We will solve this problem using the future value of an annuity formula since you are making monthly deposits into an account that earns interest.

Step 1: Use the Future Value of an Annuity Formula

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

• $FV = 300,000$ (future value)
• $P =$ monthly deposit (what we are solving for)
• $r = 10\%$ (annual interest rate) → $0.10$ in decimal
• $n = 12$ (compounded monthly)
• $t = 20$ years

Step 2: Plug in the Values

$300,000 = P \times \frac{(1 + 0.10/12)^{12 \times 20} - 1}{0.10/12}$

First, compute the terms inside the fraction:

• $r/n = 0.10 / 12 = 0.0083333$
• $nt = 12 \times 20 = 240$
• $(1.0083333)^{240} \approx 7.328$

Now, calculate the fraction:

$\frac{7.328 - 1}{0.0083333} = \frac{6.328}{0.0083333} \approx 759.36$

Step 3: Solve for $P$

$P = \frac{300,000}{759.36} \approx 395.84$

So, you would need to deposit $395.84 per month.

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Part B: How Much Interest Will You Earn?

To find the total interest earned, we first calculate the total amount deposited:

$\text{Total Deposits} = P \times 12 \times 20 = 395.84 \times 240 = 95,000$

Total future value is $300,000, so the interest earned is:

$\text{Interest} = 300,000 - 95,000 = 205,000$

So, the total interest earned is $205,000.

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Final Answers:

a) You need to deposit $395.84 per month.
b) You will earn $205,000 in interest.

Would you like a breakdown of any step?

Related Questions:

1. What if the interest rate were 8% instead of 10%?
2. How much would you need to deposit if you had only 15 years instead of 20?
3. What is the total amount you would save if you deposited a lump sum instead of monthly contributions?
4. How does increasing the monthly deposit affect the total interest earned?
5. How would the answer change if interest was compounded quarterly instead of monthly?

Tip:

The sooner you start saving, the more you benefit from compound interest. Even small increases in deposits can lead to significantly larger returns over time! 🚀