Integers are divided into several types

- Natural numbers: The collection of all counting numbers are known as natural numbers. It is denoted as N ex: {1, 2, 3, 4, 5}
- Whole numbers: The collection of natural numbers along with zero is known as whole numbers. It is denoted by W ex:{0, 1, 2, 3, 4, 5}

**Properties of addition and subtraction of integers**

**Closure under addition and subtraction:**a and b, a+b, and a-b are integers for every integer.-
**Commutative property for addition:**For every integer a and b, a+b=b+a are integers. **Associative property for addition:**For every integer a, b and c,(a+b)+c=a+(b+c) are integers.

**Properties of multiplication of integers**

- Closure under multiplication: For every integer a and b, axb=I
- Commutative property of multiplication: For every integer a and b, axb=bxa
- Multiplication by zero: For every integer a, ax0=0xa=0
- Multiplicative identity: For every integer a, ax1=1xa=a, here 1 is the multiplicative identity for integers
- Associative property of multiplications: For every integers a, b and c, (axb)xc=ax(bxc)
- Distributive property of integers: For every integer a0 is not defined a1=a ex: (-a)(-3)=2 is an integer
- (-3)(-a)=1/3 is not an integer

There is an elevator which descends into a mine shaft at the rate 6m/min. If that descent starts from 10m above the ground level, then find how long will it take to reach 350m? **Sol:** The initial height of the elevator=10m

**So now we are going to discuss a few problems related to this chapter **

Final depth of elevator =-350m[distance descended is denoted by a negative integer]

The total distance to descended by the elevator =(-350)-(10)=-360m

Then,

Time taken by the elevator to descend -6m=1min

So, time taken by the elevator to descend -360m=(-360)(-60)

=60minutes

=1hour

**So now we are going to give some problems try to solve them which are very easy**

A teacher has conducted a class test for all students for every correct answer she awarded (+3) marks and for every incorrect answer she awarded (-2) marks and no marks for not attempting any question. a) A student radhika scored 20 marks how many questions she had attempted incorrect, if she got 12 correct answers? b) Mohini scored -5marks in this test, though she got 7 correct answers. Then how many she had attempted incorrectly?

- The temperature at 12noon was 100C above zero. Until midnight if it decreases at the rate of 20C per hour, so at what time would the temperature be 80C below zero. Find the temperature at midnight?
- Write any five pairs of integers (a, b) such that ab=-3. One such pair is (6, -2) because 6(-2)=(-3)
- Fill in the blanks
- 369________=369
- (-75)_______=-1
- (-206)_____=1
- -87______=87
- _______1=-87
- ______48=-1
- 20______=-2
- ______(4)=-3

**Solutions:**

** 2) Marks awarded for 1 correct answer =+3**

Marks awarded for 1 wrong answer =-2

- Radhika scored 20 marks

Then,

Total marks awarded for 12 correct answers=12×3=36

Marks awarded for every incorrect answers=Total score-Total marks awarded for 12 correct answers

=20-36

=-16

So, the number of incorrect answers made by radhika=(-16)(-2)=8 - Mohini scored -5marks

Then,

Total marks awarded for 7 correct answers=7×3=21

Marks awarded for every incorrect answers=Total score-Total marks awarded for 12 correct answers

=-5-21

=-26

So, the number of incorrect answers made by Mohini=(-26)(-2)

=13 **Temperature at the beginning at 12noon=100C**

Rate of change of temperature= -20C per hour

Then,

Temperature at 1 PM=10+(-2) =10-2=80C

Temperature at 2 PM =8+(-2)=8-2=60C

Temperature at 3PM =6+(-2)=6-2=40C

Temperature at 4PM =4+(-2)=4-2=20C

Temperature at 5PM =2+(-2)=2-2=00C

Temperature at 6PM =0+(-2)=0-2=-20C

Temperature at 7PM =-2+(-2)=-2-2=-40C

Temperature at 8PM =-4+(-2)=-4-2=-60C

Temperature at 9PM =-6+(-2)=-6-2=-80C

Therefore, at 9PM the temperature will be 80C below zero

Then,

The temperature at midnight is 12 PM

Change in temperature in 12 hours=-20Cx12=-240C

So, at midnight temperature will be=10+(-24)=-140C

So, at midnight temperature will be 140C below 0- (15, -5)

Because, 15(-5)=(-3)

(-15, 5)

Because, (-15)(5)=(-3)

(18, -6)

Because, 18(-6)=(-3)

(-18, 6)

Because, (-18)6=(-3)

(21, -7)

Because, 21(-7)=(-3) - a) Let us assume the missing integer be x,

=369x=369

=x=(369/369)

x=1

3691=369 - Let us assume the missing integer be x,

=(-75)-x=-1

x=(-75/-1)

x=75

=(-75)-75=-1 - Let us assume the missing integer be x,

(-206)x=1

x=(-206/1)

x=-206

=(-206)(-206)=1 - Let us assume the missing integer be x,

(-87)x=87

x=(-87)/87

x=1

=(-87)(-1)=87 - Let us assume the missing integer be x,

(x)1=-87

x=(-87)x1

x=-87

=(-87)1=-87 - Let us assume the missing integer be x,

(x)48=-1

x=(-1) x48

x=-48

=(-48)48=-1

- a) Let us assume the missing integer be x,
- Let us assume the missing integer be x,

20x=-2

x=(20)/(-2)

x=-10

=(20)(-10)=-2 - Let us assume the missing integer be x,

(x)4=-3

x=(-3) x4

x=-12

=(-12)4=-3

**Conclusion**

These are some of the problems related to this chapter which are very easy to solve. These are very important and are very useful in many kinds of competitive exams. Maths is a very easy subject, no need to get panic if we understand the concept and learn the formulas perfectly it’s a very easy subject.