A sum of money in 3 years become 1344 and in 7 years it become Rs 1536 . What is the principal sum where simple. rate of interest is to be charged ?

Options

A. 4000
B. 1500
C. 1200
D. 2800

Answer

To find the principal sum when given the amounts after 3 years and 7 years under simple interest, we can follow these steps:

### Step 1: Define the Variables
Let the principal amount be \( P \) and the rate of interest be \( R \).

### Step 2: Write the Equations for Amounts
From the problem, we know:
- The amount after 3 years is \( A_3 = P + \text{SI}_3 = 1344 \)
- The amount after 7 years is \( A_7 = P + \text{SI}_7 = 1536 \)

### Step 3: Express Simple Interest
The simple interest for any time period can be calculated using the formula:
\[ \text{SI} = P \times \frac{R}{100} \times T \]
Where \( T \) is the time in years.

Thus, we can express the simple interest for 3 years and 7 years as:
- \( \text{SI}_3 = P \times \frac{R}{100} \times 3 \)
- \( \text{SI}_7 = P \times \frac{R}{100} \times 7 \)

### Step 4: Substitute Simple Interest into the Amount Equations
Substituting the expressions for simple interest into the amount equations:
1. \( P + P \times \frac{R}{100} \times 3 = 1344 \)
2. \( P + P \times \frac{R}{100} \times 7 = 1536 \)

### Step 5: Simplify the Equations
Rearranging both equations gives:
1. \( P(1 + \frac{3R}{100}) = 1344 \) (Equation 1)
2. \( P(1 + \frac{7R}{100}) = 1536 \) (Equation 2)

### Step 6: Subtract the Two Equations
Now, we can subtract Equation 1 from Equation 2:
\[ P(1 + \frac{7R}{100}) - P(1 + \frac{3R}{100}) = 1536 - 1344 \]
This simplifies to:
\[ P \left( \frac{7R}{100} - \frac{3R}{100} \right) = 192 \]
\[ P \left( \frac{4R}{100} \right) = 192 \]

### Step 7: Solve for \( P \)
From the equation above:
\[ P \cdot \frac{4R}{100} = 192 \]
\[ P = \frac{192 \times 100}{4R} \]
\[ P = \frac{19200}{4R} = \frac{4800}{R} \]

### Step 8: Calculate Simple Interest for 4 Years
The simple interest for 4 years can be calculated as:
\[ \text{SI}_4 = 192 \]
From this, we can find the simple interest for 1 year:
\[ \text{SI}_1 = \frac{192}{4} = 48 \]

### Step 9: Find Simple Interest for 3 Years
Now, we can find the simple interest for 3 years:
\[ \text{SI}_3 = 3 \times \text{SI}_1 = 3 \times 48 = 144 \]

### Step 10: Substitute Back to Find Principal
Substituting the value of \( \text{SI}_3 \) back into Equation 1:
\[ P + 144 = 1344 \]
\[ P = 1344 - 144 = 1200 \]

### Final Answer
The principal sum is \( \boxed{1200} \).

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