The topic is ratio. So today we'll discuss like how we use ratio and where we use ratio. Important thing you need to understand, ratio is something what we use everywhere and a lot of places all around you. Let's say you need to find the ratio of your wealth to your friend's wealth or you need to find the ratio of like your score to your friend's score.
So in all of these cases what we use ratio. So ratio is nothing but a comparison. A comparison which basically includes two values because we can only compare the values.
values not the quality so remember one thing ratio is only defined for the quantifiable things if i'm say i'm a very good guy and you're also a good guy so i cannot make a ratio very good to good because these are basically qualitative so what i need to understand ratio is defined on for quantities only the next thing we need to understand in ratio so exactly what is ratio where we use ratio and what exactly are some famous ratio and where we can see all the ratio in this universe also so So let's begin. So you can see like I have already discussed where we can use ratio. So ratio we can use in everywhere like anywhere you need comparison we use ratio. Anywhere you need something like let's say comparing your salary to your friend's salary, comparing your height to your friend's height, comparing your weight to your friend's weight.
So anywhere when there is quantities defined and similar kind of thing we use ratio. Sometime we also use ratio in another thing also which is basically the weight height ratio. Because weight height ratio is something which is not the exactly ratio but we are basically using weight and height for a particular thing so that is kind of an exception in case of the general concept of ratio now let's talk about something which is a very important ratio we all know the ratio is known as golden ratio you can read this about in like angels and demons you can read it this golden ratio in basically the vinci code so in all of the books also you can read this kind of ratio and it's actually everywhere think about your hand the ratio of this length to this length in your hand this length to this length line this is the golden ratio think about a butterfly a butterfly's wings are basically also in the golden ratio think about an egg the egg is basically the top and the bottom is also in the golden ratio and think about something else in universe everything we are having in golden ratio not only that if you see about the mona lisa the picture made by like leonardo da vinci one of the best creation that is also basically being maintained by golden ratio like this height like the official expression all of these things we are basically using this golden ratio ratio everywhere. So this is where we can use ratio.
And this is one of the most famous ratio. And the ratio is around one point six one eight. So if you see about a butterfly, so you can understand the ratio of this one is basically Section A. So you can see this portion is basically portion B and this portion is basically around portion A.
And this portion, I can say that this is if I write A by B. So this is also given you one point six one eight. And this is one of the magical ratio we can have, which is we known as golden ratio.
ratio. So this is one of the most utility part in a very famous ratio which is golden ratio. Now we will talk about what is the ratio, the definition of ratio and how we define ratio. So think about the definition.
The definition of ratio is basically something is like the comparison what I have already told you. So how we define ratio let's say a quantity. Let's say a is height.
So a is height is basically 20 centimeter and let's say b is height. So b is height is how much? That is let's say 10 cm. So what is the ratio?
Ratio of A to B is basically 20 upon 10 which is also same as 2 upon 1. So this is one simple example of ratio, how we can use ratio. So this ratio of A upon B is also can be written as A is to B. So this is also about the ratio.
So the ratio we can write as A upon B, we can write as A is to B also. So one thing you need to understand, this B quantity is equal to B. So this is quantity cannot be 0. If the b quantity is basically becoming 0, the problem we majorly face, this will become undefined. So please understand this quantity, you should not make it 0. Am I clear?
So this is the definition of ratio when we compare. So ratio of a b is basically a to b or I can write this is a is to b. So this is to is basically the sign of ratio. Now we come to the properties. Now think about the properties of ratio.
So ratio, if I If I write, let's say 1 upon 2 or 1 is to 2, this is same as 2 is to 4. And I've already said 1 is to 2 can also be written as 1 upon 2. So this 1 is to 2 is also same as 2 is to 4. So these are basically the same ratio. And remember one thing, generally we always use the ratio in the simplest form, in the simplest, like simplest natural number form. So if the ratio is given.
like 2 is to 4, I can write it as 1 is to 2. And similarly from this, we can define this thing also, a upon b. Now if I say a upon b, say a upon b is basically same as k upon kb. So I can multiply same thing if k not equals to 0. if k is 0 because this thing will be undefined so that's why k has to be not equal to 0 but you multiply with anything the values won't change similarly the same example let's say 3 by 5 I am multiplying it's 3 into 2 by 5 into 2 so you can see there is no change in the values similarly in division also so if I a upon B is same as a divided by K upon B divided by K when I have a division the change of value is none so I can write again the same thing let's say 4 upon 6 is nothing but 4 divided by 2 upon 6 divided by 2 which is 3 which is sorry 2 upon 3 so this thing you can see like that is basically for the ratio so this is 2 upon 3 so this is basically the properties of ratio the next thing we can have that is a upon b equals to this thing so what is a upon b equals to d so this is called a constant thing so i can say if i have a upon b equals to d let's see if i say a up then i can say also a equals to bd please understand this and next is a is to b is same as c should be what does it mean it means a upon b is same as Cd upon D.
So we can say A and CdD are proportional. So if I say A and CdD are proportional, that means A is to B is same as Cd is to D. This is just the basic of ratio.
So let's say if I say 1 is to 2 is same as 2 is to 4, I can write it like 1 is to 2, like 1 is to 2 is same as 2 is to 4. Am I clear? Now if I can write like X is to Y is same as 1 is to 2. So from this thing I can write x upon y is nothing but 1 upon 2. So this is about the ratio. This is about like the x, y and 1, 2 are proportional.
So from ratio we define also proportional. I hope you have understood this. Next thing we are going to discuss about the comparison of ratio.
Like how we can compare ratio. Important thing let's say you have too many fractions or too many ratios. How you compare these fractions or these ratios?
Because ratios if you have this kind of division form it becomes a fraction. You understand? Because half is basically... basically nothing but a fraction.
So next thing we will understand how we compare these things. So comparison of ratio. So first let's think about this.
If I need to compare these numbers 3, 4, 5, 6, 7, 8, 9, 10. So think about an easier number first. Think about a number which is 2 upon 3 and let's say 4 upon 7. How we compare these numbers? So if we have 2 upon 3 and 4 upon 7, how we can compare these two numbers? Let's think about it.
The first thing you need to need to understand the very common method is basically called cross multiplication method. You just multiply like this. So 7 into 2 is basically how much? 14 and 3 into 4 is basically 12. Now as because 14 is more than 12, I can say 2 by 3 is more than 4 upon 7. Think about it, you just need to make the numerator. So the numerator is basically 2 and the denominator is 7. So I'm multiplying the numerator of the first one with the denominator of the second one.
So So 2 x 7 that gives you 14. So I'm writing it down also. 2 x 7, what is it? It's giving you 14. And 3 x 4 is basically giving you 12. So if I am comparing 14 and 12, so 14 has to be more than 12, right?
If 14 is more than 12, so I can safely say 2 x 3 is more than 4 x 7. So this is one way we can understand the comparison, right? The second way we can also do it by LCdM method, right? Now in case of LCdM method, it's the same thing. We just need to find the LCdM.
the denominators. So what is the LCdM of denominators? That is 3, 7. The LCdM is 21. So it's 2 by 3. I have to make the denominator 21. What I need to make the denominator in 21. So if I am making the denominator 21, so I have to multiply into 7 with these two things. So it will become 14 upon 21. So I can say 2 upon 3 is nothing but 14 upon 21. Similarly, do about 4 by 7. Now I know I have to make it 21. because 21 is the LCdM. Now what I need to do, I just multiply with 3 in numerator as well as denominator.
So this becomes 12 upon 21. Now compare between these two things. 14 by 21 is more than 12 upon 21. So hence, 2 by 3 is more than 4 upon 7. So this is a generic method. But we are not going to learn this method.
I'm just giving you this method just as the basic, like how we generally compare ratio. Now think about it. Like if it has like 5 kind of ratios or 7 ratios, how we can deal with that? We cannot make LCdM with everything.
And this thing only, this cross multiplication only happens for two quantity if there is two quantity then we can only use this cross multiplication but think about for five quantities you need to make it for first two the next two the next two so this is a very hectic process so we are not going to do that now what we can do let's see now now if it is a comparison of ratio let's think about this numbers three four five six seven eight nine ten right now in case of three four five six seven eight nine ten the first few things you need to understand so this numbers are a particular pattern right and what is this pattern you can see the difference of these numbers like denominator and numerator is same what is 4 minus 3 1 what is 6 minus 5 1 what is 8 minus 7 1 what is 10 minus 9 1 so in every case the denominator is 1 more than the numerator so the difference is 1 so i can say the difference is same and all of these things are proper fraction what do you mean by proper fraction if a by b is less than 1 and more than 0 we can call it is a proper fraction so all of these things are nothing but a proper fraction so remember in this case the number which looks the bigger is actually has the highest value so 9 by 10 looks bigger right and 3 by 4 looks the smaller right so the one who is basically the highest thing that is 9 upon 10 is looks the biggest so it is the highest value next one this is the first next one this is second this is third and the remaining is fourth so i can write 9 upon 10 is more than 7 upon 8 is more than 5 upon 6 is more than 3 upon 4. Understood, I guess. So just understand this thing. If the difference is same and all of these are proper fraction, the thing which looks bigger, that is the highest number. The things which look smaller, that is the smallest number.
You can think about it also. 9 upon 10 is 90%. 7 upon 8 is how much?
87.5%. 5 upon 6 is around 84%. 3 upon 4 is 75%. So you can see that we can also use this thing also.
Sometime we can use percentage also, but in ratio, I'm just giving you the methods for ratio. So the next thing, obviously, like the concept behind this, this is the concept. The concept is if A upon B is less than one, that means it's a proper fraction, right?
It's a proper fraction. So if A upon B is less than one, so I can say A plus X upon B plus X is more than A by B, where X is more than zero, very important thing. So just take this example.
So like... just take this two things, 7, 8 and 9 upon 10. So I can say 7 plus 2 upon 8 plus 2, right? 7 plus 2 upon 8 plus 2 has to be more than 7 upon 8. Am I clear? Now what is A?
This one is A, this is B and this is X and this is X. So I can say A upon B is basically more than a plus x upon b plus x is basically more than a upon b. So 7 plus 2 by 8 plus 2 is more than 7 upon 8. So I can write 9 upon 10 is more than 7 upon 8. So by using this logic also I can say this is in this order.
So it is like from decreasing order 9 upon 10, 7 upon 8, 5 upon 6, 3 upon 4. So that's how it happens. Am I clear? I hope you guys have understood this.
Now the next thing in comparison of ratio. Yes, obviously it can be improper fraction also. also. Now what is improper fraction?
Improper fraction is something where the denominator is more than numerator. So if I say a by b is more than 1, you understand what I'm saying? A upon b is more than 1. So this thing is basically known as improper fraction. Now in case of improper fraction, now think about these numbers. Just the same kind of numbers, just the same kind of numbers.
Like the difference is how much? 7 minus 5 is 2. 11 minus 9 is 2. So in every case, the difference between numerator and denominator is same. So you just need to check whether the difference is same or not.
Now in this case, the reverse will happen. What will happen? The number which looks smaller has basically the highest value.
So here, 5 by 3, 5 upon 3 looks the smallest, right? It looks smallest. So it has the highest value. So I can write 5 upon 3 is the highest. So what next?
Next looks smaller is 7 upon 5. So 7 upon 5 is the highest value. 1 5 is the next highest what's next is 11 upon 9 and what's next that is 13 upon 11 so i've arranged already in basically which order descending order so i hope you have understood so in case of if it is proper fraction the highest looking thing has the highest value in case of improper fraction the smaller thing has the highest value the smaller looking thing the smaller looking thing has the highest value am i clear now what is the logic behind it let's check Now the logic behind is this, if A is equal to 0, then A is equal to 0. a upon b is more than 1. What if a upon b is more than 1? Then a plus x upon b plus x is less than a upon b. Now in case of improper fraction, now I'm checking the same thing. Let's think about a is basically 7 and b is basically 5 and let's say x is basically 4. Just think about this.
Now I can write 7 plus 4. Upon 5 plus 4 is basically less than 7 upon 5. By this same formula, a plus x upon b plus x is less than a upon b. So I can write 11 upon 9. is less than 7 upon 5. So, hence you have understood one thing very clearly. So, what I have written in the previous slide, you can see. So, this 7 upon 5 is actually more than 11 upon 9. So, hence it has been concluded.
Now, next thing. about some numbers which has not this which doesn't have this kind of a common property like this where the difference is same which doesn't follow in this kind of thing like any arbitrary number i'll give you one example if you think about this kind of numbers how you can grade it how you can think about which one is highest which one is lowest or how you can arrange it either increasing order decreasing order it looks difficult because if you want to think about you just want to do like what the lcm method or the cross multiplication method you understand how much you need to calculate If you want to calculate cross multiplication also, you have to calculate 43 into 29 and 24 into 37. And similarly, you do for the other fractions as well. So, it's not that easy.
So, to make it easier, what we can do here? Just check this. Now, the first thing I'm going to say, it is a very important method and please learn it because this will help you to solve this question within 30 seconds. Trust me in this, within 30 seconds. Think about this fraction.
You can see all of these fractions. So, just make a change. This is 20. 29 upon 34. So you can see all these are proper fractions, but they have nothing in common. The difference is not same in everywhere.
So if you want to solve it, if you want to understand it in increasing order or decreasing order, or if you want to find the highest number or the least number, how can you do that? Because if you just want to do the cross multiplication method like this or this, it's a very difficult thing because 43 and 29 will take at least another 30 seconds, 35 seconds just to multiply these two things only. And the remaining two we can add on like in numerous times. You understand?
me. So for this thing, I'm giving you a method which makes this thing happen in a very shorter amount of time, very short, even less than 30 seconds if you practice. So think about this method and see it. So first thing we are going to do, we do numerator upon denominator minus numerator. This is the first thing we are going to use.
What? Numerator upon denominator minus numerator. Right?
Cdlear? So what is numerator in the first case? 29. So 29 upon 34 minus 20. I'm just writing every step so that you understand what is happening. Next is 37, 43 minus 37. Next is 13, 17 minus 13. Next is 11, 14 minus 11. So I have done numerator upon denominator minus numerator for all the cases. So now how much it is?
29 upon 5. How much is that? How much it is? 37 upon 6. How much it is? 13 upon 4. How much it is?
11 upon 3. Now think about it. It becomes so easy to calculate now. What is 29 upon 5? That is 5.8 exactly. What is 37 upon 6?
That is 6.16. What is 13 upon 4? That is 3.25. What is 11 upon 3?
That is 3.67. So these are the numbers I have already got. Think about it.
Which one is the highest? 6.16 is the highest. So this is the first number.
This is the highest number. Next is 5.8, second highest. Next is third highest, which is 3.67. So this is third.
Next is fourth highest is 3.25. So this has to be fourth. So I can arrange it accordingly. It's that simple.
It's that simple. So think about it. So let's start writing down the same thing.
So it is basically 29. First is I'm coming with 37. is the highest number 37 upon 43 the second highest is 29 upon 34 the next is 13 upon 17 and the next is 11 upon 14 now it's easy right it's that easy so you can do it in 30 seconds and you can get the answer exact answer so just use this method so the next thing i'm going to use the same thing for a number which is not proper fractions right now think about this kind of fraction think about this fraction which is 34 upon 29 all these are improper fractions we need to arrange this. Right? Either it increasing or decreasing order. So now we are going to use the same thing.
I'm not changing the formula for every question. I'm just using the same thing. Just use the different approach, right? But the formula has to be same.
That is numerator upon denominator minus numerator. Now here you can say denominator is less than numerator. So it has to have a minus thing in front of it. So I can write 34 upon 29. minus 29 minus 34 Next is 43 upon 37 minus 43 Next is 17 upon 13 minus 17 Next is 14 upon 11 minus 14 So think about it All these numbers, I've written it down So this is next So I can write the same thing as minus because this is minus 5. So minus goes out. So it's minus 34 upon 5. Next one is minus 43 upon 6 minus 17 upon.
4 and minus 14 upon 11 right sorry 14 upon 3 am i clear so all of these things are negative why it is negative because the denominator is less than numerator so it becomes negative so now think about like what is the value of this thing so the value of this thing is basically nothing but 6.8 next is 43 upon 6 that is minus 7.16 next is 17 upon 4 which is minus 4.25 Next is 14 upon 3 which is minus 4.67. Approximately 14 upon 3 is like minus 4.67. Now what is the highest number? Because you know about a negative number.
The smaller the number is, the highest has its value. You understand me? The smaller the number in case of negative. So what is the smallest in negative?
Like the smallest denomination in negative is 4.25. So minus 4.25 has the highest value. So I can say this is...
like the first number in case of when we are arranging it in decreasing order. So this is the highest number. This is the first. Next is this one. This is second.
Next is this one. This is third. And next is minus 7.16. This is fourth. So I can arrange it accordingly.
So how I can arrange it? So I can write it down in descending order, which is basically just the first number is 17 upon 13. The next number is 14 upon 11. The next number is 34 upon 29 and the next number is 43 upon 37. So, I have written all the numbers exactly in the descending order. So, this is basically the descending order.
Am I clear guys? I hope you have understood what I am basically trying to do. I am trying to use this method to solve all of these things in a very, very shorter method. And trust me, all of these things will take less than 30 seconds. Less than 30 seconds.
So now we need to have a question where we need to use both of these things right? Both like some can be like in increasing order, some can be in decreasing order. And we need to use both of these things together, especially like some of the numbers can be proper fraction. Some numbers can be improper fraction. And if you have all of these things together, then what we can do?
So let's check the next slide. Now think about it, we have total 5 numbers. So out of this, these 2 proper fractions, this is improper, this is proper, this is again improper. So one thing I can say, improper fraction is more than 1. So improper is more than 1, a upon b is more than 1. That is improper and the proper fraction a upon b is less than 1. So I can say the improper fractions have to be more than the proper fraction.
fractions obviously the improper fractions have to be more than the proper fractions so if I need to arrange it in descending order just divide it in two part first write down all the improper fractions next write down the proper fractions so if I'm writing the improper fractions this is 19 upon 15 and 27 upon 22 and now I am writing the proper fractions which is 27 upon 32 11 upon 14 and 17 upon 21 so these three are basically all of these three things are proper and these two are improper fractions now think about it we have to do this thing I mean this two independently, we have to arrange these three independently. Then we can write down in the same order because the improper fractions have to be more than the proper fractions. Now if I arrange these two things, so what is like I have to do the same thing numerator upon denominator minus numerator. In this case, this is 19 upon minus 4. I'm writing it directly now.
19 upon minus 4. Next is 27 upon minus 5. Why I'm writing directly? Because I have already used the same technique in the previous one. Next is it is 27 upon 5, 11 upon 3 and 17 upon 4. So I've write down all these numbers directly.
Now if it is 19 upon 4, you can see one thing clearly here. This is basically... Minus 19 upon 4 is nothing but minus which is 4.75.
The value of this is minus 5.4. So, hence I can have minus 4.75 minus 5.4. Next, we have the same thing here which is 5.4. Next is 3.67 and next is 4.25. Now I have told you we need to arrange this separately.
Remember seeing negative do not think like this things are smaller than this. So we are arranging the... improper fractions separately and the proper fraction separately and we know improper has to be more than the proper.
So first I am arranging the improper fractions. So which one is more from this thing? I can say minus 4.75 is more.
So 19 upon 15 is the highest Next will be 27 upon 5, 27 upon 22. Because minus 5.4 is a smaller number than minus 4.75. So it is 27 upon 22. The next few things what we can add here, now arrange these three things. So what is the highest number?
That is 27, 5.4 is the highest. So the highest number is 27 upon 32 here. And next is 4.25. So in case of 4.25, this number is nothing but 27. that is 17 upon 4. So I have write down 17 upon 4 and the next is 11 upon 3 obviously.
So I can arrange it like this way. Now this is basically in the order and I am using the same order here also. So I can say 19 upon 15 is more than 27 upon 22 is more than all the proper fractions and what is the highest of the proper fraction?
It is 27 upon 32. Next is 17 upon 4 and next is 11 upon 3. So this is how we can arrange things. So you understand in case of a mixed fractions where you have a proper fraction. you have improper fractions all these kind of fractions together first you need to separate it in proper fractions and improper fractions arrange improper fractions differently arrange proper fractions differently then use the improper fractions first because they are have the highest value they have the higher value than the proper fractions write down the improper fractions then write down the arrangement of the proper fractions and that you know that's how you have got the exact arrangement in decreasing order in case of arranging proper and improper of fractions.
I hope you understood this. This is basically this comparison. This is all about our first video and next we will come with the second part of ratio.
I hope you guys liked it.