A certain sum amounts to Rs. 900 at simple interest in 5 years. It amounts to Rs. 1020 at the same rate at simple interest in 7 years. Find the sum (in Rs.).

Options

A. 500
B. 550
C. 650
D. 600

Explanation

To solve the problem step by step, we will use the formula for simple interest and the information given in the question.

### Step 1: Understand the given information
We know:
- Amount after 5 years (A1) = Rs. 900
- Amount after 7 years (A2) = Rs. 1020
- Time for the first amount (t1) = 5 years
- Time for the second amount (t2) = 7 years

### Step 2: Set up the equations
Using the formula for the amount in simple interest:
\[ A = P + SI \]
where \( SI = \frac{P \times R \times T}{100} \)

We can express the amounts as:
1. For 5 years:
\[ 900 = P + \frac{P \times R \times 5}{100} \]
This can be rewritten as:
\[ 900 = P \left(1 + \frac{5R}{100}\right) \] (Equation 1)

2. For 7 years:
\[ 1020 = P + \frac{P \times R \times 7}{100} \]
This can be rewritten as:
\[ 1020 = P \left(1 + \frac{7R}{100}\right) \] (Equation 2)

### Step 3: Subtract the two equations
Now, we will subtract Equation 1 from Equation 2:
\[ 1020 - 900 = P \left(1 + \frac{7R}{100}\right) - P \left(1 + \frac{5R}{100}\right) \]

This simplifies to:
\[ 120 = P \left(\frac{7R}{100} - \frac{5R}{100}\right) \]
\[ 120 = P \left(\frac{2R}{100}\right) \]

### Step 4: Solve for \( \frac{PR}{100} \)
From the equation above:
\[ 120 = \frac{2PR}{100} \]
Dividing both sides by 2:
\[ 60 = \frac{PR}{100} \]
Thus, we have:
\[ PR = 6000 \] (Equation 3)

### Step 5: Substitute \( PR \) back into one of the original equations
Now we can substitute \( PR \) into Equation 1:
\[ 900 = P \left(1 + \frac{5 \times 6000}{100}\right) \]
Calculating \( \frac{5 \times 6000}{100} \):
\[ \frac{30000}{100} = 300 \]
So,
\[ 900 = P (1 + 300) \]
\[ 900 = P \times 301 \]

### Step 6: Solve for \( P \)
Now, we can solve for \( P \):
\[ P = \frac{900}{301} \]
Calculating this gives:
\[ P \approx 600 \]

### Final Answer
The sum (principal amount) is Rs. 600.

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