Questions: A professional photographer wants to have 16,000 in three years to refurbish a studio. How much money must the photographer deposit at the beginning of each month into an account that earns 5% annual interest compounded monthly to reach the goal? (Round your answer to the nearest cent.)

Solution Steps

To solve this problem, we need to determine the monthly deposit required to reach a future value of $16,000 in three years, with an annual interest rate of 5% compounded monthly. This is a future value of an annuity problem. We will use the future value of an annuity formula, which is:

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

where:

• \( FV \) is the future value ($16,000),
• \( P \) is the monthly deposit,
• \( r \) is the monthly interest rate (annual rate divided by 12),
• \( n \) is the total number of deposits (months).

We will rearrange the formula to solve for \( P \).

The problem is about calculating the monthly deposit required to reach a future value of $16,000 in three years, with an annual interest rate of 5% compounded monthly. This is a future value of an annuity problem.

• Future Value (\(FV\)): $16,000
• Annual Interest Rate: 5% or 0.05
• Time Period: 3 years
• Monthly Interest Rate (\(r\)): \(\frac{0.05}{12} = 0.004166666666666667\)
• Total Number of Deposits (\(n\)): \(3 \times 12 = 36\)

The future value of an annuity formula is:

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

Rearrange the formula to solve for the monthly deposit (\(P\)):

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

Substitute the known values into the formula:

\[ P = \frac{16000 \times 0.004166666666666667}{(1 + 0.004166666666666667)^{36} - 1} \]

Calculate the result:

\[ P \approx 412.87 \]

Final Answer