A machine depreciates at the rate 7.99% of the value at the beginning of a year. The machine was purchased for  ₹ 25,00,000 and the scrap value realized when sold was  ₹ 10,00,000. Find the number of years the machine was used.

Select one:
12
9
11
10

We can solve this problem using the formula for depreciation, which follows an exponential decay pattern:

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

Where:
- \( V_n \) is the final value (scrap value) after \( n \) years.
- \( P \) is the initial value (₹ 25,00,000).
- \( r \) is the depreciation rate (7.99% or 0.0799).
- \( n \) is the number of years.

Given:
- \( V_n = ₹ 10,00,000 \)
- \( P = ₹ 25,00,000 \)
- \( r = 0.0799 \)

Substitute the values into the formula:

\[
10,00,000 = 25,00,000 \times (1 - 0.0799)^n
\]

\[
10,00,000 = 25,00,000 \times (0.9201)^n
\]

Now, divide both sides by 25,00,000:

\[
\frac{10,00,000}{25,00,000} = (0.9201)^n
\]

\[
0.4 = (0.9201)^n
\]

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

\[
\ln(0.4) = n \times \ln(0.9201)
\]

\[
n = \frac{\ln(0.4)}{\ln(0.9201)}
\]

Let's calculate \( n \):

\[
n \approx \frac{\ln(0.4)}{\ln(0.9201)} \approx \frac{-0.9162907}{-0.0830806} \approx 11.03
\]

Thus, the closest whole number is **11** years.

So, the correct option is:

**11**.

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

Here are some related questions:
1. How does a small change in the depreciation rate affect the number of years?
2. What is the impact on the scrap value if the initial purchase price changes?
3. Can this method of depreciation be applied to other assets?
4. How does this type of depreciation compare with other methods, like straight-line depreciation?
5. What would be the number of years if the scrap value was ₹5,00,000 instead?

**Tip:** When dealing with depreciation, always round to the nearest whole year to match real-world accounting practices.

Learn how to calculate the number of years a machine was used based on depreciation rate. Detailed step-by-step solution using exponential decay and logarithms, with initial value ₹25,00,000 and scrap value ₹10,00,000.

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