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A carтАЩs value depreciates by 10% annually. If its present value is тВ╣486,000, what was its value 2 years ago?
тВ╣6,00,000
тВ╣5,80,000
тВ╣5,90,000
тВ╣5,40,000
тВ╣6,00,000
Use the ratio method: 10% depreciation means the value becomes 9/10th of the previous year. To go back 2 years, multiply by the inverse square: 486,000├Ч(10/9)├Ч(10/9).
Present value of the car = тВ╣486,000, rate of annual depreciation = 10%, time period = 2 years.
V=P├Ч(1тИТ100RтАЛ)n
Use the ratio method: 10% depreciation means the value becomes 9/10th of the previous year. To go back 2 years, multiply by the inverse square: 486,000├Ч(10/9)├Ч(10/9).
Students often calculate 10% of the current value and add it back twice instead of applying compound depreciation, which ignores the base value reduction.
Set up the equation
Let the original price be P. Since the depreciation is 10% annually, the value after 2 years is given by the formula V=P(1тИТ100RтАЛ)n. Substituting R=10 and n=2, we get the equation.
486,000=P├Ч(1тИТ10010тАЛ)2
Simplify the depreciation factor
Calculate the factor by which the value decreases: 1тИТ0.1=0.9. Squaring this gives 0.92=0.81.
486,000=P├Ч(0.9)2=P├Ч0.81
Calculate the initial value P
Isolate P by dividing 486,000 by 0.81. Dividing 486 by 81 gives 6, so 486,000/0.81=600,000.
P=0.81486,000тАЛ=600,000
A is correct because calculating the value two years ago using the depreciation formula 486,000/(0.9)2 results in тВ╣6,00,000.
This concept is identical to compound interest calculations where the interest rate is treated as a negative growth rate; mastering this helps in solving population growth/decline problems as well.