Class 7 Maths Chapter 1 HOTS Questions - Integers

HOTS Question and Answers

Q1: 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

Q2: Evaluate each of the following:

(a) (–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.

(b) 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.

(c) (–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.

(d) (– 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.

(e) 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.

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

(g) (–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.

(h) [(–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.

(i) [(– 6) + 5)] ÷ [(–2) + 1]
Ans:
The given question can 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.

Q3: 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 eight °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.

Q4: An elevator descends down into a mine shaft at the rate of 6 m per min. If the descent starts from 10 meters above the ground level, how much time will it take to reach – 350 m?
Ans:
From the above question,
The initial height of the elevator becomes = 10 m
Final depth of elevator is = – 350 m … [the distance descended is denoted by a negative integer]
The total distance to descend by the elevator becomes = (-350) – (10)
= – 360 m
Hence
Time taken by the elevator to descend (negative) -6 m is = 1 min
So, the total time taken by the elevator to descend – 360 m becomes = (-360) ÷ (-60)
= 60 minutes
= 1 hour

Q5: A cement company earns a profit of around ₹ 8 per bag of white cement that is sold and simultaneously a loss of ₹ 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 = ₹ 8 per bag.
So,
The cement company earns a total profit on selling 3000 bags of white cement = 3000 × ₹ 8
= ₹ 24000
And also the,
Loss on selling one bag of grey cement is = – ₹ 5 per bag.
Hence,
Loss on selling the 5000 bags of the grey cement = 5000 × – ₹ 5
= – ₹ 25000
Total loss or profit earned by these cement companies is = profit + loss.
= 24000 + (-25000)
= – ₹1000
Hence, a loss of ₹ 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?
Answer:-
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 = ₹ 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) × ₹ 8
= ₹ 8x
Loss on selling one bag of grey cement becomes = – ₹ 5 per bag.
Then,
Loss on selling 6400 bags of grey cement becomes = 6400 × – ₹ 5
= – ₹ 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.

Q6: 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.

Q7: Fill in the following blanks:

(a) 369 ÷ _____ = 369
Ans:
Let us assume that the missing integer is x,
Now,
= 369 ÷ x = 369
= x = (369/369)
= x = 1
Hence, put the valve of x in the blank place.
= 369 ÷ 1 = 369

(b) (–75) ÷ _____ = –1
Ans:
Let us assume that the missing integer is x,
Hence,
= (-75) ÷ x = -1
= x = (-75/-1)
= x = 75
Now, put the above valve of x in the blank place.
= (-75) ÷ 75 = -1

(c) (–206) ÷ _____ = 1
Ans:
Let us assume that the missing integer is x,
So,
= (-206) ÷ x = 1
= x = (-206/1)
= x = -206
Now, put the above valve of x in the blank place.
= (-206) ÷ (-206) = 1

(d) – 87 ÷ _____ = 87
Ans:
Let us assume that the missing integer is x,
So,
= (-87) ÷ x = 87
= x = (-87)/87
= x = -1
Now, put the above valve of x in the blank place.
= (-87) ÷ (-1) = 87

(e) _____ ÷ 1 = – 87
Ans:
Let us assume that the missing integer is x,
Now,
= (x) ÷ 1 = -87
= x = (-87) × 1
= x = -87
So, put the valve of x in the blank.
= (-87) ÷ 1 = -87

(f) _____ ÷ 48 = –1
Ans:
Let us assume that the missing integer is x,
So,
= (x) ÷ 48 = -1
= x = (-1) × 48
= x = -48
Now, put the above valve of x in the following blank.
= (-48) ÷ 48 = -1

Q8: In the following class test, (+ 3) marks are given for every correct answer, (–2) marks are given for every the incorrect answer and no marks are given for not attempting any question.
(i) Radhika scored 20 marks. If she has got around 12 correct answers, then how many questions has she attempted that are incorrect?
(ii) Mohini scores –5 (negative) marks on this test, and though she has got seven correct answers. How many questions has she attempted incorrectly?
Ans:
From the above question,
Marks awarded for 1 correct answer is = + 3
Marks awarded for one wrong answer is = -2
(i) Radhika, in the test, scored 20 marks
So,
Total marks awarded for every 12 correct answers is = 12 × 3 = 36
Marks awarded for every incorrect answer = Total score – Total marks awarded for 12 correct questions.
Answers
= 20 – 36
= – 16
So, the number of incorrect answers done by Radhika = (-16) ÷ (-2)
= 8

(ii) Mohini scored a total of -5 marks
Then,
Total marks awarded for her 7 correct answers is = 7 × 3 = 21
Marks awarded for her incorrect answers = Total score – Total marks awarded for the 12 correct answers.
= – 5 – 21
= – 26
Hence, the number of incorrect answers made by Mohini = (-26) ÷ (-2)
= 13.