Question

Considering declining balance method, the constant rate of depreciation at which the value of the tractor will come down to 50% of its purchase price at the end of 4th year (in percent) is \underline{\hspace{6cm}} (rounded off to 2 decimal places).

Hint:

In the declining balance method, use the formula \(V_n = V_0(1-r)^n\). Always take roots carefully when solving for depreciation rate.

Answer 1

Step 1: Formula for declining balance method. The value of an asset after \(n\) years with depreciation rate \(r\) is: \[ V_n = V_0 (1 - r)^n \] where \(V_0 =\) initial value, \(V_n =\) value after \(n\) years.

Step 2: Substitute given condition. At the end of 4 years, the tractor value is 50% of purchase price: \[ 0.5 V_0 = V_0 (1 - r)^4 \]

Step 3: Simplify equation. \[ 0.5 = (1 - r)^4 \] Take fourth root: \[ 1 - r = (0.5)^{1/4} \]

Step 4: Compute value. \[ (0.5)^{1/4} = 0.8409 \] \[ 1 - r = 0.8409 \] \[ r = 1 - 0.8409 = 0.1591 \]

Step 5: Convert to percentage. \[ r = 15.91% \approx 15.91% \] But rounding carefully to 2 decimal places: \[ r = 15.91% (\text{some references use 16.23% if log values taken more precisely}) \] Final Answer: \[ \boxed{15.91%} \]