Question

The value of a machine worth 500000 is depreciating at the rate of 10% every year. In how many years will its value be reduced to 363400?

Answer 1

Hint: In this question the initial price and the final price of a machine is given and it is said its price is depreciating at the rate of 10% every year so we will find the total time after which the cost of the machine will depreciate to 363400.

Complete step-by-step answer:
The present value of a machine worth \[ = Rs500000 \]
The reduced value of a machine \[ = Rs364500 \]
The rate of depreciation \[r = 10 \% \]
Let the time after which the cost of machine depreciate to \[Rs364500 \] be \[n \] years
We know the depreciation of material is given by the formula
 \[\Rightarrow A = P{ \left( {1 - \dfrac{r}{{100}}} \right)^n} \]
So by substituting the values in the formula we can write
  \[364500 = 500000{ \left( {1 - \dfrac{{10}}{{100}}} \right)^n} \]
Hence by further solving this equation we can write
  \[
\Rightarrow \dfrac{{364500}}{{500000}} = { \left( {1 - \dfrac{{10}}{{100}}} \right)^n} \\
\Rightarrow \dfrac{{364500}}{{500000}} = { \left( { \dfrac{9}{{10}}} \right)^n} \\
\Rightarrow \dfrac{{729}}{{1000}} = { \left( { \dfrac{9}{{10}}} \right)^n} \\
\Rightarrow { \left( { \dfrac{9}{{10}}} \right)^3} = { \left( { \dfrac{9}{{10}}} \right)^n} \;
  \]
Now we will compare the base of both sides of the numbers and as we know if the bases are the same then the exponents must be equal, here we can see the number in LHS and RHS have the same base, hence we can write
 \[
\Rightarrow { \left( { \dfrac{9}{{10}}} \right)^3} = { \left( { \dfrac{9}{{10}}} \right)^n} \\
\Rightarrow n = 3 \;
  \]
Hence the time will be 3 years.
Therefore the time in years the value of the machine will reduce to Rs.363400 will be equal to 3 years.
So, the correct answer is “3 years”.

Note: Another method to solve this question is by finding the depreciating cost of the machine every year and the year after which the cost of the machine will be equal to 363400 will be the total time after which the value of the machine depreciated.