If a = -12 and b = -10, verify that : a + b = b + a.
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.
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.
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.
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 ?
Answer
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
Answer
(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.
Answer
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.
Answer
(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)
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 ?
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
Answer
Solving,
(-20) + (-8) ÷ (-2) × 3
= (-20) + 4 × 3
= (-20) + 12
= -8
∴ (-20) + (-8) ÷ (-2) × 3 = -8
Evaluate :
(-5) - (-48) ÷ (-16) + (-2) × 6
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
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
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)}]
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 - )}]
Answer
Solving,
48 - [18 - {16 - (5 - )}]
= 48 - [18 - {16 - (5 - 5)}]
= 48 - [18 - {16 - 0}]
= 48 - [18 - 16]
= 48 - 2
= 46
∴ 48 - [18 - {16 - (5 - )}] = 46
Evaluate :
-8 - {-6 (9 - 11) + 18 ÷ -3}
Answer
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) - (3 × 8 ÷ 4 + 1)
Answer
Solving,
(24 ÷ - 12) - (3 × 8 ÷ 4 + 1)
= (24 ÷ 3 - 12) - (24 ÷ 4 + 1)
= (8 - 12) - (6 + 1)
= (-4) - 7
= -11
∴ (24 ÷ - 12) - (3 × 8 ÷ 4 + 1) = -11
Find the difference between 8 and -8.
Answer
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.
Answer
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
Answer
(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 :
Find :
(i) A - B
(ii) D + C
(iii) C + A
(iv) B - C
Answer
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
Answer
(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) ?
Answer
Required = 35 - (-35) = 35 + 35 = 70
∴ 35 exceeds (-35) by 70.
How much is -12 less than 3 ?
Answer
Required = 3 - (-12) = 3 + 12 = 15
∴ -12 is less than 3 by 15.
Subtract (-34) from the sum of 57 and (-51).
Answer
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.
Answer
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.
Answer
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)
Answer
(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.
Answer
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.
Answer
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.