Class 7 Concise Mathematics · Chapter 1
Integers: Complete Step-by-Step Solutions
Master multiplication, division, signs, BODMAS, brackets, properties and real-life integer problems through clear worked solutions.
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Complete worked solutions
Exercise 1(A)
Multiplication, distributive property, signs and comparison
Evaluate :
(i) 427 × 8 + 2 × 427
(ii) 394 × 12 + 394 × (-2)
(iii) 558 × 27 + 3 × 558
Show step-by-step answer
Using the distributive property a × b + a × c = a × (b + c) :
(i) Solving,
427 × 8 + 2 × 427
= 427 × (8 + 2)
= 427 × 10
= 4270.
∴ 427 × 8 + 2 × 427 = 4270.
(ii) Solving,
394 × 12 + 394 × (-2)
= 394 × {12 + (-2)}
= 394 × 10
= 3940.
∴ 394 × 12 + 394 × (-2) = 3940.
(iii) Solving,
558 × 27 + 3 × 558
= 558 × (27 + 3)
= 558 × 30
= 16740.
∴ 558 × 27 + 3 × 558 = 16740.
Evaluate :
(i) 673 × 9 + 673
(ii) 1925 × 101 - 1925
Show step-by-step answer
(i) Solving,
673 × 9 + 673
= 673 × (9 + 1)
= 673 × 10
= 6730.
∴ 673 × 9 + 673 = 6730.
(ii) Solving,
1925 × 101 - 1925
= 1925 × (101 - 1)
= 1925 × 100
= 192500.
∴ 1925 × 101 - 1925 = 192500.
Verify :
(i) 37 × {8 + (-3)} = 37 × 8 + 37 × (-3)
(ii) (-82) × {(-4) + 19} = (-82) × (-4) + (-82) × 19
(iii) {7 - (-7)} × 7 = 7 × 7 - (-7) × 7
(iv) {(-15) - 8} × -6 = (-15) × (-6) - 8 × (-6)
Show step-by-step answer
(i) 37 × {8 + (-3)} = 37 × 8 + 37 × (-3)
L.H.S. = 37 × {8 + (-3)} = 37 × 5 = 185
R.H.S. = 37 × 8 + 37 × (-3) = 296 + (-111) = 296 - 111 = 185
∴ L.H.S. = R.H.S.
Hence, verified.
(ii) (-82) × {(-4) + 19} = (-82) × (-4) + (-82) × 19
L.H.S. = (-82) × {(-4) + 19} = (-82) × 15 = -1230
R.H.S. = (-82) × (-4) + (-82) × 19 = 328 + (-1558) = 328 - 1558 = -1230
∴ L.H.S. = R.H.S.
Hence, verified.
(iii) {7 - (-7)} × 7 = 7 × 7 - (-7) × 7
L.H.S. = {7 - (-7)} × 7 = (7 + 7) × 7 = 14 × 7 = 98
R.H.S. = 7 × 7 - (-7) × 7 = 49 - (-49) = 49 + 49 = 98
∴ L.H.S. = R.H.S.
Hence, verified.
(iv) {(-15) - 8} × -6 = (-15) × (-6) - 8 × (-6)
L.H.S. = {(-15) - 8} × (-6) = (-23) × (-6) = 138
R.H.S. = (-15) × (-6) - 8 × (-6) = 90 - (-48) = 90 + 48 = 138
∴ L.H.S. = R.H.S.
Hence, verified.
Evaluate :
(i) 15 × 8
(ii) 15 × (-8)
(iii) (-15) × 8
(iv) (-15) × -8
Show step-by-step answer
(i) 15 × 8 = 120.
(ii) 15 × (-8) = -120.
(iii) (-15) × 8 = -120.
(iv) (-15) × (-8) = 120.
Evaluate :
(i) 4 × 6 × 8
(ii) 4 × 6 × (-8)
(iii) 4 × (-6) × 8
(iv) (-4) × 6 × 8
(v) 4 × (-6) × (-8)
(vi) (-4) × (-6) × 8
(vii) (-4) × 6 × (-8)
(viii) (-4) × (-6) × (-8)
Show step-by-step answer
(i) 4 × 6 × 8 = 24 × 8 = 192.
(ii) 4 × 6 × (-8) = 24 × (-8) = -192.
(iii) 4 × (-6) × 8 = (-24) × 8 = -192.
(iv) (-4) × 6 × 8 = (-24) × 8 = -192.
(v) 4 × (-6) × (-8) = (-24) × (-8) = 192.
(vi) (-4) × (-6) × 8 = 24 × 8 = 192.
(vii) (-4) × 6 × (-8) = (-24) × (-8) = 192.
(viii) (-4) × (-6) × (-8) = 24 × (-8) = -192.
Evaluate :
(i) 2 × 4 × 6 × 8
(ii) 2 × (-4) × 6 × 8
(iii) (-2) × 4 × (-6) × 8
(iv) (-2) × (-4) × 6 × (-8)
(v) (-2) × (-4) × (-6) × (-8)
Show step-by-step answer
(i) 2 × 4 × 6 × 8 = 8 × 48 = 384.
(ii) 2 × (-4) × 6 × 8 = (-8) × 48 = -384.
(iii) (-2) × 4 × (-6) × 8 = (-8) × (-48) = 384.
(iv) (-2) × (-4) × 6 × (-8) = 8 × (-48) = -384.
(v) (-2) × (-4) × (-6) × (-8) = 8 × 48 = 384.
Determine the integer whose product with '-1' is :
(i) -47
(ii) 63
(iii) -1
(iv) 0
Show step-by-step answer
We will use the property:
(-1) × a = -a
Since multiplying by -1 changes only the sign and not the value of an integer, the required integer has the same value as the given product but with the opposite sign.
(i) Product = -47
(-1) x 47 = -47
Hence, required integer = 47.
(ii) Product = 63
(-1) x -63 = 63
Hence, required integer = -63.
(iii) Product = -1
(-1) x 1 = -1
Hence, required integer = 1.
(iv) Product = 0
(-1) x 0 = 0
Hence, required integer = 0.
Eighteen integers are multiplied together. What will be the sign of their product, if :
(i) 15 of them are negative and 3 are positive ?
(ii) 12 of them are negative and 6 are positive ?
(iii) 9 of them are positive and the remaining are negative ?
(iv) all are negative ?
Show step-by-step answer
The sign of the product depends on the number of negative integers. If the number of negative integers is even, the product is positive; if it is odd, the product is negative.
(i) Number of negative integers = 15, which is odd.
∴ The product will be negative.
(ii) Number of negative integers = 12, which is even.
∴ The product will be positive.
(iii) Number of negative integers = 18 - 9 = 9, which is odd.
∴ The product will be negative.
(iv) Number of negative integers = 18, which is even.
∴ The product will be positive.
Find which is greater ?
(i) (8 + 10) × 15 or 8 + 10 × 15
(ii) 12 × (6 - 8) or 12 × 6 - 8
(iii) {(-3) - 4} × (-5) or (-3) - 4 × (-5)
Show step-by-step answer
(i) (8 + 10) × 15 = 18 × 15 = 270
8 + 10 × 15 = 8 + 150 = 158
Since 270 > 158,
∴ (8 + 10) × 15 is greater.
(ii) 12 × (6 - 8) = 12 × (-2) = -24
12 × 6 - 8 = 72 - 8 = 64
Since 64 > -24,
∴ 12 × 6 - 8 is greater.
(iii) {(-3) - 4} × (-5) = (-7) × (-5) = 35
(-3) - 4 × (-5) = -3 - (-20) = -3 + 20 = 17
Since 35 > 17,
∴ {(-3) - 4} × (-5) is greater.
State true or false :
(i) product of two different integers can be zero.
(ii) product of 120 negative integers and 121 positive integers is negative.
(iii) a × (b + c) = a × b + c
(iv) (b - c) × a = b - c × a.
Show step-by-step answer
(i) True. If one of the two different integers is 0, their product is 0. For example, 0 × 5 = 0.
(ii) False. The number of negative integers is 120, which is even, so the product is positive (positive integers do not change the sign).
(iii) False. By the distributive property, a × (b + c) = a × b + a × c, not a × b + c.
(iv) False. By the distributive property, (b - c) × a = b × a - c × a, not b - c × a.
Complete worked solutions
Exercise 1(B)
Division of integers and BODMAS
Divide :
(i) 117 by 9 (ii) (-117) by 9 (iii) 117 by (-9) (iv) (-117) by (-9) (v) 225 by (-15) (vi) (-552) ÷ 24 (vii) (-798) by (-21) (viii) (-910) ÷ 26
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Like signs give a positive quotient; unlike signs give a negative quotient.
Evaluate: (-234) ÷ 13
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Evaluate: 234 ÷ (-13)
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Evaluate: (-234) ÷ (-13)
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Evaluate: 374 ÷ (-17)
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Evaluate: (-374) ÷ 17
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Evaluate: (-374) ÷ (-17)
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Evaluate: (-728) ÷ 14
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Evaluate: 272 ÷ (-17)
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Find the quotient in each division: (i) 299 ÷ 23 (ii) 299 ÷ (-23) (iii) (-384) ÷ 16 (iv) (-572) ÷ (-22) (v) 408 ÷ (-17)
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Divide: (i) 204 by 17 (ii) 152 by -19 (iii) 0 by 35 (iv) 0 by (-82) (v) 5490 by 10 (vi) 762800 by 100
Show step-by-step answer
State, true or false :
(i) 0 ÷ 32 = 0
(ii) 0 ÷ (-9) = 0
(iii) (-37) ÷ 0 = 0
(iv) 0 ÷ 0 = 0
Show step-by-step answer
(i) True. Zero divided by any non-zero integer is 0.
(ii) True. Zero divided by any non-zero integer is 0.
(iii) False. Division of any integer by zero is not defined.
(iv) False. Division of zero by zero is not defined.
Evaluate :
(i) 42 ÷ 7 + 4
(ii) 12 + 18 ÷ 3
(iii) 19 - 20 ÷ 4
(iv) 16 - 5 × 3 + 4
(v) 6 - 8 - (-6) ÷ 2
(vi) 13 - 12 ÷ 4 × 2
(vii) 16 + 8 ÷ 4 - 2 × 3
(viii) 16 ÷ 8 + 4 - 2 × 3
(ix) 16 - 8 + 4 ÷ 2 × 3
(x) (-4) + (-12) ÷ (-6)
(xi) (-18) + 6 ÷ 3 + 5
(xii) (-20) × (-1) + 14 ÷ 7
Show step-by-step answer
Solving using the BODMAS rule (Division, Multiplication, Addition, Subtraction) :
(i) Solving,
42 ÷ 7 + 4
= 6 + 4
= 10
∴ 42 ÷ 7 + 4 = 10
(ii) Solving,
12 + 18 ÷ 3
= 12 + 6
= 18
∴ 12 + 18 ÷ 3 = 18
(iii) Solving,
19 - 20 ÷ 4
= 19 - 5
= 14
∴ 19 - 20 ÷ 4 = 14
(iv) Solving,
16 - 5 × 3 + 4
= 16 - 15 + 4
= 20 - 15
= 5
∴ 16 - 5 × 3 + 4 = 5
(v) Solving,
6 - 8 - (-6) ÷ 2
= 6 - 8 - (-3)
= 6 - 8 + 3
= 9 - 8
= 1
∴ 6 - 8 - (-6) ÷ 2 = 1
(vi) Solving,
13 - 12 ÷ 4 × 2
= 13 - 3 × 2
= 13 - 6
= 7
∴ 13 - 12 ÷ 4 × 2 = 7
(vii) Solving,
16 + 8 ÷ 4 - 2 × 3
= 16 + 2 - 6
= 18 - 6
= 12
∴ 16 + 8 ÷ 4 - 2 × 3 = 12
(viii) Solving,
16 ÷ 8 + 4 - 2 × 3
= 2 + 4 - 6
= 6 - 6
= 0
∴ 16 ÷ 8 + 4 - 2 × 3 = 0
(ix) Solving,
16 - 8 + 4 ÷ 2 × 3
= 16 - 8 + 2 × 3
= 16 - 8 + 6
= 22 - 8
= 14
∴ 16 - 8 + 4 ÷ 2 × 3 = 14
(x) Solving,
(-4) + (-12) ÷ (-6)
= (-4) + 2
= -2
∴ (-4) + (-12) ÷ (-6) = -2
(xi) Solving,
(-18) + 6 ÷ 3 + 5
= (-18) + 2 + 5
= -18 + 7
= -11
∴ (-18) + 6 ÷ 3 + 5 = -11
(xii) Solving,
(-20) × (-1) + 14 ÷ 7
= 20 + 2
= 22
∴ (-20) × (-1) + 14 ÷ 7 = 22
The product of two integers is -90. If one of them is 6, find the other integer.
Show step-by-step answer
Let the other integer be x.
∴ The other integer is -15.
The product of two integers is 210. If one of them is -15, find the other.
Show step-by-step answer
Let the other integer be x.
∴ The other integer is -14.
Complete worked solutions
Exercise 1(C)
Simplification with brackets and vinculum
Evaluate :
18 - (20 - 15 ÷ 3).
Show step-by-step answer
Solving,
18 - (20 - 15 ÷ 3)
= 18 - (20 - 5)
= 18 - 15
= 3
∴ 18 - (20 - 15 ÷ 3) = 3
-15 + 24 ÷ (15 - 13).
Show step-by-step answer
Solving,
-15 + 24 ÷ (15 - 13)
= -15 + 24 ÷ 2
= -15 + 12
= -3
∴ -15 + 24 ÷ (15 - 13) = -3
Evaluate: 35 - {15 + 14 - (13 + 2 - 1 + 3)}
Show step-by-step answer
∴ The value is 23.
Evaluate: 27 - {13 + 4 - (8 + 4 - 1 + 3)}
Show step-by-step answer
∴ The value is 18.
Evaluate: 32 - [43 - {51 - (20 - 18 - 7)}]
Show step-by-step answer
∴ The value is 31.
46 - [26 - {14 - (15 - 4 ÷ 2 × 2)}]
Show step-by-step answer
Solving,
46 - [26 - {14 - (15 - 4 ÷ 2 × 2)}]
= 46 - [26 - {14 - (15 - 2 × 2)}]
= 46 - [26 - {14 - (15 - 4)}]
= 46 - [26 - {14 - 11}]
= 46 - [26 - 3]
= 46 - 23
= 23
∴ 46 - [26 - {14 - (15 - 4 ÷ 2 × 2)}] = 23
45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]
Show step-by-step answer
Solving,
45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]
= 45 - [38 - {60 ÷ 3 - (6 - 3) ÷ 3}]
= 45 - [38 - {60 ÷ 3 - 3 ÷ 3}]
= 45 - [38 - {20 - 1}]
= 45 - [38 - 19]
= 45 - 19
= 26
∴ 45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}] = 26
Evaluate: 17 - [17 - {17 - (17 - 17 - 17)}]
Show step-by-step answer
∴ The value is 0.
Evaluate: 2550 - [510 - {270 - (90 - 80 + 7)}]
Show step-by-step answer
∴ The value is 2307.
Evaluate: 30 + [-2 × (25 - 13 - 3)]
Show step-by-step answer
∴ The value is 0.
88 - {5 - (-48) ÷ (-16)}
Show step-by-step answer
Solving,
88 - {5 - (-48) ÷ (-16)}
= 88 - {5 - 3}
= 88 - 2
= 86
∴ 88 - {5 - (-48) ÷ (-16)} = 86
Evaluate: 9 × (8 - 3 + 2) - 2(2 + 3 + 3)
Show step-by-step answer
∴ The value is 11.
Evaluate: 2 - [3 - {6 - (5 - 4 - 3)}]
Show step-by-step answer
∴ The value is 1.
Complete worked solutions
Exercise 1(D)
Word problems and integer applications
The sum of two integers is -15. If one of them is 9, find the other.
Show step-by-step answer
Let the other integer be x.
9 + x = -15
x = -15 - 9
x = -24
∴ The other integer is -24.
The difference between integers x and -6 is -5. Find the values of x.
x - (-6) = -5 or -6 - x = -5
Show step-by-step answer
The difference between x and -6 is -5.
x - (-6) = -5 or -6 - x = -5
x + 6 = -5 or -x = -5 + 6
x = -5 - 6 or -x = 1
x = -11 or x = -1
∴ x = -11 or -1
The sum of two integers is 28. If one integer is -45, find the other.
Show step-by-step answer
Let the other integer be x.
(-45) + x = 28
x = 28 - (-45)
x = 28 + 45
x = 73
∴ The other integer is 73.
The sum of two integers is -56. If one integer is -42, find the other.
Show step-by-step answer
Let the other integer be x.
(-42) + x = -56
x = -56 - (-42)
x = -56 + 42
x = -14
∴ The other integer is -14.
The difference between an integer x and (-9) is 6. Find all possible values of x.
Show step-by-step answer
The difference between x and -9 is 6.
x - (-9) = 6 or -9 - x = 6
x + 9 = 6 or -x = 6 + 9
x = 6 - 9 or -x = 15
x = -3 or x = -15
∴ x = -3 or -15
Write all the integers between -15 and 15, which are divisible by 2 and 3.
Show step-by-step answer
An integer divisible by both 2 and 3 is divisible by 6.
The multiples of 6 lying between -15 and 15 are :
-12, -6, 0, 6, 12
Write all the integers between -5 and 5, which are divisible by 2 or 3.
Show step-by-step answer
The integers between -5 and 5 are -4, -3, -2, -1, 0, 1, 2, 3, 4.
Integers divisible by 2 : -4, -2, 0, 2, 4
Integers divisible by 3 : -3, 0, 3
Combining (taking each integer once), the integers divisible by 2 or 3 are :
-4, -3, -2, 0, 2, 3, 4
Find the result of subtracting the sum of all integers between 20 and 30 from the sum of all integers from 20 to 30.
Show step-by-step answer
Sum of all integers from 20 to 30 (both included) :
= 20 + 21 + 22 +... + 30
= 275
Sum of all integers between 20 and 30 (both excluded) :
= 21 + 22 +... + 29
= 225
Required result = 275 - 225 = 50
Hence, the required result is 50.
Add the product of (-13) and (-17) to the quotient of (-187) and 11.
Show step-by-step answer
Hence, the required result is 204.
The product of two integers is -180. If one of them is 12, find the other.
Show step-by-step answer
Let the other integer be x.
∴ The other integer is -15.
(i) A number changes from -20 to 30. What is the increase or decrease in the number ?
(ii) A number changes from 40 to -30. What is the increase or decrease in the number ?
Show step-by-step answer
(i) Change = 30 - (-20)
= 30 + 20
= 50
Since the change is positive,
∴ There is an increase of 50.
(ii) Change = -30 - 40
= -70
Since the change is negative,
∴ There is a decrease of 70.
Complete worked solutions
Exercise 1(E)
Properties, mixed practice and real-life questions
If a = -12 and b = -10, verify that : a + b = b + a.
Show step-by-step answer
Given a = -12 and b = -10.
L.H.S. = a + b = (-12) + (-10) = -22
R.H.S. = b + a = (-10) + (-12) = -22
∴ L.H.S. = R.H.S.
Hence, verified.
If a = -7, b = -5 and c = 8, verify that : a + (b + c) = (a + b) + c.
Show step-by-step answer
Given a = -7, b = -5 and c = 8.
L.H.S. = a + (b + c) = -7 + {(-5) + 8} = -7 + 3 = -4
R.H.S. = (a + b) + c = {(-7) + (-5)} + 8 = -12 + 8 = -4
∴ L.H.S. = R.H.S.
Hence, verified.
If a = 7, b = 5 and c = -8, verify that : a - (b - c) ≠ (a - b) - c.
Show step-by-step answer
Given a = 7, b = 5 and c = -8.
L.H.S. = a - (b - c) = 7 - {5 - (-8)} = 7 - (5 + 8) = 7 - 13 = -6
R.H.S. = (a - b) - c = (7 - 5) - (-8) = 2 + 8 = 10
Since -6 ≠ 10,
∴ L.H.S. ≠ R.H.S.
Hence, verified.
The difference between two integers x and -12 is -15. Find the value(s) of x.
Show step-by-step answer
The difference between x and -12 is -15.
x - (-12) = -15 or -12 - x = -15
x + 12 = -15 or -x = -15 + 12
x = -15 - 12 or -x = -3
x = -27 or x = 3
∴ x = -27 or 3.
An aeroplane is 800 m vertically above the head of a boy. After sometime, it is 1125 m vertically above the head of the same boy. What is the change in the height of the aeroplane ?
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Initial height = 800 m
Final height = 1125 m
Change in height = 1125 - 800 = 325 m
Since the change is positive, the height has increased.
∴ The change in the height of the aeroplane is an increase of 325 m.
Write a pair of integers whose : (i) sum = -7 (ii) difference = -7
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(i) A pair of integers whose sum is -7 :
(-3) + (-4) = -7
∴ The required pair is (-3, -4).
(ii) A pair of integers whose difference is -7 :
2 - 9 = -7
∴ The required pair is (2, 9).
Write two integers each of which is smaller than -3 and their difference is greater than -3.
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Let the two integers be -4 and -5. Both are smaller than -3.
Their difference = (-4) - (-5) = -4 + 5 = 1, which is greater than -3.
∴ The required integers are -4 and -5.
Evaluate :
(i) (-1) × (-1) × (-1) ×.......... 60 times.
(ii) (-1) × (-1) × (-1) × (-1) ×.......... 75 times.
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(i) Here, (-1) is multiplied 60 times. Since 60 is even, the product is positive.
(-1) × (-1) × (-1) ×... 60 times = 1
(ii) Here, (-1) is multiplied 75 times. Since 75 is odd, the product is negative.
(-1) × (-1) × (-1) ×... 75 times = -1
Evaluate :
(i) (-2) × (-3) × (-4) × (-5) × (-6)
(ii) (-3) × (-6) × (-9) × (-12)
(iii) (-11) × (-15) + (-11) × (-25)
(iv) 10 × (-12) + 5 × (-12)
Show step-by-step answer
(i) There are 5 negative integers (odd), so the product is negative.
(-2) × (-3) × (-4) × (-5) × (-6)
= -(2 × 3 × 4 × 5 × 6)
= -720
∴ (-2) × (-3) × (-4) × (-5) × (-6) = -720
(ii) There are 4 negative integers (even), so the product is positive.
(-3) × (-6) × (-9) × (-12)
= 3 × 6 × 9 × 12
= 1944
∴ (-3) × (-6) × (-9) × (-12) = 1944
(iii) Using the distributive property :
(-11) × (-15) + (-11) × (-25)
= (-11) × {(-15) + (-25)}
= (-11) × (-40)
= 440
∴ (-11) × (-15) + (-11) × (-25) = 440
(iv) Using the distributive property :
10 × (-12) + 5 × (-12)
= (-12) × (10 + 5)
= (-12) × 15
= -180
∴ 10 × (-12) + 5 × (-12) = -180
(i) If X × (-1) = -36, is X positive or negative ?
(ii) If X × (-1) = 36, is X positive or negative ?
Show step-by-step answer
(i) X × (-1) = -36
⇒ X = (-36) ÷ (-1) = 36
∴ X = 36, which is positive.
(ii) X × (-1) = 36
⇒ X = 36 ÷ (-1) = -36
∴ X = -36, which is negative.
Evaluate :
(-20) + (-8) ÷ (-2) × 3
Show step-by-step answer
Solving,
(-20) + (-8) ÷ (-2) × 3
= (-20) + 4 × 3
= (-20) + 12
= -8
∴ (-20) + (-8) ÷ (-2) × 3 = -8
Evaluate :
(-5) - (-48) ÷ (-16) + (-2) × 6
Show step-by-step answer
Solving,
(-5) - (-48) ÷ (-16) + (-2) × 6
= (-5) - 3 + (-12)
= -5 - 3 - 12
= -20
∴ (-5) - (-48) ÷ (-16) + (-2) × 6 = -20
Evaluate :
16 + 8 ÷ 4 - 2 × 3
Show step-by-step answer
Solving,
16 + 8 ÷ 4 - 2 × 3
= 16 + 2 - 6
= 18 - 6
= 12
∴ 16 + 8 ÷ 4 - 2 × 3 = 12
Evaluate :
16 ÷ 8 × 4 - 2 × 3
Show step-by-step answer
Solving,
16 ÷ 8 × 4 - 2 × 3
= 2 × 4 - 6
= 8 - 6
= 2
∴ 16 ÷ 8 × 4 - 2 × 3 = 2
Evaluate :
27 - [5 + {28 - (29 - 7)}]
Show step-by-step answer
Solving,
27 - [5 + {28 - (29 - 7)}]
= 27 - [5 + {28 - 22}]
= 27 - [5 + 6]
= 27 - 11
= 16
∴ 27 - [5 + {28 - (29 - 7)}] = 16
Evaluate: 48 - [18 - {16 - (5 - 4 + 1)}]
Show step-by-step answer
∴ The value is 46.
Evaluate :
-8 - {-6 (9 - 11) + 18 ÷ -3}
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Solving,
-8 - {-6 (9 - 11) + 18 ÷ -3}
= -8 - {-6 × (-2) + (-6)}
= -8 - {12 - 6}
= -8 - 6
= -14
∴ -8 - {-6 (9 - 11) + 18 ÷ -3} = -14
Evaluate: (24 ÷ 12 - 9 - 12) - (3 × 8 ÷ 4 + 1)
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∴ The value is -11.
Find the difference between 8 and -8.
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The difference between two integers 8 and -8 :
Difference = 8 - (-8) = 8 + 8 = 16
or
Difference = -8 - 8 = -16
∴ The difference between 8 and -8 is 16 or -16.
Subtract the sum of -107 and 72 from the sum of 55 and -32.
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Sum of 55 and -32 = 55 + (-32) = 23
Sum of -107 and 72 = (-107) + 72 = -35
Required result = 23 - (-35) = 23 + 35 = 58
∴ The required result is 58.
Write three consecutive integers
(i) succeeding -14
(ii) preceding -22
(iii) which are even and succeed 24
(iv) which are odd and precede 8
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(i) Three consecutive integers succeeding -14 are :
-13, -12, -11
(ii) Three consecutive integers preceding -22 are :
-23, -24, -25
(iii) Three consecutive even integers succeeding 24 are :
26, 28, 30
(iv) Three consecutive odd integers preceding 8 are :
7, 5, 3
Points A, B, C and D are marked on a number line as shown below :
Points A, B, C and D are marked on a number line as shown below : Find :
(i) A - B (ii) D + C (iii) C + A (iv) B - C
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From the number line, A = 3, B = 8, C = -9 and D = -3.
(i) A - B = 3 - 8 = -5
(ii) D + C = (-3) + (-9) = -12
(iii) C + A = (-9) + 3 = -6
(iv) B - C = 8 - (-9) = 8 + 9 = 17
At a place, the temperature on Monday was 30°C. It rose by 3°C on Tuesday and then dropped by 8°C on Wednesday. Find the temperature of this place on
(i) Tuesday (ii) Wednesday
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(i) Temperature on Tuesday = 30°C + 3°C = 33°C
∴ The temperature on Tuesday was 33°C.
(ii) Temperature on Wednesday = 33°C - 8°C = 25°C
∴ The temperature on Wednesday was 25°C.
How much does 35 exceed (-35) ?
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Required = 35 - (-35) = 35 + 35 = 70
∴ 35 exceeds (-35) by 70.
How much is -12 less than 3 ?
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Required = 3 - (-12) = 3 + 12 = 15
∴ -12 is less than 3 by 15.
Subtract (-34) from the sum of 57 and (-51).
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Sum of 57 and (-51) = 57 + (-51) = 6
Required result = 6 - (-34) = 6 + 34 = 40
∴ The required result is 40.
Write a pair of negative integers whose difference is 8.
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Consider the negative integers -2 and -10.
Difference = (-2) - (-10) = -2 + 10 = 8
∴ The required pair is (-2, -10).
Write a positive integer and a negative integer whose sum is -15.
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Consider the positive integer 5 and the negative integer -20.
Sum = 5 + (-20) = -15
∴ The required pair is (5, -20).
Evaluate :
(i) (-2) × (-2) × (-5) × 7
(ii) (-1) × (-5) × (-4) × (-6)
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(i) There are 3 negative integers (odd), so the product is negative.
(-2) × (-2) × (-5) × 7
= -(2 × 2 × 5 × 7)
= -140
∴ (-2) × (-2) × (-5) × 7 = -140
(ii) There are 4 negative integers (even), so the product is positive.
(-1) × (-5) × (-4) × (-6)
= 1 × 5 × 4 × 6
= 120
∴ (-1) × (-5) × (-4) × (-6) = 120
In a class test, containing 20 questions, 5 marks are given for every correct answer and -3 marks are given for each incorrect answer. A student of this class attempted all the questions out of which only 12 were correct. Find the score of this student.
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Number of correct answers = 12
Number of incorrect answers = 20 - 12 = 8
Marks for correct answers = 12 × 5 = 60
Marks for incorrect answers = 8 × (-3) = -24
Total score = 60 + (-24) = 60 - 24 = 36
∴ The score of this student is 36.
Smitha starts moving from a point A and takes 20 steps towards North, each step being 40 cm in length. Then she moves by taking 30 steps towards South, each step being 28 cm long. If Smitha finally reaches at point B, find the distance between A and B.
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Smitha starts moving from a point A and takes 20 steps towards North, each step being 40 cm in length. Then she moves by taking 30 steps towards South, each step being 28 cm long. If Smitha finally reaches at point B, find the distance between A and B. Taking the direction towards North as positive and towards South as negative.
Let Smitha start from point A towards North and reach point C.
Distance moved towards North = 20 × 40 = 800 cm
Smitha now starts from point C towards South and reach point B.
Distance moved towards South = 30 × 28 = 840 cm
Net displacement from A = 800 + (-840) = -40 cm
The negative sign shows that B is 40 cm to the South of A.
∴ The distance between A and B is 40 cm.
Complete worked solutions
MCQs
Objective questions with complete working
The addition of -30, -15, -10, 20 and 15 is :
-20
25
-15
15
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Solving,
(-30) + (-15) + (-10) + 20 + 15
= -55 + 35
= -20
Hence, option 1 is the correct option.
Two integers a and b are such that a ÷ b = -2; then (a, b) is :
(-8, -4)
(-2, 4)
(6, -3)
(10, 5)
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Checking each option for a ÷ b = -2 :
For (6, -3) : 6 ÷ (-3) = -2
Hence, option 3 is the correct option.
15 ÷ (-5) - (2 × -3) is equal to :
9
-9
3
-3
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Solving,
15 ÷ (-5) - (2 × -3)
= -3 - (-6)
= -3 + 6
= 3
Hence, option 3 is the correct option.
7 - 8 ÷ (-2) + 3 × (-4) is equal to :
1
-13
13
-1
Show step-by-step answer
Solving,
7 - 8 ÷ (-2) + 3 × (-4)
= 7 - (-4) + (-12)
= 7 + 4 - 12
= -1
Hence, option 4 is the correct option.
-[8 - {11 + 30 ÷ (4 - 7 - 5)}] is equal to:
-18 18 16 -14
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Hence, option 2 is correct.
The number that must be subtracted from -2 to get -15 is :
-17
17
-13
13
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Let the number to be subtracted be x.
-2 - x = -15
-x = -15 + 2
-x = -13
x = 13
Hence, option 4 is the correct option.
{-2 ÷ (-12)}{-3 ÷ (-13)}{-4 ÷ (-14)} is equal to:
(-4) × (-9) × (-16) 4 × 9 × 16 (-1/4) × (-1/9) × (-1/16) 1/(4 × 9 × 16)
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Hence, option 2 is correct.
(-45) × 0 + (-45) ÷ 0 is equal to :
1
0
-90
none of these
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The term (-45) ÷ 0 involves division by zero, which is not defined. Hence, the value of the expression is not defined.
Hence, option 4 is the correct option.
5 ÷ [5 + {5 - (5 - 5 - 5)}] is:
0 1 -1 5
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Hence, option 2 is correct.
12 ÷ 29 × 214 is:
32/9 243/2 8 none of these
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Hence, option 2 is correct.
(-3) × [5 + (-9)] is equal to :
-12
12
6
-6
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Solving,
(-3) × [5 + (-9)]
= (-3) × (-4)
= 12
Hence, option 2 is the correct option.
Liquid at 15°C is cooled to temperature -5°C, the fall in temperature is :
10°C
-10°C
20°C
-20°C
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Fall in temperature = 15°C - (-5°C) = 15°C + 5°C = 20°C
Hence, option 3 is the correct option.
The number 25 is decreased by 6 in every second. The number by which 25 will decrease in 6 seconds :
5
-5
30
36
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Decrease in 6 seconds = 6 × 6 = 36
Hence, option 4 is the correct option.
A = -9, B = -12 and C = 10; then A + B - C is equal to :
31
-31
-11
11
Show step-by-step answer
Solving,
A + B - C
= (-9) + (-12) - 10
= -21 - 10
= -31
Hence, option 2 is the correct option.
Complete worked solutions
Statement I–II
Test statements using integer properties
Statement 1: The additive inverse of (-a) is a.
Statement 2: The multiplicative inverse of an integer a is 1a.
Both statements are true. Both statements are false. Statement 1 is true, and statement 2 is false. Statement 1 is false, and statement 2 is true.
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Statement 1 is true because (-a) + a = 0.
Statement 2 is false as written because the multiplicative inverse is not defined for a = 0; it exists only when a ≠ 0.
Hence, option 3 is correct.
Statement 1: 16 ÷ (8 ÷ 4) = (16 ÷ 8) ÷ 4.
Statement 2: Subtraction of integers is not associative.
Both statements are true. Both statements are false. Statement 1 is true, and statement 2 is false. Statement 1 is false, and statement 2 is true.
Show step-by-step answer
Since 8 ≠ 12, Statement 1 is false. Statement 2 is true because subtraction is not associative.
Hence, option 4 is correct.
Complete worked solutions
Assertion–Reason
Evaluate assertions and their reasons
Assertion (A) : 155 × 277 + 155 × (-45) = 155 × [277 + (-45)]
Reason (R) : Distributive property holds for any three integers. i.e. a × (b + c) = a × b + a × c, for all integers a, b, c.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.
Show step-by-step answer
Assertion (A) is the distributive property applied to 155, 277 and (-45), so it is true.
Reason (R) correctly states the distributive property, which holds for all integers, so it is true. Also, R is the correct explanation of A.
So, both A and R are true.
Hence, option 3 is the correct option.
Assertion (A) : (-59) × (-47) ≠ (47) × (59)
Reason (R) : For any integers a and b, a × b = b × a
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.
Show step-by-step answer
Assertion (A) : (-59) × (-47) = 59 × 47 = 2773 and (47) × (59) = 2773. Since both are equal, the statement (-59) × (-47) ≠ (47) × (59) is false. So A is false.
Reason (R) : Multiplication of integers is commutative, i.e. a × b = b × a. So R is true.
So, A is false and R is true.
Hence, option 2 is the correct option.
Assertion (A) : The product of two integers is -775. If one of them is 25, the other integer is 31.
Reason (R) : If a and b are two integers, then a ÷ b or b ÷ a may or may not be an integer.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.
Show step-by-step answer
Assertion (A) : The other integer = (-775) ÷ 25 = -31, not 31. So A is false.
Reason (R) : For two integers a and b, the quotient a ÷ b or b ÷ a may or may not be an integer. So R is true.
So, A is false and R is true.
Hence, option 2 is the correct option.
Frequently asked questions
Integers: quick revision
Use these short explanations before attempting the exercises.
What is an integer?+
An integer is a whole number that may be negative, positive or zero. Examples include -5, 0 and 12.
How do we decide the sign of a product of integers?+
Count the negative factors. An even number of negative factors gives a positive product, while an odd number gives a negative product.
What is the distributive property of integers?+
For integers a, b and c, a × (b + c) = a × b + a × c. It also works with subtraction: a × (b - c) = a × b - a × c.
Can an integer be divided by zero?+
No. Division by zero is not defined. However, zero divided by any non-zero integer is zero.
What order should be followed while simplifying integer expressions?+
Use BODMAS: brackets first, then orders if present, followed by division and multiplication from left to right, and finally addition and subtraction from left to right.
Which Class 7 integer exercises are included here?+
The post includes Exercise 1(A), 1(B), 1(C), 1(D), 1(E), multiple-choice questions, Statement I-II questions and Assertion-Reason questions.