Integers

NCERT Exercise 1.1

Question 1: Following number line shows the temperature in degree celsius (°C) at different places on a particular day.

Picture taken from NCERT Book

Observe this number line and write the temperature of the places marked on it.

Answer:

Place	Temperature (°C)
Lahulspiti	-8
Srinagar	-2
Shimla	5
Ooty	14
Bengaluru	22

(a) What is the temperature difference between the hottest and the coldest places among the above?

Answer: Temperature at hottest place = 22°C

Temperature at coldest place = -8°C

Difference `= 22°C – (- 8°C) = 22°C + 8°C = 30°C`

(b) What is the temperature difference between Lahulspiti and Srinagar?

Answer: Difference of temperature between Lahulspiti and Srinagar

`= 5°C – (- 8°C) = 5°C + 8°C = 12°C`

(c) 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: Temperature at Srinagar is in negative and temperature at Shimla is in positive. Sum of temperatures at two places

`= 5°C + ( - 2°C) = 5°C - 2°C = 3°C`

It is clear that sum of temperatures at Srinagar and Shimla is less than the temperature at Shimla. But the sum is more than the temperature at Srinagar.

Question 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: Total marks

`= 25 – 5 – 10 + 15 + 10`
`= 25 + 15 + 10 – 5 – 10`
`= 50 – 5 – 10`
`= 45 – 10 = 35`

Question 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 `= - 5°C`

Temperature on Tuesday `= - 5°C - 2°C = - 7°C`

Temperature on Wednesday `= - 7°C + 4°C = - 3°C`

Question 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?

Picture taken from NCERT Book

Answer: Vertical distance between plane and submarine `= 5000 m + 1200 m = 6200 m`

Question 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.

Answer:

Deposit	Withdrawal
2000	1642
Balance	2000 - 1642 = 358

Balance after withdrawal = Rs. 358

Question 6: 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?

Picture taken from NCERT Book

Answer: Distance traveled towards west will be represented by negative integer.

Final position = Distance to east + Distance to west

`= 20 km + (- 30 km)`
`= 20 km – 30 km = - 10 km`

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

Question (i)

5	-1	-4
-5	-2	7
0	3	-3

Answer: Sum of row 1 `=5-1-4=0`

Sum of row 2 `=-5-2+7=0`

Sum of row 3 `=0+3-3=0`

Sum of column 1 `=5-5+0=0`

Sum of column 2 `=-1-2+3=0`

Sum of column 3 `=-4+7-3=0`

Sum of diagonal Left to Right `=5-2-3=0`

Sum of diagonal Right to Left `=-4-2+0=-6`

In this case, sum of numbers in one diagonal is different than other sums. So, this is not a magic square.

Question (ii)

1	-10	0
-4	-3	-2
-6	4	-7

Answer: Sum of row 1 `=1-10+0=-9`

Sum of row 2 `=-4-3-2=-9`

Sum of row 3 `=-6+4-7=-9`

Sum of column 1 `=1-4-6=-9`

Sum of column 2 `=-10-3+4=-9`

Sum of column 3 `=0-2-7=-9`

Sum of diagonal Left to Right `=1-3-7=-9`

Sum of diagonal Right to Left `=0-3-6=-9`

In this case, sum of numbers in each row, each column and each diagonal is same. So, this is a magic square.

Question 8: Verify `a – (– b) = a + b` for the following values of a and b.

(i) `a = 21`, `b = 18`

Answer: `a – (- b) = 21 – (- 18)`

`= 21 + 18 = 39`

(ii) `a = 118`, `b = 125`

Answer: `a – (- b) = 118 – ( - 125 )`

`= 118 + 125 = 243`

(iii) `a = 75`, `b = 84`

Answer: `a – (- b) = 75 – ( - 84)`

`= 75 + 84 = 159`

(iv) `a = 28`, `b = 11`

Answer: `a – ( - b) = 28 – ( - 11)`

`= 28 + 11 = 39`

Question 9: Use the sign of >, < or = in the box to make the statements true.

(a) `(– 8) + (– 4)□ (–8) – (– 4)`

Answer: LHS `= - 8 – 4 = - 12`

RHS `= - 8 – 4 = - 12`

Hence, LHS = RHS

(b) `(– 3) + 7 – (19) □ 15 – 8 + (– 9)`

Answer: LHS `= - 3 + 7 – 19 = 4 – 19 = - 15`

RHS `= 15 – 8 – 9 = 7 – 9 = - 2`

Hence, LHS < RHS

Answer: LHS `= 23 – 41 + 11 = - 18 + 11 = - 7`

RHS `= 23 – 41 – 11 = - 18 – 11 = - 29`

Hence, LHS > RHS

(d) `39 + (– 24) – (15) □ 36 + (– 52) – (– 36)`

Answer: LHS `= 39 – 24 – 16 = 15 – 15 = 0`

RHS `= 36 – 52 + 36 = 72 – 52 = 20`

Hence, LHS < RHS

(e) `– 231 + 79 + 51 □ –399 + 159 + 81`

Answer: LHS `= - 239 + 79 + 51 = - 239 + 130 = - 109`

RHS `= - 399 + 159 + 81 = - 399 + 240 = - 159`

Hence, LHS > RHS

Question 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?

Answer: Distance covered in first jump = 3 steps

Distance covered in second jump = - 2 steps

So, distance covered after every two jumps = 3 – 2 = 1 step

After 6th pair of jumps the monkey will reach the 7th step.

As 9 – 7 = 2

So, monkey needs to cover two steps now.

In the next jump, monkey will reach the ninth step.

So, monkey needs to jump `2xx6+1``=12+1=13` times to reach the ninth step.

(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?

Answer: In this case, monkey covers 2 steps after every two jumps.

At this rate, monkey would reach 7th step (from bottom) after 6th jump.

Then he needs to cover just two steps which can be easily covered in the next jump.

So, number of jumps required to reach the top step = 6 + 1 = 7

(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`

Answer: `- 3 + 2 – 3 + 2 – 3 + 2 – 3 ``+ 2 – 3 + 2 – 3 + 2 – 2 = 8`

(b) `4 – 2 + ... = 8`.

Answer: `4 – 2 + 4 – 2 + 4 – 2 + 2 = 8`

Answer: In (ii), the sum 8 represents going up by eight steps.