A machine depreciates at the rate 8% of the value at the beginning of a year. The machine was purchased for  ₹ 2,00,000 and the scrap value realized when sold was  ₹ 1,31,816.3. Find the number of years the machine was used.


Select one:
4
6
5
7

To solve this problem, we need to determine the number of years the machine was used given the depreciation rate, initial value, and final scrap value.

The depreciation rate is 8% per year, which means each year the machine's value is reduced by 8% of its value at the beginning of that year.

Let's use the formula for depreciation, which follows the pattern of exponential decay:

\[
V_n = P \times (1 - r)^n
\]

Where:
- \( V_n \) is the final value (scrap value) after \( n \) years.
- \( P \) is the initial value (₹ 2,00,000).
- \( r \) is the depreciation rate (0.08).
- \( n \) is the number of years.

Given:
- \( V_n = ₹ 1,31,816.3 \)
- \( P = ₹ 2,00,000 \)
- \( r = 0.08 \)

Substitute these values into the formula:

\[
1,31,816.3 = 2,00,000 \times (1 - 0.08)^n
\]

\[
1,31,816.3 = 2,00,000 \times (0.92)^n
\]

Now, divide both sides by 2,00,000:

\[
\frac{1,31,816.3}{2,00,000} = (0.92)^n
\]

\[
0.6590815 = (0.92)^n
\]

To solve for \( n \), we take the natural logarithm (ln) of both sides:

\[
\ln(0.6590815) = n \times \ln(0.92)
\]

\[
n = \frac{\ln(0.6590815)}{\ln(0.92)}
\]

Let's calculate \( n \):

\[
n \approx \frac{\ln(0.6590815)}{\ln(0.92)} \approx \frac{-0.416926}{-0.083382} \approx 5
\]

Thus, the number of years the machine was used is **5 years**.

So, the correct option is:

**5**.

Would you like more details, or do you have any questions?

Here are some related questions:
1. How does the depreciation rate affect the number of years?
2. What if the depreciation rate was 10% instead of 8%?
3. How would the calculation change if the initial value was different?
4. What is the difference between straight-line depreciation and the method used here?
5. How does exponential decay apply to real-world financial scenarios?

**Tip:** Understanding the formula for depreciation can help you in evaluating the long-term value of investments or machinery in various business contexts.

Learn how to calculate the number of years a machine was used based on depreciation. This problem involves exponential decay and the formula V_n = P * (1 - r)^n, where P is the initial value and r is the depreciation rate. Suitable for high school students studying finance or mathematics.

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