A sum of money, invested with simple interest, amounts to `₹23,400` in `5` years and to `₹26,640` in `8` years. Find the sum of money and the rate of interest.

Answer

To solve the problem step-by-step, we will first define the variables and then use the information provided to create equations based on the simple interest formula.

### Step 1: Define Variables
Let:
- \( P \) = Principal amount (the sum of money we need to find)
- \( R \) = Rate of interest per annum (in percentage)

### Step 2: Set Up the Equations
From the problem, we know:
1. The amount after 5 years is ₹23,400.
2. The amount after 8 years is ₹26,640.

Using the formula for simple interest:
\[ \text{Amount} = \text{Principal} + \text{Simple Interest} \]
The simple interest can be calculated as:
\[ \text{Simple Interest} = P \times \frac{R}{100} \times T \]
where \( T \) is the time in years.

#### Equation 1 (for 5 years):
\[ 23400 = P + P \times \frac{R}{100} \times 5 \]
This simplifies to:
\[ 23400 = P + \frac{5PR}{100} \]
Rearranging gives:
\[ 23400 - P = \frac{5PR}{100} \quad \text{(Equation 1)} \]

#### Equation 2 (for 8 years):
\[ 26640 = P + P \times \frac{R}{100} \times 8 \]
This simplifies to:
\[ 26640 = P + \frac{8PR}{100} \]
Rearranging gives:
\[ 26640 - P = \frac{8PR}{100} \quad \text{(Equation 2)} \]

### Step 3: Subtract Equation 1 from Equation 2
Now, we will subtract Equation 1 from Equation 2:
\[ (26640 - P) - (23400 - P) = \frac{8PR}{100} - \frac{5PR}{100} \]
This simplifies to:
\[ 26640 - 23400 = \frac{(8R - 5R)P}{100} \]
Calculating the left side:
\[ 3240 = \frac{3PR}{100} \]

### Step 4: Solve for \( PR \)
Rearranging gives:
\[ 3PR = 3240 \times 100 \]
\[ PR = \frac{324000}{3} = 108000 \quad \text{(Equation 3)} \]

### Step 5: Substitute \( PR \) into Equation 1
Now, we will substitute \( PR \) back into Equation 1:
\[ 23400 - P = \frac{5 \times 108000}{100} \]
Calculating the right side:
\[ 23400 - P = 5400 \]
Rearranging gives:
\[ P = 23400 - 5400 = 18000 \]

### Step 6: Find the Rate \( R \)
Now, substitute \( P \) back into Equation 3 to find \( R \):
\[ 18000R = 108000 \]
\[ R = \frac{108000}{18000} = 6 \]

### Final Answers
- Principal (Sum of Money) = ₹18,000
- Rate of Interest = 6%