A sum of money invested at simple interest triples itself in 8 years at simple interest. Find in how many years will it become 8 times itself at the same rate?

Options

A. 24 years
B. 28 years
C. 30 years
D. 21 years

Answer

To solve the problem step by step, we will first determine the rate of interest based on the information provided and then use that rate to find out how long it will take for the investment to become 8 times itself.

### Step 1: Understand the Problem
We know that a sum of money triples itself in 8 years. This means that if we let the principal amount (the initial sum of money) be \( P \), then the amount after 8 years is \( 3P \).

### Step 2: Calculate Simple Interest
The formula for simple interest is:
\[ \text{Simple Interest (SI)} = \text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)} / 100 \]
From the information given:
- Amount after 8 years = \( 3P \)
- Therefore, Simple Interest (SI) = Amount - Principal = \( 3P - P = 2P \)

### Step 3: Set Up the Equation
Using the simple interest formula:
\[ 2P = P \times R \times 8 / 100 \]
We can simplify this equation by dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = R \times 8 / 100 \]

### Step 4: Solve for Rate (R)
Rearranging the equation to find \( R \):
\[ R = \frac{2 \times 100}{8} = 25\% \]
Thus, the rate of interest is \( 25\% \).

### Step 5: Determine Time for Amount to Become 8 Times
Now we want to find out how long it will take for the principal to become 8 times itself. If the principal is \( P \), then the amount will be \( 8P \).

The Simple Interest needed to reach \( 8P \) is:
\[ \text{SI} = 8P - P = 7P \]

### Step 6: Set Up the New Equation
Using the simple interest formula again:
\[ 7P = P \times R \times T / 100 \]
Dividing both sides by \( P \):
\[ 7 = R \times T / 100 \]

### Step 7: Substitute the Rate (R)
We know \( R = 25\% \):
\[ 7 = 25 \times T / 100 \]

### Step 8: Solve for Time (T)
Rearranging to find \( T \):
\[ T = \frac{7 \times 100}{25} = 28 \]
Thus, it will take **28 years** for the investment to become 8 times itself.

### Final Answer
The answer is **28 years**.

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