A sum of money lent out at simple interest amounts to `₹1900` in `1` year and to `₹2800` in `4` year. Find the rate of interest and the sum of money.

Options

A. `"R= 13.75%, P= ₹ 1760"`
B. `"R= 18.75%, P= ₹ 1600"`
C. `"R= 13.75%, P= ₹ 1300"`
D. `"R= 19.75%, P= ₹ 1680"`

Answer

To solve the problem step by step, we will find the principal amount (P) and the rate of interest (R) using the information given about the amounts after 1 year and 4 years.

### Step 1: Define the Variables
Let:
- P = Principal amount (the sum of money)
- R = Rate of interest (in percentage)

### Step 2: Write the Equations for the Amounts
From the problem, we know:
1. After 1 year, the amount is ₹1900.
2. After 4 years, the amount is ₹2800.

Using the formula for Simple Interest:
- Amount (A) = Principal (P) + Simple Interest (SI)
- SI = (P * R * T) / 100

For 1 year:
\[ A_1 = P + SI_1 \]
\[ 1900 = P + \frac{P \cdot R \cdot 1}{100} \]
This simplifies to:
\[ 1900 = P + \frac{PR}{100} \]
Let this be Equation (1).

For 4 years:
\[ A_2 = P + SI_2 \]
\[ 2800 = P + \frac{P \cdot R \cdot 4}{100} \]
This simplifies to:
\[ 2800 = P + \frac{4PR}{100} \]
Let this be Equation (2).

### Step 3: Rearranging the Equations
From Equation (1):
\[ 1900 = P + \frac{PR}{100} \]
Rearranging gives:
\[ \frac{PR}{100} = 1900 - P \]
Let this be Equation (3).

From Equation (2):
\[ 2800 = P + \frac{4PR}{100} \]
Rearranging gives:
\[ \frac{4PR}{100} = 2800 - P \]
Let this be Equation (4).

### Step 4: Subtract Equation (3) from Equation (4)
Now, we will subtract Equation (3) from Equation (4):
\[ \frac{4PR}{100} - \frac{PR}{100} = (2800 - P) - (1900 - P) \]
This simplifies to:
\[ \frac{3PR}{100} = 900 \]

### Step 5: Solve for PR
Multiplying both sides by 100:
\[ 3PR = 90000 \]
Dividing by 3:
\[ PR = 30000 \]
Let this be Equation (5).

### Step 6: Substitute PR into Equation (3)
Now we will substitute the value of PR into Equation (3):
\[ \frac{30000}{100} = 1900 - P \]
This simplifies to:
\[ 300 = 1900 - P \]
Rearranging gives:
\[ P = 1900 - 300 \]
\[ P = 1600 \]

### Step 7: Find the Rate of Interest (R)
Now that we have P, we can find R using Equation (5):
\[ PR = 30000 \]
Substituting P:
\[ 1600R = 30000 \]
Dividing both sides by 1600:
\[ R = \frac{30000}{1600} \]
\[ R = 18.75 \]

### Final Answer
- Principal Amount (P) = ₹1600
- Rate of Interest (R) = 18.75%