Question

A car which costs \[\$ 3,50,000\] depreciates by 10% every year. What will be the worth of the car after three years?
A. 2,55,150
B. 1,65,150
C. 2,25,350
D. 1,55,350

Answer 1

Hint: Here we will find the worth of the car after three years by using the concept of Compound Interest. We will substitute the value of the original price of the car, the rate at which the value is decreasing, and the time period in the compound interest formula. We will then solve it further to find the new cost of the car.

Formula used:
We will use the formula Compound Interest \[ = P{\left( {1 - \dfrac{R}{{100}}} \right)^n}\], where \[P\] is present value, \[R\] is the interest rate and \[n\] is the number of time.

Complete step-by-step answer:
The Actual amount and the rate of decrement is given as,
\[P = \$ 3,50,000\]
\[R = 10\% \]
It is given that we have to find the worth of a car after three years so, \[n = 3\].
Substituting all the above value in the formula Compound Interest \[ = P{\left( {1 - \dfrac{R}{{100}}} \right)^n}\], we get
Compound Interest \[ = 350000{\left( {1 - \dfrac{{10}}{{100}}} \right)^3}\]
Taking LCM inside the bracket, we get
\[ \Rightarrow \] Compound Interest \[ = 350000{\left( {\dfrac{{100 - 10}}{{100}}} \right)^3}\]
Subtracting the terms in the numerator inside the bracket, we get
\[ \Rightarrow \] Compound Interest \[ = 350000{\left( {\dfrac{{90}}{{100}}} \right)^3}\]
Simplifying the bracket term we get,
\[ \Rightarrow \] Compound Interest \[ = 350000{\left( {\dfrac{9}{{10}}} \right)^3}\]
Rewriting the equation, we get
\[ \Rightarrow \] Compound Interest \[ = 350000 \times \dfrac{{9 \times 9 \times 9}}{{10 \times 10 \times 10}}\]
Multiplying the terms, we get
\[ \Rightarrow \] Compound Interest\[ = \$ 2,55,150\]

So, the price of a car after three years is $2, 55,150.
Hence, option (A) is the correct answer.

Note: Compound Interest is used on the total interest plus original value of a product before it. There is another method to find the worth of a car after three years by finding the decrement in value of car each year. We will find the Interest in the car for each year as
Compound Interest \[ = \] Principal value \[ \times \] Interest Rate \[ \times \] Number of year
Substituting \[\$ 3,50,000\] for principal value, \[10\% \] for interest rate and 1 for number of years in the above equation, we get
Compound Interest for 1st year \[ = 350000 \times \dfrac{{10}}{{100}} \times 1\]
Multiplying the terms, we get
\[ \Rightarrow \] Compound Interest for 1st year\[ = \$ 35000\]
Now we will subtract the interest value to find the Principal amount for the 2nd year as the price is decreasing per year. Therefore,
Principal amount for 2nd year \[ = 350000 - 35000 = \$ 315000\]
Now we will find the interest for 2nd year.
Compound Interest for 2nd year\[ = 315000 \times \dfrac{{10}}{{100}} \times 1\]
Multiplying the terms, we get
\[ \Rightarrow \] Compound Interest for 2nd year\[ = \$ 31500\]
Now we will again subtract the interest value to find the Principal amount for the 3rd year. Therefore, we get
Principal value for 3rd year\[ = 315000 - 31500 = \$ 283500\]\[\]
Compound Interest for 3rd year \[ = 283500 \times \dfrac{{10}}{{100}} \times 1\]
Multiplying the terms, we get
\[ \Rightarrow \] Compound Interest for 3rd year \[ = \$ 28350\]
Now we will again subtract the interest value to find the value of the car after three years. Therefore, we get
Value of car after three year \[ = 283500 - 28350\]
\[ \Rightarrow \] Value of car after three year \[ = \$ 2,55,150\]
So, the price of a car after three years is $2, 55,150.