Transcript for:
GCSE Maths Revision Topics

okay so in this video we're going to be having a look at the higher paper 2 for the edexcel advanced information revision list now if you haven't already seen one of these videos obviously i have released the revision list here that you can see on the screen for all of those three papers so we're going to be looking at this one today where we're looking at the paper two topics and if you want you can access these revision lists and they are in the description for the video right at the top which i'm going to show you in just a moment now as i've said with all of the previous videos and hopefully you've already seen paper one that you do need to do a little bit more revision than just watching this video now next to each of the topics i will link the full lesson for each topic that goes into a little bit more depth towards each of these so do make sure that you check those out as well throughout the course of the video i will just go over one example one type of question on each one but of course every single one could be a slightly different form of question so make sure you're using your revision guides make sure you're listening to what your teachers are asking you to revise and make sure you are using the full videos as well particularly for any topics that you're not 100 confident on so this should give you a broad overview of some of these topics and before we get started i'll quickly show you how you can find all of those things so when you're on one of these videos if you look in the description you can see just at the top on the right hand side here it says the hire and foundation revision checklist so if you click on that link you can download whichever exam board you're doing you can download those checklists that i just showed you on the screen just below that whether you're doing the foundation or higher tier you can see there that i'll link in paper one paper two and paper three and you can access any of those throughout the video if you pause it you can click onto the chapters onto the bottom left of the video and that'll bring up all of the different chapters so that you can go through and have a look at each individual topic again if you look in the description and scroll down to the bookmarks you can see next to each of those topics there's a full lesson on each of those topics so if you're not 100 sure you can click onto one of those and you can watch a full lesson looking over that topic so we go hopefully that's enough information hopefully this video is useful and helpful if it is don't forget to like the video don't forget to subscribe so you get all the updates for all the future topics coming up in these gcses and don't forget to leave me a comment and let me know how you're getting on ok so let's get started [Music] [Music] okay some error interval 6.4 has been rounded to one decimal place and like the error interval so the error interval we always tend to put either 6.4 or a letter in the middle depending on what the question is and we have our arrows always pointing to the left so we're going to find the upper and lower bound now 6.4 if it's been rounded to one decimal place it couldn't have been six point four five as soon as that five goes after that four it would have rounded up to six point five so we put that on the right hand side as our biggest possible number that it couldn't have been six point four five so it can't be equal to that number so we leave that symbol as it is it's got to be less than 6.45 now what we did there is we added on this extra 0.05 and all we're going to do for the smaller one is take that away so it would be 6.35 but it can actually be equal to 3.34 6.35 so i'm going to add my little equal to symbol on my inequality just there so it can always be equal to that lower bound because that would in fact round up to 6.4 if we rounded it to one decimal place on to the next one this one's slightly different in the wording so this one says it's been truncated to one decimal place and truncated basically just means it's been chopped off at that after this first decimal regardless of how it rounded so if we have a look at writing this is an error interval again we write these in the same way the impossible number that it couldn't have been it couldn't have been 6.5 so my top one here is 6.5 it had to be 6.4 something if it was just chopped off let's imagine just a random how this works so it could have been 6.4932 but it was just chopped here and we just wrote what was at the start okay but it couldn't have possibly been 6.5 because otherwise we'd have written 6.5 and the smallest it could have been it's just 6.4 with no digits after it couldn't have been 6.3 though because again if we truncate that it would have just been 6.3 so two different scenarios there one way you've got to add on that extra five at the end and take it up take the same size five off the start and truncating there where we just got to think about the fact that it was just chopped off and no rounding at all and the final little one we're going to have a look at here is using the calculator so it's important to know where lots of digits are pick quite a complicated one here but you just need to have a play around with you com with your calculator to make sure you can find your square root your cube root your squared your cubed your fraction button your sin your cos your tan and figuring these out so i pick quite a complicated one here so to get your cube root button depending on what calculator you have but the majority of them you still have to press the same things we press shift and then we go for let's get a different color here we go for the square root button and that allows us to get a cube root so on your calculator a cube root will pop up and then we're going to use the fraction button so the fraction button is just next to it in this diagram here the fraction button and an empty fraction is going to pop up so on the top we're going to just type in on the top of that fraction there 4.3 multiplied by press your tan button and then press 39 and normally this will go in a bracket so remember to close the bracket off there you don't put this little degree symbol in you just put 39 the calculator knows that you're talking about degrees so let's get that up on my calculator we've got 4.3 multiplied by tan the brackets opens for me 39 close the brackets press your down button on the calculator to go into the bottom of the fraction and type this sum in here 23.4 take away 6.06 so 23.4 take away 6.06 and make sure at this point you write all the numbers down that equals zero point five eight five five nine three four two three three now some calculators may go beyond this depending on what the answer is on the um on the calculator but it tends to be that you never really have to go beyond nine digits because not all calculators will go go that far so you are limited to really to the amount that some people will be able to write depending on what calculator they use now the question will probably usually say then to round your answer to one or two decimal places so if we just have a quick think if we were to round this to one dp to one decimal place and we could chop that after the five there so one decimal place would round up to 0.6 and we would round that to 0.6 if we had this to two decimal places or beyond let's have a think after two decimal places let's pick a different color it would chop it after the eight and it would be naught point five nine because it's a five after that one okay so you could have different ways of rounding this but just make sure you write down all the digits on your calculator display before actually rounding into algebra we're looking at index laws to start with so we know the index laws we've looked at this when we have powers that are getting multiplied together when they have the same base number these powers can be added together so two and three would make five we also had the process when they're being divided so x to the power of seven divided by x squared we take away the powers and we get x to the power of five and we also have powers in brackets so if we have x squared to the power of five we multiply the powers so we get x to the power of 10 in that particular example now having a look at this one we've got 30 x squared y to the power of 3 on the top and 6 x y squared on the bottom so we're doing a divide now we've got these numbers on the top and the bottom 30 on the top 6 on the bottom so numbers always just divide numbers so 30 divided by 6 gives us 5. now individually looking at the letters we've got an x squared and on the on the bottom we've got x now it's not written in there but that is a little power of 1 there that's not written so if we subtract the powers 2 take away 1 gives us x to the power of 1. i'm not going to write that in i'm just going to leave it as x because that means x to the power of 1. moving on to the y's we have y to the power of 3 y squared on the bottom and 3 take away 2 gives us just y on the bottom there so our final answer would be 5xy okay looking at this one then we've got powers in brackets so we're going to be multiplying powers here but we've got this number two in the start as well so numbers always get treated like numbers and that whole thing is being cubed so we need to work out what 2 cubed is 2 cubed is 2 times 2 times two which is eight so we're gonna have eight at the start of this one multiplying the powers for the x then x to the power of two that's going to get times by three so that's gonna become x to the power of six and y there is a power of one which not written so 1 times 3 gives us y to the power of 3 and there's our final answer there onto expanding brackets this first bracket we are multiplying everything inside the bracket by b so b times 3b well not forgetting there's a 1 in front of that 1 times 3 is 3 and b times b is b squared on to the next one 1b times 7 well 1 times 7 is 7 and there's just that one b that we're going to leave in there and we're just going to copy that symbol which is a plus so keeping that as a plus on to expanding below we have 3y times 5 3 times 5 is 15 and we've got the 1y there so we'll leave that as a y and then 3y times the 4y well 3 times 4 is 12 y times y is y squared and then again copying that symbol as a negative expanding and simplifying two brackets so we'll expand these separately so let's have a look at the first one so 3 times 5 is 15 and 3 times 4 is 12. so we have 15 x plus 12. on to the next one got to be careful with this one because it's saying take away which means this is a negative two at the start of the bracket so we're gonna times both these pieces by negative two so negative two times three x is negative six x and negative two times negative two gives us positive or plus four now we just need to tidy all this up so we have a 15x take away a 6x and that gives us 9 x's in total and we have a 12 add a 4 and 12 add 4 is 16 that's positive 16 so we get 9x plus 16. moving over to expanding and simplifying a double bracket so x times x gives us x squared x times negative 5 gives us negative 5x 3 times x is positive 3x and then positive 3 times negative 5 is negative 15 and then again we just need to tidy this up because we've got these two x pieces in the middle so we've got x squared still at the start but we've got negative 5x add 3x which in total is negative 2x negative 5 up 3 and then we've got the negative 15 at the end onto the one below so we've got some numbers in front of the x's here so i'm going to do the same process let's see if we can move some of this out of the way just about okay so 2x times 3x 2 times 3 is 6 x times x is x squared so we'll get 6x squared 2x times 4 is 8x that's negative 4 so it's negative eight x negative three times three x is well three times three is nine so negative nine x and then negative three times negative four will give us positive twelve three times four is twelve and it's a negative times a negative and then again just tidying this up we have 6x squared at the start and we've got negative 8 take away another 9x so negative 8 take away another 9x is negative 17x and then we've got that plus 12 at the end and there's our final answer and we've simplified it there with the simplifying in the middle final answer okay so factorizing we're going to take out a factor and put this into brackets so we have a look at both these pieces 3x and 15 we just think what number goes into both of those they both divide by 3 so we're going to divide this by 3. so 3 on the outside of the bracket that's the factor we're taking out 3x divided by 3 is just x copy the symbol with a plus and 15 divided by 3 is 5 and you can check your answer there you can expand that back out just to check that you do get 3x 15. now this next one says factorize fully and that's because there's more than just one thing that comes out of the bracket in this one so in that one above there was just a three that came out so here we're going to have two things and let's have a look at both these pieces we've got 35x and 21x squares so they both have an x in so there's definitely going to be an x coming out just need to think with 35 and 21 what's the biggest number that goes into both of those that is seven so the highest common factor of those two numbers is seven they also both have an x in so we can take x out as well and then follow the same process so 35 x divided by seven x or what do we times seven x by to get thirty five x we'd have to times seven x 5 but we don't want to put an x in there because we don't want to make that go up to an x squared copy the symbol we times 7 by 3 to get 21 but we want that to be an x squared so i'm going to stick another x in there with it that would enable us to get an x squared and we expand that and there's our final answer factorized fully just to note if you didn't take the 7x out and you just took a 7 out for example and this wouldn't be factorized fully you'd have got 7 brackets 5x minus 3x squared and you can see in that bracket there there's still an x in both these pieces there's a little hint there that you didn't fully factorize it okay so on to factorizing a quadratic this is the opposite of expanding a double bracket so we know our answer is going to look like already it's going to be in a double bracket both of the next at the start and when we're doing this we've just got to find the pair of numbers that are going to go into our bracket so looking at the 12 we know it's going to be a pair of numbers that multiply to make 12. so i always write these down we can have 1 and 12 2 and six or three and four we just need to have a look at these numbers there because we've got a seven x in the middle so they have to add up to make seven x so one and twelve there's no way i can make seven two and six i could either make eight by adding them together or i could make four or minus four if i do a minus with one of them so i'm not gonna make seven but i can make it with three and four i can have plus three and plus four and that would make seven so plus three and plus four and that's fact that's not a four plus three and plus four and that would be factorized having a look at the one below we've got some negative symbols going on there so we've got a slightly different process but our answer is going to look the same it's still going to be in a double bracket it's still going to be x at the start and we're still going to have two numbers in here with some symbols so 28 the numbers at times to make 28 are 1 and 28 2 and 14. what else could we have four and seven and let's have a look if these can make 28 so what we got we've got 3 in the middle so we want to make minus 3 well we're not going to do that with 1 and 28 that's not going to work 2 and 14 we could make 16 or we could make 12 that's not going to work but 4 and 7 we could do plus 4 take away seven i'd make minus three so plus four and negative seven and that's our double bracket they're finished so on to solving equations so when we've got this word solve it means it wants us to find the value of this x piece just here so we've just got to reverse the process of what's going on so at the moment it says 3 times a number 3 times x plus 5 is 26. well if we get rid of this plus 5 by taking away 5 from both sides that tells us that 3 times a number 3x must equal 21 and that means three times a number so we can reverse this process now by dividing by three which i like to write as a little fraction so divide by three and 21 divided by three is seven so we would write x equals seven and there's our final answer so look at this next one so 5x minus 2 equals 21. well this time to reverse this we've got 5 times the number take away 2. so if we add the 2 on to both sides we get 5x equals 23. and here comes the part where i explain why i like to write it as a fraction as above rather than just write and divide by 3 because in this circumstance 23 doesn't divide perfectly by 5. so if we go divide by 5 again and divide by 5 we have our answer there written for us 23 over 5 so x equals 23 over 5. okay which we could convert back into a mixed number five goes in four times of the remainder of three so i could write its four with that remainder of three-fifths there or i could even write it as a decimal i could type that into my calculator and turn it into a decimal but it's fine to leave it as a fraction four and three fifths is my answer so here's one we've got x on both sides of the equation so in order to make this look like one of the equations from before we want to get rid of x from one of the sides now it doesn't matter which one we choose i always always choose to go for the smallest x so we've got a five x on one side and a three x on the other side so three being the smaller one so we can take away three x from both sides that's smaller x so if we take away 3x from both sides well 5x take away 3x leaves us with 2x we've still got that negative 3 there and that now equals positive 9 so i won't bother putting the positive symbol however if there was a negative there i would keep that in just to emphasize that it is a negative 9 that's really important so now we've got an equation just like the question before so again we can add 3 to both sides and we get 2x equals 12 and then we can divide that by 2 again i like writing that as a fraction and 12 divided by two is six so we can write that as a whole number x equals six and chris are work colleagues adam is eight years older than chris brian is twice as old as adam the sum of their ages is 92 how old are they all okay so we can do this using equations as well now it says adam is eight years older than chris and brian is twice as old as adam now it doesn't say anything about how much older chris is than anyone else so if let's call chris x what would adam be now i says adam is eight years older than chris so adam would be eight older than that exactly eight so that'll be whatever chris's age is x plus an extra eight then it says brian is twice as old as adam well this is adam here x plus eight so brian is twice as old as that so twice as old as that would be two lots of that so two lots of x plus eight we could even expand that out now so two lots of x plus eight is two x plus 16. so we have their three ages now in terms of algebra we have chris being x adam being x plus eight and brian being two x plus sixteen it says the sum of their ages is ninety two so some beings add them together so if we add them all up and see what we get we can have x plus the x plus eight for adam plus the 2x plus 16 for brian and it says that they all equal 92. so if we start to simplify this let's have a look we've got x x and 2x which is 4x and we have the 8 and the 16 which is 24 so 4x plus 24 equals 92. and now we can just solve this just like before so i can take away 24 from both sides take away 4 is 88 take away 20 is 68 so we get 4x equals 68 and then we just need to divide by 4. so dividing that by 4 let's think gives us 17. so x equals 17. so it does say how old are they all so we could finish this off and we could just say well chris was x so chris is 17 adam was x plus eight so adam would be 17 plus eight which is 25 and then brian is twice as old as adam so brian would be 50 twice as old as 25 double 25 a nice little quick check there you could just add those all up 50 plus 25 is 75 plus the 17 is 92 so i'm happy that that works it meets all those criterias so the perimeter of the quadrilateral is double the perimeter of the triangle work out the perimeters now the lengths of each shape are given to us as expressions so we could get an expression for each of the perimeters to start with so let's have a look at that quadrilateral let's write that down over here so the quadrilateral we've got lots of pieces here so let's count up these x's to start with we've got 2x plus this one's 4x 7x plus another one is 8x so we have eight x's and the numbers we've got a negative three and a negative one and in total that is negative four so that's the quadrilateral eight x minus four and that's an expression for the perimeter then we've got the triangle let's have a look what we've got there we've got one x two x so two x and we've got three four five plus the 4 is 9 so 2x plus 9. now it says the perimeter of the quadrilateral is double the perimeter of the triangle so if we double the perimeter of the triangle then that would be equal to the perimeter of the quadrilateral okay so if we double this one then those perimeter those perimeters those expressions for the printers would be equal and if we double that we get 4x plus 18. now because those two perimeters are now equal we can set them equal to each other using an equation so we can say this one here the perimeter of the quadrilateral now 8x minus 4 is equal to this one here double the perimeter of the triangle so that's equal to 4x plus 18. we can now just solve that like an equation so we can get rid of the smaller x from both sides so minus the 4x so over here we have 4x we have 4x minus 4 equals 18. we can now add 4 to both sides so get rid of this plus 4 so plus 4 and we get let's do this over here we get 4x equals 22 and then we've got to finish that off to find out the value of 1x so dividing by 4 x will be 22 over 4 which is okay if we have a calculator but four goes into 22 five times with a remainder of two so it'll be five and two quarters and five and two quarters is five and a half so our final answer there let's bring that up to here would be five and a half or five point five and again it would have probably said in the question it was all in centimeters or it was all in meters so we just have a look at what units we'd apply there but 5.5 is our value of x now it says to work out the perimeter of the triangles so we can sub these back in if we want so 5.5 would be here in place of this one would give us 8.5 so that would be 8.5 5.5 on the x plus 2 here would give us 5.5 plus 2 which is 7.5 and then this one over here is 4. so if we add those all up 8.5 plus 7.5 is 16 plus the 4 gives us a perimeter of 20 for the triangle we could have done the same with the quadrilateral but it said the quadrilateral is double the triangle so this one here must be 40. and again we could have just sub that into the quadrilateral there and halved it for the triangle we've got our two perimeters again just not forgetting what units it was not that i've put any units in this one okay so when looking at the equation of lines and specifically when we are looking at parallel lines we're looking to compare the gradient now when we have some line equations like this and it asks two of them are parallel explain which two we need to get them all into the form y equals mx plus c and then we'll have a look at that value of m which is the gradient so we're specifically looking at this number here that goes in front of x so if we look at the first one we can see that that's 2y so if we divide everything by 2 that is going to put it into the form y equals so when we divide by 2 we get 1y is equal to 3x plus and we've got to do 5 divided by 2. now as that's not really looking at the parallel line or whether it is going to be parallel we don't need to be too concerned with getting that perfect in terms of a decimal so we can just leave it as 5 over 2. it's fine to leave it like that anyway but that's absolutely fine just to leave it and not worry about having to type into your calculator or convert it now for the next one you can see that that 3x is on the left hand side so to get that onto the right hand side we're going to subtract 3x from both sides and that will leave us with y is equal to a minus 3x and then we've got the plus 5. so as you can see already those two have different gradients for the first one we have got 3x and for the second one we have a negative 3x so we need to have a look at this final one now for this final one it's a little bit trickier because you've got negative y on the left hand side so to make that become positive y the first thing that we can do is add the y to the right hand side so if we add the y over we would have 3x is equal to and you can put that as 5 plus y or you could write it as y plus 5 as that is a positive 5. now it doesn't matter if the y is on the right hand side of the equation so now we can get rid of the 5 from both sides and if we minus 5 from both sides of the equation we would have 3x minus 5 is equal to y and of course that can be written the other way around so we could write that as y is equal to 3x minus 5. so there we go we have all of our equations done and as you can see the third line which has a gradient of three is matching the line one which has also has a gradient of three so in this question here we would say that line one and line three are parallel and the reason that they are parallel as it says explain which two is because they have the same gradient and they both have a gradient of three quadratic inequalities so we're going to solve this now obviously these um represent a parabola a curve a quadratic graph again a quadratic graph here normally when we're solving them we are solving to find these two roots here or solutions and you can be asked to represent uh to find them or to estimate them from a graph if you've been given a picture they're referred to as the roots but when we're solving an inequality we're looking at something slightly different now this first one here they're both exactly the same we're gonna have a look at the two different scenarios this first one says less than zero so essentially it's saying is what part of this curve is less than zero if you have a look this part of the curve is what's below ground what i like to refer to a less than zero z and b zero being the x axis there so really all we're going to say is well what part of the curve is between these two points below zero so if we actually just factorize this to find out what these two roots are so putting it into a double bracket again it might have a coefficient bigger than zero but i'm just going for a nice easy one here just to show you the process so the factors are 24 we can have four and six two and twelve three and eight so i'm just gonna not write them all down because i've already spotted it's three and eight but that would be plus three minus eight and that would give us minus five so our roots here our solutions are x is negative three and x equals eight so if we were to label those on the graph here this would be minus three on the x-axis and this would be eight now the bit is below the curve is between negative three and eight so i'd have to write this as um one of these types of inequalities okay where we say it's between two points and that would be my answer there x is between negative three and eight so the part the graph that is under zero or less than zero is between those two numbers now that would change slightly for this one down the bottom because i've just realized they're pointing the same way that wasn't meant to be pointing that way let's change that that's meant to say bigger than zero okay so there we go sorry about that let's change that x squared minus 5x minus 24 is bigger than zero so this would be something slightly different now because the bit that's bigger than zero and if i just redraw this graph there we go eight here negative three here the bit that's actually bigger than zero we've got one part over here and we've got a completely separate part over here so we've got two separate parts of the line and that means i can only write this as two separate inequalities now this part on the left is all this part of the curve is going to numbers less than three so my solutions there would be well anything that's less than three sorry negative three so x is less than negative three and that's one of my solutions for that part of the line but i've also got the other part and that is going to all the numbers bigger than eight it's going this way so my other solution would be x has also got to be bigger than eight so i've got two two ways of two solutions here to write with my inequalities i've got all the x values less than negative three and all the x values bigger than eight so that's solving a quadratic inequality just got to think about which part of the graph are you actually looking at okay so we've got a functions question so we've got f of x is two x minus four and g of x is x squared plus five now the first one we're gonna have a look at is this here so working out an inverse function so an inverse function just means basically doing it in reverse so the function of f is two x minus four it times this number by two and then takes away four there's a nice way of doing this algebraically we can change that f of x piece there for a letter i'm going to just change it for the letter y so y equals 2x minus 4. and i'm just going to rearrange it to make x a subject which is hopefully quite nice and easy in comparison to the one we looked at earlier so add 4 to both sides you get y plus four equals two x and then divide both sides by two you get y plus four over two equals x there we go i'm just going to rewrite that with x instead of y and that'll be the reverse function now the inverse function so instead of being y plus 4 it's x plus 4 so x plus 4 over 2 and that equals the inverse function of f so f minus 1 x there you go and that's how you do your inverse function and that can be true that can be done for any any type of function here just swap the x and y rearrange it to make it a subject and then rewrite it okay on to the next one it says workout gfx so you could be asked to put a number in this is actually asking us to put a number in it saying gfx which means i have to sub one of the functions into the other so if you read this it means what is the function of g when you put f into it so what is g when you put f into it so well g is up here so we've got to do is put f into it so if i stick f into there i know the way that i do it is i stick the function of f in a bracket and i'm just going to replace the x value here with that bracket and if i do that i get 2x minus 4 in brackets squared because it's x squared and then plus 5. so really there is an expression for that but i'm going to completely expand it and simplify it all down so we have a double bracket we have 2x minus 4 and another 2x minus 4. and then once we've expanded that we just need to remember to add 5. so if we expand that and again i'm going to skip some steps here i'm going to 2x times 2x which is 4x squared and then we've got 2x times -4 and 2x times -4 again so minus 8x minus 8x is minus 16x and then 4 times 4 gives us 16 but we're going to add 5 to that so that'll give us 21 so plus 21 and there's your expression for that you could have had it had this written in a different way you could have had f gx and in that case you'd do it the other way around it says what's the function of f when you put g into it so you put your bracket around g and slot it into the x place in f so you can have two different ones there as well i just got to make sure that you do them in the right order you could also be asked to sub a number in so it might say something along the lines of f let's do it the other way around f g i don't know two and in that case that means what's f when g is two so you put two into g and if you put two into g you get two squared plus five which is nine and then you put nine into f so what is f when g is two well when g is two we get the answer nine so you stick nine into f so putting nine into f which is our first function there you do two times nine or two lots of nine take away four two lots of nine is eighteen take away four is fourteen so that would be our answer there if it asked us to sub a number in but these ones are a bit harder when you have to put a function in and remember what order to do it in but just remember the way you read it so for this one we looked at what is g when you put f the f function in and there's some functions for you okay so for this graph here all i'm going to do is i'm going to have a look at the x values i'm going to plug them into my little formula so just like when we're drawing normal line equations it says here to get y we do sine x so let's pick an x value let's start with zero so if i type in sine zero on my calculator i get the value of zero so i'm going to plot that as a coordinate so that's 0 0 and that is going to go there then i'm just going to move along the x axis so i'm going to do 30 and then i'm going to do 60 90 120 i'm going to go all the way up to 360. i'm going to try and do them as quickly as i can so just on my calculator now i'm just typing in sine 30 obviously inside the bracket there and i get a half i'm going to plot that as best as i can it's quite nice on this graph here and that's zero going back in we get sine 60. okay and that gives us a nasty quantum nasty number there we get root three over two obviously which you might know from the exact values of trig if you check that video out but as a decimal when i convert that into a decimal there it gets me 0.86 now i can't get absolutely perfect but i can get it near enough i'm going to put it around here 0.86 and i'm going to go back in and put sine 90 in there we go sine 90 gives us a value of 1. and there is our first little part of this curve that's going to go up there now when i go for the next one i'm going to go for 120 if i go back in i'll type in sign 120. there we go i get 0.86 i actually get the same value as when i put in sign 60. now again we'll discuss that a little bit more in a second okay moving along uh sine 150 that gives me a value of a half again there we go and then sine 180 gives me a value of zero right so we've got like this sort of like hill shaped little curve there kind of looks like a little wave at the moment which you might recognize from some of your science graphs as well so on to the next one here putting in uh 210 let's have a look let's put that in 210 and we get negative a half so that's negative a half just here and then when i put the next one and i get negative 0.86 there we go so we're getting these same values as we got up there but in the negative versions now when we put 270 in we get minus one and then we get the negative 0.86 again for 300. again we get a half for 330. i'm just typing in this last one here 360. and there we go we get a zero for that one right i've typed those in quite quickly obviously just take your time working through those as a calculator and you will see we'll get these values here and obviously we want to just join that up with a nice smooth curve there we go and it makes this like almost like wave shape there we go and that's our sign graph there so obviously there's a few little bits along there that are worth discussing one of those being some of the fact that we get these same values at different places we actually get some sort of a almost like a little reflection line within the curve if i was to draw like a little reflection line up here it's almost like a mirror image there on the other side of each part of the curve just on that little positive slope there obviously it changes as we get into the negative but we could almost think of it like that there's something worth noting with these values the fact that the value of 30 there sine 30 was the same as sine 150 um let's just draw that out we've got sine 30. that equals a half and you also had sine 150 that equal to half as well there we go and if you have a look these values all these angles here as we set out to look into the sort of trigonometry a little bit further those two angles 30 150 add up to 180. so these sort of supplementary angle pieces have the same values then that actually applies to any of these if i was to look at sine 20 and sine 160 they both have the same value as well let's just type those in sine 20 gives us a value 0.34 with a few more decimals and sine 160 also gives us that same value 0.34 so you feel free to have a little go with that and have a little play around with some of those values but it's because we get this sort of mirror image on either either side of the earth the trigonometric graph there okay but that is the sine graph another thing that you might note as well is that it does go through the origin and i kind of always remember and it's a little bit silly but sin has that word in there and i always kind of remember sin is in the origin so it's just a little silly way of remembering it but sin is in the origin could just help to determine or did this this would distinguish between the sine and the cosine graph okay but sin is in the origin and then we can have a look at the cosine graph now and actually go about drawing that as well so let's have a look at that one okay so for this one it says on the grid draw the graph of y equals cos x and again it's between naught and 360. so we're just going to plug our values in on the calculator again i'm going to do that for cos x now let's have a look so cos zero this time gives us a different value we get one okay and this is obviously what distinguishes this one between the cos graph and the sign graph this one is not in the origin it starts at one there and as we start putting values you're going to see a quite a similar pattern to the one we had before um but it's obviously just starting from one this time so if we put cos 30 in let's have a look i put two brackets in by accident so cos 30. we get this 0.86 value again that we got before so 0.86 and it's just there and we put cos 60 in and we get half cos 90 gets us zero and then it starts to dip below again if i put cos 120 in that now dips down to minus a half then we get minus 0.86 then we get minus one and we get minus 0.86 again minus a half zero a half 0.86 and because 360 finishes on one there we go so we get this same similar shaped curve but it's obviously starting at one which has just sort of shifted that curve along a little bit there we go and that is our cosine graph there okay but again it's just plugging numbers into a calculator plotting the points nice and carefully joining up with a nice smooth curve okay so don't be put off by these types of questions literally it's just typing numbers into a calculator and plotting these coordinates obviously on both of these graphs here one other thing that you might have noticed is it always fluctuates between one and down here at minus one and that's also another key point here with our trigonometric graphs is that it does fluctuate between one minus one but that is our sine and cosine graph and now we're going to have a look at our tangent graph and see how that differs from these two because there's not much similarity at all between these two and the tangent graph it looks completely different so let's have a look at that one now okay so here we go we're going to draw the graph of tan x okay between naught and 360 again you're going to see this one's quite unique in terms of how it looks compared to the others so to start with i'm going to put tan zero in on the calculator again so just to confirm on my calculator i am literally just typing in tan zero and pressing equals and getting my answer and now for that one there i get zero so tan zero starts on zero i'm just gonna put that coordinate on again now if we'll go for 10 30. it's going to move along the axis there i'm getting 0.57 i'm going to try and plot that as best i can again we are only doing a sketch here so 0.57 somewhere around about there it's not perfect let's go back in 1060 and we get root three let's convert that to a decimal we get one point seven three so one point seven three is somewhere here and then we'll go for tan ninety and a calculator says maths error there we go so i can't actually put 1090 in and obviously if you've checked out that video on the exact values of trigonometry you'll know that tam 90 is undefined again i'm not going to discuss the ins and outs of why it's undefined because you'll be able to have a look at that in the exact trigonometric values video which is obviously in the description but for that one there we can't plot a coordinate so i'm going to skip out 90 i'm going to go over to the 120. so 10 120 and that gives me negative 1.73 so for 90. it's down at negative 1.73 just there okay so we're kind of thinking hold on how does this form a pattern this is looking a bit strange at the moment i've just plotted that in the wrong point so it's definitely looking strange that's under the 120. there we go that's 120 there back in let's change that for 150 see where this is going so it's minus 0.57 and that's gonna be there then we've got 180 to put in so tan 180 that gives me the value of zero there we go on to 210 and that gives me positive 0.57 so we're starting to see a bit of a pattern here we've got that 0.57 going in again and then we put the next one in we get our value up here there we go so we're starting to see a bit of a pattern here we've got an almost bit of a wave and now we're moving on to the 270 so we put 270 in again and again we get a maths error there we go so we're gonna have to skip 270 out okay we can't do 270 it matches what happens on the 90 there so we're going to go over to the 300 so tan 300 and again i get negative 1.73 which is here and on the next one we get our minus 0.57 and on the next one we get zero right there we go so that is our tan graph it's a little bit odd but when we actually join this up we get a sort of wave here we get a wave down here and we get another wave there and it's a little bit weird there we go if i was to actually draw lots of these out you'd see that this line would continue down there this line would continue up here and we get this sort of pattern of waves sort of going along the x-axis there but that is our ten graph it looks a bit strange it looks a lot different to the others it sort of seems a bit counter-intuitive when you do draw these on you know if you if you were to do this on the calculator because you have these sort of gaps from here to here and you're kind of thinking well hold on how do i join these up but they don't join up at all okay they're sort of uh just skipped out there because we have an undefined value of 90 so it all starts again okay there is lots of mathematical language behind that but to be honest all we need to think about in this video is how we go about drawing it how we would answer this question on the exam and there we go there it is just plugging those numbers into a calculator and having a little bit of an understanding of what the graph actually looks like so there we go that is our sig coz and tan graphs and how to draw them hopefully seems quite nice and simple just plugging the numbers into the calculator and plotting them on the graph and then actually just thinking about recognizing them so we had the syn graph which was that nice wave fluctuating fluctuating between one and minus one that started in the origin we had the cos graph which was pretty much exactly the same as the same graph but it just started on one and formed that nice little wave between one and minus one again and then we've got our tan graph here which is obviously quite different but again just a nice little method for actually drawing them and plotting this and just making sure we've got nice smooth curves obviously just making sure that you do have a nice smooth curve that goes through all the points so not like my little rubbishy one that i've done here where we've actually missed out a coordinate there you'd want to have nice smooth curves going through all of those but right that is our three graphs and we're going to use some of the ideas from these graphs to actually have a look at some of these quite nasty questions and see i just have a little talk about how we can be able to sort of approach those in an exam so let's have a look at our first one okay so same question that we were just looking at but we've got some questions here that we're gonna have a look at um actually answering here so it says the diagram shows them the uh part of the curve with the equation y equals f of x and here it is in the diagram the maximum point in the curve is 2 3 as we've already seen write down the coordinates of the maximum point with these different equations i've got quite a lot here all the different scenarios we could actually have a look at so for part a there okay it says f of x minus two so that's inside the bracket so it affects the x coordinate and it does the opposite of what we'd expect so it doesn't subtract two it adds two okay so it just does the opposite so the x coordinate being 2 there we would add 2 to the x coordinate and that would make it 4 and the y coordinate is not going to change that would stay as 3. okay so that's the opposite of what we'd expect when we add 2 to the x coordinate look at the next one let's try to change to a different color here right so f of x minus one now that minus one is outside of the bracket so it's outside the bracket it affects y and it does exactly what we'd expect so minus one we would expect that to subtract one from the y coordinate which it does it just changes the y intercept actually if you're familiar with your equations of lines and your coordinates and coordinate geometry you should hopefully know that number at the end is the y the y intercept there so it's taking away one from the y intercept essentially just moving the graph down one but we'll have a look at that again a little bit later but minusing one from the y-coordinate the y-coordinate currently is three so if we take one away from that the x-coordinate doesn't change so two but the y-coordinate is also going to drop down there to 2 it's going to subtract 1 from that okay so that's how we can have different scenarios there with just adding and subtracting numbers and how that affects the x and y coordinate now we have a look at question c part c there so y equals f of 2x so 2 times x normally when you're timesing by 2 you'd expect that to double something but as it's inside the bracket affecting x it does the opposite so rather than multiplying by 2 it divides it by 2 or halves it there so if we look at the x coordinate the x coordinate is currently 2 so half of that halving it because it's inside the bracket and changing the x coordinate but does the opposite so half of 2 is 1 and the y coordinate remains unchanged and that remains as 3. there we go so that's that one let's have a look at the other scenario where it's outside the bracket so three lots of f of x for part d here let's have a look at what this does so it's outside the bracket it affects y we're multiplying it by three so it does what we'd expect it makes it three times bigger but it affects the y coordinate this time because it's outside the x bracket so three lots of that y coordinate the y coordinate is currently three so the x coordinate is unchanged and the y coordinate gets multiplied by 3 and becomes 9. right there we go so that is our scenarios where we are multiplying by a number so in terms of the x there inside the bracket 2x does the opposite so it divided by two but outside the bracket it multiplied the y coordinate by three okay and on to these negative ones here which don't really follow the same trend but partly what it does so we'll have a look at this one here so we've got minus x in there now it doesn't do the opposite of what we'd expect because the opposite of minus x would just be nothing okay we wouldn't actually change anything at all it'd just be positive x it'd be the same equation okay but we follow the rest of the rules for this one it's just uh it goes a little bit out the window and we look at our minuses here so we've got a minus in there all that does is it swaps the symbol of the x coordinate it's still affecting x it's inside the bracket it just doesn't do the opposite this time okay so it just changes the the two as the x coordinate to minus two so the x coordinate is going to change it becomes -2 and the y-coordinate remains unchanged stays as three okay so this one is a negative in there that's the only scenario where it doesn't actually do the opposite or anything like that we just have to worry about is it next to the x or is it changing the y and it just flips the sign so the last one here we've got y equals minus f of x and again the minus those outside the brackets so it changes the y coordinate and again it just swaps the sign so instead of being positive 3 there is the y coordinate it's going to be negative 3. so we have positive 2 still and negative 3. okay so again just following all those little rules is it next to x or is it x to the y or outside of the x bracket that tells us which coordinate f of x when it comes to adding subtracting multiplying okay we can think about the opposite or the whether it does what we'd expect the only scenario where it doesn't is where we've got this little minus symbol in there and that just changes the the positive or negative symbol of the the corresponding coordinate whether it's x or y whether it's inside the bracket or outside the bracket okay let's have a look at one more okay so i've got five questions here they're not in any particular order of difficulty they're just some sort of different trigonometric questions where we're looking at these graphs and sort of thinking about how we would approach them but feel free for any of these five questions just pause the video obviously if you've got the worksheet printed off you can have a go at the questions see what you get see if you can sort of apply any of these little calculator methods uh to answering them and then obviously i'll go over all the answers anyway so we can have a look at all these sorts of questions so it says here the graph of y equals sine x for x values from minus 270 to plus 270 is shown below and obviously that is shown on the graph so we've got minus 270 all the way over to 270 over here it says on the same axis draw the graph of and we've got y equals 1 minus sine x for values from minus 270 to plus 270. so that's the graph we're going to have to draw there and that doesn't look very nice now this is all sort of this question here is kind of very much linked into graph transformations we're actually transforming this graph and we just need to figure out how this graph is transformed but actually we can actually do this on a calculator now this particular graph transformation here has got two different transformations going on and we could rearrange that we don't have to write one minus sine x we could write negative sine x plus one and just sort of rearrange those if we wanted to so we could do it like this we could write minus sine x plus one oh i missed my bracket there let's put that back in there we go plus one and think about what these transformations actually uh represent so that negative in front of the sine x flips the y coordinates from being positive to being negative obviously check out my video and i'll in fact i'll link that in the description below on graph transformations as well because it's probably worth checking out because it does link very strongly into these trigonometric graphs so that negative there relates to a flip or a change of the symbols in front of the y-coordinates we've then got the plus one at the end and again that's outside the bracket it changes the y-coordinate it changes the y-intercept it moves the whole graph up by one so you could actually take a graph transformation approach to this you could try flipping it over and then shifting it up one and we could probably do that quite nice and quick i'm gonna do it in two different ways though but just if you are quite keen on your graph transformations you could first flip it over and just think about that by flip oh that's not my pen let's just change that we could take all of these little coordinates here this one this one this one and this one i'm not bothered with the zeros because they'll stay where they are and we could just flip them down change the coordinates there and that would move the coordinates here and we actually end up with and i'll try and draw it there we go we end up with that and then we need to shift it all up by one so take all those coordinates shift them all at one that would shift it at one that would shift it up one let's have a look these zero coordinates need shifting up one as well try and shift them all at one it's not the nicest now when you start drawing all over this there we go shift that all at one and it makes this sort of curve here there we go so drawing it using a graph transformation approach let's just get rid of that red one there we go that is where our curve ends up following this graph transformation okay not the nicest though that's quite nasty actually it's from a very sort of horribly difficult graph transformation question having to sort of flip it over the x-axis there and then actually shift it at one as well but i'm going to take a little bit of a different approach but just so you know you could actually take that approach there using graph transformations so obviously check that video out if you're not sure on that you can always have a think about how what i've actually done there once you've had a look at that now instead of doing that i'm just going to take this nice calculator approach because this was actually a calculated question as well so rather than doing that i'm just going to plug my numbers into this formula here so i'm going to start with negative 270 so i'm going to do one take away and i'm going to do sine minus 270. there we go and that's going to give me my first value so 1 take away sine and i'll put minus 270 in the bracket and i get the value of zero there we go so that's my zero then i'm just going to change that for the negative 180 so i put minus 180 i'm just going back into my calculator and changing that value that gives me that value there back in negative 90. there we go that gives me two so negative 90 is up here at two i'll put zero in and that gives me one there we go put 90 in and that gives me sorry i'll put that in 1 minus 90. yeah that gives me zero there we go then put 180 in the last one nearly done that gives me one where's that gone one is there and then 270 there we go and that gives me two and that is there and again just drawing that up with a nice smooth curve as best you can there we go and there is our transformed graph but again two different methods that you could take to approach that obviously if you're really good with your graph transformations you could have flipped it over first change the y coordinates made them positive negative and the negative one's positive and then obviously shifting it up one doing that transformation with the one at the end changing the y-intercept um or you could just take this approach just plug in the numbers into the calculator plot them all nice and neat and connect them all up together so it's up to you which one you prefer but just a couple of different ways that you can actually approach that question right let's have a look at our next one okay so this says here is the graph of y equals cos x between the values minus 180 and 180 and again we can see that on the graph it just stretched from -180 to 180. it says on the grid sketch the graph of y equals negative 2 cos x but again between these values okay so taking a little approach to this then i'm just going to take the calculator approach now again you could take a graph transformation approach to this but let's have a look at the end and see what that graph transformation is we'll take that calculator method to start with and just start plugging the values into this formula so we've got y equals negative 2 cos x on my calculator just going to type in negative 2 because i'm going to start with that first value on the x there which is negative 180 so minus 2 cos and then negative 180 in the bracket and it gives me the value of two there we go that equals two and i'm just going to do that same approach for all of them so that two goes up to there and there we go so back into the calculator i'm not going to sort of uh do the all the little sub values in between i'm just going to go straight for negative 90 and see where that is so negative 90 ends up at zero there we go that stays where it is i'm going to go for zero now so back in negative two cos zero and that gives me a value of negative two and let's go back for ninety negative 2 cos 90 gives me a value of 0 and negative 2 cos 180 gives me a value of 2 there we go so that value of 2 is there and again we just need to join that up with a nice smooth curve there we go and that's our point drawn there so again that's one method that you could approach if i swap to a different color here we'll have a think about the graph transformation as well so negative there flips the coordinates over so i could have actually started by flipping those coordinates over so let's go about doing that we've got this one here which is going to flip to the other side there zero state that one there stays where it is this one here on one flips down to minus one which can just about see it's about there somewhere we've got the one on the axes then which stays where it is and we've got this one down here which flips up above so the minus part okay just like before changes the symbols in front of the y coordinates so we can draw that in okay i'm going to get rid of this in a sec and just about draw that in there we go so that's what the negative part does and then you've got the 2 there and again this is all linked to graph transformations but the 2 is going to double the y coordinates okay outside the bracket not affecting exit affects the y coordinate does what we expect 2 multiplies the y coordinates by 2. so this coordinate here that's currently at 1 goes up to 2 which is there 0 stays as it is this one here which is at minus one goes down to the minus two which we've got here on our our actual graph again zero stays where it is on the next one and then again finishing it off with this one that one there goes up to the two so you can actually take a graph transformation approach again again i think it is actually a bit harder than just typing in on the calculator but just so you know there are multiple methods that you could use here there isn't just one way to solve these sorts of questions but there we go that is our final answer there with our nice green line that we've drawn in i think just taking a calculator approach was a lot easier but obviously if this was a non-calculated question and this was actually a calculator question but if it was non-calculator you would have to know those graph transformations there but this would be a particularly nasty question if it was a non-calculator one right okay so let's have a look at our next one so mary buys a car for four thousand pounds each year it depreciates by twenty percent work out the value of the car in three years so that word depreciate means it falls in value now again with depreciation it's similar to compound interest it falls in value but it falls in value based on its new price so if we do this without a calculator we can do it both ways year one and the four thousand pounds is what it's worth twenty percent of that we can work out ten percent is 20 is 800 and if we take away that 800 there so 4 000 take away 800 we end up with 3 200. so we don't keep losing 800 pounds a year now in the second year it's worth three thousand two hundred ten percent of that is three hundred and twenty twenty percent of that is double that six hundred and forty and we could subtract six hundred and forty so 3200 take away 640 will leave us with our answer for the final year take away 640 there gives us 2560. and we can finish this off i should say year two we can finish this off on year three so our value is two thousand five hundred and sixty ten percent of that is two hundred and fifty six so twenty percent will be five 512 and then we can subtract 512 so 2560 take away 512 leaves us with the final value there of 2000 and 48. 2048 pounds and that'd be our final answer again we could do this with a calculator if something loses 20 in value then it's only actually worth 80 percent of it of its cost the previous year so if it's worth 80 percent the decimal that i can use for 80 percent is 0.8 if it's losing value we won't want a one-point number there we got 0.8 so we could just times it by 0.83 times or times it by 0.8 to the power of 3. so i could just type into my calculator four thousand times zero point eight to the power of three and i'll get the answer there 2048 as my final answer and let's test that out four thousand times zero point eight to the power of three and we get 2048 so that's depreciation share some money in the ratio four to seven james receives 21 pounds more than emily how much do they share between them so emily to james is in the ratio four to seven now it says that james receives 21 pounds more and if we look at these two numbers here from four to seven is an extra three parts there goes an extra three so that extra 21 pounds is three parts so if three parts of this ratio equals 21 pounds then one part must be to divide that by three must be seven pounds and from there we can work out how much they both receive if it was four to seven let's get rid of this little bit down here and each part is now worth seven pounds or four times seven gives us 28 pounds for emily and seven times seven gives us 49 pounds for james so emily emily receives 28 and james receives 49. it says how much does station do they share between them so if we can add these together it gives us the total amount so 49 plus 28 gives us seven and seven so 77 pounds there we go so they share 77 pounds between them a to b is in the ratio three to five and b to c is in the ratio two to one work out the ratio a to b to c so if we write out a to b to c and start to write what we've got so far now it says a to b is in the ratio three to five and then it gives us b to c so there's an overlap in these there's a b in both so i can't put another number just over the top of the five so i'm going to start a new level i'm going to write the other one underneath so b to c is 2 to 1. now looking at this if i can make these middle numbers the same i can merge these ratios together i can squish them together essentially so let's think 2 and 5 can make 10. so if we times the top 1 by 2 that would make that five a ten and we'd have to times the bottom one by five so if we times the top one by two that three becomes a six and the five becomes a ten looking at the one below times that by five two times five makes ten which we've already got in the middle and one times five is five and there's my ratio a to b to c six to ten to five just have a look to see if you can simplify in any of these questions but it doesn't actually ask us to simplify it there so a mobile phone costs 480 pounds in the uk and the same phone costs 600 in america the exchange rate is one pound to 1.29 where offers the cheapest price so we have two options here we could put them both into pounds or we could put them both into dollars i'm going to decide to put them both into pounds so as we can see we've already got the cost in the uk is 480 pounds but we're going to have to convert this 600 so looking at our exchange rate the actual number itself we get a bigger number in dollars so we know that our answer here is going to have to get smaller so we only have one decision to make and that is either we times or we divide by this 1.29 okay so it's times or divide okay but we know logically we're gonna get a smaller number in pounds so let's go for this when we do convert back we are going to divide so it's going to be 600 divided by 1.29 and we'll do that on the calculator 600 divided by 1.29 gives us not a very nice number here 465 pounds and 12 p to the nearest penny now it doesn't matter about all the decimals after that because at the end of the day we're looking at which offers the cheapest value and as you can see they're 465 pound 12 is cheaper than 480. obviously there is another way of doing this we could convert them both into dollars so instead of doing that what i could do is times this number here by 1.29 and 480 times 1.29 if again we work that on the calculator gives us an answer of in dollars 619 and 20 cents and again that's more than 600 so again we would state that it's cheaper in america three tins of beans and four jars of jam weigh two thousand and eighty grams and the total weight of five tins of beans is one thousand eight hundred work out the weight of one tin of beans and one jar of jam well straight away we can work out the weight of one tina beans from this because it says five tins of veins is one thousand eight hundred so if we do one thousand eight hundred and divide it by five we get the answer three hundred and sixty there we go we might have to do a bit of bus stop for that but we can do that fives into 1 800 5 into 18 goes three remainder three fives into 30 goes six and then zero so 360 grams and that is for the tin of beans so tin of beans there we go it then says three tins of beans and four jars of jam weigh 2080 back in the first line so if we work out the weight of three tins of beans now or three of those would be 360 360 and 360. if we add those all up we get 910 1080 so that's the weight of those tins of beans so if we subtract that away from this original amount so 2080 take away 1080 that leaves us with a thousand grams so those four jars of jam so four jars equals a thousand grams and we want to know the cost of one so we can divide that by four and it gives us one jar equaling 250 grams so there we go one jar is 250 and one tina beans was 360. we've got some proportion here six taps take three hours to fill a tank with water how long will it take nine taps to fill the same tank and then we're gonna state any assumptions that we've made so if six taps take three hours it will take six times as long as long as that for one tap so one tap would take three times six which would be 18 hours it takes six times longer so one tap it takes 18 hours so from here if we know how many taps we've got we can just divide this 18 hours by however many taps if we have two taps we can divide it by two which would give us nine hours we could do three taps divided by three would give us six hours but this particular question here said that there's nine taps how long would it take nine taps so if we do 18 hours divided by the nine taps it gives us a total answer of two hours so these questions are all about working out first how long would it take one tap or or whatever this questions are actually about normally so it says state any assumptions that you've made well i've assumed here that all these taps are running at the same rate okay that's my assumption all taps run at the same rate there we go it's obviously in that original statement there six taps take three hours while one of those tabs might be running a lot faster than the other and that would change our answer so we are assuming that all the taps are running at the same rate okay onto some direct and inverse proportion so there's two formulas that we can use here it says a is inversely proportional to b so it's either going to be a equals k lots of b or a equals kb or it's going to be a equals k over b so these are the two that you have to remember now with inverse proportion it's this one here where k is being divided by b so we're going to use that formula so just make sure you write them down straight away and then it gives us some values to put in so it says 15 for a so 15 equals k over and it says b is four so we can solve this for k we can times both sides by four to isolate k there and we get 60 equals okay or k equals 60. and now what we can do is we can put that 60 back into the original formula here and that will give us our formula so a equals 60 over b and that is a formula for a in terms of b and sometimes you might just be asked just to write a formula for a in terms of b and there it is but this question says find the value of a when b equals 12. so what i've got to do is stick b into b equals 12 into my formula down here so if we do that i'm going to do it to the side we get a equals 60 over 12 and 60 divided by 12 equals 5 so our final answer there is 5. so every time you just think which formula we're going to use plug your pieces in find your value of k and then reuse that in your formula for whatever values you need okay so this question says directly proportional to so we're going to be using the other formula this time it's a equals kb but you've got to read the question carefully because this one says directly proportional to the square root of b so i need to put a square root in with my v so using the same formula but just putting my square root remember it could say squared or cubed or cube rooted or anything but that's my formula i'm going to use then it says when a is 18 b equals 16. so if we put those values in a is 18 so 18 equals k and i'm gonna put a times sign in here the square root of 16 and you can do that in your head hopefully the square root of 16 is four so i'm just going to put k times four now again we can actually find the value of k here we can divide both sides by four so if i divide both sides by four we get a decimal here because it's 18 over four so 18 over four equals k now we simplify that down let's have a look divide top and bottom by 2 we get 9 over 2 which we can either leave as that or we can turn into 4.5 i might just leave it as 9 over 2 for the moment see where we end up with this but we've got our value of k so if we put that back into our formula now so just putting that back up here for our value of k we get a equals 9 over 2 root b there we go and there is our formula for a in terms of b a is 9 over 2 or 4.5 times root b then it says find the value of b when a equals 2. so if we put those into our formula there a equals 2 so 2 equals 9 over 2 root b there we go right so we need to rearrange this so i need to divide both sides by 9 over 2 or divide both sides by 4.5 if we've got a calculator that's fine if we don't know we can just do 2 divided by 9 over 2 remember in fraction rules that would be 2 over 1 and you would times it by the flipped version 2 over 9 and that would give us 4 over 9. so 4 over 9 equals root b and if you've got a calculator you can just type that straight in so 4 over 9 equals root b in order to find out what b is we just need to square both sides and when you square both sides there 4 squared is 16 9 squared is 81. there we go we get 16 over 81 so our final answer there is a fraction there we go just following those uh normal fraction rules there that would have been absolutely fine with a calculator but just a different way to approach this if it is non-calculator okay so when looking at pressure force and area you will be given the formula so pressure is equal to force divided by area and this question says find the pressure exerted by a force of 900 newtons on an area of 60 centimeters squared and it says to give your answer in newtons per meter squared now if you were to just plug these numbers into the formula we would get that pressure is equal to 900 the force divided by 60 and that will give us an answer of 15. now the problem with that is that answer there is in newtons per centimeter squared but it says here to give your answer in newtons per meter squared now i think the easiest way to go about this is just that rather than using the 60 centimeter squared and putting 60 down here we just convert that 60 centimeter squared into meter squared now in order to do that if we think about the conversion from centimeters to meters a hundred centimeters in one meter so you divide by a hundred so when converting the area unit for centimeters squared to meter squared instead of dividing by 100 we just divide by 100 squared so if we go about doing that and you can type that in your calculator 60 divided by 100 squared and on a lot of calculators that's going to give you the answer as 6 times 10 to the power of -3 now hopefully you can get that into a decimal you just need to hop the decimal three places to the left over the six and that's actually equal to naught point no not six and again a lot of calculators will just give you that value instead so in order to go about doing this then all we need to do is take our meter squared value and put it into our formula like we did before so we just do 900 and instead we'd now divide it by 0.006 and that now gives us an answer in newtons per meter squared and it gives us an answer of 150 000 so there's our final answer that is in newtons per meter squared and that would be our final answer for this question but of course just being careful on these questions that you read the units that it wants it in and i would advise changing the area units rather than trying to change the newtons per centimeter squared into newtons per meter squared okay so translating a shape by a vector now that top number in a vector means left and right and the bottom number means up and down okay thinking about positive numbers moving towards these positive numbers on the axes and negative numbers moving towards the negative numbers as well so four moves it right towards the negative numbers by four so all i ever do is pick a point you can pick any and i move it right by four so one two three four it's going to end up there the next thing i do is look at the other number which is minus two now negative two moves down towards the negative numbers so one two down and that means that new point is going to end up just there so all i have to do is draw this triangle in exactly as it is from that top point so it goes down three and across two and there we go we can just draw it in like that not forgetting as well you might actually be asked to describe a translation in which case you'd say it just like it says above you say translate the shape by a vector i need to say what the vector is okay so reflect the shape in the line x equals one now this is your x axis and this is your y axis so x equals one is right there and the only line that you can do from that point other than going across the x axis and it doesn't see the x say the x axis is to go up and down so x equals one is this line here if you think about any coordinates on that line the x coordinate is always one so that's our reflection line there now we're going to reflect it in that line which is quite nice and easy to do pick a point and we just go one two to the line so one two away and just follow that process for all the points so picking this point that is four away so another four away gets me to there and the same one at the top that's four away and four away gets me to there and then just joining it all in obviously using a pencil for this one and that'll be my triangle i haven't drawn that in particularly neatly there we go okay not forgetting as well though you could have a line it could say the line i don't know y equals three which would be across at y equals three which would be across here there's another couple of lines that we could have as well we could have this one y equals x the line y equals x is the diagonal line here where all the x and y coordinates are equal so we could have that one as well where we have a diagonal there's one more that we could have as well let's pick a different color for this we could have y equals negative x which is very similar to the one above but it's pointing in the opposite direction and if i do that in blue as well that's going down this way so we could also have to reflect in a diagonal line but again you just follow the same process counting diagonally okay so rotate the shape 90 degrees clockwise about the point -1 0. so first thing to do is obviously to locate that point which is minus 1 across not up and down so minus 1 0 is just there the next bit this is a lot easier if we have some tracing paper so you stick your tracing paper nice and flat over your shape trace it in so trace over your shape stick your pen on that rotation point and just rotate the tracing paper not moving it away from the pen so we would rotate it 90 degrees so it would be facing this way and just making sure it stays nice and flat and it would rotate the shape let's have a look it would go to here on your tracing paper and then you would just lift your tracing paper up nice and carefully and draw in at that spot and again the same process if you have to describe one of these you stick your tracing paper over the top and just move your pen into different points pick in places like here and here and moving it around until you get the one that matches when you switch it twist all your paper around 90 degrees and remember and obviously in your description you need to put the amount of degrees the direction that it's gone in and that rotation points along with the words rotation okay so enlarge the shape by a scale factor of two from the point minus four minus three so let's identify that point first minus four minus three and then we're just going to enlarge it by a scale factor of two so what i do is i pick a point on the shape normally the closest one let's say this one here and i just think all right how do i get there from the point so we go two across and one up and i'm just going to do exactly the same again so two across and one up and that there is a scale factor of two the first movement to the shape is my first scale vector the second is my second scale factor there so if i had a scale factor of three i'd do the same again i'd go two across one up and that'll be a scale factor of three but this one has just set a scale factor of two so i'm gonna leave it there now if i have a look at the shape that's drawn originally it's two along and it's three up and this is going to enlarge it by scale factor of two which doubles those numbers so it's no longer going to be three it's going to be six and it's no longer going to be two it's going to be four so from this point here that we've got to all i need to do is redraw that shape in remembering it was the bottom left corner this one here so we'll just need to go four across one two three four and six up one two three four five six and then join it all up nice and neat with a pencil and a ruler and there it is enlarged by a scale factor of two just remember if you have to describe one of these it's a really nice way of doing it as well you just pick the two corners get your ruler and a pencil join them up really nice and neat pick another one match it with the appropriate one and then again join that up really nice and neatly and it will show you down here look where that enlargement point came from and you can find the scale factor by just looking at the sides so the side length is 6 here and the side length is 3 there and 6 divided by 3 gives us a scale factor of two so if it was already drawn in we found our enlargement point we just say it's an enlargement a scale factor two and from this point down here just like it did in the question enlargement scale factor two and from this particular point okay so we've got a transformation here it says enlarge shape p with a scale factor of negative a half with the center of enlargement zero zero so always mark out your center of enlargement first and an enlargement with a negative scale factor and a fraction here means that we just got to do this very very carefully now the first things first you need to pick a point on the shape and just figure out how you get from the center of enlargement to that point there and if i just count that that's one two to the right and one two three up now all i need to do to do this and it's going to be quite small in this diagram is i just count in the opposite direction by whatever the scale factor is that's what the negative is so rather than going from from 0 0 rather than going right and up i'm going to go left and down but it's a scale factor of a half so i need to also halve those distances just as if it was negative two i would double those distances but for negative a half so i'm gonna go left one rather than two and i'm going to go down rather than going up three i need to go down one point five or one and a half so one and a half gets me to there there we go so that's one and a half i'm gonna do the next one in a different color so i'm gonna pick this point here and let's just think how we get to that let's get rid of some of these markings here there we go so to get to that we go one two three four across and one two three up so i'm just gonna half that again i'm gonna go two left rather than four right so one two and then that one and a half down which again gets me down to there there we go and that is that point then again i just need to repeat the process for the last one you might be able to start to do these in your head once you've had some practice but if we do the last point here being the top one let's just see how we get there so it's one two across just like it was to that red one and then we got to go up one two three four five six seven so half of that's going to be 3.5 so i'm going to go one to the left okay so i need to go down 3.5 one two three and a half just there there you go obviously you can do all this in pencil to keep this all nice and tidy but when you've got a negative scale factor you're just going to go backwards in the opposite direction by whatever that scale factor is obviously join it all up nice and neat and there we go and that is that one there just always go back and double check so it was 7 up so we went 3.5 down that's absolutely fine the other one was three up and we went 1.5 down half of that and the other one there perfect okay so you can see with these it does actually get bigger and it gets bigger or smaller and it rotates 180 degrees and that's how a lot of negative enlargement does to a shape okay so we're going to have a look at another one here okay so it says describe the single transformation that maps shape b onto shape a this is quite nice you hopefully you can tell that it's an enlargement okay because obviously one's got bigger than some bigger and smaller and we're going from b to a now what you can do is you can get your ruler and i've got to think i've got to do what you're going to have to draw and do this quite carefully but you get your ruler and you join up the similar points now the similar points are here and here okay obviously it's been rotated 180 degrees now if you get your ruler and join that up you end up with a little line like that and then you just do that for all the other points so i can probably get away with just doing two we'll have to wait and see so if i join up these two similar points here there we go they join up just like that and what you'll find is there's this little crossover point and that tells you where the center of enlargement is so i know that it's an enlargement we've already got that so i'm gonna have to state it's an enlargement okay so it's an enlargement i've got to get what the scale factor is i haven't got that quite yet but i'll have a look we know it's negative as it's getting this 180 flip so you know it's negative we'll come up with a number in a sec and i know where the center of enlargement is so center of enlargement let me just write down what that coordinate is and it's it's 2 2 so center enlargement 2 2 now to count that scale factor we're just going to count this side here maybe and this side here and just see how much bigger it's got so that's gone from a length of one to a length of three so that is a negative three scale factor so it's an enlargement scale factor negative three center of enlargement two two i can use that approach for any of these enlargements whether it's negative or not also you might just want to note here that if it had been from a to b it would have been the other way around it'd have gone from three down to one and my scale factor instead if i was going from a to b it'd be exactly the same description there but my scale factor would be negative a third because it's getting a third of the size going the other way okay so two different ones but the description would be exactly the same there okay so we're gonna look at some circle theorems for the purpose of keeping this video as short as we can i'm only gonna go over a couple of them and just how to approach the question if you wanna have a look at every single circle theorem how to tackle every type of question you have to look at my circle theorems video now some information to go along with this i'm going to say that ap and bp are tangents there we go so when it comes to tangents they meet at equal length that's one of our circle theorems so these two lines here are equal length meaning that a p b that triangle there is an isosceles meaning that the base angles these two are the same so there we go we can start to figure out some of the angles here because if we do 180 the angles in a triangle take away 86 it leaves us with 94. and that we can split in half to share it between our two triangles there so 94 divided by 2 is 47. so both the angles at the bottom of the triangle there are 47 there we go and as with all these questions it always says to state your reasons so you would say tangents meter equal length therefore this triangle is an isosceles okay and writing that down okay on to the next bit we've also got and it doesn't doesn't say it here but o is the center of the circle now o to b there is a radius looking at this line just here and a radius meets a tangent at 90 degrees meaning you can always draw this on the diagram when you see a tangent meter in a radius we've got a right angle just here so we can work out the value of x because if the full angle is 90 we're gonna do 90 take away the 47 and that leaves us with 43 degrees the important part with this question is to make sure you write down all those reasons so we would write the tangents meter equal length isosceles base angles are the same and then our final reason for this bit of the working out the tangent meets the radius at 90 degrees and therefore we could do 90 take away 47. okay a different circle theorem question with some different uh circle theorems within this one so if you have a look we've got these points a d c b around the circle and that forms a quadrilateral and these are called the cyclic quadrilaterals and the rule here is the opposite angles add up to 180 so we've got the 70 over there so opposite that is the 9 is the y and opposite angles have to add up to 180 so we could do 180 take away 70. and it gives us our answer of 110 degrees and our reason for that which we would have to write down again is the opposite angles in a cyclic quadrilateral add up to 180 the next one here is to find this angle x and that's going to involve one of our other circle theorems and when you've got these points made at the center i'm just having a look at points d and b i always do this with a little highlighter but if i make this angle 70 here i can also from the same two points make this angle here at the center and that's one of our other circle theorems angles at the center are double angles made at the circumference from the same two points so to work out angle x and this first one we did was y to work out angle x we would just do 70 times 2 double it and that would give us 140 degrees and again that would be accompanied by the reason angles at the center are double angles at the circumference when they're made from the same arc so our two answers here are 110 degrees and 140. again just a few little bits of circle theorems there to be having a thinking about and making sure that you are writing down all those reasons that's absolutely crucial for these okay so when looking at the volume of a composite shape there's lots of shapes that we could have but specifically if we're looking at a composite shape when looking at volume we're going to be looking at that cross section and that cross section of the shape is going to be made out of two different shapes so the cross section on this shape is this particular one just here which has five sides so in order to figure out the area of that the area of the cross section we're gonna have to split it up into different shapes and of course there's lots of different ways to do that you could split it going down this way to make a rectangle and a trapezium you could split it going across this way to make a rectangle and a trapezium and of course you could split it in lots of different ways to make two rectangles and a triangle if that's something that you prefer now whichever method you choose i'm going to go with that first one splitting it down the top we just need to be careful to get all the correct lengths on that particular 2d face just to make sure that we get the correct area so if i draw that to the side and absolutely fine for you to do this as part of your working out i'm going to split it up so i can see all the lengths so we have a height of eight we've got a little width of three up there we've got three over here and the rest we're going to have to split ourself so obviously the height over here is also eight this is also three and because that base length down there is eight this would be five so if we go about working out the area of each of those the rectangle on the left is going to be three times eight so that's going to have an area of 24 then we've got a trapezium on the right now the area of trapezium is add together the parallel sides so 8 plus 3 divide that by 2 and then multiply that by the distance between them so we're going to multiply that by 5. now if we work that out and you can type that in on your calculator we get 27.5 so adding those together would give us the overall area and if we work that out as well 24 add 27.5 is going to equal 51 so that's the area of the cross section now we have the area of the cross section just like any 3d prism we multiply it by the depth or the distance that that cross section goes through the shape which in this case is 12. so to finish this question off we take the area of the cross section 51.5 multiply it by the depth of 12 and that gives us an overall volume of 618 centimeters cubed and that would be our final answer for the area sorry for the volume of this composite prism that we have here okay so we're gonna have to work out some missing lengths in this triangle now it's not a right angled triangle so to find missing lengths and angles we can use either the sine rule or the cosine rule and these are rules that you're going to have to remember but we'll have a look at one to start with and how we know when to use it now the first thing you do is you try and identify in the triangle first of all what we're looking for which is a b so let's label that x now what i look for straight away is do i have pairs of opposites so i have this pair of opposites here and i've got both of those and then i've also got x opposite to this angle so i don't have x but it's in one of my pairs of opposites so when we've got this scenario where there's two pairs of opposites we can use the sine rule and we only ever need part of the sine rule so i'm just going to use a over sine a equals b over sine b also equals c over sine c but we only ever use i only ever need to use two of them here so let's have a look we need to label this up and i'm going to completely ignore the letters that are on the actual triangle itself i'm going to label this angle a which it already is and this and this side little a opposite that and then this angle b and the one opposite little b and all i need to do now is stick all the numbers into the formula so let's have a look a is 12 so it's air 12 over sine the one opposite that 55 is equal to b which is our x over sine b obviously you should already know there's two for two variations of this formula we could have it flipped over so we could have sine a over a equals sine b over b but this is our one for side length and we know that because our unknown piece is on the top so we're able to isolate this now quite easily so sorry i've written b there it should be 20. there we go sorry there we are 20. so what we need to do is times both sides now by sine 20. what you could do is work this out on your calculator and times your answer by sine 20 but i'm going to multiply it straight over so i can type it all in one go so times by sine 20 and if we do that we get 12 sine 20 it goes on to the top there over sine 55. there we go and all we have to do is type that into your calculator obviously just being careful that you put these angles in brackets some calculators are going to need you to do that so if we type that all into the calculator again not forgetting you could just work out 12 over sine 55 first and then times it by sine 20 but i'm just going to go for it like this so 12 sine 20 on the top closing your bracket over sine 55 and on the calculator just writing down what you got you've got 5.0103 five three nine nine eight now a question would normally say how to round this so if we imagine it was two decimal places for this one it would be five point zero one and it's a length so centimeters okay so just obviously just be careful the question says let's have a look at an idea where we've got to find the angle okay so in this question let's have a look work out the size of angle bac so let's identify that that is here bac okay so we're going to use the formula the other way around this time so sine a over a equals sine b that does not say thing b sine b over b okay so plugging in our numbers let's just label it up so let's call this one a as the a is next to it that's fine and again i'm just going to write over this one i'm just going to put b and b okay just because the way i've written my formula so then sticking in all the numbers what have we got sine a is sine x so we have sine x over 20 equaling sine 43 over 14. okay so exactly the same approach as we did before we can isolate the sine x by timesing both sides by 20. and again you could work that right hand side out and times it by 20 but i'm just going to go stick it up the top there so we end up with sine x equals 20 sine 43 over 14. if we type that into the calculator now what do we get 20 sine 43 over 14 and we get an answer here let's write it over here so we get sine x equals 0.97428 and a few more decimals and obviously just like normal trigonometry when you're doing socatoa to get the actual x here we have to the inverse of sine so we're leaving that number on your calculator you do shift sign which gets you signed -1 type in that answer or just press your answer button so shift sign answer press equals and i get an answer here of 76.9779 and again a question would ask us to round it here uh just depends so let's just go to the nearest degree we go for 77 degrees obviously just making sure what the question says here but let's just round it to one the nearest degree there so 77 okay so that's how you use the sine rule right let's see how this question is different then so work out the length of a b so this one over here now straight away looking at this look we've got a pair of opposites there but i don't have any other pairs of opposite opposites i've not got anything opposite my 15 i've not got anything opposite the 12. so i can't actually use the sign rule here and that is your clue that is your hint here that the sine rule is not going to work we're going to have to use the cosine rule so another rule that you need to know so the cosine rule is e squared equals b squared plus c squared minus 2bc cos a okay so a being the side we're looking for so we'll label this little a and this one big a ignoring the letters on the triangle and then labeling the other two sides and they can be b and c however you like and from there all you've got to do is stick these numbers in it's actually quite simple to use when you know it so a squared equals b squared which is 15 squared plus c squared which is 12 squared minus and i'm going to stick this bit in brackets 2 times 15 times 12 cos a which is down here which is 20. there we go right so sticking that all in the calculator let's have a look what we get so 15 squared plus 12 squared minus 2 times 15 [Music] times 12 cos 20. press equals and i get a squared equals 30.7106 and a few more decimals now obviously that is a squared and we don't want a squared we don't know what a is so we just need to square root both sides now so square root leaving the answer in the calculator square root answer and we get a equals 5.541719 and again obviously you'd be asked to round this in a particular way in the question let's go for two decimal places so a equals five point five four centimeters all right there we go and there's using a bit of the cosine rule okay so working out the size of angle bac which again is this one at the top and again just having a look there are definitely no opposites because we've got no other angles but uh the angle that we're looking for is going to be our a and it is opposite nine there we go so the others can be b and c again now obviously here we're looking for an angle so it's up to you if you choose to learn the formula i tend to find that i just learn this formula a squared equals b squared plus c squared minus two b c cos a and then i'm quite happy just rearranging that to get cos a on its own so in order to do that i'm going to get this whole minus 2bc cos a i'm going to add it to the other side so we get a squared plus 2bc cos a equals b squared plus c squared now i can get rid of that a squared so we can minus a squared from both sides so minus a squared and you get 2bc cos a equals b squared plus c squared minus a squared now and now you can finish off this rearrangement you can divide by 2 bc just to leave you with cos a so cos a equals b squared plus c squared minus a squared over 2 bc it's up to you you can choose to just learn that formula if you want but that's the formula we're going to use to find an angle so plug in all these numbers then just into my formula there we'll get cos a equals b squared plus c squared so 10 squared plus 5 squared minus a squared so minus 9 squared all over 2 2 times b times c so 2 times 10 times 5. nice and easy typing that into the calculator so fraction button 10 squared plus 5 squared minus 9 squared all over 2 times 10 times 5. and that equals let's have a look so cos a equals 0.44 and then same process again we need to do the inverse of cos so cos minus 1 of your answer and you get let's have a look shift cos answer and i get 63.896 there we go degrees and again we could round that so we could just say 63 point and let's just go to one decimal place 63.9 degrees again just reading the question and that's how to use the cosine rule for finding angles okay so we're going to look at some statistics this is going to be quite a short one and we've got a few questions to have a look at and we're going to start with a box plot so on here we've been given some information about some uh time in seconds that 15 people wait to be served at a garden center and it wants us to draw a box plot for this part a so we've given a list of numbers for those minutes and you just got to make sure they're in order because we need to find a few values we need to find the lowest value the highest value the median and the two quartiles the lower and upper quartile so we can quite easily see that they are in order so we can get the lowest and the highest value from either side and they're going to form the ends of our box plot so just very carefully on your scale here drawing the ends in here and 44 just make sure the scale goes up in ones it does i'm gonna do as best as i can here and that's our 44 there now we need to find the median so with 15 people we just want to find where that halfway point is so all you got to do is add one and half it so 15 plus 1 is 16 divided by 2 and that's going to be our eighth person so we just need to go along the box plot find where the eighth person is so one two three four five six seven eight so 25 is going to be our median that's quite nice it's halfway between 20 and 30 drawing and nice and neat there to form our median now we just need to find where the quartiles are so on that lower half okay the lower part or the lower quarter here if i try and highlight that we've got these numbers here and there are seven numbers there if you just count them up so one two three four five six seven so again finding the halfway point we can do seven plus one and divide it by two which gives us four so we wanna find where the fourth person is for that lower quartile so five nine eleven fourteen there is our lower quarter there so marking that on the fourteenth person one two three four and then looking at the higher quarter there 27 27 28 30. okay so once you've marked all of those on they've got to join up your little box there we go making it like the one that you can see below and then join up these parts here and there's your box plot drawn okay moving on to the bit below now i haven't added the question in here but quite often it says to compare the box plots so it says here the box popular below shows the distribution of times served a different garden center and it would obviously you would usually say to compare them now there are two things that you want to compare when you're comparing a box plot one of them is the median so if we have a look our median here is sitting on 25 and down it's sitting here on 21 now i don't have to mention any numbers but i would mention number one i would say that rose or just i'll just call it rose rose has a higher median wait time okay so a higher median weight time okay make sure that you mention some context obviously don't just say it has a higher median what's the story actually about it's about the wait time at a garden center so roses are higher median wait time the next thing we have a look at is the distance between the quartiles and again i don't have to mention any numbers but i just have to think this is the distance between the quartiles it's called the interquartile range and on the one below here is the distance now hopefully you can quite see there visually that on the greens garden center there is a higher interquartile range meaning that the data is more spread out and you want to mention that word spread out so number two i did my comparison here greens has a higher i'm going to abbreviate it to iqr into quartile range meaning the data is more spread out or meaning the wait times are more spread out there you go more spread out and mention that word there spread there you go some more spread out and that is drawing a box plot and also make a comparison between the two let's move on to our next one okay i'm looking at some ratio in proportion now so this says james has a full bag as a bag of full bag full of counters he takes a random a sample of 20 counters marks them all and puts them back into the bag and then he takes 30 counters at random now to the 35 of them are marked estimate how many counters are in the bag so look at capture recapture here now if we just make a few statements because we're going to assume that the sample is proportional to the next one that he took as well or to the rest of the counters in the bag so initially he takes a sample of 20 counters out of we don't know that's what we're trying to work out so let's call that 20 over x on the next time he takes a sample of 30 and 5 of them are marked so we'll say that that's now 5 out of 30. so out of the 35 were marked but previously it was 20 that were marked out of something we don't know but we are going to assume that these are proportional so in order to work out how many that x value is we just have to think okay well how do we get from 5 to 20 and we do that if these fractions are equivalent by multiplying by four so in order to get that number on the bottom there we would multiply this by four and that would give us 120. so our answer would be 120 counters okay obviously we are making some assumptions here we are assuming that no additional counters were added to the bag in between the two samples and also them were taken out as well and obviously there's little silly things like we're assuming that the marks haven't actually come off the counters or anything like that in this time frame okay so that is capture recapture there i'm moving on to a venn diagram now this symbol here just means all the numbers that we're going to put into the venn diagram so we've got the numbers 1 to 15 and then it says prime numbers and even numbers and what i'm going to do is i'm going to write down the prime numbers to start with so we have 2 3 5 7 11 and 13. and we could write down all the even numbers as well but i'm hoping we can recognize them quite easily now i'm going to have a look at is which numbers are in both now the only even number in that prime list is the two so that's going to go in the middle because that says it's in both so i'm going to cross the two off and cross it off from my main list it then says obviously we've got the prime numbers in set s so i need to put all the rest into set s so 3 5 7 11 13. again i'm going to cross them all off from the main list so 3 5 7 11 13 and then all the even numbers are going to go in the right circle there so we've already got rid of two so 4 6 8 10 12 14 and again just crossing those all off from the main list list so 4 6 8 10 12 14 and the remaining numbers they're gonna go around the outside so i've got one cross that off stick it on the outside nine cross it off stick it on the outside and 15 cross that off and stick it on the outside now we're asked some questions about the venn diagram it says list the members of s u e now that u stands for the union and the union is everything within the circles so that's the members of s the members of e and that number in the middle there so if i write them all down all the numbers there for part b are from left to right we've got 5 11 7 3 13 2 and then all those even numbers 4 6 8 10 12 and 14. let's have a look at part c write down p s n e now that p at the start there to sound stands for probability so write down the probability that it's an s n e now that n is a different symbol again and the n stands for the intersection so this is asking me what's the probability of picking the number that's in the intersection now in the intersection we only have this number two so there's one number so for that part c there i'd say one number out of because it's a probability and there are 15 numbers in total so the probability there will be one out of 15 for part c again it could say s u e and that would have been again we've got to count all the numbers in the union let's have a look there's 3 6 9 12 numbers there so that'll be 12 out of 15. now the final part here it says find the probability a number is in set e dash and that little dash just means not in so e dash means not in e so e dash not in e now if we have a look there are quite a few numbers that are in e so all of these numbers here are in e so all we need to do is have a look at how many numbers are not in e so let's pick a different color let's underline all the numbers that are not in e one two three four five six seven eight so the probability of picking a number that's not in e is eight again out of fifteen so again just remember that a little dash means not in that set the n means the intersection where they cross over and the u stands for the union which is all the numbers within all of the circles so the next one we've got a lot of information here so it says sophia asked 50 people which drinks they like best from tea coffee and milk and all 50 people that she asked liked at least one of the drinks then it starts to talk about all these different ideas about what drinks people liked and whether they liked all three whether they like tea and coffee so we're going to go through all of them there but ultimately it says she selects that random one of the 50 people work out the probability that this person likes t so we need to figure out from all this information how many of the people there like tea and it doesn't explicitly tell us how many like tea so when we've got these three options like this the best way for us to organize this is in a venn diagram specifically a three-way venn diagram so we need to draw i'll do this as best as i can a venn diagram that's going to allow us to have three elements in there and you just need to make sure with a venn diagram you draw a little box around the outside there we go now i want to keep this all nice and neat and tidy so i'm going to label them i'm going to call this one the milk this one the tea and this one down here the coffee i'm just going to go through the list and just start to tick things in when i know that i've got them there now the first piece of information it tells us and if we go through this step by step it says all 50 people like at least one drink so that means we know that there's going to be nobody in the outside so zero on the outside next thing we've got is 19 people like all three drinks and that's a key piece of information there because we can definitely put that in straight away the people that likely all three are right smack bang in the middle there so we've got 19 in the middle so we've dealt with that one you can tick them off as you go or highlight them so that next one says it says 16 people i'm going to swap to a different color here it says 16 people like tea and coffee but don't like milk so the people that like tea and coffee are going to be in the crossover between the t and the c circles but they do not like milk so that's fine we can put them in there 16 that like tea and coffee but do not like milk there we go tea and coffee but don't like milk it then says 21 people that's trustworthy 21 people like coffee and milk oh i'm just going to get rid of that because i've got too many things going in there i've got 21 people like coffee and milk so 21 people that like coffee and milk well let's have a look coffee and milk is it is this crossover here but also includes that 19. it doesn't say 21 people like coffee and milk but don't like tea so if 21 people like coffee and milk we've already got 19 in there that like coffee and milk so that's going to be an extra two that makes that up to 21 there in that little crossover with coffee and milk 19 of them also like tea but then 21 in total like coffee and milk so that's that one dealt with how many have we done now we've done all four of those there's lots on here on to the next one we've got 24 people like tea and milk so 24 people like tea and milk let's have a look what tea and milk is this one here but again it doesn't say that they don't like coffee so 24 people like tea and milk 19 are already in there so that's going to be an additional five people there and that makes up 19 and five that's 24 people in total that like tea and milk on to the next one we've got 40 people that like coffee we've already got three people in our coffee circle there we've got the two we've got the 19 and we've got the 16. so in total that adds up to let's have a look 2 plus 19 is 21. at the 16 is 37 and it says 40 people like coffee so it's going to be an extra three that make that coffee circle up to 40. next one this one's nice easy one to finish we've got one person likes only milk so in the milk we've got one person and then we're given no more information but there is this missing number here in the tea now just remember it did say in the question that she has 50 people so if we add up all the numbers we've got we've got 1 plus 5 plus 2 plus 19 plus 16 plus 3 and all of that adds up to make 46. and in total we have to have 50 people in our venn diagram so just in the t1 there that's an additional four people to make sure that we have 50 so we just did 50 take away 46 which left us with those remaining four people so we filled in our venn diagram now it says severe selects at random one of the 50 people work out the probability that this person likes t well that doesn't say just likes t it says the probability that they're like t so we've got a few of the numbers here that we can actually consider we've got the four the 5 the 19 and the 16. so the props for the amount of people that like t if we add them all together 5 plus 4 is 9 plus the 19 is 28 plus the 16. what does that give us 19 plus 16 plus 5 plus 4 gives us a total of 44 people that like t so there's 44 people that like t out of a total of 50 people so it's 44 out of 50. 44 of those 50 people like t and the rest don't like tea you've got the one person that only likes milk you've got the two people that like milk and coffee and the three people that only like coffee so 44 over 15 you don't need to simplify that and that's our venn diagram done just make sure you go through and tick them all off as you go so you haven't missed any so look at another one of these okay so this venn diagram is a little bit more complicated it says 82 students whereas what their favorite fruits were 39 like apples 50 like bananas 39 for oranges 21 like apple and bananas 18 like bananas and oranges and we've got some more information there but the one we're looking for is the amount of people that like all of them and this doesn't actually tell us the amount of people like or that like all of them so we're going to have to take a little bit of a different approach here but just say how many of the students like apples and oranges but not bananas so we're going to look for those people in a sec now if i draw a little venn diagram for this one again i'm going to need a little bit more space i might actually make that a little bit bigger let's make this one a bit bigger okay so we've got our venn diagram again nice little boxer on the outside and let's have a look at what needs to go in here so 82 students were asked what their favorite fruits were so we know that 82 people are going to have to go into our venn diagram now it says um that we've got lots of different bits of information haven't we here so let's have just label it up so we keep it nice and clear apples bananas and oranges down here not to mistake that for zero people there that's oranges now we don't know how many people in the middle so we're going to apply a little bit of algebra here i'm going to say okay well there is an amount of people in the middle and we'll call that x now we can start to have a look at some of the other bits of information i'm going to disregard the apples bananas and oranges a bit to start with but what i am going to do is just label that up here just so remember so apples is 39 so we know 39 in total in that circle bananas is 50. so in this total of 50 going into that circle and oranges is 39 so we know there's a total of 39 going into the oranges there okay so i'm going to take those all off i've drawn them on the venn diagram i'll remember that later now it says there we go 21 light apples and bananas so 21 like apples and bananas so that is going to be 21 in total including that crossover that's going in these two just here where the apples and bananas cross over so if i knew what the middle number was there i'd do 21 i take away the value of x in the middle but i can't i don't actually know what x is so i'm just going to write in here that's 21 minus x for that one okay so that's that one done for the moment in terms of algebra then we've got 18 that like um bananas and oranges so bananas and oranges is just here but again we don't know what that middle one is it doesn't say like like bananas oranges but not apples so that's going to be 18 take away the middle number which again is x 18 minus x onto that last one we've done this this one now so 19 light apples and oranges and again that's this one here apples and oranges but again we don't know the middle number so we'll call that 19 minus x there we go so there's our bits of algebra let's take that off and then it says 22 like exactly two of the fruits now the people that liked exactly two of the fruits are the first one we did 21 minus x i like apples and bananas we've got 18 minus x i like bananas and oranges and we've got 19 minus x like apples and oranges so what we can actually do is we can create a bit of an equation here now if we add those all together that has to equal 22 because it says 22 here like exactly two of the fruits so if we add them all up let's have a look let's just add them all up we've got 21 minus x we've got 19 minus x and we've got 18 minus x and if we add them all together let's see what we get so 18 plus 19 plus 21 gives us a total of 58. so we have 58 minus 3 x's so minus 3x so actually we can create a bit of an equation here i'm running that space a little bit so i'm going to do it over here but we have this equation we've got 58 minus 3x and it says it has to equal 22 people so actually we can just go about solving this now so actually what i can do and we can rearrange it if we like but i'm just going to minus 58 from both sides so 22 minus 58 gives us negative 36 so we have negative 3x equals negative 36. and if we do negative 36 and divide it by negative 3 that gives us a value of x which is 12. so we know that x that one in the middle now has to be 12. so actually what we can do we can start to get rid of some of these numbers and actually sorry to get rid of some of these x's and replace them with numbers so if i get rid of the x in the middle and we'll replace that with the 12 because we now know that 12 is in the middle we can then do 21 minus x above that which leaves us with 9 so we can get rid of 21 minus x oh do you got rid of the whole circle there we can get rid of 21 minus x and we can just put 21 minus 12 which is 9 so there's 9 going in there we can get rid of 19 minus x 19 minus 12 leave us with seven so we can get rid of that and we just got seven in there and then we've got 18 minus x on the right down 18 minus 12 leaves us with six so we can have six in there there we go six now we might be able to answer the question from here but let's go ahead and fill in the rest of the venn diagram as well so we know that in a there's 39 in total and we've got 9 12 and 7 that's already in there and 9 plus 12 plus 7 is 28 so we need an extra 11 in the apples there to make it up to 39. moving on to the bananas again we've got 9 12 and 6 and 9 plus 12 plus 6 is 27 we want to get to 50 so that's an extra 23 and then in the bottom one we've got seven plus 12 plus six adding up to 25 we need 39 in there down the bottom so that's going to be an additional 14 right there we go so that's our venn diagram filled in now the question said not to finish what it was actually not to forget what it was actually asking us here it said how many of the students like apples and oranges but not bananas now the ones that like apples and oranges but not bananas are here let's highlight it these ones here like apples and oranges but they don't like bananas so we've got seven people there so we've got a total of seven people out of the 82 people in total and that will be our final answer there we go so you can apply this to a venn diagram that's not a triple venn diagram where you've just got a normal two two circle venn diagram as well but when it doesn't give you that crossover number there you do have to think about how you can apply algebra to this so that's quite a complicated one there but just a little example of how you can approach some of these harder venn diagram questions okay well done that's the end of paper two welding for making it to the end of this video don't forget to check in the description to have a look at paper three for when that's coming out don't forget to subscribe to the channel to get all of the notifications for the upcoming videos for the gcses don't forget to leave me a comment and let me know if there's any topics or specific types of questions you'd like me to go over in a lesson but otherwise i'll see you for the next video [Music] over there [Music]