22 JULY 2026 14:30 - 16:00 TECHNICAL MATHEMATICS GRADE 12

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[00:28:57][music] [music] [music] >> [music] >> Heat. Heat. [music] [music] In mathematics lesson, my name is Matu.

[00:29:49]This lesson is brought to you by Hen Department of Education in collaboration with Cybon Discovery Center.

[00:29:58]So, grade 12, our topic for today is finance, growth, and decay.

[00:30:07]Um we're looking at the re uh revision of finance.

[00:30:14]We have done it in grade 10 and grade 11. Now in grade 12 there are things that we need to emphasize on. And remember this topic in the exam it carries plus minus uh 15 marks. So without any waste of time, let's uh look into our information sheet. Very important for learners to know their information sheet and how to pick the formulas from the information sheet.

[00:30:46]Make it your friend. So we have the following uh formulas that they work uh for us in finance. So looking at the the first one the first formula there we can see that is for logarithms. Basically it assist us whenever we do we dealing with formulas like compound interest. If we have to make n the subject of the formula, then we can approach the formula of a to the exponent of x= to b.

[00:31:27]We can make x the subject of the formula which is the exponent as x is equal to log base a of b. So you can see that it can be easy for us to make uh n the subject of the formula using the logarithms.

[00:31:47]So we also need to know how to identify these formulas. So simple interest, compound interest and straight line depreciation and reducing balance h method. And also we look into the effective and nominal interest rate formula. We're going to speak more of this formula during our our lesson.

[00:32:14]Now finance growth and decay. There are two main types of interest interest rates that you were introduced uh to in the previous grades namely simple interest and compound interest. uh if the interest is calculated using uh the original amount of money saved or borrowed then it is called simple interest. So the minute you read the statement and then you hear of the original money or the amount that is borrowed, you know that you will be dealing with simple interest. But simple interest is for short-term loans or they can call them higher purchase uh accounts. So sometimes when you read the question they might not indicate the simple interest but the minute they speak to higher purchase then you know that you need to use a simple interest formula and also the investments correct. Then with a compound interest if interest is calculated on original hump plus interest already end uh it is then called your compound interest and the compound interest is used with long-term loans and investment saving money over a long over a period a long period uh 5 years or more with a compound interest rate uh well above inflation rates. Correct. Uh will help you uh to accumulate wealth. Compounding can be your friend if you are saving money over a long term period. So now another weight that you need to look into inflation.

[00:34:22]Sometimes when you read your statement or your scenario you hear the word inflation rate. They might not um necessarily mention compound interest but if you hear the inflation rate then you know that we are dealing with something that has to do with compound.

[00:34:42]The same way I I mentioned with your higher purchase, they might not indicate the simple interest but they can indicate higher purchase. Then you know that you need to use your simple interest uh formula. So those are the two things that you need to understand when we are dealing with simple interest and compound interest. And then uh compound interest it can also be your worst enemy. If you are paying the bank h you are paying back a bank loan over a long uh term period. It means you will pay h more money to the bank. So those are the things that you need to understand about simple interest and compound interest. Then we have their formulas for simple interest is the one that is indicated as a = p into 1 + i * n and for the compound interest it's your a = p into 1 + i to the exponent of n. Then P it represents the present value of the investment or loan which is your original amount at the beginning.

[00:36:04]Then your A it represent your accumulated amount or future value of the investment or loan after the nth period. Correct? And then your N it talks to a time period. It can be in months. It can be in years. And then the R which is your interest rate as a percentage but normally in our formula we use it as I which is the interest rate as decimal. Then we also have our I.

[00:36:40]If we we have I we we we are going to see R / 100 which is giving us our I.

[00:36:49]And if we have our R and we want to or we have our I want to go to R we then going to multiply by 100. So now uh those are the things that we need to know about the two formulas.

[00:37:08]We need to also understand the manipulation of the formulas.

[00:37:13]Remember sometimes they won't only ask you to calculate the accumulated amount. They can give you accumulated amount and ask you to calculate the original uh money before the investment happened then which is P. You need to know how to make P the subject of the formula. Or they can give you the accumulated amount, the initial amount, the number of years and they ask you to calculate the interest rate and then or sometimes they can ask you to calculate the period taken correct which is n then you need to know how to manipulate your formulas and that's what I will be dealing with now so that you understand how to make other variables the subject subject of the formula.

[00:38:06]Starting with the first one, we have to make P the subject of the formula which is the simplest one where I will have a = P into 1 + N * I. Then in that case I will need to make my P the subject of the formula. Then I have to get rid of um what is inside the bracket 1 + n I we divide both sides by that then they will cancel each other.

[00:38:46]Therefore now your p it will be a all over 1 + i in n. Correct? So that's how simple you will make P the subject of the of the formula. Then we look into the second one.

[00:39:08]This one the method of getting n and I is the same procedure. So now we are having a = p into 1 + n * i. Right? Then now we are going to make n or i the subject of the formula. The first thing that we need to get rid of is p. Then I can divide both sides by by p. So on the right hand side the p will divide each other. Then I will have a over p equal to then your bracket will fall off. I will have 1 + n i. Then from there you can see that there is an addition sign between one and n i. Then I can simply transpose my -1 to the right hand side. Then I will have a / p minus one equal to n i. So in this case if we want to make our n the subject of the formula we can divide both sides by i.

[00:40:36]Therefore your n then it will be a / p minus one all divided by i or in case we want to make our i the subject of the formula then we are going to divide at this stage we are going to divide both side by n which is going to give you a over pus1 all divided by by N. Hence I said your N and I in this case you follow the same procedure up until the stage where you have to decide either you're going to divide by I or you're going to divide by N to make either N or I the subject of the of the formula. So that is the procedure of making P N or I the subject of the formula. looking into our simple interest formula. So now um let's look into our compound interest formula.

[00:41:42]There we also need to make P the subject of the formula and the subject of the formula and I the subject of the formula. So how do we go about this with making p the subject of the formula I think is the most simplest one where I will just have a = p into 1 + i to the exponent of n.

[00:42:08]Then we have to make p the subject of the formula. I will just get rid of uh the brackets. So it will be 1 + i to the exponent of n 1 + i to the exponent of n. Then they can divide each other. Therefore p it will be a all over 1 + i to the exponent of n. So that's how you will manipulate that formula to get p. So now in a case where we need to find n we need to make n the subject of the formula. So the from the formula given I'm having a = p into 1 + i to the exponent of n. The first thing that we going to do is to divide both side by p.

[00:43:18]Then we are going to have a over p equal to the p. Then we take each other out into 1 + i to the exponent of n.

[00:43:32]Then the procedure when we get to this stage it says for us to make n the subject of the formula we need to introduce the logarithms.

[00:43:43]So one can say log of a / p all equal to the log of 1 + i to the exponent of n.

[00:44:01]Correct.

[00:44:02]Then if we want to take it a long route, we can say the rules of logs is I can take the exponent and multiply my coefficient with then it means I will have log a / p = n * the log of 1 + i. Then we can see that when we arrive at this stage now we need to make I I mean N the subject of the formula. Then we will simply divide both side by the log of 1 + i and also this side we divide by the log of 1 + i. Then they can take each other out. Then we will have n equal to the log of a over p all over the log of 1 + i.

[00:45:04]Then that's how you are going to make n the subject of the formula.

[00:45:10]Or if in a case where you have a / p equal to 1 + i to the exponent of n. Remember I spoke about h the formula where we had a to the exponent of x is equal to um b.

[00:45:38]All right. So in that case now we have to make a x the subject of the formula.

[00:45:45]We said we can have it as x is equal to the log of a then b then which is what is going to happen here. If we want to take it in a short route, we can say log of 1 + i. That is the base and a over over p. So using that formula from your information sheet, it guides you on how to make n the subject of the formula. Then in a case where you just need to take it step by step, you can follow the first procedure that I have shown you. But they will both of them they will give you the same answer. So now we have made P the subject of the formula. We have made N the subject of the formula. Then let's look at how do we then make I the subject of the formula. So from the formula given p into i to the exponent of n we need to make now our n I mean our i the subject of the formula. We get rid of p both side. We divide by p both side. Then from there we're going to have a all over p into 1 + i to the exponent of n. Now in this case we need to get rid of the exponent of n. Remember we're looking for i. So I will introduce the nth root both side where the nth root will take care of your exponent of n. Then we are going to have the nth root of a over p = 1 + i. Then we can see how I got rid of p and how I got rid of the exponent of n. We introduced um n both side. The square root of n both side. Remember even when you have x² = 4. Then we need to get rid of four.

[00:48:05]We always introduce the square root both sides so that the square root can take care of our square thing. So that is the same method we are going to apply when we have to get rid of the the N. Then for me to make I the subject of the formula I will just transpose one.

[00:48:29]Therefore my I will be the nth root of A over P -1.

[00:48:39]Remember my minus one is not inside the square root. is outside is outside the square root. So you need to be careful of of that as well. So now that we have dealt with how to make a pi and n the subject of the formula for both simple interest and compound interest.

[00:49:03]If we are going to deal with the depreciation where the formula has um instead of positive it will have negative between your one and I the procedure is still the same. We are going to deal with the same procedure.

[00:49:20]The only thing is that at the end we need to divide by1.

[00:49:24]If we are using a the formula like or we are using a formula like this one those are the DK formulas the depreciation formula. So how to make a P and I mean I mean P I and N the subject of the formula you follow the same process. At the end of the day in this case you are just going to get rid of a -1 which is the coefficient of I.

[00:50:01]Now talking to depreciation in depreciation in many situation equipments can lose h its value over a given time period.

[00:50:18]and the motor vehicle computers lose value over time due to wear or tear or become outdated.

[00:50:28]Correct? When uh this happens, we then say that the equipment equipment depreciating in value over time. Then we have what we call book value and scrap value. Book value is the value of the equipment at the given time after depreciation has taken place and the scrap value is the book value of the equipment at the end of its useful uh life. So the two they are both represented by the value of P. I mean the value of A sorry the value of A. In this case if you have an iPhone 17 Pro Max which is roughly 30 to 35,000 after 5 years you want to sell it. You can't sell it more than the amount that you bought it with. You can't sell it more than 35,000. It will have to be less than 35,000.

[00:51:43]So it means that iPhone 17 Pro Max it has depreciated in value. So now two types of uh depreciation we have linear depreciation and reducing balance depreciation.

[00:51:59]So now the linear and reducing balance depreciation we can also interpret them graphically with the linear it it's a blue line there where it happens in a linear it means the constant h depreciation correct the amount of depreciation is the same over time then with your reducing balance depreciation we can see that it's reducing exponentially.

[00:52:34]Correct? So now we need to now understand when they are talking about a linear depreciation and reducing balance depreciation how does it look like in terms of of your your graph or graphically.

[00:52:51]So now we do have formulas of linear depreciation and uh reducing balance depreciation. So this one here is for linear and then we also have the one for um reducing balance.

[00:53:14]So now you read the statement again and you hear where they talking about the linear depreciation and the reducing balance method depreciation then you you you will know which formula to pick because in finance you don't need to guess which formula I have to use by reading the statement.

[00:53:37]The statement will then tell you which formula must be used to answer a particular question. So in this case we have a which is a book or a scrap value and we have P which is the present value before the equipments can depreciate. Then we also have the depreciation rate which is I N which is time period. Then what happens in this formulas the value of P now it's big and the value of A it is small it means before depreciation the values are expensive and after then they become cheaper which is a different from growth. growth remember we start investing a small amount of money which is the value of P then accumulating we accumulate a huge amount of money which is the value of A so you also need to understand the two like that we're saying in depreciation our present value it's bigger than the accumulated or the scrap value correct now uh we look into the following uh questions.

[00:55:06]The first question it says, a mechanic bought a trolley jet for 10,000 and 63 rand and paid a deposit of 1,000 rand. The balance uh owing was to be paid over 24 months in equal monthly installment of 464 uh 48 cent or 48 cent how to get using simple interest. So now when you read the statement you can hear that the formula that we need to use in this case it's a simple interest formula and if you go to your information sheet the simple interest formula it looks like like that.

[00:56:03]So now there is no need for us to guess which formula am I going to use because the statement as we reading it tells us about the simple interest formula. But when you read the statement again, you can hear that you have the amount of a a trolley jack which in this case it will act as your P and then your deposit is the amount that you paid before you can even take uh that trolley jack home.

[00:56:36]Correct? And then it is paid over 24 months. 24 months which is two years.

[00:56:46]Then from there we look into our our question. Our question says uh we need to calculate the total amount paid over 24 months. The total amount paid over 24 months. And there are two types of amount that is are paid here. The first one is a deposit 1,000 deposit that is paid and also the monthly equal payment of 4,648 cent. So now we are just going to start by saying uh 4,00 I mean 400 464 comma 48 464 guys. we multiply it by 24.

[00:57:40]If you multiply by 24, 464A 48 * 24, it will give you your 11,47A 52. We then add our,000 of deposit. If we add our 1,000 of deposit then we are having 12,147 rand 52 cent. So that is the total amount paid and you can check that if you bought it cash the trolley jack was going to be 10,63 rand but now you didn't buy it cash you bought it over a short loan which is a simple interest higher purchase then that is the amount that we are going to to pay after two years for that trolley check. So now uh that's how you answer these type of questions and you read your statement. In most cases your answers are from the statement. So now uh we look into uh 1.2 where now they saying to us that we need to calculate the interest rate charged per anap interest rates charged per per anap. So you look into our statement there we need to take out or find a P I and N.

[00:59:26]So in this case we can see that our accumulated amount the amount that the amount that is paid it's 12,147 rent 52 cent correct then now what is the present value correct it is 1063 minus the deposit you minus the deposit because that is the amount that you are owing correct you pay deposit from that 10,000 so the amount that is now is owed is 9,63 so that's what you need to understand when we dealing with higher purchase if you pay deposit we also subtract the money that you have paid h from that 10,000 then your I it's what we don't know but 24 months equivalent to two years correct so those are the things that you need to identify for us to calculate the interest rate then it becomes simple because we know that a = p 1 + i in n then our a which is 12,14752 cent equal to our P is 9,63 rent into 1 + I multiply by 2 years then we can see that now our I it's an unknown variable we need to make I the subject of the formula We are going to uh follow the procedure that we dealt with. We are going to divide both side by 9,63 9,63.

[01:01:39]Then from there you will have 12,14752 cent divided by 9 63 equal to 1 + 2 I correct then we can transpose our one then we are going to have 1247 divided by 9,63 3 - 1 = 2 I then in this case we need to make I the subject of the formula we are just going to divide both side by two then two cancel two therefore your I in this case then it will be um it will 0 comma 17 02 right then you have to multiply it by 100 so that it can give us your 17 02%.

[01:02:57]So our interest there it was 17 comma 02%.

[01:03:04]So that's how you make I the subject of the formula and you calculate the interest rate charged per per term. I hope it was um clear and straightforward.

[01:03:16]Then we look into the second one.

[01:03:22]We still dealing with uh simple interest. Oh, this one is compound interest. Sorry. So we read the statement again. It says a car engine overheated at an increasing compound rate increasing compound rate.

[01:03:43]So now the minute you hear the compound rate then it tells you that we are going to use a compound interest formula. A compound interest formula. So like I said the minute you read the statement it then tells you about which formula are we going to to use then our compound interest rate there is 26%. per hour due to a leak in one of the water pipes. Right. The initial temperature before the engine overheated it was uh 75° C.

[01:04:29]You can hear that they saying before the over heating which is P. It was the initial temperature P. Then the question says we need to determine the temperature of the engine at the end of 3 hours.

[01:04:49]If it continue to overheat at the same rate. So that 3 hours done its period it's time period 3 hours time period n.

[01:05:03]Then that we have identified the formula that we need to use. We are just going to say a = p into 1 + i to the exponent of n. Then you get one mark for choosing the correct formula because we have five formulas of finance. You just need to choose one out of those five formulas and the formula needs to be correct. Then we identified the formula through the statement compound rate. We know that we are going to use a compound interest and it was increasing.

[01:05:47]It was increasing which is growth. Then now we are just going to have our 75 into 1 + in this case our I is going to be 26 / 100 to the exponent of 3 always I we divide it by 100 I we divide it by 100 then the answer there it is going to uh give youund 50 150 comma 03° Celsius. We don't penalize for units but that's how uh how much the temperature will be after 3 hours.

[01:06:39]So simple stuff. The second mark it can be for substitution and the last mark for for answer. If you can check in most cases technical maths for you to choose the correct formula we give you a mark and that's how we pick up formulas there. So I also think that this one it was also straightforward on how to pick up the formula and analyze the the the scenario then you can be able to determine the correct answer that is required.

[01:07:15]Okay. Then we look into the third one which is the third example there. Now in this case they're talking about uh the following. It reads as follows.

[01:07:29]In 20 in 2018 engineering equipment costed 260,000 in 2018. Correct. We're talking about the engineering equipment.

[01:07:44]Now the 3.1 it says if the equipment bought in 2018 depreciated now the weight depreciated it means now we are dealing with some things that are losing value the equipment was 260,000 in 2018 and it depreciated to 25% of its original value, we need to calculate the current value of the equipment.

[01:08:22]So if the equipment was 260, we just need to get 25% of that 260,000.

[01:08:33]How I'm going to do that? You can say um 260,000.

[01:08:42]You can either multiply it by 0 comma 25 or 25%.

[01:08:48]If you multiply it by 25% then what are we going to get there? We are going to get 65,000.

[01:09:02]We are going to get 65,000 which is 20% I mean 25% of that amount.

[01:09:12]And that's how simple you you answer this type of questions. One mark.

[01:09:19]Like I said, the other one can say 260,000 multiply by 0a 25 0a 25.

[01:09:38]Let's write it nicely there.

[01:09:47]So you can say 0a 25 you're still going to get 65,000 then that is one mark.

[01:09:57]So now looking at this type of questions we have the value of p which is the original amount of the equipment before depreciating.

[01:10:10]Now after it has depreciated to it or of its original value now the amount that we are getting here it's a it means it's a book value of those engineering equipment correct so now we looking into 3.2 to the question continues. It says we now have to look for the what the equipment depreciated at the rate of 14% per using reducing balance method.

[01:10:48]Reducing balance method remember there is a difference between linear depreciation and reducing balance depreciation. So now reducing balance is the one where we have 1 - i to the exponent of n reducing balance. Correct? Then now I have now h shown you how to pick up which formula are you going to use. You get it from the statement. As you read the statement, the statement will then tell you which formula you need to use.

[01:11:26]Then it says, determine how long to the nearest year it took for the equipment to depreciate to 25% of its original value. Correct? So now if we look into the previous slide, we had those amount.

[01:11:50]We have I, we have N. So now I is what we don't know how long correct to the nearest year it means we must look for N. Now the rate of depreciation in this case is 14%.

[01:12:08]which we always divide it by 100 is going to give us 0,4 right and the original amount of the equipment it was 260,000 then it depreciated to 65,000 and now you can see that P is greater than A in this case and now we need to Now make h n the subject of the formula which I said whether it was positive or negative the method of making and the subject of the formula is the same. So it means we are going to have our 65,000 equal to 260,000 into one I mean minus not plus min - 0a 1 4 to the exponent of n for us to make n the subject of the formula We know that we are going to divide both sides by 260,000.

[01:13:33]Then we get rid of that 260,000.

[01:13:36]1065,000 divided by 260,000 equal to 1 - 0.14 to the exponent of n. Then I said in this case uh we can introduce the logs.

[01:14:00]You can say log of 65,000 all divided by 260,000 equal to the log of 1 - 04 to the exponent of n. And in this case we see our n can be multiplied there. Then we're going to have your lock of 65,000 all divided by 260,000 and ultip by log 1 - 0 4 and that's what we are going to to divide by both side divide both side by log of 1 - 0.14 and log of 0.14.

[01:15:07]So the minute we divide them then our n it will be in this case our n it will be 9.

[01:15:21]year to the nearest year then we can say this it will be 10 years correct.

[01:15:35]So that's how we we calculate the n reading of the statement picking up the correct formula the correct substitution and the manipulation of that formula how to make n the subject of the formula very crucial and learners need to understand that okay uh when we move forward Now we are going to be talking to different time periods.

[01:16:12]Different time periods. What it is is affecting it's affecting our I which is the interest and N which is the time period and the symbol for this we can use it as M different compounding periods we can indicate it as M we can indicate it as M. So uh the interest compounded interest compounded interest can be charged differently within a year. The first one it says annually annually means once a year. It means your value of M it will just be one. And what does it do to I? The interest rate is given h and the number of years as given. It means once interest will remain interest and number of years remain as given. So now the minute they are saying semianually semianually it means the interest now is charged twice I mean twice a year. We charging that interest twice a year which is M.

[01:17:34]Our M now becomes two. And what does it do to the interest? We divide the interest by two. We multiply the number of year by two.

[01:17:47]And another one of semiannually they can also give it in a form of half half yearly.

[01:17:58]They can also have yearly half yearly is also two is the same as semianual semi and half one in the same the same thing correct. So now quarterly quarterly we have four quarters in a year which is the interest it will be charged four times a year. So your m in this case it will be four then you divide the interest by four and you multiply the number of years by four and that's how it works in terms of h compounding period. So now by by monthly it's six it means it's every other month correct 6 months then we do the same interest divided by six and number of years multiplied by six then we also have semionthly semionthly it means in a month you will be paying a interest I mean interest will be charged twice in a month. So you can see that it going to be 24 because we have 24 h months in a year and if it's twice a month then I mean 12 months in a year and if it's charged h each month which is twice then it will be 24 then bi-weekly bi-weekly it's 26 remember we have 52 weeks in a year then you just divide it by two you are going to get 26 which is bi weekly then weekly it's 52 like I indicated and daily we are having 365 days in a year so that's how uh we need to understand this thing so now This thing is an on a compound compound h remember we are having a compound interest formula as p 1 + i to the exponent of n.

[01:20:21]So in this case it means for us to use the different compounding we are going to take the compound interest formula but where there is I we divide by M where there is N we multiply by by M.

[01:20:36]Now the compounding uh compounding interest or compounding period it also work on nominal and effective interest rate. We can see that on our formula we are having the value of m which is going to be the compounding uh periods.

[01:21:04]Right? So now this formula is for effective and nominal interest rate or we can say nominal and effective interest rate which is I effective equal to 1 + i / m to the exponent of m -1.

[01:21:27]So in this case again they can give you either the nominal rate and they ask you to calculate the effective interest rate which is straightforward and simple. But now if they've given you the effective interest rate and they ask you to make the nominal interest rate the subject of the formula then we can see that we also need to understand how are we going to manipulate that formula for us to arrive at I norm being the subject of the formula. So now we are going to deal with that the I effective equal to 1 + your IO which is the nominal / m to the exponent of m -1 then we need to make iO formula the first thing that I will do is to transpose my1 to the right hand side we are going to get I effective + 1 equal to 1 + iM to the exponent of m.

[01:22:53]And in this case we can see that we need to get rid of our m. How do we get rid of m? We are going to introduce the mth root the very same way we getting rid of the n in compound interest formula and the mth root will get rid of that m and we are going to get the mth root of i effective + 1 = 1 + i / m.

[01:23:37]Then we are going to take this one. We transpose it as well. It will become negative. But now your m root of i effective + one that is inside there the square root -1 = i all divided by m. So in this case we can see that for us to make the subject of the formula the correct thing to do is to multiply both side by n then m will divide m there. Therefore the I norm it will then be m * the m root of i effective + one - one + one minus one. So that's how uh we are going to to make the iO.

[01:24:51]So it's very important for us to know how to manipulate um these formulas.

[01:24:59]So now let's look into the questions where we are dealing with the nominal and effective interest rate.

[01:25:09]Then can see that the first one there it says we must convert a nominal interest rate a nominal rate of 18% per anom compounded monthly to an effective interest rate. So we have I know which is 18% we can divide it by 100 it will give us 08 and the formula it's I effective equal to 1 + I all / m to the exponent of m -1 one but another interesting thing is that as we read our statement they are saying the nominal a nominal rate of 18% per compounded monthly monthly it means each and every month the interest is charged which is m equal to 12 because we have 12 months in one year then our m is 12 and in this case we are just going to say 1 plus 0 08 / 12 to the exponent of 12 - 1. Then you can punch that in your uh calculator.

[01:26:47]And as we punch that in our calculator we are getting 0 um 1 9 56 correct. Then from there we can multiply it by 100. Therefore the I effective it will be 19 comma 5 6%.

[01:27:18]19A 56%.

[01:27:22]And that's how we we get that one.

[01:27:27]Simple and straightforward.

[01:27:30]Normally they will say three marks in the exam. one mark for formula, correct substitution, and the answer, three marks for free. If they generous, they can give it four months and they check if you got the compounding h interest correctly or compounding period correctly in terms of your monthly, weekly, daily and stuff.

[01:27:57]So now uh we now want to convert from effective interest rate to nominal interest rate. So in this case they just said convert an effective interest rate of 13.5% per anom on a to a nominal interest uh nominal rate sorry nominal rate per anom compounded say meward m = 2 I effective = 13a 5 we can divide by 100 we are going to get 0.135 correct. So now uh we have our I effective which is equal to 1 + 0a 135 / 2 to the exponent of 2 - 1. Then in this case we are going to to transpose the positive one.

[01:29:20]Then we are going to have sorry no no no made a mistake in terms of substitution.

[01:29:27]We are looking for I know I effective I effective it's 0.135.

[01:29:40]I norm it's unknown. Sorry about that. I nom it's unknown. So we are going to have 0.1 35 + 1 = 1 + IM / 2 to the exponent of 2.

[01:30:00]Then from there we're going to have 1a 1 35 = 1 + i all divided by two to the exponent of of two. So in this case what are we going to do? We are going to introduce the square root both side. Then our square roo<unk> of 1 135 = 1 + I know all divided by by two. So now looking into that we know that we are going to transpose here we are going to have the square root of 1a35 minus 1 equal to iome / 2 which we can simply multiply by two both side then your nome it will be um 13 comma 07 we multiplied which is 13 07% sorry 13 0% 13 07%.

[01:31:28]In this case it means your IO which was 0a 1 307 we multiplied by by 100 and that's how you make your I know the subject of the formula. This type of questions it can take four marks or three marks depending on the the examiner. So now you will know how to make effective interest rate the subject of the formula and the subject of the formula.

[01:32:01]So now when we deal with the the compounding growth on a timeline, remember we spoke about it where we are using a different interest or we dealing with your multiple deposit or withdrawals during the the investment. So this is the types of a formula that we are going to use. We saying combining growth h is growth that takes place exponentially and then I spoke about division of uh i and multiplication of n and then a it's still the accumulated or final amount p it's your principal or initial amount i it's interest n is number of years and m is different period or compounding period. So now we have a scenario this type of questions they carry plus - 5 marks or plus - 6 marks. So we read the statement we then analyze it on a time line. Let's read the statement together. saying h Samuel opened a savings account to save for a boat cruise that he wants to go on h on at the end of 5 years he made an initial deposit of 20,000 correct the first bullet says the interest rate for the first two years was 6% per compounded monthly h the second bullet it says At the end of uh the first two years, he deposited a further amount of 5,000.

[01:33:59]And then it says the interest rate changed to 5% per anom compounded half yearly. Determine uh showing all calculations whether he he will have enough money uh enough money in savings account for the boat boat cruise which will cost 85,000 rand. So you can see that a lot is happening in this scenario. So it's only advisable that learners use a timeline to interpret the the scenario so that we can arrive at the correct answer without any challenge. So now the first thing that we're going to to do here is to draw our our timeline is to have our our timeline Correct. Let's have a proper line for for our timeline. Remember, we are talking about 5 years. So we're going to have n we start at zero 1 2 3 4 and five. So now we have five years and what is happening in our five years?

[01:35:37]They're saying the initial deposit it was 20,000. It means at the beginning somewhere deposited 20,000 into the account. Correct? Then the first bullet it says for the first two years it means from year 1 to year two the interest there it was 6%. We divide it by 100 we are going to get 0.06.

[01:36:09]And they said uh it was compounded monthly. Compounded monthly which is our M in the very same uh the first two years it was 12. It means the interest was charged each and every uh month right and they say the second bullet now at the end of the first two years right what happened now the interest or he deposited a further 5,000 it means at the end of two years what is happening here 5,000 was also deposited into that amount that account sorry and the interest rate now changed.

[01:37:00]It means from the second year up until the end of the investment our interest now changed from 6 to 5%.

[01:37:12]Which we also divided by 100s. We are going to get what 0 5. Then from there different compounding in this case is compounded half year. Half yearly we spoke about it we saying is true.

[01:37:30]Then in your timeline you can see that I need to calculate the first two years. At the end of the 2 year, I add 5,000 and the interest now changed from 6% to 5%. For the remaining 3 years. So your timeline now help you to analyze your scenario better. So how are we going to answer this one?

[01:38:04]Uh we are just going to say first two years first two years we had an amount of 20,000 that was deposited into 1 plus our interest it was 0 comma 06 six, right? All divided by 12 because it was compounded monthly. Then for 2 years, we also multiplied by by 12.

[01:38:47]Remember what I said, you divide I by 12 and you multiply N by 12, right? And then now we are going to get our amount.

[01:39:01]Our amount in this case it will be um 22,000. Let's punch it in the calculator.

[01:39:11]So we are going to have 1 + 0.06 / 12 to the exponent of 2 * by by 12. So we are going to get 22,53 comma 1 9 1 9 55 52 then that is the amount that we get at the end of 2 years. But the very same amount or at the end of 2 years 5,000 was also deposited.

[01:39:54]Then it means we are going to add what?

[01:39:57]We are going to add 5,000.

[01:40:02]We are going to add 5,000 to that amount. Then what are we going to get?

[01:40:08]We are going to add 5,000.

[01:40:11]Then we are going to get 27,543 1 955 52.

[01:40:22]So that is the amount that we are getting h at the end of two years for the first two years. Then the interest rate changed from 6 to 5 and then the compounding changed from 12 to half yearly which is two. Then from there we need to calculate for the remaining 3 years. From year two to year five, we only have 3 years and we are going to see remaining 3 years.

[01:41:03]Then in the remaining 3 years we have 27,543 9 552 into 1 + 0.05 05 because the interest has changed. We divide by two. We have 3 years. We multiply it by by two.

[01:41:27]So now at the end of the the investment we are going to have an amount [clears throat] of 31, um 941, 66.

[01:41:44]And remember what the question said. The question said will he we must calculate whether he will have enough money in the savings account for a boat cruise and then the boat cruise cost 35,000 and according to our calculation we are arriving at 31,941 r66.

[01:42:11]Then you have to make a conclusion. You have to make a conclusion to say um he will not he will not have enough money.

[01:42:27]Not have enough money.

[01:42:38]That would be a conclusion. he will not have enough money and that's how we we look into it and simple and straightforward because he said we must determine whether Samuel will have enough money whether it can be or it may not be. So in this case he will not have enough money for a boat cruise and that's how simple we answer this type of questions. looking at our timeline.

[01:43:12]Very important.

[01:43:14]Now let's look at um another one.

[01:43:20]Uh number six is read as follows. An amount of 20,000 is invested into an account that offers the interest rate of 10% per anom compounded monthly. Right. At the end of 18 months, the interest rate changed to 8% per anom compounded quarterly. The second bullet, it says the interest rate then remained unchanged for the remaining years.

[01:43:55]The amount of 3,000 was withdrawn from the account at the end of the third year.

[01:44:05]determine the amount of money in the investment account at the end of the fourth year.

[01:44:16]So now our timeline it will be for four years.

[01:44:22]Now I know when you read and you you look into months you just get confused but I will explain on how to deal with that. Now we have a timeline for 4 years.

[01:44:43]We have N1 I mean zero, N1, N_sub_2, N3 and lastly N4.

[01:44:59]In this case, initially the amount of 20,000 was deposited.

[01:45:08]Correct? So now let's now read and interpret.

[01:45:13]So now they said the interest rate is 10% per anom compounded monthly. Correct? Now at the end of 18 months, 18 months if you look into it in months, remember uh one year it is 12 months and two years it has 24 months.

[01:45:44]If we looking it in terms of months then when they are talking about 18 months 18 months basically it says 1 year uh 6 months correct which is 1a 5 in terms of years 1 comma 5 1 5 because 18 months it will be between 1 and the first year and the second year which is going to B N S 1 comma 5 18 months 1 comma 5 correct 18 months 1 comma 5 so that's how we we deal with that now it means from zero to your 18th month which is 1a 5 years we are having an interest of 10%.

[01:46:50]we divide it by 100 we are getting 0a 1 and our m in that case is going to be 12 because is compounded monthly right so now we also see at the end of the third year the interest remain unchanged right for the remaining years but it says an amount of 3,000 it was withdrawn from the account at the end of third year. It means from 18 months which is 1a 5 years to the end of 3 years we must also calculate that period. Why? Because in this year at the end of third year the amount of 3,000 it was withdrawn. We now subtracting 3,000 from that account.

[01:47:52]Correct? Then from there we are just going to calculate for the last year. So now the interest there is just 8% we divide it by 100 we get 0, 08. and the compounding it is quarterly which is four.

[01:48:18]And that's how we interpret our timeline. It is simple and straightforward.

[01:48:28]Simple and straightforward. So now let's calculate this one. Remember the first one is going to be for the first uh for the first 18 months which is 1 and a half year.

[01:48:48]Then we say first 18 months we are having an amount of 20,000 plus our our interest it was 0a 1 all over 12 then 0 1.5 * by by 12 as well correct then the amount that we are going to get there it will be 20,000 into 1 + 0a 1 divided by 12 to the exponent of 1.5 * by by 12.

[01:50:07]Then we get an amount of um 23,00 222A 2 4 6 remember that is the amount that you get where at the end of 18 months which is 1 5 year correct that amount We are still going to put it in the account from five I mean 18 months to the end of 3 years and the end of 3 years you can see that your end there it will be also 1 comma 5 because from 1a 5 to 3 you are getting 1.5 correct. Correct. Yes. Now the interest is changed to 8% compounded quarterly.

[01:51:14]Correct. So now we are going to calculate uh till the end of uh the third year.

[01:51:22]Then we're going to say the next um the next one 5 years which is the next 18 month we are having what we are having 23,000 um let's fix that 23 2222 2 2466 into 1 + interest has changed to what? 0.0 8 divided by 4. We also have 1a 5 * 4. So now this one it will be the end of third year is the same the next 1.5 years or the end of third year and the amount that we are going to get there is going to be um we have 23,222 Then the interest now is 0.08 / 4.

[01:52:48]Then we also multiply by 4.

[01:52:55]Then we are going to get 26,1 52, 0241.

[01:53:05]That is the amount that we are getting at the end of the third year. And what is happening at the end of the third year? I mean the the third year 3,000 was withdrawn according to bullet number three. 3,000 was withdrawn. So now it means there we are going to subtract what? We are going to subtract 3,000 rand.

[01:53:32]Then when we subtract 3,000 rand, what are we left with? Now we subtract 3,000, we are going to be left with 23,152.

[01:53:48]That is the amount that we are going to be left with at the end of the third year after withdrawing 3,000.

[01:53:57]Then from there we now need to calculate the remaining year which is only one one year. The remaining one year the interest remains as 8% compounded quarterly. So now we just going to say remaining remaining year or remaining one year. We know that in this case we are just going to sing a it's equal to 23,000 152 comma 021 into 1 + 0 08 / 4 to the^ of 1 * 4 then the amount that we get there it will be 25,000 and 60 49 that's what we are going to to get here they just told us that we must calculate the money in the investment at the end of four year And that's what we are going to get at the end of 4 years.

[01:55:18]So that's how simple we we calculate this using the timeline.

[01:55:26]So you need to be able to draw up your timeline while reading the scenario so that you can do proper calculations without any any sweat. So that's how we calculate that one.

[01:55:44]Okay. And that's how we deal with the timeline.

[01:55:50]All right. Then that brings us to the end of our lesson and then the summary.

[01:55:57]We dealt with a simple and compound interest where we talking about the growth and we spoke also about the depreciation. Remember there is different between linear depreciation and um reducing balance depreciation.

[01:56:17]And we also dealt with different time periods where we are dealing with our number line. And we also spoke to the annual effective annual and I mean nominal and effective interest uh rate.

[01:56:35]So that is the summary for our lesson.

[01:56:40]The next uh lesson it will be on solving 2DN 3D triangles in trigonometry which is going to happen next week. So now that will be the end. Thank you and go get those marks for finance. Thank you.