Alright so let's see how you guys did. How much would you need to invest today if instead you could only earn 6% interest now let's think logically before we dive into all the numbers let's think at a 10% interest rate. We need to deposit $9917 at a 6% interest rate. What do you guys think? Do you think it's gonna be a higher number than 9009 17 Or a lower number than 9009 17? Let's think about it for a second. It's a lower interest rate. Right? So you're not gonna earn as much interest. Let's go ahead and dive in and let's see what happens. So in this case everything staying the same we still have the same future value of 12,000 we have the same number of periods and we're still looking for a present value. However we've got a new interest rate right? The interest rate r. Is now 6% instead of 10%. So let's go ahead and draw our timeline real quick right here And it's 012 years from now. So today is zero and then we've got one and two years from now and instead we're earning 6% interest. Now I'm gonna put it in a different color. So it stands out 6% interest Rather than 10% interest right? But we're still looking for a future value. This future value is still 12,000 right? And we want to know what it's worth today. So we're gonna bring it back in time and find out what it's worth today. Okay so let's go ahead and use our formula just like we did before we had a present value equals future value divided by one plus R. To the end. And notice not much is changing here. We've got 12,000 in the numerator. But now instead of 1.10 it's going to be one point oh six right? One plus 6% which is point oh six. So we're gonna have 1.06 instead of 1.1 in the denominator and it's still gonna be two years. So we're gonna square that denominator 1.06 to the second power. And that's easy enough. Right? We should do that part first. So let's go ahead and do 1.06. Where did it go? One point oh six times one point oh six. And that gives us a denominator. I'm not gonna round till the very end. Our denominator is gonna be 1.1236. That's one point oh six squared right, 1.6 times one point oh six gives us 1.1236. Our numerator is gonna be 12,000. So let's go ahead and do that 12,000 Divided by 1.1236. And we get 10,600 and we'll run it to 10,680. So that means today we need to deposit $10,680. So that in two years we'll have $12,000. That should make sense. Right? We're earning less interest so we're gonna need more money now to get to the $12,000 in two years right Before we were earning more interest. So more of that 12,000 was made up of interest. But now we need to deposit more so that we can accumulate up to 12,000 just the same. So let's do just like we did above and let's see what happens. Let's increase our balance with the compounding equation. So what we're gonna do is we're gonna take the 10,006 80 and we're gonna multiply it by one point oh six. Right? This is this will tell us what it's worth after one year we're earning 6% interest so it's gonna be worth 6% more. One plus 6%. 1 point oh six. That was our compounding equation, right? We multiplied by the one plus art of the end. So 6 10,080 times one point oh six. So I'm going around here to $11,321 after one year. And then if we were to multiply it by 1.06 again for the second year. Well there we go. We're up to our 12,000 again. Remember we got a little bit of rounding errors because we just we were rounding numbers but that's exactly what happened To have $12,000 in two years at a 6% interest rate. Well, we're gonna have to deposit $10,680, right? And this is what can be confusing to students, right? Because the only thing that changed here is the interest rate. Alright. But we're still looking for that same future value. We still have that same future value and we're looking for a present value. Okay? Uh So that's about it for this problem. Let's go ahead and move on to one more topic about time value of money.