The price of an article increases by `20%` every year. If the difference between the price of the article at the end of the third year and at the end of the fourth year is Rs. 324, what is the price of the article at the end of the second year?

Options

A. Rs. 1,350
B. Rs. 1,180
C. Rs. 1,260
D. Rs. 1,250

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the initial price of the article
Let the initial price of the article be \( X \).

### Step 2: Calculate the price at the end of each year
The price of the article increases by 20% each year. Therefore:
- At the end of the first year:
\[ \text{Price after 1 year} = X \times 1.2 \]
- At the end of the second year:
\[ \text{Price after 2 years} = X \times 1.2^2 = X \times 1.44 \]
- At the end of the third year:
\[ \text{Price after 3 years} = X \times 1.2^3 = X \times 1.728 \]
- At the end of the fourth year:
\[ \text{Price after 4 years} = X \times 1.2^4 = X \times 2.0736 \]

### Step 3: Set up the equation for the difference in prices
According to the problem, the difference between the price at the end of the fourth year and the price at the end of the third year is Rs. 324. Therefore, we can write:
\[ (X \times 1.2^4) - (X \times 1.2^3) = 324 \]

### Step 4: Factor out the common terms
Factoring out \( X \times 1.2^3 \) from the left-hand side gives:
\[ X \times 1.2^3 \times (1.2 - 1) = 324 \]
This simplifies to:
\[ X \times 1.2^3 \times 0.2 = 324 \]

### Step 5: Solve for \( X \)
Now, we can rearrange the equation to solve for \( X \):
\[ X \times 1.728 \times 0.2 = 324 \]
\[ X \times 0.3456 = 324 \]
\[ X = \frac{324}{0.3456} = 936 \]

### Step 6: Calculate the price at the end of the second year
Now that we have \( X \), we can find the price at the end of the second year:
\[ \text{Price after 2 years} = 936 \times 1.44 = 1345.44 \]

### Final Answer
Thus, the price of the article at the end of the second year is Rs. 1345.44.

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