How do you add integers with the same sign?

Add the absolute values and keep the common sign. Example: (-3) + (-5) = -(3+5) = -8.

What is the additive inverse?

The additive inverse of a number is what you add to it to get zero. For 5, it is -5; for -7, it is 7.

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NCERT Solutions for Class 6 Maths Chapter 6 — Integers

Q: How do you add integers with different signs?

Subtract the smaller absolute value from the larger, and keep the sign of the number with larger absolute value.

Whole numbers (0, 1, 2, 3, ...) are not enough to describe many real situations — temperatures below zero, debts, depths below sea level. Integers extend whole numbers to include negative numbers. In this chapter, you learn to represent integers on a number line, compare them, and perform addition and subtraction. Every solution below follows a step-by-step approach so you understand the logic, not just the answer.

Key Concepts

Concept	What It Means	Example
Integers	The collection of all negative numbers, zero, and positive numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...	Temperature: +5 (above zero) or -3 (below zero)
Positive Numbers	Numbers greater than zero. Written with or without a + sign.	+3, 7, 15
Negative Numbers	Numbers less than zero. Always written with a - sign.	-2, -8, -100
Number Line Representation	A straight line with zero at the centre, positive numbers to the right, and negative numbers to the left.	-3 is 3 steps to the left of 0; +4 is 4 steps to the right
Addition of Integers	Adding integers using the number line or rules: same sign → add and keep the sign; different signs → subtract and keep the sign of the larger absolute value	(-3) + (-5) = -8; (-8) + (+3) = -5
Subtraction of Integers	Subtracting an integer is the same as adding its additive inverse: a - b = a + (-b)	6 - (-4) = 6 + 4 = 10
Additive Inverse	The opposite of a number. When you add a number and its additive inverse, the result is zero.	Additive inverse of 7 is -7; of -3 is 3
Properties of Addition on Integers	Closure property (sum of integers is an integer), Commutative property (a + b = b + a), Associative property ((a + b) + c = a + (b + c)), Identity element (0 + a = a + 0 = a)	(3 + 5) = (5 + 3) = 8

Exercise 6.1 — Introduction to Integers and the Number Line

Question 1: Write opposites of the following: (a) Increase in weight (b) 30 km north (c) 80 m east (d) Loss of Rs 700 (e) 100 m above sea level

Solution

Step 1: The opposite of a quantity means reversing the direction or sign.

Answer

(a) Increase in weight → Decrease in weight
(b) 30 km north → 30 km south
(c) 80 m east → 80 m west
(d) Loss of Rs 700 → Profit of Rs 700
(e) 100 m above sea level → 100 m below sea level

Question 2: Represent the following numbers as integers with appropriate signs: (a) An aeroplane is flying at a height two thousand metres above the ground. (b) A submarine is moving at a depth eight hundred metres below sea level. (c) A deposit of Rs 2,000. (d) A withdrawal of Rs 500.

Solution

Step 1: "Above" and "deposit" are positive. "Below" and "withdrawal" are negative.

Answer

(a) Height two thousand metres above the ground → +2000
(b) Depth eight hundred metres below sea level → -800
(c) Deposit of Rs 2,000 → +2000
(d) Withdrawal of Rs 500 → -500

Question 3: Represent the following numbers on a number line: (a) +5 (b) -10 (c) +8 (d) -1 (e) -6

Solution

Step 1: Draw a horizontal line. Mark 0 at the centre.
Step 2: Mark positive numbers to the right of 0 and negative numbers to the left.
Step 3: Place each integer at the correct position:
(a) +5: Mark 5 steps to the right of 0.
(b) -10: Mark 10 steps to the left of 0.
(c) +8: Mark 8 steps to the right of 0.
(d) -1: Mark 1 step to the left of 0.
(e) -6: Mark 6 steps to the left of 0.

AnswerEach integer is placed on the number line — positive to the right of zero, negative to the left.

Question 4: Arrange the following integers in ascending order: -5, 3, -8, 0, 2, -1

Solution

Step 1: On the number line, the further left a number is, the smaller it is.
Step 2: Compare: -8 < -5 < -1 < 0 < 2 < 3.

AnswerAscending order: -8, -5, -1, 0, 2, 3

Question 5: Arrange the following integers in descending order: -2, 5, -9, 0, 3, -7

Solution

Step 1: On the number line, the further right a number is, the larger it is.
Step 2: Compare: 5 > 3 > 0 > -2 > -7 > -9.

AnswerDescending order: 5, 3, 0, -2, -7, -9

Question 6: Which is greater in each of the following? (a) 2, -7 (b) -3, -8 (c) 0, -2 (d) -5, 3

Solution

Step 1: On the number line, the number further to the right is greater.

Answer

(a) 2 > -7 → 2 is greater (positive is always greater than negative)
(b) -3 > -8 → -3 is greater (-3 is closer to zero, so it is larger)
(c) 0 > -2 → 0 is greater (zero is greater than any negative number)
(d) 3 > -5 → 3 is greater (positive is always greater than negative)

Exercise 6.2 — Addition of Integers

Question 1: Using the number line, find: (a) 3 + 5 (b) (-3) + 5 (c) 3 + (-5) (d) (-3) + (-5)

Solution

(a) 3 + 5:

Step 1: Start at 0 on the number line. Move 3 steps to the right to reach 3.
Step 2: From 3, move 5 more steps to the right.
Step 3: You reach 3 + 5 = 8.

Answer3 + 5 = 8

(b) (-3) + 5:

Step 1: Start at 0. Move 3 steps to the left (for -3) to reach -3.
Step 2: From -3, move 5 steps to the right (for +5).
Step 3: You reach -3 + 5 = 2.

Answer(-3) + 5 = 2

(c) 3 + (-5):

Step 1: Start at 0. Move 3 steps to the right to reach 3.
Step 2: From 3, move 5 steps to the left (for -5).
Step 3: You reach 3 - 5 = -2.

Answer3 + (-5) = -2

(d) (-3) + (-5):

Step 1: Start at 0. Move 3 steps to the left to reach -3.
Step 2: From -3, move 5 more steps to the left (for -5).
Step 3: You reach -3 - 5 = -8.

Answer(-3) + (-5) = -8

Question 2: Add without using the number line: (a) (-7) + (-9) (b) (+4) + (-7) (c) (-3) + (+9) (d) (+10) + (-10)

Solution

Rule: Same sign → add absolute values, keep the sign. Different sign → subtract smaller absolute value from larger, keep sign of the number with larger absolute value.

(a) (-7) + (-9):

Step 1: Both are negative (same sign).
Step 2: Add absolute values: 7 + 9 = 16.
Step 3: Keep the negative sign.

Answer(-7) + (-9) = -16

(b) (+4) + (-7):

Step 1: Different signs. Find absolute values: |4| = 4, |-7| = 7.
Step 2: Subtract: 7 - 4 = 3.
Step 3: The number with larger absolute value is -7, so the answer is negative.

Answer(+4) + (-7) = -3

(c) (-3) + (+9):

Step 1: Different signs. Absolute values: |3| = 3, |9| = 9.
Step 2: Subtract: 9 - 3 = 6.
Step 3: The number with larger absolute value is +9, so the answer is positive.

Answer(-3) + (+9) = 6

(d) (+10) + (-10):

Step 1: Different signs. Absolute values are equal: |10| = |10| = 10.
Step 2: Subtract: 10 - 10 = 0.

Answer(+10) + (-10) = 0

Question 3: Fill in the blanks: (a) (-3) + ___ = 0 (b) ___ + (-7) = 0 (c) 15 + ___ = 0 (d) (-9) + ___ = 0

Solution

Rule: A number + its additive inverse = 0. The additive inverse of a is -a.

(a) (-3) + ___ = 0:

Step 1: The additive inverse of -3 is 3.

Answer(-3) + 3 = 0

(b) ___ + (-7) = 0:

Step 1: The additive inverse of -7 is 7.

Answer7 + (-7) = 0

(c) 15 + ___ = 0:

Step 1: The additive inverse of 15 is -15.

Answer15 + (-15) = 0

(d) (-9) + ___ = 0:

Step 1: The additive inverse of -9 is 9.

Answer(-9) + 9 = 0

Question 4: Find the sum: (a) 137 + (-54) (b) (-82) + 37 (c) (-312) + (+39) + (-192)

Solution

(a) 137 + (-54):

Step 1: Different signs. |137| = 137, |-54| = 54.
Step 2: Subtract: 137 - 54 = 83.
Step 3: The larger absolute value is 137 (positive), so the result is positive.

Answer137 + (-54) = 83

(b) (-82) + 37:

Step 1: Different signs. |-82| = 82, |37| = 37.
Step 2: Subtract: 82 - 37 = 45.
Step 3: The larger absolute value is 82 (negative), so the result is negative.

Answer(-82) + 37 = -45

(c) (-312) + (+39) + (-192):

Step 1: Group the negative numbers: (-312) + (-192) = -(312 + 192) = -504.
Step 2: Add the result with the positive number: (-504) + (+39).
Step 3: Different signs. |504| = 504, |39| = 39. Subtract: 504 - 39 = 465.
Step 4: The larger absolute value is negative, so the result is negative.

Answer(-312) + (+39) + (-192) = -465

Exercise 6.3 — Subtraction of Integers

Question 1: Find: (a) 9 - (-6) (b) (-3) - 7 (c) (-5) - (-8) (d) 12 - 15 (e) (-10) - (-3)

Solution

Rule: To subtract an integer, add its additive inverse. a - b = a + (-b). In other words, change the sign of the number being subtracted and then add.

(a) 9 - (-6):

Step 1: Change the sign of -6 to get +6.
Step 2: 9 + 6 = 15.

Answer9 - (-6) = 15

(b) (-3) - 7:

Step 1: Change the sign of 7 to get -7.
Step 2: (-3) + (-7) = -(3 + 7) = -10.

Answer(-3) - 7 = -10

(c) (-5) - (-8):

Step 1: Change the sign of -8 to get +8.
Step 2: (-5) + 8 = 3.

Answer(-5) - (-8) = 3

(d) 12 - 15:

Step 1: Change the sign of 15 to get -15.
Step 2: 12 + (-15). Different signs. |12| = 12, |15| = 15. Subtract: 15 - 12 = 3. The larger absolute value is negative.

Answer12 - 15 = -3

(e) (-10) - (-3):

Step 1: Change the sign of -3 to get +3.
Step 2: (-10) + 3. Different signs. |10| = 10, |3| = 3. Subtract: 10 - 3 = 7. The larger absolute value is negative.

Answer(-10) - (-3) = -7

Question 2: Fill in the blanks: (a) (-6) - ___ = -13 (b) ___ - 7 = 3 (c) ___ - (-5) = -3

Solution

(a) (-6) - ___ = -13:

Step 1: Let the blank be x. So (-6) - x = -13.
Step 2: (-6) - x = -13 means -6 + (-x) = -13.
Step 3: -x = -13 + 6 = -7.
Step 4: x = 7.

Answer(-6) - 7 = -13

(b) ___ - 7 = 3:

Step 1: Let the blank be y. So y - 7 = 3.
Step 2: y = 3 + 7 = 10.

Answer10 - 7 = 3

(c) ___ - (-5) = -3:

Step 1: Let the blank be z. So z - (-5) = -3.
Step 2: z + 5 = -3 (since subtracting a negative is adding a positive).
Step 3: z = -3 - 5 = -8.

Answer(-8) - (-5) = -3

Question 3: The temperature at 12 noon was 10 degrees C above zero. If it decreases at the rate of 2 degrees C per hour until midnight, at what time would the temperature be 8 degrees C below zero? What would be the temperature at midnight?

Solution

Step 1: Starting temperature = +10 degrees C.
Step 2: Rate of decrease = 2 degrees C per hour.
Step 3: Temperature to reach = -8 degrees C.
Step 4: Total drop needed = 10 - (-8) = 10 + 8 = 18 degrees C.
Step 5: Time required = 18 / 2 = 9 hours.
Step 6: 12 noon + 9 hours = 9 PM.
Step 7: Temperature at midnight (12 hours after noon): Total drop = 12 x 2 = 24 degrees C. Temperature = 10 - 24 = -14 degrees C.

AnswerThe temperature reaches 8 degrees below zero at 9 PM. The temperature at midnight would be -14 degrees C.

Question 4: In a class test, (+3) marks are given for every correct answer and (-2) marks for every incorrect answer. No marks are deducted for unattempted questions. (a) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly? (b) Mohini scored -5 marks. If she has got 3 correct answers, how many questions has she attempted incorrectly?

Solution

(a) Radhika:

Step 1: Marks for 12 correct answers = 12 x (+3) = +36.
Step 2: Total marks scored = 20.
Step 3: Marks lost due to incorrect answers = 36 - 20 = 16.
Step 4: Each incorrect answer loses 2 marks. Number of incorrect answers = 16 / 2 = 8.

AnswerRadhika attempted 8 questions incorrectly.

(b) Mohini:

Step 1: Marks for 3 correct answers = 3 x (+3) = +9.
Step 2: Total marks scored = -5.
Step 3: Marks lost = 9 - (-5) = 9 + 5 = 14.
Step 4: Each incorrect answer loses 2 marks. Number of incorrect answers = 14 / 2 = 7.

AnswerMohini attempted 7 questions incorrectly.

Question 5: A shopkeeper earns a profit of Rs 1 by selling one pen and incurs a loss of 40 paise per pencil. (a) In a particular month, she incurs a loss of Rs 5. In this period, she sold 45 pens. How many pencils were sold? (b) In the next month, she earns neither profit nor loss. If she sold 70 pens, how many pencils were sold?

Solution

(a) Loss of Rs 5, 45 pens sold:

Step 1: Profit from 45 pens = 45 x Rs 1 = Rs 45.
Step 2: Total loss = Rs 5, so: Profit from pens - Loss from pencils = -5.
Step 3: 45 - Loss from pencils = -5.
Step 4: Loss from pencils = 45 + 5 = Rs 50.
Step 5: Loss per pencil = 40 paise = Rs 0.40.
Step 6: Number of pencils = 50 / 0.40 = 125.

Answer125 pencils were sold.

(b) No profit no loss, 70 pens sold:

Step 1: Profit from 70 pens = 70 x Rs 1 = Rs 70.
Step 2: Total = 0, so: Profit from pens - Loss from pencils = 0.
Step 3: 70 - Loss from pencils = 0.
Step 4: Loss from pencils = Rs 70.
Step 5: Number of pencils = 70 / 0.40 = 175.

Answer175 pencils were sold.

Common Mistakes

1. Treating subtraction the same as addition. Subtracting a negative number is NOT making it more negative. Remember: a - (-b) = a + b. For example, 5 - (-3) = 5 + 3 = 8, NOT 2.

2. Confusing the sign rules for addition. When adding numbers with different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value. (-8) + 3 = -5, not -11 or +5.

3. Writing -5 > -3. On the number line, -3 is to the RIGHT of -5, so -3 is greater. Negative numbers closer to zero are larger. The correct statement is -3 > -5.

4. Forgetting that zero is neither positive nor negative. Zero is an integer, but it is not positive or negative. It separates positive and negative integers on the number line.

5. Adding absolute values incorrectly. When both numbers are negative, add the absolute values and keep the negative sign. (-7) + (-4) = -(7 + 4) = -11, NOT -3 or +3.

6. Not converting units properly in word problems. In profit/loss problems, always check if the units match. Rs 1 and 40 paise must be in the same unit (Rs 1 and Rs 0.40) before calculating.

Exam Checklist

[ ] I can represent integers on a number line — positive to the right, negative to the left of zero.
[ ] I can write opposites of real-world quantities (profit/loss, above/below, north/south).
[ ] I can arrange integers in ascending and descending order correctly.
[ ] I can compare integers and identify which is greater (remember: -3 > -7).
[ ] I can add integers using the number line method.
[ ] I can add integers without the number line using sign rules (same sign, different sign).
[ ] I know that adding a number and its additive inverse always gives zero.
[ ] I can subtract integers by converting to addition: a - b = a + (-b).
[ ] I can solve word problems involving profit/loss and temperature changes using integers.
[ ] I can identify the additive inverse of any integer and use it to verify subtraction answers.

Frequently Asked Questions

Integers are the collection of all negative numbers, zero, and positive numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...

NCERT Solutions for Class 6 Maths Chapter 6 Integers | KalviGavani

NCERT Solutions for Class 6 Maths Chapter 6 — Integers

Key Concepts

Exercise 6.1 — Introduction to Integers and the Number Line

Exercise 6.2 — Addition of Integers

Exercise 6.3 — Subtraction of Integers

Common Mistakes

Exam Checklist

Frequently Asked Questions