Finance. Suppose that $ 5 comma 400 is invested at 4.5 % annual interest​ rate, compounded monthly. How much money will be in the account in​ (A) 3 ​months? (B) 25 ​years?
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Part 1
​(A) Amount after 3 ​months: ​$
​(Round to the nearest​ cent.)

To solve this problem, we use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for in years.

Given:
- $ P = 5400 $ dollars
- $ r = 4.5\% = 0.045 $
- $ n = 12 $ (since the interest is compounded monthly)
- $ t = \frac{3}{12} = 0.25 $ years for part (A)
- $ t = 25 $ years for part (B)

### Part (A): Amount after 3 months

\[ A = 5400 \left(1 + \frac{0.045}{12}\right)^{12 \times 0.25} \]

Let's calculate this step-by-step.

1. Calculate the monthly interest rate:

\[ \frac{0.045}{12} = 0.00375 \]

2. Calculate the exponent:

\[ 12 \times 0.25 = 3 \]

3. Calculate the amount:

\[ A = 5400 \left(1 + 0.00375\right)^3 \]
\[ A = 5400 \left(1.00375\right)^3 \]
\[ A \approx 5400 \times 1.01128 \]
\[ A \approx 5459.01 \]

So, the amount after 3 months is approximately $5459.01.

### Part (B): Amount after 25 years

Let's use the same formula but with $ t = 25 $:

\[ A = 5400 \left(1 + \frac{0.045}{12}\right)^{12 \times 25} \]

1. Calculate the exponent:

\[ 12 \times 25 = 300 \]

2. Calculate the amount:

\[ A = 5400 \left(1 + 0.00375\right)^{300} \]
\[ A = 5400 \left(1.00375\right)^{300} \]

Now we need to calculate $ \left(1.00375\right)^{300} $:

\[ \left(1.00375\right)^{300} \approx 3.05097 \]

So,

\[ A \approx 5400 \times 3.05097 \]
\[ A \approx 16475.24 \]

So, the amount after 25 years is approximately $16,475.24.

### Summary:

- Amount after 3 months: $ \approx \$5459.01 $
- Amount after 25 years: $ \approx \$16475.24 $

Do you want more details or have any questions?

Here are some questions you might want to ask next:
1. How is the compound interest formula derived?
2. What happens if the interest rate changes?
3. How would daily compounding affect the final amount?
4. Can you show a graph of the investment growth over time?
5. What is the difference between simple and compound interest?

Tip: Always make sure to convert the interest rate to a decimal and the time period to years when using the compound interest formula.

Learn how to calculate compound interest with monthly compounding. Solve the problem of determining the amount of money accumulated after 3 months and 25 years with an initial investment of $5400 at an annual interest rate of 4.5%. Get step-by-step solutions and understand the compound interest formula.

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