These are the two equations we came up with. With substituting, as I said, the i and the A with the Ps. So the first equation will give you P, which if instead of doing, or what we went through one by one. Calculating P 1, P 2, P 3, all the way to P 36, we can use the first equation. We can say okay, if A I give you is $968 for an substitute A here, 968. The i, we can't have it as the point O, O 833. The N is 36 months and P then has to give you $30,000. And this is the opposite. If I can tell you, okay, this is the $30,000 and you have an i equal to 0.833% per month. And you have an n or number of interest period which is 36 months, what will be the equivalent A for the next 36 months? What you will do is to apply 50,000 for the P here, the i the 0.833, the n is 36 all the way, and you have to get with the number $968. So these are the two equations that we will be using along moving forward in case we have a connections between, and we want to understand or calculate that uniform series with a present worth. Now, things I want you to remember about the uniform cities values are the following. One, all payments must be equal. If you want to use the previous two equations that I just highlighted, that you have to have in the cash flow diagram. Let's say you have all the way here to n. And this is 0, 1, 2, all the way. So this is, that you are drawing here, they are exactly the same value. And that's all equal to A. A 1, 2, 3, all the way to A n. However, if you have another cash flow diagram here to n. And you have, let's say, one value like this. Another value like that. A third value like this. A fourth value like that. As you can see, it's different and they are unequal. So you can't use this as a uniform series as you can use from the previous slide with the two equations that I highlighted. In this case, for case let's say number one here, we can use, if we want to find equivalent P value, of all these A values, we can use the equation from the previous one, and just find the P. But we have such a situation like this. And we want to find the equivalent value of all these installments in the present worth in option number two here or case number two. In this case, what we will do is to deal with all these installments as future values. And we want to calculate each one present value here. Similar to what we did here when we try to explain the equation where we came with the equations between linking the A and the P. So in this case, you have how many formulas, you have the F formula from F1 that you will turn it to P. F2 you will turn into another P, F3 to P until Fn to another P and then you'll sum up all the P's to find the total P value of all these future values. So this is one important point. All payments must be equal. Also, payments must not be interrupted. So for example, if you have, let's say case number three here. And you have also from zero all the way to n, and you have 1, 2, 3, 4, and then you can say you have A, A, and then you have A here and A here. So in this case, you have a missing A in term number three or pair number three. So that will not be, or we will not be able to implement or use the equations that I just highlighted that will link the A to the P. So make sure that there is no missing payments in the middle, so as to have uniform cities like guest number one here. So let's say this is case number three. So make sure, again, that all payments must be equal, not like the situation we have in case number two, and they must not be interrupted like missing the payment in one of the periods along the line. Also, all payments must carry the same interests. So if I give you situation here, and I will tell you okay, that first three period of time here will be highlighting i1 and then the rest will be highlighting i2. Then we can't deal with all the As the same with using the same equation and convert it to P. But in this case, we have two questions. One question is this part, and then, using the i1, and the second part of the question is to use this A is using i2. The simpler answer for that if you want to find the P value of this situation, we can then convert the first three months using i1 to P1 and then, we convert the As here. Even if they are the same A or thus equivalent value, but using i2 to find the value here, using the P value. And then from the P value here, we deal with this as a future value to find the P 2 here, and then we sum up these two to find the P total. So this is a more complicated scenario, but I will give you more questions or exercise about this for later, in case you have two i's, not just a one same i. So it's very important to have all payments must carry the same interest rate here. So what did we learn so far? We learned the introduction to the basics and the main elements of the cash flow diagram. So we highlighted the P and the F, which is the present and the future. And we've connected using the equation from the previous modules. Second, what we learned is, if we want to find that equivalent uniform series for present worth value we have, today. And we came up with two equations. If we want to find that A, uniform series, giving P, the present worth. Or, we can say okay, I can pay around up to $1,000 for the next 36 months for a specific car that I want to buy. What's the interest rate? Let's say you have an interest rate of 12%, compounded monthly. So how much then, the worth of the car you want to pay today? So, then you can use also the equation that we came up with. We just explained before. Which is P. To find the P value equal to the E value plus or times the terms highlighting the interest per period of time. So for next, we will go move forward to study how we can find the equivalent future value of a uniform of series values. And instead of looking at uniform series values, finding the present worth, we want now to find the future value for uniform series values.