The simple interest on a certain amount of money is 27% of the principal amount. The time for which the money is lent is one-third of the numerical value of the rate per cent per annum. What is the rate per cent?

Options

A. `15%`
B. `12%`
C. `9%`
D. `6%`

Answer

To solve the problem step by step, we will use the formula for simple interest and the information provided in the question.

### Step 1: Understand the given information
We know that:
- Simple Interest (SI) = 27% of Principal (P)
- Time (T) = (1/3) * Rate (R)

### Step 2: Set up the equations
Let the Principal (P) = 100 (for simplicity, we can assume any amount, but 100 makes calculations easier).
Then, Simple Interest (SI) = 27% of 100 = 27.

Now, we can express the time in terms of the rate:
Let Rate (R) = R% per annum.
Then, Time (T) = (1/3) * R.

### Step 3: Use the formula for Simple Interest
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 27 = \frac{100 \times R \times \left(\frac{1}{3}R\right)}{100} \]

### Step 4: Simplify the equation
The equation simplifies to:
\[ 27 = \frac{100 \times R \times \frac{1}{3}R}{100} \]
This simplifies further to:
\[ 27 = \frac{R^2}{3} \]

### Step 5: Solve for R
Multiplying both sides by 3 gives:
\[ 81 = R^2 \]
Taking the square root of both sides:
\[ R = \sqrt{81} \]
Thus,
\[ R = 9 \]

### Conclusion
The rate of interest is **9%**.