A sum of money becomes eight times of itself in 3 years at compound interest. The rate of interest per annum is

Options

A. 100%
B. 80%
C. 20%
D. 10%

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Problem
We need to find the rate of interest per annum when a sum of money becomes eight times itself in 3 years at compound interest.

### Step 2: Define the Variables
Let:
- Principal amount = P
- Amount after 3 years = 8P (since it becomes eight times itself)
- Rate of interest per annum = R
- Time = T = 3 years

### Step 3: Use the Compound Interest Formula
The formula for the amount A in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the known values:
\[ 8P = P \left(1 + \frac{R}{100}\right)^3 \]

### Step 4: Simplify the Equation
Since P is common on both sides, we can cancel it out (assuming P ≠ 0):
\[ 8 = \left(1 + \frac{R}{100}\right)^3 \]

### Step 5: Rewrite 8 as a Power of 2
We can express 8 as:
\[ 8 = 2^3 \]
Thus, we have:
\[ 2^3 = \left(1 + \frac{R}{100}\right)^3 \]

### Step 6: Equate the Bases
Since the bases are the same, we can equate the terms:
\[ 2 = 1 + \frac{R}{100} \]

### Step 7: Solve for R
Now, isolate R:
\[ \frac{R}{100} = 2 - 1 \]
\[ \frac{R}{100} = 1 \]
Multiplying both sides by 100 gives:
\[ R = 100 \]

### Conclusion
The rate of interest per annum is **100%**.

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