Suppose you want to have $300,000 for retirement in 25 years. Your account earns 10% interest compounded monthly.

a) How much would you need to deposit in the account each month?

$
333.52
Incorrect

b) How much interest will you earn?

$
199,944


To solve this problem, we’ll use the future value of an ordinary annuity formula, given that interest is compounded monthly:

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

where:
- $ FV $ is the future value you want to achieve, which is $300,000,
- $ P $ is the monthly deposit (which we need to find),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of deposits.

Let's go through each part.

### Part (a): Monthly Deposit Calculation
1. **Convert the annual interest rate to a monthly rate**:
   \[
   r = \frac{10\%}{12} = 0.10 / 12 = 0.008333\ldots
   \]

2. **Determine the total number of deposits (months)**:
   \[
   n = 25 \times 12 = 300
   \]

3. **Rearrange the formula to solve for $ P $**:
   \[
   P = \frac{FV \cdot r}{(1 + r)^n - 1}
   \]

4. **Plug in the values**:
   \[
   P = \frac{300,000 \cdot 0.008333}{(1 + 0.008333)^{300} - 1}
   \]

Calculating this will yield the monthly deposit required.

#### Solution:
\[
P \approx 334.57
\]

So, you would need to deposit approximately **$334.57** each month to reach $300,000 in 25 years.

---

### Part (b): Interest Earned
To find the interest earned, we’ll calculate the total amount deposited over time and subtract it from the future value.

1. **Total contributions**:
   \[
   \text{Total Deposits} = P \times n = 334.57 \times 300 \approx 100,371
   \]

2. **Total interest earned**:
   \[
   \text{Interest Earned} = FV - \text{Total Deposits} = 300,000 - 100,371 \approx 199,629
   \]

So, the total interest earned would be approximately **$199,629**.

---

Let me know if you want more details on any part, or have any questions!

Here are five related questions to consider:
1. What if the interest were compounded annually instead of monthly?
2. How much more would you need if the goal were $500,000 instead of $300,000?
3. What would the monthly deposit be if the interest rate were 8% instead of 10%?
4. How does increasing the deposit frequency (e.g., bi-weekly deposits) impact the required monthly deposit?
5. What if the deposit period were only 20 years instead of 25?

**Tip:** For retirement savings, compounding frequency significantly affects the future value, so monthly compounding grows the amount faster than annual compounding!

Suppose you want to have $300,000 for retirement in 25 years. Your account earns 10% interest compounded monthly.

a) How much would you need to deposit in the account each month?

b) How much interest will you earn?

Find out how much you need to save monthly to reach a $300,000 retirement goal in 25 years with a 10% annual interest rate compounded monthly. This problem includes calculations for monthly deposits and total interest earned, using the annuity future value formula. Ideal for students and adults interested in financial planning and compound interest calculations.

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