Suppose you want to have $700,000 for retirement in 35 years. Your account earns 5% interest. a) How much would you need to deposit in the account each month? S b) How much interest will you earn?

This problem involves calculating the monthly deposit required to accumulate $700,000 in 35 years with a 5% interest rate, compounded monthly. To solve it, we’ll use the future value of an annuity formula for monthly contributions:

### Future Value of Annuity Formula:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- $FV$ is the future value (in this case, $700,000),
- $P$ is the monthly deposit (which we are solving for),
- $r$ is the monthly interest rate,
- $n$ is the total number of deposits (number of months).

### Step 1: Given Information
- The annual interest rate is 5%, so the monthly interest rate $r = \frac{5\%}{12} = \frac{0.05}{12} \approx 0.004167$.
- Time is 35 years, so the total number of months $n = 35 \times 12 = 420$.
- Future value $FV = 700,000$.

### Step 2: Solve for $P$

Rearranging the formula to solve for $P$:
\[
P = \frac{FV \times r}{(1 + r)^n - 1}
\]
Substitute the values and calculate.

Let me compute that for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can use the provided formula to calculate the monthly deposit $P$:

\[
P = \frac{700,000 \times 0.004167}{(1 + 0.004167)^{420} - 1}
\]

After finding the monthly deposit, you can compute the total amount deposited over 35 years, then subtract it from $700,000 to find the total interest earned.

Feel free to ask if you want to discuss the calculations further, and I can help with the steps in more detail. 

Would you like me to help guide the manual calculation?

Here are 5 related questions for further practice:
1. What if the interest rate were 6% instead of 5%?
2. How would the monthly deposit change if you needed $1,000,000 in 35 years?
3. What happens if you shorten the time period to 30 years?
4. How does compounding quarterly affect the amount of interest earned?
5. Can you calculate the total interest if you deposit a lump sum instead?

**Tip**: Always double-check the formula and units when calculating annuity problems.

Suppose you want to have $700,000 for retirement in 35 years. Your account earns 5% interest. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

Calculate how much you need to deposit monthly to accumulate $700,000 in 35 years at a 5% interest rate. This detailed solution explains the future value of annuities and includes key formulas, suitable for high school students studying financial math.

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