NCERT Solutions for Class 7th: Ch 1 Integers Math

NCERT Solutions for Class 7th: Ch 1 Integers Maths

Page No: 3

Try these

We have done various patterns with numbers in our previous class. Can you find a pattern for each of the following? If yes, complete them:

(a) 7, 3, – 1, – 5, _____, _____, _____.
(b) – 2, – 4, – 6, – 8, _____, _____, _____.
(c) 15, 10, 5, 0, _____, _____, _____.
(d) – 11, – 8, – 5, – 2, _____, _____, _____.
Make some more such patterns and ask your friends to complete them.

Answer

(a) 7, 3, -1, -5, -9, -13, -17
(b) -2, -4, -6, -8, -10, -12, -14
(c) 15, 10, 5, 0, -5, -10, -15
(d) -11, -8, -5, -2, -1, -4, -7

Page No: 4

Exercise 14.1

1. 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?

Answer

(a) The temperatures of the places marked on number line are:
Lahulspiti : -8⁰C
Srinagar :  -2⁰C
Shimla :  5⁰C
Ooty :  14⁰C
Bangalore : 22⁰C

(b) Temperature at the hottest place = 22⁰C
Temperature at the coldest place = -8⁰C
The temperature difference between these places = 22⁰C - (-8⁰C) = 30⁰C

(c) The temperature at Lahulspiti = -8⁰C
The temperature at Srinagar = -2⁰C
The temperature difference between these= -8⁰C - (-2⁰C) = -6⁰C

(d) Temperature of Srinagar= -2⁰C
Temperature of Shimla = 5⁰C
Temperature of Srinagar and Shimla taken together = -2⁰C + 5⁰ = 3⁰C
Therefore:  3⁰C < 5⁰C
Yes, we can say that temperature of Srinagar and Shimla taken together is less than the temperature at Shimla.
However, 3⁰C > -2⁰C
Thus, the temperature at Srinagar is not less than the temperature at Shimla.

2. 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?

Answer

Jack’s scores in five successive rounds were 25, -5, -10, 15 and 10.
His total at the end = 25 – 5 – 10 + 15 + 10 = 35

3. 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?

Answer

Temperature on Monday= -5⁰C 
Temperature on Tuesday = -5⁰C - 2⁰C = -7⁰C
Temperature on Wednesday= -7⁰C + 4⁰C = -3⁰C
Hence, the temperature on Tuesday and Wednesday was -7⁰C and -3⁰C.

4. 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?
Answer

Height of plane is above the sea level = 5000 m
A submarine is floating below the sea level = -1200m
Vertical distance between them = 5000 m - (-1200)m  = 5000 m + 1200m = 6200m

5. 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.

Answer

If withdrawal of amount from the account is represented by a negative integer, then the amount deposited will be represented by a positive integer.
A/q,
Amount deposited by Mohan = Rs. 2,000
Amount withdrawn = Rs. (-1642)
Balance in Mohan’s account = 2000 + (-1642) = 2000 - 1642 = 358

6. 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?
Answer

If the distance towards east is represented by a positive integer then the distance travelled towards west will be represented by negative integer.
Distance travelled towards east direction = 20 km
Distance travelled towards west direction = 30 km
Distance travelled from A = 20 + (-30) = -10 km

7. In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square.
Answer

In square (i) each row, column and diagonal have the same sum i.e., 0. However, sum of one of its diagonals is not 0. 
Therefore, square (i) is not a magic square.
In square (ii) each row, column and diagonal have the same sum i.e., -9.
Therefore, square (ii) is a magic square.

8. 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

Answer

(i) a = 21, b= 18
→ a - (-b) = 21 - (-18) = 21 +18 = 39
→ a + b = 21 + 18 = 39
∴ a - (-b) = a + b = 39
  
(ii) a= 118, b= 125
→ a - (-b) = 118 - (-125) = 118 + 125 = 243
→ a + b = 118 + 125 = 243
∴ a - (-b) = a + b = 243

(iii) a=75, b= 84
→ a - (-b) = 75 - (-84) = 75 + 84 = 159
→ a + b = 75 + 84 = 159
∴ a - (-b) = a + b = 159

(iv) a=28, b= 11
→ a - (-b) = 28 - (-11) = 28 + 11 = 39
→ a + b = 28 +11 = 39
∴ a - (-b) = a + b = 39

9. 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

Answer

(a) LHS = (–8) + (–4) = -8-4 = -12
RHS = (–8) – (–4) = -8+4 = -4
∴ -12 < -4

(b) LHS = (–3) + 7 – (19) = -15
RHS = 15 – 8 + (–9) = 15 - 8 - 9 = -2
∴ -15 < -2

(c) LHS = 23 – 41 + 11 = -7
RHS = 23 – 41 – 11 = 29
∴ -7 > -29

(d) LHS = 39 + (-24) - (15) = 39 - 24 - 15 = 0
RHS = 36 + (-52) - (-36) = 36 - 52 + 36 = 20
∴ 0 < 20

(e)  LHS = –231 + 79 + 51 = -101
RHS = –399 + 159 + 81 = -159
∴ -101 > -159

10. A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the first step). The water level is at the ninth step.

(i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level?

(ii) After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step?

(iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his moves in part (i) and (ii) by completing the following; (a) – 3 + 2 – ... = – 8 (b) 4 – 2 + ... = 8. In (a) the sum (– 8) represents going down by eight steps. So, what will the sum 8 in (b) represent?

Answer

(a) The monkey was at step = 1
After first jump he will be at the step = 1+3 = 4
After second jump he will at the step = 4 + (-2) = 2
After third jump he will be at the step = 2 + 3 = 5
After fourth jump he will be at the step = 5 + (-2) = 3
After fifth jump he will be at the step = 3 + 3 = 6
After sixth jump he will be at the step = 6 + (-2) = 4
After seventh jump he will be at the step = 4 + 3 = 7
After eighth jump he will be at the step = 7 + (-2) = 5
After ninth jump he will be at the step = 5 + 3 = 8
After tenth jump he will at the step = 8 + (-2) = 6
After eleventh jump he will at the step = 6 + 3 = 9
Therefore, after 11th jump the monkey will reach the water level.

(b) The monkey was at the step 9.
After first jump he will be at the step = 9 + (-4) = 5
After second jump he will be at the step = 5 + 2 = 7
After third jump he will be at the step = 7 + (-4) = 3
After fourth jump he will be at the step = 3 + 2 = 5
After fifth jump he will be at the step = 5 + (-4) = 1
Therefore, after fifth jump he will reach back the top step.

(c) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, then his moves in part (i):
-3 + 2 -3 + 2 -3 + 2 -3 + 2 -3 + 2 -3 = -8
moves in part (ii):
4 -2 +4 -2 +4 = 8
Moves in part (ii) represent going up by 8 steps.

Page No: 8

Try these

1. Write a pair of integers whose sum gives 
(a) a negative integer
(b) zero
(c) an integer smaller than both the integers.
(d) an integer smaller than only one of the integers.
(e) an integer greater than both the integers.

Answer

(a) 2 + (-8) = -6
(b) 4 + (-4) = 0
(c) -2 + -4 = -6
(d) 4 + (-2) = 2
(e) 2 + 5 = 7

2. Write a pair of integers whose difference gives 
(a) a negative integer. 
(b) zero.
(c) an integer smaller than both the integers. 
(d) an integer greater than only one of the integers. 
(e) an integer greater than both the integers.

Answer

(a) 2 - 8 = -6
(b) 4 - 4 = 0
(c) 2 - 7 = -5
(d) 9 - 3 = 6
(e) 4 - (-3) = 7

Page No: 9

Exercise 1.2

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

Answer

(a) -8 + (+1) = -7
(b) -12 – (-2) = -10
(c) 5 + (-5) = 0

2. (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.

Answer

(a) -2 - (-10) = 8
(b) -8 +3 = -5
(c) -2 - (+1) = -3

3. 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?

Answer

Team A scored = -40, 10, 0
Total scored by Team A= -40 +10 +0 = -30
Team B scored = 10, 0, - 40
Total scored by Team B = 10 +0 + (-40) = -30
Hence, both teams scored equal.

4. Fill in the blanks to make the following statements true:
(i) (–5) + (............) = (– 8) + (............)
(ii) –53 + ............ = –53
(iii) 17 + ............ = 0
(iv) [13 + (– 12)] + (............) = ............ + [(–12) + (–7)]
(v) (– 4) + [............ + (–3)] = [............ + 15] + ............

Answer

(i) (-5) + (-8) = (-8) + (-5)
(ii) -53 + 0 = -53
(iii) 17 + (-17) = 0
(iv) [13 + (-12)] + (-7 ) = 13 + [(-12) + (-7)]
(v) (-4) + [15 + (-3)] = [-4 + 15] + (-3)

Page No: 10

Try these

Find:
4 × (– 8), 
8 × (–2), 
3 × (–7), 
10 × (–1) using number line.

Answer

4 × (– 8) means -8 + (-8) + (-8) + 8) = -32

8 × (–2) means -2 + (-2) + (-2) + (-2) + (-2) + (-2) + (-2) + (-2) = -16

3 × (–7) means -7 + (-7) + (-7) = -21

10 × (–1) means -1 + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) = -10

Find:
(i) 6 × (-19) 
(ii) 12 × (-32) 
(iii) 7 × (-22)

Answer

(i) 6 × (-19) = -19 + (-19) + (-19) + (-19) + (-19) + (-19)= -114

(ii) 12 × (-32) = -32 + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) = -144096

(iii) 7 × (-22) = -22 + (-22) + (-22) + (-22) + (-22) + (-22) + (-22)= -154

Page No. 11

Try these

1. Find:
(a) 15 × (–16)
(b) 21 × (–32)
(c) (– 42) × 12 
(d) –55 × 15 

Answer

(a) 15 × (–16) = 16 × (–15) = -240
(b) 21 × (–32) = 32 × (–21) = -672
(c) (–42) × 12 = -504
(d) –55 × 15 = -825

2. Check if
(a) 25 × (–21) = (–25) × 21 
(b) (–23) × 20 = 23 × (–20) 
Write five more such examples.

Answer

(a) Yes.
25 × (–21) = (–25) × 21 = -525

(b) Yes.
(–23) × 20 = 23 × (–20) = -460

Five examples: (i) 15 × (–16) = 16 × (–15)
(ii) 14 × (–18) = 18 × (–14)
(iii) 25 × (–16) = 16 × (–25)
(iv) 28 × (–15) = 15 × (–28)
(v) 8 × (–16) = 16 × (–8)

Page No. 21

Exercise 1.3

1. Find each of the following products:
(a) 3 × (–1) 
(b) (–1) × 225 
(c) (–21) × (–30) 
(d) (–316) × (–1) 
(e) (–15) × 0 × (–18) 
(f) (–12) × (–11) × (10) 
(g) 9 × (–3) × (– 6) 
(h) (–18) × (–5) × (– 4) 
(i) (–1) × (–2) × (–3) × 4 
(j) (–3) × (–6) × (–2) × (–1)

Answer

(a) 3 x (–1) = –3

(b) (–1) x 225 = –225

(c) (–21) x (–30) = 630

(d) (–316) x (–1) = 316

(e) (–15) x 0 x (–18) = 0

(f) (–12) x (–11) x (10) = 132 x 10 = 1320

(g) 9 x (–3) x (–6) = 9 x 18 = 162

(h) (–18) x (–5) x (–4) = 90 x (–4) = –360

(i) (–1) x (–2) x (–3) x 4 = (–6 x 4) = –24

(j) (–3) x (–6) x (2) x (–1) = (–18) x (–2) = 36

2. Verify the following:
(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]
(b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]

Answer

(a) 18 x [7 + (–3)] = [18 x 7] + [18 x (–3)]
⇒ 18 x 4 = 126 + (–54)
⇒ 72 = 72
⇒ L.H.S. = R.H.S. Hence verified.

(b) (–21) x [(–4) + (–6)] = [(–21) x (–4)] + [(–21) x (–6)]
⇒ (–21) x (–10) = 84 + 126
⇒ 210 = 210
⇒ L.H.S. = R.H.S. Hence verified.

3. (i) For any integer a, what is (–1) × a equal to?
(ii) Determine the integer whose product with (–1) is 
(a) –22 
(b) 37 
(c) 0

Answer

(i) (–1)×a=–a, where aa is an integer.

(ii) (a) (–1)×(–22)=22
(b) (–1)×37=–37
(c) (–1)×0=0

4. Starting from (–1) × 5, write various products showing some pattern to show (–1) × (–1) = 1.

Answer

(–1)×5=–5(–1)×4=–4
(–1)×3=–3(–1)×2=–2
(–1)×1=–1(–1)×0=0
(–1)×(–1)=1

Thus, we can conclude that this pattern shows the product of one negative integer and one positive integer is negative integer whereas the product of two negative integers is a positive integer.

5. Find the product, using suitable properties: 
(a) 26 × (– 48) + (– 48) × (–36) 
(b) 8 × 53 × (–125) 
(c) 15 × (–25) × (– 4) × (–10) 
(d) (– 41) × 102 
(e) 625 × (–35) + (– 625) × 65
(f) 7 × (50 – 2) 
(g) (–17) × (–29) 
(h) (–57) × (–19) + 57

Answer

(a) 26×(–48)+(–48)×(–36)
⇒ (–48)×[26+(–36)] [Distributive property]
⇒ (–48)×(–10)
⇒ 480

(b) 8×53×(–125)
⇒ 53×[8×(–125)] [Commutative property]
⇒ 53×(–1000)
⇒ –53000

(c) 15×(–25)×(–4)×(–10)
⇒ 15×[(–25)×(–4)×(–10)] [Commutative property]
⇒ 15×(–1000)
⇒ –15000

(d) (–41)×(102)
⇒ –41×[100+2] [Distributive property]
⇒ [(–41)×100]+[(–41)×2]
⇒ –4100+(–82)
⇒ –4182

(e) 625×(–35)+(–625)×65
⇒ 625×[(–35)+(–65)] [Distributive property]
⇒ 625×(–100)
⇒ –62500

(f) 7×(50–2)
⇒ 7×50–7×2 [Distributive property]
⇒ 350–14=336

(g) (–17)×(–29)
⇒ (–17)×[(–30)+1] [Distributive property]
⇒ (–17)×(30)+(–17)×1
⇒ 510+(–17)
⇒ 493

(h) (–57)×(–19)+57
⇒ (–57)×(–19)+57×1
⇒ 57 x 19 + 57 x 1
⇒ 57 x (19 + 1) [Distributive property]
⇒ 57 x 20 = 1140

6. 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?

Answer

Given: Present room temperature = 40°C
Decreasing the temperature every hour = 5°C
Room temperature after 10 hours = 40°C + 10 x (–5°C)
= 40°C – 50°C
= –10°C
Thus, the room temperature after 10 hours is –10°C after the process begins.

7. In a class test containing 10 questions, 5 marks are awarded for every correct answer and (–2) marks are awarded for every incorrect answer and 0 for questions not attempted.
(i) Mohan gets four correct and six incorrect answers. What is his score?
(ii) Reshma gets five correct answers and five incorrect answers, what is her score?
(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?

Answer

(i) Mohan gets marks for four correct questions = 4 x 5 = 20
He gets marks for six incorrect questions = 6 x (–2) = –12
Therefore, total scores of Mohan = (4 x 5) + [6 x (–2)] = 20 – 12 = 8
Thus, Mohan gets 8 marks in a class test.

(ii) Reshma gets marks for five correct questions = 5 x 5 = 25
She gets marks for five incorrect questions = 5 x (–2) = –10
Therefore, total score of Resham = 25 + (–10) = 15
Thus, Reshma gets 15 marks in a class test.

(iii) Heena gets marks for two correct questions = 2 x 5 = 10
She gets marks for five incorrect questions = 5 x (–2) = –10
Therefore, total score of Resham = 10 + (–10) = 0
Thus, Reshma gets 0 marks in a class test.

8. A cement company earns a profit of Rs 8 per bag of white cement sold and a loss of Rs 5 per bag of grey cement sold. 
(a) 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?
(b) What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags.

Answer

Given: Profit of 1 bag of white cement = Rs. 8
And Loss of 1 bag of grey cement = Rs. 5

(a) Profit on selling 3000 bags of white cement = 3000 × 8 = Rs. 24,000
Loss of selling 5000 bags of grey cement = 5000 × Rs. 5 = Rs. 25,000
Since Profit < Loss
Therefore, his total loss on selling the grey cement bags = Loss – Profit
= 25,000 – 24,000 = Rs. 1,000
Thus, he has lost of Rs. 1,000 on selling the grey cement bags.

(b) Let the number of bags of white cement be x.
According to question, Loss = Profit
∴ 5 x 6,400 = x × 8
⇒ x = (5×6400)/8 = 5000 bags
Thus, he must sell 4000 white cement bags to have neither profit nor loss.

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