Class 6 Maths - Integers NCERT Solutions

Chapter 6: Integers (NCERT Solutions)

Exercise 6.1 (Integers & Number Line)

Q1. 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

(a) Decrease in weight

(b) 30 km south

(d) Gain of Rs. 700

(e) 100 m below sea level

Q2. Represent the following numbers as integers with appropriate signs:
(a) An aeroplane is flying at a height of 1500 m above the ground.
(b) A submarine is at a depth of 1200 m below sea level.
(c) A deposit of Rs. 2500 in a bank.
(d) A withdrawal of Rs. 700 from a bank.

(a) +1500 (above ground is positive)

(b) −1200 (below sea level is negative)

(d) −700 (withdrawal is negative)

Q3. Represent the following on a number line:
(a) −5 (b) 4 (c) −8 (d) −1 (e) −6

On a number line, positive integers are to the right of zero and negative integers are to the left of zero.

Number line: ← −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 →

(a) −5 is 5 units to the left of 0.

(b) 4 is 4 units to the right of 0.

(d) −1 is 1 unit to the left of 0.

(e) −6 is 6 units to the left of 0.

Q4. Adjacent to each of the following integers, write the integer that comes just after it:
(a) −4 (b) −3 (c) +5 (d) 0

(a) −4 → −3

(b) −3 → −2

(d) 0 → +1

Q5. Write all integers between −6 and 6.

−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5

(Total of 11 integers between −6 and 6, not including −6 and 6 themselves.)

Q6. Using the number line write the integer which is:
(a) 3 more than 5 (b) 5 more than −5 (c) 6 less than 2 (d) 3 less than −2

(a) 5 + 3 = 8

(b) −5 + 5 = 0

(d) −2 − 3 = −5

Q7. Write all the negative integers greater than −6.

Negative integers greater than −6 are: −5, −4, −3, −2, −1.

(On a number line, "greater" means to the right, so −5 > −6, etc.)

Q8. Arrange the following integers in ascending order:
−5, −3, 0, −1, 4, −2, 2, −6

Ascending order (smallest to largest):
−6, −5, −3, −2, −1, 0, 2, 4

Exercise 6.2 (Addition & Subtraction of Integers)

Q1. Using the number line, add the following integers:
(a) 9 + (−6) (b) 5 + (−11) (c) (−1) + (−7) (d) (−5) + 10 (e) (−1) + (−2) + (−3) (f) (−2) + 8 + (−4)

(a) 9 + (−6) = 9 − 6 = 3

(b) 5 + (−11) = 5 − 11 = −6

(d) (−5) + 10 = 10 − 5 = 5

(e) (−1) + (−2) + (−3) = −(1+2+3) = −6

(f) (−2) + 8 + (−4) = 8 − 2 − 4 = 2

Q2. Find the sum of: (a) 137 + (−354) (b) (−52) + 52 (c) (−312) + 39 + 192 (d) (−50) + (−200) + 300

(a) 137 − 354 = −217

(b) (−52) + 52 = 0 (additive inverse)

(d) −50 + (−200) + 300 = −250 + 300 = 50

Q3. Find the sum: (a) (−7) + (−9) + 4 + 16 (b) 37 + (−2) + (−65) + (−8)

(a) Group positives and negatives:
(4 + 16) + (−7 − 9) = 20 + (−16) = 4

(b) 37 + (−2) + (−65) + (−8):
Positives: 37. Negatives: −2 − 65 − 8 = −75.
37 + (−75) = −38

Exercise 6.3 (Subtraction of Integers)

Q1. Subtract: (a) 35 − (−15) (b) (−15) − (−18) (c) (−20) − 13 (d) 23 − (−12) (e) (−32) − (−40)

Rule: a − (−b) = a + b

(a) 35 − (−15) = 35 + 15 = 50

(b) (−15) − (−18) = −15 + 18 = 3

(d) 23 − (−12) = 23 + 12 = 35

(e) (−32) − (−40) = −32 + 40 = 8

Q2. Fill in the blanks with >, < or = :
(a) (−3) + (−6) ___ (−3) − (−6)
(b) (−21) − (−10) ___ (−31) + (−11)
(c) 45 − (−11) ___ 57 + (−4)
(d) (−25) − (−42) ___ (−42) − (−25)

(a) LHS = −9. RHS = −3 + 6 = 3. So −9 < 3.

(b) LHS = −21 + 10 = −11. RHS = −31 − 11 = −42. So −11 > −42.

(d) LHS = −25 + 42 = 17. RHS = −42 + 25 = −17. So 17 > −17.

Q5. A plane is flying at the height of 5000 m above sea level. At a particular point it is exactly above a submarine floating 1200 m below sea level. What is the vertical distance between them?

Height of plane = +5000 m. Depth of submarine = −1200 m.

Distance = 5000 − (−1200) = 5000 + 1200 = 6200 m.

Q6. Mohan deposited Rs. 2000 in his bank account and later withdrew Rs. 1698. What is his balance?

Balance = Deposit − Withdrawal = 2000 − 1698 = Rs. 302.

Class 6 Maths - Integers Practice Questions

Chapter 6: Integers (Practice Questions)

RD Sharma / HOT Practice

Q1. Write all integers between −9 and −3 (exclusive). How many are there?

Integers between −9 and −3: −8, −7, −6, −5, −4. Total = 5 integers.

Q2. Arrange in descending order: −7, 3, −12, 0, 5, −1, 9

Descending (largest to smallest): 9, 5, 3, 0, −1, −7, −12.

Q3. Calculate: (−145) + 78 + (−35) + 100

Group positives: 78 + 100 = 178.
Group negatives: −145 + (−35) = −180.
Total = 178 + (−180) = −2.

Q4. The temperature in Srinagar this week:
Mon: −5°C, Tue: −3°C, Wed: −8°C, Thu: 2°C, Fri: 0°C, Sat: −4°C, Sun: −1°C.
On which day was it coldest? Which day was the warmest?

Arranging: −8 < −5 < −4 < −3 < −1 < 0 < 2.
Coldest: Wednesday (−8°C).
Warmest: Thursday (+2°C).

Q5. Find the value of: 15 − (−8) − (−3) + (−6)

= 15 + 8 + 3 + (−6)
= 15 + 8 + 3 − 6
= 26 − 6 = 20.

Q6. What must be subtracted from −9 to get 4?

Let the number be x. Then: −9 − x = 4.
x = −9 − 4 = −13.
So, −13 must be subtracted from −9 to get 4.
Verify: −9 − (−13) = −9 + 13 = 4. ✓

Q7. What must be added to −7 to obtain 11?

Let the number be x. Then: −7 + x = 11.
x = 11 − (−7) = 11 + 7 = 18.
Verify: −7 + 18 = 11. ✓

Q8. A diver descends 12 m below sea level. He then rises 5 m. What is his final position?

Start: −12 m (below sea level).
Rise: +5 m.
Final position: −12 + 5 = −7 m (7 m below sea level).

Q9. A shopkeeper makes a profit of Rs. 250 on Monday, a loss of Rs. 120 on Tuesday, a profit of Rs. 350 on Wednesday, and a loss of Rs. 80 on Thursday. What is his total profit or loss?

Total = +250 + (−120) + 350 + (−80)
Positives: 250 + 350 = 600.
Negatives: −120 + (−80) = −200.
Net = 600 − 200 = Rs. 400 profit.

Q10. Verify: (−8) + [5 + (−3)] = [(−8) + 5] + (−3)

LHS = (−8) + [5 + (−3)] = (−8) + 2 = −6.
RHS = [(−8) + 5] + (−3) = (−3) + (−3) = −6.
LHS = RHS = −6. ✓ (This proves the associative property of addition.)

Q11. Evaluate: (−999) + 999

Every integer has an additive inverse. −999 and 999 are additive inverses of each other.
(−999) + 999 = 0.

Q12. Write the additive inverse of: (a) −57 (b) 83 (c) 0 (d) −100

(a) Additive inverse of −57 = 57

(b) Additive inverse of 83 = −83

(d) Additive inverse of −100 = 100

Q13. The sum of two integers is −40. If one integer is 16, find the other.

Other integer = (−40) − 16 = −56.
Verify: 16 + (−56) = −40. ✓

Q14. Is subtraction of integers commutative? Show with an example.

No. Subtraction is NOT commutative.
Example: 5 − 3 = 2, but 3 − 5 = −2.
Since 2 ≠ −2, subtraction of integers is not commutative.

Q15. Find: (−5) + (−4) + (−3) + ... + (−1) + 0 + 1 + 2 + ... + 5

Each positive integer cancels its negative counterpart:
(−5+5) + (−4+4) + (−3+3) + (−2+2) + (−1+1) + 0 = 0+0+0+0+0+0 = 0.

Q16. On a quiz show, a player scores +4 for each correct answer and −2 for each wrong answer. If a player answers 8 questions correctly and 3 incorrectly, what is the total score?

Correct: 8 × 4 = +32.
Wrong: 3 × (−2) = −6.
Total score = 32 + (−6) = 26.

Q17. Which is greater: (−4) + 3 or (−4) − 3?

(−4) + 3 = −1.
(−4) − 3 = −7.
Since −1 > −7, (−4) + 3 is greater.

Q18. Evaluate: |−15| − |−7| + |3|

Absolute value: |−15| = 15, |−7| = 7, |3| = 3.
= 15 − 7 + 3 = 11.

Q19. Fill in the blanks: The sum of an integer and its _____ is always zero.

The sum of an integer and its additive inverse (opposite) is always zero.
Example: 15 + (−15) = 0.

Q20. The elevation of a place A is 5200 m above sea level. The elevation of place B is 1300 m below sea level. What is the difference in their elevations?

Elevation of A = +5200 m. Elevation of B = −1300 m.
Difference = 5200 − (−1300) = 5200 + 1300 = 6500 m.

Class 6 Maths - Integers Summary

Chapter 6: Integers (Concepts & Summary)

1. What are Integers?

Integers = Whole numbers + Negative numbers = { ..., −3, −2, −1, 0, 1, 2, 3, ... }
Positive integers: 1, 2, 3, ... (right of 0 on number line)
Negative integers: −1, −2, −3, ... (left of 0 on number line)
0 is neither positive nor negative.
Opposite (Additive Inverse): For any integer n, its opposite is −n. Their sum = 0.
Absolute Value: The distance of an integer from 0. |−5| = 5, |5| = 5.

2. Comparing & Ordering Integers

On a number line, numbers increase from left to right.
Every positive integer is greater than every negative integer.
Among negative integers, the one with the smaller absolute value is greater. e.g., −2 > −5.
0 is greater than every negative integer and less than every positive integer.
Ascending order: Smallest to largest. e.g., −5, −3, 0, 2, 4.
Descending order: Largest to smallest. e.g., 4, 2, 0, −3, −5.

3. Addition of Integers

Same signs: Add absolute values, keep the sign.
e.g., (−4) + (−5) = −9; (+3) + (+7) = +10.
Opposite signs: Subtract the smaller absolute value from the larger, keep sign of the larger.
e.g., (−8) + 5 = −3; 10 + (−4) = +6.

Properties of Addition:

Commutative: a + b = b + a
Associative: (a + b) + c = a + (b + c)
Identity: a + 0 = a
Additive Inverse: a + (−a) = 0

4. Subtraction of Integers

Key Rule: To subtract an integer, add its additive inverse.
a − b = a + (−b)
e.g., 8 − (−3) = 8 + 3 = 11
e.g., (−5) − 4 = −5 + (−4) = −9
e.g., (−6) − (−2) = −6 + 2 = −4
Subtraction is NOT commutative: a − b ≠ b − a (in general)
Subtraction is NOT associative: (a − b) − c ≠ a − (b − c) (in general)

5. Real-life Applications of Integers

Situation	Positive (+)	Negative (−)
Altitude	Above sea level	Below sea level
Temperature	Above 0°C	Below 0°C
Money	Profit / Deposit	Loss / Withdrawal
Direction	North / East / Up	South / West / Down

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