A certain sum amounts to Rs. 4,600 after 5 years and to Rs. 6,000 after 8 years at the same rate of simple interest per annum. What will be the simple interest on a sum of Rs. 8,500 for `6 1/2` years at the same rate?

Options

A. Rs. 10,515
B. Rs. 10,215
C. Rs. 11,375
D. Rs. 11,460

Answer

To solve the problem step by step, we will first determine the principal amount (P) and the rate of interest (R) using the information given about the amounts after 5 and 8 years. Then, we will calculate the simple interest on Rs. 8,500 for 6.5 years at the same rate.

### Step 1: Set up the equations based on the information given.

From the problem, we know:
- Amount after 5 years = Rs. 4,600
- Amount after 8 years = Rs. 6,000

Using the formula for the amount in simple interest:
\[ \text{Amount} = \text{Principal} + \text{Simple Interest} \]
We can express this as:
1. \( 4600 = P + \frac{P \cdot R \cdot 5}{100} \) (Equation 1)
2. \( 6000 = P + \frac{P \cdot R \cdot 8}{100} \) (Equation 2)

### Step 2: Rearranging the equations.

From Equation 1:
\[ 4600 - P = \frac{P \cdot R \cdot 5}{100} \]
Multiplying both sides by 100:
\[ 100(4600 - P) = 5PR \]
\[ 460000 - 100P = 5PR \] (Equation 3)

From Equation 2:
\[ 6000 - P = \frac{P \cdot R \cdot 8}{100} \]
Multiplying both sides by 100:
\[ 100(6000 - P) = 8PR \]
\[ 600000 - 100P = 8PR \] (Equation 4)

### Step 3: Solve the two equations.

Now we have two equations (Equation 3 and Equation 4):
1. \( 460000 - 100P = 5PR \)
2. \( 600000 - 100P = 8PR \)

We can subtract Equation 3 from Equation 4:
\[ (600000 - 100P) - (460000 - 100P) = 8PR - 5PR \]
This simplifies to:
\[ 600000 - 460000 = 3PR \]
\[ 140000 = 3PR \]
Thus, we can express \( PR \) as:
\[ PR = \frac{140000}{3} \] (Equation 5)

### Step 4: Substitute back to find P and R.

Now we can substitute \( PR \) back into either Equation 3 or Equation 4. Let's use Equation 3:
\[ 460000 - 100P = 5 \left(\frac{140000}{3}\right) \]
\[ 460000 - 100P = \frac{700000}{3} \]
Multiplying everything by 3 to eliminate the fraction:
\[ 1380000 - 300P = 700000 \]
Rearranging gives:
\[ 1380000 - 700000 = 300P \]
\[ 680000 = 300P \]
Thus:
\[ P = \frac{680000}{300} = 2266.67 \]

### Step 5: Find the rate of interest (R).

Using Equation 5:
\[ PR = \frac{140000}{3} \]
Substituting \( P \):
\[ 2266.67 \cdot R = \frac{140000}{3} \]
\[ R = \frac{140000}{3 \cdot 2266.67} \]
Calculating gives:
\[ R \approx 20.6\% \]

### Step 6: Calculate the simple interest on Rs. 8,500 for 6.5 years.

Using the formula for simple interest:
\[ \text{SI} = \frac{P \cdot R \cdot T}{100} \]
Where:
- \( P = 8500 \)
- \( R = 20.6 \)
- \( T = 6.5 \)

Substituting the values:
\[ \text{SI} = \frac{8500 \cdot 20.6 \cdot 6.5}{100} \]
Calculating gives:
\[ \text{SI} = \frac{8500 \cdot 20.6 \cdot 6.5}{100} = 1123.1 \]

### Final Answer:
The simple interest on a sum of Rs. 8,500 for 6.5 years at the same rate is approximately Rs. 1,123.1.

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