Question

A machine depreciates at 10% p.a. for first two years and then 7% p.a. for the next three
years, depreciation being calculated by the diminishing value method. If the value of the
machine be Rs 10,000 initially find the average rate of depreciation and the depreciated
value at the end of the fifth year.
Ans) 8.21%. Solve and show me​

Answer 1

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In first case,

$V _{n_1 } =Rs. \: 10000 \\ r_{1} = 10\% \\ n_{1} = 2 \: years$

$\therefore \: \: V_{0_1} = V_{n_1}(1 - \frac{r_1}{100} ) {}^{n_{1}} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = 10000(1 - \frac{10}{100} ) {}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = 8100$

So, Value of the machine after 2 years, = Rs. 8100.

In the second case ,

$V _{n_2 } =Rs. \: 8100 \\ r_{2} = 7\% \\ n_{2} = 3 \: years$

$\therefore \: \: V_{0_2} = V_{n_2}(1 - \frac{r_2}{100} ) {}^{n_{2}} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = 8100(1 - \frac{7}{100} ) {}^{3} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: =6515.29b \\ \: \: \: \: \: \: \: \: \: \: \: \: \: =6515 \: \: (Approx.)$

Hence,

The depreciated value at the end of fifth years = Rs. 6515

Now,

$V _{n_1 } =Rs. \: 10000 \\ V_{0} =Rs. \: 6515 \\ n_{1} = 5 \: years$

Let ,

The rate of interest = r

$\therefore \: \: V_{0} = V_{n}(1 - \frac{r}{100} ) {}^{n} \\ = > 6515 = 10000(1 - \frac{r}{100} ) {}^{5} \\ = > r = 8.2\%$

So,

The average rate of depreciation = 8.2 %

And the depreciated value = Rs. 6515 at the end of the fifth year.

Answer 2

The average rate depreciation 7% p.a (approx)

Given:

A machine depreciates at 10% p.a. for first two years

and then 7% p.a. for the next three years  

The initial value of Machine = 10000

To find:

The average rate of depreciation and the depreciated value at the end of the fifth year

Solution:  

Machine rate after 2 years

The rate of depreciates  = 10% p.a for 2 years

The machine value after 2 years

= $10000(1-\frac{10}{100})^{2}$

= $10000(\frac{90}{100})^{2}$

= $10000(\frac{9}{10})(\frac{9}{10} )$ = 8100 Rs

Machine rate next 3 years

Rate of depreciation = 7% p.a for 3 years

Here initial machine value will be 8100

The machine value after 3 years

= $8100(1-\frac{7}{100})^{3}$

= $8100(\frac{93}{100})^{3}$

=  6,515.25

The final value of machine after 5 years = 6,515.32

The decreased value of machine = 10000 - 6,515.25 = 3484.75

The percentage of depreciation = $\frac{3484.75}{10000} (100)$ = 34.847%

Therefore,

The average rate depreciation =   34.847% / 5 = 7 (approx)

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