[Music] Hello to this video I suggest you review the entire course on fraction operations the purpose of this sequence is to remind you and explain to you the most important elements of this chapter more precisely we will talk about addition of subtraction of fractions of multiplication of one verse two fractions and finally of the division of two fractions to prepare for a test or even an exam this will obviously not be enough you will still have to practice by doing many exercises in any case for the short let's go so we will start by talking about addition of fractions and therefore of subtraction of fractions because we will see that for additions and subtractions well the rules are strictly the same we will start from an example of an example we have two fractions with the same denominator this is the simplest case we will see it we will then see how it happens when the data the denominators are different and we would like to do the sum of a quarter +24 so we can't do simpler no doubt you know the answer but it's not serious I would just like to very quickly show you geometrically what this means when we talk about sharing so for that well I will geometrically schematize a car that is to say one part out of four that is why I have here a disc which is divided into four pieces and I have assured in red a piece on katché so represent a quarter of my disc in this case I want to add we said two cars so there I make the same division into quarters and I have assured therefore towards two pieces out of four I therefore have two cars well if I go the sum of one piece and two pieces what happens and obviously I obtain three pieces in red and in glass so on the far right we have there the offense result which is therefore a +2/4 one piece more pieces or three pieces out of four and we can see clearly here that we actually have insults and the three late so that is geometric but now mathematically what have we done in fact well if we look at it more closely we have made in water in the numerator the sum of the 2 numerator I made a + 2 and at the bottom or denominator I did not touch I left in quarters it is normal if to one because I add two because and well I obtain at the end what difference it is a bit like if we count pizzas if there is a pizza I add two pizzas the end I have three pizzas and well we count the difference as we count the objects and well it is in this way that we can interpret it in mathematical terms we will have we will do so when we add two fractions which have the same denominator we will make the sum of the numerator a + b and we will keep the denominator which will be said that this is why the formula which is there in a general way we can write it and it tells us that ensures of plus b on of so I have at the bottom the same denominator of is equal to a + b on of and suddenly for a subtraction will have almost the same formula if we subtract ensures of - b on haybions will make the difference of the numerator to - b and we will keep on we will say when we subtract two fractions which have the same denominator well we subtract the numerator between to - b and we keep the denominator then that is good it is even very good it is easy we understand well we add of difference between them give of difference we add eighths between them not that would give eighths we would add shots between that would give thirds but there each time we find ourselves adding two fractions which have the same denominator but fractions which we add do not always necessarily have the same denominator how can we do well in this case the two formulas which are here no longer work since we no longer have the same d so how we are going to get out of it and well the technique will consist of modifying either one fraction or the other fraction ourselves the two we will see it in order to obtain the sum or the difference of two new fractions which both have the same denominator, that is to say that we cannot directly use a formula when we have fractions which do not have the same denominator, we must modify them or modify at least one of them to better understand we will deal with two very simple examples here is the first one so we have carried out here 4/3 + 2 9th we can clearly see here that we have two fractions which do not have the same denominator for one c3 for the other these nine so no question for the moment of being able to use one of these two formulas we said that in this case we will modify at least one of the two fractions to make that then we are two fractions which is the same denominator so let's look at the denominators a little because that is where the work will be done is there a way to modify this 3 so that it arrives at nine to modify this 9 so that it arrives at 3 then yes because nine is a multiple of 3 so suddenly if I multiply three by three well we can clearly see that I have 9 so I could do this I could do three times 3 which will give me nine but be careful if we multiply the denominator so that it keeps the same fraction so that we keep equal fractions we must do the same in the numerator we say that we do not change a fraction when we multiply its numerator and its denominator by the same number which means that if I want to have the right to multiply the denominator by three I must do the same in the numerator and god multiply and also the numerator by three so there no problem I do not modify the fraction it remains the same it is just a different division but these are numbers which are equal and what will that give savalan let's do the 4 x 3 12 in the numerator 3 x 3 obviously 9 in the denominator since that is what we wanted and well there what happens we now obtain two fractions which have the same denominator the mend is here well there what do we said we add the numerators 12 and 2.14 and we don't touch the denominator a guy of nine and well finished we added our 4/3 plus come second example a third minus five because this time it is a subtraction of fractions and we notice that we cannot apply our formula either since the denominators are 3 and 4 they are not the same they are different so how to do how could we modify at least one of the two fractions to have fractions with the same denominator so I'm going to do something that is not right well the g3 the ag 4 which would mean that I could do plus one year at the highest at the bottom like that I would have 4 in the denominator except that this be careful it is forbidden the property tells us we do not change a fraction when we multiply its numerator and its denominator by the same number when we multiply suddenly when we divide also but in no case when we add a number that is false if I add here healthier and more 1 I am absolutely not certain to have a fraction which is equal besides here it is not true so what we must find is not a common number to add or remove it is indeed a multiplier coefficient a number that we are going to multiply but the problem is that here 4 is not a multiple of 3 and the inverse is also true like earlier where we had nine which was a multiple of 3 so I do not see there a simple number that I could multiply by 3 which will give me 4 and yes I had said earlier that from time to time it is the two fractions that must be modified that is to say that it would be necessary to modify the two denominators so that in the end I have the same denominator so there is a little thing which is quite simple which sometimes finds its limits you will be able to see it in other videos where we go a little further here I simply present the basic properties and the elementary methods but this what works is to multiply the denominators between them necessarily if here I do 3 x 4 and there I do 4 x 3 and well I will have on the two denominators 12 and suddenly with this little trick there we are assured of obtaining two fractions same denominator so let's go I will therefore do x 4 on this denominator there is multiplied by three on this denominator there we said we do not change a fraction when we multiply at the top and at the bottom by a mail number so if I did x 4 here at the bottom I have to do x 4 in the numerator also here I multiplied by three in the denominator so I have to multiply by three in the numerator from there we will therefore obtain two new fractions let's go then once 4 it gives 4 3 x 4 it gives 12 - 5 x 3 it gives 15 4 x 3 it gives 12 and the wonderful we have two fractions which have the same denominator well there again it's normal we arranged for that and 12 and well as we do everything we apply our property which tells us that we take the difference of the two numerators that is to say 4 - 15 this time which will give me minus 11 and we keep the denominator we are in 12th or remaining 12th out of 12 so that's it for additions and subtractions of fractions puts these two formulas aside we will remember well that to necessarily apply one of these two formulas we must have fractions with the same denominator if this is not the case we arrange for it to be so we now move on to multiplications of fractions so here is a first example of multiplication of a very simple fraction 1/2 multiplied by three fifths and I can already tell you that for a product of fractions so when we multiply two fractions not only is it not necessary that they have the same denominator but in addition I strongly advise you not to put them with the same denominator one these sources of errors 2 is useless and 3 is going to terribly complicate the expression of our fraction which means that then we will have to simplify the fraction so when we do the product of two fractions no we do not put them in the same denominator the principle is the following we multiply in line that is to say that we will multiply numerator between them and multiply the denominators between them we understand well that there is no need for it to be the same denominator so we must not confuse the two methods here for example when I have a half multiplied by three fifths what am I going to do well I will do an x 3 I multiply the numerator on 2 x 5 I multiply the denominators well well we can finish an x 3 that makes 3 2 times 5 that makes 10 result three tenths and we can write in a general way that when we multiply the fac the fraction ensures b part it is sure of and well that gives us times it is in water the numerator multiply on sometimes from below the denominators multiply so be careful from time to time there are small situations which are not so rare where it would be necessary to do something before multiplying I just want to show an example but again I invite you to join the track to deal with other examples which are a little more complex so here is my example 7 on 18 x 81 on 56 are already numbers which are a little bigger if I apply the rule which we have just seen what am I going to do I am going to multiply so the numerator between them that is to say I am going to do this x 81 on 18 x 56 hockey so there we see that mentally it is not obvious let's admit that we get there we compose these two products if I do this x 80 then I pass on the calculations we should find 565 if I do 18 x 56 even more complicated we would find 1008 ok so that is the answer except that for the break-in when we does fraction calculations we will always present the result in a simple form 767 over 1008 it's not very meaningful especially since in reality behind that hides a fraction which can be written in front of much simpler so that would mean that now we would have to simplify this fraction and simplify its song 567 over 1800 and not easy quite simply because by multiplying here and racking our brains doing it is by doing these two products we have lost very important information which is that between the numerator and the denominator here well I have common factors which means that I could simplify before here there the expression of my fraction and so that's not how we do it when we have this type of calculation to do what we do is that at this level so I go back a little we start by simplifying our fraction and for that we will have to show these common factors so let's go so I just copied for the moment showed two fractions and the two multiplication symbols the idea is actually to make common factors appear at the top and bottom so that we can get rid of them to simplify the fraction well let's look here g7 have 7 it's a prime number anyway I'm not going to be able to decompose it so I can leave this one let's look at the bottom if there is a way to make a factor 7 appear then 18 no it's not a multiple of 7 on the other hand 56 yes this time 8 56 which means that here I can instead of 56 I can write this x 8 in the next step we can already here simplified by seven in the numerator and the denominator but what I can do even better the ag 18 and there I have 81 diagonally 18,81 it makes us think a bit of the 9 table which means that it is possible to decompose 81 ans product of nine times something and the same for 18 so 81 ckoi bass and two times 9 and 18 c koi bass and 9 x 2 2 x 9 is there what happens gc factors 7 which I can get rid of as we said before and I also have these factors 9 which I can get rid of which means that now what we have left well in the numerator I just have the factor 9 so I write it down nine on and denominators which what I have left factor of l factor 8 that is to say twice 8 16 and well finished the result of your products of two fractions and 9/16 and we see that here we can do everything mentally quite simply because we are not shown too high as just now with our 567 and 1008 we were then stuck but we immediately simplified as much as we could as long as it was visible so we will now tackle the notion of division of two fractions but before that we will need to talk about the inverse of the notion of inverse we will see it in reality the division of fraction is totally linked to the multiplication of fractions and for that you have to understand what the inverse of a man is so it's quite simple inverse like a worm turned around and we're going to see some examples to understand what the inverse of a love is if I take for example the number 3 what is the inverse of 3 well the inverse of 3 is one in three it's a third the inverse of two would be one in two the inverse of 5 would be one in five but then in this case what is the inverse of 1/2 well the same I return the inverse of 1/2 would be 1st May 2 1st sacékripa of 1st well the inverse of 1/2 it's quite simply them because two on one so we understand that this little table I can read it in both directions inverse 2/3 and 1/3 but I can also say the inverse of 1/3 and 3 the inverse of 2 and a half or the inverse of 1/2 and 2 is totally symmetrical so the inverse of this 12th I think you understood this twelfth I invert and well it will be 12/7 and then in general kahn would be the inverse 2x and well quite simply one over x and looking at all that we will be able to establish a property on the inverses by multiplying each time a number and its inverse if I do three times a third well that will give me one that gives me three thirds three times a third three thirds that is to say if I do a half times two half of 2m well that also gives me one and if I do this twelfth times 12 7th so this 12th with its inverse g7 on the rise is at the bottom 12 years at the 12 at the bottom I will have 1 over 1 left that is to say one and if I do x x 1 over x well x x 1 over x that will give me x over x that is to say also and you understood that we can say that of a number are inverses one of the other if their product is equal to 1 and it also works the other way around if we have two numbers whose product is equal to 1 and well these two numbers are inverses of one another from there let's talk now about quotient we are slowly getting closer to the notion of dividing two fractions we have understood now how we can make inverses we are going to need them let's look now at what role an inverse plays when we make a quotient of two numbers so the first deposit I am going to talk about will be two divided by five so if I do 2 / 5 well that gives 0.4 can do it on the calculator we see indeed that it gives 0.4 I am going to do another calculation now I am going to do two but x the inverse of 5 so one pays 2.5 what is it we said earlier a fifth that gives what well let's go to the calculator even if these are calculations which are very simple to do but I want to move on here to mental calculation to be well concentrated on the essential I therefore do twice 1 / 5 and that gives us 0.24 let's see another example to now carry out 4 / useful a little simpler this one besides because 4 is half of 8 so we understand well that when we do 4 / 8 we find 0.5 and I will now do 4 x the inverse of 8 then the inverse 2.8 always the same it is an eighth I will therefore do four times an eighth and I find and yes obviously 0.5 and well that is very interesting because we see that we can go from a division to a multiplication while fortunately keeping the same result and we will say that divided by a number it is the same thing as x its inverse which means that in a way I could get rid of the division completely and each time I have a division I could make a multiplication how is it good the number which divides becomes its inverse divided by four it is the same as x 1 because / 7 it is the same as x 1 7th and cetera in general / a number is multiplied by its inverse you see where I'm going now when we have to do divisions of fractions and well let's not lose so much let's go we would like to divide two thirds by five because and we are going to establish the last formula the one for the division of fractions we have just said that divided by a number returns to x its inverse so let's already identify the one that divides is indeed the one that divides its five because I am going to put it in color and well divided by five because divided two thirds by five because that amounts to multiplying two thirds by what we are talking about inverse of 5 because the inverse of 5h inverses 4 over 5 and there I come back to a multiplication of fractions which means that finally the multiplication the formula for the multiplication of fractions will be useful to us when we have to divide a fraction we will say in the general case / b / it's sure of it is equal to assure b x of the sources and I return the second I return the one that divides so be careful not the first not the two just the second case the soft property 10 / a number returns to x its inverse so we must not to be mistaken in any case there we have almost finished by it is now enough to multiply we had seen just now that the multiplication was done online either twice 4 8 3 x 5 15 and we have the result two thirds / 5 because it is equal to 8 over 15 so here is the last formula on fraction operations to summarize we will remember that when we have to do an addition and a subtraction which have the same denominator well we add the numerator we keep the denominator if they do not have the same denominator well no modify at least one so that they have the same denominator we will remember that for the multiplication especially not we do not touch the denominator we simply do an online calculation the numerator between the nominees the denominators between them be careful from time to time when we have standards which are a little big yvan better simplified to that's all and finally we will remember that for the division well it is a hidden multiplication it is enough just to take the inverse of the fraction which divides this sequence is finished