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#### Page No 63:

#### Question 1:

#### Answer:

(i) A decrease of 8

(ii) A gain of Rs 7

(iii) Losing a weight of 5 kg

(iv) 10 km below the sea level

(v) 5^{o}C above the freezing point

(vi) A withdrawal of Rs 100

(vii) Spending Rs 500

(viii) Going 6 m to the west

(ix) The opposite of 24 is -24.

(x) The opposite of -34 is 34.

#### Page No 63:

#### Question 2:

(i) A decrease of 8

(ii) A gain of Rs 7

(iii) Losing a weight of 5 kg

(iv) 10 km below the sea level

(v) 5^{o}C above the freezing point

(vi) A withdrawal of Rs 100

(vii) Spending Rs 500

(viii) Going 6 m to the west

(ix) The opposite of 24 is -24.

(x) The opposite of -34 is 34.

#### Answer:

(i) +Rs 600

(ii) -Rs 800

(iii) -7^{o}C

(iv) -9

(v) +2 km

(vi) -3 km

(vii) + Rs 200

(viii) -Rs 300

#### Page No 64:

#### Question 3:

(i) +Rs 600

(ii) -Rs 800

(iii) -7^{o}C

(iv) -9

(v) +2 km

(vi) -3 km

(vii) + Rs 200

(viii) -Rs 300

#### Answer:

(i) -5

(ii) -2

(iii) 0

(iv) 7

#### Page No 64:

#### Question 4:

(i) -5

(ii) -2

(iii) 0

(iv) 7

#### Answer:

(i)0, -2

0 > -2

This is because zero is greater than every negative integer.

(ii) -3, -5

-3 > -5

Since 3 is smaller than 5, -3 is greater than -5.

(iii) -5, 2

2 > -5

This is because every positive integer is greater than every negative integer.

(iv) -16, 8

8 > -16

This is because every positive integer is greater than every negative integer.

v) -365, -913

-365 > -913

Since 365 is smaller than 913, -365 is greater than -913.

vi) -888, 8

8 > -888

This is because every positive integer is greater than every negative integer.

#### Page No 64:

#### Question 5:

(i)0, -2

0 > -2

This is because zero is greater than every negative integer.

(ii) -3, -5

-3 > -5

Since 3 is smaller than 5, -3 is greater than -5.

(iii) -5, 2

2 > -5

This is because every positive integer is greater than every negative integer.

(iv) -16, 8

8 > -16

This is because every positive integer is greater than every negative integer.

v) -365, -913

-365 > -913

Since 365 is smaller than 913, -365 is greater than -913.

vi) -888, 8

8 > -888

This is because every positive integer is greater than every negative integer.

#### Answer:

i) -7 < 6

This is because every positive integer is greater than every negative integer.

ii) -1 < 0

This is because zero is greater than every negative integer.

iii) -27 < -13

Since 27 is greater than 13, -27 is smaller than -13.

iv) -26 < 17

This is because every positive integer is greater than every negative integer.

v) -603 < -317

Since 603 is greater than 317, -603 is smaller than -317.

vi) -777 < 7

This is because every positive integer is greater than every negative integer.

#### Page No 64:

#### Question 6:

i) -7 < 6

This is because every positive integer is greater than every negative integer.

ii) -1 < 0

This is because zero is greater than every negative integer.

iii) -27 < -13

Since 27 is greater than 13, -27 is smaller than -13.

iv) -26 < 17

This is because every positive integer is greater than every negative integer.

v) -603 < -317

Since 603 is greater than 317, -603 is smaller than -317.

vi) -777 < 7

This is because every positive integer is greater than every negative integer.

#### Answer:

i) 1, 2, 3, 4, 5

ii) -4, -3, -2, -1

iii) -2, -1, 0, 1, 2

iv) -6

#### Page No 64:

#### Question 7:

i) 1, 2, 3, 4, 5

ii) -4, -3, -2, -1

iii) -2, -1, 0, 1, 2

iv) -6

#### Answer:

i) 0 < 7

This is because 0 is less than any positive integer.

ii) 0 > -3

This is because 0 is greater than any negative integer.

iii) -5 < -2

Since 5 is greater than 2, -5 is smaller than -2.

iv) -15 < 13

This is because every positive integer is greater than every negative integer.

v) -231 < -132

Since 231 is greater than 132, -231 is smaller than -132.

vi) -6 < 6

This is because every positive integer is greater than every negative integer.

#### Page No 64:

#### Question 8:

i) 0 < 7

This is because 0 is less than any positive integer.

ii) 0 > -3

This is because 0 is greater than any negative integer.

iii) -5 < -2

Since 5 is greater than 2, -5 is smaller than -2.

iv) -15 < 13

This is because every positive integer is greater than every negative integer.

v) -231 < -132

Since 231 is greater than 132, -231 is smaller than -132.

vi) -6 < 6

This is because every positive integer is greater than every negative integer.

#### Answer:

i) -7 < -2 < 0 < 5 < 8

ii) -100 < -23 < -6 < -1 < 0 < 12

iii) -501 < -363 < -17 < 15 < 165

iv) -106 < -81 < -16 < -2 < 0 < 16 < 21

#### Page No 64:

#### Question 9:

i) -7 < -2 < 0 < 5 < 8

ii) -100 < -23 < -6 < -1 < 0 < 12

iii) -501 < -363 < -17 < 15 < 165

iv) -106 < -81 < -16 < -2 < 0 < 16 < 21

#### Answer:

i) 36 > 7 > 0 > -3 > -9 > -132

ii) 51 > 0 > -2 > -8 > -53

iii) 36 > 0 > -5 > -71 > -81

iv) 413 > 102 > -7 > -365 > -515

#### Page No 64:

#### Question 10:

i) 36 > 7 > 0 > -3 > -9 > -132

ii) 51 > 0 > -2 > -8 > -53

iii) 36 > 0 > -5 > -71 > -81

iv) 413 > 102 > -7 > -365 > -515

#### Answer:

i) 4 more than 6

We want an integer that is 4 more than 6. So, we will start from 6 and proceed 4 steps to the right to obtain 10.

ii) 5 more than -6

We want an integer that is 5 more than -6. So, we will start from -6 and proceed 5 steps to the right to obtain -1.

iii) 6 less than 2

We want an integer that is 6 less than 2. So, we will start from 2 and proceed 6 steps to the left to obtain -4.

iv) 2 less than -3

We want an integer that is 2 less than -3. So, we will start from -3 and proceed 2 steps to the left to obtain -5

#### Page No 64:

#### Question 11:

i) 4 more than 6

We want an integer that is 4 more than 6. So, we will start from 6 and proceed 4 steps to the right to obtain 10.

ii) 5 more than -6

We want an integer that is 5 more than -6. So, we will start from -6 and proceed 5 steps to the right to obtain -1.

iii) 6 less than 2

We want an integer that is 6 less than 2. So, we will start from 2 and proceed 6 steps to the left to obtain -4.

iv) 2 less than -3

We want an integer that is 2 less than -3. So, we will start from -3 and proceed 2 steps to the left to obtain -5

#### Answer:

i) False

This is because 0 is greater than every negative integer.

ii) False

0 is an integer as we know that every whole number is an integer and 0 is a whole number.

iii) True

0 is an integer that is neither positive nor negative. So, the opposite of zero is zero.

iv) False

Since 10 is greater than 6, -10 is smaller than -6.

v) True

This is because an absolute value is a positive number. For example, -2 is an integer, but its absolute value is 2 and it is greater than -2.

vi) True

This is because all negative integers are to the left of 0.

vii) True

This is because natural numbers are positive and every positive integer is greater than every negative integer.

viii) False

This is because the successor of -187 is equal to -186 (-186 + 1). In succession, we move from the left to the right along a number line.

ix) False

This is because the predecessor of -215 is -216 (-216 - 1). To find the predecessor, we move from the right to the left along a number line.

#### Page No 64:

#### Question 12:

i) False

This is because 0 is greater than every negative integer.

ii) False

0 is an integer as we know that every whole number is an integer and 0 is a whole number.

iii) True

0 is an integer that is neither positive nor negative. So, the opposite of zero is zero.

iv) False

Since 10 is greater than 6, -10 is smaller than -6.

v) True

This is because an absolute value is a positive number. For example, -2 is an integer, but its absolute value is 2 and it is greater than -2.

vi) True

This is because all negative integers are to the left of 0.

vii) True

This is because natural numbers are positive and every positive integer is greater than every negative integer.

viii) False

This is because the successor of -187 is equal to -186 (-186 + 1). In succession, we move from the left to the right along a number line.

ix) False

This is because the predecessor of -215 is -216 (-216 - 1). To find the predecessor, we move from the right to the left along a number line.

#### Answer:

i) The value of |-9| is 9

ii) The value of |-36| is 36

iii) The value of |0| is 0

iv) The value of |15| is 15

v) The value of |-3| is 3

$\therefore $ -|-3| = -3

vi) 7 + |-3|

= 7 + 3 (The value of |-3| is 3)

= 10

vii) |7 - 4|

= |3|

= 3 (The value of |3| is 3)

viii) 8 - |-7|

= 8 - 7 (The value of |-7| is 7)

= 1

#### Page No 64:

#### Question 13:

i) The value of |-9| is 9

ii) The value of |-36| is 36

iii) The value of |0| is 0

iv) The value of |15| is 15

v) The value of |-3| is 3

$\therefore $ -|-3| = -3

vi) 7 + |-3|

= 7 + 3 (The value of |-3| is 3)

= 10

vii) |7 - 4|

= |3|

= 3 (The value of |3| is 3)

viii) 8 - |-7|

= 8 - 7 (The value of |-7| is 7)

= 1

#### Answer:

i) Every negative integer that is to the right of -7 is greater than -7.

So, five negative integers that are greater than -7 are -6, -5, -4, -3, -2 and -1.

ii) Every negative integer that is to the left of -20 is less than -20.

So, five negative integers that are less than -20 are -21, -22, -23, -24 and -25.

#### Page No 68:

#### Question 1:

i) Every negative integer that is to the right of -7 is greater than -7.

So, five negative integers that are greater than -7 are -6, -5, -4, -3, -2 and -1.

ii) Every negative integer that is to the left of -20 is less than -20.

So, five negative integers that are less than -20 are -21, -22, -23, -24 and -25.

#### Answer:

i) On the number line, we start from 0 and move 9 steps to the right to reach a point A. Now, starting from A, we move 6 steps to the left to reach point B.

B represents the integer 3.

$\therefore $ 9 + (−6) = 3

(ii) On the number line, we start from 0 and move 3 steps to the left to reach point A. Now, starting from A, we move 7 steps to the right to reach point B.

B represents the integer 4.

$\therefore $ (−3) + 7 = 4

(iii) On the number line, we start from 0 and move 8 steps to the right to reach point A. Now, starting from A, we move 8 steps to the left to reach point B.

B represents the integer 0.

$\therefore $ 8 + (−8) = 0

(iv) On the number line, we start from 0 and move 1 step to the left to reach point A. Now, starting from A, we move 3 steps to the left to reach point B.

B represents the integer −4.

$\therefore $ (−1) + (−3) = −4

(v) On the number line, we start from 0 and move 4 steps to the left to reach point A. Now, starting from A, we move 7 steps to the left to reach point B.

B represents the integer −11.

$\therefore $ (−4) + (−7) = −11

(vi) On the number line, we start from 0 and move 2 steps to the left to reach point A. Now, starting from A, we move 8 steps to the left to reach point B.

B represents the integer −10.

$\therefore $ (−2) + (−8) = −10

(vii) On the number line, we start from 0 and move 3 steps to the right to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 4 steps to the left to reach point C.

C represents the integer −3.

$\therefore $ 3 + (−2) + (−4) = −3

(viii) On the number line, we start from 0 and move 1 step to the left to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 3 steps to the left to reach point C.

C represents the integer −6.

$\therefore $(−1) + (−2) + (−3) = −6

(ix) On the number line, we start from 0 and move 5 steps to the right to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 6 steps to the left to reach point C.

C represents the integer −3.

$\therefore $ 5 + (−2) + (−6) = −3

#### Page No 68:

#### Question 2:

i) On the number line, we start from 0 and move 9 steps to the right to reach a point A. Now, starting from A, we move 6 steps to the left to reach point B.

B represents the integer 3.

$\therefore $ 9 + (−6) = 3

(ii) On the number line, we start from 0 and move 3 steps to the left to reach point A. Now, starting from A, we move 7 steps to the right to reach point B.

B represents the integer 4.

$\therefore $ (−3) + 7 = 4

(iii) On the number line, we start from 0 and move 8 steps to the right to reach point A. Now, starting from A, we move 8 steps to the left to reach point B.

B represents the integer 0.

$\therefore $ 8 + (−8) = 0

(iv) On the number line, we start from 0 and move 1 step to the left to reach point A. Now, starting from A, we move 3 steps to the left to reach point B.

B represents the integer −4.

$\therefore $ (−1) + (−3) = −4

(v) On the number line, we start from 0 and move 4 steps to the left to reach point A. Now, starting from A, we move 7 steps to the left to reach point B.

B represents the integer −11.

$\therefore $ (−4) + (−7) = −11

(vi) On the number line, we start from 0 and move 2 steps to the left to reach point A. Now, starting from A, we move 8 steps to the left to reach point B.

B represents the integer −10.

$\therefore $ (−2) + (−8) = −10

(vii) On the number line, we start from 0 and move 3 steps to the right to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 4 steps to the left to reach point C.

C represents the integer −3.

$\therefore $ 3 + (−2) + (−4) = −3

(viii) On the number line, we start from 0 and move 1 step to the left to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 3 steps to the left to reach point C.

C represents the integer −6.

$\therefore $(−1) + (−2) + (−3) = −6

(ix) On the number line, we start from 0 and move 5 steps to the right to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 6 steps to the left to reach point C.

C represents the integer −3.

$\therefore $ 5 + (−2) + (−6) = −3

#### Answer:

(i)

(−3) + (−9)

= −3 − 9

= −12

(ii)

(−7) + (−8)

= −7 − 8

= −15

(iii)

(−9) + 16

= −9 + 16

= 7

(iv)

(−13) + 25

= −13 + 25

= 12

(v)

8 + (−17)

= 8 − 17

= −9

(v)

2 + (−12)

= 2 − 12

= −10

#### Page No 68:

#### Question 3:

(i)

(−3) + (−9)

= −3 − 9

= −12

(ii)

(−7) + (−8)

= −7 − 8

= −15

(iii)

(−9) + 16

= −9 + 16

= 7

(iv)

(−13) + 25

= −13 + 25

= 12

(v)

8 + (−17)

= 8 − 17

= −9

(v)

2 + (−12)

= 2 − 12

= −10

#### Answer:

(i)

$\begin{array}{l}-\text{365}\\ \frac{\text{}-\text{87}}{-\text{452}}\end{array}$

−365

−365 −87 −365 −87 -365 and -87 are both negative integers. So, we add 365 and 87, and put the negative sign before the sum.

(ii)

$\begin{array}{l}-\text{687}\\ \frac{\text{}-\text{73}}{-\text{760}}\end{array}$

-687 and -73 are both negative integers. So, we add 365 and 87, and put the negative sign before the sum.

(iii)

$\begin{array}{l}-\text{1065}\\ \frac{\text{}-\text{987}}{-2052}\end{array}$

-1065 and -987 are both negative integers. So, we add 1065 and 987, and put the negative sign before the sum.

(iv)

$\begin{array}{l}-\text{3596}\\ \frac{-\text{1089}}{-4685}\end{array}$

-3596 and -1089 are both negative integers. So, we add 3596 and 1089, and put the negative sign before the sum.

#### Page No 68:

#### Question 4:

(i)

$\begin{array}{l}-\text{365}\\ \frac{\text{}-\text{87}}{-\text{452}}\end{array}$

−365

−365 −87 −365 −87 -365 and -87 are both negative integers. So, we add 365 and 87, and put the negative sign before the sum.

(ii)

$\begin{array}{l}-\text{687}\\ \frac{\text{}-\text{73}}{-\text{760}}\end{array}$

-687 and -73 are both negative integers. So, we add 365 and 87, and put the negative sign before the sum.

(iii)

$\begin{array}{l}-\text{1065}\\ \frac{\text{}-\text{987}}{-2052}\end{array}$

-1065 and -987 are both negative integers. So, we add 1065 and 987, and put the negative sign before the sum.

(iv)

$\begin{array}{l}-\text{3596}\\ \frac{-\text{1089}}{-4685}\end{array}$

-3596 and -1089 are both negative integers. So, we add 3596 and 1089, and put the negative sign before the sum.

#### Answer:

i)

$\begin{array}{l}-206\\ \frac{+\mathrm{98}}{-108}\\ \mathrm{Since}\mathrm{we}\mathrm{are}\mathrm{adding}\mathrm{a}\mathrm{negative}\mathrm{number}\mathrm{with}\mathrm{a}\mathrm{positive}\mathrm{number},\\ \mathrm{we}\mathrm{shall}\mathrm{subtract}\mathrm{the}\mathrm{smaller}\mathrm{number},\mathrm{i}.\mathrm{e}.98\mathrm{from}\mathrm{the}\mathrm{greater}\mathrm{number},\mathrm{i}.\mathrm{e}.206\\ 206-98=108\\ \mathrm{Since}\mathrm{the}\mathrm{greater}\mathrm{number}\mathrm{is}\mathrm{negative},\mathrm{the}\mathrm{sign}\mathrm{of}\mathrm{the}\mathrm{result}\mathrm{will}\mathrm{be}\mathrm{negative}.\phantom{\rule{0ex}{0ex}}\mathrm{So},\mathrm{the}\mathrm{answer}\mathrm{will}\mathrm{be}-108\end{array}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

ii)

$\begin{array}{l}\text{}178\\ \frac{\text{-69}}{\text{1}0\text{9}}\end{array}$

$\begin{array}{l}\text{Sinceweareaddinganegativenumberwithapositivenumber,}\\ \text{weshallsubtractthesmallernumber,i.e.69,fromthegreaternumber,i.e.178}\\ \text{178-69=109}\\ \mathrm{Sincethegreaternumberispositive,thesign}\text{oftheresultwillbepositive.}\text{}\end{array}\phantom{\rule{0ex}{0ex}}\mathrm{So,}\mathrm{the}\mathrm{answerwillbe109}\phantom{\rule{0ex}{0ex}}$

(iii)

$\begin{array}{l}\text{}312\\ \text{-}\frac{103}{\underset{}{209\text{\hspace{0.17em}}}}\\ \text{}\end{array}$

Since we are adding a negative number with a positive number,

we shall subtract the smaller number, i.e. -103, from the greater number, i.e. 312

312 - 103 = 209

Since the greater number is positive, the sign of the result will be positive.

So, the answer will be 209

$\begin{array}{l}\left(iv\right)\\ -493\\ \frac{+289}{-204}\end{array}$

Since we are adding a negative number with a positive number,

we shall subtract the smaller number, i.e. 289, from the greater number, i.e. 493.

493 - 289 = 204

Since the greater number is negative, the sign of the result will be negative.

So, the answer will be -204

#### Page No 68:

#### Question 5:

i)

$\begin{array}{l}-206\\ \frac{+\mathrm{98}}{-108}\\ \mathrm{Since}\mathrm{we}\mathrm{are}\mathrm{adding}\mathrm{a}\mathrm{negative}\mathrm{number}\mathrm{with}\mathrm{a}\mathrm{positive}\mathrm{number},\\ \mathrm{we}\mathrm{shall}\mathrm{subtract}\mathrm{the}\mathrm{smaller}\mathrm{number},\mathrm{i}.\mathrm{e}.98\mathrm{from}\mathrm{the}\mathrm{greater}\mathrm{number},\mathrm{i}.\mathrm{e}.206\\ 206-98=108\\ \mathrm{Since}\mathrm{the}\mathrm{greater}\mathrm{number}\mathrm{is}\mathrm{negative},\mathrm{the}\mathrm{sign}\mathrm{of}\mathrm{the}\mathrm{result}\mathrm{will}\mathrm{be}\mathrm{negative}.\phantom{\rule{0ex}{0ex}}\mathrm{So},\mathrm{the}\mathrm{answer}\mathrm{will}\mathrm{be}-108\end{array}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

ii)

$\begin{array}{l}\text{}178\\ \frac{\text{-69}}{\text{1}0\text{9}}\end{array}$

$\begin{array}{l}\text{Sinceweareaddinganegativenumberwithapositivenumber,}\\ \text{weshallsubtractthesmallernumber,i.e.69,fromthegreaternumber,i.e.178}\\ \text{178-69=109}\\ \mathrm{Sincethegreaternumberispositive,thesign}\text{oftheresultwillbepositive.}\text{}\end{array}\phantom{\rule{0ex}{0ex}}\mathrm{So,}\mathrm{the}\mathrm{answerwillbe109}\phantom{\rule{0ex}{0ex}}$

(iii)

$\begin{array}{l}\text{}312\\ \text{-}\frac{103}{\underset{}{209\text{\hspace{0.17em}}}}\\ \text{}\end{array}$

Since we are adding a negative number with a positive number,

we shall subtract the smaller number, i.e. -103, from the greater number, i.e. 312

312 - 103 = 209

Since the greater number is positive, the sign of the result will be positive.

So, the answer will be 209

$\begin{array}{l}\left(iv\right)\\ -493\\ \frac{+289}{-204}\end{array}$

Since we are adding a negative number with a positive number,

we shall subtract the smaller number, i.e. 289, from the greater number, i.e. 493.

493 - 289 = 204

Since the greater number is negative, the sign of the result will be negative.

So, the answer will be -204

#### Answer:

$\begin{array}{l}\left(\text{i}\right)\text{137and}-\text{354}\\ -\text{354}\\ \frac{+137}{-217}\\ \\ \left(\text{ii}\right)\text{1}00\text{1and}-\text{13}\\ 1001\\ \frac{-13}{988}\\ \\ \text{(iii)}-\text{3}0\text{57and199}\\ -\text{3}0\text{57}\\ \frac{\text{199}}{-2858}\\ \\ \left(\text{iv}\right)\text{}-\text{36and1}0\text{27}\\ \text{3}0\text{57}\\ \frac{-\text{36}}{3021}\\ \\ \left(\text{v}\right)\text{}-\text{389and}-\text{1}0\text{32}\\ -\text{1}0\text{32}\\ \frac{-\text{389}}{-1421}\\ \\ \left(\text{vi}\right)\text{}-\text{36and1}00\\ \text{1}0\text{0}\\ \frac{-\text{36}}{64}\\ \\ \text{(vii)3}00\text{2and}-\text{888}\\ \text{3}00\text{2}\\ \frac{-\text{888}}{2114}\end{array}$

(viii) −18, + 25 and −37

25 + (−18) + (−37)

= 25 – (18 + 37)

= 25 – 55

= –30

(ix) −312, 39 and 192

39 + 192 + (−312)

= 39 + 192 - 312

= 231 −312

= −81

(x) −51, −203, 36 and −28

36 + (−51) + (−203) + (−28)

= 36 − (51 + 203 + 28)

= 36 – 282

= −246

#### Page No 68:

#### Question 6:

$\begin{array}{l}\left(\text{i}\right)\text{137and}-\text{354}\\ -\text{354}\\ \frac{+137}{-217}\\ \\ \left(\text{ii}\right)\text{1}00\text{1and}-\text{13}\\ 1001\\ \frac{-13}{988}\\ \\ \text{(iii)}-\text{3}0\text{57and199}\\ -\text{3}0\text{57}\\ \frac{\text{199}}{-2858}\\ \\ \left(\text{iv}\right)\text{}-\text{36and1}0\text{27}\\ \text{3}0\text{57}\\ \frac{-\text{36}}{3021}\\ \\ \left(\text{v}\right)\text{}-\text{389and}-\text{1}0\text{32}\\ -\text{1}0\text{32}\\ \frac{-\text{389}}{-1421}\\ \\ \left(\text{vi}\right)\text{}-\text{36and1}00\\ \text{1}0\text{0}\\ \frac{-\text{36}}{64}\\ \\ \text{(vii)3}00\text{2and}-\text{888}\\ \text{3}00\text{2}\\ \frac{-\text{888}}{2114}\end{array}$

(viii) −18, + 25 and −37

25 + (−18) + (−37)

= 25 – (18 + 37)

= 25 – 55

= –30

(ix) −312, 39 and 192

39 + 192 + (−312)

= 39 + 192 - 312

= 231 −312

= −81

(x) −51, −203, 36 and −28

36 + (−51) + (−203) + (−28)

= 36 − (51 + 203 + 28)

= 36 – 282

= −246

#### Answer:

(i) −57 + 57 = 0

So, the additive inverse of −57 is 57.

(ii) 183 − 183 = 0

So, the additive inverse of 183 is −183.

(iii) 0 + 0 = 0

So, the additive inverse of 0 is 0.

(iv) −1001 + 1001 = 0

So, the additive inverse ofâ€‹ −1001 is 1001.

(v) 2054 − 2054 = 0

So, the additive inverse ofâ€‹ 2054 is −2054

#### Page No 68:

#### Question 7:

(i) −57 + 57 = 0

So, the additive inverse of −57 is 57.

(ii) 183 − 183 = 0

So, the additive inverse of 183 is −183.

(iii) 0 + 0 = 0

So, the additive inverse of 0 is 0.

(iv) −1001 + 1001 = 0

So, the additive inverse ofâ€‹ −1001 is 1001.

(v) 2054 − 2054 = 0

So, the additive inverse ofâ€‹ 2054 is −2054

#### Answer:

(i) The successor of 201:

201 + 1 = 202

(ii) The successor of 70:

70 + 1 = 71

(iii) The successor of −5:

−5 + 1 = −4

(iv) The successor of −99:

−99 + 1 = −98

(v) The successor of −500:

−500 + 1 = −499

#### Page No 68:

#### Question 8:

(i) The successor of 201:

201 + 1 = 202

(ii) The successor of 70:

70 + 1 = 71

(iii) The successor of −5:

−5 + 1 = −4

(iv) The successor of −99:

−99 + 1 = −98

(v) The successor of −500:

−500 + 1 = −499

#### Answer:

(i) The predecessor of 120:

120 − 1 = 119

(ii) The predecessor of 79:

79 − 1 = 78

(iii) The predecessor of −8:

−8 − 1 = −9

(iv) The predecessor of −141:

−141 − 1 = −142

â€‹(v) The predecessor of −300:

−300 − 1 = −301

#### Page No 69:

#### Question 9:

(i) The predecessor of 120:

120 − 1 = 119

(ii) The predecessor of 79:

79 − 1 = 78

(iii) The predecessor of −8:

−8 − 1 = −9

(iv) The predecessor of −141:

−141 − 1 = −142

â€‹(v) The predecessor of −300:

−300 − 1 = −301

#### Answer:

(i) (−7) + (−9) + 12 + (−16)

= 12 − (7 + 9 + 16)

= 12 − 32

= −20

(ii) 37 + (−23) + (−65) + 9 + (−12)

= 37 + 9 − (23 + 65 + 12)

= 46-100

= −54

â€‹(iii) (−145) + 79 + (−265) + (−41) + 2

= 79 +2 − ( 145 + 265 + 41)

= 81 − 451

= −370

(iv) 1056 + (−798) + (−38) + 44 + (−1)

= 1056 + 44 − (798 + 38 + 1)

= 1100 − 837

= −263

#### Page No 69:

#### Question 10:

(i) (−7) + (−9) + 12 + (−16)

= 12 − (7 + 9 + 16)

= 12 − 32

= −20

(ii) 37 + (−23) + (−65) + 9 + (−12)

= 37 + 9 − (23 + 65 + 12)

= 46-100

= −54

â€‹(iii) (−145) + 79 + (−265) + (−41) + 2

= 79 +2 − ( 145 + 265 + 41)

= 81 − 451

= −370

(iv) 1056 + (−798) + (−38) + 44 + (−1)

= 1056 + 44 − (798 + 38 + 1)

= 1100 − 837

= −263

#### Answer:

Let the distance covered in the direction of north be positive and that in the direction of south be negative.

Distance travelled to the north of Patna = 60 km

Distance travelled to the south of Patna = -90 km

Total distance travelled by the car = 60 + (â€‹-90)

= -30 km

The car was 30 km south of Patna.

#### Page No 69:

#### Question 11:

Let the distance covered in the direction of north be positive and that in the direction of south be negative.

Distance travelled to the north of Patna = 60 km

Distance travelled to the south of Patna = -90 km

Total distance travelled by the car = 60 + (â€‹-90)

= -30 km

The car was 30 km south of Patna.

#### Answer:

Total cost price = Price of pencils + Price of pens

= 30 + 90 + 25

= Rs 145

Total amount sold = Price of pen + Price of pencils

= 20 + 70

= 90

Selling price - costing price = 90 $-$ 145

= $-$55

The negative sign implies loss.

Hence, his net loss was Rs 55.

#### Page No 69:

#### Question 12:

Total cost price = Price of pencils + Price of pens

= 30 + 90 + 25

= Rs 145

Total amount sold = Price of pen + Price of pencils

= 20 + 70

= 90

Selling price - costing price = 90 $-$ 145

= $-$55

The negative sign implies loss.

Hence, his net loss was Rs 55.

#### Answer:

(i) True

For example: - 2 + (-1) = -3

(ii) False

It can be negative or positive.

For example: -2 + 3 = 1 gives a positive integer, but -5 + 2 = -3 gives a negative integer.

(iii) True

For example: 100 + (-100) = 0

(iv) False

For example: (-5) + 2 + 3 = 0

(v) False

|-5| = 5 and | -3 | = 3, 5 > 3

(vi) False

|8 − 5| = 3

|8| + |−5| = 8 + 5

= 13

$\therefore $|8 − 5|$\ne $|8| + |−5|

#### Page No 69:

#### Question 13:

(i) True

For example: - 2 + (-1) = -3

(ii) False

It can be negative or positive.

For example: -2 + 3 = 1 gives a positive integer, but -5 + 2 = -3 gives a negative integer.

(iii) True

For example: 100 + (-100) = 0

(iv) False

For example: (-5) + 2 + 3 = 0

(v) False

|-5| = 5 and | -3 | = 3, 5 > 3

(vi) False

|8 − 5| = 3

|8| + |−5| = 8 + 5

= 13

$\therefore $|8 − 5|$\ne $|8| + |−5|

#### Answer:

(i) *a* + 6 = 0

=> a = 0 − 6

=> a = − 6

(ii) 5 + *a* = 0

=> a = 0 − 5

(iii) *a* + (−4) = 0

=> a = 0 − (−4)

=> a = 4

(iv) −8 + *a* = 0

=> a = 0 + 8

=> a = 8

#### Page No 70:

#### Question 1:

(i) *a* + 6 = 0

=> a = 0 − 6

=> a = − 6

(ii) 5 + *a* = 0

=> a = 0 − 5

(iii) *a* + (−4) = 0

=> a = 0 − (−4)

=> a = 4

(iv) −8 + *a* = 0

=> a = 0 + 8

=> a = 8

#### Answer:

(i) −34 − 18

= −52

(ii) 25 − (−15)

= 25 + 15

= 40

(iii) −28 from −43

= −43 − (−28)

= −43 + 28

â€‹= −15

(iv) 68 from −37

= −37 − 68

= −105

â€‹(v) 219 from 0

= 0 − 219

= −219

(vi) −92 from 0

= 0 − (−92)

= 0 + 92

= 92

(vii) −135 from −250

= −250 − (−135)

â€‹= −250 + 135

= −115

(viii) −2768 from −287

= −287 − (−2768)

â€‹= 2768 −â€‹ 287

= 2481

(ix) 6240 from −271

= −271 − (6240)

= −271 − 6240

= −6511

(x) −3012 from 6250

= 6250 − (−3012)

= 6250 + 3012

â€‹= 9262

#### Page No 70:

#### Question 2:

(i) −34 − 18

= −52

(ii) 25 − (−15)

= 25 + 15

= 40

(iii) −28 from −43

= −43 − (−28)

= −43 + 28

â€‹= −15

(iv) 68 from −37

= −37 − 68

= −105

â€‹(v) 219 from 0

= 0 − 219

= −219

(vi) −92 from 0

= 0 − (−92)

= 0 + 92

= 92

(vii) −135 from −250

= −250 − (−135)

â€‹= −250 + 135

= −115

(viii) −2768 from −287

= −287 − (−2768)

â€‹= 2768 −â€‹ 287

= 2481

(ix) 6240 from −271

= −271 − (6240)

= −271 − 6240

= −6511

(x) −3012 from 6250

= 6250 − (−3012)

= 6250 + 3012

â€‹= 9262

#### Answer:

Sum of −1050 and 813:

−1050 + 813

= −237

Subtracting the sum of −1050 and 813 from −23:

−23 − (−237)

= −23 +237

= 214

#### Page No 70:

#### Question 3:

Sum of −1050 and 813:

−1050 + 813

= −237

Subtracting the sum of −1050 and 813 from −23:

−23 − (−237)

= −23 +237

= 214

#### Answer:

Sum of 138 and −250:

138 + (−250)

= 138 − 250

= −112

Sum of 136 and −272:

= 136 + (−272)

= 136 − 272

= −136

Subtracting the sum of −250 and 138 from the sum of 136 and −272:

−136 − (−112â€‹)

= −136 + 112â€‹

= −24

#### Page No 70:

#### Question 4:

Sum of 138 and −250:

138 + (−250)

= 138 − 250

= −112

Sum of 136 and −272:

= 136 + (−272)

= 136 − 272

= −136

Subtracting the sum of −250 and 138 from the sum of 136 and −272:

−136 − (−112â€‹)

= −136 + 112â€‹

= −24

#### Answer:

Adding 33 and −47:

â€‹33 + (−47)

= 33 − 47

= −14

Subtracting −84 from −14:

−14 − (−84)

= −14 + 84

= 70

#### Page No 70:

#### Question 5:

Adding 33 and −47:

â€‹33 + (−47)

= 33 − 47

= −14

Subtracting −84 from −14:

−14 − (−84)

= −14 + 84

= 70

#### Answer:

Difference of −8 and −68:

−8 − (−68)

â€‹= −8 + 68

= 60

Adding -36 to 60:

−36 + 60

= 24

#### Page No 70:

#### Question 6:

Difference of −8 and −68:

−8 − (−68)

â€‹= −8 + 68

= 60

Adding -36 to 60:

−36 + 60

= 24

#### Answer:

(i) [37 − (−8)] + [11 − (−30)]

= (37 + 8) + (11 + 30)

= 45 + 41

= 86

(ii) [−13 − (−17) + [−22 − (−40)]

= (−13 +17) + (-22 + 40)

= 4 + 18

= 22

#### Page No 70:

#### Question 7:

(i) [37 − (−8)] + [11 − (−30)]

= (37 + 8) + (11 + 30)

= 45 + 41

= 86

(ii) [−13 − (−17) + [−22 − (−40)]

= (−13 +17) + (-22 + 40)

= 4 + 18

= 22

#### Answer:

No, they are not equal.

34 − (−72)

= 34 + 72

â€‹= 106

(−72) − 34

= −72 − 34

= −106

Since 106 is not equal to −106, the two expressions are not equal.

#### Page No 70:

#### Question 8:

No, they are not equal.

34 − (−72)

= 34 + 72

â€‹= 106

(−72) − 34

= −72 − 34

= −106

Since 106 is not equal to −106, the two expressions are not equal.

#### Answer:

Let the other integer be *x. *

According to question, we have: *x* + 170 = −13

=> *x* = −13 − 170

=> *x* = −183

Thus, the other integer is −183.

#### Page No 70:

#### Question 9:

Let the other integer be *x. *

According to question, we have: *x* + 170 = −13

=> *x* = −13 − 170

=> *x* = −183

Thus, the other integer is −183.

#### Answer:

Let the other integer be *x*.

According to question, we have:*x* + (−47) = 65

=> x − 47 = 65

=> x = 65 + 47

=> x = 112

Thus, the other integer is 112.

#### Page No 70:

#### Question 10:

Let the other integer be *x*.

According to question, we have:*x* + (−47) = 65

=> x − 47 = 65

=> x = 65 + 47

=> x = 112

Thus, the other integer is 112.

#### Answer:

(i) True

An integer added to an integer gives an integer.

(ii) True

An integer subtracted from an integer gives an integer.

iii) False

−8 − (−7)

= −8 + 7

= −1

Since 14 is greater than 1, −1 is greater than −14.

iv) True

−5 − 2 = −7

Since 8 is greater than 7, −7 is greater than −8.

− 7 > −8

â€‹v) False

L.H.S.

(−7) − 3 = −10

R.H.S.

(−3) − (−7)

= (−3) + 7

= 4

$\therefore $ L.H.S. $\ne $ R.H.S.

#### Page No 71:

#### Question 11:

(i) True

An integer added to an integer gives an integer.

(ii) True

An integer subtracted from an integer gives an integer.

iii) False

−8 − (−7)

= −8 + 7

= −1

Since 14 is greater than 1, −1 is greater than −14.

iv) True

−5 − 2 = −7

Since 8 is greater than 7, −7 is greater than −8.

− 7 > −8

â€‹v) False

L.H.S.

(−7) − 3 = −10

R.H.S.

(−3) − (−7)

= (−3) + 7

= 4

$\therefore $ L.H.S. $\ne $ R.H.S.

#### Answer:

Let us consider the height above the sea level as positive and that below the sea level as negative.

$\therefore $ Height of point A from sea level = 5700 m

Depth of point B from sea level = -39600 m

Vertical distance between A and B = Distance of point A from sea level - Distance of point B from sea level

= 5700 - (â€‹-39600)

= 45300 m

#### Page No 71:

#### Question 12:

Let us consider the height above the sea level as positive and that below the sea level as negative.

$\therefore $ Height of point A from sea level = 5700 m

Depth of point B from sea level = -39600 m

Vertical distance between A and B = Distance of point A from sea level - Distance of point B from sea level

= 5700 - (â€‹-39600)

= 45300 m

#### Answer:

Initial temperature of Srinagar at 6 p.m. = 1°C

Final temperature of Srinagar at midnight = −4°C

Change in temperature = Final temperature - Initial temperature

â€‹= (−4 − 1)°C

= −5°C

So, the temperature has changed by −5°C.

The negative sign indicates that the temperature has fallen.

So, the temperature has fallen by 5°C.

#### Page No 72:

#### Question 1:

Initial temperature of Srinagar at 6 p.m. = 1°C

Final temperature of Srinagar at midnight = −4°C

Change in temperature = Final temperature - Initial temperature

â€‹= (−4 − 1)°C

= −5°C

So, the temperature has changed by −5°C.

The negative sign indicates that the temperature has fallen.

So, the temperature has fallen by 5°C.

#### Answer:

(i) 15 by 9

= 15 × 9

= 135

(ii) 18 by −7

= –(18 × 7)

= –126

(iii) 29 by –11

= –(29 × 11)

= –319

(iv) –18 by 13

= –(18 × 13)

= –234

(v) –56 by 16

= –(56 × 16)

= –896

(vi) 32 by –21

= –(32 × 21)

= –672

(vii) –57 by 0

= –(57 × 0)

= 0

(viii) 0 by –31

= –(0 × 31)

= 0

(ix) –12 by –9

= (–12) × (– 9)

= 108

(x) (–â€‹746) by (–8)

= (–746) × (–8)

= 5968

(xi) 118 by −7

â€‹= 118 × (-7)

= –826

(xii) −238 by −143

= (−238) × (−143)

= 34034

#### Page No 73:

#### Question 2:

(i) 15 by 9

= 15 × 9

= 135

(ii) 18 by −7

= –(18 × 7)

= –126

(iii) 29 by –11

= –(29 × 11)

= –319

(iv) –18 by 13

= –(18 × 13)

= –234

(v) –56 by 16

= –(56 × 16)

= –896

(vi) 32 by –21

= –(32 × 21)

= –672

(vii) –57 by 0

= –(57 × 0)

= 0

(viii) 0 by –31

= –(0 × 31)

= 0

(ix) –12 by –9

= (–12) × (– 9)

= 108

(x) (–â€‹746) by (–8)

= (–746) × (–8)

= 5968

(xi) 118 by −7

â€‹= 118 × (-7)

= –826

(xii) −238 by −143

= (−238) × (−143)

= 34034

#### Answer:

(i) (–2) × 3 × (–4)

= [(–2) × 3] × (–4)

= (–6) × (–4)

= 24

(ii) 2 × (–5) × (–6)

= [2 × (–5)] × (–6)

= (–10) × (–6)

= 60

(iii) (–8) × 3 × 5

= [(–8) × 3] × 5

= (–24) × 5

= –120

(iv) 8 × 7 × (–10)

= [8 × 7] × (–10)

= 56 × (–10)

= –560

(v) (–3) × (–7) × (–6)

= [(–3) × (–7)] × (–6)

= 21 × (–6)

= –126

(vi) (–8) × (–3) × (–9)

= [(–8) × (–3)] × (–9)

= 24 × (–9)

= –216

#### Page No 73:

#### Question 3:

(i) (–2) × 3 × (–4)

= [(–2) × 3] × (–4)

= (–6) × (–4)

= 24

(ii) 2 × (–5) × (–6)

= [2 × (–5)] × (–6)

= (–10) × (–6)

= 60

(iii) (–8) × 3 × 5

= [(–8) × 3] × 5

= (–24) × 5

= –120

(iv) 8 × 7 × (–10)

= [8 × 7] × (–10)

= 56 × (–10)

= –560

(v) (–3) × (–7) × (–6)

= [(–3) × (–7)] × (–6)

= 21 × (–6)

= –126

(vi) (–8) × (–3) × (–9)

= [(–8) × (–3)] × (–9)

= 24 × (–9)

= –216

#### Answer:

(i) 18 × (–27) × 30

= (–27) × [18 × 30]

= (–27) × 540

= –14580

(ii) (–8) × (–63) × 9

= [(–8) × (–63)] × 9

= 504 × 9

= 4536

(iii) (–17) × (–23) × 41

= [(–17) × (–23)] × 41

= 391 × 41

= 16031

(iv) (–51) × (–47) × (–19)

= [(–51) × (–47)] × (–19)

= 2397 × (–19)

= – 45543

#### Page No 73:

#### Question 4:

(i) 18 × (–27) × 30

= (–27) × [18 × 30]

= (–27) × 540

= –14580

(ii) (–8) × (–63) × 9

= [(–8) × (–63)] × 9

= 504 × 9

= 4536

(iii) (–17) × (–23) × 41

= [(–17) × (–23)] × 41

= 391 × 41

= 16031

(iv) (–51) × (–47) × (–19)

= [(–51) × (–47)] × (–19)

= 2397 × (–19)

= – 45543

#### Answer:

(i) __L.H.S__.

=18 × [9 + (–7)]

= 18 × [9 – 7]

= 18 × 2

= 36

__R.H.S__.

=18 × 9 + 18 × (–7)

= 162 – (18 × 7)

= 162 – 126

= 36

$\therefore $ L.H.S = R.H.S

Hence, verified.

(ii) (–13) × [(–6) + (–19)] = (–13) × (–6) + (–13) × (–19)__L.H.S.__

= (–13) × [(–6) + (–19)]

= (–13) × [–6 – 19]

= (–13) × (–25)

= 325__R.H.S__.

= (–13) × (–6) + (–13) × (–19)

= 78 + 247

= 325

$\therefore $ L.H.S = R.H.S

Hence, verified.

#### Page No 73:

#### Question 5:

(i) __L.H.S__.

=18 × [9 + (–7)]

= 18 × [9 – 7]

= 18 × 2

= 36

__R.H.S__.

=18 × 9 + 18 × (–7)

= 162 – (18 × 7)

= 162 – 126

= 36

$\therefore $ L.H.S = R.H.S

Hence, verified.

(ii) (–13) × [(–6) + (–19)] = (–13) × (–6) + (–13) × (–19)__L.H.S.__

= (–13) × [(–6) + (–19)]

= (–13) × [–6 – 19]

= (–13) × (–25)

= 325__R.H.S__.

= (–13) × (–6) + (–13) × (–19)

= 78 + 247

= 325

$\therefore $ L.H.S = R.H.S

Hence, verified.

#### Answer:

× | –3 | –2 | –1 | 0 | 1 | 2 | 3 |

–3 | 9 | 6 | 3 | 0 | –3 | –6 | –9 |

–2 | 6 | 4 | 2 | 0 | –2 | –4 | –6 |

–1 | 3 | 2 | 1 | 0 | –1 | –2 | –3 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | –3 | –2 | –1 | 0 | 1 | 2 | 3 |

2 | –6 | –4 | –2 | 0 | 2 | 4 | 6 |

3 | –9 | –6 | –3 | 0 | 3 | 6 | 9 |

#### Page No 73:

#### Question 6:

× | –3 | –2 | –1 | 0 | 1 | 2 | 3 |

–3 | 9 | 6 | 3 | 0 | –3 | –6 | –9 |

–2 | 6 | 4 | 2 | 0 | –2 | –4 | –6 |

–1 | 3 | 2 | 1 | 0 | –1 | –2 | –3 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | –3 | –2 | –1 | 0 | 1 | 2 | 3 |

2 | –6 | –4 | –2 | 0 | 2 | 4 | 6 |

3 | –9 | –6 | –3 | 0 | 3 | 6 | 9 |

#### Answer:

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

__True__

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

__False__

The product of two negative integers is always a positive integer.

(iii) The product of three negative integers is a negative integer.

__True__

(iv) Every integer when multiplied by (–1) gives its multiplicative inverse.

__False__

Every integer when multiplied by (1) gives its multiplicative inverse.

#### Page No 73:

#### Question 7:

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

__True__

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

__False__

The product of two negative integers is always a positive integer.

(iii) The product of three negative integers is a negative integer.

__True__

(iv) Every integer when multiplied by (–1) gives its multiplicative inverse.

__False__

Every integer when multiplied by (1) gives its multiplicative inverse.

#### Answer:

(i) (–9) × 6 + (–9) × 4__Solution:__

Using the distributive law:

(–9) × 6 + (–9) × 4

= (–9) × (6+9)

= (–9) × 10

= –90

(ii) 8 × (–12) + 7 × (–12)__Solution__:

Using the distributive law:

8 × (–12) + 7 × (–12)

= (–12) × (8+7)

= (–12) × 15

= –180

(iii) 30 × (–22) + 30 × (14)__Solution__:

Using the distributive law:

30 × (–22) + 30 × (14)

= 30 × [(–22) + 14]

= 30 × [–22 + 14]

= 30 × (–8)

= –240

(iv) (–15) × (–14) + (–15) × (–6)__Solution:__

(–15) × (–14) + (–15) × (–6)

Using the distributive law:

= (–15) × [ (–14) + (–6)]

= (–15) × [–14 – 6]

= (–15) × (–20)

= 300

(v) 43 × (–33) + 43 × (–17)__Solution:__

43 × (–33) + 43 × (–17)

Using the distributive law:

= (43 ) × [–(33) + (–17)]

= (43 ) × [–33 – 17]

= 43 × (–50)

= –2150

(vi) (–36) × (72) + (–36) × 28__Solution__

(–36) × (72) + (–36) × 28

Using the distributive law:

= (–36) × (72 + 28 )

= (–36) × 100

= –3600

(vii) (–27) × (–16) + (–27) × (–14)__Solution:__

(–27) × (–16) + (–27) × (–14)

Using the distributive law:

= (–27) × [(–16) + (–14)]

= (–27) × [–16 –14]

= (–27) × [–30]

= 810

#### Page No 75:

#### Question 1:

(i) (–9) × 6 + (–9) × 4__Solution:__

Using the distributive law:

(–9) × 6 + (–9) × 4

= (–9) × (6+9)

= (–9) × 10

= –90

(ii) 8 × (–12) + 7 × (–12)__Solution__:

Using the distributive law:

8 × (–12) + 7 × (–12)

= (–12) × (8+7)

= (–12) × 15

= –180

(iii) 30 × (–22) + 30 × (14)__Solution__:

Using the distributive law:

30 × (–22) + 30 × (14)

= 30 × [(–22) + 14]

= 30 × [–22 + 14]

= 30 × (–8)

= –240

(iv) (–15) × (–14) + (–15) × (–6)__Solution:__

(–15) × (–14) + (–15) × (–6)

Using the distributive law:

= (–15) × [ (–14) + (–6)]

= (–15) × [–14 – 6]

= (–15) × (–20)

= 300

(v) 43 × (–33) + 43 × (–17)__Solution:__

43 × (–33) + 43 × (–17)

Using the distributive law:

= (43 ) × [–(33) + (–17)]

= (43 ) × [–33 – 17]

= 43 × (–50)

= –2150

(vi) (–36) × (72) + (–36) × 28__Solution__

(–36) × (72) + (–36) × 28

Using the distributive law:

= (–36) × (72 + 28 )

= (–36) × 100

= –3600

(vii) (–27) × (–16) + (–27) × (–14)__Solution:__

(–27) × (–16) + (–27) × (–14)

Using the distributive law:

= (–27) × [(–16) + (–14)]

= (–27) × [–16 –14]

= (–27) × [–30]

= 810

#### Answer:

(i) 85 by **–**17

= $\frac{-85}{17}$

= –5

(ii) –72 by 18

=$\frac{-72}{18}$

= –4

(iii) –80 by 16

= $\frac{-80}{16}$

= –5

(iv) –121 by 11

=$\frac{-121}{11}$

= –11

(v) 108 by –12

= $\frac{108}{-12}$

= –9

(vi) –161 by 23

= $\frac{-161}{23}$

= –7

(vii) –76 by –19

=$\frac{-76}{-19}$

= 4

(viii) –147 by –21

= $\frac{-147}{-21}$

= 7

(ix) –639 by –71

$\begin{array}{l}=\frac{-639}{-71}\\ =9\end{array}$

(x) –639 by –71

$\begin{array}{l}=\frac{-639}{-71}\\ =9\end{array}$

(x) –15625 by –125

$\begin{array}{l}=\frac{-15625}{-125}\\ =125\end{array}$

(xi) 2067 by –1

$\begin{array}{l}=\frac{2067}{-1}\\ =-2067\end{array}$

(xii) 1765 by –1765

$\begin{array}{l}=\frac{1765}{-1765}\\ =-1\times \frac{1765}{1765}\end{array}\phantom{\rule{0ex}{0ex}}=-1\times 1\phantom{\rule{0ex}{0ex}}=-1$

(xiii) 0 by –278

$\begin{array}{l}=\frac{0}{-278}\\ =0\end{array}$

(xiv) 3000 by –100

$\begin{array}{l}=\frac{3000}{-100}\\ =-30\end{array}$

#### Page No 75:

#### Question 2:

(i) 85 by **–**17

= $\frac{-85}{17}$

= –5

(ii) –72 by 18

=$\frac{-72}{18}$

= –4

(iii) –80 by 16

= $\frac{-80}{16}$

= –5

(iv) –121 by 11

=$\frac{-121}{11}$

= –11

(v) 108 by –12

= $\frac{108}{-12}$

= –9

(vi) –161 by 23

= $\frac{-161}{23}$

= –7

(vii) –76 by –19

=$\frac{-76}{-19}$

= 4

(viii) –147 by –21

= $\frac{-147}{-21}$

= 7

(ix) –639 by –71

$\begin{array}{l}=\frac{-639}{-71}\\ =9\end{array}$

(x) –639 by –71

$\begin{array}{l}=\frac{-639}{-71}\\ =9\end{array}$

(x) –15625 by –125

$\begin{array}{l}=\frac{-15625}{-125}\\ =125\end{array}$

(xi) 2067 by –1

$\begin{array}{l}=\frac{2067}{-1}\\ =-2067\end{array}$

(xii) 1765 by –1765

$\begin{array}{l}=\frac{1765}{-1765}\\ =-1\times \frac{1765}{1765}\end{array}\phantom{\rule{0ex}{0ex}}=-1\times 1\phantom{\rule{0ex}{0ex}}=-1$

(xiii) 0 by –278

$\begin{array}{l}=\frac{0}{-278}\\ =0\end{array}$

(xiv) 3000 by –100

$\begin{array}{l}=\frac{3000}{-100}\\ =-30\end{array}$

#### Answer:

(i) 80 ÷ (__–16__) = –5

(ii) (–84) ÷ (__12__) = –7

(iii) (__–125__) ÷ (–5) = 25

(iv) (__0__) ÷ (372) = 0

(v) (__–186__) ÷ 1 = –186

(vi) (__–34__) ÷ 17 = –2

(vii) (__–165__) ÷ 165 = –1

(viii) (__–73__) ÷ –1 = 73

(ix) 1 ÷ (__–1__) = –1

#### Page No 75:

#### Question 3:

(i) 80 ÷ (__–16__) = –5

(ii) (–84) ÷ (__12__) = –7

(iii) (__–125__) ÷ (–5) = 25

(iv) (__0__) ÷ (372) = 0

(v) (__–186__) ÷ 1 = –186

(vi) (__–34__) ÷ 17 = –2

(vii) (__–165__) ÷ 165 = –1

(viii) (__–73__) ÷ –1 = 73

(ix) 1 ÷ (__–1__) = –1

#### Answer:

(i) True

(ii) False

This is because we cannot divide any integer by 0. If we do so, we get the quotient as infinity.

(iii) True

(iv) False

This is because the division of any two negative integers always gives a positive quotient.

(v) True

(vi) True

(vii) True

(viii) True

(ix) False

This is because the division of any two negative integers always gives a positive quotient.

#### Page No 75:

#### Question 1:

(i) True

(ii) False

This is because we cannot divide any integer by 0. If we do so, we get the quotient as infinity.

(iii) True

(iv) False

This is because the division of any two negative integers always gives a positive quotient.

(v) True

(vi) True

(vii) True

(viii) True

(ix) False

This is because the division of any two negative integers always gives a positive quotient.

#### Answer:

(b) –4 < –3

Since 4 is greater than 3, –4 is less than –3.

#### Page No 75:

#### Question 2:

(b) –4 < –3

Since 4 is greater than 3, –4 is less than –3.

#### Answer:

(c) –5

2 less than –3 means the following:

= –3 – 2

= –5

#### Page No 76:

#### Question 3:

(c) –5

2 less than –3 means the following:

= –3 – 2

= –5

#### Answer:

c) –1

4 more than –5 means the following:

= –5 + 4

= –1

#### Page No 76:

#### Question 4:

c) –1

4 more than –5 means the following:

= –5 + 4

= –1

#### Answer:

(a) –9

2 less than −7 means the following:

= −7 − 2

= −9

#### Page No 76:

#### Question 5:

(a) –9

2 less than −7 means the following:

= −7 − 2

= −9

#### Answer:

(b) 10

7 + |-3|

= 7 + (+ 3) (The absolute value of −3 is 3.)

= 7 + 3

= 10

#### Page No 76:

#### Question 6:

(b) 10

7 + |-3|

= 7 + (+ 3) (The absolute value of −3 is 3.)

= 7 + 3

= 10

#### Answer:

(c) –77

(−42) + (−35)

= −42 − 35

= −77

#### Page No 76:

#### Question 7:

(c) –77

(−42) + (−35)

= −42 − 35

= −77

#### Answer:

(b) –31

(−37) + 6

= −37 + 6

= −31

#### Page No 76:

#### Question 8:

(b) –31

(−37) + 6

= −37 + 6

= −31

#### Answer:

(c) 22

49 + (−27)

= 49 − 27

â€‹= 22

#### Page No 76:

#### Question 9:

(c) 22

49 + (−27)

= 49 − 27

â€‹= 22

#### Answer:

(c) –17

In succession, we move from the left to the right of the number line.

#### Page No 76:

#### Question 10:

(c) –17

In succession, we move from the left to the right of the number line.

#### Answer:

(b) –17

To find the predecessor of a number, we move from the right to the left of a number line.

#### Page No 76:

#### Question 11:

(b) –17

To find the predecessor of a number, we move from the right to the left of a number line.

#### Answer:

(a) 5

If we add the additive inverse of a number to the number, we get 0.

−5 + 5 = 0

#### Page No 76:

#### Question 12:

(a) 5

If we add the additive inverse of a number to the number, we get 0.

−5 + 5 = 0

#### Answer:

(b) –7

−12 − (−5)

= −12 + 5

= −7

#### Page No 76:

#### Question 13:

(b) –7

−12 − (−5)

= −12 + 5

= −7

#### Answer:

(b) 13.5 − (−8)

= 5 + 8

= 13

#### Page No 76:

#### Question 14:

(b) 13.5 − (−8)

= 5 + 8

= 13

#### Answer:

(c) –55

Let *x* be the other integer.*x* + 30 = –25

$\Rightarrow $ *x* = –25 – 30

$\Rightarrow $ *x* = –55

#### Page No 76:

#### Question 15:

(c) –55

Let *x* be the other integer.*x* + 30 = –25

$\Rightarrow $ *x* = –25 – 30

$\Rightarrow $ *x* = –55

#### Answer:

(a) 25

Let the other integer be *x**x* + (-5) = 20

$\Rightarrow $*x* - 5 = 20

$\Rightarrow $*x* = 25

â€‹

#### Page No 76:

#### Question 16:

(a) 25

Let the other integer be *x**x* + (-5) = 20

$\Rightarrow $*x* - 5 = 20

$\Rightarrow $*x* = 25

â€‹

#### Answer:

(b) −21

Let the other integer be *x*.*x* + 8 = −13

=> *x* = −13 − 8

=> *x* = −21

#### Page No 76:

#### Question 17:

(b) −21

Let the other integer be *x*.*x* + 8 = −13

=> *x* = −13 − 8

=> *x* = −21

#### Answer:

(b) 8

0 − (−8)

= 0 + 8

= 8

#### Page No 76:

#### Question 18:

(b) 8

0 − (−8)

= 0 + 8

= 8

#### Answer:

(c) 0

8 + (−8)

= 8 − 8

= 0

#### Page No 76:

#### Question 19:

(c) 0

8 + (−8)

= 8 − 8

= 0

#### Answer:

(c) 1

(−6) + 4 − (−3)

= −6 + 4 + 3

= −6 + 7

= 1

#### Page No 76:

#### Question 20:

(c) 1

(−6) + 4 − (−3)

= −6 + 4 + 3

= −6 + 7

= 1

#### Answer:

(c) 10

6 − (−4)

= 6 + 4

= 10

#### Page No 76:

#### Question 21:

(c) 10

6 − (−4)

= 6 + 4

= 10

#### Answer:

(a) –20

(−7) + (−9) + 12 + (−16)

= −7 − 9 + 12 −16

= −20

#### Page No 76:

#### Question 22:

(a) –20

(−7) + (−9) + 12 + (−16)

= −7 − 9 + 12 −16

= −20

#### Answer:

(c) –12

–â€‹4 –â€‹ 8

= –â€‹12

#### Page No 76:

#### Question 23:

(c) –12

–â€‹4 –â€‹ 8

= –â€‹12

#### Answer:

(c) 3

We have:

−6 − (−9)

= −6 + 9

= 3

#### Page No 76:

#### Question 24:

(c) 3

We have:

−6 − (−9)

= −6 + 9

= 3

#### Answer:

(c) 15

We have:

10 − (−5)

â€‹= 10 + 5

= 15

#### Page No 77:

#### Question 25:

(c) 15

We have:

10 − (−5)

â€‹= 10 + 5

= 15

#### Answer:

(b) –54

We have:

(−6) × 9

= −(6 × 9â€‹)

= −54

#### Page No 77:

#### Question 26:

(b) –54

We have:

(−6) × 9

= −(6 × 9â€‹)

= −54

#### Answer:

(a) –90

(−9) × 6 + (−9) × 4

Using distributive law:

(−9) × (6 + 4)

= (−9) × (10)

= −90

#### Page No 77:

#### Question 27:

(a) –90

(−9) × 6 + (−9) × 4

Using distributive law:

(−9) × (6 + 4)

= (−9) × (10)

= −90

#### Answer:

(b) –4

36 ÷ (−9)

$\frac{36}{-9}\phantom{\rule{0ex}{0ex}}=\frac{36}{9\times (-1)}\phantom{\rule{0ex}{0ex}}=\frac{1}{(-1)}\times \frac{36}{9}\phantom{\rule{0ex}{0ex}}=-1\times 4=-4$

#### Page No 78:

#### Question 1:

(b) –4

36 ÷ (−9)

$\frac{36}{-9}\phantom{\rule{0ex}{0ex}}=\frac{36}{9\times (-1)}\phantom{\rule{0ex}{0ex}}=\frac{1}{(-1)}\times \frac{36}{9}\phantom{\rule{0ex}{0ex}}=-1\times 4=-4$

#### Answer:

The numbers ...–4, –3, –2, –1, 0, 1, 2, 3, 4... are integers.

The group of positive and negative numbers including 0 is called integers.

–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5

#### Page No 78:

#### Question 2:

The numbers ...–4, –3, –2, –1, 0, 1, 2, 3, 4... are integers.

The group of positive and negative numbers including 0 is called integers.

–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5

#### Answer:

(i) 0, –3

0

This is because 0 is greater than any negative integer.

(ii) –4, –6

–4

Since 6 is greater than 4, –4 is greater than –6.

(iii) –99, 9

9

This is because every positive integer is greater than any negative integer.

(iv) –385, –615

–385

Since 615 is greater than 385, –385 is greater than –615.

#### Page No 78:

#### Question 3:

(i) 0, –3

0

This is because 0 is greater than any negative integer.

(ii) –4, –6

–4

Since 6 is greater than 4, –4 is greater than –6.

(iii) –99, 9

9

This is because every positive integer is greater than any negative integer.

(iv) –385, –615

–385

Since 615 is greater than 385, –385 is greater than –615.

#### Answer:

We can arrange the given integers in the increasing order in the following manner:

–36, –18, –5, –1, 0, 1, 8, 16

#### Page No 78:

#### Question 4:

We can arrange the given integers in the increasing order in the following manner:

–36, –18, –5, –1, 0, 1, 8, 16

#### Answer:

(i) 9 – |–6|

= 9 – (6)

= 9 – 6

= 3

(ii) 6 + |–4|

= 6 + (4)

= 6 + 4

= 10

(iii) –8 – |–3|

= –8 – 3

= –11

#### Page No 78:

#### Question 5:

(i) 9 – |–6|

= 9 – (6)

= 9 – 6

= 3

(ii) 6 + |–4|

= 6 + (4)

= 6 + 4

= 10

(iii) –8 – |–3|

= –8 – 3

= –11

#### Answer:

Four integers less than –6 (i.e. four negative integers that lie to the left of –6) are –7, –8, –9 and –10.

Four integers greater than –6 (i.e. four negative integers that lie to the right of –6 ) are –5, –4, –3 and –2.

#### Page No 78:

#### Question 6:

Four integers less than –6 (i.e. four negative integers that lie to the left of –6) are –7, –8, –9 and –10.

Four integers greater than –6 (i.e. four negative integers that lie to the right of –6 ) are –5, –4, –3 and –2.

#### Answer:

(i) 8 + (–16)

= 8 – 16

= –8

(ii) (–5) + (–6)

= –5 – 6

= –11

(iii) (–6) × (–8)

= (6 × 8)

= 48

(iv) (–36) ÷ 6

$\frac{-36}{6}\phantom{\rule{0ex}{0ex}}=(-1)\times \frac{36}{6}\phantom{\rule{0ex}{0ex}}=-6$

(v)

30 – (–50)

= 30 + 50

= 80

(vi) (–40) ÷ (–10)

$\begin{array}{l}=\frac{-40}{-10}\\ =\frac{(-1)\times 40}{(-1)\times 10}\\ =4\end{array}$

(vii) 8 × (–5)

= –(8 × 5)

= –40

(viii) (–30) – 15

= –30 – 15

= –45

#### Page No 78:

#### Question 7:

(i) 8 + (–16)

= 8 – 16

= –8

(ii) (–5) + (–6)

= –5 – 6

= –11

(iii) (–6) × (–8)

= (6 × 8)

= 48

(iv) (–36) ÷ 6

$\frac{-36}{6}\phantom{\rule{0ex}{0ex}}=(-1)\times \frac{36}{6}\phantom{\rule{0ex}{0ex}}=-6$

(v)

30 – (–50)

= 30 + 50

= 80

(vi) (–40) ÷ (–10)

$\begin{array}{l}=\frac{-40}{-10}\\ =\frac{(-1)\times 40}{(-1)\times 10}\\ =4\end{array}$

(vii) 8 × (–5)

= –(8 × 5)

= –40

(viii) (–30) – 15

= –30 – 15

= –45

#### Answer:

Let the integer be *x*.

$\therefore $ 34 + *x* = –12

or *x* = –12 – 34

or *x* = –46

Therefore, the other integer is –46.

#### Page No 78:

#### Question 8:

Let the integer be *x*.

$\therefore $ 34 + *x* = –12

or *x* = –12 – 34

or *x* = –46

Therefore, the other integer is –46.

#### Answer:

(i) (–24) × (68) + (–24) × 32

= –(24) × (68+32)

= –24 × 100

= –2400

(ii) (–9) × 18 – (–9) × 8

= –(9 ) × [18 – 8]

= –9 × 10

= –90

(iii) (–147) ÷ (–21)

$\begin{array}{l}=\frac{-147}{-21}\\ =\frac{(-1)\times 147}{(-1)\times 21}\\ =\frac{(-1)}{(-1)}\times \frac{147}{21}\\ =7\end{array}$

(iv) 16 ÷ (–1)

$\begin{array}{l}=\frac{16}{-1}\\ =\frac{16\times (-1)}{(-1)\times (-1)}\\ =16\times (-1)\\ =-16\end{array}$ {Multiplying the numerator and the denominator by (–1)}

#### Page No 78:

#### Question 9:

(i) (–24) × (68) + (–24) × 32

= –(24) × (68+32)

= –24 × 100

= –2400

(ii) (–9) × 18 – (–9) × 8

= –(9 ) × [18 – 8]

= –9 × 10

= –90

(iii) (–147) ÷ (–21)

$\begin{array}{l}=\frac{-147}{-21}\\ =\frac{(-1)\times 147}{(-1)\times 21}\\ =\frac{(-1)}{(-1)}\times \frac{147}{21}\\ =7\end{array}$

(iv) 16 ÷ (–1)

$\begin{array}{l}=\frac{16}{-1}\\ =\frac{16\times (-1)}{(-1)\times (-1)}\\ =16\times (-1)\\ =-16\end{array}$ {Multiplying the numerator and the denominator by (–1)}

#### Answer:

(b) −88

The successor of −89 is â€‹−88. The successor of a number lies towards its right on a number line. â€‹

−88 lies to the right of â€‹−89.

#### Page No 78:

#### Question 10:

(b) −88

The successor of −89 is â€‹−88. The successor of a number lies towards its right on a number line. â€‹

−88 lies to the right of â€‹−89.

#### Answer:

(b) â€‹−100

The predecessor of a number lies to the left of the number.

â€‹â€‹−100 lies to the left of −â€‹99. Hence, â€‹â€‹−100 is a predecessor of −â€‹99.

#### Page No 78:

#### Question 11:

(b) â€‹−100

The predecessor of a number lies to the left of the number.

â€‹â€‹−100 lies to the left of −â€‹99. Hence, â€‹â€‹−100 is a predecessor of −â€‹99.

#### Answer:

(c) â€‹23

Additive inverse of a number added to the number gives 0.

−23 + 23 = 0

Hence, 23 is the additive inverse of −23.

#### Page No 78:

#### Question 12:

(c) â€‹23

Additive inverse of a number added to the number gives 0.

−23 + 23 = 0

Hence, 23 is the additive inverse of −23.

#### Answer:

(b) >

Here, L.H.S. = (−13 + 6)

= −7

R.H.S. = −25 − (−9)

= −25 + 9

â€‹ = −16

−7 > −16

L.H.S. > R.H.S.

#### Page No 78:

#### Question 13:

(b) >

Here, L.H.S. = (−13 + 6)

= −7

R.H.S. = −25 − (−9)

= −25 + 9

â€‹ = −16

−7 > −16

L.H.S. > R.H.S.

#### Answer:

(c) 20 *x* + (−8) = 12

=> *x* − 8 = 12

=> *x* = 12 + 8

=> *x* = 20

#### Page No 78:

#### Question 14:

(c) 20 *x* + (−8) = 12

=> *x* − 8 = 12

=> *x* = 12 + 8

=> *x* = 20

#### Answer:

(c) -2

5 more than (−7) means 5 added to (−7).

5 + (−7)

= 5 − 7

= −2

#### Page No 78:

#### Question 15:

(c) -2

5 more than (−7) means 5 added to (−7).

5 + (−7)

= 5 − 7

= −2

#### Answer:

(d) −47

Let the number to be added to 16 be *x*.*x* + 16 = (−31)

=> x = (−31)−16

=> x = −47

#### Page No 78:

#### Question 16:

(d) −47

Let the number to be added to 16 be *x*.*x* + 16 = (−31)

=> x = (−31)−16

=> x = −47

#### Answer:

(d) −70

−36 â€‹− 34

= −70

#### Page No 78:

#### Question 17:

(d) −70

−36 â€‹− 34

= −70

#### Answer:

(i)

Let the required number be *x*.

−23 − *x *= 15

=> −23 = 15 + *x*

=> 15 + *x* = −23

=> *x* = −15 −23

=> *x* = −38

(ii)

The largest negative integer is -1.

(iii)

The smallest positive integer is 1.

(iv)

(−8) + (−6) − (−3)

= (−8) + (−6) +3

= −8 â€‹−6 + 3

= −11

(v)

The predecessor of −200:

(−200 − 1)

= −201

#### Page No 79:

#### Question 18:

(i)

Let the required number be *x*.

−23 − *x *= 15

=> −23 = 15 + *x*

=> 15 + *x* = −23

=> *x* = −15 −23

=> *x* = −38

(ii)

The largest negative integer is -1.

(iii)

The smallest positive integer is 1.

(iv)

(−8) + (−6) − (−3)

= (−8) + (−6) +3

= −8 â€‹−6 + 3

= −11

(v)

The predecessor of −200:

(−200 − 1)

= −201

#### Answer:

(i) T

(ii) F

−(−36) − 1

= 36 − 1â€‹

= 35

(iii) F

This is because −10 is less than −6.

(iv) T

(v) T

−|−15|

= â€‹−(15)

= −15

â€‹(vi) F

|−40| + 40

= 40 + 40

= 80

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