To determine how long it will take for the value of an investment to depreciate to a quarter of its original value, given a depreciation rate of 8.2% per annum on the reducing balance method, follow these steps:
1. Understand the Problem:
- We need to find the time `t` in years.
- The original value of the investment can be considered as 1 (normalized value).
- The final value will be 0.25 of the original value (quarter of the original value).
- The depreciation rate is 8.2% per annum.
2. Express in Mathematical Terms:
- The problem involves exponential decay, described by the formula:
[tex]\[
V_t = V_0 \times (1 - r)^t
\][/tex]
- Where:
- [tex]\( V_t \)[/tex] is the final value.
- [tex]\( V_0 \)[/tex] is the original value.
- [tex]\( r \)[/tex] is the annual depreciation rate.
- [tex]\( t \)[/tex] is the time in years.
3. Substitute Known Values:
- Let [tex]\( V_0 = 1 \)[/tex]
- Let [tex]\( V_t = 0.25 \)[/tex]
- Let [tex]\( r = 0.082 \)[/tex] (8.2% expressed as a decimal)
- The equation becomes:
[tex]\[
0.25 = 1 \times (1 - 0.082)^t
\][/tex]
- Simplifies to:
[tex]\[
0.25 = (0.918)^t
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
- Take the natural logarithm (ln) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
\ln(0.25) = t \times \ln(0.918)
\][/tex]
[tex]\[
t = \frac{\ln(0.25)}{\ln(0.918)}
\][/tex]
5. Find and Round the Value:
- Calculating the natural logarithms and solving for [tex]\( t \)[/tex], we find:
[tex]\[
t \approx 16.203
\][/tex]
6. Round to the Nearest Year:
- The nearest whole number is 16 years.
So, it will take approximately 16 years for the value of the investment to depreciate to a quarter of its original value based on an annual depreciation rate of 8.2%.
