Caroline is opening a CD to save for college. She is...

Caroline is opening a CD to save for college. She is considering a 3 -year $\mathrm{CD}$ or a 3$\frac{1}{2}$ -year CD since she starts college around that time. She needs to be able to have the money to make tuition payments on time, and she does not want to have to withdraw money early from the CD and face a penalty. She has $\$ 19,400$ to deposit. a. How much interest would she earn at 4.2$\%$ compounded monthly for three years? Round to the nearest cent. b. How much interest would she earn at 4.2$\%$ compounded monthly for 3$\frac{1}{2}$ years? Round to the nearest cent. c. Caroline decides on a college after opening the 3$\frac{1}{2}$ -year $\mathrm{CD},$ and the college needs the first tuition payment a month before the $\mathrm{CD}$ matures. Caroline must withdraw money from the CD early, after 3 years and 5 months. She faces two penalties. First, the interest rate for the last five months of the CD was lowered to 2$\%$ . Additionally, there was a $\$ 250$ penalty. Find the interest on the last five months of the CD. Round to the nearest cent. d. Find the total interest on the 3$\frac{1}{2}$ year CD after 3 years and 5 months. e. The interest is reduced by subtracting the $\$ 250$ penalty. What does the account earn for the 3 years and 5 months? f. Find the balance on the CD after she withdraws $\$ 12,000$ after 3 years and five months. g. The final month of the CD receives 2$\%$ interest. What is the final month's interest? Round to the nearest. What is the final month's interest? Round to the nearest cent. h. What is the total interest for the 3$\frac{1}{2}$ year $\mathrm{CD} ?$ i. Would Caroline have been better off with the 3 -year CD? Explain?

Caroline is opening a CD to save for college. She is considering a 3 -year $\mathrm{CD}$ or a 3$\frac{1}{2}$ -year CD since she starts college around that time. She needs to be able to have the money to make tuition payments on time, and she does not want to have to withdraw money early from the CD and face a penalty. She has $\$ 19,400$ to deposit.
a. How much interest would she earn at 4.2$\%$ compounded monthly for three years? Round to the nearest cent.
b. How much interest would she earn at 4.2$\%$ compounded monthly for 3$\frac{1}{2}$ years? Round to the nearest cent.
c. Caroline decides on a college after opening the 3$\frac{1}{2}$ -year $\mathrm{CD},$ and the college needs the first tuition payment a month before the $\mathrm{CD}$ matures. Caroline must withdraw money from the CD early, after 3 years and 5 months. She faces two penalties. First, the interest rate for the last five months of the CD was lowered to 2$\%$ . Additionally, there was a $\$ 250$ penalty. Find the interest on the last five months of the CD. Round to the nearest cent.
d. Find the total interest on the 3$\frac{1}{2}$ year CD after 3 years and 5 months.
e. The interest is reduced by subtracting the $\$ 250$ penalty. What does the account earn for the 3 years and 5 months?
f. Find the balance on the CD after she withdraws $\$ 12,000$ after 3 years and five months.
g. The final month of the CD receives 2$\%$ interest. What is the final month's interest? Round to the nearest. What is the final month's interest? Round to the nearest cent.
h. What is the total interest for the 3$\frac{1}{2}$ year $\mathrm{CD} ?$
i. Would Caroline have been better off with the 3 -year CD? Explain?

Key Concepts

Compound Interest

Compound interest is the process where interest is earned on both the initial principal and the accumulated interest from previous periods. This concept is central in financial mathematics because it shows how investments grow over time when the interest earned is reinvested, leading to exponential growth rather than linear growth as seen in simple interest.

Compounding Frequency

Compounding frequency refers to the number of times interest is calculated and added to the principal over a specific period. For instance, when interest is compounded monthly, the annual interest rate is divided into 12 parts and applied each month, affecting both the total return and the time intervals for the growth of the investment.

Time Conversion in Compound Interest

Time conversion in compound interest problems involves aligning the time periods used in the interest rate with the actual duration of the investment. This often requires converting years into months or other units to match the compounding frequency, ensuring that the formula's exponent correctly reflects the total number of compounding periods.

Certificate of Deposit (CD)

A Certificate of Deposit (CD) is a financial product offered by banks, where money is deposited for a fixed period with a predetermined interest rate. CDs are known for their safety and typically provide higher interest rates compared to regular savings accounts, but they may impose penalties if funds are withdrawn before the term expires.

Early Withdrawal Penalties

Early withdrawal penalties are fees or reductions in interest rate applied when an investor withdraws funds from a financial product like a CD before the maturity date. These penalties serve as a deterrent against premature withdrawals and can affect the total interest earned, as the investor might incur additional costs or reduced growth on the investment.

Balance Calculation Post Withdrawal

After an early withdrawal, calculating the remaining balance on an investment involves accounting for the interest accrued up to that point, any penalties incurred, and any partial withdrawals made. This calculation is crucial for determining how much of the original principal and earned interest remains available after all adjustments have been applied.

Transcript

00:03 Okay, we are here to help caroline work through this complicated situation.

00:08 Starting with part a, she's investing $19 ,400 for three years at a 4 .2 % interest rate compounded monthly, and because it's monthly, n equals 12.

00:19 So we're going to figure out the amount of interest she would earn.

00:22 First, let's find the final balance using the equation b equals p times 1 plus r over n to the nt.

00:31 Okay, we'll substitute all of our values in there, 19 ,000.

00:36 And for r, we're going to use 0 .042.

00:39 We need to convert that percent to a decimal.

00:42 And for n, we're using 12.

00:43 And we're raising it to the 12 times 3, which is the 36th power.

00:48 So let's see what we get for that.

00:53 So the final balance there is $22 ,000 and $23.

00:58 Well, we want to know the interest, so we need to subtract the original principle from that amount to get the interest.

01:05 So we're going to subtract $19 ,400 from that amount.

01:08 And the interest is $2 ,600 .23.

01:14 Okay, now let's work on part b to figure out the amount of interest you would earn with the other cd.

01:20 So the other cd is also compounded monthly, so n is still 12, same interest rate, longer period of time.

01:33 Okay, so we're substituting our numbers into the same formula, and we have 19 ,400 times 1 plus 0 .042 over 12, raised to the 12 times 3 .5.

01:49 And we end up with $22 ,466.

01:57 And 30 cents.

01:59 Now that's the final balance, and we want to know the interest.

02:02 So we're going to subtract the original $19 ,400 from that.

02:06 And the interest would be $3 ,066 .30.

02:15 Okay, now we're on part c.

02:17 And some things have changed for caroline, and now she's earning her 4 .2 % interest for the first three years, but then she's only earning 2 % for the last 5%.

02:27 Months.

02:28 And we want to calculate the amount of interest she earns in those last five months.

02:32 But we have to start by figuring out what she's starting with at the beginning of that time period.

02:37 So we need to know how much money she had in her account at the end of three years.

02:41 So let's use the formula b equals p 19 ,400 times 1 plus 0 .042 over 12 to the three times 12 to figure out how much she had in her account at the end of three years.

02:58 And you know that is exactly what we did for part a.

03:01 Come to think of it, we got $22 ,000 and $23.

03:07 So why not take advantage of that? okay, so what's going to happen then is for the last five months of the cd, she's going to be earning interest on that particular principle.

03:21 So we're going to use that as our new value of p over here for the final part of the cd.

03:26 So now we're going to have b equals $22 ,000 and $23 times 1 plus .02 because now she's only earning 2 % divided by 12.

03:40 And for how long? well, she's doing this for five months.

03:44 So that's five 12ths of a year times 12 times per year.

03:48 So that's really just an exponent of five.

03:50 It's going to be five compoundings.

03:53 And let's calculate that amount.

03:57 So the final amount now, final amount, now is $22 ,184.

04:03 And $18.

04:05 Okay.

04:06 So the question is, how much interest did she earn in those last five months? so we need to subtract from this amount the principal.

04:14 So we need to take $22 ,184 and $18 and subtract $22 ,000 and $23.

04:23 And we end up with $183 .85.

04:29 That's the amount of interest she earned in that last five -month period.

04:36 Now in part d, we want to know how much interest she earns altogether.

04:40 So the first three years, she earned $2 ,600 and 23 cents, and in the last five months, she earned $183 .85.

04:49 Those were our answers from parts a and c, and if we add those together, we'll get the total interest.

04:55 So over the three -and -a -half -year period, she actually earned $2 ,784, and $8 .5 of interest...

Please give Ace some feedback