The simple interest on a certain sum of money is `(3)/(16)` of the principal and the number of months is equal to the rate per cent. Find the rate per annum and the time period.

Answer

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

### Step 1: Understand the given information
We know that:
- The simple interest (SI) is \( \frac{3}{16} \) of the principal (P).
- The number of months (T) is equal to the rate of interest (R) in percent.

### Step 2: Set up the equation for Simple Interest
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
According to the problem, we can express the simple interest as:
\[ SI = \frac{3}{16} P \]

### Step 3: Substitute the values into the formula
Since we know that \( SI = \frac{3}{16} P \), we can substitute this into the simple interest formula:
\[ \frac{3}{16} P = \frac{P \times R \times T}{100} \]

### Step 4: Cancel out P from both sides
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{3}{16} = \frac{R \times T}{100} \]

### Step 5: Substitute T with R
Since the number of months is equal to the rate of interest (R), we can replace T with R in the equation:
\[ \frac{3}{16} = \frac{R \times R}{100} \]
This simplifies to:
\[ \frac{3}{16} = \frac{R^2}{100} \]

### Step 6: Cross-multiply to solve for R
Cross-multiplying gives us:
\[ 3 \times 100 = 16 \times R^2 \]
This simplifies to:
\[ 300 = 16R^2 \]

### Step 7: Solve for R^2
Now, divide both sides by 16:
\[ R^2 = \frac{300}{16} \]
This simplifies to:
\[ R^2 = 18.75 \]

### Step 8: Take the square root to find R
Now, take the square root of both sides:
\[ R = \sqrt{18.75} \]
Calculating this gives:
\[ R \approx 4.33 \]

### Step 9: Determine the time period in months
Since \( T = R \), the time period in months is approximately:
\[ T \approx 4.33 \text{ months} \]

### Step 10: Convert months to years and months
To express this in years and months:
- 4 months and approximately 10 days (since 0.33 of a month is about 10 days).

### Final Answer:
- Rate per annum (R) is approximately \( 4.33\% \).
- Time period (T) is approximately \( 4 \) months.

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