A sum of money becomes triple of itself in 8 yr at SI. In twelve years, it will become how many times of itself at the same rate?

Options

A. 6 times
B. 4 times
C. 5 times
D. none of the above

Answer

To solve the problem step by step, we will use the concept of Simple Interest (SI) and the relationship between time, principal, and interest.

### Step 1: Understand the problem
We know that a sum of money becomes triple of itself in 8 years at Simple Interest. This means that the final amount (A) is 3 times the principal (P) after 8 years.

**Hint:** Remember that the formula for Simple Interest is \( A = P + SI \) and \( SI = \frac{P \times r \times t}{100} \).

### Step 2: Set up the equation
From the information given:
\[ A = 3P \]
Since \( A = P + SI \), we can write:
\[ 3P = P + SI \]
This implies:
\[ SI = 3P - P = 2P \]

**Hint:** The interest earned (SI) is the difference between the total amount and the principal.

### Step 3: Calculate the rate of interest
We know that the interest earned in 8 years is \( SI = 2P \). Using the formula for Simple Interest:
\[ SI = \frac{P \times r \times t}{100} \]
Substituting \( SI \) and \( t \):
\[ 2P = \frac{P \times r \times 8}{100} \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = \frac{r \times 8}{100} \]
Now, solving for \( r \):
\[ r = \frac{2 \times 100}{8} = 25\% \]

**Hint:** The rate of interest can be calculated by isolating \( r \) in the equation.

### Step 4: Determine the amount after 12 years
Now, we need to find out how many times the principal will become after 12 years at the same rate of interest (25%).

Using the Simple Interest formula again:
\[ SI_{12} = \frac{P \times r \times t}{100} \]
Substituting \( r = 25\% \) and \( t = 12 \):
\[ SI_{12} = \frac{P \times 25 \times 12}{100} = \frac{300P}{100} = 3P \]

### Step 5: Calculate the total amount after 12 years
Now, the total amount after 12 years will be:
\[ A_{12} = P + SI_{12} = P + 3P = 4P \]

### Step 6: Find how many times of itself
To find how many times the principal \( P \) becomes:
\[ \text{Times} = \frac{A_{12}}{P} = \frac{4P}{P} = 4 \]

Thus, the sum of money will become **4 times** of itself in 12 years.

**Final Answer:** The correct option is B) 4 times.

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