GCSE MATHS: Interest and Depreciation LIVE LESSON 6 EDITED

hey guys just before we get into the lesson I just want to let you know that you can catch these Live on YouTube uh every Tuesday from 6 p.m until 7 P.M in the UK time and that means you can be more interactive you can ask your questions and they'll be answered okay so without further Ado let's get into the lesson okay welcome back to the sixth GCSE uh maths lesson that we're going to be doing so as per request we're going to start doing interest and depreciation which are very very tightly knit um topics okay so it's essentially the exact same equation in both senses so we should probably talk about interest and depreciation to start with so I think the easier one to start off with is going to definitely be interest so when we talk about interest we have two types of Interest we have compound interest and we also have something called simple interest and honestly it can be a bit tricky to kind of wrap your head around these two so with compound interest uh actually let's start with simple interest with simple interest the amount of money that you gain stays the same each and every year okay so we should also mention that interest generally refers to bank accounts or some kind of investment okay so that's where you'll see the words interest okay you also see interest on loans and things like that but for the purposes of kind of GCSE maths it's normally bank account so it's saving interest so like two percent interest per year or whatever or that they invest it in some kind of savings account or stocks or whatever it is okay so essentially interest is the amount of money so if we were to just quickly write this interest is money gained per year okay so that's all it is in a nutshell now there's two types of interest you have compounding you have simple interest with simple interest the amount is fixed okay so for example if I invested one thousand pounds and there's two percent simple interest then what I do to work out how much I get per year is I just work out two percent of a thousand so one percent you divided by one hundred so that'd be ten so it's two times ten is twenty pounds so every single year you get 20 pounds into your bank account or portfolio or whatever you want to call it okay 20 pounds every year so after 10 years so all you do is you do 10 times 20 equals 200 you've gained 200 pounds perfect so in total you just have 1200 pounds it's called Simple because it doesn't change okay once you know how much you get per year that amount is fixed compound interest is a bit more interesting that was an accidental plan I promise the amount you get changes every single year okay so to explain this so first off the amount changes okay so let's say once again use the exact same example we have a thousand pounds and we have two percent interest now these numbers are going to get very ugly very quickly okay after the first year it's the exact same thing right so two percent is equal to 20 pounds right you work it out the same way so after one year you have 1020 pounds now to work out how much you get for the next year is you need to work out two percent of 1020. okay so that would be 20 pounds and 40 Pence and then you add that on so in the second year you now have 1040.40 and then to work out for the third year you do the same thing you take 20 sorry 2 of 1040 pounds and forty pounds and add that on and so on so the amount you get increases every single year okay that's pretty tricky to to kind of work out right you don't like if I asked you okay after 10 years how much do I have you don't want to do this 10 times it would be here forever so thankfully there is actually an equation we can use to work it out okay now if something increases by two percent it's now at 102 percent of the original value right because 100 is its normal value so if I've increased it by two percent it's now at 102 of its original value so the what I can do to work out the amount that we have and let's say it's after 10 years we take our original investment of a thousand pounds times it by now two percent would be a hundred and two percent which as a decimal is 1.02 so you put 1.02 to the power of the number of years okay so if you want an actual equation for this you have final equals initial so final amount of money equals the initial amount of money times the percentage to the power of the number of years okay so Evie does that make sense it's very tricky to get to wrap your head around but once we do some questions I promise it'll be a lot easier but this is why compound interest can be a bit tricky okay but why hope you take from this is this part here if something increases by two percent it's at 102 of its original value the increases by 10 it's at 110 and so on right you're just passing a minus thing from 100 okay are we all okay with that so far once you do some questions I promise it would be a lot easier although I think the first question on this is a bit tricky let's get started Okay cool so we have Jesse invests 8 000 pounds for n years so some number of years in a savings account to find the value V of her investment after n years uses the formula V equals 8 000 times 1.025 n now if you remember my equation that I uh wrote just now it was amount equals well done perfect EV well done so initial times 1.02 it wasn't 1.025 I got confused there but even you're 100 correct to the power of the number of years right that looks exactly like her formula but 8 000 pounds is the initial so they've just replaced initial with eight thousand pounds good so the initial amounts eight thousand pounds the percentage is 1.025 and as Evie said that means the interest is 2.5 all we look at is the bit after the one 0.025 as a percentage is two and a half percent perfect okay find the total amount of Interest okay so the actual value right using V is going to be eight thousand Times by 1.025 it says in three years so my number of years is three okay so we're going to do here is I've got my calculator here we have eight thousand times 1.025 to the power of 3. now this is what her investment is worth after three years exponentials um as in e to the power of something so i e is the power of 2x or whatever not in this lesson if so and the final value of the investment is 8615 0.13 so you might notice it says one two five but with money we always round to two decimal places right you don't have 0.5 of a penny okay so this is kind of like that Eevee right because I can write this as eight thousand one point zero two five to the power of three so it is very similar to a times B to the power of x but it's similar it's this is more like an introduction to it but I will cover it a bit later okay but this is actually an exponential equation like you just said so well done yeah I will this is an exponential equation in and of itself right but are you talking about like drawing them and things like that or just in general yeah now if we look at the original question it says the amount of interest earned so then yes I will eventually so to work out the amount of Interest all we do is we subtract the initial investment so the actual amount of Interest so far interest is 615 pounds 13 Pence well done Lila exactly right okay exactly right so not bad three marks I think that's pretty good you can do it all on a calculator fairly quickly fairly easily so we have Perry again so invest 25 000 pounds three years 2.7 compound interest so again if we use that equation the final amount will be the initial investment of 25 000 pounds again we're just using the equation I used here initial it is 25 000. times now the percentage as a decimal right it's 2.7 interest so it's going up to 102 .7 so to work it out all we need to do is divide that by 100 which hopefully you can see would be 1.027 right if you're not sure this will only come up on a calculator paper by the way so with compound interest it has to be a calculator paper so don't worry if you're unsure then three years is to the power of the number of years and that's it so once again whip out the calculator by changing just from left to right we have 2.7 we have 25 one two three and again rounded to two decimal places so I'll add that just for a second just to show you what I got on the calculator start to two decimal places but again if it's asking specifically for the amount of interest you subtract the initial investment okay so we're going to take away so for the actual interest what we're effectively doing is 27 080.17 minus the initial amount if they ask you what the final value is this would be our answer but if they ask us for the interest we have to subtract 25 000 so that should be 2080 pounds and 17 pens okay again three marks I think that's pretty quick it's not bad right not too bad in my opinion anyway leave that there for a second perfect again just make sure you stop me the second you find something you're a bit stuck on okay good so this gets a bit trickier but I'm going to show you two ways to do it faster and easier okay um but I'm going to show you a really long way first okay and then I'm going to show you two ways that I'll make it you can do it in one line okay but I need to show you it a long tedious way first okay but if you're ever unsure do it the really long way so we have a house for 350 000 pounds okay first year it increased by three percent then it decree then increases by two percent again in the second year then it decreases by five percent when something depreciates it means it loses value so in this case you can think of it as the opposite to interest okay but depreciate in general means to lose value so in this case it's the opposite to increased okay so in terms of instead of it being interested to depreciation okay work out the value of the house at the end of three years so the first way I'm going to show you which is the longest way is to just work out how much it changes by each year so if I write this out as first year we have a 350 000 pound house and it increases by three percent so that's the same as it being a hundred and three percent of its initial value right so I'm going to times it by that as a decimal which is 1.03 so again while I type it in if you're unsure please do just stop me 350 one two three multiplied by 1.03 you get 360 500 pounds stuff 360 500. now the second year this is where things get a bit more interesting the house price increased by two percent so we're going to times it by you know it's 102 right that's all good so 1.02 but keep in mind it's increased from this value not the 350 000. so it's increased Again by two percent so instead of 350 000 I have 360 500 Times by 1.02 right so the the house increased by three percent up to that level then increased by two percent so it's two it's increased by two percent from that value not the original one okay so again what we can do is we do 360 000 500 uh hang on a minute I wrote that incorrectly there we are 360. oh my goodness 360 500. multiplied by 1.02 so 367 710 at 10 pounds okay so again I'm showing you this long way and I know that it's um it's long but trust me there's a reason why I'm making you do it the Long Way first okay so again if it's it's now depreciated by five percent right it's gone down by five percent now there are actually two ways to do this as well funnily enough but if it's decreased by five percent can you see that it would now be a 95 of its original value right so when it increased by three percent good perfect EV well done oh Lila as well perfect yeah so 100 if it increased by three percent it goes to 103 if it decreases it goes down right 0.95 so I'm going to Times by 0.95 which is a number less than one which means the number should decrease okay I'm going to show you so three six seven seven one zero and we get 349 324 pounds and 50 pence okay so that's our answer that's three marks okay bear in mind this is three marks so three steps makes sense I'm gonna show you a new way right so it was kind of tedious when we had to keep rewriting the number right here's a tip we can use the answer button the ANS button here and it saves whatever our last answer was okay so what I can do now is I could do 350 000 times 1.03 if I started from the beginning equals then I just press times again and it gives me answer times or I can just do answer times the next one 1.02 equals then I press times again 0.95 equals and I get the same answer all I'm doing is pressing times again and again and again right so the second method is using the answer button and all it saves you doing is writing out the same thing again and again so we're all okay with the second method so Lila Evie is that all okay with the second method oh Evie that is exactly what the last method is this is the fastest method perfect so either you just said the perfect thing if we're just timesing three things in a line what I can do is exactly like Evie just said so I can do 350 000 whoops the first year I Times by 1.03 the second year I Times by 1.02 the third year Times by 0.95 that will get me the exact same answer by doing it all in one line exactly as Evie just said okay so really well done good job of explaining that so Times by 1.03 Times by 1.02 Times by 0.95 and we get the exact same answer and I've done it in one line really well done well spotted and that is the fastest method and the one that I use okay I had to show you the long method so you could see that the bottom method works if I just wrote the last line it might have confused some people okay but I'm hoping that makes sense I'm really impressed by the way well done really well done is anyone unsure of this third method because it's what I'm going to be using from now on because it does it in one step which gives you three marks so you can do it in about 20 seconds 25 seconds give me a few seconds perfect really well done okay so here's a more standard question okay you can definitely see something like this coming up in the non-calculator oh sorry the calculator paper so again we have Leanne invests 2 500 pounds for four years three percent compound interest so I'm going to write out the equation again but you probably don't need it right the final amount is the initial multiplied by the interest to the power of the number of years so in this case we want the final amount the initial is 2500 the interest is three percent uh I actually don't know the answer Lyla I will definitely check it though this is 1.03 gives a three percent increase and it's four years so if I sub all of that in let's see what we get 2500 Times by 1.03 to the power of 4. 2813.77 well done Lila perfect and that's two marks not too bad not too bad at all okay so again this is going to be fairly repetitive so I want to test you guys right so again we have 9500 Times by the percentage increase 1.018 let's have five it's just a couple of questions that might be a bit repetitive just to make sure you're 100 okay uh just a piece of advice just in general with GCSE maths is whilst you're doing it in a lesson it's very easy to say okay I get it that's good but you will forget it over time so make sure you're kind of keeping consistent with doing questions outside of the lessons or just going over your notes 386.34 okay not bad right perfect okay I'm really impressed that you guys are managing this so for this question you can do it I'm going to show you the really shortcut way actually I'm going to show you the slightly longer way right so we have a very similar question to the previous question where I showed you the shortcut okay but instead it's depreciation so if it depreciates by 30 percent that's the same as being 70 of its value so 0.7 decreases by 14 that would be 86 which is 0.86 so if we did this all in one line you have 18 000 multiplied by one year of decreasing by 30 percent multiplied by 0.86 and how many years does it decrease by 14 was the second and third year so it'll be two years and in total we have three years done is anyone unsure of that so again I've done it in the shortcut method that I showed you earlier so if it doesn't make sense now would be the perfect time for you to let me know and we can go over the shortcut way of again wait so Evie is that you don't understand or is it okay it's okay cool perfect so again hopefully you can see how like powerful that method is it's a three mark question which normally means you need like three lines of working out and you need to do all this other stuff you can do it in one step so it's a really good way to save time in the exam so you can use it for the harder questions at least that's the idea okay so that everyone's okay with that interest can be a tricky topic so really well done if you're understanding it but again keep up with it okay so we've practiced that a bit now here the next two questions involve something that's slightly different okay so we have Nick bought a new car it doesn't tell us the value each year it depreciates by 12 percent work out the number of years it takes for the car to half in value so give me a second just to at that ah Lila impressive okay I'll show you a trick on that once again I'm going to show you two methods okay but this is one example of trial and Improvement which is something that's in every single spec so Edexcel AQA cie doesn't matter what you're doing trial and Improvement is a key thing that they test across all boards okay and this is an example of it really well done Lila in short you start counting from the first equals okay so 12 depreciates so that would be 0.88 right now each time we apply this so after the first year it's gone to 0.88 of its initial value the second year the way I work out what percentage it's going to be is I do 0.88 squared because it's 88 of 88 percent and that gives me to 0.77 okay so I need to keep going through to half in value it needs to go down to 0.5 okay third year I'm doing 0.88 cubed 0.68 and again if you want you can actually jump right so I could go straight to year five and see what it looks like but just for demonstration I'm going to keep going until I get below 0.5 that doesn't count as 0.59 so zero point and Dot I expect this one to be the correct one oh no nevermind there's another one three keep going okay so this is a fairly tedious method right 0.46 so in short it takes six years because that is the first year that it's below 0.5 so are you guys okay with that okay I'm going to show you a slightly easier method it's easier on thumbs so you'd have to write it all out every time is that okay yeah perfect okay now here's my shortcut super easy method and once again as always it uses the answer button okay 0.88 so after the first year how much what's the percentage I press equals it's 0.88 okay that's one year okay so if you want you can keep a tally but I'm going to keep my thumbs up instead right the next year I'm just taking that original amount and timesing by 0.88 what's that gonna give me the second year and if you notice it's the same the next year is going to be again 88 so I press equals three times four times five times six times that's it you just count the number of times you press equal so it's the ANS button and you just count the number of equal time uh count the times you press equals are you guys okay with that method and it hopefully the reason why I wrote all of this out is that you could keep track that it does actually work okay now every year it matched up perfectly yeah perfect and Lila did that answer your question about when you start counting you count from literally the first one when you save it as the answer button perfect that's exactly what I like to hear okay next question is very similar once again it's trial and Improvement so trial an improvement and once again there are multiple methods on doing this okay in the same vein as above there's a longer way and there's a shorter way and once again I'm going to start with the longer way because then you appreciate the short way okay uh I'm not sure how to pronounce his name actually I think it's fion but I'm not 100 sure invest 5 600 pounds in a savings account and she gets two percent per annum so per year compound interest after n years she has 6061 pounds and 62 pence in her account work at the value of n cool so first year she's going to have 5600 Times by 1.02 we all agreed so five seven one two now because it's compound interest we now take that new amount all right so five seven one two and Times by 1.02 again you can also do the thing where you square it and then Cube it and then to power four right like increase the power that also works so if instead of changing this first number you just put a power here that will give you the same answer okay you get five eight two six point two four three years again I take that value there's a reason why I'm doing it this way though because I want you to see that we can just use the answer button okay 5924 sorry four two even I'm really hoping it's the next one I'm I'm actually 100 sure it's the next one five nine four two point there we go to your right so take so n 4 . now once again right and you could have done this with powers by the way so you could have done for the second year you could have done this right to the power of 2 equals see what that equals change it to a three and if you can see it's giving me the exact same answer right I actually wrote down the wrong amount but yeah it gives me the same answer so what you can do again is do five thousand six hundred times by 1.02 that gives me the first year then if you just press times 1.02 that gives me the second year third year fourth year I get my answer that easy you can save a lot of time doing it that way okay and once again you can also count from the first equal sign right are you are you all okay with that whoops I'm good you don't so your math teacher's slightly incorrect right so you actually don't need method marks so what I might do right is I could have gone straight from the first year and guessed the third year and then went to the fourth year right that is fine okay because it's trial and Improvement so you don't need to write the third year by all you could have gone from one to two and then straight down to four okay you don't need to do it I guarantee it so you don't need to write all of them down I'm writing all of them down just so you can see that the quick method works the same as the um slow method it gives you the same results each time I don't want you to just believe me I want to show you okay give me a few more seconds for any other questions we don't have many more questions because this is a fairly short topic but the last thing is you always have again this is across every single exam board is you have something called Best Buy where basically you need to compare two things and see which one's better value for money so it could also be value for money is what it could be called okay which is useful in real life okay really useful in real life okay and you need to compare which one will give you the best value right so how much money you get so I want to change something in this question instead of some money just for demonstration can I change this to two thousand pounds and I'm going to do this question again but without money okay I hope that that isn't complicated just for now I want to show you an example with money specifically okay so as you're going to invest 2 000 pounds for five years she can either choose investment a or investment B yeah investment a is 2.7 compound interest investment B is 2.8 simple interest so compound interest we're experts at that now let's just work this out you would have two thousand pounds times 1.027 and I'll put a bracket to power five right so let's do that we have two thousand multiplied whoops sorry I pressed X instead of times 1.027 to the power of 5 right and let me just double check that yeah that's good two two eight four point nine eight so just so you know the best value for money in this case is whichever one has the most amount of money after five years okay now I would like to point out that compound interest is better when you have bigger sums of money okay I've chosen a very small amount of money for their system to be clear 2.8 simple interest so the way I work this out is I work out what 2.8 percent of 2000 is right best way to do that is you take that as a percentage uh sorry a decimal and times it by two thousand right I'm hoping it's not too bad of a number I don't think it will be 56 that's a really easy number so we haven't done simple interest a hundred thousand percent it will definitely it has to um this is this is a bit Advanced but just who knows compound interest right if it's a lower rate is worse in the short term better in the long term okay so at the beginning obviously after the first year because the interest rate is lower you have less money but it builds up over time you won't need to do either it gives us five years okay and it will always give you a time frame okay so all you need to do is work out after five years how much money would be in the account if you did with compound interest and how much would it be with simple interest okay whichever number is bigger that's the one you go with but very good questions okay if you're talking about for you personally what would be better then you talk about long-term short term and things like that but in exam questions they give you a time frame okay really good question though so with simple interest we said at the beginning of the lesson that the amount doesn't change so what I'm effectively doing is I'm taking my two thousand pounds and I'm adding 56 for every year that I'm investing so it's five years I'm adding 56 five times what whoops now that is incredibly tedious so instead of adding I don't know why I wrote twenty thousand sorry instead of adding five times what you can do is do five times 56. and those two things are equal to each other okay hopefully that should make sense but again if you're not sure please let me know we get 2 280. uh yeah perfect okay generally speaking compound interest is better but only if the interest rate itself is is comparable if it's similar okay so which one's better value for money or in this case which one's better which investment should Alice choose it's investment a so the way I like to do it is I always imagine myself in this situation would I rather have 2284 pounds and 98 Pence or just 2280 but obviously more money right four marks so quite a lot but it's not that bad but are you guys okay with that perfect hopefully yeah hopefully I've explained it well that's good and this is where it gets interesting okay so this is also something they like to do so they're both compound interest so Matt wants to invest eight grand for three years there's two Banks but the interest rate is fixed for Bank a which is 1.2 every year but for Bank B it's higher in the first year lower in the next two so which bank will give Matt the most interest so once again I'm going to split my thing in half and this is exactly what I recommend you do in the exam and work out the difference right so again Bank A's easy I don't doubt that you guys can do this right take the 8 000 pounds times it by the percentage increase so it's 1.012 to the power of the number of years which is three let's do this nice and quick and again round it to two decimal places so we get eight two nine one point four seven okay now we're going to use EV suggestion for this next one right we're going to chain it along so we have the 8 000 pounds and for the first oh god I've got eighty thousand eight thousand pounds for the first year the interest is 1.02 then it goes down to 1.01 for every additional year how many years do we have left we're doing it for three years one year is two percent so we have two more years left okay so what I'm going to do is 7.02 wait you don't technically need the brackets to be honest 1.01 and then make that to the power two eight three two four point zero two so which bank should he go with or which bank it has the most interest when it's Bank B again that's four marks so your papers for most exam boards are out of 80. so that's 4 out of 80 you get for this and we did it in two lines of on our calculator four marks so these These are really good questions to kind of Master and I'm hoping that you guys are actually getting pretty comfortable with it already but please keep up with it okay you have to keep doing questions every now and then because it's very easy to make mistakes especially when you do things like chaining this along okay any questions about that think it's all okay wonderful okay again great to hear it again this is a very similar question to what we've just done very very similar okay so Melvin invest five grand two and a half percent every year and Charlie invest 4500 paying three percent every year work out the difference in money it's the same thing but imagine this is Bank a bank B that's the only difference right they've just changed it to people so we do Melvin we only need a bit of space and Charlie so let's work out the difference right it's not not too tricky so Melvin he has five thousand pounds the interest rate is 1.025 number of years is five years there we go same as what we've done before okay five one two three times by e five six five seven point zero four and with Charlie it's four thousand five hundred but it's a higher interest rate so let's see if it's any better so what I'm going to do is just edit my equation again using the arrow keys just so I don't have to do the same thing over and over again five two one six so it seems like Melvin was better so he invested more money to begin with with a lower interest rate but seems like he's made it out on top but the question actually actually says work out the difference in money so we need to do now is do five six five seven point zero four subtract five two one six 0.73 let's do it so five six five seven point zero four subtract five two one six point seven three three that's it 440 pounds and 31 Pence and that's four marks again look how quickly we get this done four marks should be roughly four minutes and yet we've only written this much and look how much space they gave you too really useful questions to kind of smash out I'll leave that on the board just for two more seconds it's a really good question Lila um more than likely it would just be one question in your calculator paper so I'll be paper two if you're doing Ed Excel paper five if you're doing AQA and so on but normally it's only one question so I have a few more kind of basically what I've done is I've actually gotten questions from every single exam board okay so this first one is Edexcel okay the next one so for some reason the labeling actually got a bit weird and I've written where I've gotten the question from so this is Edexcel paper to 2020 higher there's one for AQA and OCR as well just to show you that it's literally every exam board okay so for this question the car is bought for 1 500 pounds but depreciates in value eight percent so in other words 0.92 because it's at 92 percent of its value every year how much is it worth after two years well one thousand five hundred Times by 0.92 how many years two years okay so the questions are all very similar always one two six nine point six and it says what is the total depreciation after two years so this number as a percentage so to change anything into a percentage what we're going to do is take the new value which is this divided by the original value Times by 100 that's it to change anything into a percentage you take the current value divided by the old value Times by 100. and then Times by 100 and you get 84.64 percent not bad and that's the actual question from Edexcel next I believe is OCR and then AQA is the last one we'll do give you a few more seconds to ask any questions you might have cool the next one again the labeling did get messed up this is OCR whoops come on be faster OCR November 2019 right perfect Joan invests six thousand pounds in a savings account the savings account pays a compound interest at a rate of 2.4 for the first year 1.7 for each extra year work out the value of Jones Investment at the end of three years the reason why it says Part B is because part A was unrelated so I I cut it out okay but if you go to OCR 2019 November you'll find this exact question okay um so again this is the exact same thing we've just been doing right we take our six thousand pounds the first year is 2.4 so that's 1.024 we do one more question after this and then we're done okay unless you guys have any questions and we'll continue for a bit then 1.07 uh zero one seven at the end of three years okay so the wording of this question is a bit interesting because that that is valid right why don't we take it from 100 right so this is the actual value of the car expressed as a percentage of the original value so what line is saying which is a hundred percent valid why don't we do this so you do 100 minus the answer 15.36 and this is actually my bad I didn't mention this in the mark scheme because the question is a bit vague they do actually reward marks for both of these okay it's yeah so as a percentage so the way I interpreted it and the way Lila interpreted it was slightly different but both are actually correct okay I've expressed the new value as a percentage of the original value which is kind of what the question's saying but Lila's pointed out that the actual depreciation would be a hundred minus that that also works so both of these are correct even though they're different answers which might seem a bit scary and weird but both of those are actually correct okay but yeah very good point and this value would be squared because you you have this interest rate for two years okay so one year of 1.024 and then you have two years of 1.017 so six three five four point six seven all good and again it's the same thing we've been doing this is completely Universal it's for all exam boards the last question is from AQA again apologies for this being a bit wonky and again just in the interest of time it's a very similar question right you're working out the compound interest so hopefully you can see that it is the same no matter what exam board you're doing it can be a bit tricky of course but it is all the same kind of stuff okay is any of that confusing okay that's good perfect let's hear it are there any questions about that so I know that um I think it was you Evie that asked about the kind of exponential equations hopefully you can see that these do take on the form of exponential equations but can I ask kind of what um what style of question we resist the thing you're talking about or is there a different style of question you wanted to go through because there is quite a bit on exponential equations and again we will cover it over the coming time using the graphs okay what I'll do is I'll note that down as something that you want to go over and I'll try and Slot it into one of the future lessons okay The Substitute values yeah no worries so modeling with exponentials basically I will try and Slot that in okay um a bit later if there are no questions about this then we'll call it equation of circle oh yeah 100 that's coming up too we're going to cover all of the GCSE spec okay but yeah absolutely and yeah so if that's all okay then we can probably call it quits there