Let's solve each part of the problem step by step.

a) Future Value of the Account

The formula to calculate the future value of an annuity (like regular deposits) with compound interest is:

$FV = P \times \frac{(1 + r)^n - 1}{r}$

Where:

• $P$ is the monthly deposit ($100)
• $r$ is the monthly interest rate (annual rate divided by 12)
• $n$ is the total number of deposits (months)

Given:

• $P = 100$
• Annual interest rate = 7% or 0.07
• $r = \frac{0.07}{12}$
• Time = 30 years
• $n = 30 \times 12 = 360$ months

Let's calculate the future value.

$r = \frac{0.07}{12} \approx 0.005833$ $FV = 100 \times \frac{(1 + 0.005833)^{360} - 1}{0.005833}$

$FV \approx 100 \times \frac{(1.005833)^{360} - 1}{0.005833}$

Now, let's compute this value:

$FV \approx 100 \times \frac{(1.005833)^{360} - 1}{0.005833}$

$FV \approx 100 \times \frac{10.474 - 1}{0.005833}$

$FV \approx 100 \times \frac{9.474}{0.005833}$

$FV \approx 100 \times 1623.111$

$FV \approx 162,311.10$

So, the amount in the account after 30 years will be approximately $162,311.10.

b) Total Amount Deposited

To find the total money deposited, simply multiply the monthly deposit by the number of months:

$\text{Total Deposits} = 100 \times 360 = 36,000$

So, the total amount you will put into the account is $36,000.

c) Total Interest Earned

The total interest earned is the future value minus the total amount deposited:

$\text{Total Interest} = FV - \text{Total Deposits} = 162,311.10 - 36,000 = 126,311.10$

So, the total interest earned will be $126,311.10.

Summary

• a) Future value of the account: $162,311.10
• b) Total amount deposited: $36,000
• c) Total interest earned: $126,311.10

Would you like further details or have any questions?

Here are some related questions:

1. How would the future value change if the interest rate was 5% instead of 7%?
2. What would happen if you deposited $200 instead of $100 each month?
3. How much interest would you earn if you only saved for 20 years?
4. How does the compounding frequency affect the total amount?
5. What would be the total amount if you only compounded annually instead of monthly?

Tip: The power of compound interest is in the time the money is allowed to grow. The longer you save, the more your money grows!