The value of machine depreciates every year at the rate of 20% of its value of the beginnning of the year (i.e the raete of depreciation is 20%). The machine eas purchased for Rs. 2,50,000 when new, and the srap value realised when sold was Rs 1,28,000. Find the numer of years that the machine was used.

Answer

To solve the problem of determining the number of years a machine was used based on its depreciation, we can use the formula for depreciation similar to the compound interest formula. Here’s a step-by-step solution:

### Step 1: Identify the Variables
- **Initial Value (P)**: The cost of the machine when new = Rs. 2,50,000
- **Scrap Value (A)**: The value when sold = Rs. 1,28,000
- **Rate of Depreciation (r)**: 20% per annum
- **Number of Years (n)**: This is what we need to find.

### Step 2: Write the Depreciation Formula
The formula for the value of the machine after n years, considering depreciation, can be expressed as:
\[ A = P \left(1 - \frac{r}{100}\right)^n \]
Substituting the known values:
\[ 1,28,000 = 2,50,000 \left(1 - \frac{20}{100}\right)^n \]

### Step 3: Simplify the Equation
Calculate \(1 - \frac{20}{100}\):
\[ 1 - 0.20 = 0.80 \]
So the equation becomes:
\[ 1,28,000 = 2,50,000 (0.80)^n \]

### Step 4: Divide Both Sides by 2,50,000
To isolate \((0.80)^n\), divide both sides by 2,50,000:
\[ \frac{1,28,000}{2,50,000} = (0.80)^n \]
Calculating the left side:
\[ \frac{1,28,000}{2,50,000} = 0.512 \]
Now the equation is:
\[ 0.512 = (0.80)^n \]

### Step 5: Rewrite 0.512 as a Power of 0.80
To solve for n, we need to express 0.512 as a power of 0.80. We can calculate:
\[ 0.80^3 = 0.512 \]
Thus, we can rewrite the equation as:
\[ 0.512 = (0.80)^3 \]
So we have:
\[ (0.80)^n = (0.80)^3 \]

### Step 6: Equate the Exponents
Since the bases are the same, we can equate the exponents:
\[ n = 3 \]

### Conclusion
The number of years the machine was used is **3 years**.

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