Question

The price of a car depreciates in the first year by \(25 \%\) in the second year by \(20 \%\) in the third year by \(15 \%\) and so on. The final price of the car after 3 years, if the present cost of the car 9 is Rs. \(10,00,000\) : (a) \(7,80,000\) (b) \(5,10,000\) (c) \(6,90,000\) (d) \(1,70,000\)

Short answer

Answer: (b) 5,10,000

Step-by-step solution

Find the initial price of the car

The present cost of the car is Rs. 10,00,000.

Calculate the first year depreciation

The price of the car depreciates by 25% in the first year. So, we need to find 25% of the initial price and then subtract it from the initial price. First-year depreciation = (25/100) * 10,00,000 = 2,50,000 Price after first year = 10,00,000 - 2,50,000 = 7,50,000

Calculate the second-year depreciation

In the second year, the car price depreciates by 20%. So, we need to find 20% of the price after the first year and then subtract it from the first year price. Second-year depreciation = (20/100) * 7,50,000 = 1,50,000 Price after the second year = 7,50,000 - 1,50,000 = 6,00,000

Calculate the third-year depreciation

In the third year, the car price depreciates by 15%. So, we need to find 15% of the price after the second year and then subtract it from the second-year price. Third-year depreciation = (15/100) * 6,00,000 = 90,000 Price after the third year = 6,00,000 - 90,000 = 5,10,000

Choose the correct answer

We have calculated the final price of the car after 3 years as Rs. 5,10,000. So, the correct answer is (b) 5,10,000.

Key concepts

Percentage Calculation

Percentage calculations are used to determine how much of a portion is taken away or added to an original amount. In the context of depreciation, a percentage represents how much value is lost from a product over time. For example, if a car depreciates by 25% in the first year, we calculate 25% of the car's initial price and subtract it from the original amount. The formula for finding the percentage of any number is: \[ \text{Percentage of a number} = \left( \frac{\text{Percentage}}{100} \right) \times \text{Number} \] For our car's depreciation, in the first year, we apply the formula as follows: - Take 25% of Rs. 10,00,000 by multiplying it by (25/100). - Subtract the resulting amount from Rs. 10,00,000 to find the new price of the car.Using this method consistently over multiple years gives us a clear picture of how its value decreases over time.

Financial Mathematics

Financial mathematics often involves understanding how values change over periods due to percentages, interest rates, or depreciation. One crucial concept is that percentages can build upon previous values rather than the original amount, particularly in depreciation. For car depreciation, this means that each year's percentage loss is calculated based on the start-of-the-year value, not the original purchase price. After knowing the price each subsequent year:

• In year one, calculate a 25% reduction.
• Use the reduced amount to then calculate a 20% reduction for year two.
• From year two's reduced amount, find a 15% downturn for year three.

This approach ensures that the depreciation reflects the car's changing value, providing a more accurate financial projection.

Cost Price Calculation

Calculating the cost price after multiple depreciations requires systematically applying percentage reductions sequentially. Start with the initial cost and successively apply each year’s depreciation percentage: 1. **Initial Price:** Begin with the car's original cost, which was Rs. 10,00,000. 2. **First Year:** Subtract 25% of the initial price to determine the price after year one. The calculation was Rs. 10,00,000 - Rs. 2,50,000 = Rs. 7,50,000. 3. **Second Year:** Apply a 20% reduction to the first year’s adjusted price (Rs. 7,50,000). The new price is Rs. 7,50,000 - Rs. 1,50,000 = Rs. 6,00,000. 4. **Third Year:** Finally, reduce the second year's price by 15%, resulting in Rs. 6,00,000 - Rs. 90,000 = Rs. 5,10,000. This step-by-step method ensures accurate calculation of the cost price after deductions.