Important Questions: Integers | Mathematics (Maths) Class 7 PDF Download

Q1: Following number line given below shows the temperature present in degree celsius at different places on a particular day.
(i) Observe the number line and write down the temperature of the places marked on it.
Ans:
By observing the above number line, we can find out the temperature of the cities as follows,
The temperature in the city of Lahulspiti is -8°C.
The temperature in the city of Srinagar is -2°C
The temperature in the city of Shimla is 5°C.
The temperature in the city of Ooty is 14°C.
The temperature in the city of Bengaluru is 22°C.

(ii) What is the temperature difference between the hottest and the coldest places among the cities stated above?
Ans:
From the above number line, we can observe that,
The temperature at the given hottest place, that is, Bengaluru, is 22°C.
The temperature at the given coldest place, that is, Lahulspiti, is -8°C
The temperature difference between the hottest and the coldest place is given as = 22°C – (-8°C)
= 22°C + 8°C
= 30° Celsius
Hence, the total temperature difference between the hottest and the coldest place is 30ºC.

(iii) What is the temperature difference between the cities of Lahulspiti and Srinagar?
Ans:
From the above-given number line,
The temperature in the city of Lahulspiti is -8°C.
The temperature in the city of Srinagar is -2°C
∴The temperature difference between the cities Lahulspiti and Srinagar is = -2ºC – (8ºC)
= – 2°C + 8°C
= 6°C

(iv) Can we say that the temperature of Srinagar and Shimla taken together is less than the temperature present at Shimla? Is it also less than the temperature present at Srinagar?
Ans:
From the above-given number line,
The temperature in the city of Srinagar =-2°C
The temperature in the city of Shimla = 5°C
The temperature of the cities Srinagar and Shimla taken together becomes = – 2°C + 5°C
= 3° degree C
 5°C > 3°C
Hence, the temperature of the cities Srinagar and Shimla taken together is indeed less than the temperature present at Shimla.
Then,
3° > -2°
And No, the temperature of the cities Srinagar and Shimla taken together is not less than the temperature of the city Srinagar.

Q2: Mohan deposits Rs. 2,000 in his bank account and then withdraws Rs. 1,642 from it the following day. Now, if the withdrawal of the amount from the account is represented by a negative integer, then how will you represent the total amount deposited? Also, Find the balance in Mohan’s account after the withdrawal.
Ans:
Withdrawal of these amounts from the account is represented by a negative integer.
Then, the deposit of the amount to the account is represented by a positive integer.
From the above question,
The total amount that is deposited in the bank account by the Mohan = Rs. 2000
The total amount that is withdrawn from the bank account by the Mohan is = – Rs. 1642
Final Balance in Mohan’s account after the withdrawal = amount deposited + amount is withdrawn
= Rs. 2000 + (-Rs. 1642)
= Rs. 2000 – Rs. 1642
= Rs. 358
Hence, the total balance in Mohan’s account after the withdrawal is Rs. 358

Q3: In the following quiz, positive marks are given for every correct answer and negative marks are given for each incorrect answer. If Jack’s scores in the quiz for five successive rounds were 25, – 5, – 10, 10, and 15 so, what was his total at the end?
Ans:
From the above question,
Jack’s scores in the five successive rounds are 25, -5, -10, 15 and 10
Hence, Their total score of Jack at the end will be = 25 + (-5) + (-10) + 15 + 10
= 25 – 5 – 10 + 15 + 10
= 50 – 15
= 35 marks
∴ Now, Jack’s total score at the end is 35.

Q4: In the city of Srinagar, temperature was – 5°C on Monday, and then it dropped by  2°C on Tuesday. What was the temperature of the city of Srinagar on Tuesday? On Wednesday, the temperature rose by 4°C. What was the temperature on this day?
Ans:
From the above question,
The temperature on Monday at Srinagar is = -5C
The temperature on Tuesday at the city of Srinagar is dropped by 2C = Temperature on Monday – 2C
= -5C – 2C
= -7 celsius
The temperature on Wednesday at the city Srinagar rose by 4C = Temperature on Tuesday + 4C.
= -7C + 4C
= -3 celsius
Thus, the temperature on days Tuesday and Wednesday was found to be -7C and -3C, respectively.

Q5: In a magic square, every row, column and diagonal has the same sum. Check which of these following is a magic square. 
Ans:
Firstly we consider the square (i)
Now By adding these numbers in each of the rows, we get,
= 5 + (- 1) + (- 4) equals to 5 – 1 – 4 = 5 – 5 = 0
= -5 + (-2) + 7 equals to – 5 – 2 + 7 = -7 + 7 = 0
= 0 + 3 + (-3) = 3 – 3 = 0
By adding these numbers in every column we receive,
= 5 + (- 5) + 0 is equal to 5 – 5 = 0
= (-1) + (-2) + 3 equals to -1 – 2 + 3 = -3 + 3 = 0
= -4 + 7 + (-3) equals to -4 + 7 – 3 = -7 + 7 = 0
By adding these numbers in diagonals, we receive,
= 5 + (-2) + (-3) is equal to 5 – 2 – 3 = 5 – 5 = 0
= -4 + (-2) + 0 is equal to – 4 – 2 = -6
Because the sum of one diagonal is not always equal to zero,
Hence, (i) is not a magic square.
Now, we should consider the square (ii)
By adding these numbers to each rows we receive,
= 1 + (-10) + 0 is equal to 1 – 10 + 0 = -9
= (-4) + (-3) + (-2) equal to -4 – 3 – 2 = -9
= (-6) + 4 + (-7) becomes equal to -6 + 4 – 7 = -13 + 4 = -9
By adding these numbers in each column we receive,
= 1 + (-4) + (-6) equals to 1 – 4 – 6 = 1 – 10 = -9
= (-10) + (-3) + 4 equals to -10 – 3 + 4 = -13 + 4
= 0 + (-2) + (-7) equals to 0 – 2 – 7 = -9
By adding these numbers in diagonals, we receive,
= 1 + (-3) + (-7) equals to 1 – 3 – 7 = 1 – 10 = -9
= 0 + (-3) + (-6) equal to 0 – 3 – 6 = -9
Hence This (ii) square is a magic square because the sum of each row, each column and the diagonal becomes equal to -9 (negative).

Q6: Verify a – (– b) is equal to a + b for the following values of alphabets a and b.
(i) a = 21, b = 18
Ans:
From the above question,
a = 21 and b = 18
So To verify a – (- b) is equal to a + b
Let us take the Left Hand Side (LHS) = a – (- b)
= 21 – (- 18)
= 21 + 18
= 39
Now, lets take Right Hand Side (RHS) = a + b
= 21 + 18
= 39
By comparing both the LHS and the RHS.
LHS = RHS
39 = 39
Hence, the value of a and b are verified.

(ii) a = 118, b = 125
Ans:
From the above question,
a = 118 and b = 125
To verify this a – (- b) = a + b
Let us take the Left Hand Side (LHS) = a – (- b)
= 118 – (- 125)
= 118 + 125
= 243
Now, take the Right Hand Side (RHS) = a + b
= 118 + 125
= 243
By comparing both the LHS and the RHS
LHS = RHS
243 = 243
Hence, the values of a and b are verified.

(iii) a = 75, b = 84
Ans:
From the above question,
a = 75 and b = 84
To verify that the a – (- b) = a + b
Let us take the Left Hand Side (LHS) = a – (- b)
= 75 – (- 84)
= 75 + 84
= 159
Now, the Right Hand Side (RHS) = a + b
= 75 + 84
= 159
By comparing both LHS and RHS, we find that,
LHS = RHS
159 = 159
Hence, the value of a and b is verified as.

(iv) a = 28, b = 11
Ans:
From the above question,
a = 28 and b = 11
To verify that a – (- b) = a + b
Let us now take Left Hand Side (LHS) = a – (- b)
= 28 – (- 11)
= 28 + 11
= 39
Now, Right Hand Side (RHS) = a + b
= 28 + 11
= 39
By comparing both the LHS and the RHS
LHS = RHS
39 = 39
Hence, the value of a and b are verified.

Q7: A water tank has stepped inside it. A monkey is sitting on the utter topmost step (which is the first step). The water level is present at the ninth step.
(i) He jumps three steps down the stairs and then successively jumps back two steps upwards. In how many jumps will the Monkey reach the following water level?
Ans:
Let us consider the steps moved down are represented by a positive integer, and then the steps moved up are represented by a negative integer.
Initially, the Monkey is sitting on the topmost step, which is the first step.
In the 1st jump monkey will be at the step = 1 + 3 = 4 steps
In the 2nd jump monkey will be at the step = 4 + (-2) = 4 – 2 = 2 steps
In the 3rd jump monkey will be at the step = 2 + 3 = 5 steps
In the 4th jump monkey will be at the step = 5 + (-2) = 5 – 2 = 3 steps
In the 5th jump monkey will be at the step = 3 + 3 = 6 steps
In the 6th jump monkey will be at the step = 6 + (-2) = 6 – 2 = 4 steps
In the 7th jump monkey will be at the step = 4 + 3 = 7 steps
In the 8th jump monkey will be at the step = 7 + (-2) = 7 – 2 = 5 steps
In the 9th jump monkey will be at the step = 5 + 3 = 8 steps
In the 10th jump monkey will be at the step = 8 + (-2) = 8 – 2 = 6 steps
In the 11th jump monkey will be at the step = 6 + 3 = 9 steps
∴Monkey took a total of 11 jumps (i.e., 9th step) to reach the water level.

(ii) After drinking water, the Monkey wants to go back. For this, the Monkey jumps four steps up and then successively jumps back two steps down in his every move. In how many total jumps will he reach back to the top step?
Ans: Let us consider the steps moved down are represented by the positive integers, and then the steps moved up are represented by the negative integers.
Initially, the Monkey is sitting on the ninth step, i.e., at the water level.
In the 1st jump monkey will be at the step = 9 + (-4) = 9 – 4 = 5 steps
In the 2nd jump monkey will be at the step = 5 + 2 = 7 steps
In the 3rd jump monkey will be at the step = 7 + (-4) = 7 – 4 = 3 steps
In the 4th jump monkey will be at the step = 3 + 2 = 5 steps
In the 5th jump monkey will be at the step = 5 + (-4) = 5 – 4 = 1 step
∴ Hence the Monkey took five jumps to reach back to the top step, i.e., the first step.

Q8: Fill in the blanks to make the following statements true:
(i) (–5) + (– 8) = (– 8) + (…………)
Ans:
Let us assume that the missing integer is x,
So,
= (–5) + (– 8) which equals to (– 8) + (x)
= – 5 – 8 = – 8 + x
= – 13 = – 8 + x
By sending – 8 from the RHS to the LHS, it becomes 8,
= – 13 + 8 = x
= x = – 5
Now substitute the x value in the place of the blank place present,
(–5) + (– 8) = (– 8) + (- 5) … [This following equation is present in the form of the Commutative law of Addition]

(ii) –53 + ………… = –53
Ans: Let us assume that the missing integer is x,
So,
= –53 + x = –53
By sending – 53 from the LHS to the RHS, it becomes 53,
= x = -53 + 53
= x = 0
Now substitute the following x value in the blank place,
= –53 + 0 = –53 … [This equation is present in the form of Closure property of Addition]

(iii) 17 + ………… = 0

Ans: Let us assume that the missing integer is x,
So,
= 17 + x = 0
By sending 17 from the LHS to the RHS, it becomes -17,
= x = 0 – 17
= x = – 17
Now substitute this x value in the blank place,
= 17 + (-17) = 0 … [This equation is present in the form of Closure property of Addition]
= 17 – 17 = 0

(iv) [13 + (– 12)] + (…………) = 13 + [(–12) + (–7)]
Ans: Let us assume that the missing integer is x,
So,
= [13 + (– 12)] + (x) = 13 + [(–12) + (–7)]
= [13 – 12] + (x) = 13 + [–12 –7]
= [1] + (x) = 13 + [-19]
= 1 + (x) = 13 – 19
= 1 + (x) = -6
By sending one from the LHS to the RHS, it becomes -1,
= x = -6 – 1
= x = -7
Now substitute the following x value in the blank place value,
= [13 + (– 12)] + (-7) equals to 13 + [(–12) + (–7)] … [This equation is present in the form of the Associative Property of Addition]

(v) (– 4) + [15 + (–3)] equals to [– 4 + 15] +…………
Ans:
Let us assume that the missing integer is x,
So,
= (– 4) + [15 + (–3)] is equal to [– 4 + 15] + x
= (– 4) + [15 – 3)] equals to [– 4 + 15] + x
= (-4) + [12] = [11] + x
= 8 = 11 + x
Now, By sending 11 from the RHS to the LHS, it becomes -11,
= 8 – 11 = x
= x = -3

Now substitute the x value in the place of the blank place,

= (– 4) + [15 + (–3)] equals to [– 4 + 15] + -3 … [The following equation is in the form of the Associative property of the Addition]

Q9: Find the product using the suitable properties:
(i) 26 × (– 48) + (– 48) × (–36)
Ans: This given equation is in the form of the Distributive law of the  Multiplication property over Addition.
= a × (b + c) becomes equal to (a × b) + (a × c)
Let, a = -48, b = 26, c = -36
So,
= 26 × (– 48) + (– 48) × (–36)
= -48 × (26 + (-36)
= -48 × (26 – 36)
= -48 × (-10)
= 480 … [∵ (- × – = +)

(ii) 8 × 53 × (–125)
Ans: The given equation is present in the form of the Commutative law of Multiplication.
= a × b = b × a
Then,
= 8 × [53 × (-125)]
= 8 × [(-125) × 53]
= [8 × (-125)] × 53
= [-1000] × 53
= – 53000

(iii) 15 × (–25) × (– 4) × (–10)
Ans: This given equation is in the form of the Commutative law of the Multiplication property.
= a × b = b × a
So,
= 15 × [(–25) × (– 4)] × (–10)
= 15 × [100] × (–10)
= 15 × [-1000]
= – 15000

(iv) (– 41) × 102
Ans: This given equation is in the form of a Distributive law of the Multiplication property over Addition.
= a × (b + c) = (a × b) + (a × c)
= (-41) × (100 + 2)
= (-41) × 100 + (-41) × 2
= – 4100 – 82
= – 4182

(v) 625 × (–35) + (– 625) × 65
Ans: This given equation is in the form of the Distributive law of Multiplication over Addition.
= a × (b + c) = (a × b) + (a × c)
= 625 × [(-35) + (-65)]
= 625 × [-100]
= – 62500

Q10. A certain freezing process requires that the room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the final room temperature 10 hours after the actual process begins?
Ans:
From the above question, it is given that
Let us take the lowered temperature as a negative integer,
Initial temperature will be= 40ºC
Change in temperature per hour is = -5ºC
Change in temperature after 10 hours will be = (-5) × 10 = -50ºC
∴The final room temperature after the 10 hours of freezing process = 40ºC + (-50ºC)= -10ºC

Q11: In a class test containing about ten questions, five marks are awarded for each correct answer and (–2) marks are awarded for every incorrect answer and 0 for questions which are not attempted.
(i) Mohan gets four correct answers and six incorrect answers on his test. What is his total score?
Ans:
From the above question,
Marks awarded for one correct answer is = 5
Hence,
The total marks awarded for his four correct answers are = four × 5 = 20 marks.
Marks awarded for 1 wrong answer = -2 (negative)
Hence,
Total marks awarded for 6 wrong answers is = 6 × -2 = -12
∴Total score obtained by Mohan = 20 + (-12)
= 20 – 12
= 8

(ii) Reshma gets five correct answers and similarly five incorrect answers; what is her total score?
Ans: From the above question,
Marks awarded for one correct answer is = 5
Hence,
Total marks awarded for 5 correct answer becomes = 5 × 5 = 25
Marks awarded for one wrong answer is = -2
Hence,
Total marks awarded for 5 wrong answer becomes = 5 × -2 = -10
∴Total score obtained by Reshma is = 25 + (-10)
= 25 – 10
= 15

(iii) Heena gets two correct answers and five incorrect answers out of the seven questions she attempts. What is her final score?
Ans:
From the above question,
Marks awarded for one correct answer is = 5
Hence,
Total marks awarded for 2 correct answer is = 2 × 5 = 10
Marks awarded for one wrong answer is = -2
Hence,
Total marks awarded for 5 wrong answer becomes = 5 × -2 = -10
Marks awarded for the questions which are not attempted is = 0
∴Total score obtained by Heena is = 10 + (-10)
= 10 – 10
= 0

Q12: A cement company earns a profit of around Rs. 8 per bag of white cement that is sold and simultaneously a loss of Rs. 5 per bag of grey cement that is sold.
(i) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
Ans:
We denote profit by a positive integer and loss by a negative integer,
So From the above question,
The Cement company earns a profit on selling one bag of white cement = Rs. 8 per bag.
So,
The cement company earns a total profit on selling 3000 bags of white cement = 3000 × Rs. 8
= Rs. 24000
And also the, 
Loss on selling one bag of grey cement is = – Rs. 5 per bag.
Hence,
Loss on selling the 5000 bags of the grey cement = 5000 × – Rs. 5
= – Rs. 25000
Total loss or profit earned by these cement companies is = profit + loss.
= 24000 + (-25000)
= – Rs.1000
Hence, a loss of Rs. 1000 will be incurred by the company.

(ii) What is the number of white cement bags that must sell to have neither a profit nor loss if the total number of grey bags sold is 6,400 bags?
Ans: We denote the profit as a positive integer and the loss as a negative integer,
From the above question,
The cement company earns the profit on selling one bag of white cement as = Rs. 8 per bag.
Now Let the number of white cement bags present be x.
Then,
The cement company earns a profit on selling these x bags of white cement as = (x) × Rs. 8
= Rs. 8x
Loss on selling one bag of grey cement becomes = – Rs. 5 per bag.
Then,
Loss on selling 6400 bags of grey cement becomes = 6400 × – Rs. 5
= – Rs. 32000
According to the above question,
Company to have neither profit nor loss, must sell,
= Profit + loss = 0
= 8x + (-32000) =0
By sending -32000 from the LHS to the RHS, it becomes 32000
= 8x = 32000
= x = 32000/8
= x = 4000
Hence, the 4000 bags of white cement should sell to have neither profit nor loss.

Q13: Evaluate each of the following:
(i) (–30) ÷ 10
Ans:
= (–30) ÷ 10
= – 3
When we divide the negative integer by a positive integer, we first divide them as whole numbers and then put the minus sign (-) before the quotient.

(ii) 50 ÷ (–5)
Ans:
= (50) ÷ (-5)
= – 10
When we divide the positive integer by a negative integer, we first divide them as whole numbers and then apply the minus sign (-) before the quotient.

(iii) (–36) ÷ (–9)
Ans:
= (-36) ÷ (-9)
= 4
When we divide the negative integer by a similar negative integer, we first divide these as whole numbers and then put the positive sign (+) before the quotient.

(iv) (– 49) ÷ (49)
Ans:
= (–49) ÷ 49
= – 1
When we divide the negative integer by a positive integer, we first divide these as whole numbers and then put the minus sign (-) before the quotient.

(v) 13 ÷ [(–2) + 1]

Ans:
= 13 ÷ [(–2) + 1]
= 13 ÷ (-1)
= – 13
When we divide the positive integer by a negative integer, we first divide these as whole numbers and then put the minus sign (-) before the quotient.

(vi) 0 ÷ (–12)
Ans:
= 0 ÷ (-12)
= 0
When we divide zero by a negative integer, it gives zero.

(vii) (–31) ÷ [(–30) + (–1)]
Ans:
= (–31) ÷ [(–30) + (–1)]
= (-31) ÷ [-30 – 1]
= (-31) ÷ (-31)
= 1
When we divide the negative integer by a negative integer, we first divide these as whole numbers and then put the positive sign (+) before the quotient.

(viii) [(–36) ÷ 12] ÷ 3
Ans: First, we have to solve these integers within the bracket,
= [(–36) ÷ 12]
= (–36) ÷ 12
= – 3
Then,
= (-3) ÷ 3
= -1
When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put the minus sign (-) before the quotient.

(ix) [(– 6) + 5)] ÷ [(–2) + 1]
Ans: The given Qcan be written as,
= [-1] ÷ [-1]
= 1
When we divide the negative integer by a negative integer, we first divide these as whole numbers and then put the positive sign (+) before the quotient.

Q14: Verify that a ÷ (b + c) is not equal to (a ÷ b) + (a ÷ c) for each of the following symbols of a, b and c.
(i) a = 12, b = – 4, c = 2
Ans:
From the above question, a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
Given, a = 12, b = – 4 (negative), c = 2
Now, consider that the LHS = a ÷ (b + c)
= 12 ÷ (-4 + 2)
= 12 ÷ (-2)
= -6
When we divide a following positive integer by any of the negative integers, we first divide them as a whole number and then put the minus sign (-) before their quotient.
Then, consider that the RHS is equal to = (a ÷ b) + (a ÷ c)
= (12 ÷ (-4)) + (12 ÷ 2)
= (-3) + (6)
= 3
By comparing the LHS and RHS, we get,
= -6 ≠ 3
= LHS ≠ RHS
Hence, the given values have been verified.

(ii) a = (–10), b = 1, c = 1
Ans:
From the above question, a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
Given, a = (-10), b = 1, c = 1
Now, consider that the LHS = a ÷ (b + c)
= (-10) ÷ (1 + 1)
= (-10) ÷ (2)
= -5
When we divide a negative integer by any other positive integer, we first divide them as a whole number and then put the minus sign (-) before the quotient.
Then, consider RHS = (a ÷ b) + (a ÷ c)
= ((-10) ÷ (1)) + ((-10) ÷ 1)
= (-10) + (-10)
= -10 – 10
= -20
By comparing LHS and RHS
= -5 ≠ -20
= LHS ≠ RHS
Hence, the given values are verified.

Q15: The temperature at 12 noon was 10 degrees C above zero. If it decreases at the rate of 2C per hour until midnight, at what time would the temperature be  8°C below zero? Also, What would be the temperature at midnight?
Ans:
From the above question, it is given that,
The temperature at the beginning, which is, at 12 noon, is = 10C
The rate of change of temperature becomes = – 2C per hour.
Then,
Temperature present at 1 PM = 10 + (-2) = 10 – 2 = 8° C
Temperature present at 2 PM = 8 + (-2) = 8 – 2 = 6° C
Temperature present at 3 PM = 6 + (-2) = 6 – 2 = 4°C
Temperature present at 4 PM = 4 + (-2) = 4 – 2 = 2°C
Temperature present at 5 PM = 2 + (-2) = 2 – 2 = 0°C
Temperature present at 6 PM = 0 + (-2) = 0 – 2 = -2°C
Temperature present at 7 PM = -2 + (-2) = -2 -2 = -4°C
Temperature present at 8 PM = -4 + (-2) = -4 – 2 = -6°C
Temperature present at 9 PM = -6 + (-2) = -6 – 2 = -8°C
∴At 9 PM, the temperature will be 8° C below zero.
Then,
The temperature at mid-night which is at 12 AM
Change in the temperature in every 12 hours = -2°C × 12 = – 24°C
So, at midnight the temperature will be = 10 + (-24)
= – 14°C
 At midnight the temperature will be 14°C below 0.