1.1 POSITIVE AND NEGATIVE INTEGERS:
When two positive integers are added we get a positive integer.
eg: +7 + 5 = + 12
When two negative integers are added we get a negative integer
eg: −8 + −5 = −13
When one positive integer and one negative integer are added we get a negative or a positive integer.
Case I : When the positive integer is bigger
+ 36+( −21) = 15
Case II : When the positive integer is smaller
(−36) + (+21) = −15
Exercise 1.1
1. Following number line shows the temperature in degree Celsius (Co) at different places on a particular day.
(a) Observe this number line and write the temperature of the places marked on it.
Solution: By observing the number line, we can find the temperature of the cities as follows,
Temperature at Lahulspiti is – 8oC
Temperature at Srinagar is –2oC
Temperature at Shimla is 5oC
Temperature at Ooty is 14oC
Temperature at Bengaluru is 22oC
(b) What is the temperature difference between the hottest and the coldest places among the above?
Solution:– From the number line we observe that,
The temperature at the hottest place Bengaluru is 22oC
The temperature at the coldest place Lahulspiti is – 8oC
Temperature difference between hottest and coldest place is = 22oC – (– 8oC)
= 22o C + 8o C = 30o C
Hence, the temperature difference between the hottest and the coldest place is 30o C.
(c) What is the temperature difference between Lahulspiti and Srinagar?
Solution:– From the given number line,
The temperature at the Lahulspiti is –8o C
The temperature at the Srinagar is –2o C
∴ The temperature difference between Lahulspiti and Srinagar is = –2o C – (−8o C)
= – 2° C + 8o C = 6o C
∴ The temperature difference between Lahulspiti and Srinagar is 6o C
(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?
Solution:– From the given number line,
The temperature at Srinagar = –2o C
The temperature at Shimla = 5o C
The temperature of Srinagar and Shimla taken together is = – 2o C + 5o C = 3o C
∴ 5o C > 3o C
So, the temperature of Srinagar and Shimla taken together is less than the temperature at Shimla.
Then, 3o > − 2o
No, the temperature of Srinagar and Shimla taken together is not less than the temperature of Srinagar.
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?
Solution:– Jack’s score in five successive rounds are 25, − 5, −10, 15 and 10
The total score of Jack at the end will be = 25 + (5) + (10) + 15 + 10
= 25 – 5 – 10 + 15 + 10 = 50 – 15 = 35
∴ Jack’s total score at the end is 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?
Solution: Temperature on Monday at Srinagar = – 5o C
Temperature on Tuesday at Srinagar is dropped by 2o C = Temperature on Monday – 2o C
= – 5o C – 2o C = – 7o C
Temperature on Wednesday at Srinagar is rose by 4o C = Temperature on Tuesday + 4o C
= – 7o C + 4o C = – 3o C
Thus, the temperature on Tuesday and Wednesday was –7o C and –3o C respectively.
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?
Solution:– Plane is flying at the height = 5000 m
Depth of Submarine = –1200 m
The vertical distance between plane and submarine = 5000 m – (– 1200) m
= 5000 m + 1200 m = 6200 m
The vertical distance between plane and submarine = 6200 m
5. Mohan deposits ₹ 2,000 in his bank account and withdraws ₹ 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.
Solution:– Withdrawal of amount from the account is represented by a negative integer.
Then, deposit of amount to the account is represented by a positive integer
Total amount deposited in bank account by the Mohan = +₹ 2000
Total amount withdrawn from the bank account by the Mohan = – ₹ 1642
Balance in Mohan’s account after the withdrawal = amount deposited + amount withdrawn
= ₹ 2000 + (–₹ 1642) = ₹ 2000 – ₹ 1642 = ₹ 358
Hence, the balance in Mohan’s account after the withdrawal is ₹ 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?
Solution:– It is given that
A positive integer represents the distance towards the east.
Then, distance travelled towards the west will be represented by a negative integer.
Rita travels a distance in east direction = 20 km
Rita travels a distance in west direction = – 30 km
∴ Distance travelled from A = 20 + (– 30)
= 20 – 30 = –10 km
Hence, we will represent the distance travelled by Rita from point A by a negative integer,– 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.
Solution:– First we consider the square (i)
By adding the numbers in each row we get,
= 5 + (– 1) + (– 4) = 5 – 1 – 4 = 5 – 5 = 0
= –5 + (–2) + 7 = – 5 – 2 + 7 = –7 + 7 = 0
= 0 + 3 + (–3) = 3 – 3 = 0
By adding the numbers in each column we get,
= 5 + (– 5) + 0 = 5 – 5 = 0
= (–1) + (–2) + 3 = –1 – 2 + 3 = –3 + 3 = 0
= –4 + 7 + (–3) = – 4 + 7 – 3 = –7 + 7 = 0
By adding the numbers in the diagonals we get,
= 5 + (–2) + (–3) = 5 – 2 – 3 = 5 – 5 = 0
= –4 + (–2) + 0 = – 4 – 2 = –6
Because sum of one diagonal is not equal to zero,
So, (i) is not a magic square
Now, we consider the square (ii)
By adding the numbers in each rows we get,
= 1 + (–10) + 0 = 1 – 10 + 0 = –9
= (–4) + (–3) + (–2) = –4 – 3 – 2 = –9
= (–6) + 4 + (–7) = –6 + 4 – 7 = –13 + 4 = –9
By adding the numbers in each columns we get,
= 1 + (– 4) + (–6) = 1 – 4 – 6 = 1 – 10 = –9
= (–10) + (–3) + 4 = –10 – 3 + 4 = –13 + 4 =−9
= 0 + (–2) + (–7) = 0 – 2 – 7 = –9
By adding the numbers in diagonals we get,
= 1 + (–3) + (–7) = 1 – 3 – 7 = 1 – 10 = –9
= 0 + (–3) + (–6) = 0 – 3 – 6 = –9
This (ii) square is a magic square, because sum of each row, each column and diagonal is equal.
8.Verify a – (– b) = a + b for the following values of a and b.
(i) a = 21, b = 18
Solution:– From the question, a = 21 and b = 18
To verify a – (– b) = a + b
Let us take Left Hand Side (LHS) = a – (– b)
= 21 – (– 18) = 21 + 18 = 39
Now, Right Hand Side (RHS) = a + b
= 21 + 18 = 39
By comparing LHS and RHS we get LHS = RHS
∴ 39 = 39
Hence, the value of a and b is verified.
(ii) a = 118, b = 125
Solution:– From the question,
a = 118 and b = 125
To verify a – (– b) = a + b
Let us take Left Hand Side (LHS) = a – (– b)
= 118 – (– 125) = 118 + 125 = 243
Now, Right Hand Side (RHS) = a + b
= 118 + 125 = 243
By comparing LHS and RHS, we get , LHS = RHS
∴ 243 = 243
Hence, the value of a and b is verified.
(iii) a = 75, b = 84
Solution:–From the question,
a = 75 and b = 84
To verify a – (– b) = a + b
Let us take Left Hand Side (LHS) = a – (– b)
= 75 – (– 84) = 75 + 84 = 159
Now, Right Hand Side (RHS) = a + b
= 75 + 84 = 159
By comparing LHS and RHS, we get, LHS = RHS
∴ 159 = 159
Hence, the value of a and b is verified.
(iv) a = 28, b = 11
Solution:– From the question,
a = 28 and b = 11
To verify a – (– b) = a + b
Let us take Left Hand Side (LHS) = a – (– b)
= 28 – (– 11) = 28 + 11 = 39
Now, Right Hand Side (RHS) = a + b
= 28 + 11 = 39
By comparing LHS and RHS, we get, LHS = RHS
∴ 39 = 39
Hence, the value of a and b is verified.
9.Use the sign of >, < or = in the box to make the statements true.
(a) (–8) + (–4) [ ] (–8) – (–4)
Solution:– Let us take Left Hand Side (LHS) = (–8) + (–4)
= –8 – 4 = –12
Now, Right Hand Side (RHS) = (–8) – (–4)
= –8 + 4 = –4
By comparing LHS and RHS, we get, LHS < RHS
So we get –12 < –4
∴ (–8) + (–4) [<] (–8) – (–4)
(b) (–3) + 7 – (19) [ ] 15 – 8 + (–9)
Solution:– Let us take Left Hand Side (LHS) = (–3) + 7 – 19
= – 3 + 7 – 19 = – 22 + 7 = – 15
Now, Right Hand Side (RHS) = 15 – 8 + (–9)
= 15 – 8 – 9 = 15 – 17 = –2
By comparing LHS and RHS, we get, LHS < RHS
So we get –15 < –2
∴ (–3) + 7 – (19) [<] 15 – 8 + (–9)
(c) 23 – 41 + 11 [ ] 23 – 41 – 11
Solution:– Let us take Left Hand Side (LHS) = 23 – 41 + 11
= 34 – 41 = – 7
Now, Right Hand Side (RHS) = 23 – 41 – 11
= 23 – 52 = – 29
By comparing LHS and RHS, we get, LHS > RHS
So we get – 7 > –29
∴ 23 – 41 + 11 [>] 23 – 41 – 11
(d) 39 + (–24) – (15) [ ] 36 + (–52) – (– 36)
Solution:–Let us take Left Hand Side (LHS) = 39 + (–24) – 15
= 39 – 24 – 15 = 39 – 39 = 0
Now, Right Hand Side (RHS) = 36 + (–52) – (– 36)
= 36 – 52 + 36 = 72 – 52 = 20
By comparing LHS and RHS, we get, LHS < RHS
So we get 0 < 20
∴ 39 + (–24) – (15) [<] 36 + (–52) – (– 36)
(e) – 231 + 79 + 51 [ ] –399 + 159 + 81
Solution:– Let us take Left Hand Side (LHS) = – 231 + 79 + 51
= – 231 + 130 = –101
Now, Right Hand Side (RHS) = – 399 + 159 + 81
= – 399 + 240 = – 159
By comparing LHS and RHS, we get, LHS > RHS
So we get –101 > –159
∴ – 231 + 79 + 51 [>] –399 + 159 + 81
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?
Solution:– i) The monkey is sitting on the top (i.e., the first step)
The position of monkey after the
1st jump monkey will be at step = 1 + 3 = 4 steps
2nd jump monkey will be at step = 4 + (-2) = 4 – 2 = 2 steps
3rd jump monkey will be at step = 2 + 3 = 5 steps
4th jump monkey will be at step = 5 + (-2) = 5 – 2 = 3 steps
5th jump monkey will be at step = 3 + 3 = 6 steps
6th jump monkey will be at step = 6 + (-2) = 6 – 2 = 4 steps
7th jump monkey will be at step = 4 + 3 = 7 steps
8th jump monkey will be at step = 7 + (-2) = 7 – 2 = 5 steps
9th jump monkey will be at step = 5 + 3 = 8 steps
10th jump monkey will be at step = 8 + (-2) = 8 – 2 = 6 steps
11th jump monkey will be at step = 6 + 3 = 9 steps
∴ Monkey took 11 jumps (i.e., 9th step) to reach the water level
Thus, required number of jumps = 11
ii) The monkey’s position is at the water level (the ninth step.)
The position of monkey after the
1st jump J1 is at 5th step ↑
2nd jump J2 is at 7th step ↓
3rd jump J3 is at 3rd step ↑
4th jump J4 is a t 5th step ↓
5th jump J5 is at 1st step ↑
Hence, the required number of jumps = 5
1st jump J1 is at 5th step ↑
2nd jump J2 is at 7th step ↓
3rd jump J3 is at 3rd step ↑
4th jump J4 is at 5th step ↓
5th jump J5 is at 1st step ↑
Thus, the required number of jumps = 5.
iii) Since, the number of steps moved by the monkey towards the top is taken as positive and the number of steps moved towards the water level is taken as negative.
In case (i), we have:
| Jumps | J1 | J2 | J3 | J4 | J5 | J6 | J7 | J8 | J9 | J10 | J11 |
| Number of steps | − 3 | + 2 | 3 | + 2 | -3 | + 2 | 3 | + 2 | 3 | + 2 | 3 |
Therefore (i) Total number of steps
= 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3
= 8 which represents the monkey goes down by 8 steps.
In case (ii), we have:
| Jumps | J1 | J2 | J3 | J4 | J5 |
| Number of steps | + 4 | 2 | + 4 | 2 | +4 |
Therefore (ii) Total number of steps.
= + 4 – 2 + 4 – 2 + 4 = 8
Here, the monkey is going up by 8 steps. Therefore in (b), the sum (+8) will represent going up by 8 steps.
Try these (page 8)
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 greater than both the integers.
e) An integer greater than both the integers.
Solution:– a) (−16) and 9
Sum: (−16) + 9 = −7 which is negative integer.
b) −18 and 18
Sum: = −18 + 18 = 0 which is zero
c) (−6) and (−8)
Sum: (−6) + (−8) = − 14
which is smaller than (−6) and (−8)
d) 4 and 7
Sum: 4 + 7 = 11
which is greater than 4 and 7
e) 19 and 21
Sum: 19 + 21 = 40
40 is greater than 19 and 21
- 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 both the integers. e) An integer greater than both the integers.
Solution:– a) 3 and 7
Difference: 3−7 = −4 which is a negative integer.
b) −9 and −9
Difference: −9 − (−9) = −9 + 9 = 0 which is zero
c) 5 and 9
Difference: 5−9 = −4 which is smaller than 5 and −9
d) 16 and −5
Difference: 16 −(− 5 )= 16 + 5 = 21 which is greater than 16 and −5
e) 15 and 6
Difference: 15 = 15 + 6 = 21
21 is greater than 15 as well as 6 .