Exercise: 1-E
Q1: If a = -12 and b = -10, verify that : a + b = b + a.
Given: a = -12, b = -10
Step 1: Find a + b
⇒ -12 + (-10) = -22
Step 2: Find b + a
⇒ -10 + (-12) = -22
Hence, a + b = b + a
Verified
Q2: If a = -7, b = -5 and c = 8, verify that : a + (b + c) = (a + b) + c.
Step 1: Find b + c
⇒ -5 + 8 = 3
Step 2: Find a + (b + c)
⇒ -7 + 3 = -4
Step 3: Find a + b
⇒ -7 + (-5) = -12
Step 4: (a + b) + c = -12 + 8 = -4
Conclusion: a + (b + c) = (a + b) + c
Verified
Q3: If a = 7, b = 5 and c = -8, verity that : a – (b – c) ≠ (a— b) — c.
Step 1: Evaluate b – c
⇒ 5 – (-8) = 5 + 8 = 13
Step 2: a – (b – c)
⇒ 7 – 13 = -6
Step 3: a – b
⇒ 7 – 5 = 2
Step 4: (a – b) – c
⇒ 2 – (-8) = 2 + 8 = 10
Conclusion: -6 ≠ 10
Verified that both sides are not equal
Q4: The difference between two integers x and -12 is -15. Find the value(s) of x.
Option 1:
Given: x – (-12) = -15
Step 1: x + 12 = -15
Step 2: Subtract 12 from both sides
⇒ x = -15 – 12 = -27
Required value of x = -27
Option 2:
Given: (-12) – x = -15
Step 1: (-12) – x = -15
Step 2: Subtract 12 from both sides
⇒ x = -12 + 15 = 3
Required value of x = 3
Q5: An areoplane 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?
Initial Height: 800 m
Final Height: 1125 m
Step 1: Change in height = Final – Initial
⇒ 1125 – 800 = 325 m
Change in height = 325 m
Q6: Write a pair of integers whose: i. sum = -7; ii. difference = -7.
Let one number be x.
Then, sum = -7 ⇒ x + y = -7
And, difference = -7 ⇒ x – y = -7
Solving:Adding both equations:
(x + y) + (x – y) = -7 + (-7)
⇒ 2x = -14 ⇒ x = -7Now substitute in x + y = -7:
-7 + y = -7 ⇒ y = 0
Required pair = (-7, 0)
Q7: Write two integers each of which is smaller than -3 and their difference is greater than -3.
Let the numbers be -8 and -4
Step 1: Check both < -3
⇒ Yes: -8 < -3 and -4 < -3
Step 2: Difference = -4 – (-8) = -4 + 8 = 4
4 > -3
Required integers = -8 and -4
Q8: Evaluate:
i. (-1) × (-1) × (-1) × …… 60 times
Even number of negative signs ⇒ Result is +1
Answer = 1
ii. (-1) × (-1) × (-1) × …… 75 times
Odd number of negative signs ⇒ Result is -1
Answer = -1
Q9: Evaluate:
i. (-2)×(-3)×(-4)×(-5)×(-6)
Even number of negative signs = 5 (odd) ⇒ Result is negative
Multiply values: 2×3×4×5×6 = 720
Answer = -720
ii. (-3)×(-6)×(-9)×(-12)
4 negative numbers ⇒ Even ⇒ Result is positive
3×6×9×12 = 1944
Answer = 1944
iii. (-11)×(-15)+(-11)×(-25)
⇒ Use distributive property:
⇒ (-11)[-15 + (-25)] = (-11)(-40) = 440
Answer = 440
iv. 10×(-12)+5×(-12)
⇒ -120 + (-60) = -180
Answer = -180
Q10:
i. If x × (-1) = -36, is x positive or negative?
x × (-1) = -36
⇒ x must be positive
x = 36 (Positive)
ii. If x × (-1) = 36, is x positive or negative?
x × (-1) = 36
⇒ x must be negative
x = -36 (Negative)
Q11: Evaluate the following expressions:
i. (-20) + (-8) ÷ (-2) × 3
Step-by-step:
Step 1: Division first: (-8) ÷ (-2) = 4
Step 2: Then multiplication: 4 × 3 = 12
Step 3: Finally: (-20) + 12 = -8
ii. (-5) – (-48) ÷ (-16) + (-2) × 6
Step 1: Division: (-48) ÷ (-16) = >3
Step 2: Multiplication: (-2) × 6 = -12
Step 3: Now expression becomes: (-5) – 3 + (-12)
Step 4: Solve step-by-step: (-5) – 3 = -8; then -8 + (-12) = -20
iii. 16 + 8 ÷ 4 – 2 × 3
Step 1: Division: 8 ÷ 4 = 2
Step 2: Multiplication: 2 × 3 = 6>
Step 3: Expression becomes: 16 + 2 – 6
Step 4: Then: 18 – 6 = 12
iv. 16 ÷ 8 × 4 – 2 × 3
Step 1: Start with division: 16 ÷ 8 = 2
Step 2: Now multiply: 2 × 4 = 8
Step 3: Next part of the expression: 2 × 3 = 6
Step 4: Final subtraction: 8 – 6 = 2
v. 27 – [5 + {28 – (29 – 7)}]
Step 1: Start from innermost bracket: (29 – 7) = 22
Step 2: Then: {28 – 22} = 6
Step 3: Now: [5 + 6] = 11
Step 4: Finally: 27 – 11 = 16
vi. \(48-\left[18-\left\{16-\left(5-\overline{4+1}\right)\right\}\right]\)
Step 1: Innermost: (4 + 1) = 5
Step 2: Then: (5 – 5) = 0
Step 3: Now: {16 – 0} = 16
Step 4: Then: [18 – 16] = 2
Step 5: Finally: 48 – 2 = 46
vii. \(-8-\left\{-6\left(9-11\right)+18\div-3\right\}\)
Step 1: (9 – 11) = -2
Step 2: Then: -6 × (-2) = 12
Step 3: 18 ÷ -3 = -6
Step 4: Inside braces: 12 + (-6) = 6
Step 5: Final expression: -8 – 6 = -14
viii. \(\left(24\div\overline{12-9}-12\right)-\left(3\times8\div4+1\right)\)
Step 1: Left part: 12 – 9 = 3 ⇒ 24 ÷ 3 = 8 ⇒ 8 – 12 = -4
Step 2: Right part: 3 × 8 = 24 ⇒ 24 ÷ 4 = 6 ⇒ 6 + 1 = 7
Step 3: Now: -4 – 7 = -11
Q12: Find the difference between 8 and -8.
Option 1:
Difference = 8 – (-8) = 8 + 8 =
16
Option 2:
Difference = (-8) – 8 =
-16
Q13: Subtract the sum of -107 and 72 from the sum of 55 and -32.
Step 1: Find sum of 55 and -32
⇒ 55 + (-32) = 23Step 2: Find sum of -107 and 72
⇒ -107 + 72 = -35Step 3: Subtract second sum from first
⇒ 23 – (-35) = 23 + 35 = 58
Answer = 58
Q14: Write three consecutive integers
i. succeeding -14
-13, -12, -11
Answer = -13, -12, -11
ii. preceding -22
-23, -24, -25
Answer = -25, -24, -23
iii. which are even and succeed 24
26, 28, 30
Answer = 26, 28, 30
iv. which are odd and precede 8
7, 5, 3
Answer = 7, 5, 3
Q15: Points A, and D are marked on a number line as shown below:
-9
-3
3
8
Values:
A = 3, B = 8, C = -9, D = -3
(a) A – B = 3 – 8 = -5
Answer = -5
(b) D + C = (-3) + (-9) = -12
Answer = -12
(c) C + A = -9 + 3 = -6
Answer = -6
(d) B – C = 8 – (-9) = 8 + 9 = 17
Answer = 17
Q16: 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:
Temperature rose by 3°C
⇒ 30 + 3 =
33°C
ii. Wednesday:
Temperature dropped by 8°C from Tuesday
⇒ 33 – 8 =
25°C
Q17: How much does 35 exceed (-35)?
Excess = 35 – (-35) =
⇒ 70
Answer = 70
Q18: How much is -12 less than 3?
3 – (-12) = 3 + 12 =
15
Q19: Subtract (-34) from the sum of 57 and (-51).
Step 1: 57 + (-51) = 6
Step 2: 6 – (-34) = 6 + 34 =
40
Q20: Write a pair of negative integers whose difference is 8.
Let’s try: (-2) and (-10)
⇒ -2 – (-10) = -2 + 10 = 8
Answer = (-2, -10)
Q21: Write a positive integer and a negative integer whose sum is -15.
Let’s choose: Positive = 5, Negative = -20
⇒ 5 + (-20) = -15
Answer = (5, -20)
Q22: Evaluate:
i. (-2) × (-2) × (-5) × 7
Step-by-step:
⇒ (-2 × -2) = 4
⇒ 4 × (-5) = -20
⇒ -20 × 7 =
-140
ii. (-1) × (-5) × (-4) × (-6)
Step-by-step:
⇒ (-1 × -5) = 5
⇒ 5 × (-4) = -20
⇒ -20 × (-6) =
120
Q23: 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 out of which only 12 were correct. Find the score of this student.
Correct = 12, Incorrect = 20 – 12 = 8
⇒ (12 × 5) + (8 × -3)
⇒ 60 + (-24) =
36 marks
Q24: 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.
North distance = 20 × 40 = 800 cm
South distance = 30 × 28 = 840 cm
Final position: 840 – 800 =
40 cm South of starting point
