ncert solutions for class 7maths chapter 1
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.
ncert solutions for class 7maths chapter 1
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.
ncert solutions for class 7maths chapter 1
FAQ’s
1. What are integers?
Integers are numbers that include all positive numbers, negative numbers, and zero. Examples: -3, -2, -1, 0, 1, 2, 3.
2. What is the importance of a number line in learning integers?
A number line helps visualize integers. Positive numbers are placed to the right of zero, and negative numbers are placed to the left.
3. What are the main operations performed on integers?
The main operations are addition, subtraction, multiplication, and division, each following specific rules for positive and negative numbers.
4. What are the key properties of integers?
Integers follow properties like Closure, Commutativity, Associativity, and Distributivity for arithmetic operations.
5. How do we represent integers in real-life examples?
Integers are used to represent situations like temperature changes, bank transactions, heights below or above sea level, and elevator movements.
