A certain sum amounts to Rs. 3,825 in 4 years and to Rs. 4,050 in 6 years. Find the rate percent and the sum.

Answer

To solve the problem step by step, we will use the information provided in the question to find the principal amount (P) and the rate of interest (R).

### Step 1: Understand the Given Information
We know:
- Amount after 4 years (A1) = Rs. 3,825
- Amount after 6 years (A2) = Rs. 4,050
- Time for A1 (T1) = 4 years
- Time for A2 (T2) = 6 years

### Step 2: Write the Formula for Amount
The formula for the amount in simple interest is:
\[ A = P + I \]
Where \( I \) (Interest) is given by:
\[ I = \frac{P \times R \times T}{100} \]
Thus, we can rewrite the amount formula as:
\[ A = P + \frac{P \times R \times T}{100} \]
This can be simplified to:
\[ A = P \left(1 + \frac{R \times T}{100}\right) \]

### Step 3: Set Up the Equations
For the first case (4 years):
\[ 3825 = P \left(1 + \frac{R \times 4}{100}\right) \] (Equation 1)

For the second case (6 years):
\[ 4050 = P \left(1 + \frac{R \times 6}{100}\right) \] (Equation 2)

### Step 4: Express Both Equations
From Equation 1:
\[ 3825 = P + \frac{4PR}{100} \]
From Equation 2:
\[ 4050 = P + \frac{6PR}{100} \]

### Step 5: Subtract the Two Equations
Subtract Equation 1 from Equation 2:
\[ 4050 - 3825 = \left(P + \frac{6PR}{100}\right) - \left(P + \frac{4PR}{100}\right) \]

This simplifies to:
\[ 225 = \frac{6PR}{100} - \frac{4PR}{100} \]
\[ 225 = \frac{2PR}{100} \]

### Step 6: Solve for PR
Multiply both sides by 100:
\[ 22500 = 2PR \]
Now divide by 2:
\[ PR = 11250 \] (Equation 3)

### Step 7: Substitute PR Back into One of the Original Equations
Using Equation 1:
\[ 3825 = P + \frac{4 \times 11250}{100} \]
Calculate \( \frac{4 \times 11250}{100} \):
\[ \frac{45000}{100} = 450 \]

Now substitute back:
\[ 3825 = P + 450 \]
Thus,
\[ P = 3825 - 450 \]
\[ P = 3375 \]

### Step 8: Find the Rate of Interest (R)
Using Equation 3:
\[ PR = 11250 \]
Substituting \( P = 3375 \):
\[ 3375R = 11250 \]
Now solve for \( R \):
\[ R = \frac{11250}{3375} \]
Calculating this gives:
\[ R = 3.33 \text{ (approximately)} \]

### Final Results
- Principal (P) = Rs. 3,375
- Rate of Interest (R) = 3.33%