# Question on arithmetic (Percentages)

A machine depreciates in value each year at the rate of 10% of its previous value. However every second year there is some maintenance work so that in that particular year, depreciation is only 5% of its previous value. If at the end of fourth year, the value of the machine stands at Rs.146,205, then find the value of the machine at the start of the first year.

I have looked up a few solution in the internet which says depreciation will be 10%-5%-10%-5% in the respective years. I cannot understand why this is the case.

Depreciation:

1st year= 10%

2nd year= 5% of (-10-10+ $\frac{10*10}{100}$ ) by succesive depreciation formula.

I cant uncerstand why this is equal to 5% . This will be equal to 5% only when the term to the right of 'of' is 100.

Where have I gone wrong. Also please show the calculation of the last two years as well.

• Initial value = A. After one year, value = 0.9A. After two years, the value is (0.95)(0.9)A = 0.855A. After third year, value = (0.9)(0.95)(0.9)A, and after 4th year, value = (0.95)(0.9)(0.95)(0.9)A = 0.731025A. If the value after 4 years is RS 146,205, then the initial value was 200,000 (=146,205/0.731025). – Andrew Gray Jun 29 '17 at 7:36

You start from a initial value $X_0$.

End first year value $X_1=(1-10\%)X_0$.

End second year value $X_2=(1-5\%)X_1$.

End third year value $X_3=(1-10\%)X_2$.

End fourth year value $X_4=(1-5\%)X_3$, i.e., $$146,205=X_4=(1-5\%)^2(1-10\%)^2X_0.$$

Therefore $$X_0=\frac{146,205}{(1-5\%)^2(1-10\%)^2}.$$

• Why is the 2nd year value decreasing by 5% and why is the third year value decreasing by 10%? – Soumee Jun 29 '17 at 7:43
• Because it is written in the text: depreciation is always $10\%$, except at "even" years which is only $5\%$. – Paolo Leonetti Jun 29 '17 at 7:44
• Ok....I thought for the 2nd year depreciation is 5% of 10%. Got wrong in understanding the language of the question. – Soumee Jun 29 '17 at 7:47

When the value depreciates $10\%$, the machine is worth $90\%=0.9$ of its old cost.

When the value depreciates $5\%$, the marching is worth $95\%=0.95$ of its old cost.

We have that $P(0.9)(0.95)(0.9)(0.95)=146205$.

The original value was thus $\boxed{200,000}$ Rs