Integers

Integers are a set of numbers that include positive numbers, negative numbers, and zero. They are represented by the symbol ‘Z’.

Key Concepts:

• Number Line: A visual representation of integers, with positive numbers to the right of zero and negative numbers to the left.
• Operations on Integers: You learn how to add, subtract, multiply, and divide integers, including rules for different combinations of positive and negative numbers.
• Properties of Integers: These include properties like closure, commutativity, associativity,and distributivity that hold true for integer operations.
• Representation of Integers: You explore different ways to represent integers, such as using a thermometer scale or a number line.

Understanding Integers is crucial as it forms the foundation for more complex mathematical concepts in the future.

Exercise 1.1 

1. Write down a pair of integers whose:

(a) sum is -7

(b) difference is -10

(c) sum is 0.

Ans : 

(a) sum is -7: -4 and -3

• -4 + (-3) = -7

(b) difference is -10: -2 and 8

• -2 – 8 = -10

(c) sum is 0: 5 and -5

• 5 + (-5) = 0

2. 

(a) Write a pair of negative integers whose difference gives 8.

(b) Write a negative integer and positive integer whose sum is -5.

(c) Write a negative integer and a positive integer whose difference is -3.

Ans : 

(a) -15 and -7

• -7 – (-15) = -7 + 15 = 8

(b) -9 and 4

• -9 + 4 = -5

(c) 2 and 5

• 2 – 5 = -3

3. In a quiz, team A scored -40, 10, 0 and team B scored 10, 0, -40 in three successive rounds. Which team scored more? Can you say that we can add integers in any order?

Ans : 

Total score for Team A:

• -40 + 10 + 0 = -30

Total score for Team B:

• 10 + 0 + (-40) = -30

Both teams scored the same, -30 points.

Yes, we can add integers in any order. This property is known as the commutative property of addition.

4. Fill in the blanks to make the following statements true:

(i) (-5) + (-8) = (-8) + (…)

(ii) -53 + … = -53

(iii) 17 + … = 0

(iv) [13 + (-12)] + (…) = 13 + [(-12) + (-7)]

(v) (-4) + [15 + (-3)] = [-4 + 15] + …

Ans : 

(i) (-5) + (-8) = (-8) + (-5) 

(ii) -53 + 0 = -53 

(iii) 17 + (-17) = 0 

(iv) [13 + (-12)] + (-7) = 13 + [(-12) + (-7)] 

(v) (-4) + [15 + (-3)] = [-4 + 15] + (-3)

Exercise 1.2

1. Find each of the following products:

(a) 3 × (-1)

(b) (-1) × 225

(c) (-21) × (-30)

(d) (-316) × (-1)

(e) (-15) × 0 × (-18)

(f) (-12) × (-11) × (10)

(g) 9 × (-3) × (-6)

(h) (-18) × (-5) × (-4)

(i) (-1) ×(-2) × (-3) × 4

(j) (-3) × (-6) × (-2) × (-1)

Ans : 

(a) 3 × (-1) = -3

(b) (-1) × 225 = -225

(c) (-21) × (-30) = 630

(d) (-316) × (-1) = 316

(e) (-15) × 0 × (-18) = 0 (Any number multiplied by 0 is 0)

(f) (-12) × (-11) × (10) = 1320

(g) 9 × (-3) × (-6) = 162

(h) (-18) × (-5) × (-4) = -360

(i) (-1) × (-2) × (-3) × 4 = -24

(j) (-3) × (-6) × (-2) × (-1) = 36

2. Verify the following:

(a) 18 × [7 + (-3)] = [18 × 7] + [18 × (-3)]

(b) (-21) × [(-4) + (-6)] = [(-21) × (-4)] + [(-21) × (-6)]

Ans : 

(a) 18 × [7 + (-3)] = [18 × 7] + [18 × (-3)]

Left Hand Side (LHS):

• 18 × [7 + (-3)] = 18 × 4 = 72

Right Hand Side (RHS):

• [18 × 7] + [18 × (-3)] = 126 + (-54) = 72

Since LHS = RHS, the equation is verified.

(b) (-21) × [(-4) + (-6)] = [(-21) × (-4)] + [(-21) × (-6)]

LHS:

• (-21) × [(-4) + (-6)] = (-21) × (-10) = 210

RHS:

• [(-21) × (-4)] + [(-21) × (-6)] = 84 + 126 = 210

Since LHS = RHS, the equation is verified.

3. (i) For any integer a, what is (-1) × a equal to?

(ii) Determine the integer whose product with (-1) is 0.

(a) -22

(b) 37

(c) 0

Ans : 

(i) For any integer a, what is (-1) × a equal to?

• (-1) × a = -a

(ii) Determine the integer whose product with (-1) is 0.

• (c) 0
  • (-1) × 0 = 0

4. Starting from (-1) × 5, write various products showing some pattern to show (-1) × (-1) = 1

Ans : 

(-1) × 5 = -5

(-1) × 4 = -4

(-1) × 3 = -3

(-1) × 2 = -2

(-1) × 1 = -1

Exercise 1.3

1. Evaluate each of the following:

(a) (-30) ÷ 10

(b) 50 ÷ (-5)

(c) (-36) ÷ (-9)

(d) (-49) ÷ (49)

(e) 13 ÷ [(-2) + 1]

(f) 0 ÷ (-12)

(g) (-31) ÷ [(-30) + (-1)]

(h) [(-36) ÷ 12] ÷ 3

(i) [(-6) + 5] ÷ [(-2) + 1]

Ans : 

(a) (-30) ÷ 10 = -3

(b) 50 ÷ (-5) = -10

(c) (-36) ÷ (-9) = 4

(d) (-49) ÷ (49) = -1

(e) 13 ÷ [(-2) + 1] = 13 ÷ (-1) = -13

(f) 0 ÷ (-12) = 0 (Any number divided by 0 is 0)

(g) (-31) ÷ [(-30) + (-1)] = (-31) ÷ (-31) = 1

(h) [(-36) ÷ 12] ÷ 3 = (-3) ÷ 3 = -1

(i) [(-6) + 5] ÷ [(-2) + 1] = (-1) ÷ (-1) = 1

2. Verify that: a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c.

(а) a = 12, b = – 4, c = 2

(b) a = (-10), b = 1, c = 1

Ans : 

Verifying the inequality: a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)

(a) 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

Since -6 ≠ 3, the inequality holds true for these values.

(b) 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) = -20

Since -5 ≠ -20, the inequality holds true for these values as well.

Therefore, for both sets of values, we have verified that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c).

3. Fill in the blanks:

(a) 369 ÷ ___ = 369

(b) (-75) ÷ ___ = -1

(c) (-206) ÷ ___ =1

(d) -87 ÷ ___ = -87

(e) ___ ÷ 1 = -87

(g) 20 ÷ ___ = -2

(h) ___ + (4) = -3

Ans : 

(a) 369 ÷ 1 = 369 

(b) (-75) ÷ 75 = -1 

(c) (-206) ÷ (-206) = 1 

(d) -87 ÷ 1 = -87 

(e) -87 ÷ 1 = -87 

(g) 20 ÷ (-10) = -2

(h) (-12) ÷ (4) = -3

4. Write five pairs of integers (a, b) such that a ÷ b = -3. One such pair is (6, -2) because 6 + (-2) = -3.

Ans : 

(6, -2) (Given)

(9, -3)

(-12, 4)

(15, -5)

(-18, 6)

(21, -7)

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 midnight?

Ans : 

Finding the time when the temperature is -8°C:

• The temperature starts at 10°C above zero, which is +10°C.
• To reach -8°C, the temperature needs to decrease by 18°C (from 10°C to 0°C, and then from 0°C to -8°C).
• If the temperature decreases by 2°C per hour, it will take 18°C / 2°C/hour = 9 hours to reach -8°C.
• So, the temperature will be -8°C at 12 noon + 9 hours = 9 PM.

Finding the temperature at midnight:

• We know it takes 9 hours to reach -8°C.
• From 9 PM to midnight, there are 3 more hours.
• In 3 hours, the temperature will decrease by 3 hours * 2°C/hour = 6°C.
• So, the temperature at midnight will be -8°C – 6°C = -14°C.

Therefore, the temperature will be -8°C at 9 PM and -14°C at midnight.

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?

Ans : 

(i) Radhika’s score:

• Marks for correct answers = 12 correct answers * (+3 marks/answer) = 36 marks
• Total marks = 20 marks
• So, marks for incorrect answers = 20 marks – 36 marks = -16 marks
• Since each incorrect answer is worth -2 marks, the number of incorrect answers = -16 marks / (-2 marks/answer) = 8 incorrect answers

Therefore, Radhika attempted 8 questions incorrectly.

(ii) Mohini’s score:

• Marks for correct answers = 7 correct answers * (+3 marks/answer) = 21 marks
• Total marks = -5 marks
• So, marks for incorrect answers = -5 marks – 21 marks = -26 marks
• Since each incorrect answer is worth -2 marks, the number of incorrect answers = -26 marks / (-2 marks/answer) = 13 incorrect answers

Therefore, Mohini attempted 13 questions incorrectly.

7. An elevator descends into a nine 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.

Ans : 

• The elevator starts at 10 meters above ground level.
• It needs to reach -350 meters, which is a total distance of 10 + 350 = 360 meters.
• The elevator descends at a rate of 6 meters per minute.

Calculating the time:

To find the time it takes, we divide the total distance by the rate of descent:

• Time = Total distance / Rate of descent
• Time = 360 meters / 6 meters/minute = 60 minutes

So, it will take the elevator 60 minutes to reach -350 meters.