Chapter 1 Integers Exercise 1.4
Question 1: Evaluate each of the following:
(a) (–30) ÷ 10
Answer:
= (-30) ÷ 10
= -3
(b) 50 ÷ (–5)
Answer:
= (50) ÷ (-5)
= -10
(c) (–36) ÷ (–9)
Answer:
= (-36) ÷ (-9)
= 4
(d) (– 49) ÷ (49)
Answer:
= (-49) ÷ 49
= -1
(e) 13 ÷ [(–2) + 1]
Answer:
= 13 ÷ [(-2) + 1]
= 13 ÷ (-1)
= -13
(f) 0 ÷ (–12)
Answer:
= 0 ÷ (-12)
= 0
(g) (–31) ÷ [(–30) + (–1)]
Answer:
= (-31) ÷ [(-30) + (-1)]
= (-31) ÷ [-30 - 1]
= (-31) ÷ (-31)
= 1
(h) [(–36) ÷ 12] ÷ 3
Answer:
= [(-36) ÷ 12]
= (-36) ÷ 12
= -3
= (-3) ÷ 3
= -1
(i) [(– 6) + 5)] ÷ [(–2) + 1]
Answer:
= [-1] ÷ [-1]
= 1
Question 2: Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c.
(a) a = 12, b = – 4, c = 2
Answer:
= a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
a = 12
b = -4
c = 2
Left Hand Side = a ÷ (b + c)
= 12 ÷ (-4 + 2)
= 12 ÷ (-2)
= -6
Right Hand Side = (a ÷ b) + (a ÷ c)
= (12 ÷ (-4)) + (12 ÷ 2)
= (-3) + (6)
= 3
Comparing LHS and RHS
= -6 ≠ 3
= LHS ≠ RHS
Therefore, the given values are verified.
(b) a = (–10), b = 1, c = 1
Answer:
= a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
a = (-10)
b = 1
c = 1
Left Hand Side = a ÷ (b + c)
= (-10) ÷ (1 + 1)
= (-10) ÷ (2)
= -5
Right Hand Side = (a ÷ b) + (a ÷ c)
= ((-10) ÷ (1)) + ((-10) ÷ 1)
= (-10) + (-10)
= -10 - 10
= -20
Comparing LHS and RHS
= -5 ≠ -20
= LHS ≠ RHS
Therefore, the given values are verified.
Question 3: Fill in the blanks:
(a) 369 ÷ _____ = 369
Answer:
Let the missing integer be x.
= 369 ÷ x = 369
= x = (369/369)
= x = 1
(b) (–75) ÷ _____ = –1
Answer:
Let the missing integer be x.
= (-75) ÷ x = -1
= x = (-75/-1)
= x = 75
(c) (–206) ÷ _____ = 1
Answer:
Let the missing integer be x.
= (-206) ÷ x = 1
= x = (-206/1)
= x = -206
(d) – 87 ÷ _____ = 87
Answer:
Let the missing integer be x.
= (-87) ÷ x = 87
= x = (-87)/87
= x = -1
(e) _____ ÷ 1 = – 87
Answer:
Let the missing integer be x.
= (x) ÷ 1 = -87
= x = (-87) × 1
= x = -87
(f) _____ ÷ 48 = –1
Answer:
Let the missing integer be x.
= (x) ÷ 48 = -1
= x = (-1) × 48
= x = -48
(g) 20 ÷ _____ = –2
Answer:
Let the missing integer be x.
= 20 ÷ x = -2
= x = (20)/ (-2)
= x = -10
(h) _____ ÷ (4) = –3
Answer:
Let the missing integer be x.
= (x) ÷ 4 = -3
= x = (-3) × 4
= x = -12
Question 4: Write five pairs of integers (a, b) such that a ÷ b = –3. One such pair is (6, –2) because 6 ÷ (–2) = (–3).
Answer:
1. (15, -5) as, 15 ÷ (-5) = (-3)
2. (-15, 5) as, (-15) ÷ (5) = (-3)
3. (18, -6) as, 18 ÷ (-6) = (-3)
4. (-18, 6) as, (-18) ÷ 6 = (-3)
5. (21, -7) as, 21 ÷ (-7) = (-3)
Question 5: The temperature at 12 noon was 10°C above zero. If it decreases at the rate of 2°C per hour until midnight, at what time would the temperature be 8°C below zero? What would be the temperature at mid-night?
Answer:
Temperature at the beginning = 10°C
Rate of change of temperature = -2°C per hour
Temperature at 1 PM = 10 + (-2) = 8°C
Temperature at 2 PM = 8 + (-2) = 6°C
Temperature at 3 PM = 6 + (-2) = 4°C
Temperature at 4 PM = 4 + (-2) = 2°C
Temperature at 5 PM = 2 + (-2) = 0°C
Temperature at 6 PM = 0 + (-2) = -2°C
Temperature at 7 PM = -2 + (-2) = -4°C
Temperature at 8 PM = -4 + (-2) = -6°C
Temperature at 9 PM = -6 + (-2) = -8°C
Therefore, at 9 PM the temperature will be -8°C.
The temperature at mid-night = -2oC × 12 = -24°C
At midnight temperature will be = 10 + (-24) = -14°C
Therefore, at midnight temperature will be -14°C.
Question 6: In a class test (+ 3) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any question.
(i) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly?
(ii) Mohini scores –5 marks in this test, though she has got 7 correct answers. How many questions has she attempted incorrectly?
Answer:
Marks awarded for 1 correct answer = + 3
Marks awarded for 1 wrong answer = -2
(i) Radhika’s score = 20 marks
Total marks awarded for 12 correct answers = 12 × 3 = 36
Marks awarded for incorrect answers = Total score - Total marks awarded for 12 correct answers
= 20 - 36
= -16
Therefore, the number of incorrect answers made by Radhika = (-16) ÷ (-2) = 8
(ii) Mohini’s score = -5 marks
Total marks awarded for 7 correct answers = 7 × 3 = 21
Marks awarded for incorrect answers = Total score - Total marks awarded for 12 correct answers
= -5 - 21
= -26
Therefore, the number of incorrect answers made by Mohini = (-26) ÷ (-2) = 13
Question 7: An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach – 350 m.
Answer:
The initial height of the elevator = 10 m
Final depth of elevator = -350 m (as distance descended is denoted by a negative integer]
The total distance to descended by the elevator = (-350) - (10)
= -360 m
Time taken by the elevator to descend -6 m = 1 min
So, time taken by the elevator to descend -360 m = (-360) ÷ (-60)
= 60 minutes
= 1 hour
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