Simple interest on a certain amount is `(9)/(16)` of the principal. If the numbers representing the rate of interest in percent and time in years be equal, then time, for which the principal is lent out, is

Options

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

Answer

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

### Step 1: Understand the Problem
We know that the simple interest (SI) is \( \frac{9}{16} \) of the principal (P). We also know that the rate of interest (R) in percent and the time (T) in years are equal.

### Step 2: Set Up the Equations
Let:
- Principal = \( P \)
- Rate of Interest = \( R = x \) (in percent)
- Time = \( T = x \) (in years)

From the problem, we have:
\[ SI = \frac{9}{16} P \]

The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]

Substituting the values of \( R \) and \( T \):
\[ SI = \frac{P \times x \times x}{100} = \frac{P \times x^2}{100} \]

### Step 3: Set the Two Expressions for SI Equal
Now we can set the two expressions for SI equal to each other:
\[ \frac{9}{16} P = \frac{P \times x^2}{100} \]

### Step 4: Cancel Out the Principal
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{9}{16} = \frac{x^2}{100} \]

### Step 5: Cross Multiply to Solve for \( x^2 \)
Cross multiplying gives us:
\[ 9 \times 100 = 16 \times x^2 \]
\[ 900 = 16x^2 \]

### Step 6: Solve for \( x^2 \)
Now, divide both sides by 16:
\[ x^2 = \frac{900}{16} = 56.25 \]

### Step 7: Take the Square Root
Taking the square root of both sides:
\[ x = \sqrt{56.25} = 7.5 \]

### Step 8: Conclusion
Since \( x \) represents both the rate of interest and the time in years, we conclude:
\[ \text{Time (T)} = 7.5 \text{ years} \]

### Final Answer
The time for which the principal is lent out is **7.5 years**.

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