A sum borrowed at 6% p.a earns an interest which is `"1/3"^(rd)` of its principal in x years. Find x.
A. `5(5)/(9)`
B. `4(2)/(9)`
C. `6(3)/(7)`
D. `5(3)/(4)`

Options

A. D
B. B
C. C
D. A

Answer

To solve the problem, we need to find the time period \( x \) in which a sum of money borrowed at 6% per annum earns an interest that is one-third of its principal.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the principal amount be \( P \).
- The rate of interest is given as \( 6\% \) per annum.
- The simple interest (SI) earned is \( \frac{1}{3} P \).

2. **Use the Simple Interest Formula:**
The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where \( R \) is the rate of interest and \( T \) is the time in years.

3. **Substitute the Known Values:**
We know that:
- SI = \( \frac{1}{3} P \)
- \( R = 6 \)
- \( T = x \)

Substituting these into the formula gives:
\[ \frac{1}{3} P = \frac{P \times 6 \times x}{100} \]

4. **Cancel the Principal \( P \):**
Since \( P \) is common on both sides and \( P \neq 0 \), we can cancel it out:
\[ \frac{1}{3} = \frac{6x}{100} \]

5. **Cross Multiply to Solve for \( x \):**
Cross multiplying gives:
\[ 100 \cdot \frac{1}{3} = 6x \]
This simplifies to:
\[ \frac{100}{3} = 6x \]

6. **Isolate \( x \):**
To find \( x \), divide both sides by 6:
\[ x = \frac{100}{3 \times 6} = \frac{100}{18} = \frac{50}{9} \]

7. **Convert to Mixed Number:**
To convert \( \frac{50}{9} \) to a mixed number:
- Divide 50 by 9, which gives 5 with a remainder of 5.
- Thus, \( \frac{50}{9} = 5 \frac{5}{9} \).

### Final Answer:
The value of \( x \) is \( 5 \frac{5}{9} \).