Welcome back. Okay, so we're in Section 3. 30.6 of the man is pre Calculus 2010 edition against the eighth edition, Page 3 12, number 56. Okay. All right, so in this question, or ask to redo question 53. But I think that's a typo. Because 56 says we're making 884 dollar per month payments on $86,000 mortgage. But in 53 they have a 30 year mortgage at 8.275%. I think e think they're actually redoing question 55 which has payments of 50. Okay, so all right, so I'm gonna take the interest rate, which is 12 8% a p r from question 55 instead. Okay. All right. Just to that. So here's the situation. Um, you could be making payments of $884. 61 cents for an $86,000 mortgage for 10 years. Okay, so let's take a look at the present value. So present value is equal to the payment per term in this case per month of one minus, uh, quantity one minus three rate per term in this case per month to the negative and the number of terms and a number of payments over that same interest rate. Right. So let's see how much we paid. If we're paying 884th dollars and 61 cents. Okay, you got one minus. Um, sorry. This is a plus one. Plus. Now it's 12% of 120.12, and there's 12 payments being done monthly. All right, so over 12 and the same thing with point 12/12 point of one to the negative end this part. So this is one point of one to the negative end, which is negatively 12 payments for 12 payments for year for 10 years, 120 payments. There you go. So this tells me how much I will have paid by then. Right? All right, so this is $884. 61 cents times one minus 1.1 to the negative. 1 20 all over one. That All right, So that's what I'm gonna calculate. Okay, here we go. Okay. So let's get a calculator. Okay. That's calculator. Was my calculator. There it is already. So let's see. So it's $884.61 times Let's see out for y equals rational one minus 1.1 to the negative 1. 20 over 0.1. And that that's close. The quantity. And that means well paid. 61,000 $61,607.78. Okay, so let's write that down. So that is approximately to the nearest sent, 61,000 $657.78. Okay, so what does that mean? Well, we were trying to pay off in $86,000 mortgage. So what's 86,000? Minus 61,000? This is dollars, right? For 7.78. And you know what? I just I don't feel like doing it this way. The suit on the calculator. Come on, calculator. So it's gonna be $86,000 mortgage minus that amount. Then we get $24,342. 22 cents. Okay, so let's see what that is. So it's gonna be 24,000 $342.22. Now, the thing is this, we're gonna now change your payments from $884.61 per month Did 1000 50 per month And see how many more payments? You know, we did 120 payments, you know, 12 payments per year for 10 years. Right? So the question is, how long is the term of the loan? It's gonna be less than 30 years that you originally were gonna dio, right? Um, so we're gonna use the present value formula again. Okay, here we go. So the present value now is a present value is the payment times one minus quantity one plus tau interest rate per payment to the negative end. Number of payments all over the interest rate. Right. And we're still talking about 12% per month, so 120.12% for years of one per month. Okay, so we wanna pay off 24,000 remaining, 342,000, 20 cents roughly. And my new payment is 1050 per month. And so the question is, can we solve for N? So it's one minus. I remember ours point 12/12 10.0.1. So one plus point of one is one point or one to the negative end all over point. 01 Now we want to sell for end. So let's divide by 2050. 366,022 cents, divided by 1050 Leaves one minus 1.1 to the negative end delightedly point a one. All right, let's one play both sides. We point a 1 to 100 others survived. By 100 you get $24,342. 22 cents over 105,000 is one minus 1.1 to the negative end. All right, so if we subtract this fractional from both sides and add this exponential to both sides, we get 1.1 to the negative end is one minus that crazy fraction already. Mhm. So now we take the log of both sides. We're gonna get negative. N l n take your lug of your favorite pace. Favorite base rate of 1.1 is Thea Ln of this mess, one minus two for 24,300 to 1 to 2 over 105,000. Okay. And so we divided by a negative. Ellen at 1.1. We get in and is negative. Ln of, um, Hang on. I wrote this a little bit sloppy. Yeah, this one is not part of the fraction. Right? So let's clean it up. It's one minus 24,000 over 105,000. There you go. Yeah. So it's Yellen of one minus 24,342 to 2 over over 105,000 over Ellen of 1.1. So and is approximately how many more payments? So we did 10 years. And now how many more years instead of 20 more years? Because it was also 30 year term mortgage in question 55. This basic question 55 that's the type of 53 was 55. All right, so let's put that in the calculator. So it's negative Ln off. Okay, So could waiting on. Should I write this down with my pen? Oh, get from it. Oh, boy. It's negative. Ellen of that over Ln of that. Okay. Okay, I get it. So let's get the calculator out calculator using this simulator on TV for plus c Super Edition. Here we go. All right. So it's gonna be Alfa y equals rational negative. Ln of one minus, Alfa y equals rational. Just to make it easy to read for you guys. Um 24,342 and 22 over 105,000. Close that parentheses, writing over the island of 1.1 the l n at 1.1. And that gives you 26.5 more payments. Roughly well to the nearest half a month. Alright, So 26.5. Roughly. All right. Good. Okay. Yeah, that's about 26.4 wives, 26.5. So instead of instead of another 20 years or 240 payments, it's only 26.5 payments. That's good. Okay, good. Nice. Already. So 26.5 payments initial. 26 2016 payments. Mm. So tormented. Tend to pay off is alright. So to amount of time to pay this off is the 10 years that we did already. So time equals the 10 years we did already. Plus well, 26.5. Let's see, divided by 12 goes to remainder 2.5. So it's gonna be 10 years plus two years. Let's see 2.5 months. Let's make it three months. So 3/12 is one quarter. It's about 12.25 years. There you go. Good. Now, Part B says, How much money did you save? All right. Well, let's find out. Let's go. Whoa. How much money did just say by doing that? Alright, so let's compare what we would have been paying on the calculator way would have been paying for 30 years. Um, 12 payments per year of 884,000. 61 cents. Right? That would do that, right? Yeah. So it would have been paying 318,000. Isn't an 86,000 that long? $318,459.60. Right? But now what we did, we did this. Instead, we did 120 payments for 10 years, times the 800 $84. 61 cents. Right. But then we did approximately, um 26.5 payments times 1000 50. So how much money did was pay now only 100 1,003,000. 20 cents. While that's a big savings. So question is what is the savings. Let's take what we would have paid for 30 years. Minus will be paid in that 12 and a quarter years and we get a savings of 184,000, $481.40. Okay. With cell phone enumerated. See you next time. Good luck with your homework. Bye bye.