Demco Products, a company that manufactures stainless steel control valves, has a fund for equipment replacement that contains $500,000. The company plans to spend $85,000 each year on new equipment.

Estimate the number of years it will take to reduce the fund to zero at an interest rate of 9.00% per year. 

 

The number of years is estimated to be between 
 and 
 .

To estimate the number of years it will take for Demco Products' fund to reduce to zero, considering a starting balance of $500,000 and an annual spending of $85,000 with an interest rate of 9% per year, we need to model this as a problem involving the depletion of a fund over time with annual withdrawals and interest compounding.

### Key Information:
- Initial fund: $ P = 500,000 $
- Annual withdrawal: $ W = 85,000 $
- Annual interest rate: $ r = 9\% = 0.09 $

### Approach:

1. **Interest Compounding**: The interest rate is applied to the remaining balance each year.
2. **Annual Withdrawal**: Each year, $85,000 is withdrawn after the interest is added.

The formula to calculate the remaining balance $ B_n $ after $ n $ years is:
\[
B_{n+1} = B_n \times (1 + r) - W
\]

Starting with $ B_0 = 500,000 $, we want to find the number of years $ n $ such that $ B_n = 0 $.

### Solution Process:

We can set up the iterative calculation for each year to see how the balance decreases over time. However, for exact calculations, we can use the formula for the future value of an annuity considering withdrawals:

The formula for the remaining fund after $ n $ years is:

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

We want to solve for $ n $ when $ B_n = 0 $:

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

Rearranging this equation for $ n $:

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

\[
P \times (1 + r)^n \times r = W \times ((1 + r)^n - 1)
\]

\[
P \times r \times (1 + r)^n = W \times (1 + r)^n - W
\]

\[
(P \times r + W) \times (1 + r)^n = W
\]

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

Finally, solving for $ n $:

\[
n = \frac{\log\left(\frac{W}{P \times r + W}\right)}{\log(1 + r)}
\]

Substituting the values $ P = 500,000 $, $ W = 85,000 $, and $ r = 0.09 $:

\[
n = \frac{\log\left(\frac{85,000}{500,000 \times 0.09 + 85,000}\right)}{\log(1.09)}
\]

Let's calculate this value.

### Calculation

Let’s compute the exact number of years it takes for the fund to reduce to zero:

\[
n = \frac{\log\left(\frac{85,000}{45,000 + 85,000}\right)}{\log(1.09)}
\]

\[
n = \frac{\log\left(\frac{85,000}{130,000}\right)}{\log(1.09)}
\]

\[
n = \frac{\log(0.6538)}{\log(1.09)}
\]

\[
n \approx \frac{-0.1843}{0.0374} \approx 4.926
\]

### Conclusion:

The number of years it will take for the fund to reduce to zero is approximately **4.93 years**. Since we typically deal with whole years in financial planning, this suggests it will take **between 4 and 5 years**.

Would you like more details on this calculation or have any questions?

### Related Questions:

1. How would the number of years change if the interest rate were 5% instead of 9%?
2. What would be the remaining balance after 3 years?
3. How would annual withdrawals affect the fund if they increased by 5% each year?
4. What if the company decided to reduce the annual withdrawal to $50,000? How many years would the fund last?
5. How would compounding semi-annually instead of annually affect the number of years?

### Tip:
When calculating financial problems with interest rates, always check whether the interest compounds annually, semi-annually, quarterly, etc., as it affects the calculation significantly.

Calculate how long it will take for Demco Products' $500,000 equipment replacement fund to deplete with annual $85,000 withdrawals and a 9% interest rate. Understand the compound interest and annuity withdrawal formula used in financial planning.

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