The value of an equipment depreciates by `20%` each year . If the difference between the prices at the end of `3^(rd) and 4^(th)` year is Rs. 3,328 then what is the price of the equipment at the end of the second year ?

Options

A. Rs. 20, 800
B. Rs. 18,600
C. Rs. 16,640
D. Rs. 17, 400

Answer

To solve the problem step by step, we will follow the depreciation formula and the given conditions.

### Step 1: Understand the depreciation
The equipment depreciates by 20% each year. This means that at the end of each year, the value of the equipment is 80% of its value at the beginning of that year.

### Step 2: Define the original price
Let the original price of the equipment be \( X \).

### Step 3: Calculate the price at the end of each year
- **End of 1st year:**
\[ \text{Price after 1 year} = X \times 0.8 \]

- **End of 2nd year:**
\[ \text{Price after 2 years} = (X \times 0.8) \times 0.8 = X \times 0.8^2 = X \times 0.64 \]

- **End of 3rd year:**
\[ \text{Price after 3 years} = (X \times 0.64) \times 0.8 = X \times 0.64 \times 0.8 = X \times 0.512 \]

- **End of 4th year:**
\[ \text{Price after 4 years} = (X \times 0.512) \times 0.8 = X \times 0.512 \times 0.8 = X \times 0.4096 \]

### Step 4: Set up the equation for the difference
According to the problem, the difference between the prices at the end of the 3rd and 4th years is Rs. 3,328. Therefore, we can write:
\[ \text{Price after 3 years} - \text{Price after 4 years} = 3328 \]
Substituting the values we calculated:
\[ X \times 0.512 - X \times 0.4096 = 3328 \]

### Step 5: Simplify the equation
Factor out \( X \):
\[ X \times (0.512 - 0.4096) = 3328 \]
Calculating the difference:
\[ 0.512 - 0.4096 = 0.1024 \]
Thus, we have:
\[ X \times 0.1024 = 3328 \]

### Step 6: Solve for \( X \)
Now, divide both sides by 0.1024 to find \( X \):
\[ X = \frac{3328}{0.1024} = 32,500 \]

### Step 7: Calculate the price at the end of the 2nd year
Now that we have \( X \), we can find the price at the end of the 2nd year:
\[ \text{Price after 2 years} = X \times 0.64 = 32,500 \times 0.64 = 20,800 \]

### Final Answer
The price of the equipment at the end of the second year is **Rs. 20,800**.

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