Divide Rs 7053 into three parts so that the amount after 2, 3 and 4 years respectively may be equal, the rates of interest being 4% per annum.

Options

A. Rs 2500, Rs 3500, Rs 1053
B. Rs 2432, Rs 2346, Rs 2265
C. Rs 2568, Rs 3200, Rs 1285
D. Rs 2360, Rs 2289, Rs 2404

Answer

To solve the problem of dividing Rs 7053 into three parts such that the amounts after 2, 3, and 4 years at 4% per annum are equal, we can follow these steps:

### Step 1: Define the Parts
Let the three parts be:
- \( P_1 \) (the first part)
- \( P_2 \) (the second part)
- \( P_3 \) (the third part)

According to the problem, we have:
\[ P_1 + P_2 + P_3 = 7053 \]

### Step 2: Calculate the Amounts
The amounts after 2, 3, and 4 years can be calculated using the formula for Simple Interest:
\[ A = P + SI \]
Where \( SI = \frac{P \times r \times t}{100} \)

1. For \( P_1 \) after 2 years:
\[ A_1 = P_1 + \frac{P_1 \times 4 \times 2}{100} = P_1 \left(1 + \frac{8}{100}\right) = P_1 \times 1.08 \]

2. For \( P_2 \) after 3 years:
\[ A_2 = P_2 + \frac{P_2 \times 4 \times 3}{100} = P_2 \left(1 + \frac{12}{100}\right) = P_2 \times 1.12 \]

3. For \( P_3 \) after 4 years:
\[ A_3 = P_3 + \frac{P_3 \times 4 \times 4}{100} = P_3 \left(1 + \frac{16}{100}\right) = P_3 \times 1.16 \]

### Step 3: Set the Amounts Equal
Since the amounts are equal:
\[ P_1 \times 1.08 = P_2 \times 1.12 = P_3 \times 1.16 \]

### Step 4: Express \( P_2 \) and \( P_3 \) in terms of \( P_1 \)
From the equality:
1. From \( P_1 \times 1.08 = P_2 \times 1.12 \):
\[ P_2 = \frac{P_1 \times 1.08}{1.12} = P_1 \times 0.9643 \quad (\text{approximately}) \]

2. From \( P_1 \times 1.08 = P_3 \times 1.16 \):
\[ P_3 = \frac{P_1 \times 1.08}{1.16} = P_1 \times 0.9310 \quad (\text{approximately}) \]

### Step 5: Substitute into the Total
Now substitute \( P_2 \) and \( P_3 \) into the total:
\[ P_1 + P_1 \times 0.9643 + P_1 \times 0.9310 = 7053 \]
\[ P_1 (1 + 0.9643 + 0.9310) = 7053 \]
\[ P_1 \times 2.8953 = 7053 \]
\[ P_1 = \frac{7053}{2.8953} \approx 2435 \]

### Step 6: Calculate \( P_2 \) and \( P_3 \)
Now we can find \( P_2 \) and \( P_3 \):
1. \( P_2 \approx 2435 \times 0.9643 \approx 2347 \)
2. \( P_3 \approx 2435 \times 0.9310 \approx 2269 \)

### Final Parts
Thus, the three parts are approximately:
- \( P_1 \approx 2435 \)
- \( P_2 \approx 2347 \)
- \( P_3 \approx 2269 \)

### Summary
The three parts into which Rs 7053 is divided are approximately Rs 2435, Rs 2347, and Rs 2269.

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