Exercise 4A
Page No 63:
Question 1:
Write the opposite of each of the following:
(i) An increase of 8
(ii) A loss of Rs 7
(iii) Gaining a weight of 5 kg
(iv) 10 km above sea level
(v) 5°C below the freezing point
(vi) A deposit of Rs 100
(vii) Earning Rs 500
(viii) Going 6 m to the east
(ix) 24
(x) −34
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) 5oC 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:
Indicate the following using ‘+’ or ‘−’ sign:
(i) A gain of Rs 600
(ii) A loss of Rs 800
(iii) 7°C below the freezing point
(iv) Decrease of 9
(v) 2 km above sea level
(vi) 3 km below sea level
(vii) A deposit of Rs 200
(viii) A withdrawal of Rs 300
ANSWER:
(i) +Rs 600
(ii) -Rs 800
(iii) -7oC
(iv) -9
(v) +2 km
(vi) -3 km
(vii) + Rs 200
(viii) -Rs 300
Page No 64:
Question 3:
Mark the following integers on a number line:
(i) −5
(ii) −2
(iii) 0
(iv) −7
(v) −13
ANSWER:
(i) -5
(ii) -2
(iii) 0
(iv) 7
Page No 64:
Question 4:
Which number is larger in each of the following pairs?
(i) 0, −2
(ii) −3, −5
(iii) −5, 2
(iv) −16, 8
(v) −365, −913
(vi) −888, 8
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:
Which number is smaller in each of the following pairs?
(i) 6, −7
(ii) 0, −1
(iii) −13, −27
(iv) −26, 17
(v) −317, −603
(vi) −777, 7
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:
Write all integers between
(i) 0 and 6
(ii) −5 and 0
(iii) −3 and 3
(iv) −7 and −5
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:
Fill in the blanks by appropriate symbol > or <:
(i) 0 …… 7
(ii) 0 …… −3
(iii) −5 …… −2
(iv) −15 …… 13
(v) −231 …… −132
(vi) −6 … … 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:
Write the following integers in the increasing order:
(i) 5, −7, −2, 0, 8
(ii) −23, 12, 0, −6, −100, −1
(iii) −17, 15, −363, −501, 165
(iv) 21, −106, −16, 16, 0, −2, −81
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:
Write the following integers in the decreasing order:
(i) 0, 7, −3, −9, −132, 36
(ii) 51, −53, −8, 0, −2
(iii) −71, −81, 36, 0, −5
(iv) −365, −515, 102, 413, −7
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:
Using the number line, write the integer which is
(i) 4 more than 6
(ii) 5 more than −6
(iii) 6 less than 2
(iv) 2 les than −3
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:
For each of the following statements, write (T) for true and (F) for false:
(i) The smallest integer is zero.
(ii) Zero is not an integer.
(iii) The opposite of zero is zero.
(iv) −10 is greater than −6.
(v) The absolute value of an integer is always greater than the integer.
(vi) 0 is larger than every negative integer.
(vii) Every negative integer is less than every natural number.
(viii) The successor of −187 is −188.
(ix) The predecessor of −215 is −214.
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:
Find the value of
(i) |−9|
(ii) |−36|
(iii) |0|
(iv) |15|
(v) −|−3|
(vi) 7 + |−3|
(vii) |7−4|
(viii) 8 −|−7|
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
∴∴ -|-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) Write five negative integers greater than −7.
(ii) Write five negative integers less than −20.
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:
Exercise 4B
Question 1:
Use the number line and add the following integers:
(i) 9 + (−6)
(ii) (−3) + 7
(iii) 8 + (−8)
(iv) (−1) + (−3)
(v) (−4) + (−7)
(vi) (−2) + (−8)
(vii) 3 + (−2) + (−4)
(viii) (−1) + (−2) + (−3)
(ix) 5 + (−2) + (−6)
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.
∴∴ 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.
∴∴ (−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.
∴∴ 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.
∴∴ (−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.
∴∴ (−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.
∴∴ (−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.
∴∴ 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.
∴∴ (−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.
∴∴ 5 + (−2) + (−6) = −3
Page No 68:
Question 2:
Fill in the blanks:
(i) (−3) + (−9) = …….
(ii) (−7) + (−8) = …….
(iii) (−9) + 16 = …….
(iv) (−13) + 25 = …….
(v) 8 + (−17) = …….
(vi) 2 + (−12) = …….
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:
Add:
(i) −365 −87 -365 -87
(ii)−73 −687 -73 -687
(iii)−1065 −987 -1065 -987
(iv) −3596 −1089 -3596 -1089
ANSWER:
(i)
−365 −87−452−365 −87−452
−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)
−687 −73−760−687 −73−760
-687 and -73 are both negative integers. So, we add 365 and 87, and put the negative sign before the sum.
(iii)
−1065 −987−2052−1065 −987−2052
-1065 and -987 are both negative integers. So, we add 1065 and 987, and put the negative sign before the sum.
(iv)
−3596−1089−4685−3596−1089−4685
-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:
Add:
(i) −206 +98 -206 +98
(ii)+178 −69 +178 -69
(iii)−103 +312 -103 +312
(iv) −493 +289 -493 +289
ANSWER:
i)
−206 +98−108 Since we are adding a negative number with a positive number,we shall subtract the smaller number, i.e. 98 from the greater number, i.e. 206206 − 98 = 108 Since the greater number is negative, the sign of the result will be negative. So, the answer will be −108−206 +98−108 Since we are adding a negative number with a positive number,we shall subtract the smaller number, i.e. 98 from the greater number, i.e. 206206 – 98 = 108 Since the greater number is negative, the sign of the result will be negative. So, the answer will be -108
ii)
178 -69 109 178 -69 109
Since we are adding a negative number with a positive number,we shall subtract the smaller number, i.e. 69, from the greater number, i.e. 178178 – 69 = 109 Since the greater number is positive, the sign of the result will be positive. So, the answer will be 109Since we are adding a negative number with a positive number,we shall subtract the smaller number, i.e. 69, from the greater number, i.e. 178178 – 69 = 109 Since the greater number is positive, the sign of the result will be positive. So, the answer will be 109
(iii)
312 -103 209 312 -103 209
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
(iv) −493+289−204(iv) −493+289−204
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:
Find the sum of
(i) 137 and −354
(ii) 1001 and −13
(iii) −3057 and 199
(iv) −36 and 1027
(v) −389 and −1032
(vi) −36 and 100
(vii) 3002 and −888
(viii) −18, + 25 and −37
(ix) −312, 37 and 192
(x) −51, −203, 36 and −28
ANSWER:
(i) 137 and −354−354+137−217(ii) 1001 and −131001−13988(iii) −3057 and 199−3057 199−2858(iv) −36 and 10273057−363021(v) −389 and −1032−1032−389−1421(vi) −36 and 100100−3664(vii) 3002 and −8883002−8882114(i) 137 and −354−354+137−217(ii) 1001 and −131001−13988(iii) −3057 and 199−3057 199−2858(iv) −36 and 10273057−363021(v) −389 and −1032−1032−389−1421(vi) −36 and 100100−3664(vii) 3002 and −8883002−8882114
(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:
Find the additive inverse of
(i) −57
(ii) 183
(iii) 0
(iv) −1001
(v) 2054
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 of −1001 is 1001.
(v) 2054 − 2054 = 0
So, the additive inverse of 2054 is −2054
Page No 68:
Question 7:
Write the successor of each one of the following:
(i) 201
(ii) 70
(iii) −5
(iv) −99
(v) −500
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:
Write the predecessor of each one of the following:
(i) 120
(ii) 79
(iii) −8
(iv) −141
(v) −300
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
(v) The predecessor of −300:
−300 − 1 = −301
Page No 69:
Question 9:
Simplify:
(i) (−7) + (−9) + 12 + (−16)
(ii) 37 + (−23) + (−65) + 9 + (−12)
(iii) (−145) + 79 + (−265) + (−41) + 2
(iv) 1056 + (−798) + (−38) + 44 + (−1)
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
(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:
A car travelled 60 km to the north of Patna and then 90 km to the south from there. How far from Patna was the car finally?
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 + (-90)
= -30 km
The car was 30 km south of Patna.
Page No 69:
Question 11:
A man bought some pencils for Rs 30 and some pens for Rs 90. The next day, he again bought some pencils for Rs 25. Then, he sold all the pencils for Rs 20 and the pens for Rs 70. What was his net gain or loss?
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:
For each of the following statements write (T) for true and (F) for false:
(i) The sum of two negative integers is always a negative integer.
(ii) The sum of a negative integer and a positive integer is always a negative integer.
(iii) The sum of an integer and its negative is zero.
(iv) The sum of three different integers can never be zero.
(v) | −5| < |−3|
(vi) |8 − 5| = |8| + |−5|
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
∴∴ |8 − 5|≠≠|8| + |−5|
Page No 69:
Question 13:
Find an integer a such that
(i) a + 6 = 0
(ii) 5 + a = 0
(iii) a + (−4) = 0
(iv) −8 + a = 0
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:
Exercise 4C
Question 1:
Subtract:
(i) 18 from −34
(ii) −15 from 25
(iii) −28 from −43
(iv) 68 from −37
(v) 219 from 0
(vi) −92 from 0
(vii) −135 from −250
(viii) −2768 from −287
(ix) 6240 from −271
(x) −3012 from 6250
ANSWER:
(i) −34 − 18
= −52
(ii) 25 − (−15)
= 25 + 15
= 40
(iii) −28 from −43
= −43 − (−28)
= −43 + 28
= −15
(iv) 68 from −37
= −37 − 68
= −105
(v) 219 from 0
= 0 − 219
= −219
(vi) −92 from 0
= 0 − (−92)
= 0 + 92
= 92
(vii) −135 from −250
= −250 − (−135)
= −250 + 135
= −115
(viii) −2768 from −287
= −287 − (−2768)
= 2768 − 287
= 2481
(ix) 6240 from −271
= −271 − (6240)
= −271 − 6240
= −6511
(x) −3012 from 6250
= 6250 − (−3012)
= 6250 + 3012
= 9262
Page No 70:
Question 2:
Subtract the sum of −1050 and 813 from −23.
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:
Subtract the sum of −250 and 138 from the sum of 136 and −272.
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 − (−112)
= −136 + 112
= −24
Page No 70:
Question 4:
From the sum of 33 and −47, subtract −84.
ANSWER:
Adding 33 and −47:
33 + (−47)
= 33 − 47
= −14
Subtracting −84 from −14:
−14 − (−84)
= −14 + 84
= 70
Page No 70:
Question 5:
Add −36 to the difference of −8 and −68.
ANSWER:
Difference of −8 and −68:
−8 − (−68)
= −8 + 68
= 60
Adding -36 to 60:
−36 + 60
= 24
Page No 70:
Question 6:
Simplify:
(i) [37 − (−8)] + [11 − (−30)]
(ii) [−13 − (−17) + [−22 − (−40)]
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:
Find 34 − (−72) and (−72) − 34. Are they equal?
ANSWER:
No, they are not equal.
34 − (−72)
= 34 + 72
= 106
(−72) − 34
= −72 − 34
= −106
Since 106 is not equal to −106, the two expressions are not equal.
Page No 70:
Question 8:
The sum of two integers is −13. If one of the numbers is 170, find the other.
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:
The sum of two integers is 65. If one of the integers is −47, find the other.
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:
Which of the following statements are true and which are false?
(i) The sum of two integers is always an integer.
(ii) The difference of two integers is always an integer.
(iii) −14 > −8 − (−7)
(iv) −5 − 2 > −8
(v) (−7) − 3 = (−3) − (−7)
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
v) False
L.H.S.
(−7) − 3 = −10
R.H.S.
(−3) − (−7)
= (−3) + 7
= 4
∴∴ L.H.S. ≠≠ R.H.S.
Page No 71:
Question 11:
The point A is on a mountain which is 5700 metres above sea level and the point B is in a mine which is 39600 metres below sea level. Find the vertical distance between A and B.
Figure
ANSWER:
Let us consider the height above the sea level as positive and that below the sea level as negative.
∴∴ 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 – (-39600)
= 45300 m
Page No 71:
Question 12:
On a day in Srinagar, the temperature at 6 p.m. was 1°C but at midnight that day, it dropped to −4°C. By how many degrees Celsius did the temperature fall?
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
= (−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:
Exercise 4D
Question 1:
Multiply:
(i) 15 by 9
(ii) 18 by −7
(iii) 29 by −11
(iv) −18 by 13
(v) −56 by 16
(vi) 32 by −21
(vii) −57 by 0
(viii) 0 by −31
(ix) −12 by −9
(x) −746 by −8
(xi) 118 by −7
(xii) −238 by −143
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) (–746) by (–8)
= (–746) × (–8)
= 5968
(xi) 118 by −7
= 118 × (-7)
= –826
(xii) −238 by −143
= (−238) × (−143)
= 34034
Page No 73:
Question 2:
Find the products:
(i) (−2) × 3 × (−4)
(ii) 2 × (−5) × (−6)
(iii) (−8) × 3 × 5
(iv) 8 × 7 × (−10)
(v) (−3) × (−7) × (−6)
(vi) (−8) × (−3) × (−9)
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:
Use convenient groupings and find the values of
(i) 18 × (−27) × 30
(ii) (−8) × (−63) × 9
(iii) (−17) × (−23) × 41
(iv) (−51) × (−47) × (−19)
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:
Verify the following:
(i) 18 × [9 + (−7)] = 18 × 9 + 18 × (−7)
(ii) (−13) × [(−6) + (−19)] = (−13) × (−6) + (−13) × (−19)
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
∴∴ 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
∴∴ L.H.S = R.H.S
Hence, verified.
Page No 73:
Question 5:
Complete the following multiplication table:
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
| −3 | |||||||
| −2 | |||||||
| −1 | |||||||
| 0 | |||||||
| 1 | |||||||
| 2 | |||||||
| 3 |
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:
Which of the following statements are true and which are fals?
(i) The product of a positive integer and a negative integer is negative.
(ii) The product of two negative integers is a negative integer.
(iii) The product of three negative integers is a negative integer.
(iv) Every integer when multiplied with −1 gives its multiplicative inverse.
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:
Simplify:
(i) (−9) × 6 + (−9) × 4
(ii) 8 × (−12) + 7 × (−12)
(iii) 30 × (−22) + 30 × (14)
(iv) (−15) × (−14) + (−15) × (−6)
(v) 43 × (−33) + 43 × (−17)
(vi) (−36) × (72) + (−36) × 28
(vii) (−27) × (−16) + (−27) × (−14)
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:
Exercise 4E
Question 1:
Divide:
(i) 85 by −17
(ii) −72 by 18
(iii) −80 by 16
(iv) −121 by 11
(v) 108 by −12
(vi) −161 by 23
(vii) −76 by −19
(viii) −147 by −21
(ix) −639 by −71
(x) −15625 by −125
(xi) 2067 by −1
(xii) 1765 by −1765
(xiii) 0 by −278
(xiv) 3000 by −100
ANSWER:
(i) 85 by –17
= −8517−8517
= –5
(ii) –72 by 18
=−7218−7218
= –4
(iii) –80 by 16
= −8016−8016
= –5
(iv) –121 by 11
=−12111−12111
= –11
(v) 108 by –12
= 108−12108−12
= –9
(vi) –161 by 23
= −16123−16123
= –7
(vii) –76 by –19
=−76−19−76−19
= 4
(viii) –147 by –21
= −147−21−147−21
= 7
(ix) –639 by –71
=−639−71=9=−639−71=9
(x) –639 by –71
=−639−71=9=−639−71=9
(x) –15625 by –125
=−15625−125=125=−15625−125=125
(xi) 2067 by –1
=2067−1=−2067=2067−1=−2067
(xii) 1765 by –1765
=1765−1765=−1×17651765= −1×1= −1=1765−1765=−1×17651765= -1×1= -1
(xiii) 0 by –278
=0−278=0=0−278=0
(xiv) 3000 by –100
=3000−100=−30=3000−100=−30
Page No 75:
Question 2:
Fill in the blanks:
(i) 80 ÷ (……) = −5
(ii) (−84) + (……) = −7
(iii) (……) ÷ (−5) = 25
(iv) (……) ÷ 372 = 0
(v) (……) ÷ 1 = −186
(vi) (……) ÷ 17 = −2
(vii) (……) ÷ 165 = −1
(viii) (……) ÷ (−1) = 73
(ix) 1 ÷ (……) = −1
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:
Write (T) for true and (F) for false for each of the following statements:
(i) 0 ÷ (−6) = 0
(ii) (−8) ÷ 0 = 0
(iii) 15 ÷ (−1) = −15
(iv) (−16) ÷ (−4) = −4
(v) (−7) ÷ (−1) = 7
(vi) (−18) ÷ 9 = −2
(vii) 20 ÷ (−5) = −4
(viii) (−10) ÷ 1 = −10
(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:
Which of the following is a true statement?
(a) −4 > −3
(b) −4 < −3
(c) −4 and −3 are non-comparable
ANSWER:
(b) –4 < –3
Since 4 is greater than 3, –4 is less than –3.
Page No 75:
Question 2:
2 less than −3 is
(a) −1
(b) 1
(c) −5
(d) 5
ANSWER:
(c) –5
2 less than –3 means the following:
= –3 – 2
= –5
Page No 76:
Question 3:
4 more than −5 is
(a) 9
(b) −9
(c) −1
(d) 1
ANSWER:
c) –1
4 more than –5 means the following:
= –5 + 4
= –1
Page No 76:
Question 4:
2 less than −7 is
(a) −9
(b) −5
(c) 5
(d) none of these
ANSWER:
(a) –9
2 less than −7 means the following:
= −7 − 2
= −9
Page No 76:
Question 5:
7 + |−3| = ?
(a) 4
(b) 10
(c) −10
(d) none of these
ANSWER:
(b) 10
7 + |-3|
= 7 + (+ 3) (The absolute value of −3 is 3.)
= 7 + 3
= 10
Page No 76:
Question 6:
(−42) + (−35) = ?
(a) −7
(b) 7
(c) −77
(d) none of these
ANSWER:
(c) –77
(−42) + (−35)
= −42 − 35
= −77
Page No 76:
Question 7:
(−37) + 6 = ?
(a) −43
(b) −31
(c) 31
(d) none of these
ANSWER:
(b) –31
(−37) + 6
= −37 + 6
= −31
Page No 76:
Question 8:
49 + (−27) = ?
(a) −73
(b) 73
(c) 22
(d) none of these
ANSWER:
(c) 22
49 + (−27)
= 49 − 27
= 22
Page No 76:
Question 9:
The successor of −18 is
(a) −19
(b) 17
(c) −17
(d) 19
ANSWER:
(c) –17
In succession, we move from the left to the right of the number line.
Page No 76:
Question 10:
The predecessor of −16 is
(a) −15
(b) −17
(c) 15
(d) 17
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:
The additive inverse of −5 is
(a) 5
(b) 0
(c) −4
(d) −6
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:
−12 − (−5) = ?
(a) −17
(b) −7
(c) 7
(d) none of these
ANSWER:
(b) –7
−12 − (−5)
= −12 + 5
= −7
Page No 76:
Question 13:
−5 − (−8) = ?
(a) 3
(b) 13
(c) −3
(d) none of these
ANSWER:
(b) 13.5 − (−8)
= 5 + 8
= 13
Page No 76:
Question 14:
The sum of two integers is −25. If one of them is 30 then the other is
(a) 55
(b) 5
(c) −55
(d) none of these
ANSWER:
(c) –55
Let x be the other integer.
x + 30 = –25
⇒⇒ x = –25 – 30
⇒⇒ x = –55
Page No 76:
Question 15:
The sum of two integers is 20. If one of them is −5 then the other is
(a) 25
(b) −25
(c) 15
(d) none of these
ANSWER:
(a) 25
Let the other integer be x
x + (-5) = 20
⇒⇒x – 5 = 20
⇒⇒x = 25
Page No 76:
Question 16:
The sum of two integers is −13. If one of them is 8 then the other is
(a) −5
(b) −21
(c) 21
(d) none of these
ANSWER:
(b) −21
Let the other integer be x.
x + 8 = −13
=> x = −13 − 8
=> x = −21
Page No 76:
Question 17:
On subtracting −8 from 0, we get
(a) −8
(b) 8
(c) none of these
ANSWER:
(b) 8
0 − (−8)
= 0 + 8
= 8
Page No 76:
Question 18:
8 + (−8) = ?
(a) 16
(b) −16
(c) 0
(d) none of these
ANSWER:
(c) 0
8 + (−8)
= 8 − 8
= 0
Page No 76:
Question 19:
(−6) + 4 − (−3) = ?
(a) −5
(b) −1
(c) 1
(d) none of these
ANSWER:
(c) 1
(−6) + 4 − (−3)
= −6 + 4 + 3
= −6 + 7
= 1
Page No 76:
Question 20:
6 − (−4) = ?
(a) 2
(b) −10
(c) 10
(d) none of these
ANSWER:
(c) 10
6 − (−4)
= 6 + 4
= 10
Page No 76:
Question 21:
(−7) + (−9) + 12 + (−16) = ?
(a) −20
(b) 20
(c) −12
(d) none of these
ANSWER:
(a) –20
(−7) + (−9) + 12 + (−16)
= −7 − 9 + 12 −16
= −20
Page No 76:
Question 22:
On subtracting 8 from −4, we get
(a) 4
(b) 12
(c) −12
(d) none of these
ANSWER:
(c) –12
–4 – 8
= –12
Page No 76:
Question 23:
On subtracting −9 from −6, we get
(a) −15
(b) −3
(c) 3
(d) none of these
ANSWER:
(c) 3
We have:
−6 − (−9)
= −6 + 9
= 3
Page No 76:
Question 24:
On subtracting −5 from 10, we get
(a) 5
(b) −15
(c) 15
(d) none of these
ANSWER:
(c) 15
We have:
10 − (−5)
= 10 + 5
= 15
Page No 77:
Question 25:
(−6) × 9 = ?
(a) 54
(b) −54
(c) none of these
ANSWER:
(b) –54
We have:
(−6) × 9
= −(6 × 9)
= −54
Page No 77:
Question 26:
(−9) × 6 + (−9) × 4 = ?
(a) −90
(b) 90
(c) −18
(d) 18
ANSWER:
(a) –90
(−9) × 6 + (−9) × 4
Using distributive law:
(−9) × (6 + 4)
= (−9) × (10)
= −90
Page No 77:
Question 27:
36 ÷ (−9) = ?
(a) 4
(b) −4
(c) none of these
ANSWER:
(b) –4
36 ÷ (−9)
36−9=369×(−1)= 1(−1)×369= −1 ×4= −436-9=369×(-1)= 1(-1)×369= -1 ×4= -4
Page No 78:
Exercise 4F
Question 1:
What are integers? Write all integers from −5 to 5.
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:
In each of the pairs given below, find the larger integer.
(i) 0, −3
(ii) −4, −6
(iii) −99, 9
(iv) −385, −615
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:
Write the following integers in increasing order:
−18, 16, 0, −5, 8, −36, −1, 1
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:
Find the value of:
(i) 9 − |−6|
(ii) 6 + |−4|
(iii) −8 − |−3|
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:
Write four integers less than −6 and four integers greater tha −6.
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:
Evaluate:
(i) 8 + (−16)
(ii) (−5) + (−6)
(iii) (−6) × (−8)
(iv) (−36) ÷ 6
(v) 30 − (−50)
(vi) (−40) ÷ (−10)
(vii) 8 × (−5)
(viii) (−30) − 15
ANSWER:
(i) 8 + (–16)
= 8 – 16
= –8
(ii) (–5) + (–6)
= –5 – 6
= –11
(iii) (–6) × (–8)
= (6 × 8)
= 48
(iv) (–36) ÷ 6
−366= (−1)×366= −6-366= (-1)×366= -6
(v)
30 – (–50)
= 30 + 50
= 80
(vi) (–40) ÷ (–10)
=−40−10=(−1)×40(−1)×10=4=-40-10=(-1)×40(-1)×10=4
(vii) 8 × (–5)
= –(8 × 5)
= –40
(viii) (–30) – 15
= –30 – 15
= –45
Page No 78:
Question 7:
The sum of two integers is −12. If one of them is 34, find the other.
ANSWER:
Let the integer be x.
∴∴ 34 + x = –12
or x = –12 – 34
or x = –46
Therefore, the other integer is –46.
Page No 78:
Question 8:
Simplify:
(i) (−24) × (68) + (−24) × 32
(ii) (−9) × 18 − (−9) × 8
(iii) (−147) ÷ (−21)
(iv) 16 ÷ (−1)
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)
=−147−21=(−1)×147(−1)×21=(−1)(−1)×14721=7=-147-21=(-1)×147(-1)×21=(-1)(-1)×14721=7
(iv) 16 ÷ (–1)
=16−1=16×(−1)(−1)×(−1)=16×(−1)=−16=16-1=16×(-1)(-1)×(-1)=16×(-1)=-16 {Multiplying the numerator and the denominator by (–1)}
Page No 78:
Question 9:
The successor of −89 is
(a) −90
(b) −88
(c) 90
(d) 88
ANSWER:
(b) −88
The successor of −89 is −88. The successor of a number lies towards its right on a number line.
−88 lies to the right of −89.
Page No 78:
Question 10:
The predecessor of −99 is
(a) −98
(b) −100
(c) 98
(d) 100
ANSWER:
(b) −100
The predecessor of a number lies to the left of the number.
−100 lies to the left of −99. Hence, −100 is a predecessor of −99.
Page No 78:
Question 11:
Additive inverse of −23 is
(a) −123-123
(b) 123123
(c) 23
(d) −23
ANSWER:
(c) 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:
If (−13 + 6) −25 − (−9)(-13 + 6) -25 – (-9), then the correct symbol in the place holder is
(a) <
(b) >
(c) =
(d) none of these
ANSWER:
(b) >
Here, L.H.S. = (−13 + 6)
= −7
R.H.S. = −25 − (−9)
= −25 + 9
= −16
−7 > −16
L.H.S. > R.H.S.
Page No 78:
Question 13:
? + (−8) = 12
(a) −4
(−20)
(c) 20
(d) 4
ANSWER:
(c) 20
x + (−8) = 12
=> x − 8 = 12
=> x = 12 + 8
=> x = 20
Page No 78:
Question 14:
The integer which is 5 more that (−7) is
(a) −12
(b) 12
(c) −2
(d) 2
ANSWER:
(c) -2
5 more than (−7) means 5 added to (−7).
5 + (−7)
= 5 − 7
= −2
Page No 78:
Question 15:
What should be added to 16 to get (−31)?
(a) 15
(b) −15
(c) 47
(d) −47
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:
When 34 is subtracted from −36, we get
(a) 2
(b) −2
(c) 70
(d) −70
ANSWER:
(d) −70
−36 − 34
= −70
Page No 78:
Question 17:
Fill in the blanks.
(i) −23 − (?) = 15.
(ii) The largest negative integer is …… .
(iii) The smallest positive integer is …… .
(iv) (−8) + (−6) − (−3) = …… .
(v) The predecessor of −200 is …… .
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 −6 + 3
= −11
(v)
The predecessor of −200:
(−200 − 1)
= −201
Page No 79:
Question 18:
Write ‘T’ for true and ‘F’ for false in each of the following:
(i) 0 is neither positive nor negative.
(ii) −(−36) − 1 = − 37.
(iii) On the number line −10 lies to the right of −6.
(iv) 0 is an integer.
(v) −|−15| = −15.
(vi) |−40| + 40 = 0.
ANSWER:
(i) T
(ii) F
−(−36) − 1
= 36 − 1
= 35
(iii) F
This is because −10 is less than −6.
(iv) T
(v) T
−|−15|
= −(15)
= −15
(vi) F
|−40| + 40
= 40 + 40
= 80
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