Question

A machine depreciates by one fourthof its value each year. If it costs​ $510 comma 000 ​new, what is its value after 8 ​years?

Answer 1

To calculate the value of a machine that depreciates by one fourth of its value each year, we can use the formula for depreciation. The machine loses 25% of its value every year, which means it retains 75% of its value from the previous year.

Let's denote the initial value of the machine as \( V_0 = 510,000 \).

The value of the machine after \( t \) years, when it depreciates by 25% each year, can be calculated using the formula:

\[
V_t = V_0 \times (1 - d)^t
\]

where \( d \) is the depreciation rate (in this case, \( d = 0.25 \)), and \( t \) is the number of years.

Now, substituting the values:

\[
V_t = 510,000 \times (1 - 0.25)^8
\]
\[
= 510,000 \times (0.75)^8
\]

Now, we need to calculate \( (0.75)^8 \):

\[
(0.75)^8 \approx 0.100112915
\]

Now, multiply by the initial value:

\[
V_t = 510,000 \times 0.100112915 \approx 51,058.78
\]

Thus, the value of the machine after 8 years is approximately **$51,058.78**.