If certain sum is doubled in 8 yr on simple interest, in how many years will it be 4 times?

Options

A. 24
B. 16
C. 32
D. 12

Answer

To solve the problem, we need to determine how long it will take for a certain sum of money to quadruple (become 4 times) under simple interest, given that it doubles in 8 years.

### Step-by-Step Solution:

1. **Understand the Problem**: We know that a certain sum (let's denote it as P) doubles in 8 years under simple interest. This means that the final amount (A) after 8 years is 2P.

2. **Calculate the Simple Interest**: The formula for the amount in simple interest is:
\[ A = P + SI \]
where \( SI \) is the simple interest. Since the amount doubles, we can express this as:
\[ 2P = P + SI \]
This simplifies to:
\[ SI = 2P - P = P \]

3. **Determine the Rate of Interest**: The simple interest (SI) earned in 8 years is equal to the principal (P). The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( R \) is the rate of interest and \( T \) is the time in years. Substituting SI = P and T = 8 years, we have:
\[ P = \frac{P \times R \times 8}{100} \]
Dividing both sides by P (assuming P ≠ 0):
\[ 1 = \frac{R \times 8}{100} \]
Rearranging gives:
\[ R = \frac{100}{8} = 12.5\% \]

4. **Calculate the Time to Quadruple**: Now we want to find out how long it will take for the principal (P) to become 4P. The amount when it quadruples is:
\[ A = 4P \]
Using the simple interest formula again:
\[ 4P = P + SI \]
This means:
\[ SI = 4P - P = 3P \]
Using the simple interest formula:
\[ 3P = \frac{P \times R \times T}{100} \]
Substituting \( R = 12.5 \):
\[ 3P = \frac{P \times 12.5 \times T}{100} \]
Dividing both sides by P:
\[ 3 = \frac{12.5 \times T}{100} \]
Rearranging gives:
\[ T = \frac{3 \times 100}{12.5} = 24 \text{ years} \]

### Final Answer:
It will take **24 years** for the sum to become 4 times.