Chapter 6: Integers

Integers

Ex- 6.1

1. Write opposites of the following:

(a) Increase in weight

(b) 30 km north

(c) 80 m east

(d) Loss of Rs 700

(e) 100 m above sea level

Solution:

We use the concept of integers to write opposites of the given statements.

a) Increase in weight – Decrease in Weight

b) 30 km north – 30 km south

c) 80 m east – 80 m west

d) Loss of Rs 700 – Profit of Rs 700

e) 100 m above sea level – 100 m below sea level

2. Represent the following numbers as integers with appropriate signs.

 (a) An airplane is flying at a height two thousand meter above the ground.

(b) A submarine is moving at a depth, eight hundred meters below the sea level.

 (c) A deposit of rupees two hundred.

(d) Withdrawal of rupees seven hundred.

Solution:

(a) + 2000

(b) – 800

(c) + 200

(d) – 700

3. Represent the following numbers on number line:

(a) +5

(b) – 10

(c) + 8

(d) – 1

(e) – 6

Solution:

• + 5

•  – 10

• + 8

• – 1

• – 6 

4. Adjacent figure is a vertical number line, representing integers. Observe it and locate the following points:

 (a) If point D is + 8, then which point is – 8?

(b) Is point G a negative integer or a positive integer?

 (c) Write integers for points B and E.

(d) Which point marked on this number line has the least value?

 (e) Arrange all the points in decreasing order of value.

Solution:

(a) If point D is +8, point F represents -8, i.e. 8 points below 0.
(b) Point G is a negative integer as its below 0.
(c) B = +4 and E = – 10
(d) Point E has the least value of -10.
(e) All the points are arranged in decreasing order as: D, C, B, A, O, H, G, F and E.

5. Following is the list of temperatures of five places in India on a particular day of the year. Place                              Temperature

 Siachin                           10°C below 0°C        ……………..

 Shimla                             2°C below 0°C         ……………..

 Ahmedabad                   30°C above 0°C       ……………..

  Delhi                              20°C above 0°C        ……………..          

 Srinagar                          5°C below 0°C          ……………..

 (a) Write the temperatures of these places in the form of integers in the blank column.

 (b) Following is the number line representing the temperature in degree Celsius. Plot the name of the city against its temperature.

 (c) Which is the coolest place?

(d) Write the names of the places where temperatures are above 10°C

Solution:

(c) Siachin is the coolest place.

(d) In Ahmedabad and Delhi, the temperature is above 10℃.

6. In each of the following pairs, which number is to the right of the other on the number line?

 (a) 2, 9 (b) – 3, – 8 (c) 0, – 1 (d) – 11, 10 (e) – 6, 6 (f) 1, – 100

Solution:

(a) 9 is to the right of 2

(b) -3 is to the right of -8

(c) 0 is to the right of -1

(d) 10 is to the right of -11

(e) 6 is to the right of -6

(f) 1 is to the right of -100

7. Write all the integers between the given pairs (write them in the increasing order.)

 (a) 0 and – 7 (b) – 4 and 4 (c) – 8 and – 15 (d) – 30 and – 23

Solution:

(a) Integers between 0 and -7

-6, -5, -4, -3, -2, -1

(b) Integers between -4 and 4

-3, -2, -1, 0, 1, 2, 3

(c) Integers between -8 and -15

-14, -13, -12, -11, -10, -9

(d) Integers between -30 and -23

-29, -28, -27, -26, -25, -24

8. (a) Write four negative integers greater than – 20.

 (b) Write four integers less than – 10

Solution:

(a) Four negative integers greater than -20 are-

-19, -18, -17, -16

(b) Four negative integers less than -10 are-

-11, -12, -13, -14

9. For the following statements, write True (T) or False (F). If the statement is false, correct the statement.

(a) – 8 is to the right of – 10 on a number line.

 (b) – 100 is to the right of – 50 on a number line.

(c) Smallest negative integer is – 1.

(d) – 26 is greater than – 25.

Solution:

(a) T

(b) F; – 100 is to the left of – 50 on number line

(c) F; greatest negative integer is – 1

(d) F; – 26 is smaller than – 25

10. Draw a number line and answer the following:

 (a) Which number will we reach if we move 4 numbers to the right of – 2.

(b) Which number will we reach if we move 5 numbers to the left of 1.

(c) If we are at – 8 on the number line, in which direction should we move to reach – 13?

(d) If we are at – 6 on the number line, in which direction should we move to reach – 1?

Solution:

 (a) 2

(b) – 4

(c) to the left

EXERCISE 6.2

1. Using the number line write the integer which is:

 (a) 3 more than 5

 (b) 5 more than –5

 (c) 6 less than 2

 (d) 3 less than –2

Solution:

(a) 3 more than 5 means adding 3 to 5 which gives us 8.

(b) 5 more than -5 means adding 5 and -5, which gives us 0.

(c) 6 less than 2 means subtracting 6 from 2, which gives us -4 as the difference.

(d) 3 less than -2 means subtracting 3 from -2, which gives us -5.

2. Use number line and add the following integers:

(a) 9 + (– 6)

(b) 5 + (– 11)

(c) (– 1) + (– 7)

 (d) (– 5) + 10

(e) (– 1) + (– 2) + (– 3)

 (f) (– 2) + 8 + (– 4)

Solution:

(a) 9 + ( -6)

First We move 9 steps to the right from 0.

And them, we move 6 steps to the left.

     We reach 3.

Therefore, 9 + (-6) = 3.

(b) 5 + (-11)

First we move 5 steps to the right from 0.

And then, we move 11 steps to the left.

We reach -6.

Therefore, 5 + (-11) = -6.

(c) (-1) +(-7)

First we move 1 steps to the left from 0.

And then, we move 7steps to the left.

We reach -8.

Therefore, (-1) + (-7) = -8.

(d) (-5) + 10

First we move 5 steps to the left from 0.

And then, we move 10 steps to the right.

We reach 5.

Therefore, (-5) + 10 = 5.

(e) (-1) + (-2) + (-3)

Frist we move 1 steps to the left from 0.

And then, we move 2 steps to the left.

And, we move 3 steps to the left.

We reach -6.

Therefore, (-1) + (-2) + (-3) = -6.

(f) (-2) + 8 + (-4)

First we move 2 steps to the left from 0.

And then, we move 8 steps to the right.

And, we move 4 steps to the left.

We reach 2.

(-2) + 8 + (-4) = 2.

3. Add without using number line:

 (a) 11 + (– 7)

(b) (– 13) + (+ 18)

(c) (– 10) + (+ 19)

(d) (– 250) + (+ 150)

(e) (– 380) + (– 270)

(f) (– 217) + (– 100)

Solution:

(a) 11 + (– 7)

= 4

(b) (– 13) + (+ 18)

= 5

(c) (– 10) + (+ 19)

= 9

(d) (– 250) + (+ 150)

= 100

(e) (– 380) + (– 270)

= -650

(f) (– 217) + (– 100)

= -317

4. Find the sum of:

 (a) 137 and – 354

 (b) – 52 and 52

 (c) – 312, 39 and 192

 (d) – 50, – 200 and 300

Solution:

• 137 + (-354)

= 137 – 354

= -217

• -52 + 52

= 0

• -312 + 39 + 192

= – 312 + 231

= – 81

• – 50 + (-200) + 300

= -50 – 200 + 300

= -250 + 300

= 50

5. Find the sum:

 (a) (– 7) + (– 9) + 4 + 16

 (b) (37) + (– 2) + (– 65) + (– 8)

Solution:

• (-7) + (-9) + 4 + 16

= -7 – 9 + 4 + 16

= – 16 + 20

= 4

• (37) + (-2) + (-65) + (-8) 

= 37 – 2 – 65 -8

= 35 – 65 – 8

= – 38

EXERCISE 6.3

1. Find

(a) 35 – (20)

             (b) 72 – (90)

             (c) (– 15) – (– 18)

             (d) (–20) – (13)

             (e) 23 – (– 12)

            (f) (–32) – (– 40)

      Solution:

• 35 – 20

= 15

• 72 – 90

= – 19

• (-15) – (-18)

= -15 + 18

= 3

• (-20) – (13)

= -20 – 13

= – 33

• 23 – (-12)

= 23 + 12

= 35

• (-32) – (- 40)

= -32 + 40

= 8

2. Fill in the blanks with >, < or = sign.

(a) (– 3) + (– 6) ______ (– 3) – (– 6)

 (b) (– 21) – (– 10) _____ (– 31) + (– 11)

(c) 45 – (– 11) ______ 57 + (– 4)

(d) (– 25) – (– 42) _____ (– 42) – (– 25)

Solution:

3. Fill in the blanks.

 (a) (– 8) + _____ = 0

 (b) 13 + _____ = 0

 (c) 12 + (– 12) = ____

 (d) (– 4) + ____ = – 12

(e) ____ – 15 = – 10

Solution:

4. Find

(a) (– 7) – 8 – (– 25)

 (b) (– 13) + 32 – 8 – 1

 (c) (– 7) + (– 8) + (– 90)

(d) 50 – (– 40) – (– 2)

Solution:

• -7 -8 – (-25)

= -7 -8 + 25

= 10

• (-13) + 32 – 8 – 1

= -13 +32 -9

= 10

• (-7) + (-8) + 25

= -7 -8 + 25

= 10

• 50 – (-40) – (-2)

= 50 + 40 + 2

= 92