NCERT Solutions: Integers (Exercise 1.1, 1.2, 1.3)

Exercise 1.1

Q1: Write down a pair of integers whose

(a) Sum is -7.

Ans:
= -4 + (-3)
= -4 + -3 (both integers are negative, so their sum is negative)
= -7

(b) Difference is -10.

Ans:
= -25 - (-15)
= -25 + 15 (subtracting a negative is the same as adding its positive)
= -10

(c) Sum is 0.

Ans:
= 4 + (-4)
= 4 - 4 (a number and its additive inverse add to zero)
= 0

Note: You can also think other combinations, it completely depends on you.
Q2: (a) Write a pair of negative integers whose difference gives 8.

Ans:
= (-5) - (-13)
= -5 + 13 (subtracting a negative equals adding its positive)
= 8

(b) Write a negative integer and a positive integer whose sum is -5.

Ans:
= -25 + 20
= -5

(c) Write a negative integer and a positive integer whose difference is -3. 

Ans: 
= -2 - 1
= -2 - 1 (subtracting a positive makes the result more negative)
= -3

Q3: In a quiz, team A scored -40, 10, 0 and team B scores 10, 0, -40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order? 

Ans: 

• Team A scored -40, 10, 0.
  ⇒ Total score of Team A = -40 + 10 + 0 = -30
• Team B scored 10, 0, -40.
  ⇒ Total score of Team B = 10 + 0 + (-40) = -30
• Both teams have the same total score (-30).
  Yes, we can add integers in any order because addition of integers is commutative: A + B = B + A.

Q4: Fill in the blanks to make the following statements true: 
(i) (-5) + (-8) = (-8) + (....)

Ans: 

Assume the missing integer is x.
Then,
⇒ (-5) + (-8) = (-8) + x
⇒ -13 = -8 + x
Add 8 to both sides: x = -13 + 8 = -5
So,
(-5) + (-8) = (-8) + (-5) ... [This illustrates the commutative law of addition]

(ii) -53 +... = -53

Ans: 

Assume the missing integer is x.
Then,
⇒ -53 + x = -53
Add 53 to both sides: x = 0
So,
⇒ -53 + 0 = -53 ... [adding zero leaves the number unchanged]

(iii) 17 +... = 0

Ans: 

Assume the missing integer is x.
Then,
⇒ 17 + x = 0
Subtract 17 from both sides: x = -17
So,
⇒ 17 + (-17) = 0 ... [a number plus its additive inverse is zero]

(iv) [13 + (-12)] + (....) = 13 + [(-12) + (-7)]

Ans: 

Assume the missing integer is x.
Then,
⇒ [13 + (-12)] + x = 13 + [(-12) + (-7)]
Compute inside brackets: [13 - 12] + x = 13 + [-19]
⇒ 1 + x = 13 - 19 = -6
Therefore x = -6 - 1 = -7
So,
⇒ [13 + (-12)] + (-7) = 13 + [(-12) + (-7)] ... [associative law of addition]

(v)  (-4) + [15 + (-3)] = [-4 + 15] + .....

Ans: 

Assume the missing integer is x.
Then,
⇒ (-4) + [15 + (-3)] = [-4 + 15] + x
Evaluate left side: (-4) + 12 = 8
Evaluate [-4 + 15] = 11, so 11 + x = 8
Thus x = 8 - 11 = -3
So,
⇒ (-4) + [15 + (-3)] = [-4 + 15] + (-3) ... [associative law of addition]

Exercise 1.2

Q1: Find the each of the following products
(a) 3 x (-1)

Ans: 

By the rule for multiplying integers:
= 3 × (-1)
= -3 ... [since + × - = -]

(b) (-1) x 225

Ans: 

= (-1) × 225
= -225 ... [since - × + = -]

(c) (-21) x (-30) 

Ans: 

= (-21) × (-30)
= 630 ... [since - × - = +]

(d) (-316) x (-1)

Ans: 

= (-316) × (-1)
= 316 ... [since - × - = +]

(e) (-15) x 0 x (-18) 

Ans: 

= (-15) × 0 × (-18) = 0
Any product that includes 0 is 0.

(f) (-12) x (-11) x (10)

Ans: 

= (-12) × (-11) × 10
First multiply the two numbers with the same sign:
= 132 × 10 ... [since - × - = +]
= 1320

(g) 9 x (-3) x (-6)

Ans: 

= 9 × (-3) × (-6)
Multiply (-3) × (-6) first (they have the same sign):
= 9 × 18 ... [since - × - = +]
= 162

(h) (-18) x (-5) x (-4) 

Ans: 

= (-18) × (-5) × (-4)
First (-18) × (-5) = 90 (positive)
Then 90 × (-4) = -360 ... [positive × negative = negative]

(i) (-1) x (-2) x (-3) x 4

Ans: 

= (-1) × (-2) × (-3) × 4
= 2 × (-3) × 4 ... [since -1 × -2 = 2]
= (-6) × 4 = -24

(j) (-3) x (-6) x (2) x (-1)

Ans: 

= (-3) × (-6) × 2 × (-1)
= 18 × 2 × (-1) ... [since -3 × -6 = 18]
= 36 × (-1) = -36

Q2: Verify the following 
(a) 18 x [7 + (-3)] = [18 x 7] + [18 x (-3)]

Ans: 

Left-hand side:
18 × [7 + (-3)] = 18 × 4 = 72
Right-hand side:
[18 × 7] + [18 × (-3)] = 126 + (-54) = 72
Therefore L.H.S. = R.H.S. Hence verified (distributive law).

(b)(-21) x [(-4) + (-6)] = [(-21) x (-4)] + [(-21) x (-6)] 

Ans: 

Left-hand side:
(-21) × [(-4) + (-6)] = (-21) × (-10) = 210
Right-hand side:
[(-21) × (-4)] + [(-21) × (-6)] = 84 + 126 = 210
Thus L.H.S. = R.H.S. Hence verified.

Q3: (i) For any integer a, what is (-1) x a equal to?

Ans: 
= (-1) × a = -a
Multiplying any integer by -1 gives its additive inverse.

(ii) Determine the integer whose product with (-1) is
(a) -22

Ans: 

Find n such that (-1) × n = -22.
Multiplying both sides by -1 gives n = 22.

(b) 37

Ans: 

Find n such that (-1) × n = 37.
Then n = -37.

(c) 0

Ans: 

Find n such that (-1) × n = 0.
Then n = 0 (only zero gives zero when multiplied by any number).

Q4: Starting from (-1) x 5, write various products showing some patterns to show (-1) x (-1) = 1

Ans: 

Observe the pattern as the second factor decreases by 1 each time:
(-1) × 5 = -5
(-1) × 4 = -4
(-1) × 3 = -3
(-1) × 2 = -2
(-1) × 1 = -1
(-1) × 0 = 0
(-1) × (-1) = 1
This pattern shows that (-1) times a positive integer is negative, and (-1) times (-1) is positive 1.

Exercise 1.3 

Q1: Evaluate each of the following
(a) (-30) ÷ 10

Ans:

When a negative integer is divided by a positive integer, the quotient is negative.
= (-30) ÷ 10 = -3

(b) 50 ÷ (-5)

Ans:

A positive integer divided by a negative integer gives a negative quotient.
= 50 ÷ (-5) = -10

(c) (-36) ÷ (-9)

Ans: 

A negative integer divided by a negative integer gives a positive quotient.
= (-36) ÷ (-9) = 4

(d) (-49) ÷ 49

Ans: 

= (-49) ÷ 49 = -1

(e) 13 + [(-2) + 1]

Ans:

Evaluate inside the bracket first:
= 13 + [(-2) + 1] = 13 + (-1) = 12

(f) 0 ÷ (-12)

Ans:

Zero divided by any non-zero integer is zero.
= 0 ÷ (-12) = 0

(g) (-31) ÷ [(-30) + (-1)]

Ans:

Compute the denominator: (-30) + (-1) = -31
So,
= (-31) ÷ (-31) = 1

(h) [(-36) ÷ 12] ÷ 3

Ans: 

First evaluate inside brackets:
= (-36) ÷ 12 = -3
Then,
= (-3) ÷ 3 = -1

(i) [(-6) + 5] ÷ [(-2) + 1]

Ans: 

Compute numerator and denominator:
Numerator = (-6) + 5 = -1
Denominator = (-2) + 1 = -1
So,
= (-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  

Ans:
We check whether a ÷ (b + c) equals (a ÷ b) + (a ÷ c).
Given 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 -6 ≠ 3, LHS ≠ RHS. Hence verified.

(b) a = (-10), b = 1 c = 1

Ans:
Given 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 -5 ≠ -20, LHS ≠ RHS. Hence verified.

Q3: Fill in the blanks
(a) 369 ÷ _____   = 369

Ans: 

Let the missing integer be x.
369 ÷ x = 369 ⇒ x = 1
So, 369 ÷ 1 = 369

(b) (-75) ÷ _____   = (-1)

Ans: 

Let the missing integer be x.
(-75) ÷ x = -1 ⇒ x = 75
So, (-75) ÷ 75 = -1

(c) (-206) ÷ _____   = 1

Ans: 

Let the missing integer be x.
(-206) ÷ x = 1 ⇒ x = -206
So, (-206) ÷ (-206) = 1

(d) (-87) ÷  _____  = 87

Ans: 

Let the missing integer be x.
(-87) ÷ x = 87 ⇒ x = -1
So, (-87) ÷ (-1) = 87

(e)  _____  ÷1 = -87

Ans: 

Let the missing integer be x.
x ÷ 1 = -87 ⇒ x = -87
So, (-87) ÷ 1 = -87

(f)  _____  ÷ 48 = -1

Ans: 

Let the missing integer be x.
x ÷ 48 = -1 ⇒ x = -48
So, (-48) ÷ 48 = -1

(g) 20 ÷  _____  = -2

Ans: 

Let the missing integer be x.
20 ÷ x = -2 ⇒ x = 20 ÷ (-2) = -10
So, 20 ÷ (-10) = -2

(h) _____   ÷ (4) = -3

Ans: 

Let the missing integer be x.
x ÷ 4 = -3 ⇒ x = -3 × 4 = -12
So, (-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)

Ans: 

(i)  (-6) ÷ 2 = -3
(ii) 9 ÷ (-3) = -3
(iii) 12 ÷ (-4) = -3
(iv) (-9) ÷ 3 = -3
(v) (-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 midnight?

Ans: Given:
Temperature at 12 noon = 10°C
Rate of change = -2°C per hour

Temperature after t hours = 10 + (-2)t

We want temperature = -8°C:
10 - 2t = -8 ⇒ -2t = -18 ⇒ t = 9 hours

So, 9 hours after 12 noon is 9 p.m. At 9 p.m. the temperature is -8°C.

Temperature at midnight (12 a.m.) is after 12 hours:
10 - 2×12 = 10 - 24 = -14°C

So, at midnight the temperature will be 14°C below zero (-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 scored -5 marks in this test, though she got 7 correct answers. How many questions has she attempted incorrectly?

Ans: Marks for each correct answer = +3
Marks for each wrong answer = -2

(i) Radhika scored 20 marks and has 12 correct answers
Total from correct answers = 12 × 3 = 36
Marks from wrong answers = Total score - marks from correct = 20 - 36 = -16
Number of incorrect answers = (-16) ÷ (-2) = 8

(ii) Mohini scored -5 marks and has 7 correct answers
Total from correct answers = 7 × 3 = 21
Marks from wrong answers = Total score - marks from correct = -5 - 21 = -26
Number of incorrect answers = (-26) ÷ (-2) = 13

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?

Ans: Initial height = +10 m
Final depth = -350 m

Total distance to descend = 10 - (-350) = 360 m

Rate = 6 m/min ⇒ Time = 360 ÷ 6 = 60 minutes = 1 hour