

(i) 10 – 2 = 8

(ii) 39 – 13 = 26 and so on.

##### Question 44:

Write a pair of integers whose difference gives an integer greater than both the integers.

##### Answer:

For this, we take both integers a and b negative.

e.g. (i) –2 – (–3) = – 2 + 3 = 1

(ii) –5 – (–8) = – 5 + 8 = 3 and so on.

##### Question 45:

A number line representing integers is given below.

–3 and –2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O?

##### Answer:

From number line, integers marked by

B, D, H, J, M and O are –6, –4, 0, 2, 5 and 7 respectively.

##### Question 46:

Arrange 7, –5, 4, 0 and –4 in ascending order and then mark them on a number line to check your answer.

##### Answer:

The ascending order of given integers is –5, –4,

0, 4 and 7.

##### Question 47:

We have done various patterns with numbers in our previous class. Can you find a pattern for
each of the following? If yes, complete this:

7, 3, –1, –5, ............, .............., ...............

##### Answer:

7, 3, –1, –5, –9, –13, –17

##### Question 48:

We have done various patterns with numbers in our previous class. Can you find a pattern for
each of the following? If yes, complete this:

–2, –4, –6, –8, ............, .............., ...............

##### Answer:

–2, –4, –6, –8, –10, –12, –14

##### Question 49:

We have done various patterns with numbers in our previous class. Can you find a pattern for
each of the following? If yes, complete this:

15, 10, 5, 0, ............, .............., ...............

##### Answer:

15, 10, 5, 0, –5, –10, –15

##### Question 50:

We have done various patterns with numbers in our previous class. Can you find a pattern for
each of the following? If yes, complete this:

–11, –8, –5, –2, ............, .............., ...............

##### Answer:

–11, –8, –5, –2, 1, 4, 7

##### Question 51:

Find using number line

(a) 4 × (–8)

(b) 8 × (–2)

(c) 3 × (–7)

(d) 10 × (–1)

##### Answer:

(a) 4 × (–8)

Therefore 10 × (–1) = –10.

##### Question 52:

Find:

(i) 6 × (–19)

(ii) 12 × (–32)

(iii) 7 × (–22)

##### Answer:

(i) 6 × (–19) = –(6 × 19) = –114

(ii) 12 × (–32) = –(12 × 32) = –384

(iii) 7 × (–22) = –(7 × 22) = –154

##### Question 53:

Find:

(a) 15 × (–16)

(b) 21 × (–32)

(c) (–42) × 12

(d) –55 × 15

##### Answer:

(a) 15 × (–16) = –240

(b) 21 × (–32) = –672

(c) (–42) × 12 = –504

(d) –55 × 15 = –825

##### Question 54:

Check if 25 × (–21) = (–25) × 21

##### Answer:

25 × (–21) = –525

and (–25) × 21 = –525

So, 25 × (–21) = (–25) × 21

##### Question 55:

Check if (–23) × 20 = 23 × (–20)

##### Answer:

(–23) × 20 = –460

and 23 × (–20) = –460

So, (–23) × 20 = 23 × (–20)

##### Question 56:

Starting from (–5) × 4, find (–5) × (–6)

##### Answer:

–5 × 4 = –20

–5 × 3 = –15 [= –20 + 5]

–5 × 2 = –10 [= –15 + 5]

–5 × 1 = –5 [= –10 + 5]

–5 × 0 = 0 [= –5 + 5]

–5 × –1 = 5 [= 0 + 5]

–5 × –2 = 10 [= 5 + 5]

–5 × –3 = 15 [= 10 + 5]

–5 × –4 = 20 [= 15 + 5]

–5 × –5 = 25 [= 20 + 5]

–5 × –6 = 30 [= 25 + 5]

##### Question 57:

Starting from (–6) × 3, find (–6) × (–7)

##### Answer:

–6 × 3 = –18

–6 × 2 = –12 [= –18 + 6]

–6 × 1 = –6 [= –12 + 6]

–6 × 0 = 0 [= –6 + 6]

–6 × –1 = 6 [= 0 + 6]

–6 × –2 = 12 [= 6 + 6]

–6 × –3 = 18 [= 12 + 6]

–6 × –4 = 24 [= 18 + 6]

–6 × –5 = 30 [= 24 + 6]

–6 × –6 = 36 [= 30 + 6]

–6 × –7 = 42 [= 36 + 6]

##### Question 58:

Find:

(i) (–31) × (–100)

(ii) (–25) × (–72)

(iii) (–83) × (–28)

##### Answer:

(i) (–31) × (–100) = 31 × 100 = 3100

(ii) (–25) × (–72) = 25 × 72 = 1800

(iii) (–83) × (–28) = 83 × 28 = 2324

##### Question 59:

Is 10 × [6 + (–2)] = 10 × 6 + 10 × (–2)?

##### Answer:

10 × [6 + (–2)] = 10 × [6 –2] = 10 × 4 = 40

and 10 × 6 + 10 × (–2) = 60 –20 = 40

So, 10 × [6 × (–2)] = 10 × 6 + 10 × (–2)

##### Question 60:

Is (–15) × [(–7) + (–1)] = (–15) × (–7) + (–15) × (–1)?

##### Answer:

(–15) × [(–7) + (–1)] = (–15) × (–8) = 120 and

(–15) × (–7) + (–15) × (–1) = 105 + 15 = 120

So, (–15) [(–7) × (–1)] = (–15) × (–7) + (–15) × (–1)

##### Question 61:

Is 10 × [6 –(–2)] = 10 × 6 –10 × (–2)?

##### Answer:

10 × [6 –(–2)] = 10 × 8 = 80

and 10 × 6 – 10 × (–2) = 60 + 20 = 80

So, 10 × [6 – (–2)] = 10 × 6 –10 × (–2)

##### Question 62:

Is (–15) × [(–7) –(–1)] = (–15) × (–7) – (–15) × (–1)?

##### Answer:

(–15) × [(–7) –(–1)] = (–15) × (–6) = 90

and (–15) × (–7) –(–15) × (–1) = 105 – 15 = 90

So, (–15) × [(–7) –(–1)] = (–15) × (–7) – (–15) × (–1).

##### Question 63:

Find:

(i) (–49) × 18

(ii) (–25) × (–31)

(iii) 70 × (–19) + (–1) × 70 using distributivity

##### Answer:

(i) (–49) × 18 = (–49) × [20 – 2]

= –49 × 20 – (–49) × 2

= (–980) – (–98) = –882

(ii) (–25) × (–31) = (–25) × [(–30) + (–1)]

= (–25) × (–30) + (–25) × (–1)

= 750 + 25 = 775

(iii) 70 × (–19) + (–1) × 70

= 70 [(–19) + (–1)]

= 70 × [(–20)] = –1400

##### Question 64:

Find product of 3 × (–1).

##### Answer:

3 × (–1) = –3

##### Question 65:

Find product of (–1) × 225.

##### Answer:

(–1) × 225 = –225

##### Question 66:

Find product of (–21) × (–30).

##### Answer:

(–21) × (–30) = 630

##### Question 67:

Find product of (–316) × (–1).

##### Answer:

(–316) × (–1)= 316

##### Question 68:

Find product of (–15) × 0 × (–18).

##### Answer:

(–15) × 0 × (–18) = [(–15) × 0] × (–18)]

= 0 × (–18) = 0

##### Question 69:

Find product of (–12) × (–11) × (10).

##### Answer:

(–12) × (–11) × 10 = [(–12) × (–11)] × 10

= 132 × 10 = 1320

##### Question 70:

Find product of 9 × (–3) × (–6).

##### Answer:

9 × (–3) × (–6) = [9 × (–3)] × (–6)

= (–27) × (–6) = 162

##### Question 71:

Find product of (–18) × (–5) × (–4).

##### Answer:

(–18) × (–5) × (–4) = [(–18) × (–5)] × (–4)

= 90 × (–4)= –360

##### Question 72:

Find product of (–1) × (–2) × (–3) × 4.

##### Answer:

(–1) × (–2) × (–3) × 4 = [(–1) × (–2)] × [(–3) × 4]

= 2 × (–12) = –24

##### Question 73:

Find product of (–3) × (–6) × (–2) × (–1).

##### Answer:

(–3) × (–6) × (–2) × (–1) = [(–3) × (–6)] × [(–2) × –1)]

= 18 × 2 = 36

##### Question 74:

Verify the following:

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

##### Answer:

LHS = 18 × [7 + (–3)] = 18 × 4 = 72

RHS = [18 × 7] + [18 × (–3)] = 126 + (–54) = 72

So, 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]

Hence, verified.

##### Question 75:

Verify the following:

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

##### Answer:

LHS = (–21) × [(–4) + (–6)] = (–21) × (–10) = 210

RHS = [(–21) × (–4)] + [(–21) × (–6)] = 84 + 126 = 210

So, (–21) × [(–4) + (–6)] = [(–21) × (–4)] + [(–21) × (–6)]

Hence, verified.

##### Question 76:

For any integer a, what is (–1) × a equals to?

##### Answer:

(–1) × a = –a

##### Question 77:

Determine the integer whose product with (–1) is

(a) –22 (b) 37 (c) 0

##### Answer:

(a) (–1) × 22 = –22

So, if we multiply (–1) with 22, then we get –22.

(b) (–1) × (–37) = 37

So, if we multiply (–1) with (–37), then we get 37.

(c) (–1) × 0 = 0

So, if we multiply (–1) with 0, then we get 0.

##### Question 78:

Starting from (–1) × 5, write various products showing some pattern to show (–1) × (–1) = 1.

##### Answer:

(–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]

(–1) × (–1) = 1 [= 0 + 1]

##### Question 79:

Find the product using suitable property:

26 × (–48) + (–48) × (–36)

##### Answer:

26 × (–48) + (–48) × (–36)

= (–48) × [26 + (–36)]

= – 48 × (–10)

= 480

##### Question 80:

Find the product using suitable property:

8 × 53 × (–125)

##### Answer:

8 × 53 × (–125) = 53 × 8 × (–125)

= 53 × [8 × (–125)]

= 53 × (–1000)

= –53000

##### Question 81:

Find the product using suitable property:

15 × (–25) × (–4) × (–10)

##### Answer:

15 × (–25) × (–4) × (–10)

= [15 × (–10)] × [(–25) × (–4)]

= –150 × 100 = –15000

##### Question 82:

Find the product using suitable property:

(–41) × 102

##### Answer:

(–41) × 102 = (–41) × (100 + 2)

= (–41) × 100 + (–41) × 2

= –4100 + (–82)= –4182

##### Question 83:

Find the product using suitable property:

625 × (–35) + (–625) × 65

##### Answer:

625 × (–35) + (–625) × 65

(–625) × 65 can be written as 625 × (–65)

$\therefore $ 625 × (–35) + (–625) × 65

= 625 × (–35) + 625 × (–65)

= 625 × [(–35) + (–65)]

= 625 × [(–100)] = –62500

##### Question 84:

Find the product using suitable property:

7 × [50 – 2]

##### Answer:

7 × [50 – 2] = 7 × 50 – 7 × 2

= 350 – 14 = 336

##### Question 85:

Find the product using suitable property:

(–17) × (–29)

##### Answer:

(–17) × (–29) = (–17) × [(–30) + 1]

= –17 × (–30) + (–17) × 1

= 510 – 17 = 493

##### Question 86:

Find the product using suitable property:

(–57) × (–19) + 57

##### Answer:

(–57) × (–19) + 57

57 can be written as (–57) × (–1)

$\therefore $ (–57) × (–19) + (–57) × (–1)

= (–57) [(–19) + (–1)]

= –57 × [(–20)] = 1140

##### Question 87:

A certain freezing process requires that room temperature be lowered from 40$\xb0$C at the rate of 5$\xb0$C every hour. What will be the room temperature 10 hours after the process begins?

##### Answer:

Current room temperature = 40$\xb0$C

In one hour room temperature lowered by = 5$\xb0$C

After 10 hours room temperature lowered by = 10 × 5$\xb0$C

After 10 hours, room temperature will be

= 40$\xb0$C – (10 × 5$\xb0$C)

= 40$\xb0$C – 50$\xb0$C = –10$\xb0$C

$\therefore $ 10 hours after the process begins room temperature is –10$\xb0$C.

##### Question 88:

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 questions not attempted.

(i) Mohan gets four correct and six incorrect answers. What is his score?

(ii) Reshma gets five correct answers and five incorrect answers, what is her score?

(iii) Heena gets two correct and five incorrect answers out of 7 questions she attempts.
What is her score?

##### Answer:

(i) Mohan got scores = (4 × 5) + [6 × (–2)]

= [20 + (–12)] = 8

$\therefore $ His score is 8.

(ii) Reshma got scores = 5 × 5 + 5 × (–2)

= 5 × [5 + (–2)]

= 5 × (3) = 15

Her score is 15.

(iii) Heena got scores = 2 × 5 + 5 × (–2)

= 5 × [2 + (–2)]

= 5 × (0) = 0

Her score is 0.

##### Question 89:

A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bags of grey cement sold. The company sells 3000 bags of white cement and 5000 bags of grey cement in a month. What is its profit or loss?

##### Answer:

Company sold sacks white cement = 3000

Profit earned by company = 3000 × ₹8 = ₹24000

Company sold sacks of grey cement = 5000

Loss suffered by company = 5000 × ₹5 = ₹25000

Loss suffered by company after selling white and grey cement = 24000 – 25000 = –1000

Company suffered ₹1000 loss in one month.

##### Question 90:

A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bags of grey cement sold. What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6400 bags.

##### Answer:

Number of grey bags sold by company = 6400

Company suffered loss after selling grey bags = 6400 × ₹5 = ₹32,000

No. of white cement bags sold by company to earn neither profit nor loss = 32000/8 =
4000

So, company sold 4000 white cement bags.

##### Question 91:

Replace the blank with an integer to make it a true statement.

(a) (–3) × _____ = 27

(b) 5 × _____ = (–35)

(c) _____ × (–8) = (–56)

(d) _____× (–12) = 132

##### Answer:

(a) (–3) × (–9) = 27

(b) 5 × (–7) = (–35)

(c) 7 × (–8) = (–56)

(d) (–11) × (–12) = 132

##### Question 92:

Find:

(a) (–100) $\xf7$ 5

(b) (–81) $\xf7$ 9

(c) (–75) $\xf7$ 5

(d) (–32) $\xf7$ 2

##### Answer:

(a) (–100) $\xf7$ 5 = –20

(b) (–81) $\xf7$ 9 = –9

(c) (–75) $\xf7$ 5 = –15

(d) (–32) $\xf7$ 2 = –16

##### Question 93:

Find:

(a) 125 $\xf7$ (–25)

(b) 80 $\xf7$ (–5)

(c) 64 $\xf7$ (–16)

##### Answer:

(a) 125 $\xf7$ (–25) = –5

(b) 80 $\xf7$ (–5) = –16

(c) 64 $\xf7$ (–16) = –4

##### Question 94:

Find:

(a) (–36) $\xf7$ (–4)

(b) (–201) $\xf7$ (–3)

(c) (–325) $\xf7$ (–13)

##### Answer:

(a) (–36) $\xf7$ (–4) = 9

(b) (–201) $\xf7$ (–3) = 67

(c) (–325) $\xf7$ (–13) = 25

##### Question 95:

Is (i) 1 $\xf7$ a = 1?

(ii) a $\xf7$ (–1) = –a? for any integer a.

Take different values of a and check.

##### Answer:

(i) 1 $\xf7$ a = 1 is not true.

(ii) a $\xf7$ (–1) = –a is true.

Students can check this with the help of their teachers.

##### Question 96:

Evaluate: (–30) $\xf7$ 10

##### Answer:

(–30) $\xf7$ 10 = –3

##### Question 97:

Evaluate: 50 $\xf7$ (–5)

##### Answer:

50 × (–5) = –10

##### Question 98:

Evaluate: (–36) $\xf7$ (–9)

##### Answer:

(–36) $\xf7$ (–9) = 4

##### Question 99:

Evaluate: (–49) $\xf7$ (49)

##### Answer:

(–49) $\xf7$ (49) = –1

##### Question 100:

Evaluate: 13 $\xf7$ [(–2) + 1]

##### Answer:

13 $\xf7$ [(–2) + 1] = 13 $\xf7$ (–1) = –13

##### Question 101:

Evaluate: 0 $\xf7$ (–12)

##### Answer:

0 $\xf7$ (–12) = 0

##### Question 102:

Evaluate: (–31) $\xf7$ [(–30) + (–1)]

##### Answer:

(–31) $\xf7$ [(–30) + (–1)] = (–31) $\xf7$ (–31) = 1

##### Question 103:

Evaluate: [(–36) $\xf7$ 12] $\xf7$ 3

##### Answer:

[(–36) $\xf7$ 12] $\xf7$ 3 = (–3) $\xf7$ 3 = –1

##### Question 104:

Evaluate: [(–6) + 5)] $\xf7$ [(–2) + 1]

##### Answer:

[(–6) + 5] $\xf7$ [(–2) + 1]= (–1) $\xf7$ (–1) = 1

##### Question 105:

Verify that a $\xf7$ (b + c) $\ne $ (a $\xf7$ b) + (a $\xf7$ c) for a = 12, b = –4, c = 2.

##### Answer:

Given that a = 12, b = –4, c = 2

LHS = a $\xf7$ (b + c)

= 12 $\xf7$ (– 4 + 2)

= 12 $\xf7$ (–2)= –6

RHS = (a $\xf7$ b) + (a $\xf7$ c)

= [12 $\xf7$ (–4)] + [(12) $\xf7$ 2]

= (–3) + (6) = 3

So, a $\xf7$ (b + c) $\ne $ (a $\xf7$ b) + (a $\xf7$ c)

##### Question 106:

Verify that a $\xf7$ (b + c) $\ne $ (a $\xf7$ b) + (a $\xf7$ c) for a = (–10), b = 1, = 1.

##### Answer:

Given that a = –10, b = 1, c = 1

LHS = a $\xf7$ (b + c)

= (–10) $\xf7$ (1 + 1)

= –10 $\xf7$ 2 = –5

RHS = (a $\xf7$ b) + (a $\xf7$ c)

= [(–10) $\xf7$ 1] + [(–10) $\xf7$ 1]

= (–10) + (–10)= –20

So, a $\xf7$ (b + c) $\ne $ (a $\xf7$ b) + (a $\xf7$ c)

##### Question 107:

369 $\xf7$ _____ = 369

##### Answer:

369 $\xf7$ 1 = 369

##### Question 108:

(–75) $\xf7$ _____ = –1

##### Answer:

(–75) $\xf7$ 75 = –1

##### Question 109:

(–206) $\xf7$ _____ = 1

##### Answer:

(–206) $\xf7$ (–206) = 1

##### Question 110:

–87 $\xf7$ _____ = 87

##### Answer:

(–87) $\xf7$ (–1) = 87

##### Question 111:

_____ $\xf7$ 1 = –87

##### Answer:

(–87) $\xf7$ 1 = –87

##### Question 112:

_____ $\xf7$ 48 = –1

##### Answer:

(–48) $\xf7$ 48 = –1

##### Question 113:

20 $\xf7$ _____ = –2

##### Answer:

20 $\xf7$ (–10) = –2

##### Question 114:

_____ $\xf7$ (4) = –3

##### Answer:

–12 $\xf7$ (4) = –3

##### Question 115:

Write five pairs of integers (a, b) such that a $\xf7$ b = –3. One such pair is (6, –2) because 6 $\xf7$ (–2) = (–3).

##### Answer:

(i) 9 $\xf7$ (–3) = (–3)

(ii) 12 $\xf7$ (–4) = (–3)

(iii) 27 $\xf7$ (–9) = (–3)

(iv) (–15) $\xf7$ 5 = (–3)

(v) (–18) $\xf7$ 6 = (–3)

Therefore, (9, –3), (12, –4), (27, –9), (–15, 5) and (–18, 6) are five such pairs.

##### Question 116:

The temperature at 12 noon was 10$\xb0$C above zero. If it decreases at the rate of 2$\xb0$C per hour until midnight, at what time would the temperature be 8$\xb0$C below zero? What would be the temperature at midnight?

##### Answer:

At 12 noon temperature = 0 + 10$\xb0$C = 10$\xb0$C

at 1 pm temperature would be = 0$\xb0$C – 2$\xb0$C = 8$\xb0$C

at 2 pm temperature would be = 8$\xb0$C – 2$\xb0$C = 6$\xb0$C

at 3 pm temperature would be = 6$\xb0$C – 2$\xb0$C = 4$\xb0$C

at 4 pm temperature would be = 4$\xb0$C – 2$\xb0$C = 2$\xb0$C

at 5 pm temperature would be = 2$\xb0$C – 2$\xb0$C = 0$\xb0$C

at 6 pm temperature would be = 0$\xb0$C – 2$\xb0$C = –2$\xb0$C

at 7 pm temperature would be = –2$\xb0$C – 2$\xb0$C = –4$\xb0$C

at 8 pm temperature would be = –4$\xb0$C – 2$\xb0$C= –6$\xb0$C

at 9 pm temperature would be = –6$\xb0$C – 2$\xb0$C = –8$\xb0$C

at 10 pm temperature would be = –8$\xb0$C – 2$\xb0$C = –10$\xb0$C

at 11 pm temperature would be = –10$\xb0$C – 2$\xb0$C = –12$\xb0$C

at 12 midnight temperature would be = –12$\xb0$C – 2$\xb0$C = –14$\xb0$C

So, at 9 pm temperature would be 8$\xb0$C below zero and temperature at midnight would be –14$\xb0$C below zero.

##### Question 117:

In a class test (+3) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any question. Radhika scored 20 marks. If she has got 12 correct answers, how many has she attempted incorrectly?

##### Answer:

Marks given for one correct answer = +3

Marks given for 12 correct answers = 3 × 12 = 36

Radhika’s score = 20

Marks received for incorrect answers = 20 –36 = –16

Marks given for one in correct answer = (–2)

So, number of incorrect answers = (–16) $\xf7$ (–2) = 8.

##### Question 118:

In a class test (+3) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any question. Mohini scores –5 marks in this test, though she has got 7 correct answers How many questions has she attempted incorrectly?

##### Answer:

Marks given for one correct answer = +3

Marks given for correct answers = 3 × 7 = 21

Mohini’s score = –5

Marks received for incorrect answers = –5 –21 = –26

Marks given for one incorrect answer = (–2)

So, number of incorrect answers = (–26) $\xf7$ (–2) = 13.

##### Question 119:

An elevator descends into a mine shaft at the rate of 6 m/minute. If its descent starts from 10 m above the ground level, how long will it take to reach –350 m.

##### Answer:

Total distance covered by elevator = 10 –(–350) = 360 m

Elevator descends at the speed = 6 m per minute

360 m descends covered in = 360 $\xf7$ 6 = 60 minutes

$\therefore $ Elevator will take 60 minute or 1 hour to reach –350 m below the ground level.