Hello everyone, welcome back to a new video.
In today's video, we are going to do ratio and proportion.
Before doing ratio and proportion, we should know the
definition of both of them.
What are they and what are their symbols? Then we will join
their properties and then we will do questions based on
them.
So first of all, let's start with the ratio.
What is ratio?
You must have read the ratio of A is to B.
In this, these two dots are the symbols of our ratio.
So what is ratio?
Let's take an example of a small class.
Let us suppose we have a class.
We have been told that the number of girls in that class is
double the number of boys.
So let us suppose we take the number of girls.
We take X.
We have been told that the number of girls is double the
number of boys.
So when the number of girls is X, then according to that,
what is the number of boys?
Then what will be the number of boys? Twice of X, that is
2X.
Now we have to find the ratio.
What will be the ratio?
This is the comparison between two quantities.
If we have two quantities, then what are the two quantities?
Number of girls and number of boys are two quantities.
So the comparison between these two quantities, if we
denote it in the form of A is to B, then what will be the
ratio?
So what are we finding in this? The ratio of number of
girls is to number of boys.
So what will happen?
If I have written A is to B, then we can write it as A by B.
We can convert this into a fraction and write it as A by B.
So here we have girls as X, so write X below girls.
Then this is the symbol of the ratio.
Then what is the number of boys? 2X.
So write 2X below boys.
So what do we have X is to 2X?
The ratio of girls is to boys.
Now we should know one more thing about the ratio.
In which form do we leave it? We leave it in the simplest
form or lowest form.
So if we have this X is to 2X, then how can we write it?
Is equal to X by 2X.
Now cancel out the common factor.
What is common? Both the sides are X.
So XX is cancelled.
What do we have? 1 by 2.
So what is the ratio of girls is to boys?
The ratio of girls is to boys is 1 is to 2.
If we have one quantity of girls, then what is the double
quantity of girls is to boys.
Let us suppose if girls are 50, then boys will be double
handed.
If girls are 25, then boys will be 50.
So this is just a comparison that what is the relation
between number of girls and number of boys.
Or what comparison we are able to find out the relation
between both the quantities.
Now one very important thing to pay attention in the ratio
is that the order matters in this.
I have written this A is to B and this is 1 is to 2.
I cannot write this in reverse.
I cannot say that this is equal to 2 is to 1.
This is not equal at all.
So if A is to B, then it will be equal to A is to B.
It can never be equal to B is to A.
Except both of them are units.
Unit means 1.
Because 1 is to 2, what will we get?
We will get half if we convert it into fraction.
And if we look at 2 is to 1, then we will get 2 by 1 that
is 2.
So half and 2 are not equal.
So what will happen in the ratio?
The order will matter.
We cannot interchange the front and the back.
We will write as it is given.
So the front quantity A is called antecedent.
And the back quantity B is called consequent.
So these are the names.
Now if we have one more thing in this.
Like we had taken x as common and cancelled out x as x.
So we had 1 is to 2 ratio.
So if we have any ratio, we can multiply or divide both the
sides.
If we multiply and divide both the sides, then there will
be no change in the ratio.
Because it will automatically get cancelled.
Because if we multiply both the sides, then we can write 2
as common and cancel out 2.
So there is no difference.
We will get A is to B and A by B.
So if we are multiplying or dividing the same thing in the
ratio, then there will be no effect in the ratio.
So this is clear.
So the ratio is clear to everyone.
We also know the names of both the terms.
Now the next thing which is important in the ratio.
What is the ratio in this?
Ratio is unitless.
We don't have any unit of the ratio.
Like we have unit of distance, unit of measurement weight.
Everything has a unit.
But in this, the ratio which we are taking out, it will get
converted into fraction.
So its unit will get cancelled out.
So the ratio will be unitless.
So ratio is always unitless.
And the two quantities which we are comparing,
A and B, Boys and Girls,
These two things, Antecedent and Consequent,
Whatever these things are representing,
What should be the unit of those two quantities?
It should be same.
Same as in,
Like if we are taking out the ratio of water and milk.
Water is 1 and milk is 3 times of it.
This is with us.
So when can we take out the ratio of this?
When both the things are in the same unit.
In which we will measure water?
It's a liquid.
So we will measure in liter.
In which we will measure milk?
We will measure in liter.
So if we have water 1 liter.
And the quantity of milk is 3 liter.
So when we will take out this ratio,
It will be 1 liter by 3 liter.
Liter by liter will get cancelled.
So our unit will get cancelled.
The ratio which is 1 is to 3.
In this, no unit is there.
This is unitless.
So we can take out the ratio of same unit quantities.
We can compare them.
Now you will not compare 1 liter water with 2 kg potatoes.
Its unit is different.
It is in kg and this is in liter.
Or you can't compare it with distance or kilometer.
You will compare it with same unit quantity.
So if we are taking out the ratio of two quantities.
What should be there?
Unit should be same.
Then only we can take out the ratio.
So till here the basic definition of ratio is over.
Next thing which is coming is
What is written after ratio?
Proportion.
So when we know the ratio,
What is proportion?
Till now what we have done is
We have done A is to B only.
Isn't it?
This is comparison of two quantities.
Yes, one more thing I will tell you.
It is not necessary that you only do A is to B.
You can do only of two quantities.
You can take one more ratio in this.
You can do A is to B is to C also.
Means you can do comparison of three quantities together.
Similarly you can do of four quantities.
So it is not necessary that you can do only of two
quantities.
You can do more than that also.
So moving on to proportion.
What will be the proportion?
Now the ratio which we have.
What is the ratio?
A is to B.
Let us suppose one ratio we have is this.
And second ratio we have is C is to D.
We have two different ratios.
If we equate these two different ratios.
Equate means equal to sign is added.
Now both are equated.
So, what will this be called? This thing will be called as
proportion.
This is nothing but proportion.
What will be the symbol of proportion?
Like the symbol of ratio, these two dots.
Similarly, if we write this in place of equal to, instead
of A is to B is equal to, if we write it like this, four
dots, means two times ratio symbol, then this will be the
symbol of proportion. So, what is this denoting?
Proportion.
We have understood proportion.
Whenever two ratios become equal, then they will be called
as proportion. Now, like in ratios, we had read the name of
A and B, antecedent and consequent.
Similarly, in proportion, they have names.
Like this A and D, whichever is in extreme, means right and
left, in front and at the end, these will be called as
extremes.
Because these two are written at the end, so these will be
called as extremes.
And what will we call B and C?
These will be called as means.
So, A and D became extremes.
B and C became means.
Now, if I solve this further, try to solve this further.
If I am solving this,
A is to B is equal to C is to D.
This is our proportion, right?
So, we will solve this.
What can we write A is to B?
A by B.
We can write fraction like this.
We can change the ratio in fraction.
Is equal to sign.
Change C is to D in fraction.
What is this? C by D.
Now, we have been studying since childhood, when we got
such things.
So, if we cross multiply, everyone must have studied this.
So, cross multiply.
How to cross multiply?
We will cross A by D.
So, what will be this?
A into D is equal to B into C.
Now, what was the name of A and D?
Extremes.
And B and C was named as means.
So, if you want to write this in words, what will be this?
What are the products of these two?
Of A and D?
Both are extremes.
So, we can write this as product of.
Product of.
This is short form of product.
Product means multiplication.
Product of extremes is equal to.
Whose equal is this?
Product of means is equal to.
Product of is equal to product of means.
So, this has become a property of yours.
In proportion.
That product of extremes is equal to product of means.
Right? So, you have done this much.
Next, we will move on to.
What are the results and properties of these?
Ratio and proportions of both.
Let's see.
Results.
What we have just studied.
What we have studied is that
If we multiply or divide by the same term in the ratio,
then there will be no effect in our ratio.
That is also a property.
Now, we will study.
What will we study?
Let's write the first property.
What did we have?
Proportion.
We had A is to B.
Sign of proportion.
C is to D.
In this, A and D were the names of extremes.
B and C were the names of means.
Right? So, let us suppose that this is D.
If there was something else in place of D.
If it was X or Y.
That means, whatever is coming in the last one.
D.
What will be called by us?
This will be called by us.
Fourth.
Proportional.
Fourth is proportional.
But in whose order is this fourth coming? According to
these three, this fourth is coming.
So, fourth proportional to.
A, B and C.
This is fourth proportional to.
A, B and C.
What is this of these three?
Fourth proportional.
Similarly, C.
What will be this C term?
This will be with us.
It is coming on the third number.
If we count.
So, this is our third proportional.
Whose third proportional?
This remaining.
A, B and D.
What is this?
Third proportional.
So, C is our third proportional.
And, D is our fourth proportional.
This is just a name.
To remember.
When we know the name of something.
Then, we remember that thing for a long time.
If we know the name.
Or, if we know an image of it.
Then, that thing remains in our mind.
For a long time.
That is why, I am making you write this name.
As such, it does not have a big role.
So, once you have a question from fourth proportional.
That, let us suppose.
Find the fourth proportional of.
Fourth proportional of.
Find.
The Fourth Proportional Of 5, 8 , 15
I hope, everyone is able to see this clearly.
What we have to tell about 5, 8 and 15?
We have to tell about fourth proportional.
First, let us write fourth proportional in this form.
What is this?
If we are writing this.
5 ratio
Where should I write?
Should I write the solution here?
Is everyone able to see?
I am writing the solution here.
5 is to 8
Then, the sign of proportional.
Then, 15 is to fourth proportional.
Let us suppose, it is d.
15 is to d.
Then, what we have to find?
We have to find d.
How will we find out?
Simple.
Apply the definition.
5 is to 8
Proportional is equal to 15 is to d.
Cross multiply.
5d is equal to 18 to 15
What is the value of d? 5 is to 3 8 is to 3 24 What is the
value of d? 4 is to 24
It was an easy question.
Everyone can do it.
It is not that easy.
But, there are many basic questions.
Related to ratio.
Till now.
Ratio and proportion are related.
If you do it,
If you give time to them,
Then, it will be solved easily and quickly.
It is not just a question of ratio and proportion.
It is a question of every topic.
The more you practice in maths, the faster and quicker you
will solve them.
This is your property.
It is your names.
Third proportional and fourth proportional.
And, a question related to them.
Now, second.
We will do the second property.
We have the second.
Mixed ratio
Or, Compound ratio
Compound ratio
What is the mix ratio and compound ratio?
We have
A is to B
And, the second ratio is
C is to D
What is the mixed ratio?
We have two ratios.
What is the mixed ratio and compound ratio?
Multiply the first ratio
A into C
And, the second ratio
B is to D
This is the mixed ratio or compound ratio.
Similarly, if we make some changes, we will get the
following properties.
For example,
A is to B and C is to D
These four quantities are different.
They are different.
So, if we write
C is to D as A is to B
We have two ratios A is to B
And, the second ratio is
A is to B instead of C is to D.
So, if we find the mixed ratio of these two ratios,
A is to A
B is to B
The ratio is B into B
That is B square.
So, A square is to B square.
So, this is the mixed ratio.
When these same quantities or same ratios are obtained, we
will get Duplicate or Duplicate
So, what will be the next step?
Next, moving on to
If we can get one more form of this, like duplicate,
Sub-duplicate, what will happen in sub-duplicate?
In this, square was there, in that it will be there, root.
So, sub-duplicate, fourth one, fourth point.
What will be the sub-duplicant we have?
If we have two ratios, A is to B, and let us suppose we
have another one, not sub-duplicate, means only one is
there, in the form of A is to B.
So, what will be the sub-duplicate of this? Under root A is
to under root B.
What is this? Under root A is to under root B.
This is sub-duplicate.
In that, A was square, in the last one, in the duplicate,
in the sub-duplicate, it will be under root.
Similarly, duplicate is done.
Now, we will have triplicate.
What will happen in this? Power 3 will be there.
So, we will have A cube is to B cube.
And similarly, sub-triplicate.
In sub-triplicate, it will be cube root of A is to cube
root of, these are our simple properties.
All these are related to what?
These are all related to mixed ratios.
We will have more cases.
So, we have compound, duplicate, triplicate, all these are
there.
Next, moving on to, if we have A ratio B.
So, if we take their inverse,
A ratio B, if we take their inverse, that is 1 by A ratio 1
by B.
So, what will happen to these two? These two ratios will be
same.
Inverse ratio will also be same.
For example, we can write A is to B as A by B.
Then, what can we write?
If we reciprocate this, 1 by A divided by, sorry, 1 by B
divided by 1 by A.
This will be equal to 1 by B.
1 by B upon 1 by A.
Because, A will go up and B will go down.
Then, if we write in ratio, 1 by B ratio 1 by A. So, what
will happen to these two? These two will be equal.
Moving on to next property.
Next, what will we get?
This is inversely.
Next, you will get, you must have heard this name a lot,
Componendo and Dividendo.
These are related to proportionals.
So, if we have these proportionals,
A is to B proportional to C is to D.
First, I will tell you term wise, Componendo, then
Dividendo, and then, mixture of these two, Componendo and
Dividendo.
So, what will happen to Componendo?
If we have A is to B, A is to B equals to C is to D.
What we have to do is, if we have these two things equal,
if we delete equal quantity from these two, then there will
be no difference.
If we have 3 is to 3, if we delete 1 from both sides, then
what will happen? 2 and 2 here also.
So, 2 is equal to 2.
So, if we remove same quantity from same thing, then they
will be equal to each other. Similarly, if I delete minus 1
from both sides, minus 1, minus 1.
So, there will be no difference in equality.
Because both are equal, we are subtracting same quantity
from both sides. So, what will happen? Solve this.
We will take LCM as B.
A minus B is equal to C minus D by D.
So, write this in proportional.
This will be A minus B is to B.
Write this in bracket so that it is not difficult to read.
Proportion sign is C minus D by ratio D.
These will be proportional to each other.
This is done.
This minus 1 thing is called
Dividendo.
We did minus 1 from both sides, so Dividendo.
We will do same for other case.
Plus 1.
If we add same quantity
in any equal thing,
A by B or C by D.
What is this? A by B is equal to
C by D. Add same quantity.
Plus 1, plus 1. Solve this.
What will come after solving?
A plus B by B is equal to C plus D by D.
If we write this in proportional,
Ratio B is proportional sign.
C plus D is ratio D.
These will be proportional.
So, this thing in which we did plus 1 is called Componendo.
And in which we did minus 1 is called Dividendo.
If we have to do both things in a question,
In many questions, it makes the question very easy.
It makes the question very easy.
If we have this thing,
A is to B proportional C is to D.
And A by B is equal to C by D.
If we have this given, we can write this as equal to A plus
B
A plus B by A minus B.
Means, these two terms A and B if we add them and divide by
their subtraction, it will become equal.
Addition of these two, divided by their subtraction.
So, this is very important thing.
In future, we will see how this question will be converted
in short and it will be solved easily.
So, this was today's related proportion and ratio related
results and their definitions.
We will do questions in their next video.
Till then, keep learning this.
Practice all the definitions I have given.
And thank you for watching the video.
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