a) The monthly deposit needed is approximately \(1,515.02\) dollars.

b) The interest earned will be approximately \(336,395.20\) dollars.

UpStudy Free Solution:

To solve this problem, we need to determine the monthly deposit needed to reach $700,000 in 20 years with a 6% annual interest rate, compounded monthly. We will use the future value of an ordinary annuity formula:

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

where:

- \(FV\) is the future value ($700,000)

- \(P\) is the monthly deposit

- \(r\) is the monthly interest rate (annual rate\( / \)12)

- \(n\) is the total number of payments (years \(\times \)12)

Step 1: Calculate the monthly interest rate and the total number of payments

Annual interest rate = 6%

\[r = \frac { 6\% } { 12} = \frac { 0.06} { 12} = 0.005\]

Total number of payments over 20 years:

\[n = 20 \times 12 = 240\]

Step 2: Use the future value formula to solve for \(P\)

\[700,000 = P \times \frac { ( 1 + 0.005) ^ { 240} - 1} { 0.005} \]

First, calculate the value of \(( 1 + 0.005) ^ { 240} \):

\[( 1 + 0.005) ^ { 240} = ( 1.005) ^ { 240} \approx 3.3102\]

Next, calculate \(\frac { ( 1 + 0.005) ^ { 240} - 1} { 0.005} \):

\[\frac { ( 1.005) ^ { 240} - 1} { 0.005} = \frac { 3.3102 - 1} { 0.005} = \frac { 2.3102} { 0.005} = 462.04\]

Now, solve for \(P\):

\[700,000 = P \times 462.04\]

\[P = \frac { 17500000} { 11551} \approx 1,515.02\]

So, the monthly deposit \(P\) is approximately $1,515.02.

Step 3: Calculate the total amount deposited and the interest earned

Total amount deposited over 20 years:

\[\text { Total Deposits} = P \times n = 1,515.02 \times 240 \approx 363,604.80\]

Interest earned:

\[\text { Interest Earned} = FV - \text { Total Deposits} = 700,000 - 363,604.80 \approx 336,395.20\]