Class 7 Maths - Integers NCERT Solutions

Chapter 1: Integers (NCERT Solutions)

Exercise 1.1

Q1. Following number line shows the temperature in degree celsius (°C) at different places on a particular day.
(a) Observe this number line and write the temperature of the places marked on it.
(b) What is the temperature difference between the hottest and the coldest places among the above?
(c) What is the temperature difference between Lahulspiti and Srinagar?
(d) Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar?

(a) By observing the number line:
Lahulspiti = -8°C
Srinagar = -2°C
Shimla = 5°C
Ooty = 14°C
Bengaluru = 22°C

(b) Hottest place (Bengaluru) = 22°C
Coldest place (Lahulspiti) = -8°C
Difference = 22 - (-8) = 22 + 8 = 30°C

(c) Temperature of Srinagar = -2°C
Temperature of Lahulspiti = -8°C
Difference = (-2) - (-8) = -2 + 8 = 6°C

(d) Temperature of Srinagar and Shimla taken together = -2 + 5 = 3°C.
Temperature at Shimla = 5°C. Since 3°C < 5°C, Yes, it is less than the temperature at Shimla.
Temperature at Srinagar = -2°C. Since 3°C is not less than -2°C, No, it is not less than the temperature at Srinagar.

Q2. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack's scores in five successive rounds were 25, -5, -10, 15 and 10, what was his total at the end?

Jack's scores = 25, -5, -10, 15, 10
Total score = 25 + (-5) + (-10) + 15 + 10
= 25 - 5 - 10 + 15 + 10
= (25 + 15 + 10) - (5 + 10)
= 50 - 15 = 35

His total score at the end was 35.

Q3. At Srinagar temperature was -5°C on Monday and then it dropped by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day?

Temperature on Monday = -5°C.
Temperature dropped by 2°C on Tuesday.
Temperature on Tuesday = -5 - 2 = -7°C.

Temperature rose by 4°C on Wednesday.
Temperature on Wednesday = -7 + 4 = -3°C.

Q4. A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them?

Height of plane above sea level = +5000 m.
Depth of submarine below sea level = -1200 m.
Vertical distance = Distance between the two = 5000 - (-1200)
= 5000 + 1200 = 6200 m.

Q5. Mohan deposits Rs 2,000 in his bank account and withdraws Rs 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohan's account after the withdrawal.

Since withdrawal is represented by a negative integer, the deposited amount is represented by a positive integer (+2000).

Deposited amount = + Rs 2000
Withdrawal amount = - Rs 1642
Balance = 2000 + (-1642)
= 2000 - 1642 = Rs 358.

Q6. Rita goes 20 km towards east from a point A to the point B. From B, she moves 30 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance travelled towards west? By which integer will you represent her final position from A?

If distance towards east is positive, then distance towards west will be represented by a negative integer.

Distance travelled east = +20 km.
Distance travelled west = -30 km.
Final position from A = 20 + (-30)
= 20 - 30 = -10 km.

Her final position from A is represented by the integer -10 (indicating 10 km towards the west).

Q7. In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square.
(Square 1 has rows: [5, -1, -4], [-5, -2, 7], [0, 3, -3])
(Square 2 has rows: [1, -10, 0], [-4, -3, -2], [-6, 4, -7])

Checking Square 1:
Rows: 5 - 1 - 4 = 0; -5 - 2 + 7 = 0; 0 + 3 - 3 = 0.
Columns: 5 - 5 + 0 = 0; -1 - 2 + 3 = 0; -4 + 7 - 3 = 0.
Diagonal 1: 5 - 2 - 3 = 0.
Diagonal 2: -4 - 2 + 0 = -6 (Not equal to 0).
Hence, Square 1 is not a magic square.

Checking Square 2:
Rows: 1 - 10 + 0 = -9; -4 - 3 - 2 = -9; -6 + 4 - 7 = -9.
Columns: 1 - 4 - 6 = -9; -10 - 3 + 4 = -9; 0 - 2 - 7 = -9.
Diagonal 1: 1 - 3 - 7 = -9.
Diagonal 2: 0 - 3 - 6 = -9.
Since all sums equal -9, Square 2 is a magic square.

Q8. Verify a - (-b) = a + b for the following values of a and b.
(i) a = 21, b = 18
(ii) a = 118, b = 125
(iii) a = 75, b = 84
(iv) a = 28, b = 11

(i) a = 21, b = 18
LHS = 21 - (-18) = 21 + 18 = 39.
RHS = 21 + 18 = 39.
LHS = RHS. Verified.

(ii) a = 118, b = 125
LHS = 118 - (-125) = 118 + 125 = 243.
RHS = 118 + 125 = 243.
LHS = RHS. Verified.

(iii) a = 75, b = 84
LHS = 75 - (-84) = 75 + 84 = 159.
RHS = 75 + 84 = 159.
LHS = RHS. Verified.

(iv) a = 28, b = 11
LHS = 28 - (-11) = 28 + 11 = 39.
RHS = 28 + 11 = 39.
LHS = RHS. Verified.

Q9. Use the sign of >, < or = in the box to make the statements true.
(a) (-8) + (-4) [ ] (-8) - (-4)
(b) (-3) + 7 - (19) [ ] 15 - 8 + (-9)
(c) 23 - 41 + 11 [ ] 23 - 41 - 11
(d) 39 + (-24) - (15) [ ] 36 + (-52) - (-36)
(e) -231 + 79 + 51 [ ] -399 + 159 + 81

(a) LHS: -8 - 4 = -12. RHS: -8 + 4 = -4. Since -12 < -4, use <.

(b) LHS: -3 + 7 - 19 = 4 - 19 = -15. RHS: 15 - 8 - 9 = 7 - 9 = -2. Since -15 < -2, use <.

(c) LHS: 23 - 41 + 11 = -18 + 11 = -7. RHS: 23 - 41 - 11 = -18 - 11 = -29. Since -7 > -29, use >.

(d) LHS: 39 - 24 - 15 = 15 - 15 = 0. RHS: 36 - 52 + 36 = -16 + 36 = 20. Since 0 < 20, use <.

(e) LHS: -231 + 130 = -101. RHS: -399 + 240 = -159. Since -101 > -159, use >.

Exercise 1.2

Q1. Write down a pair of integers whose:
(a) sum is -7
(b) difference is -10
(c) sum is 0

(a) Sum is -7: Example: (-4) and (-3). Since (-4) + (-3) = -7.
(b) Difference is -10: Example: (-20) and (-10). Since (-20) - (-10) = -20 + 10 = -10.
(c) Sum is 0: Example: 5 and (-5). Since 5 + (-5) = 0.

Q2.
(a) Write a pair of negative integers whose difference gives 8.
(b) Write a negative integer and a positive integer whose sum is -5.
(c) Write a negative integer and a positive integer whose difference is -3.

(a) Example: (-2) and (-10). Difference = (-2) - (-10) = -2 + 10 = 8.

(b) Example: (-8) and 3. Sum = (-8) + 3 = -5.

(c) Example: (-2) and 1. Difference = (-2) - 1 = -3.

Q3. 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 we say that we can add integers in any order?

Total score of Team A = (-40) + 10 + 0 = -30

Total score of Team B = 10 + 0 + (-40) = 10 - 40 = -30

Comparing the scores: -30 = -30. So, both teams scored the same.

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

Q4. 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] + ......

(i) (-5) + (-8) = (-8) + (-5) [Commutative property]

(ii) -53 + 0 = -53 [Additive identity]

(iii) 17 + (-17) = 0 [Additive inverse]

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

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

Exercise 1.3

Q1. 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]

(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

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

(h) [(-36) ÷ 12] ÷ 3 = (-3) ÷ 3 = -1

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

Q2. Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c.
(a) a = 12, b = -4, c = 2
(b) a = -10, b = 1, c = 1

(a) a = 12, b = -4, c = 2
LHS = a ÷ (b + c) = 12 ÷ [(-4) + 2] = 12 ÷ (-2) = -6
RHS = (a ÷ b) + (a ÷ c) = [12 ÷ (-4)] + [12 ÷ 2] = -3 + 6 = 3
Since LHS ≠ RHS (-6 ≠ 3), verified.

(b) a = -10, b = 1, c = 1
LHS = a ÷ (b + c) = (-10) ÷ (1 + 1) = (-10) ÷ 2 = -5
RHS = (a ÷ b) + (a ÷ c) = [(-10) ÷ 1] + [(-10) ÷ 1] = -10 + (-10) = -20
Since LHS ≠ RHS (-5 ≠ -20), verified.

Q3. Fill in the blanks:
(a) 369 ÷ ______ = 369
(b) (-75) ÷ ______ = -1
(c) (-206) ÷ ______ = 1
(d) -87 ÷ ______ = 87
(e) ______ ÷ 1 = -87
(f) ______ ÷ 48 = -1
(g) 20 ÷ ______ = -2
(h) ______ ÷ (4) = -3

(a) 369 ÷ 1 = 369

(b) (-75) ÷ 75 = -1

(c) (-206) ÷ (-206) = 1

(d) -87 ÷ (-1) = 87

(e) -87 ÷ 1 = -87

(f) -48 ÷ 48 = -1

(g) 20 ÷ (-10) = -2

(h) -12 ÷ (4) = -3

Q4. Write five pairs of integers (a, b) such that a ÷ b = -3. One such pair is (6, -2) because 6 ÷ (-2) = (-3).

Five pairs are:
1. (9, -3) because 9 ÷ (-3) = -3
2. (-9, 3) because -9 ÷ 3 = -3
3. (12, -4) because 12 ÷ (-4) = -3
4. (-12, 4) because -12 ÷ 4 = -3
5. (15, -5) because 15 ÷ (-5) = -3

Q5. 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 mid-night?

Initial temperature at 12 noon = +10°C
Target temperature = -8°C (8°C below zero)
Total decrease required = 10 - (-8) = 18°C
Rate of decrease = 2°C per hour.
Time taken = Total decrease / Rate of decrease = 18 / 2 = 9 hours.
So, the time will be 12 noon + 9 hours = 9 PM.

Temperature at mid-night (12 midnight) i.e., after 12 hours:
Total decrease in 12 hours = 12 × 2 = 24°C.
Temperature at mid-night = 10 - 24 = -14°C.

Q6. 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?

(i) Total marks scored by Radhika = 20
Marks obtained for 12 correct answers = 12 × 3 = 36
Let the number of incorrect answers be x.
Marks for incorrect answers = x × (-2) = -2x
Total marks = 36 - 2x = 20
2x = 36 - 20 = 16
x = 8.
Radhika attempted 8 questions incorrectly.

(ii) Total marks scored by Mohini = -5
Marks obtained for 7 correct answers = 7 × 3 = 21
Let the number of incorrect answers be y.
Marks for incorrect answers = y × (-2) = -2y
Total marks = 21 - 2y = -5
2y = 21 - (-5) = 26
y = 13.
Mohini attempted 13 questions incorrectly.

Q7. An elevator descends into a mine 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?

Initial position = +10 m.
Final position = -350 m.
Total distance to be covered = 10 - (-350) = 10 + 350 = 360 m.
Rate of descent = 6 m/min.
Time taken = Distance / Rate = 360 / 6 = 60 minutes (or 1 hour).
It will take 1 hour to reach -350 m.

Class 7 Maths - Integers Practice Questions

Chapter 1: Integers (Practice Questions)

RD Sharma / Extra Practice

Q1. The sum of two integers is -15. If one of them is 9, find the other integer.

Let the required integer be x.
x + 9 = -15
x = -15 - 9 = -24

Q2. Subtract -25 from the sum of 15 and -38.

Sum of 15 and -38 = 15 + (-38) = -23.
Now subtract -25 from -23:
-23 - (-25) = -23 + 25 = 2

Q3. Evaluate: (-16) × 12 + (-16) × 8

Using distributive property: a × b + a × c = a × (b + c)
(-16) × [12 + 8] = (-16) × 20 = -320

Q4. Find the product using suitable properties: 8 × 53 × (-125)

Rearranging using commutative property:
= 53 × [8 × (-125)]
= 53 × (-1000) = -53000

Q5. Simplify: (-1) × (-1) × (-1) ... 50 times.

When (-1) is multiplied an even number of times, the product is 1.
Since 50 is an even number, the product is 1.

Q6. Simplify: (-1) × (-1) × (-1) ... 171 times.

When (-1) is multiplied an odd number of times, the product is -1.
Since 171 is an odd number, the product is -1.

Q7. Find the value of: [32 + 2 × 17 + (-6)] ÷ 15

= [32 + 34 - 6] ÷ 15
= [66 - 6] ÷ 15
= 60 ÷ 15 = 4

Q8. What will be the sign of the product if we multiply 90 negative integers and 9 positive integers?

The product of an even number (90) of negative integers is positive.
The product of positive integers is always positive.
Positive × Positive = Positive.

Q9. By what number should (-240) be divided to obtain 16?

Let the number be x.
(-240) ÷ x = 16
x = (-240) / 16 = -15

Q10. Simplify: (-15) × (-14) + (-15) × (-6)

Using distributive property:
= (-15) × [(-14) + (-6)]
= (-15) × (-20) = 300

Q11. The product of two integers is -160. If one of them is 20, find the other.

Let the other integer be y.
y × 20 = -160
y = -160 / 20 = -8

Q12. Write an integer which is 4 less than its additive inverse.

Let the integer be x. Its additive inverse is -x.
According to the condition: x = -x - 4
x + x = -4
2x = -4 ⇒ x = -2

Q13. Compare: (-3) × (-4) × (-5) and (-3) + (-4) + (-5)

Product = (-3) × (-4) × (-5) = 12 × (-5) = -60.
Sum = (-3) + (-4) + (-5) = -12.
Since -12 > -60, the sum is greater than the product.

Q14. Find the value of: [(-10) + (+16)] ÷ [(-2) + (+1) + (+3)]

Numerator: (-10) + 16 = 6.
Denominator: -2 + 1 + 3 = 2.
Result = 6 ÷ 2 = 3.

Q15. At a certain freezing process requires that room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

Initial temperature = 40°C.
Change in temperature per hour = -5°C.
Change in 10 hours = 10 × (-5) = -50°C.
Final temperature = 40 + (-50) = -10°C.

Q16. Simplify: 25 - [20 - {10 - (7 - 5 + 3)}]

= 25 - [20 - {10 - (2 + 3)}]
= 25 - [20 - {10 - 5}]
= 25 - [20 - 5]
= 25 - 15 = 10

Q17. A monkey is climbing up a pole 10 meters high. It climbs 3 meters in one minute but slips down 2 meters in the next minute. In how many minutes will it reach the top?

Effective climb in 2 mins = 3 - 2 = 1 meter.
In 14 mins it climbs 7 meters.
In the 15th minute, it climbs 3 meters. 7 + 3 = 10 meters (reaches the top).
So, it takes 15 minutes to reach the top.

Q18. Evaluate: (-100) ÷ (-10) ÷ (-5)

Solve left to right:
= [(-100) ÷ (-10)] ÷ (-5)
= 10 ÷ (-5) = -2

Q19. Find the multiplicative inverse of -7.

The multiplicative inverse of any non-zero integer a is 1/a.
So, the multiplicative inverse of -7 is -1/7.

Q20. If a = -8, b = -7, c = 6, verify that (a + b) + c = a + (b + c).

LHS = ((-8) + (-7)) + 6 = (-15) + 6 = -9.
RHS = (-8) + ((-7) + 6) = (-8) + (-1) = -9.
Since LHS = RHS, it is verified (Associative Property).

Class 7 Maths - Integers Summary

Chapter 1: Integers (Concepts & Summary)

1. Basic Concepts

Integers are a bigger collection of numbers which is formed by whole numbers and their negatives. {..., -3, -2, -1, 0, 1, 2, 3, ...}
Zero is an integer which is neither positive nor negative.
On a number line, numbers to the right of zero are positive, and numbers to the left are negative. The value increases as we move right and decreases as we move left.
For any integer a, we have a + (-a) = 0 and (-a) + a = 0. So, a and (-a) are additive inverses of each other.

2. Properties of Addition and Subtraction

Closure Property: Integers are closed under addition and subtraction.
If a and b are integers, then a + b is an integer and a - b is an integer.
Commutative Property: Addition is commutative for integers.
For any two integers a and b, a + b = b + a. (Note: Subtraction is NOT commutative).
Associative Property: Addition is associative for integers.
For integers a, b and c, (a + b) + c = a + (b + c).
Additive Identity: Zero is the additive identity for integers.
For any integer a, a + 0 = a = 0 + a.

3. Multiplication of Integers

Product of two positive integers is a positive integer. ( (+) × (+) = (+) )
Product of a positive integer and a negative integer is a negative integer. ( (+) × (-) = (-) )
Product of two negative integers is a positive integer. ( (-) × (-) = (+) )
Product of an even number of negative integers is positive.
Product of an odd number of negative integers is negative.

4. Properties of Multiplication

Closure Property: Integers are closed under multiplication.
For any two integers a and b, a × b is an integer.
Commutative Property: Multiplication is commutative for integers.
For any two integers a and b, a × b = b × a.
Multiplication by Zero: Product of an integer and zero is zero.
a × 0 = 0 × a = 0.
Multiplicative Identity: 1 is the multiplicative identity for integers.
a × 1 = 1 × a = a.
Associative Property: Multiplication is associative.
(a × b) × c = a × (b × c).
Distributive Property: Multiplication distributes over addition and subtraction.
a × (b + c) = a × b + a × c
a × (b - c) = a × b - a × c

5. Division of Integers

When a positive integer is divided by a negative integer, the quotient obtained is a negative integer.
When a negative integer is divided by a positive integer, the quotient obtained is a negative integer.
When a negative integer is divided by a negative integer, the quotient obtained is a positive integer.
Division by zero is meaningless (not defined): a ÷ 0 is not defined.
Zero divided by a non-zero integer is zero: 0 ÷ a = 0 (for a ≠ 0).
For any integer a, a ÷ 1 = a.
Integers are not closed under division. Example: (-5) ÷ (-2) is not an integer.

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