Hello everyone, welcome back to a new video.
In the last video, we have done the average.
Today, we will do the cases in the next video.
If you have not seen the video on average, then please
watch it first and then watch this video.
So, as I told you in the last class, average is sum of all
the observations upon total number of observations.
Sum is denoted by S.
Average is denoted by A.
A is equal to S upon n.
n is the number of observations.
So, we will do the cases related to average.
In the last video, we have done that if we add all the
observations with a particular number.
Means, if we add all the observations with a particular
number, subtract, multiply or divide.
We have done this much and then we have done a small
question on average.
Today, what we will do is, suppose we have some data.
In that data, we were not doing anything.
We were just adding, increasing or decreasing the marks.
We were not doing anything in the number of observations.
But, let us suppose, now what we will do is, if we make
some changes in the observations. Means, whatever is the
total number of observations, we will make some changes in
it. What will happen to it?
Because of which, what will happen in our average?
There will be a difference.
So, how will we do those questions? What formula will we
use for them? We will see that today.
So, first of all, what do we have?
We have a data.
In that data, what do we have? Suppose, we have n
observations.
We have n number of observations.
If we have n number of observations, then what will be the
total number of observations in it?
It will be n only.
Because we have n number of observations.
So, first one is x1, second one is x2, third one is x3 and
so on till n.
Now, what we have to do is, suppose, their average will be
a.
What will happen to it?
Their sum will be a.
a would be x1 plus x2 plus xnth sum divided by n.
But, what we have to do is, we have to replace one of these
observations.
Suppose, we have this x2 observation.
We have to replace it with another one.
Either, we have to remove it.
Or, we have to bring one more member in it.
Suppose, we have to bring one more member in it.
xn plus 1.
Then, what will be the difference in our average?
Suppose, we remove this xn member.
Then, what will be the difference in our new average?
Or, we replace something.
Suppose, we replace x2 with x100.
Then, what did we replace x2 with?
We replaced it with x100.
Then, what will be the difference in our new average?
What will be the increment and decrement?
We will learn that in today's class.
And, we will also do related questions.
So, suppose, we have a set of group.
Suppose, we have a set of group.
Now, I am giving you an example of a group.
I am writing all these cases.
But, you can use these cases in anything.
For example, you can use these cases in marks.
You can use these cases in age.
You just have to be clear about the concept.
That, if we remove one observation.
Either, we remove one observation from the data.
Or, we bring a new observation.
If we remove one observation from the data.
For example, we have 10 observations.
If we remove one observation from the data.
Then, we have 9 observations left.
So, there will be some difference in our average.
So, we will see that.
If we bring one more observation of x11.
Then, we have total observation of x11.
Then, we will have difference in our average.
If we are replacing x10 with x100.
Then, there will be some difference in our average.
There can be increment or decrement.
It can increase or decrease.
So, whatever similar questions are there.
You can correlate them with these questions.
And, apply the formula.
So, now we will see.
I am talking about the group.
So, suppose.
If we have first case.
When a person joins the group.
If we have data or a group.
Then, one observation is added.
So, in the case of group.
Suppose, a person joins a group.
Means, it is added.
Then, what will happen?
What will be the difference in our average?
If we already have data.
In that, a new member is added.
A new observation is added.
Then, what will be our average?
It can be same or increase or decrease.
It will be same in very less cases.
But, what will happen?
It can be increase or decrease.
If it is same, then there is no problem.
But, let us suppose.
We have two categories.
That is, first category is.
If we have.
What will be the average?
It will be decrease.
If average is decreased.
So, first case will be.
If new average.
New average will be whatever will come.
If new average is increased.
Means, what is happening in new average?
If we are having increment in new average.
Then, what can we find out?
First case will be this.
What will be the second case?
If new average is decreased.
Means, what is happening in new average?
It is less than the first average.
So, it is decreasing.
So, we have these two cases.
Either it is increasing or decreasing.
So, let us suppose.
We had old average A.
And, suppose new average is increased by 2.
So, A plus 2.
So, what will be the increment?
In this case, if we have.
Let me write here.
If old average was A.
New average is A plus 2.
So, what is the increment in this?
Increment is only 2.
Increment is 2 in average.
So, we will call this as increment in average.
So, we will use this.
So, what are we finding out in this?
Our formula is.
Suppose, when a person joins the group.
So, what do we have to find out?
We have to find out age.
Either it is age question.
I am relating it to age question.
If we have age question.
How will we understand this?
We will understand it by age question.
Suppose, we have a set of data.
In that, we have given some ages.
We will sum all those ages.
Divide it by total members.
So, what will we get?
Average age of that group.
Now, if new member joins.
So, he will also have an age.
And, if he joins.
So, number of members will also increase.
So, what will be the difference?
So, we will find out.
Age of new member.
The formula to find out age of new member.
Either his age can be given.
So, you can see this in co-relate question.
Whatever question you get.
What you have to find out.
In that, the formula will be used.
So, age of new member would be.
What will be the age of new member?
Previous average.
Increment.
See, what is the case of this? Means, new average is
increased.
This will be given in the question.
New average is increasing so many times.
Or, it is decreasing.
All this will be given.
So, in previous age.
Minus.
Remove the bracket from here.
Remove the bracket from previous.
What will be this?
Previous age.
Minus.
Remove the bracket.
Increment.
In average.
Means.
Previous average.
average the total number of people including the new
joinees into total number of people including new joinees
into total number of people including new joinees into
total number of people including new joinees into total
number of people including new
joinees into total number of people including total number
of people including new joinees into total number of people
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new joinees into total number of people including new
joinees into total number of people including new joinees
into total number of
people including new joinees into total number of people
including if new average is increased if there is an
increment in average if new average is same is decreased
means there is a decrement in the number of people
including new joinees into total
number of people including new joinees into total number of
people including new joinees into total number of people
including new joinees into total number of people including
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of people including new join average into remaining people
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remaining two means if we are taking average runs of a
cricketer what is that? means he is
giving 32 runs in 10 innings
32 runs in 10 innings so how many runs must he make in his
next inning? means in new match how many runs should be
made? so as to increase so as to increase what to do?
increase its average runs by 4 what we have to do is
increase its average by 4 just
now I told you that join and leave group
we will try to relate the same question to the previous
average.
What is previous average?
For 32-10 innings.
Which means, previous average is 32.
and till now how many members were there in the starting of
the group suppose there is n, so new person used to join in
that but now our innings, means how many we have played
means now presently, whose previous average has come out in
that our n is how
much? he has played 10 innings so till now he has played 10
innings now we will add one more inning in this because we
have to tell about next inning so this is the case of joins
when a new member was joining in that so in this also we
have to add one
inning so now we will take out for new inning means which
case is this?
A part, joins the group case and what we have to do? what
we have to increase the average? we have to increase the
average so whose case is this? this is the first case of A
part
A part is being added, joined, new inning is being played
and increment should be there in new average so we will use
this formula so do it previous average is this and till now
our n is 2 now our new n will be we have to find out for
11th inning so so
as to increase the average by 4 if we have new average what
we have to do for new average? we have to increase 4 in
previous average that is 32 plus 4 so this is our new
average but what we want? we want increment we have to do
for these two averages
we have to see the difference if there is decrement then
see decrement still we have to see the difference just
subtract the small one from the big one so what will
happen? till now previous one was small new one was big so
the average value is increasing
so our increment will be in this and that was given to us
we had to find increase in average so increment will be 4
which is given to us we have not found out this this is
already given to us so increment is given previous average
is given total number
is also given earlier it was 10 now new innings total
number of innings is total number of innings what we used
to do? we used to find out total number of members
including new joinee we used to include that so in this
also we have to find out for new
innings we have to include that so now total number of
innings is 10 plus 1 that is this was our 11th innings so
now what we will do instead of that? we will use 11 so just
put up the formula so what we have to see? how many runs
must he make in his next
innings? means in new age formula we have to find out his
new runs so you apply the formula of that formula will be
applied directly so we require runs how many runs must he
make? so write runs runs he should make this will be equal
to previous average
same, we will do the formula of new age age of new member
means we are looking at new runs so previous average
previous average is how much? 32 in this question previous
average is 32 in case of increment so we will write this
formula as plus plus increment
is from how much? increment is from 4 4th increment into 4
will be total number of people including new joiners so
total number of innings from 10 to 11 so into 11 so what is
this? 32 plus 44 that is 676 so what we have how many runs
he should make? he
should make 76 runs in 11th innings in order to have
average increment 4 means if we are increasing average
increment 4 so he has to make this many runs in next
innings ok so with whom we relate this? we relate with that
formula so we have done this easy
in your mind you should have very clear formula ok next
move on to what we have? C part
C part A B we have left the group and joined the group
A part was join
B part was left the group now we have new joiner but old
joiner is also leaving so what is that? we have replaced
means we have replaced one joiner with another joiner so
how we will find new joiner's age? we will see in this so
what we will see before
starting this case you have made a mistake in last formula
what is that? if we were doing B part you also didn't tell
me how will you tell? you are on that side and I am on this
side but you didn't see when a person leaves the group then
how we will find
new member's age?
I also forgot while teaching are you seeing? so what will
happen? age of the member who will we find the age of? age
of the member just correct this case age of the member
who left the group similarly this one will also change what
will happen? age of member who left the group new member is
not joining so why will we talk about him? we will talk
about the person on left so age of the member who left this
is the formula
there is no change in the formula the formula is same now
come to third case on replacement case that is when a
person replaces another person means we are taking one
observation from the data and putting another observation
instead of it there will be
two changes in the averages there will be two changes
either the average will increase one case will increase and
the other case will decrease if the average will decrease
so there will be two cases similarly like the last case
like AB so when a person
replaces another person in this case total number of
observation was also increasing or decreasing but in this
case total number of observation or total number of persons
in the group is same because one is going and the other is
replacing so they will
be equal so what will be their formula? if we have to find
age of new member new member means new person or new member
would be age of new person would be age of the member who
left the group who left the group now this is an increment
so put a plus sign
plus bracket now what is happening? increment in average
into number of people now number of people before replacing
or after replacing will be same because one is replaced so
into number of people will be same age in average.
So, age of new member, age of new member would be age of
left member or plus or minus, minus decrement in average
into total number of people.
Easy. It is also related to them.
So, now see what we will do related to this.
We will do one question so that it will be clear.
See this question.
Read it.
Out of 10 teachers, out of 10 teachers, how many teachers?
10 teachers.
Out of 10 teachers of a school, one teacher retires, one
teacher what happens? Retires and at his place, at his
place what comes? New teacher comes, whose age is 25 years.
One retires, so he brings the young generation that is 25
years old teacher.
The average age of the teacher, whoever would have been the
teacher earlier, their average age would have been, means
we would have taken sum of their age and what would have
been? Divided by 10.
We have 10 teachers.
Suppose we have 10 teachers, 1, 2, up to 10 teachers.
Everyone would have a particular age.
Suppose A1, A2 and so on
A10 would be the age. So, what would be their average? Sum
all these ages divided by 10.
When was this? When one of our teachers was a retired
teacher.
Now what we have? Suppose the teacher was a retired
teacher, teacher number 10, don't give him, give him 1.
Poor guy is also retiring.
So, give him number 1.
So, who are we replacing A1 with?
Let us suppose, at his place, a new teacher has come.
Suppose she has come.
So, what are we doing at her place? Instead of A1, we have
a new member, 11th, whose age is 25. The retired member
would have more age.
Now, she has 25.
So, what will be the new average?
Now, instead of their age, we will keep their age 25 and
then we will sum with the age of other 9 teachers.
And who will we divide? Even now, we will divide by 10
because even now, we have total number of teachers, which
is 10 only.
So, now we have to find out the average age of the teacher
is reduced by 3 years.
Means, the new average is less than the old average. If it
is reduced, then whose case will be formed? It will be a
case of decrement.
Reduced by 3 years. The age of retired teacher is.
So, the retired teachers who have gone, their age has been
asked. We have to tell that.
So, how will we do? If we apply this formula, this one, age
of new member, because we have to take the case of
decrement, second case.
So, what will happen in the case of decrement?
Using the formula, age of new member, new teacher is equal
to age of old member.
Means, all the old members have become old.
Age of the teacher left, who left?
Means, the retired teachers. Age of the teacher who left.
And in the case of decrement, there is minus sign. Minus,
in bracket, decrement in average.
How much is the average reduced? By 3 years.
So, 3 decrement into, how many number of teachers are there?
There were 10 earlier, still there are 10. So, into 10.
So, what is this? Age of new teacher is given, 25 years.
So, 25 is equal to, we have to find out the retired
teacher. So, consider it as x.
x minus 3 into 10, that is 30.
So, what is the value of x?
We have the value of x.
It will be plus here, 25 plus 30, that is 55.
So, what is the age of our retired teacher? 55 years.
Its answer is 55 years. That's why he retired.
So, I hope you have understood the cases of A, B, C.
You have understood the subparts of increment and decrement.
And how I have applied these age questions in cricket
questions.
And similarly, you can apply it in other questions as well.
And if you are able to try and practice, if you are not
able to, then don't worry, there will be more formulas of
average.
We will do it from them.
Thank you for watching the video.
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