If the simple interest on a certain sum for 1.5 years at 8% p.a, is, Rs 360. Find the compound interest on the same sum for 5 the same rate of interest but double the time?

Options

A. Rs 778
B. Rs 778.13
C. Rs 779
D. Rs 779.13

Answer

To solve the problem step by step, we will first determine the principal amount using the given simple interest, and then we will calculate the compound interest for double the time at the same rate.

### Step 1: Calculate the Principal Amount
We know the formula for Simple Interest (SI):
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest (Rs 360)
- \(P\) = Principal amount (unknown)
- \(R\) = Rate of interest (8%)
- \(T\) = Time (1.5 years)

Substituting the known values into the formula:
\[ 360 = \frac{P \times 8 \times 1.5}{100} \]

### Step 2: Simplify the Equation
First, calculate \(8 \times 1.5\):
\[ 8 \times 1.5 = 12 \]
Now, substitute this back into the equation:
\[ 360 = \frac{P \times 12}{100} \]

### Step 3: Solve for Principal (P)
To isolate \(P\), multiply both sides by 100:
\[ 360 \times 100 = P \times 12 \]
\[ 36000 = P \times 12 \]
Now, divide both sides by 12:
\[ P = \frac{36000}{12} = 3000 \]

### Step 4: Calculate Compound Interest
Now we need to find the Compound Interest (CI) for the same principal amount over double the time (3 years) at the same rate (8%).

The formula for Compound Interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \(A\) = Amount after time \(T\)
- \(P\) = Principal (Rs 3000)
- \(R\) = Rate of interest (8%)
- \(T\) = Time (3 years)

Substituting the known values:
\[ A = 3000 \left(1 + \frac{8}{100}\right)^3 \]
\[ A = 3000 \left(1 + 0.08\right)^3 \]
\[ A = 3000 \left(1.08\right)^3 \]

### Step 5: Calculate \(1.08^3\)
Calculating \(1.08^3\):
\[ 1.08^3 \approx 1.259712 \]
Now substitute this back into the amount formula:
\[ A \approx 3000 \times 1.259712 \approx 3779.136 \]

### Step 6: Calculate Compound Interest (CI)
Now, we can find the Compound Interest using the formula:
\[ CI = A - P \]
Substituting the values we calculated:
\[ CI \approx 3779.136 - 3000 = 779.136 \]

### Final Answer
The compound interest on the same sum for 5 years at the same rate of interest is approximately Rs 779.136.

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