The value of a machine depreciated by 10% per year during the first two years and 15% per year during the third year. Express the total depreciation of the machine, as percent, during the three years.

Let initial value of machine be ₹ x.

For first year :

P = ₹ x

R = 10%

T = 1 year

Depreciation = $\dfrac{P \times R \times T}{100} = \dfrac{x \times 10 \times 1}{100} = \dfrac{x}{10}$.

Value at end of 1 year = P - Depreciation = $x - \dfrac{x}{10} = \dfrac{9x}{10}$.

For second year :

P = ₹ $\dfrac{9x}{10}$

R = 10%

T = 1 year

Depreciation = $\dfrac{P \times R \times T}{100} = \dfrac{\dfrac{9x}{10} \times 10 \times 1}{100} = \dfrac{9x}{100}$.

Value at end of second year = P - Depreciation

= $\dfrac{9x}{10} - \dfrac{9x}{100} = \dfrac{90x - 9x}{100} = \dfrac{81x}{100}$.

For third year :

P = ₹ $\dfrac{81x}{100}$

R = 15%

T = 1 year

Depreciation = $\dfrac{P \times R \times T}{100} = \dfrac{\dfrac{81x}{100} \times 15 \times 1}{100} = \dfrac{243x}{2000}$.

Value at end of third year = P - Depreciation

= $\dfrac{81x}{100} - \dfrac{243x}{2000} = \dfrac{1620x - 243x}{2000} = \dfrac{1377x}{2000}$.

Total depreciation = Initial value - Value at end of third year

$= x - \dfrac{1377x}{2000} \\[1em] = \dfrac{2000x - 1377x}{2000} \\[1em] = \dfrac{623x}{2000}.$

Percent depreciated

$\dfrac{\text{Total depreciation}}{\text{Initial value}} \times 100 \\[1em] = \dfrac{\dfrac{623x}{2000}}{x} \times 100 \\[1em] = \dfrac{623x}{x \times 2000} \times 100 \\[1em] = \dfrac{623}{20} \\[1em] = 31.15\%$

Hence, total depreciation of machine = 31.15%.