Simple interest on a certain sum at a certain rate is `9/16` of the sum. If the number representing rate percent and time in years be equal, then the time is-

Options

A. `5 1/2`years
B. `6 1/2`years
C. `6 1/4`years
D. `7 1/2`years

Answer

To solve the problem step by step, we need to use the formula for Simple Interest (SI), which is given by:

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount (the sum)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 1: Understand the given information
We know that the Simple Interest is \( \frac{9}{16} \) of the sum (Principal amount, \( P \)). Therefore, we can write:

\[ SI = \frac{9}{16} P \]

### Step 2: Set up the equation using the SI formula
From the formula of Simple Interest, we can substitute \( SI \):

\[ \frac{9}{16} P = \frac{P \times R \times T}{100} \]

### Step 3: Cancel \( P \) from both sides
Assuming \( P \neq 0 \), we can divide both sides by \( P \):

\[ \frac{9}{16} = \frac{R \times T}{100} \]

### Step 4: Use the condition that rate and time are equal
According to the problem, the rate \( R \) and time \( T \) are equal. Let’s denote them both as \( T \):

\[ R = T \]

Substituting \( R \) with \( T \) in the equation gives:

\[ \frac{9}{16} = \frac{T \times T}{100} \]

### Step 5: Simplify the equation
This simplifies to:

\[ \frac{9}{16} = \frac{T^2}{100} \]

### Step 6: Cross-multiply to solve for \( T^2 \)
Cross-multiplying gives:

\[ 9 \times 100 = 16 \times T^2 \]

\[ 900 = 16 T^2 \]

### Step 7: Solve for \( T^2 \)
Now, divide both sides by 16:

\[ T^2 = \frac{900}{16} \]

### Step 8: Simplify the fraction
Simplifying \( \frac{900}{16} \):

\[ T^2 = 56.25 \]

### Step 9: Take the square root to find \( T \)
Taking the square root of both sides:

\[ T = \sqrt{56.25} \]

Calculating the square root:

\[ T = 7.5 \]

### Conclusion
Thus, the time \( T \) is \( 7.5 \) years, which can also be expressed as \( 7 \frac{1}{2} \) years.