At what per cent annual compound interest rate a certain sum amounts to its 27 times in 3 yr?

Options

A. 100
B. 150
C. 175
D. 200

Answer

To solve the problem of finding the annual compound interest rate at which a certain sum amounts to 27 times in 3 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the amount (A) after 3 years is 27 times the principal (P). Therefore, we can express this as:
\[ A = 27P \]

2. **Using the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where:
- \(A\) = Amount after time \(t\)
- \(P\) = Principal amount
- \(r\) = Rate of interest per annum
- \(t\) = Time in years

3. **Substituting the Known Values**:
We substitute \(A = 27P\) and \(t = 3\) into the formula:
\[ 27P = P \left(1 + \frac{r}{100}\right)^3 \]

4. **Dividing Both Sides by \(P\)**:
Assuming \(P \neq 0\), we can divide both sides by \(P\):
\[ 27 = \left(1 + \frac{r}{100}\right)^3 \]

5. **Taking the Cube Root**:
To isolate \(1 + \frac{r}{100}\), we take the cube root of both sides:
\[ 1 + \frac{r}{100} = 27^{\frac{1}{3}} \]
Since \(27 = 3^3\), we find:
\[ 27^{\frac{1}{3}} = 3 \]
Thus:
\[ 1 + \frac{r}{100} = 3 \]

6. **Solving for \(r\)**:
Now, we can solve for \(r\):
\[ \frac{r}{100} = 3 - 1 = 2 \]
Multiplying both sides by 100 gives:
\[ r = 200 \]

### Final Answer:
The annual compound interest rate is **200%**.