This tutorial offers a rigorous yet accessible breakdown of foundational mathematics, prioritizing conceptual depth over mere rote memorization. It is an exemplary model of pedagogical clarity that effectively bridges the gap between textbook theory and practical problem-solving.
Install our extension to search inside any video instantly.
Review Exercise 1 - 9th Class Math | Punjab BoardAdded:
Assalamu Alaikum, how are you all? Hope I do it to all of you who are absolutely You will be fine. So today we are ninth graders.
Review Exercise One Completed I will solve it this way. But at this time At the top of the screen you guys will see a wallpaper You will see that many questions are answered.
You can see the timing. So question number The one you can see at the top of the screen Is. Read this and what is being said in it Is? Four options are given against each Statement. He is saying that every There are four options against the statement.
You will be given or have been given. Ok?
What do you have to do? is in circle the correct Option. The correct option you have or the one The correct answer should be circled.
Meaning to circulate. Well then? So Like this statement in part number one So now this thing is against this statement Which option is correct? So how do you know Will work? So look at Square 7, what is this?
Is? So is this an integer? integer so this Of course it can't be because I told you Zero and all positive numbers are mentioned behind And all negative numbers are integers.
Are. So any number in this shape is never an integer. So integer so this Cannot be done. Ok? Similarly, ration number Move forward towards. Is this a ration number?
Can it? So for that we do this.
The easiest way is to convert it to decimal.
Convert it into. Ok? and then look Take. So let's convert it to decimal.
Bring a calculator. So look, we have this The calculator has arrived. Square Root of Seven So I'm going to take the intake and this on this calculator.
Its square root will come. Ok? Your Whatever scientific calculator you have, expect it.
I hope you will learn how to The square root of any number is taken.
Ok? I'll do it this way, I'll put it down so that according to this number we Keep talking. So what did you learn?
You learned that ration numbers are Which ones are there? One is non Terminating What else was learned? And that There are recurring ones. So if these two qualities These are the characteristics or the characteristics If it is present inside any number then we will call it It will say ration number. Ok? So non What is called terminating? Non A terminating number is a number that has The numbers after decimal do not end.
You are happening. So you see this number is not over Are happening. Limited numbers appear on the calculator Are. But it may continue beyond this also.
So our first option is that this non is the terminating number. Ah, the second option If you see, it is recurring. Recurring means that It happens that after the decimal there are some numbers which are You are repeating it again and again. Whether it's a number Repeating or multiple numbers You are repeating it again and again. So in that you A special pattern will be visible as the numbers repeat.
From. So if you put in it after the decimal If you look at it, you will see any number pattern.
It doesn't seem to be repeating again and again.
So it means this is not recurring either. So Both these features are because it does not have So this cannot be a ration number also.
So now we move on to irrational numbers.
So any number is irrational when It happens? When that time is non-terminating.
So I wrote this here.
Terminating and other features What happens? It should also be non-recurring. If If you have two of these qualities, you are an irrational number.
You can say. Ok? So if we look at the numbers So this is non-terminating. Non Terminating means the number continues after the decimal You are leaving. So they keep going Are. So this is non-terminating. Now Is this non-recurring? So yes, absolutely this is non-recurring because there is no decimal after the decimal.
Any number that repeats in any pattern It is not happening again and again. So because these two Specialties are present in it. So that's why I I can say that this is an irrational number.
Ok? So between rational and irrational You can see the difference here that the ration number is non-terminating and an irrational number is also non-terminating. But ration The number will be recurring and irrational.
The number will be non-recurring. If you If I show you above, you will also get a picture.
You should remember it. So I got this from your book.
I took a picture. So you can see it here as well The irrational number will be non-terminating and It will be non-recurring. Whereas the ration number which is It is non-terminating and recurring.
I hope you saw it well from behind.
You people must be understanding what I am saying.
Ok? So let's eliminate all these steps.
Are. We will circulate it. But as if I told you about the four options.
You should know. So natural number If you see, you have small natural numbers.
Learned in classes. Represented by n We do. numbers that start with one and they move on and on are called natural numbers. so his It has nothing to do with this whatsoever.
So there can't be a fourth number either. so therefore Now you know that irrational numbers are Why is it? Because the rest of the numbers are not like this You know that too. So I hope you talk You must be understanding. So let's go to the second one.
Are. So the second statement to you guys It is visible at the top of the screen. into what Is it written? Pi and e are two numbers that Now which ones are they? Is this a natural number?
Are? are integers? Ration or irrational Are. Ok? So again natural number so It cannot be because that 1 2 3 4 5 6 It is set. Ok? are not even integers Can because they are positive, negative and zero There are numbers with. Ok? So this is You look different just by your appearance. Pie and e. Well, my advice to you is that this Remember it by word of mouth only. Ok? some things You should always keep in mind that pie and e which are always irrational numbers.
This can be a general objective or MCQ.
Is. Ok? So without solving it You should know what Pi and e are.
There are irrational numbers. but I still I'll tell you, even if you don't remember.
How will you crack it? So look at this There cannot be both options. We know Let's go towards A and B C, then nature ration Number. So pi is a ration number. So if I If I bring you a calculator, here it is.
We have a calculator. If on the calculator If I press pi here, what happens after pi?
Do you know? decimal numbers after the point If you look at them, they are non-terminating.
Are. Ok? So what am I here for Will I write? So these are non-terminating and After that, a specific number of patterns are added to it.
Whatever is there is not being repeated again and again. So What this shows is that this non-recurring Is. So I wrote this non-recurring Number. So now in whom do these two characteristics exist?
in which the number repeats again and again In which you are never ending and in which you repeat It is not happening. So just in the last question I had talked about irrational numbers in detail.
It is a speciality that irrational numbers These things are found inside. Whereas What does a ration number contain? Non Terminating but recurring There are. So this way you mean that Draw conclusions as to why everything is right and why is it wrong. Ok? So from here We realized this is irrational. ration Not there. Ok? So just like pie Found out. Similarly, you can find out e.
yes. If yes, then learn to put e on your calculator.
Please take it. Ok? On this calculator, it's straight E is written here. If you press e, it will The number comes up. Ok? So you can also get this number See. It will have both these features.
The number after the decimal point is not repeated If it is repeating then it is also non-recurring and Neither are they ending, so non-terminating.
Also there. So then this is the same thing inside again e Two specialities are found. Ok? So this This way you can also know your statement.
It is said that ration number is there and from again these two So it would be better if you remember it.
So part number three you guys have to screen It is visible above. read this then in What is being said? If n is not a Perfect square if n is perfect square na ho den n what will happen to what is? The ration number will be Will it be irrational, natural, or integer?
will be? Ok? So first of all the question There is a need to understand. So look at n which is Basically representing a number which is not a perfect square. Now The question arose as to what those numbers would be.
which are not perfect squares? So Look, three, this is not a perfect square.
Five is not a perfect square. intake No, it is not 11. A lot of these kinds of There are numbers which are not perfect squares. Now The question arose, how did you know, Sir?
Not a perfect square? Because A perfect square is a number that that the result is two identical numbers To multiply. Now I am 16 I can say it is perfect. Ok?
Because multiplying two identical numbers Coming after doing it. Similarly, four which is This is also a perfect square. Because to If you multiply by two, you will get four.
Similarly, you will multiply eight by eight.
So 64 will come. So these are all the numbers are perfect squares. And all these numbers What are they? Not a perfect square Is. What do you want to learn from here?
From here onwards you have to learn that Perfect if inside the square root Not a square number like here Pay square 3, so which one is this?
Will it be a number? It will be irrational. Ok? and if Like this number will also be irrational.
Why? because inside the square root I am telling you again. Perfect Square If the number is not there then it is irrational.
Why? I will tell you this also now. right now you Listen to this. and while inside the square root If a number has a perfect square, then it There is a ration number. How? Because its What's inside you is not finished with the sign of the radical Can do it. Inside this you will see the sign of the radical You can finish it. Like I write 16 4 to the power of 2 and then that's square will cancel out with the square root So four will come. So you see this number Which one is it? Ration number is. Ok? So So whenever the perfect inside the square root If the number is square, then it is the ration number.
will come equal to. Ok? 16 plus you You can keep any number, keep four. So See, 2 * 2 = 4 and square Cancels out from the square root so two comes out.
But in this case the perfect square They don't happen. This should be erased. now then I think you must have understood. like this In the case where the square root is three, you can use Cannot eliminate the sign of radical. So Therefore it is irrational. So this is the way.
Questions based on perfect squares It has been told. So you will remember this what if inside the square root If the number is a perfect square then it will be will be called a ration number and if the square root of There will not be a number with a perfect square inside So that will be called an irrational number. Why?
Because inside the perfect square we can eliminate the sign of the radical and A number that is not a perfect square Can't finish it. So I hope this You people must have understood the thing. So this From this point of view I can say that Option D is our correct option.
Ok? So part number four for you guys is visible at the top of the screen. into what Speaking? Square 3 + Square 5 This statement is equivalent to what?
Will it happen? The whole number will be equal to the integer key.
Will it be rational or irrational?
So see, solve this type of question.
to give you a principle to follow will be. That is, if two numbers of the same type You are getting added. Ok? Like these numbers and Both these numbers are of the same type and if If these are being added then the answer that comes That too will be like this. this thing Let me explain with an example. Like taking five I'm here and I'll take three. Ok? So its If the answer is eight, then see what is five.
There is a ration number. Three is one ration There is a number. So whenever the ration number contains ration If the number is added then the answer will also be ration.
Ok? Like natural number in natural number If there is an ad then the response will also be natural. So this Keep in mind the principle that two are of the same kind If there is a number, the answer should be of the same type or of the same kind.
Must be related to. So this point of If we look at this thing from the view, then there are two The numbers are here. One this and one this. So Are these two numbers the same? So Yes, these are the same number. How did you know?
If you look at three here, that is You have read it in the previous question.
What? that when there is a square root inside Number which is not a perfect square. Correct Is? Like three is a perfect square.
If not, what is that number called?
is called an irrational number. Ok? So three Whatever is there is also irrational from this point of view From. So five is also a perfect square.
Not there. So from this point of view, this also What happened? Irrational number happened. Ok? So Now when two irrational numbers are added then Whatever the answer to this, but that relation Who will you keep it from? I will keep it from irrational only. So this This is how we solved this question.
and we will say that this number is It is irrational. Ok? So part number five You can see it on the top of the screen.
If you read this, what is being said?
For all x belongs to r and x is equal to x is called means this statement is What are you saying? First of all, this has to be understood.
For all means all the numbers i.e. x is represents all the numbers that That belongs to r. Now whatever is What is? Representing real numbers Is. This symbol means Belongs to. Belongs to means to belong Keeps it. Ok? So this whole part What does it mean? This whole part means that It so happens that x represents all such numbers.
Used to be. That is, for all x of this entire part.
Ok? x represents all such numbers is doing which belong to r.
And what was said next in the statement?
that x = x now this property is That means which property is it? what to it Will you say? Then four options have been given.
You have to tell me which of these. Ok? So I hope the question is understood will be. So basically properties Speaking of, what do we have? Here But let's write x = x here. Ok?
Now look at the reflexive, you will see The answer will be revealed automatically. Because reflexive reflect you reflexive to You can keep this in mind also. Reflect Like you stand in front of the mirror. Correct Is? So when I stand in front of the mirror If you are, then understand yourself like this Reflection is visible. mind like this Keep it in me. So keep it in your mind Different people have different ways Is. So reflect in the mirror's point of I keep it from the view only. So Reflect's What does it mean? As if we were standing in front of a mirror If we exist, we can only see ourselves. Ok?
Similarly, any number is equal to It happens? Ok? He was his own equal Is. Ok? That means each number has its own It is equal. This property was called goes the reflexive property. You came Hope you have understood that this The reflexive property is because Every number is equal to itself.
Is. Ok? So part number six you guys is visible at the top of the screen. read from this So what is being said in this? Let A B C Belongs to R if A B C which is present here They are visible if they are related to R.
R means real number then A > B and B is > C which implies that A > C is Called dash property. So who is this property?
Is it C? So you would have written the property here.
Are. We have property A if it is bigger What else is mentioned from B to End? B if is greater than C. Ok? So then what it means This will be which implies that A is will be greater than C. Now which property is this?
Is? Everyone has to see it one by one.
If we look at trichotomy, then trichotomy This cannot be property. Because What does the trichotomy property say? that this It states that a number is either equal to itself or one number is different from another number may be larger or one number may be larger than the other May be small. Ok? So, in which three Trichotomy in various possibilities Comes. So I can keep you in mind like this There will be three types of possibilities in Ho Try. Correct Is? So here we only have two types of Possibility has been stated. Ok? so this this It can't happen. So know about trichotomy Missed it. Then look further Transitive. This is what is said inside the transitive It is said that if one number is and if the other number is greater than If the first number is greater than the third number, then the first number is That too will be bigger than the third one. This Let me explain with an example. like if I Should I make it four here, the bigger one is three From. Ok? And a three which is bigger is two From. Ok? So then it means that Four will also be bigger than two. I think Inside the properties, you must have learned that If a number is greater than a number such that If itself is greater than any other number then obviously The thing is, I think this is for me Should not be told. Just identify the property.
Is. If you want to understand then do the previous exercise I will go and see you. So basically like this What we learned about transitive It is done inside. So therefore it is transitive The property will be called what it is. So here I am I can circulate this on B option To. This cannot be an additive either. of additives There will always be an add symbol inside.
It cannot be multiplicative either because There must be some multiplication here.
It is no more. Ok? So part number seven you People are visible on the screen.
What is being said in this? 2 to the power x * a to the power x = 64 if this equation is Then x = what? So x is equal to what?
will be? You have to tell me this. will be equal to x, Will its equal be this or this? Then What will the value be equal to? Now the value of x Ah, how would I know? x is present here Here. Then find out their value It will cost x. Ok? So this is solved again will be. So here we will write 2 power x * 8 Even power x = 64. So this is our equation Near. So let's solve this. Then x The value of will come to us. So solve Will go to do it. So 2 to the power x is like that Let's do that. The rest is eight, 64. This Let's make their factors. Ok?
Because whatever the exponent There are questions in which we factor Make it. You understood this much Will happen. So 2 * 4 = 8, 2 * 2 = 4 and 2 * 1 = 2 so three times two is two. So what do I do to this Can I write? 2 to the power of 3 and x already is present. So this created power outside. Correct Is? If we factorize 64, it becomes 64.
Went. 2 * 3 = 6, 2 * 2 = 4, 2 * 1 = 2, 2 1 * 6 = 12, 2 * 8 = 16, 2 * 4 = 8 and 2 * 2 = 4 and 2 * 1 = 2 so how many fours are there We have? They have come to us 1 2 3 4 5 6 So if two are mixed together six times If we multiply it, it becomes 64. So This means I can write 2 to the power of six.
Am. Ok? So I wrote this down for 64.
place 2 to the power of 6. Ok? Ok? Now The meaning of an exponent is that any How many times should these numbers be multiplied by each other?
Used to be. Now if we move ahead then this and To simplify further, this x is like that It will remain so. May the power of power always be She goes? Multiply. So what is the power of 2?
Will it happen? 3x and here 2 to the power of 6 is like that It is going on like that. Now the base is the same. When If the base is the same then the power would be added.
Are. So 2 to the power now becomes x + 3x. And 2 to the power of 6 over here. Now these two are x.
That is, what will be the answer to x + 3x? 4x Since both are values of x, add Are happening. So now what do I do with the power to Can I write? 2 to the power 4x = 2 6 now here There is a rule of math. When the equation That is, base one on both sides of the equal sign Whatever it is, their powers are one They become equal to each other. Ok? Or Then you can also understand it like this If the bases on both sides of the equation are equal to each other If they are equal then their exponents will also be one.
Will be equal to the other. Ok? This rule is yours Keep this in your mind. So this rule According to what can I write now? I I can write here a 4x = 6 this is my The equation that is near is achieved through this. This further Let's solve it by drawing a line here. So 4x = 6 From here we can know the value of x Have to do it. So four is multiplying here Is. have to move to the other side of the equation Because we are finding the value of x. So It will divide on the other side. Now this two Let's go to So 2 * 2 = 4 and 2 * 3 = 6 How much is x equal to? 3 / 2 So this is our The value of x is passed. So let's see Where is it in the question? So 3/2 first There is only option. So you circulate this.
So this way you have the value of x.
I told you. So I hope this question You must have understood what it is. of the next Let's go to the side. so part number at you guys is visible at the top of the screen. This Read what is being said in this? Late a b belongs to r then a = b and b = a is Called dash property. So which one is this Do you have property? This is what we have to tell. then a and b Who are they related to? r From. Then a will be equal to b. and b It should be equal to a. So, now which one is it?
Will there be property? The first option we have is reflexive. ah reflexive so this It cannot happen because I have reflected I just told you according to the mirror last The property has the reflexive property It happens that every number is equal to itself.
Is. Ok? So this cannot be it. So Because in this we talked about the order of numbers.
It is being said that a is the first number and b is the next And here b is before and a is after. So This is not reflexive. We know this It's gone. After that, if we proceed further Symmetric can be asymmetric. Why?
Because in symmetric there is order She goes. Ok? like inside symmetric The same thing is said that if a number which If it is equal to the second number then it is equal It remains if its order is changed.
So if you put a equal to b then you You can also write it like this b = a. Correct Is? This property is inside the queries It is commonly used. So this point of From the view I found out that this is asymmetric There is property. So part number nine you People are visible on the screen. So This is the statement. Whose equivalent will this be?
Out of the four? We have to tell this. So this If you try solving it, you will know. So Let's solve it. Square is 75 and Is 27. Ok? So let's factorize Are. This is the first step. So When we factor 75, we get a three This sorry goes 3 * 2 = 6 forests forward If it goes then it will become 15. 3 * 5 = 15 now this Let's go to five. 5 * 5 = 25 5 * 1 = 5 So basically 5 * 5 * 3 came. So I this I can write 5 ^ 2 * 3 these things now I hope you have understood.
So here I will write 3, sorry 5.
Power 2 * 3 then plus 27 Do factorization. 27 = 3 * 9 = 27 3 * 3 = 9 and 3 * 1 = 3 so write 3 I can be here. So write 3 to the power of 3.
Took. Ok? But now what do I do about it Will I do it? I don't write 3 3, instead look at 3 3 I will write. So I can't cancel it.
Because if I write two to the power of 3 I can cancel this with this.
Ok? So because 3 * 3 * 3 I I will delete this. This confuses you Can. Ok? So this three times three If we multiply it, we get 27. So I can write it like this also. 3 3 27 and I can write it like this also. 3 ^ 2 * 3 = 27 then its factors will be written like this So that we can see that these are the symbols of the root.
Can finish it. So let's write it like this Let's take it. 3^2 sorry not to write here We have to write above. Ok? So come here will go into the square root of 3 * 3 so now look at The squared values are the roots of I will finish it. So what will come out?
Five will come and three will remain inside. here Also, the square root of the value is mutually exclusive.
Will end in. So the one inside Square 3 will remain. And now you have this Have you learnt anything? that if these two If so, we can share their numbers with each other.
You can add it. So how much is five and three?
Will you go? Eight. And this is the same. Hope I do, now I need to tell you the details.
No. All the exercises are over. good If you have asked questions before then you understand the matter Will be. So this is our answer. So check this out So this answer is present inside option D.
Ok? So that's why our option is D.
Will be OK. So part number 10 for you guys is visible at the top of the screen. So asked It is being said that what is its product Will you come? A prime number will come, an odd number will come, Ration or Irrational. Ok? So the product Who is called? Whenever the number two It is multiplied like 2 * 2.
The result that comes in return is called the product.
It is said. Ok? So this number is What will it be called? It will be called a product. So this This is kind of an expression. This is an expression.
The multiplication of these two expressions What will the answer that comes after this be called?
It will be called a product. Ok? So as long as we If we don't multiply them then we will know It won't work, no matter what kind of answer it is.
There is a number. So let's write and multiply Get it done. So 3 + * 3 - 5 so this is our I have a question. solve this question Do it. So you see a formula in mind.
will come with your a + b and a - b which What is it equal to? of a² - b². So Look, there are three here too. Three here too Is. Here also the square is 5. here too Square is 5. and plus minus in between There is a difference. So it means that this one We can apply the formula. Ok?
So let's apply this formula here.
I hope you have understood now. And If the formula is applied, the square of 3 will become Will go. Minus Square 5 Hole's Square It will be made. We will proceed further. So 3 of Square is 9. Square Square The route will be cancelled. So five will come.
So 9 - 5 equals two will come back to four. So Now Four came to us with the answer. So four is What is that? A Is this a prime number? So You should know what is called a prime number.
Are? So look at the prime number which is short I write pn. Ok? or the complete I just write it down. So look at prime number one There are numbers which are either divisible by themselves There are. Ok? I write it myself.
Itself means on oneself. So Prime numbers are numbers that are divisible by themselves.
Are they pay divided or one pay divided There are. Apart from this, any number third Numbers are not divisible. So four is Does it divide on itself? Yes on myself It happens. If you write four here and If we write four here then 4 * 1 = 4 Will go. Ok? So it happened on four as well. So what Does this happen in the forest? So one on divide if Let's try it here. Four If I divide one by one, then one by four If I multiply it, I will get four.
Will go. So there was a division on the forest also. So Does this happen to anyone else? Yes to Pay Too It happens. If I put the four in, two pay If I divide it, what will be 2 * 2? Four Will be done. So what are we to learn now?
found? What we learned was that this prime There is no number. Why? Why? because this There is division on one side and there is division on four sides also.
It is happening and along with it two pay is also happening.
If it were a prime number, it would be only on itself and It would have been in the forest. Ok? So that's why four is That is not a prime number. Ok? Prime I will give you an example of a number.
Am. Ok? So that like three which is is a prime number. You can pay three for one Yes, divide. Ok? And Three himself is called Three You can pay. 3 * 1 = 3. Other than this Can't do it on anyone else. Ok? So now You must have understood the matter well. So we What did you find out? We found out that Prime This is not the number. Now let us look at the odd numbers.
What are odd numbers? Odd number primes They erase the number. It is needed now Not there. Odd numbers are numbers that are two There should not be a pay divide. So four is two Pay gets divided easily. Ok? 2 * 2 = 4 so this means it is not an odd number either Is. If we go further, then irrational number. Correct Is? So is this an irrational number? Four. So a is not irrational because a is a number such that They are irrational in the form of p/q If it were not there then because this is one below four.
can be written as p / q It is in shape. You can understand this. so therefore This is not irrational. this point of view You can also see it from here and I have given it in detail.
It has been told earlier that if it is terminated From the point of view of non-terminating If you see, four means the number after which there is something If not, it is decimal. decimal k If you keep repeating zero again and again then There will be no difference above this four. So From this point of view it is non-terminating Happened. Ok? And recurring happened along with it. Correct Is? If it is a recurring number and non-recurring number then The terminating number is irrational.
It doesn't happen, it is a ration. Ok?
Whatever is irrational is non-recurring.
Is. So from this point of view we have both I told you the way you Looks better. So from this point of view you You can say that this is a ration number.
Four. Ok? Sorry, I deleted it. So What was the result here?
It was four. Four is a rational number. then restored I hope you understood all the MCQs.
Must have come. A mistake was made unknowingly So I apologize. But you must understand the concept Must have found it. Ok? So part number two Let's go to the side. So question number two Part number one, which is for you guys to see on the screen It is visible above. First question Let's read what it says inside Is? It is being said in this that if A is This is equal to 3/2, B which is equal to 5/3 be equal to C which is equal to 7 / 5 yes. Then verify that by proving it Show me. Verify that both of these And those who are within the equation of each other Are equal. Ok? So basically both of these The ends are equal to each other. By proving that Have to show. And how will this be proved? This a, Put the values of b and c inside this equation I will do it and then prove it.
How will this happen now? So let's solve the question If I do it, I will understand it simultaneously. Correct Is? So to solve it, first of all here I will write the solution. After writing the suggestion Rewrite the question once again Are. So the question we have is a * b + c And what do we have on the other end? ab + ab is it okay? So you should know this thing It should be in advance that this left hand side is It is called end because it is on the left hand side.
It is on the right hand side and it is on the right hand side This end is called that because it is on the right hand side.
It happens towards. Ok? So name them and so that we can tell that both And they are equal to each other. So Let me do that. both hands I do it together. Ok? so one Let's draw a straight line. So this is what I Drew a straight line. Ok? now here We'll do the left hand side. right hand here Will do the side. Both together. You can also do this You can. Turn the left hand side up, down Do it on the right hand side and do it simultaneously You can. This is how it will happen. in the same frame You will see both ends. Ok? So What do we have on the left hand side? Left On the hand side we have a * b + c okay Is? Ok? and on the right hand side we have What is? On the right hand side we have a ab + ac. Ok? Now the values inside these We will put it inside both of them. Ok? So Let's solve this end first.
Let me solve this end. So a values Let's put it first. So values Will put. a is equal to 3/2k. After that The bracket will open. B which is equal to 5/3 Is. Then C which is what we have for 7/5 Is. So I wrote this 7/5. first this end Let's solve it. Ok? so that one Let the end be complete. So now this situation How will you deal? first of the brackets The first bracket inside the There is data, let's solve it. Then Let's multiply that with the outside. So Therefore 3/2 will remain the same. And here But now we will take their LCM. Ok?
You should know this thing. like 5/3 is + 7 / 5. Whenever any such If there is plus or minus in the meantime, then You take its LCM. and how the LCM Is it taken? Both lower denominators Such factors are known for those who are Are. Like three and five, then three and I will write five. 3 * 1 = 3 5 same as And then if it goes to five, one will remain the same.
5 * 1 = 5 so when we have two like this below If we become one then what we are or what we currently are We get numbers by comparing them with each other.
Let's multiply it. Ok? So 5 * 3 How much will it be? 15 So our here What will become of the LCM? 15 So this Thus LCM is taken. So let us go So this is Ruffle. Ok? solve that side Are doing. So let's write down 15.
So here I wrote 15 and 15 to write How do you solve it further? This one below There is an LCM, we keep it inside.
Ok? and deal with it once and deal with it once Get it done. How? One by one. 15 inside kept. This three which is the denominator, take it out Keep it and divide it. So what?
Will you go? 3 * 5 = 15 so this comes to 15. which is above The answer comes like this five has come.
We can calculate this with the numerator above.
Let's get it multiplied. Ok? So 5* 5 How much will we have? 25. So this In this way, what number did we get on one side?
Near? 25. So let me write 25 here. This We will get the second part dealt with like this. A 15 Will keep it inside. And what will A do? This one We will keep the denominator out. So I kept five Outside. So what is 5 * 3? 15. So what The answer came with this answer with the number above I will get it multiplied. So what is 7 * 3?
will be? 21 so there's a plus in the middle so plus Enter and write 21 here, close the bracket Gave. Ok? This was a rough area. erase this We give. Let's move ahead. If we move forward then 3/2 Same as it is. After that how many A 21 and 25 Are there? 25 21 so this five and one 6 2 and 2 46 so here it comes 46 / 15. Now this this I can write the situation like this also.
Both these numbers are still being multiplied.
Are. And even if I write it like this, no one It is not an issue that multiplies by 46.
/ 15 Both have the same purpose. just like this What will happen if we write from it? In cancellation It will be easy. Looks more easy by the look. Correct Is? So now see cancel now when this Whenever such situations arise, the numbers go up and down If this happens then this number can cut it. These The number can cut it. So let's see someone The number cancels someone. Yes he does.
to which is ah 2 * 1 = 2 2 * 2 = 4 and 2 * 3 = 6 23 saved. And these three are also cancelled.
Does down to 15. 3 * 1 = 3 and 3 * 5 = 15 So what do we have left now? When this is on the 23rd I multiply by one so 23* 1 = 23 will become. On the five when I came from one If I do it then it will become 5 * 1 = 5. So basically How much did we get on the left hand side?
He writes it down. So here I will write A left hand side he came to us 23/ 5. So, as you know, the left hand side Did you know the right hand side in the same way?
do. Ok? So let's find out.
When I go to find out about this, I will Now you will have to put the values first. So Let's have a what do we have? 3/2 so Here it is written 3 / 2 followed by multiply b Who is he equal to? 5/3 of. So here 5/ 3 I'm looking at the values from here.
Ok? After that there is a plus in between so Let's add a plus. a that we have How much is it? 3/2 so again 3/2 c which How much do we have? 7/5 So here I will write 7 / 5 now I will pass it on.
Cancellation happens yes. Three Three It gets cancelled. So what is left behind?
Will you go? 5 * 1 = 5 and only two remains below If it is there then we will write two. Ok? 2 * 1 = 2 or You can understand it like this also. then here What is getting cancelled? something here It is not getting cancelled. So what is 7 * 3?
Went? 21 and below multiply both If you do it then what is 5 * 2? 10. Further we will proceed. So now take their LCM You take it. If you take LCM then again the same The thing I told you about was that the ones below which are denominators like what here Is? Two and 10 so here pay two and 10 is written.
Ok? LCM of these denominators Is taken. After that, 2 on the table of A to * 1 = 2, 2 * 5 = 10 then after that five pe goes. 5 * 1 = 5 so below come two ones Went. So this means that now as many as these two If there are numbers here, multiply them.
So 5 * 2 = 10 so our LCM will come Now let's look at multiplying these two.
10 will come later. Ok? Now take it from here Let's remove it. beans after removal You will keep the same thing 10 inside. Ok? And We will keep this outside. So we will keep two here.
Out of. So what is 2 * 5? 10. So The answer is this answer with the number above.
We will get it multiplied as well. So 5 * 5 How much will it be? 25 and then plus inside it is ahead. Then one more thing, delete it.
I will give you the first one when these two numbers are here If the number and the number here are the same then the above Write the number as it is. Ok?
I told him this thing but it just seems like We have been doing it till now. If you do it anyway This number in, this number out, even then The same answer will come. So, let's say 10 * 1 = 10.
and the answer is one when it is divided by 21 If we multiply it, it will be 21 * 1 = 21.
Is. So 21 will come here. Ok? So Same thing happened. After that add both of them Do it. So 21 and 25, so again five and one.
Two 6 and two 4 will make 46. So here 46 came and what is coming down? 10 So here But we will write 10. Ok? after this Let's delete this. Further after deletion What will happen if we solve it further? 2 * 5 = 10 and 2 * 2 = 4, 2 * 3 = 6 so What is basically here? 23 / 5 Our Came closer. And what is this that we have? Right Hand side. So now you can see that here On your left hand side, 23/5 is the same Your right hand side is 23/5 so because Now both these ends are equal. So that's why I I can write a statement here. So this I wrote the statement. Hanes It Is Verified consideration. Hens means therefore this That has been proven, what is that thing? that this The end is equal to this end. Ok?
So you do this, but before this you also do one more thing I will write it here. I'll take this down.
You write this here. So okay? Left The left side is equal to the right hand side. And After that, write this statement here.
Please give it. So that is much better. So I hope you understand the main concept.
Every teacher's method will be different. So much All the children will see this. So I hope Yes, you must have understood. of the next part Let's go to the side. So part number two for you guys is visible at the top of the screen. Previous It is just like the part. Both ends which are They want to prove that they belong to each other.
Is it equal or not? So values put It will be simple like the previous question. So Let's write the query first.
In the question we have a + b * c and the second The side is ac + bc. Ok? So now out of this A left hand side will be formed. a right The hand side will be formed. And we turn the page in half.
Divide it into parts. So this page Divided it in half. So What is happening on our left hand side?
Is? The left hand side is being built near us a + b and outside is c and right hand side What do we have brewing? He is becoming ac + bc. So let's put the values Are. Put after looking at the values from above.
So we have a equal to 3/2. So I wrote 3/2 here. + b we have Equivalent to 5/3. So he wrote it down. and out which is c. What is the value of c in our Near? 7 / 5 So I wrote this 7 / 5.
Ok? Now if we move forward, then that will be the first We will solve the bracket part. So Let's get down to what we have? Three and two.
So we will find the factors of three and two.
So three and two means 3 * 1 = 3 not two go. I will write it on the table of three.
Like that. Then three. Now to pay divide When the same as before and 2 * 1 = 2 then below One forest has arrived. Now multiply both of these Do it. This is our LCM. So 3 * 2 how much is a? Six. So now I'm here What can I write? Six. After that it Let's erase it. The next thing is the same as the last You have learned in the question. will keep this Inside. We will keep this outside. So let's keep it in We had six. Kept out to. So 2 * 3 How much is it? Six. The answer is three.
Get it multiplied with the above.
Will give. So what is 3 * 3? Nine. So Did the first part. Similarly, the second part will that do. If there is a plus in the middle, then a plus Put. Now we will keep the six inside again.
Will keep three out. So I kept the six Inside. If we exclude three then 3 * 2 = 6 which is the answer Have two. The number above it, which is five We will multiply it by that. So what is 5 * 2?
Will be done? 10 So this brings us to 10.
Removed it, put the brackets and take it out If you have 7/5 then that is the same fill time Write it like that. Let's move further.
So now I want to tell you what is 10 and 9?
19 / 6 * 7 / 5 This will come to us. Now If these are up or down, they can be cancelled.
each other. Let's see. not also It does not get cancelled. all from each other There are different numbers. So the number above Let's multiply. So here we come Pass calculator multiplied by 19 So I got 133, so I wrote 133 here.
By 5 * 6 = 30. So this is like that I wrote 30 like that. Ok? So this is our What came near? This came to us from the left Hand side. Ok? Similarly, right hand We will find out the side. So let's look at the values.
Let's write. So what are our values?
Near? The values a has are equal to 3 / 2 Is. c How much is it? Equal to 7 / 5 Is. Plus and b is equal to what? 5 / Equal to 3. And C is again 7/5 Equal. If we proceed further then there will be cancellation.
There is nothing above or below. So 7 * 3 = 21 5 * 2 = 10 plus a cancels out here From 5. So 7/3 will be left. Ok? Further We will take this forward. So what will happen now?
We will take the LCM of 10 and three. So 10 and this Three. So A will take 5 * 2 = 10 turns.
10 and three are two different numbers. Ok?
So let's try it. 2 * 5 = 10, three Same as it is, then 5 * 1 = 5, three like Like that, then 3 * 1 = 3. So, now down One each is saved, so now all these numbers Multiply it. So 5 * 2 = 10 10 * 3 = 30 so 30 we get the LCM. Correct Is? So 30 if I take the LCM here now I'll draw a line and write 30 here.
So what will we do now? 30 will be kept inside and If there are 10 outside, we will keep 10 outside. That means that This is 30 inside and 10 outside. Now let's divide them Are. So what is 10 * 3? 30 then The answer lies with the one above We will multiply it. So 21 with three How much will it be if we multiply it? So I need a calculator (21 * by 3) 63 will come. Ok? would have put it up Are. After adding it up, it came to 63, so 63 I will write it down. Again, we will save it like this. on the 30th Will keep it inside. Whom to exclude? will keep three Outside. Kept this inside. This was kept outside. So 3* 10 is 30. So now multiply by 10.
Will give it with consumption. So what is 10 7?
Will you go? 70 So it came to us. Ok?
Deleted it. Add these two using a calculator Do. So 63 is already written. 70 Let's just add to it. So it will become 133. So what should I write here? 133 / 30. Ok? And what did this bring us?
This came to us on the right hand side. Ok?
So what will I do now? A here on the left hand side right hand side each other It is equal. So I'll write it here.
Left hand side is equal to right hand Side. Because these two ends are opposite to each other Are equal. So now the statement or Let me write the conclusion. So this is what I Wrote the conclusion. once it is verified That this end proves to be equal to this end Happened. Ok? So this question, which That too is over. So question number three.
You can see it on the top of the screen.
What is being said in this that if a equal be b equal to 4/3 be c equal to 5/2 7/4 then verify the associative property of real numbers. What does it even mean If a b c is equal to this then you are real The number one property that you have come after reading If it is an associative property, you can verify it or Prove it. Ok? Now Associative You have read property in two ways.
With Respect to Edition and With Respect Two multiplication. So point of addition You will also verify it through the view.
Associative Property and Multiplication From a point of view as well. Ok? So the question It's a little long. This is how we do it.
Divide it into two parts. One The bar associative property is expressed as Let us verify it accordingly. And In the second part its associative Property in terms of multiplication Let's verify it and see. Ok? So Let's get started. So first of all you I will write the solution here. to write the solution Later, in terms of editions like you did and In terms of multiplication, the first edition If there is a word of then in terms of edition Let's verify the associative. then write Let's take the heading. So I wrote this heading.
Took. Associative Property with Respect to Edition. Now what is this property? it He hasn't told. You must remember it by heart Needed What is called associative property?
Is? In which the order does not matter. That The property is called. like if a On one side let a be a + b + c and on the other side let a + b + c then the associative property tells us that That is, you can add any two first.
do it. For example, on this side of the equation, if these Add two and on the other side of the equation If I add these two, it gives equality The above doesn't matter. A small We also do it by example so that many children It should be easy to understand. as if For example, we're going to do this: number But this is a fractional number, so Let's do a small thing like 2 + 3 + 4. Here Also 2 + 3 + 4 is ok? a is equal to this.
b is equal to this. c is equal to this. here also a b c okay? Now you see, if I first Should I solve both of these from here or else Let me solve both of these from here and here If I solve both of these before then Question Pe Fark or Equation Ke Par Fark It won't matter. So even if I do Let me show you what's going on here.
will go 3 and two how many five then + 4 back There's two left here, 4, 5, 6, 7, so look, five.
and 4 9 and seven and 2 9 so look at associative The property says that inside the addition You can add any two before that.
It doesn't matter, okay, I get the point.
If yes, then let us prove it by writing Let's take it here, so I have done this associative I have written the property, now this left hand is in it side and this is the right hand side both Like the previous question, do one by one proof.
Will take. Ok? then split the page in half Let's share it. So it splits the page in half.
I divided it into. Now I write here Left hand side. So the left hand side What do we have? that is a + b + c so First add these two from here.
Will take. On the other hand, others do what they do.
Are. So on the right hand side we have a + b + Then we add both of these in c.
Ok? So let's put values in it now.
do. So the values are written above.
Look at them and write them down. a we have 4/3 equals a + b we have 5/2 Equal and here these brackets come like this Will go. plus c how much do we have? 7 / 4 Equal to. Ok? So let's do this first Let's solve it. So I will solve this. So Solve the bracketed part first.
We do. So the three and two below Find out the factors. So 3 * 1 = 3 to no go. So write it as it is. 2 * 1 = 2 So 3 and 2 give two numbers. to both of these If we multiply then 3 * 2 = 6 so LCM How much did we get? The LCM of these two Six came. So I will write it here Six. Now we'll put the six inside and Three will be kept out. Ok? Remember or not?
Let's put this inside and the number over here Keep it outside. So what will be the answer? 3 * 2 = 6 so whatever the answer is, write this answer above Multiply it by the number. So 4 * 2 = 8 so in the first part we have At came. Then plus in the middle so plus After that, do the same again I will keep six inside, I will keep two outside now 2 * 3 = 6 will become ok then I will tell it again hmm six kept inside two kept this outside Three answer came, now I call this three the one above If I multiply the number by five, I get 5* 3 is what 15 becomes, so I'll write it here.
15 and this bracket is going on so Let's leave it as it is here.
/ 4 You could have written it as it is and put it here Delete it from. If we move further then add If you add eight to 15, you get 23 It will be made. So here we go, 23 down.
Six and here is plus and here is 7/4 what shall we do now? LCM again Will take. Ok? So let's do the LCM again.
Let's take it. So now we will take LCM again.
So when we know the factors of six and four If we do it then 2 * 3 = 6 2 * 2 = 4 then 3 * 1 = 3 to the same as it is because it is not divisible Would. Pay to sorry to was not written correctly 2 * 1 = 2 so both came to one. Now these Multiply it. 3 * 2 = 6 6 * 2 = 12 So what will come of the 12 that I have?
The LCM of both of them will come. So here I wrote 12, this has to be erased. erase After that the same division again inside 12 Will write. If we write this six outside then it will be 6 * 2 How much will it be? 12 so two is left behind. This If we multiply two with this then we get 23 * How much will it be 2? 46 will come here 46 and then plus then 12 inside Will keep the four down the bottom out So 12 inside and four outside 4 * 3 = 12 How much is 7 * 3 when we multiply three?
It will become 21, so write 21 here, after this If we move further, then add both of them.
46 21 so six and one 7 4 and 2 6 so 67 will become 67 / 12 over here now ah this Both are not divided up and down.
Further. So, this is what we have, left The hand side has arrived. So, you write here.
Give me the left hand side. Ok? And now we Let's move to the right hand side. So, Will have to check the values again. So this You can see the values. So, equal to Sign inserted. a we have 4 / 3, so here Wrote 4/3 on it. Opened the bracket. B our The pass is 5/2. + C we have 7 / 4. Correct Is? Now we will take their LCM. So let's go to this Let's solve it further. 4/3 same as before And here the bracket is open, take its LCM So what will be the sum of 2 and 4? 2 and 2 of 4 * 1 = 2 2 * 2 = 4 and 2 * 1 = 2 so 2 * 2 = 4 means 2 * 2 = 4 so the four LCM comes Is. So here we wrote four. After that If we move forward, we will keep the four inside.
Ok? and keep the number here Outside. So what will be the answer? 2 * 2 = 4 so When two is multiplied by five, it becomes 5 * 2 How much will it be? 10 So here comes 10. Now The number here and the number here, that is Whatever denominator here is its When the denominator and the bottom number are the same So you would have written the number above as it is.
Are. So you can take it as it is Even if you divide it, you will get the same answer.
It will continue as it has been doing till now. So this Bracket close, go ahead and move 4 by three Used to be. So this is exactly 10 and 7 17 / 4 has arrived. LCM will happen again. So four more These are three two numbers. Ok? 2 * 2 = 4 Three of the same 2 * 1 = 2 Three of the same So 3 * 1 = 3, so what is the number? 2 * 2 = 4 4 * 3 = 12 so what do we get 12 Will you go? LCM. So here it is written 12 Let's erase it. We will do division again. 12 What if we keep three out here?
Will you go? 3 * 4 = 12 sorry 3 * 4 = 12 so Four answers will come here. Four to Four If we multiply it, what will 4 become?
16 plus then now we will keep 12 inside. I I can do it verbally also but you people So that it becomes a firm habit. Then these four If taken out then 4 3 12 and three from 17 How much will it be if we multiply it? 51 So here it will come 51 and here it will be Remove it. When we add 16 to 51, 16 added to 51 is six and one seven, five and one six.
So here it will come to 67 By 12, so what did you guys get?
right hand side and remove it from here now you can see that your The left hand side and the right hand The sides are equal to each other. Correct Is? You can also circulate them like this or Then there is another way also. You give it to Name two equation number one and give it to Name equation number two and tell me here.
Now this left hand side which is right hand The side is equal to. So here I have it This is how it was explained from equation one and two.
There was one and this was two. Both of these in front Keeping this in mind, I can say that now this left The hand side is on the right hand side Is equal. So in this way the associative law which It has been proved. The final conclusion here Write it down. So here I finally have it Wrote the conclusion. once it is verified That's a + b + c and this means both sides First solved this end and then this end Even after solving this, both the ends will remain equal.
Went. So this is how you can prove it.
Ok? So I hope it's easy There was a question. Just remember the property Needed Similarly, you can find out other properties.
will that do. So I have this property here written for multiplication Associative. Ok? So proof this too.
Is. So what will we do again from Again?
What does this property say here?
Like if from the point of view of multiplication If we talk to a * b * c then the same thing is a * b * c and if a * b * c is on the other side also then from here You will solve these two first and then here If you solve both of these then from this Equality doesn't matter.
The answer comes the same. like 2 * 3 * Should I do 4? Ok? also from the example You will remember the property again. 3 * 4 So now see if we solve it in this order What will be 3 * 2? 6 * 4 here 2 * 4 * 3 = 12 so 6 * 4 is also 24.
12 * 2 is also 24. So basically this The property itself speaks for itself. So Now, in terms of multiplication here Write the associative property of Will give. So here I have associative Wrote down the property. Ok? This thing to you Again, like a * b, write it as ab.
You can. So that's why multiply here Did not put it. Ok? So now a and b are It is being multiplied. But first the page You will have to divide it in half. So this page is halfway Divided it. left hand side our What will pass? left hand side our will pass a b. First, both of these Let's get it multiplied on the side. and here What do we have on the right hand side?
Will you go? The right hand side will be A B C and From here we get B C multiplied.
Ok? So let us now consider the values Let's put it. The values above will be A B And for C, this is A 4 / 3, so here you Write A 4/3 and B how much do we have?
So these two numbers are multiplied together.
They are being multiplied. so therefore Now we will add the multiplier. Ok? Or else You will have to insert brackets. and then b What we have is 5 / 2, so write it here.
We will take 5/2 so this comes to 5/2 and C is our Whatever is nearby, now the brackets have come, so brackets Let's put it in. Ok? A C we have How much is it? C we have 7 / 4. So here Now let's write 7/4. So I wrote this I have 7 / 4 so let's solve this.
There is up and down cancellation in multiplication Can. So 2 * 1 = 2 2 * 2 = 4 So above What is the sum of two numbers 5 * 2? 10 then 10 will come here and 3 * 1 will come down = 3 and what is outside? uh now these brackets After finishing this we put this sign.
So it is 7/4, so I wrote 7/4. Correct Is? After that, if we move forward, then the one above The number will be multiplied from above. So 7 10 7 How much will it be? What is 70 and 4 * 3 Will you go? 12 can be further divided.
This means 2 * 3 = 6 and 2 * 5 = 10 2 6 = 12 Whatever is beyond this cannot be divided.
So what will come to us in 35/6?
The left hand side will come up. Ok? exactly this We will find out the right hand side. Correct Is? So now we can see the value of a from here.
Can you? Like what is the value of a?
This is the value of b. This is the value of c.
Ok? So look from above or from here Look at it. No need to top up again and again Is. So we have a which is equal to Came? It was 4/3. So a is written as 4 / 3 and b is written as Whose do we have? 5 bye I think Let's just look above. 5 / 2 So come here will go 5 / 2 then it is being multiplied Next number. C How much do we have? Ah C We have 7/4, so you write it here.
I will take 7/4. Ok? solve it now Take. And if we solve this, it will be 4/3.
If we multiply it like this, 5 * 7 is 35 If it happens then 35 comes and 4 * 2 = 8 So here comes the eight. Ok? This You can also write it in brackets and Write in the multiplier so that 35 by 8 You may feel better about cancellation. So 4 * 1 = 4 4 * 2 = 8 This is cut. The rest is not cut Can. So 35 divided by how much comes down to Will you go? 3 * 2 = 6 will come. Ok? So this In a way, you guys have the right hand side which that too has come 35/6 so both ends We have proved that we are equal to each other.
Given. Ok? So that's why it's here now We will write the conclusion. So here I have But I wrote the conclusion here on my left hand.
The side is equal to the right hand side.
We forgot one thing here. Here it Let's call it equation three and Then we name it Equation Four.
Because we have named it one two.
Gave. So you can do it differently.
to these questions. separate it and Separate it. But because the question is collected Is. If you ask both things at the same time, So I also did it together. Ok?
You guys should also use some common sense.
do. The main thing is to understand and follow your You can do this to the teacher also. Ok? So his Later we write the conclusion here. So this I have written the conclusion here. Hens It It is verified that this thing has been proved has the associative property that is with Respect to Multiplication of Each Other It is equal. Ok? So this was the question It's simple, just a little longer than it looks.
But if you understand it then you can solve it easily.
Will take. Ok? Let's move on to the next one. So Question number four is for you guys on the screen.
It is visible above. read this then What is being said? Is Zero a Ration Number? Is zero a rational number?
Explain what your answer is Please explain and tell me. This is in the question Speaking. Ok? So let's go to this To solve this, first of all we go here I will write the solution on it. Now explain this There can be different ways of doing it.
But for me, I did what I thought was right.
I am doing it anyway. So first a straight Whatever you tell in the answer because he Whether the question has two parts or not Is. Then explain whatever the answer is.
Do it. So first let us tell you the answer.
I mean that zero is a rational number. So this I wrote an answer straight forward.
Yes zero is a rational number. So this question We have given the answer. Now it remains Please explain the explanation. So let's go Let's write down the heading. So this is what I wrote down the heading of the explanation and Before writing the data inside Explain, I Let me tell you conceptually that I What am I going to write? So look, we have this It is said that zero is a ration number.
So first of all you should remember this What is called ration number? So ration number If you bring this to mind, you will get it This will give any number which is equal to p / q It should be written in shape and where the q is It should not be equal to zero. So these two conditions would be There is a possibility of any number being a ration number.
Ok? So if zero is a rational number then He will have to pass both these conditions. you this You can understand. So zero, which is both terms.
Can pass. How? Like you are 0/1 Will write. Ok? So whose equal will it be?
It will become equal to zero. You are on the calculator Even if you try it, you will get the same answer.
If we make it 0/1 then this will become zero. i.e Multiply or divide any number by zero If it happens then the answer becomes zero. So Now if you look at it from this point of view, What did you understand that this zero is a This pic you can call it and this is below The denominator is q. Correct Is? And the first condition was met. Second condition What was I in? q in the denominator, which is It should not be equal to zero. so you can see that here we have q i.e. The denominator is kept equal to one. Zero K Did not keep it equal. So this is what zero is One will be called a ration number. Ok? So you You might have understood a little bit. Now a little of it Let us explain and write some data. So I have made some statements in the explanation.
I wrote it. Let's read it once. Zero is A ration number. Ah zero is a ration number.
Because it can be written in the form This can be written as p / q Which is the definition of Ah ration number.
So because zero can be written in this form And inside this shape, P and Q are Who do you belong to? Let's keep it from Z.
Of course, it is a good thing if you write this line.
If you don't write, even this much will suffice for you.
will be in the form of P / Q and then Delete this and Q is not equal to 0.
I must write this thing down. These are two conditions.
Ok? So I'll tell you here What is written? So we will do that The explanation is the same as it is now.
I have told you by example that zero We can write it in the form of p / q. and q Which is not even equal to zero. We do so We do. You could have written an example here as well.
Are. Let's talk later. So here I have But I wrote the example. Which ones in the example Can we write the number? A such that A 0 / 1 We can also write 0 in this form.
Because the answer to this is also zero.
Is. Ok? So this is a ration number. Correct Is? And apart from this, you can also write like this You can. Now to write examples also There can be different methods. Ok? So You can follow any method of any teacher.
You can. You can also write examples like this and Or write any number below 0 and If you make it equal to 0 then this example also It would be better if you could write it like this Because now you see that this is our if p So this is our q, so the first condition is met.
and the second condition is what in the denominator of That is, q, which number is this one here This number is not equal to zero because I have written seven here, so when If you solve this on a calculator If the answer is zero then it means zero We can also write it in this form and Because we can write in this form. So That is why it is called ration number. Ok?
So you can write a lot of zeros like this.
Yes, for example, you can also write it like this Follow your teacher's method.
That would be much better. And the thing about it What is this? So look at the place of P and Q which Whatever number it is, it will be an integer. Z is what it is Basically, I will explain it a little higher. Z Which set does it represent?
Represents a set of integers.
What numbers are integers? 0 + - 1 + - 2 + - 3 so it goes on like this Are. So if you look carefully, this will also what is an? is an integer because zero is is present in the set. The intake is also is an integer. + 7 which will be next in this is also an integer. So that's why here in the definition It is written in the form of p / q and The relation between p and q within this p/q Let z denote the set of integers And q is not equal to. I hope you understand It might have gone a little bit. So you already have She must know maths. So I hope I understand He must have arrived. So let's move on to the next question.
Let's go. So question number five for you guys.
is visible at the top of the screen. This Read. So what is being said in this?
State Trichotomy Property of Real Numbers. What does it mean? of real numbers Write the trichotomy property, statement Do. Ok? So the trichotomy property you People should know. So let's go to the most I will write the solution here first. Solution After writing, I will write it here.
So I wrote this trichotomy property.
Ok? You can turn to the last page of your book.
You can go and see. This is the property. And What is said in this? for all values of AB means whatever is inside this statement You are seeing A B, that ab which is Basically they belong to real numbers.
This is the meaning of the above line. This which It is a symbol which means belongs to.
To belong. Ok? a then A and B are real Relates to numbers. and this What does the property say? that either There will be any two real numbers which have these three there will be a relationship from within at one time or So one number will be greater than the other number.
Or one number will equal the other.
Or one number is smaller than the other.
Can. Ok? So this property is yours You can also understand it by going to the previous exercise.
There I explained in detail with examples.
Is. So I just asked you to write it here. So this I wrote it down and this property says the same thing.
that these three lie between two real numbers Only one of these relationships can exist at a time.
Is. Ok? There can be only one connection. So Let's move on to the next question. So Question number six you guys screen It is visible above. So what did it say?
Going to find two ration numbers. You two Know Rational Numbers Between Four Ends Five. Between four and five. So It is easy to understand the question that there are two such Make points to understand. one to four Name it, one to five. So that's four more Understand that this is the number line between five.
someone from among them someone from among them Please tell me two numbers. four and five There are a lot of numbers in between. As If we talk in decimal then 4.1 4.2 4.3 4.4 So you just have to know two numbers.
Ok? So how do you know those two numbers now?
do? In the previous exercise 1.1 you You must have learnt that you are a second-rate person.
To find the intermediate value, we Let us find out the average. Because of the average What does it mean? The meaning of average is Middle value. Ok? So any two numbers To find the value between We will find out the average of those two numbers.
Then we will know. Now we know the average What is the formula for doing it? that he who There are two numbers, the middle number is known.
are doing it by adding them to pay and dividing them Two. Ok? So you might have remembered something.
I will solve the rest and then I will tell you the rest.
You will also understand. So to solve First of all, we will write the solution here.
What is the first thing you do after writing the solution?
do? I want to know the average. So here I wrote down the statement average of 4 and Five. So the formula to find the average What happens? that both the numbers which are I want to know the intermediate numbers. to both Add and divide to pay. So add these If we divide it by two, we get 9/2 Is. Ok? You can convert this into decimal form.
You can also ask questions after doing it. but my Let it remain in this form. So Much better. Ok? So now we move forward What to do? Now we have the middle number What did you find out? Here if this In the meantime, this number, what number here?
Did we get it? We get 9/2. You calculator You can also calculate and see that 9 / 2 Who shows what is there to whom? in the midst of these Whether it comes or not. Ok? So 9 / 2 so You see here it comes to 4.5, so four more This is the number that comes among five. Correct Is? If we look at it in decimal then 4.5 is now the next How will you find out the number? Now because two If we had to know the numbers then either these Find out the number between the two.
Will it come here or you can do some other work?
You can say that the number between these two Find out who will come here. then either You can do it too. But how does he know that do? If you know both of these So then again from this number and this number We will find out the average and the number in between It will be known. Ok? So I hope Yes, you must be understanding. so again Write a statement. So this again I wrote the statement. Now average of 9/ 2 and 5. Now the average of these two is middle We will get the point. So how is their average?
Will you find out? Like the one above. Both numbers Write 9 / 2 and what is the second number? +5 more Divide these into two pieces. So do it here Let's divide by two. Ok? Now most What will you do first? LCM of the above Will take. Ok? So how will its LCM be found?
So look down where nothing happens What happens? There is a forest. Ok? So now Let's turn it down a little bit because This should be in front of it right here. so yes This is now right in front of him. Now I I can solve this further. So look at 9/ 2 + 5 / 1 then when we take its LCM then I will draw a line below. LCM of 2 and 1 How much does it cost? So look, 2 and 1 are 2 * 1 = 2 and one is already there so two alone This becomes a factor. So the LCM we have Two has come. What are the next steps?
Are there? So I told you guys The number here and the number here when one As it happens, the number above remains the same.
It is written like that. But you make it You can do it your way too. as of right now Till I explained to you, keep it inside here.
And I'm going to put the two here outside and If I divide it then it will become 2 * 1 = two.
So whatever answer comes above, write this answer above.
We will multiply it by the number. Correct Is? So what will 9 * 1 become? Nine will come We have. Ok? So what am I here for Will I write it down? If there is a plus in between nine Plus put. Then deal with it Have to get it done. So let's keep two from again Inside. So I kept these two inside. The ones here We kept the number outside. So what does the forest need?
Should I multiply it or should I get two? So to If I multiply it, it will become two. So what The answer is this answer from the number above If we multiply, what will be 5 * 2 Will you go? 10 So this came to us. So Basically when I calculated the LCM of this part You can see it here, I have it What came? here comes. So now to the next step I can write this instead. Ok? So Let's write this. So again equal We will put the sign here. And what about here Is? 9 + 10 / 2 and then what is already below That too. Ok? Equal again remained on top.
This should be inserted later so that the matter is correct.
are. Ok? what shall we do now? A to We will solve this further later. erase this We give. There is no need for this now. So this It disappeared from here. did this after it was erased What will become of the top and top? how many are 10 and 9 Are there? 19 will come up to 19/2 and If there is already two below then it will come like this. Correct Is? Now you should know this thing.
If you don't know then I will tell you.
Where there is nothing there is a forest below.
So when there is a fraction above and below There should be a fraction. Ok? So the number below When it goes up, right? taking up If you go, it reverses. Ok? So What will it become now? This will be done. Let's go to the forest I will write it later or if you write it then there is no problem.
It is not there. No, but you could have done it verbally.
Are. I don't write. Ok? So what is this Will be done? Now this will happen. this further up I will have to do it. So this will be 19/2 The multiplier here will be 1/2. So now you You can see that this number here is reversed.
Went. Ok? What was happening earlier? These It was getting divided. Ok? A lot of people this They do it like this also. 9/2/2 then 19/2 If we change divide to multiply It will become 1/2. So it is the same thing in both.
The way you understand this thing, it is as easy as It will take. Ok? Let's move ahead. Now let's move ahead So whatever is being cancelled, nothing is happening.
So the bottom two numbers will be multiplied.
Four will come. If we do it from the forest on 19th then our The result will be 19 * 1 = 19. So here I'll write 19 and write four below.
So what we have is the second number.
Will go. So let's check the calculator too.
does it even come between them or No. So here comes the calculator dividing 19.
If I make a four, I get 4.75.
So now you can see the calculator here.
I will remove it. 4.75 has come. So 4.5 and 5 4.75 is present in between. So Here it is 19 by four, so write four here.
We give. So these two numbers are You may have between four and five.
So you can make these your answers. Correct Is? This was to explain. delete this From here. So this is your first number. Correct Is? And this is your second number. Ok?
How did you come? Found out. final conclusion Write it down. So write this final conclusion.
Gave. Two Rational Numbers Between 4 and 5 R 9/2 and 19/4. Ok? So I hope Yes, I understand this question as well.
Must have gone. So part of question number seven Number one you guys see at the top of the screen Used to be. So what should he do? Simplifi Have to do it. Simplify means easy form To convert into. Ok? to solve After. So what will we write first?
Suction as always. After writing the suggestion Rewrite the question once again. So The sign of the radical is in question. Index It's five. x to the power is 15. y to the power Is 35. and down in the denominator we have What is the power of g? It is 20. So We have this. Now we have simplified it Have to do it. So look up and down whoever is there Also the variables are not the same. so therefore up I can't do it down below. Then simply now we Eliminate the sign of the radical. So radical To eliminate the sign we do this Convert this from radical form to exponential convert it into form. Then power We should multiply what everyone has.
Will give. So that's a seemingly easy method. So Let's solve it that way. Ok?
So what will you do? This is called exponential If you change it to this, it will come here x to the power 15 y to the power 35 and divide by z To the power how much? 20 and these When converted into exponential it becomes 1/5.
Ok? After that, if we move forward, then now we How will you solve this? Now this is 1/5 That is everyone's power, isn't it? So this We get it multiplied with everyone.
I don't think it explains much. do it now You should because I have given you this exercise.
I have tried every step. And here y is. Ok? So these are multiplying Are. Even if you put it among yourselves, there is no issue.
Even if you don't put it, it's not an issue. So 35 Also it got multiplied by 1/5. and below We have g in the fraction. with the G 20 is the power so 1/5 with this also It got multiplied. the thing after that There is division, yes it happens. 5 * 3 how much It happens? 15 This also got cut. 5 7 35 this too is cut and 5 * 4 = 20 so all three powers are cut Are. Ok? So here I would write four.
Am. So now what will be left behind?
We will be left with three three left behind.
If it was then three would come. How many times is the power of y left?
Will you follow? How many power of seven and z are left Will you go? Four. So this is how you guys have this The question is over. here It would be better if you add multiplier. This way Even if you write it down, because the variables are If they are written together, it means that They are being multiplied. Ok? So Part number two is at the top of the screen for you guys.
It is visible. Now we have to simplify it.
So first of all we will write the solution here. suction After writing, repeat the question again.
Do it. So the question in radical form Is. The index is three. The power of 27 is How much is it? It is 2x. Ok? We call this radical You can also solve it in the form of.
Because your exercise has radical losses have also been mentioned. But most children Teachers or most children will find it easy After converting to exponential. So So let's go, this is also an easy way. We this Let's change it into exponential.
From radical form to exponential when If you change, what do you do? Inside would have written the same values as they are Are. Ok? So 2x here and here What will come out? The power of this hole is It will go to 1/3, okay? This is the number here He comes down. After this further What will happen if we go? Factorise 27 Here you go. Ok? So we will make factors of 27.
So A wrote 27. 3 * 9 = 27, 3 * 3 = 9 end 3 * 1 = 3 so write three three times Can. So 3 to the power is 3. Outside we have 2x and beyond that we have What is? 1/3 Further if we take it forward The power of power should multiply among themselves.
She goes. So what will be the power of three?
A 3 * 2 = 6 will become 6x. outside we have What is? 1/3 So write it as it is.
If you take this a step further, your It will be solved. Delete it here From. Ok? So what will we do now? See again What will be this power of power? mutually It will be multiplied. So what is this Will you go? 3 6x * 1 / 3 So now this cancels out Is. 3 * 2 what happened? Six to 3 How much power is saved? 2x saved. And this is you People will get their answer. Ok? So part Number three, you guys look at the screen.
It is coming. Now let us simplify this Is. So to simplify, first So I wrote the solution. How to do this now Solve? So let's write down the question. To all What is the power of 6 * 3 and hole first? n + 2 and divide by 3 power n + 1 - 3 power n now What do we do? If you look carefully Here 3 is n. Here also there is 3 n and above can also be 3 n. Ok? So if I because here on both sides in the middle There is a minus. There are 3 n on both sides. If I common If I take it, the 3 n above can be cancelled.
Ok? So what do I have to do now? These Will have to do it differently. And here also 3 You will have to bring power n. Ok? So this You have learned things in the past. hopes I hope you will understand with a little explanation.
With. So let the six remain as it is.
and then multiply 3 to the power of n Let's take 3 and raise it to the power of 2. Now Many children are getting confused about this. So See, if the power of 3 is written as n + 2 then This can be done differently also. 3 of Power because the bases were the same, that's why the power There is an ad. Then take them to the previous step.
So we can write 3 power 2 like this.
Because when you solve it, this is The answer will come. Ok? What did I do? This Changed in this step. So this is our I had this nearby. It got changed in this. Its If you move further, then the same thing will happen below.
You can do it separately. So here's the line I pulled it and could have raised it to the power of 3.
Am. Multiply 3 to the power of one once I take it. It doesn't matter whether you write forest or not.
It would have happened. So let's write it down. Let us tell you It will work, I have separated n and one.
What will happen next after this? 3 Power n. Just like you planned do. The one inside these two is 3 n which Yes, I will take common. When 3 to the power n is common If I take it, if I write it above now, then six is like that.
Let it remain like that. Let's write down 3 n.
Ok? And here it's 3 2, so I'm going to divide this by I can also write nine because 3 * 3 = 9 It happens. So 3 n we took out so inside What will be left for us? here I am Let me tell you. A: I will change the color and write this first.
I take it. 3^n - 1 right? Now tell me Am. Look, a here was 3^n. 3 here too Power was n. So I threw it out. Ok? So What do I have left inside? survived inside 3 1 ok? And where nothing remains The forest survives there. Ok? So now hope I do it, you guys must have understood. like now I planned this 3 to the power of n It will be cancelled and now the question will be raised.
Your problem will be solved easily. Ok? So this that six is being multiplied by 9 and six Are. If we multiply them together then it becomes 9 * 6 = 54. So 54 was written. Below are 3 of the Ah 3 If the power is 1 then it remains three and in three If one goes away then two will be left behind. So 54 / 2 Came. Now it can be cancelled. 2 * 2 = 4 2 * 7 = 14 so we will be left with 27 and This will become the answer after being simplified.
Ok? Question number eight for you guys is visible at the top of the screen. Read this So what is being said in this? The Sum of Three executive odd integers is 51. What Does it make sense? k three consecutive odd integers When we add these or their sum will be 51 Is equal to. Find the Three Integers Find those three integers. translation You may have understood to some extent, but Talking about the concept, first of all You should know about odd numbers.
Who is called? Ok? and integers Who is called? I know about integers Not all positive numbers, not all negative numbers And zeros are integers. Correct Is? So now this is being said in the question There are three odd numbers whose sum is 51.
If you know it, please find out and tell me that these three What are those three odd numbers? Ok? So First of all, you should know this What is the odd number called? So Look, let's talk about it a little bit.
Over the concept. So see if I number Should I write 1 2 then 3 4 5 6 7 8 9 sorry little Write away. 9 10 11 12 13 14 for For example, let's write this many numbers, let's write 15 also.
Am. There are enough. Ok? Now who has the odd number?
It is said? Numbers that are too deep to divide If they are not, they are called odd numbers.
Like I had read in the lower classes. Joe to Pay Those numbers which are divided are called even numbers.
goes. So which three do we need? Us I need three. Let's make a box here.
We need three odd numbers whose sum is 51.
So now what is the odd number in this line?
The first is an odd number. After that the third odd number Is. Fifth is the odd number. Seventh odd number Is. Ninth 11th 13th. This way it goes on We are going on and on. So now you are here What are the notes that leave a number An odd number is present after it. So one thing you Note that now we need this blue color.
There are circles of which you have to tell the number What number do we add three to here to get 51?
Come. So now like this kind of exercise There was a question inside also. a special kind There is a pattern to find out odd numbers.
How? See, like you need three odd numbers.
First let me write it here.
Wait. Ah first odd number from black color here I write on it. Ah first, second, this third.
Ok? So, now you see that ah if the first Let's keep number one, okay? So, after the forest To find the second odd number, we need What will need to be done? So, inside one if two add If you do this, you will get another odd number.
Will go. And if you add four inside one If you give it, you will get the third odd number.
Will go. Let's check it. So check if Look at this, what is two and one?
Three. And what is four and one? Five.
So if you look here, The first is one, the second is three and the third is five Is. So here also one, three and five are our It has come closer. And what does consequential mean?
It happens? Continuous So look at these three odes.
The numbers are consistent. So this way Consecutive odd numbers can be found.
Any three consumables. Ok? As You can find out three from anyone. like 9 10 9 11 13 these are the three elements of this formula Can be found out with help. Yes Absolutely. For example, put the first odd number as nine.
Ok? And so what will be the second odd number?
We will add two within nine. Third number What will happen? We will add four inside nine.
So see, I will add two. And here comes four ads.
I will do it. So now I have all three.
Will go. The first one is nine. The second is 11.
The third one is 13. So now you are here You can see 9 11 13 9 11 13. So like this What do you guys want to learn from here?
Learning is the thing. Blue color only here Let me tell you that if inside the first odd number If we add two, the second one will always come and we If we add four, we get the third odd number.
Will come. Ok? So, we use this rule will use this to solve the question To do. How will they be used now?
If I solve it then you people will understand.
I will come. There is nothing more difficult No. So look at the best way to solve it First I will write the solution here. Solution After writing three consecutive us digits Needed So write the statements here.
Are. So I wrote this statement first.
Odd Integer. Now what is the first odd integer?
will be. Now we don't know who was the first.
It is like that. So whatever you don't know in maths What do we write on it? We are always there Let us consider it as x. So here You must write late also because this is us I am assuming from my side. So lie down Whose first odd number was kept equal to? x K. So What will the second odd number be equal to? So this I wrote the statement and it was a simple matter.
If we add two inside the first odd number So the second one will come and if we add four then The third one will come. Well then? So we have this I don't know which is the first number. But we This is known because the executive constantly If it is an odd number, we will always add two.
So the second odd number will come and four add If you do, the third one will come. So that's why we We will write x + 2 here on our own. This Let's write another statement like this. So this What will be the third odd integer? So the first If we add four inside x + 4 then we have The third odd number that comes near you will come. Correct Is? Is? So these odd numbers that we have in our Assumed from the side. What is the situation now?
It is being said that those three odd numbers are their The sum is 51. Ok? So according to Let us write down the given condition. So I wrote this I have given according to given conditions and given First odd number according to the condition. Its Add in the second odd number, which is x + 2 And add the third odd number inside it, which is x + 4 we put. These three are what I Write here from the point of view of the edition For. And the person asking the question has given the equivalent of this How much did you tell? 51 so we will write 51 here.
Ok? Now open the brackets. Then x + x + 2 + x + 4 because there is a plus sign outside There will be no changes. 51 came. How many x are there now?
Total is 3x. So this is 3x. 4 and 2 How much is it? six. So 3x + 6 over here Will go. And here it will come 51. further we We will take this forward. So we get the value of x I have to find out because I know the first odd number.
Once it happens, the rest will also be known. So So ah, if we move 3 x 6 over there, 51 and when this +6 goes to the other side it becomes -6 Will go. So here it becomes -6. So what Left? 3x = 45 If 6 are removed from 51, 45 remains.
Now we're multiplying by three here x With. This also has to be taken to the other side.
So what will happen if he goes to the other side?
It will be divided. Ok? so 3 1 3 one here 15 is left over, 3 * 5 = 15, so basically x Whatever it is, how much did we get, 15 came, so x We found out that it is equal to 15.
So where is the x in the query?
If we look inside the question, we have this x If it was kept then we came to know this from the question This is the first odd according to the given conditions.
Which integer is it, it is 15. Now 15 Deposit two inside and another one will come. Four Deposit it and the third one will come. Ok? so the same Follow the formula. Ok? So here We write it down. So I wrote here 100. After writing 100, you would have written the statement.
Are. So I wrote these statements.
Because we have to find out three. So the first We put it equal to x. So the value of x So from here we came to know. Since it is 15.
So the second one has to be found out so inside x If we add two, we will have the next one.
Will go. So if you deposit two within 15 So 15 16 17 will come to us. Similarly If we want to find the third odd integer then If you add four inside, the third will come.
Ok? So if we add four within 15 If we give it, we will get 19. Ok?
So basically this was the first, the second, the third Was. We just put in the value of x. Correct Is? So what three integers did we get?
15 + 17 + 19 so even when you add it, you can see Let's see whether 51 comes or not. So this The calculator is here 15 + 17 + 19 so now you Look, here it is 51. So this means that Our question is exactly the correct number.
We know sorry it is 51 so absolutely correct Found out the numbers. Ok? So the question The method of solving can be any.
Ok? But my main aim is to To make it capable that you can use every method You can understand it well. So you can do it this way too You can. This is also a perfectly fine method. Three When I asked for the number, I got three numbers.
We let you know. So question number nine You can see it on the top of the screen.
If you read this, what is being said in it?
Abdullah picked up 96 balls. Abdullah has 96 balls which are picked up and placed Into Two Buckets. And give them two ah buckets Placed or inserted inside. You can understand anything Are. Ok? So, let it be said that Has been. What are you saying next? One Bucket Has 28 More Balls. uh what's inside a bucket There are 28 more balls. Then the other bucket? Second Regarding the bucket. What do you say in the question now?
What are you asking us? Sorry.
How many balls were in each bucket? Now both How many balls will be inside each one?
You tell me this. Ok? So the overall translation So I heard it, but let me explain it with a concept.
Bucket means you can do bucket, talkery.
Anything that can hold a ball Could. Ok? So, for example, this is the first Let's take this bucket and let's take this other bucket.
Let's take it. Ok? What did Abdullah do?
Is? Abdullah bowled a total of 96 balls.
Have picked up. Now what did he do after picking up 96 Did? Some put balls inside it and some Put the balls inside it. Ok? with and I told you that I am not Telling how many balls are in the first bucket How many balls are there in the second bucket?
So you have to tell me. I'll just tell you this I give you one of these two inside which There are 28 more balls than the other one.
Ok? So now you must have understood the thought.
We have to know and tell here that every How many balls are there in the bucket that He should also justify the situation that a There may be 28 more inside the basket. So easy It's a small question. There is not much issue in this.
If I solve it, you will understand.
So to solve it, first of all We write the solution. So here I am Wrote the suggestion. then here Statements will have to be written. Inside the question The data that has been reported. So first of all, the total He writes down the balls that have been mentioned. So this I wrote the statement, the first statement is this Total balls will be made because Abdullah How many balls have you talked about in total? 96 balls have talked about. So next to the total ball here we I will write it on it, I can also write 96 balls Whether you write further or not, it will still work. Correct Is? After that what is the next situation given that I have talked about two buckets. a bucket and this See another bucket. So you can do this in one bucket.
You can also write a statement from the point of view yes. You can also name a first and a second yes. Ok? So let's just say that How much is there in each one? So we make this Statement by first and second Let's make it. So I write it down. So this I wrote the statement. a number of balls In the first bucket. Ok? Now this is for us I don't know how much is in the first bucket and how much is in the second.
How much is there in it? So whatever is inside math He doesn't know that he would have assumed something from his side.
We do it. Ok? So here you x I will write it down. So in the first bucket we do duty Let's say there are x balls. Similarly, the second bucket How many balls are there in it? So this is what I Wrote the statement quoting the second bucket That number of balls in the second bucket. So now He did not tell this in the second bucket How much is it but it is told that for one bucket Inside there are 28 peacock balls which are one of There are more inside than the other. So if This is the first bucket. Ok? So the second Now 28 more balls will have to be bowled inside. So the first If there are 28 balls more than x then x + 28 Let's write it here. Ok? Understand this You people should come. So look at me I will try again. those who understand The video has arrived, so you can forward it first We don't have the number of balls in the bucket.
Address. I kept the x. Now how much is in the second bucket We don't even know if there are balls. But Because in the question he told that the second 28 more balls in the bucket than the first one Are. So this is the first ball inside the first bucket.
I wrote down the number of balls.
Ok? And also added 28 so that Now we don't know what's in the second bucket.
We'll find out how many balls there are now.
But whatever it is, there will be 28 balls in it.
If there are more, then this statement is correct Now, from this point of view, you will see You can do that inside this bucket i.e. seconds The first one inside the bucket is 28 There are too many balls. So this statement means that I made the first and second ones. Hope I understand the question as per will be. Ok? So now the situation of the question What can we write according to this? I I will write the statement. So this is what I Statement written according to a given Condition. So the condition given to us is that a How many balls are there in total? 96 and the two buckets Has been done. How many balls are in the first bucket?
That is x. So you would write x here. And How many balls are in the second bucket? this is We don't know. But if 28 inside the first If you deposit then the total will be in the second bucket A number of balls will be made. So therefore x + 28 I can write about the second bucket of balls Instead of numbers. Ok? So here I am again I will write it here. put brackets here too Let's take it. Let it be easy for you x + 28.
So this is how the question will proceed.
The same thing has been said in the question that the total There are 96 balls. This is in the first bucket if x If there is a number then how much will be there in the second one? x + 28 Because 28 is more. Ok? and both Only if we add, it will become 96. Ok? So I hope this makes sense.
So let's clean it up a bit and write it down. So The first bucket contains x numbers and the second bucket contains The number of balls that will be formed inside the bucket is x + 28 will be made. And what is this total equal to?
Will it happen? Because Abdullah faced 96 balls.
If it was then it will become 96. Ok? Now we make this You can calculate the value of x.
By doing. So if we open the brackets then x like this If the plus sign is outside then the sign inside If we do not change then x + 28 remains the same and 96 will come here. Now this is 2x so they If you add it, it will become 2x. 28 here plus then we have to find the value of x Because this is what is revealing the number.
So that's why we will take it there on 28th. when it's there If you go here, it is plus, so go there It will become minus. And further will follow So how much is left when 28 is taken out of 2x = 96 Will you go? So one cannot go from six to eight.
A gain or a carry is 16 Out of these, 8 were lost and 8 were saved. a carry here If I did, eight survived. If 2 out of 8 go then six So 68 comes here. Now to here It is being multiplied. on the other side of this to find the value of x Can. So what is 68 by? Two. Now If this two pay is divided then divide it.
2 * 3 = 6 2 * 4 = 8 so the value of x How much did it cost basically? The value of x comes to our Pass 34 and this x stands for what?
Is he representing? This is the x If you look at the question, x is the number of Balls means representing the number of balls doing. Inside whom? of the first bucket Inside. So inside the first bucket we find There are 34 balls left. Ok? So the second bucket Now this can also be easily known. in 34 Add 28. Ok? So I hope You must be understanding. So now you are here I will write on it. So let me tell you one more thing Was. These are the values we have for x and y Sorry x, I kept it on my side. So you are here Whatever it is, I will definitely write it later. whatever the thing If you do it from your side then lie down there Let's write. Ok? So now let's say we wrote 100.
100 means how much in the first bucket Will there be balls? So let's write the statement.
So I wrote this statement here.
Number of balls in the first bucket. a so that x It was kept equal to. So the value of x is 34.
Is. So there are 34 balls in the first bucket.
Ok? And that too in the second bucket Let's write the statement. So this is what I Wrote the statement. a number of balls in Second bucket. So now in the second bucket There are 34 balls in the first and 28 balls in the second.
There are more. So 34 34 plus 28.
will that do. or from this point of view If so, what was x equal to? ah sorry First, it was set equal to x. First bucket.
The number of balls in the second bucket is x + was kept equal to 28. So therefore you Understand it in some other way. So what am I here for Will I write? x + 28 is okay? Because this is the first Showing the number of buckets is x and 28 Because it was too much. So what is here Will you go? You have found 34 values for x.
Are. So 34 + 28 will come. and in the end How much will it cost? 34 plus 28 equals eight And 4 12 a 3 and 1 4 5 6 so how much is this Will you go? 62 so what's in the second bucket is 62 Balls are present. Ok? So after that, whatever Yes, let's delete it. So I hope Yes, you people should understand this question.
I must have come. a bit technical But it is very easy. this to someone else You could have done it this way also. But this small If there is a method, children prefer the shorter method.
Are. That's why I did it using this method Gave. Question number 10 for you guys is visible at the top of the screen. Read this So what is being said in this? Salma Invested ₹350,000 In a bank. What does it mean that Salma A bank has so much investment money Inside. Ok? You should know this that whenever a person invests money So because the word invest comes here Did you go? So in return, that is why investing so that he gets more I can get the money later. Ok? He again The bank is telling you how much it will give.
In the question. A which paid simple and Profit at the rate of 7 * 1/4% per annum. Now The bank is telling him that if you I am telling Salma that there is so much money in our bank.
that I will invest this much money in the bank So we will give you simple profit based Will give money. What does enum mean?
Year by year. And what rate will be? The rate will be this much percent. 7 * 1/4. Correct Is? I will tell you in simple words Look here you have made simple profits.
One has to be careful because its formula which Yes, it is different. Profit interest in There are different formulas. different There is a variety. Like you did in the past Word of compound interest in the question Must have read it. Ok? So the formula for that is It is different. Simple profit is here The formula will be different. So restore whatever The bank is telling him that if you If you are keeping the money then we will return it to you after 1 year.
Yes, you can increase the money by this much percentage.
Ok? The matter is not finalised yet, the bank of. Right now, we have talked about only one year.
Ok? What is he saying next after that?
After 2 years 2 years later. Ok? That means now What do you mean? Let's read it first. The rate was Increased. The rate will be increased.
How much will it increase? to 8% per annum reduced to 8% Will go. Ok? That is, he is saying that If you keep it for 1 year, then you will That means you will get money at the rate of 7 * 1/4%.
You will get benefit from this. But if you 1 If you keep it for more than a year then we will condition or stipulate that Will benefit you more. and how much Will it be beneficial? As written here After 2 years, that is, if you have invested money for 2 years Did not withdraw from the bank. Ok? to the rate to Then we increase the percentage of profit.
7 * 1/4 to 8%. Ok? So if Salma did not withdraw money from the bank for 2 years If he is allowed to stay in the same place then after 2 years he will be Start getting benefits as per percentage Will go. This bank is telling him. Now the last What am I saying in the question?
Find the amount you owe Tell me. She had what she had at the end of 7 years, at the end of 7 years, that is, 7 years after. Ok? So what does that mean for 7 years?
How much money will Salma have now? it You tell. So, to the extent of translation, I understand I have arrived. Overall, this is in the question It is said that Salma had given ₹3,50,000 Whatever you have, invest it in the bank. The bank is calling him If you do not withdraw the money for 2 years then According to this percentage, we will give you simple will give profit and if you continue even after 2 years If you keep money in the bank, we will make your profit.
Will increase it by 8%. Now the questioner is speaking That is right according to this, you understand this Let's say she doesn't withdraw money from the bank for 7 years.
And if he does not withdraw the money for 7 years How much money does he have from the bank?
It is in his hands to tell this. Ok? So now You must have understood the question. This Concept solving also means according to the concept Let's do it right now. Like before this Up to 7 years have been given in the question No. Ok? So look at seven to seven years.
Salma's money will remain with the bank.
Ok? Now because two conditions have been given, right?
He. The percentage in the first two years You will get benefit at the rate of 7 1/4% And this much benefit was received for 2 out of 7 years.
Then again, if you subtract 2 from 7, you get 5 Years were saved. Then the remaining 5 years will be There was a profit of 8% in that. So now 7 How much money will there be after a year? 7 years later his Whatever money he has, whatever money he has He had planted it in the beginning. Ok? ₹3,50,000 Firstly, it will be money. then after 2 years How much money will there be after that? of this percent We will make money accordingly. Ok? So that The amount will come. Then how much money after 5 years Will it be done? According to this percentage we will calculate the amount Will make. So they will write that amount here and After this the total amount of money will be taken out.
Which Salma will get after 7 years. Correct Is? To put it simply, I want to say this How much profit will you make in the first 2 years?
Whatever happened, he will find it out. Then profit in 5 years We will find out how much it cost. Both of them here I will write on it. Add the amount that is inside it and then by collecting these three you I will withdraw the money. Ok? So now I Fundamental core you should understand the question Went. Now I will solve it and you will understand.
Will remain. So, to solve it, first of all Here we will write the solution and use it to solve it.
There can be different methods. Ok?
But you should understand everything. Correct Is? So let's first look at the main data, which is the question.
that they write the form of the statement In. So only two things are important in the question One is that what is the total time period If time is in question then total years And what does he have to do after 7 years? Find The amount he needs to know How much money will I get? So these two things are That question will be my main highlight.
According to. Ok? so that's why I ah time I will write the period here, seven years more After that total amount of money after 7 Years. We are going to find out that. So Let's put a question mark here. So Two things happened. Now as I am Said that he would get Rs 3500 plus in the first two years.
I will get the benefit accordingly. then the next You will get this benefit in 5 years.
So that's why the question is also divided into two parts.
Will divide. Profit for the first 2 years I will withdraw the amount first. then for the next 5 years Will withdraw the profit amount of Rs. So let's go Let's write the heading here. So this is what I Write the heading here Finding Profit for 2 years. Now click here to find out your profit.
What will be the pay formula? He is the one you know Should be. Only then will you be able to move forward. So Call it profit, call it interest, call it anything.
You can. The main thing is to solve the question.
Here the formula will be PTR. Ok?
Now what will P mean? You last You have learnt it in the questions. Meaning of P The principal amount will be Ok? Principal What is the amount? Ok? I am like this I do it, I keep writing. Then I Let me tell you. The main thing here is this formula You have to keep PTR in mind. Correct Is? So let's make statements according to this Let's write. So I have these three here Wrote the statements. Now I can tell you Hmm, what is PTR called? So P here means principal Amount. Ok? So here I wrote P Took. What is the principal amount?
I will tell you again. an amount that The whole story is going on above or on whomever The initial amount of money That is called the principal amount. As Salma had invested Rs 3500 on this.
Profits had started coming in. So here it is This will be called our principal amount. Ok?
So write down ₹3,50,000. Ok?
So here I have written ₹3 lakh along with it.
You can. You can also write it at the end, without You can also write for rupees. Now it depends It depends on you that whatever question you have How he looks fine. If you write It's a good thing. So this ₹3,50,000 is ours The principal amount has been paid. and in the formula What do you want? Like I said here The formula is PT R. Ok? Then from P Principal amount. What will happen to T? Time.
Now how much time are we talking about? We We are talking about 2 years. because he The profit rate for the two years in question is different.
It is kept. So Time with a capital T I will represent you. And here the time Yes, he is 2 years old. So here pay two years Write empty two, write two years. So his What does R mean at the end? the rate at which We will get profit accordingly. So the rate This is what we have: 7 * 1/4. So let's go to this Let's calculate. So here I am I will write it as R. Let R represent rate.
Let's take it. And what is the profit rate? 7 * 1/4% Ok? How much will this amount be now?
Let's eliminate this fractional form.
So to eliminate this fractional form Bring a calculator for it. So here we come Calculator. Is 1/4. By dividing this 1/4 Write it in decimal. It's easy for you Will go. So this one divided by 4.
I will get 0.25. So inside the intake If we deposit 0.25 then the value we will have If it comes, how much will it come? I now this I will write it here and put equal in front of it.
Just do it 7.25 and definitely charge this percentage Please put it because to finish it 1 /100 has to be written. Ok? So here it is Let's delete the formula from. So hope I do it, you must have understood till here.
I just converted it into decimal form.
Did. Now you should remember the formula by heart.
Needed So you can write here we know that Are. So let's do this now on top. Above After doing this I wrote here we know That. And now after writing We Know That I will write the formula. And what is the formula Will we have it? Profit. Ok? Ah Is equal to P * T by R. Correct Is? So what is the value of P? Above ₹3,50,000 then here we will write ₹3,50.
Ok? After that T is two, so two Wrote it. R is 7.25 so 7.25 And I put the percentage sign here.
Ok? We will solve this further. Further If we solve then when 3500 is divided by two If you multiply it, it will be double the value.
Will go. So double 3.5 becomes 7. So I What should I write here? 7 lakhs then this I wrote the amount of Rs 7 lakh. Ok?
After that solve both of these and come here.
Went. What's next? See 7.25% Is. Now how will you solve this? Rough Area Let me tell you. So look at any number what would you do if you wanted to eliminate decimals from Are? A forest is placed beneath it. After that Whatever digits there were, they would write zeros in their place.
Are. Ok? So after that there are two digits.
Two zeros will appear below. So at 7.25 if I want to solve it. Ok? So We will discuss the percentage later. 7.25 If I want to solve this then instead of this I can write below Will go to 100. So how much will this equal?
725 / 100 so this is how I end up with the decimal I will do it. Ok? So here I write I will take a 725 /100 ok? After that multiply again.
Now I have given the percentage in the previous question.
I told you, but I'll tell you again.
Those who are looking at this question that percentage The sine of is always equal to 1 / 100 of.
So whenever you want to finish it, then In its place, write 1/100 and multiply it.
Do whatever comes along with the number here.
But. Ok? Why does this happen In detail I have already told you in the previous question. So So let's save time here and this Instead of percentage, we will write 1/100. Correct Is? And let's delete it from here. Now How will you solve this further? So see All these are multiplying among themselves.
Are. If you understand them in such a way that There are 200 below. If this value is above then whatever the values above and below are Can cancel each other out through the table Are. Ok? So therefore this denominator which Yes, both of them can cancel this also.
This can also be cancelled. So here We can reduce the zeros. Such as There are two zeros here. With their help, here Two zeros were reduced. Then two zeros here too Are. So with its help two zeros and one cancel out.
Will go. There are ups and downs. So the one below It doesn't matter whether you write the forest or not.
It would have happened. So the remaining amount What will we be left with? Only 70 and 725.
Now multiply them together. Correct Is? When we multiply these, it will come Gaya calculator Ah 70 * By 725 So we get ₹750 Ok? So I'll write it here Value in Rupees Answer always with unit Let's write. So therefore ₹5750 So this is my final amount of profit.
Took it out. For how long? within 2 years Salma earned this much money from the bank Are. Similarly, you could have taken out the next 5 years.
If so, we will go down again and all this We will repeat the process. So the heading Let's write. So I have made this heading here.
I wrote it down. Finding Profit for Five Years.
Now the same thing will happen for five years also.
Write the principal amount, write the time, Write the rate according to which the benefit is being given Yes, bank. So let's write the statements.
So I wrote down these three things.
What is the principal amount? He So the amount at the beginning of the question, Salma The ₹3,50000 paid by him will remain the same. Ok?
So here in the principal amount ₹3,50,000 will come. Ok? So here I wrote Rupees ₹3,50,000 Wrote it. Ok? After that, the time duration Now we have a capital T representing Are. So that's 5 years. So here 5 Wrote years. Ok? After that the rate How much are we being given? rate If I tell you in the question, then the next 5 The bank is offering 8% for the year Is offering. Ok? So therefore the rate Represented by a capital R. So here Wrote R on it and wrote 8% here.
Ok? So now apply the simple formula Will take. Or we know that he will write it. So this I wrote, we know that. And what are we?
I know that the profit that is there, ah What will be the formula? Its formula will be P * T * R then put the values. So If the principal amount was ₹3,50,000 then the same Will remain. Ok? So I wrote this ₹3,50,000.
After that the time will be five now and The rate will be 8%. Ok? This If you increase further then 3500 will get Rs.
Let's multiply it. So it came We have calculators 350 Multiply by 5 and this gives us ₹1750000 So I wrote this here ₹1750000 After that to eliminate the multiplier 8% So divide by 100 will be written as 1/100 This percentage sign is also gone. These Two zeros would cancel these two zeros Are. What is left behind?
7500 Okay, two zeros are cut off, two are still left.
and multiply by 8 one then multiply do it, the same answer comes, okay so now This will bring our answer here.
17000 Make it and make it 500 and multiply it by 8 So ₹140000 came back So here I enter the unit ₹140 Whatever I have, I got profit in the next 5 Of the year. So, basically we have two profits have arrived. The profit for 5 years is this much and This is the profit we have for 2 years.
And now what is the total amount after years He will tell you how much he will get.
So, I have written these statements here.
What will the heading be? Finding Total Money. Now We are finding out. So, know the total money What statement would you write to do this?
Total money after 7 years. Now after 7 years How much amount was made? So his principal The amount was 3500, one because he He will get it back if he gave it to the bank.
How much profit did he make in 2 years?
We have received a profit of Rs 50750 So here we will write 50750 And what was the profit for the next 5 years?
1440000 So here we will write 1440000, now these three If you add this then this calculator is ₹140000 It is already written, add it inside it.
Let's take ₹750 And add ₹350 to it.
Okay, so this is the amount I got ₹54750 So I'll write this down here.
54000 ₹750 So this is basically the amount that Salma will get.
You will get it after 7 years, okay after 7 years.
Is? So this is how you solved the question.
Took. I am speaking again, solve the question Every teacher has different ways of doing it.
Of. So the main thing is to understand the concept. If Once you understand the concept, you can You can also solve the question using this method.
Ok? So with this the exercise ends.
If any mistake is made in the exercise Yes, please forgive me for that. would have tried That there should be no mistake. but still so long Exercise can happen anywhere. Balance Thank you very much for watching the video Thanks. The main purpose of making the video is People have to be made capable of understanding the things that You don't understand tuition in school If it had come here, you would have been able to learn from it and move forward.
Ok? So take great care of yourself.
Read with full concentration and move ahead.
Inshallah see you in the next video Is. Take care of yourself. Allah Hafiz.
Related Videos
A Brutal Radical Expression Made Easy! The Shortcut Changes Everything.
tamoshop
112 views•2026-06-02
V : jee main /advance class 11 mathematics : Binomial Theorem class-1 ( 29 may 2026 )
dcamclassesiitjeemainsadva9953
125 views•2026-05-29
Is This Pentomino Tileable?
3cycle
241 views•2026-05-30
This Sudoku Has Many Lines!!
CrackingTheCryptic
2K views•2026-05-29
Olympiad Mathematics | Indian Can You Solve This One?
PhilCoolMath
268 views•2026-06-02
Olympiad Mathematics | Indian | Can You Solve This?
PhilCoolMath
669 views•2026-06-02
Can you get the Correct answer for this Math Quiz?
Fendora01
24K views•2026-05-29
NUMBERBLOCKS COUNT THE TOTAL SUM OF TEN NUMBERS | ADD SMALL TO BIGGEST NUMBER | hello george
hellogeorge2294
5K views•2026-05-28
