Apison Durcnased & 5130,082 hme 10 gaisa06 pavinq 2093 down and gigning 30-Year mondade 20-year morgage at 5.190 compounded monthly: How much interest wal relin...

Savings and interest. A sum of \$ 1000$ is deposited in a savings account for which interest is compounded monthly. The future value $A$ is a function of the annual percentage rate $r$ and the term $t,$ in months, and is given by $$A(r, t)=1000\left(1+\frac{r}{12}\right)^{12 t}$$ a) Determine $A(0.05,10)$ b) What is the interest earned for the rate and term in part (a)? c) How much more interest can be earned over the same term as in part (a) if the APR is increased to $5.75 \% ?$

For this question who were told to use the results in Exercise 85 to find eight. So let's first write down the result from 85. So Exercise 85 tells us the accumulated value A is equal to well times one plus R over 12 to the power of 12 p minus one times one plus problem for us. So now we just need to identify what all those variables are. So for this particular question, uh, investor, there's depositing $100 on the first day of each month. So therefore, that's R P principle, so it's equal to $100. Now the account pays an annual interest rate of 2% compounded monthly, So our is 2%. So don't forget to write that and decimal format was 0.2 okay. And the last thing that we need to know is what T is or more like, what 12 t is right. So we're told that the balance in the account for five years is, um, 100 times one plus 0.2 of the 12 1 all the way up to 60 right, So the last power is 60 so Therefore, 12 T is equal to 60 which makes sense because they're a total of 16 30 60 com pounding period, since we're doing this for five years and each year has 12 months. So now we have everything we need to find a so A is going to be equal to 100 times. This will be one plus 0.0 2/12 power, 60 minus one times one plus 12 over zero point direction. So now you just have to punch this into a calculator. So let's see. That's going to be one plus to the power of 60. Subtract one and then multiplied by one plus 12 over 0.2 and you end up with a total, UH, $6315. Andi 24 cents.

And so for looking to see how much more money to Shawn's initial deposit will have earned, then Jessica's We first have to find the amount for their final that so to Shawn's final remount, and we want to subtract Jessica's final amount. So first of all, it's going to find to Shawn's final now, and we know that could be ah, found by the equation one hundred, because that's the initial amount on then one plus the rate which the rate it says it's two point five percent. So two point five percent express of the decimal. It's your point zero to five to the time, which we know is going to be ten years. And then Jessica's Final Mount is just gonna be the same one hundred dollar initial deposit. But then we know that her rate is only two percent, so it's going to be your points. You're two faced the same amount of time. Ten. So then way do that. That's going to be so the left side. If we do that, it's going to be one hundred and twenty eight dollars and one cent. And then for the right side for Jessica's amount, you know that's going to be one hundred and twenty one dollars and ninety cents. So if we subtract them, I get six dollars and eleven cents. So then that means that this is a difference between her Shawn's initial deposits, earnings as a procedure. Jessica's initial deposit. Ernie's after ten years. So we know we consider it up like that. Just a circle back, because our equation is gonna be final. Siegel to initial again times one plus the rate expressed as he does, long razed to the time.

Okay so far. That's problem. No nine history. Okay. We actually already know that a. You close to $50,000. We want to find P. Okay. So we know that the annual rate is 3.5%. Okay. It's compound. Um Mostly. Okay. So this is I. Yeah. And then he goes to And it goes to after 10 years. Okay. 10 years times 12 months are months. Is one year. It goes to 120. Okay. So now we can find p. p. equals two. Hey, times are Times are three times high. Yeah, Divided by one plus I to the N -1. Okay so which goes to. Okay so now let's solve for this equation and solve for this expression. So we get around. Okay. Okay so code and Alice should should the place $5,548.6 in in a saving account. Each mouth. Okay. So they can Get the 50,000 down payment in 10 years.

Okay, so we're given that if 6000 is deposited in the account, savings are paying 5% compound accordingly. That means we have our principal. P is equal to six Upton, and we have a great of 5% which is 0.5 And it's compound it cordially. Which means that our end value is for then what amount will be in the account after 10 years. So we have that p is equal to for a Z z p times one plus I threw her of end. Actually Sorry, this is actually 10 years and then compounded quarterly means we have our baits over for So that's 6000 sometimes one plus 0.5 over for the power of tech. Let's put that into our cat. Come here. One plus 10.5 divided by foreigners. Your power of 10 use mi 6793 0.62