In how many years will the interest on a certain sum of money at 5% p.a. be three-eights of itself? (Express your answer in years and months)

Options

A. 8 years 5 months
B. 7 years 6 months
C. 7 years 5 months
D. 8 years 6 months

Answer

To solve the problem, we need to find out how many years it will take for the interest on a certain sum of money at 5% per annum to be three-eighths of the principal amount itself.

Let's break it down step by step:

### Step 1: Understand the relationship between Principal, Interest, and Time
The formula for Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Set up the equation
According to the problem, we want the interest to be three-eighths of the principal:
\[ \text{SI} = \frac{3}{8} P \]

### Step 3: Substitute the values into the formula
We know that the rate \( R = 5\% \). Substituting this into the SI formula gives us:
\[ \frac{3}{8} P = \frac{P \times 5 \times T}{100} \]

### Step 4: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{3}{8} = \frac{5T}{100} \]

### Step 5: Cross-multiply to solve for \( T \)
Cross-multiplying gives:
\[ 3 \times 100 = 5T \times 8 \]
\[ 300 = 40T \]

### Step 6: Solve for \( T \)
Now, divide both sides by 40:
\[ T = \frac{300}{40} = 7.5 \text{ years} \]

### Step 7: Convert to years and months
The decimal 0.5 years is equivalent to 6 months. Therefore:
\[ T = 7 \text{ years and } 6 \text{ months} \]

### Final Answer
The interest on a certain sum of money at 5% p.a. will be three-eighths of itself in **7 years and 6 months**.

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