Operations with Integers — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Operations with Integers" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Operations with Integers" — 8 important questions with detailed answers for CBSE bo…
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NCERT Solutions for Class 7 Mathematics — Operations with Integers. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Operations with Integers — All 8 Questions Solved
Q1. What are integers? Explain with 5 examples and show them on a number line.
Definition of Integers:
Integers are whole numbers including zero, positive numbers, and negative numbers. They do NOT include fractions or decimals.
Set of integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
5 Examples of Integers:
1. -5 (negative integer)
2. 0 (zero)
3. 7 (positive integer)
4. -12 (negative integer)
5. 100 (positive integer)
Number Line Representation:
← Negative ─────── Zero ────── Positive →
... -5 -4 -3 -2 -1 0 1 2 3 4 5 ...
Key Points:
• Integers on the right of 0 are positive (greater than 0)
• Integers on the left of 0 are negative (less than 0)
• 0 is neither positive nor negative
• As you move right, integers increase in value
• As you move left, integers decrease in value
Q2. Add: (-15) + (-8) and (-20) + 13. Explain your steps.
Problem 1: (-15) + (-8)
Step 1: Identify the signs
Both numbers are negative.
When adding two negative integers, add their absolute values and keep the negative sign.
Step 2: Add absolute values
|-15| = 15
|-8| = 8
15 + 8 = 23
Step 3: Apply the negative sign
(-15) + (-8) = -23
Answer: -23
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Problem 2: (-20) + 13
Step 1: Identify the signs
One number is negative, one is positive.
When adding integers with different signs, subtract the smaller absolute value from the larger and use the sign of the number with the larger absolute value.
Step 2: Compare absolute values
|-20| = 20
|13| = 13
20 > 13, so the answer will be negative.
Step 3: Subtract
20 - 13 = 7
Step 4: Apply the sign of the larger absolute value
(-20) + 13 = -7
Answer: -7
Q3. Subtract: 12 - (-8) and (-5) - 6. Show the steps clearly.
Problem 1: 12 - (-8)
Step 1: Convert subtraction to addition
Subtracting a negative is the same as adding its positive.
12 - (-8) = 12 + 8
Step 2: Add
12 + 8 = 20
Answer: 20
Rule: a - (-b) = a + b
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Problem 2: (-5) - 6
Step 1: Convert subtraction to addition
Subtracting a positive is adding its negative.
(-5) - 6 = (-5) + (-6)
Step 2: Add two negative numbers
Add their absolute values: 5 + 6 = 11
Keep the negative sign: -11
Answer: -11
Rule: a - b = a + (-b)
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General Rule for Subtraction of Integers:
Subtracting an integer is the same as adding its additive inverse.
If we subtract (-x), it becomes +x.
If we subtract (+x), it becomes -x.
Q4. Multiply: (-6) × 5, (-4) × (-9), and 0 × (-15). Explain the sign rules.
Problem 1: (-6) × 5
Step 1: Multiply absolute values
6 × 5 = 30
Step 2: Apply sign rule
Negative × Positive = Negative
(-6) × 5 = -30
Answer: -30
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Problem 2: (-4) × (-9)
Step 1: Multiply absolute values
4 × 9 = 36
Step 2: Apply sign rule
Negative × Negative = Positive
(-4) × (-9) = +36 or 36
Answer: 36
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Problem 3: 0 × (-15)
Step 1: Apply zero property
Any number multiplied by 0 equals 0, regardless of sign.
0 × (-15) = 0
Answer: 0
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Sign Rules for Multiplication of Integers:
(+) × (+) = (+) ... Positive × Positive = Positive
(+) × (-) = (-) ... Positive × Negative = Negative
(-) × (+) = (-) ... Negative × Positive = Negative
(-) × (-) = (+) ... Negative × Negative = Positive
Zero Property:
0 × a = 0 for any integer a
Q5. Divide: (-36) ÷ 6, (-48) ÷ (-8), and 0 ÷ (-5). State the sign rules for division.
Problem 1: (-36) ÷ 6
Step 1: Divide absolute values
36 ÷ 6 = 6
Step 2: Apply sign rule
Negative ÷ Positive = Negative
(-36) ÷ 6 = -6
Answer: -6
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Problem 2: (-48) ÷ (-8)
Step 1: Divide absolute values
48 ÷ 8 = 6
Step 2: Apply sign rule
Negative ÷ Negative = Positive
(-48) ÷ (-8) = +6 or 6
Answer: 6
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Problem 3: 0 ÷ (-5)
Step 1: Apply zero property
Zero divided by any non-zero integer equals 0.
0 ÷ (-5) = 0
Answer: 0
Note: We CANNOT divide by 0. Division by zero is undefined.
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Sign Rules for Division of Integers:
(+) ÷ (+) = (+) ... Positive ÷ Positive = Positive
(+) ÷ (-) = (-) ... Positive ÷ Negative = Negative
(-) ÷ (+) = (-) ... Negative ÷ Positive = Negative
(-) ÷ (-) = (+) ... Negative ÷ Negative = Positive
Zero Property:
0 ÷ a = 0 for any non-zero integer a
Division by zero is undefined.
Q6. Name and verify the commutative property for addition and multiplication using integers -3, 5.
Commutative Property: When we add or multiply two integers, the order does not matter.
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Commutative Property of Addition:
For any integers a and b: a + b = b + a
Verification with -3 and 5:
Left side: (-3) + 5 = 2
Right side: 5 + (-3) = 2
Both sides equal 2 ✓
Therefore, (-3) + 5 = 5 + (-3)
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Commutative Property of Multiplication:
For any integers a and b: a × b = b × a
Verification with -3 and 5:
Left side: (-3) × 5 = -15
Right side: 5 × (-3) = -15
Both sides equal -15 ✓
Therefore, (-3) × 5 = 5 × (-3)
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Commutative Property of Subtraction:
Subtraction is NOT commutative because a - b ≠ b - a in general.
Example:
(-3) - 5 = -8
5 - (-3) = 8
-8 ≠ 8, so NOT commutative ✗
---
Commutative Property of Division:
Division is NOT commutative because a ÷ b ≠ b ÷ a in general.
Example:
(-3) ÷ 5 = -0.6
5 ÷ (-3) = -5/3 ≈ -1.67
They are NOT equal, so NOT commutative ✗
Q7. Verify the associative property for multiplication using integers 2, (-3), and 4.
Associative Property: When we add or multiply three integers, the way we group them does not change the result.
Associative Property of Multiplication:
For any integers a, b, and c: (a × b) × c = a × (b × c)
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Verification with 2, (-3), and 4:
Left side: (2 × (-3)) × 4
Step 1: 2 × (-3) = -6
Step 2: (-6) × 4 = -24
Left side = -24
Right side: 2 × ((-3) × 4)
Step 1: (-3) × 4 = -12
Step 2: 2 × (-12) = -24
Right side = -24
Both sides equal -24 ✓
Therefore, (2 × (-3)) × 4 = 2 × ((-3) × 4)
---
Conclusion:
The associative property of multiplication holds for integers.
We can group the factors in any way, and the product remains the same.
---
Note on Subtraction:
Subtraction is NOT associative because (a - b) - c ≠ a - (b - c) in general.
Example:
(10 - 5) - 2 = 5 - 2 = 3
10 - (5 - 2) = 10 - 3 = 7
3 ≠ 7, so NOT associative ✗
Q8. What is the additive identity for integers? Prove it using (-7) as an example.
Additive Identity:
The additive identity is the number that, when added to any integer, gives the same integer back.
For integers, the additive identity is 0.
Property: For any integer a:
a + 0 = a and 0 + a = a
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Proof using (-7):
Step 1: Add 0 to (-7)
(-7) + 0 = -7
Step 2: Check if the result equals the original number
Result = Original number? Yes, -7 = -7 ✓
Step 3: Verify the reverse (commutative property)
0 + (-7) = -7 ✓
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Verification:
Let's try another example with 5:
5 + 0 = 5 ✓
0 + 5 = 5 ✓
And with (-12):
(-12) + 0 = -12 ✓
0 + (-12) = -12 ✓
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Conclusion:
Zero (0) is the additive identity for integers because adding 0 to any integer leaves that integer unchanged. This property holds for ALL integers, both positive and negative.
Importance: Understanding additive identity helps in solving equations. If we can add 0 (or its equivalent form) to both sides of an equation, we don't change the equation's truth value.
Operations with Integers — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is the sum of -5 and 8?
- A. -13
- B. 3
- C. -3
- D. 13
Answer: B. 3 — When adding integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value. |8| - |-5| = 3, and 8 is positive.
MCQ 2. Subtract -7 from 10.
- A. 3
- B. -3
- C. 17
- D. -17
Answer: C. 17 — Subtracting a negative integer is the same as adding its positive counterpart. So, 10 - (-7) = 10 + 7 = 17.
MCQ 3. What is the product of -6 and 4?
- A. -24
- B. 24
- C. 10
- D. -10
Answer: A. -24 — When multiplying integers with different signs, the product is always negative. So, -6 × 4 = -24.
MCQ 4. Divide -42 by -7.
- A. -6
- B. 6
- C. 7
- D. -7
Answer: B. 6 — When dividing two integers with the same sign, the quotient is always positive. So, -42 ÷ -7 = 6.
MCQ 5. Which of the following is the additive identity for integers?
- A. 1
- B. -1
- C. 0
- D. any positive integer
Answer: C. 0 — The additive identity is a number that, when added to any integer, leaves the integer unchanged. For integers, this number is 0 (a + 0 = a).
MCQ 6. Which of the following is the multiplicative identity for integers?
- A. 0
- B. 1
- C. -1
- D. 10
Answer: B. 1 — The multiplicative identity is a number that, when multiplied by any integer, leaves the integer unchanged. For integers, this number is 1 (a × 1 = a).
MCQ 7. Simplify: 5 - (-3) + (-8)
- A. 0
- B. -6
- C. 10
- D. 6
Answer: A. 0 — First, change subtraction of a negative to addition: 5 + 3 + (-8). Then, perform additions: 8 + (-8) = 0.
MCQ 8. Which property is shown by the equation: -15 + 7 = 7 + (-15)?
- A. Associative Property
- B. Commutative Property
- C. Distributive Property
- D. Additive Identity
Answer: B. Commutative Property — The Commutative Property of Addition states that changing the order of addends does not change the sum (a + b = b + a).
MCQ 9. Which property is shown by the equation: (3 × -4) × 5 = 3 × (-4 × 5)?
- A. Commutative Property
- B. Distributive Property
- C. Associative Property
- D. Multiplicative Identity
Answer: C. Associative Property — The Associative Property of Multiplication states that the grouping of factors does not change the product ((a × b) × c = a × (b × c)).
MCQ 10. Calculate: 12 ÷ (-3) + 5 × (-2)
- A. -14
- B. -6
- C. 6
- D. 14
Answer: A. -14 — First, perform division and multiplication according to BODMAS: 12 ÷ (-3) = -4, and 5 × (-2) = -10. Then add the results: -4 + (-10) = -14.
MCQ 11. A submarine is at 300 meters below sea level. It ascends 120 meters. What is its new position?
- A. -180 m
- B. -420 m
- C. 180 m
- D. 420 m
Answer: A. -180 m — Below sea level is represented by a negative integer. So, the initial position is -300 m. Ascending means adding, so -300 + 120 = -180 m.
MCQ 12. Using the distributive property, simplify: -8 × (10 - 2)
- A. -64
- B. -96
- C. 64
- D. 96
Answer: A. -64 — Apply the distributive property: -8 × 10 - (-8) × 2 = -80 - (-16) = -80 + 16 = -64. Alternatively, -8 × 8 = -64.
MCQ 13. What is the additive inverse of -13?
- A. 13
- B. 0
- C. -1/13
- D. 1/13
Answer: A. 13 — The additive inverse of an integer is the number that, when added to the integer, results in 0. The additive inverse of -13 is 13, because -13 + 13 = 0.
MCQ 14. Which of the following statements is true?
- A. -5 > 2
- B. -10 < -12
- C. 0 > -3
- D. 4 < -1
Answer: C. 0 > -3 — On a number line, numbers to the right are greater. 0 is to the right of -3, so 0 > -3.
MCQ 15. A shopkeeper gains ₹5 on each pen and loses ₹2 on each pencil. If he sells 10 pens and 15 pencils, what is his total profit or loss?
- A. Profit of ₹20
- B. Loss of ₹20
- C. Profit of ₹70
- D. Loss of ₹70
Answer: A. Profit of ₹20 — Profit from pens = 10 × 5 = ₹50. Loss from pencils = 15 × (-2) = -₹30. Total = 50 + (-30) = ₹20. Since the result is positive, it's a profit.
MCQ 16. What is the value of |-9|?
- A. -9
- B. 9
- C. 0
- D. 1/9
Answer: B. 9 — The absolute value of a number is its distance from zero on the number line, always a non-negative value. So, |-9| = 9.
MCQ 17. If a = -3, b = 4, and c = -2, what is a × b + c?
- A. -14
- B. -10
- C. 10
- D. 14
Answer: A. -14 — Substitute the values: (-3) × 4 + (-2) = -12 + (-2) = -14.
MCQ 18. What integer must be multiplied by -9 to get 72?
- A. 8
- B. -8
- C. -9
- D. 9
Answer: B. -8 — Let the integer be x. Then -9 × x = 72. To find x, divide 72 by -9. x = 72 ÷ (-9) = -8.
MCQ 19. The temperature in Delhi was -5°C on Monday. It dropped by 3°C on Tuesday and then rose by 7°C on Wednesday. What was the temperature on Wednesday?
- A. -1°C
- B. 1°C
- C. -15°C
- D. 5°C
Answer: A. -1°C — Monday: -5°C. Tuesday (dropped by 3°C): -5 - 3 = -8°C. Wednesday (rose by 7°C): -8 + 7 = -1°C.
MCQ 20. Which integer lies exactly between -3 and 1 on the number line?
- A. -2
- B. 0
- C. -1
- D. 2
Answer: C. -1 — On the number line, the integers between -3 and 1 are -2, -1, 0. The middle integer is -1.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.