Question

A manufacturer reckons that the value of the machine, which cost him \[Rs.\ 15625\] , will depreciate each year by \[20\%\]. Find the estimated value at the end of \[5\] years.

Answer 1

Hint:In this question, we need to find an estimated value at the end of \[5\] years. Also given the value of the machine costs \[Rs.\ 15625\], will depreciate each year by \[20\%\]. There is a direct formula to find the depreciated value of the machine . Then we need to substitute all the known values in the formula to find the depreciated value. With the help of depreciation rate, we can easily find the depreciated value of the machine at the end of \[5\] year

Formula used:
\[A = P\left( 1 + \dfrac{r}{100} \right)^{n}\]
Where, \[P\] is the principal, \[r\] is the rate of interest, \[n\] is the number of years and \[A\] is the depreciated value.

Complete step by step answer:
Given, principal, \[P = Rs.15625\]
Rate of interest, \[r\% = - 20\%\]........(Since the rate of interest is depreciated)
Number of years , \[n = 5\]. Let \[A\] be the depreciated value of the machine at the end of \[5\] years.Now the relation among \[A,\ P,\ r\%\] and \[n\] is given by,
\[A = P\left( 1 + \dfrac{r}{100} \right)^{n}\]
Now on substituting all the known values, we get,
\[\Rightarrow \ A = 15625\left( 1 + \left( - \dfrac{20}{100} \right) \right)^{5}\]
On removing the parentheses, we get,
\[\Rightarrow \ A = 15625\left( 1 - \dfrac{20}{100} \right)^{5}\]

On simplifying, we get,
\[\Rightarrow \ A = 15625\left( 1 - \dfrac{1}{5} \right)^{5}\]
On taking LCM ,
We get,
\[\Rightarrow \ A = 15625\left( \dfrac{5 – 1}{5} \right)^{5}\]
Again on simplifying,
We get,
\[\Rightarrow \ A = 15625\left( \dfrac{4}{5} \right)^{5}\]
Now on expanding the powers,
We get,
\[\Rightarrow \ A = 15625 \times \dfrac{4}{5} \times \dfrac{4}{5} \times \dfrac{4}{5} \times \dfrac{4}{5} \times \dfrac{4}{5}\]

On multiplying ,
We get,
\[\Rightarrow \ A = \dfrac{15625 \times 1024}{3125}\]
On simplifying,
We get,
\[A = 5 \times 1024\]
On further simplifying,
We get,
\[\therefore A = 5120\]

Thus we get the depreciated value of the machine at the end of \[5\] years as \[Rs.\ 5120\].

Note:In order to solve these types of questions, we have to note that if the value is depreciated then the rate of interest will be negative. We need to be careful while calculating the depreciated value if asked for more years and also we can calculate the estimated value by calculating the depreciated value of a machine at the end of each year. Compound interest is nothing but the interest calculated for the principal and the interest accumulated over a period of years before.