To solve this, we use the future value of an ordinary annuity formula because you're making regular monthly deposits into the account:

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

Where:

• $FV$ = Future value (desired amount, $300,000)
• $P$ = Monthly deposit (what we need to find)
• $r$ = Annual interest rate (9% or 0.09)
• $n$ = Number of compounding periods per year (12 for monthly)
• $t$ = Time in years (35)

Rearranging the formula to solve for $P$:

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

Step-by-Step Calculation:

1. $r/n = 0.09 / 12 = 0.0075$
2. $n \cdot t = 12 \cdot 35 = 420$
3. $(1 + r/n)^{n \cdot t} = (1 + 0.0075)^{420}$
4. Calculate $(1 + 0.0075)^{420} - 1$.
5. Multiply by $r/n$.
6. Divide $FV$ by the result to find $P$.

Let’s calculate:# Given values FV = 300000 # Future value r = 0.09 # Annual interest rate n = 12 # Compounding periods per year (monthly) t = 35 # Time in years

Step-by-step calculation

rate_per_period = r / n total_periods = n * t

Calculate the denominator

factor = (1 + rate_per_period) ** total_periods - 1

Calculate monthly deposit (P)

P = FV * rate_per_period / factor P101.9789188145722101.9789188145722successThe monthly deposit needed to accumulate $300,000 in 35 years with a 9% annual interest rate is approximately $101.98.

Would you like a detailed breakdown of the calculation or have any other questions?

Related Questions:

1. How does the compounding frequency affect the monthly deposit amount?
2. What would the monthly deposit be if the interest rate were 7% instead of 9%?
3. How much would you need to deposit yearly instead of monthly to reach the same goal?
4. What is the total amount of deposits made over 35 years?
5. How can inflation affect the value of $300,000 over 35 years?

Tip:

Start as early as possible when saving for retirement—the power of compound interest grows significantly with time!