A machine depreciates at the rate 9.25% of the value at the beginning of a year. The machine was purchased for  ₹ 4,00,000 and the scrap value realized when sold was  ₹ 2,00,000. Find the number of years the machine was used.

Select one:
6.84
6.64
7.14
7.36

The problem requires us to determine the number of years the machine was used, given its depreciation rate, initial value, and scrap value. The depreciation follows a geometric progression, where each year's value is reduced by 9.25% of the value at the beginning of that year.

### Step-by-Step Solution:

1. **Initial Value (P):** ₹ 4,00,000
2. **Scrap Value (S):** ₹ 2,00,000
3. **Depreciation Rate (r):** 9.25% per year

The value of the machine at any year can be described using the formula for depreciation:

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

where:
- \(V_n\) is the value after \(n\) years.
- \(P\) is the initial value.
- \(r\) is the depreciation rate.
- \(n\) is the number of years.

Given that the scrap value is ₹ 2,00,000, we set up the equation:

\[
2,00,000 = 4,00,000 \times (1 - 0.0925)^n
\]

Simplify the equation:

\[
\frac{2,00,000}{4,00,000} = (1 - 0.0925)^n
\]

\[
\frac{1}{2} = (0.9075)^n
\]

Now, take the natural logarithm (log) on both sides:

\[
\ln\left(\frac{1}{2}\right) = n \times \ln(0.9075)
\]

Solve for \(n\):

\[
n = \frac{\ln\left(\frac{1}{2}\right)}{\ln(0.9075)}
\]

Calculating:

\[
n \approx \frac{-0.6931}{-0.0971} \approx 7.14 \text{ years}
\]

### Final Answer:
The machine was used for approximately **7.14 years**.

### Correct Option:
The correct option is **7.14**.

Would you like more details or have any questions?

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**Related Questions:**
1. What would be the value of the machine after 3 years?
2. How much would the machine be worth after 5 years if the depreciation rate was 8% instead of 9.25%?
3. If the machine's scrap value was ₹ 1,50,000, how many years would it have been used?
4. What would be the depreciation rate if the machine's value after 6 years was ₹ 2,50,000?
5. How does changing the depreciation rate to 10% affect the number of years?

**Tip:** Depreciation calculations are crucial in determining the lifespan and resale value of assets over time, especially for financial planning.

Learn how to calculate the number of years a machine was used based on its initial value, depreciation rate, and scrap value. This problem involves understanding geometric progression and exponential decay in financial calculations. Suitable for Grades 10-12.

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