Hello everyone, welcome back to another video.
Today we are going to do the formulas of compound interest.
In the last video, we saw a couple of formulas of simple
interest.
Mainly there was only one, PRT divided by 100.
And then we did a couple of questions on the basis of that.
Today what we are going to do is a little tough
comparatively of simple interest.
That is compound interest.
I had told you the difference between compound interest and
simple interest in the first video.
So what is compound interest?
The interest which is above the interest is our compound
interest.
The principal value in this, changes after some time.
And what will that change be?
The amount which we used to get, like the simple interest
which we got for one year, that we used to get in
principal, we used to get the amount.
So that amount will be our principal value for the second
year.
Then the interest which we will calculate in that, we will
not do that on the first principal value, but on the new
principal value, that is the amount which we got after one
year, we will calculate the interest on that.
So first of all, let us see what is the formula of compound
interest.
Before finding the formula of compound interest, let us see
the formula of the amount which was from A.
So the formula of amount will be P into 1 plus R by 100
whole power T.
We had taken capital T, so let us take capital.
So what is this?
Amount is equal to P for principal value,
R is our rate of interest per annum, and this is our time
in years.
These are the names of all these terms.
This is our amount.
So what will be the amount?
Similarly, amount was our SI plus principal.
So what is SI?
Amount minus principal.
Similarly, what will be CI?
CI would be amount minus principal.
This would be our compound interest formula.
That is CI minus principal value.
Amount minus principal, that would be our compound interest
formula.
So first of all, we will find out the amount.
What we used to do in the last simple interest formula?
We used to find out the simple interest first, then we used
to find out the amount.
In this formula, we have the formula of amount.
We are finding out the amount first, then what will be CI?
Amount minus, we will minus the principal value from it.
So that will be our total compound interest for the T years.
Now, in the first year, the simple interest will be the
same compound interest.
In the second year, our simple interest was same.
Every year, constant value was same.
But compound interest used to increase.
So the compound interest will always increase with our
simple interest.
Or it will be equal to our simple interest.
So this is our compound interest formula.
Now, the cases that come on the basis of this formula are
Compounded half yearly.
This was our simple interest formula when rate of interest
was per annum.
But in some cases, we get compounded half yearly.
Or our interest is compounded half yearly.
Means, in half year, interest is added on interest.
In half year, interest is changing.
Our principal will change and new principal value will come.
This is happening on half yearly basis.
Similarly, it can happen on quarterly basis.
So, to do that type of things,
I will tell you a simple formula.
You can also remember this formula.
Now, what is n here? n is the number of cycles in one year.
What is n? n is the number of cycles in one year.
In one year.
What are the cycles?
Let me tell you.
Suppose, the letter is given to you.
Compounded half yearly.
So, this 8% rate of interest was of per annum.
So, what will be the rate of interest of half year?
So, the value of amount will be
P1 plus half yearly.
So, we will divide r by 2.
Because, r is for per annum.
So, divided by 2.
Divided by 100.
So, it becomes 200.
What is the power?
Now, when the rate of interest was per annum.
So, the rate was half.
So, what we have to do with time?
We have to double the time.
Because, in one year, if we are looking at half yearly.
So, in one year, 6 months.
How can we divide one year in 6 months?
Divide it into two parts.
2, 1, 6 months.
1, 6 months.
Then 6 months.
So, these two together becomes 2.
So, for this, we used to divide time by 2.
Okay? We divide 8% by 2.
And, what do we do with time period?
We multiply it by 2.
So, this is nothing.
In this formula, if we multiply value of n by 2.
In half yearly, if we multiply value of n by 2.
And, put it in this formula.
Then, we will get the amount of half yearly.
Similarly, if we have compounded quarterly.
Quarterly.
Means, what will be the value of n? 4.
Quarterly means, gap of 3 months.
So, how will it be 3 months?
It will be divided into 4 parts.
Now, we have one year.
It is divided into 4 parts.
So, what will be the amount in this?
If we have 8% per annum.
Means, the rate of interest of the whole year.
So, for 3 months, we have to divide it into 4 parts.
Now, what will be the value?
R by 4.
So, this is the principle.
Plus, R by 4.
Divided by 100.
And, what will be the whole power?
If we have to divide quarterly in 4 years.
So, we will multiply it by 4.
Because, it is divided into 4 portions.
So, it will be multiplied by 4t.
So, if you would have put the value of n in this formula.
If you would have put the value of n as 4.
So, you would have got this formula of compounded quarterly.
So, you can remember this formula.
Or, you can remember this basic topic.
That, if we are dividing our R percentage.
If we are dividing by 2 in half yearly.
So, we will multiply time by 2.
Similarly, if we are dividing R by 4 in quarterly.
So, we will multiply time by 4.
So, this is the formula of compounded half yearly and
compounded annually.
Now, when the amount has come.
So, obviously, you will get the CI.
How will you get the CI? If you will do the amount minus
principle.
So, you will get the CI.
Now, we have done a case in simple interest.
That, if you have different rate of interest for different
time periods.
So, it can be the same in compounded interest.
That, your rate of interest is different for different
years.
For first year, it is different.
For second year, it is different.
For third year, it is different.
So, what will be the formula of compounded interest?
When we have different rate of interest for each year.
What is the rate of interest for each year? It is different.
Each year or every year.
So, R1 is the rate of interest for first year.
first year rate of interest, R2% is our second year rate of
interest, similarly R3% will be our third year rate of
interest and so on.
So, then what will be our amount?
We don't have to do anything.
We have to do the same.
The principal value will remain the same.
Then 1 plus R1 by 100.
Whole power.
If it is for one year, then whole power will be 1.
We will do into 1 plus R2 by 100.
Whole power.
In this also 1 will be there because we are looking for one
year.
The difference between these two is one year.
So, we are looking for one year.
Into R3 by 100.
Whole power 1.
So, this will be your amount.
This is the formula of the amount.
When we have different rates of interest for each year.
Okay? Note this down. Next, we will move on to a new topic.
That is, if we have been given a difference between CI and
SI.
We can also get such cases.
When we have been given a case of difference.
That is, what is there in compound interest and simple
interest?
There is a difference.
Their subtraction is given.
So, obviously, compound interest will always be greater
than simple interest.
So, what difference will be given to us?
CI minus SI.
Or we will have to subtract it.
So, the cases that we have given.
Cases of difference.
Difference of CI and SI.
Now, the first year, when it is of one year.
So, what did I tell you?
In the first year, both simple interest and compound
interest are equal.
So, if both are equal, then their difference will be zero.
Now, when it comes to the difference between CI and CI.
That thing starts coming from where?
In the second year.
That is, in the second year, what will happen to you?
Compound interest and simple interest will be different
from each other.
If they are different, then we can subtract them.
In the first year, it was zero.
So, what is there in this?
In two years.
So, who are we talking about?
We are talking about the two-year case.
So, what will be the difference?
So, what you can do?
You can remember this formula directly.
But, let me tell you a little proof of this.
If you do not remember the formula.
And basically, you remember the formula of CI and SI.
Then you can also get it from them.
But, what happens in the exam?
We do not have so much time in the exam.
That we can solve the proof completely.
But, if we have time and we do not remember the rest of the
things.
We have time for this.
Then we can do this.
Otherwise, you will have to remember the formula.
Okay? So, if we subtract CI for two years.
So, what is CI?
CI will come to us.
P.
Sorry.
Not CI.
Whose formula is this?
First, we have to subtract the amount for CI.
So, the formula for the amount will be P.
1 plus R by 100.
Whole power.
For two years, the value of T has become 2.
Whole power 2.
T.
So, what is the simple interest?
Amount minus principle.
That is P.
1 plus R by 100.
Whole power 2.
Minus P.
This would be our formula.
Now, solve this.
How will we solve this?
So, open the square root.
So, P.
1 plus.
A plus B whole square.
So, what is the property of A?
A square plus B square plus 2AB.
So, A becomes 1.
And this becomes A.
And this becomes B.
So, what is this?
Your A.
A square plus B square.
Plus 2AB.
Minus.
Minus P.
So, open this individually.
Or leave it as it is.
Now, what are we solving next?
For SI.
Value of SI.
What is the value of SI?
It is simple.
PRT by 100.
We have been given T for two years.
So, PR.
Into 2.
2 by 100.
Now, we have to find the difference between the two.
What do we have to find?
Difference.
That is CI minus SI.
So, if we minus it.
First, let us make this smaller.
So, what will this be?
P plus PR by 100 whole square.
Plus 2PR by 100.
Minus P.
We have distributed P individually.
We have multiplied it by every term.
P cancels with P.
P minus P.
Now, we have only these two terms.
So, if we divide both by CI minus SI.
Then, we will get P.
R by 100 whole square.
Plus 2PR by 100.
Minus SI.
So, what will we get after this?
Minus 2PR by 100.
Now, both of these are negative.
So, both of these will cancel out.
What is left with us?
CI minus SI would be PR by 100 whole square.
This will be the formula for the difference between CI and
SI in two years.
You can remember this directly.
How can you remember this directly?
What was the amount?
P1 plus R by 100 whole square.
If you subtract 1 plus.
What will be left?
R by 100 whole square.
You can remember it like this.
As you remember the formula.
What was the proof?
It was simple.
In the formula, we subtracted both CI and SI.
Now, what will be the result of the same case?
The same case can be of three years.
If it is not of two years.
Now, what is the difference of three years? So, what will
be the result of case 2? Case 2.
CI minus SI.
Note the difference.
In three years.
We are subtracting for three years.
So, what will be the result for three years?
CI.
What is it?
Amount minus principle.
So, what is the amount?
The formula for the amount is P1 plus R by 100 whole power
T.
What is T? 3.
So, 3 minus P.
This is our CI.
What will be SI?
PRT.
What is T? 3 years.
So, PR into 3.
So, 3 PR divided by 100.
Okay? Subtract both.
See what is coming.
So, CI minus.
Solve CI.
So, P.
P.
How can you solve this?
Apply the property of cube.
Whole cube of A plus B is equal to.
A cube plus B cube plus.
3 AB A plus B.
Or you can open the square one.
And multiply it by 1 plus R plus 100.
We can write it as 1 upon R plus 100 whole square.
Into 1 plus R by 100.
Generally, children do not remember the property of cube.
So, either you will remember.
Because this is very important.
But if you do not remember.
Then you can solve it by shortening it.
Okay? You should focus more on this method. Rather than
using the formula.
Because in exam, you do not know.
When you forget the formula.
So, you can use the step by step.
Which you have studied earlier.
How can you do that?
That also you should learn.
Mainly in the practice session.
When you are practicing.
Okay? So, what will be this?
P.
A square plus B square.
Plus 2 AB.
Into 1 plus R by 100.
Minus P.
Now P.
Multiply 1 by the whole bracket.
So, 1 plus R by 100.
Now multiply this by the whole bracket.
R square by 100.
So, this would be R square by 100.
R by 100 whole square.
Plus R by 100 whole.
It will be a cube.
Plus 2 R by 100.
So, this will be.
Plus 2 R by 100.
Plus 2 R by 100 whole square.
2 R by 100 whole square.
Minus P.
This P is in minus.
Now, this is our C.I.
What do you have to do with Ci?
You have to subtract it. So, Ci minus Si, do it in this.
So, put this value in minus Si.
Minus 3Pr by 100.
Okay? So, now you multiply P individually with all the
terms inside.
So, what will you get?
You will get this.
If you are taking out the difference, then P, P plus Pr by
100.
You solve it yourself and match the final answer.
Plus P, R by 100 whole square.
Plus P, R by 100 whole cube.
Plus 2Pr by 100.
Plus 2Pr by 100 whole square.
Minus P and minus 3Pr by 100.
Okay? Now, this P minus P cancel out.
Now, which of the same things are seen from them?
Let's see the term of R square.
R upon 100 square.
One term is this.
The other term is this.
So, what is this? 3P, 2 plus 1, 3.
3Pr by 100 whole square.
Now, let's see the cube one.
So, we have this one.
And we have this one.
This is not a cube.
There is only one cube.
That is, we have only this cube.
So, this will be plus P, R by 100 whole cube.
Now, let's see the term of Pr.
The term of Pr is this.
1 plus 2, 3.
And this is minus 3.
So, these two together will cancel out.
Now, we have only this result.
That is, 3P into R by 100 whole square plus P, R by 100
whole cube.
What is common in this?
P is common.
And R square by 100 square is common.
Whole square is common.
What is left in the bracket? 3 plus R by 100.
So, what is this?
CI minus SI formula for 3 in 3 years.
What is the formula?
PR by 100 whole square into 3 plus R by 100.
So, you have to remember this directly.
You have to remember this formula directly.
Basically, the formula of compound interest will be related
to population.
Let's see the case of population.
I will tell you some formulas in this video.
In the next video, we will do questions on these formulas.
So, what is the case of population? Just like our amount is
made with the help of principal and interest.
Similarly, we are getting the new amount with the help of
the previous income.
So, let's suppose our population is P in starting.
Then, what will be the next year? Suppose, it is P1 in
starting.
Starting population.
Or population in start.
After some time, in few years, that is T years, our
population will increase.
There was a rate of interest.
But, there is no interest from any rate or rate percent.
So, from any rate, our population will either increase or
decrease.
So, let's take 8% from any rate.
Either, it will increase or decrease.
Now, it will increase or decrease with this percentage.
Then, in T years, the population which has increased or
decreased at this rate, that is our new population, P2.
So, this is our present population.
Earlier, our population was P1.
So, what was the present money that we used to get?
It was the amount that we got back.
Earlier, the initial money was principal.
So, we will go on the same scale.
So, our initial population will play the role of principal.
The present population will play the role of amount.
And, T and R percent are same.
Now, there is a case of increment or decrement.
In that case, there was no increment.
In that case, your CI will increase.
So, in this case, either your increase or decrease is from
R percent.
So, two cases are formed.
One is the case of increase, one is the case of population
increase and one is the case of population decrease.
So, let us see the case of increase.
What was the formula of amount?
Amount is equal to P1 plus R by 100, whole power T.
This was the formula.
So, in this also, what should we do instead of A?
New population present P2 is equal to
Initial population P1 is equal to P1 into 1 plus R by 100,
whole power T. In this case, if there is an increment, then
there is a positive sign.
In the case of population decrease, there is nothing to do.
Put a negative sign in the middle.
So, this will be P2 is equal to P1 1 minus R by 100, whole
power T. This would be in the case of population decrease.
When your population is decreasing.
So, the formula of compound interest, the formula of
amount, we correlated it in the population questions.
In the next video, we will do the remaining formulas.
And, basically, we will do questions based on these.
Till then, you practice simple interest.
And, the formulas of compound interest,
I have just proved it to you.
What you have to do is, you have to remember the formula of
difference.
Thank you for watching this video.
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