A certain amount of sum is invested at simple interest. If the sum becomes k times itself in 16 years and 2k times itself in 40 years, in how many year will it become 4k times of itself?

Options

A. 96 years
B. 88 years
C. 80 years
D. 64 years

Answer

To solve the problem, we need to analyze the information given about the investment and the simple interest.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a principal amount \( P \).
- After 16 years, the amount becomes \( kP \).
- After 40 years, the amount becomes \( 2kP \).

2. **Finding the Simple Interest Rate**:
- The formula for the amount \( A \) in simple interest is given by:
\[ A = P + SI \]
- Where \( SI \) (Simple Interest) can be calculated as:
\[ SI = P \cdot r \cdot t \]
- Here, \( r \) is the rate of interest (in decimal) and \( t \) is the time in years.

3. **Setting Up the Equations**:
- From the first condition (after 16 years):
\[ kP = P + P \cdot r \cdot 16 \]
Simplifying this gives:
\[ kP - P = 16Pr \implies (k - 1)P = 16Pr \implies r = \frac{k - 1}{16} \]

- From the second condition (after 40 years):
\[ 2kP = P + P \cdot r \cdot 40 \]
Simplifying this gives:
\[ 2kP - P = 40Pr \implies (2k - 1)P = 40Pr \implies r = \frac{2k - 1}{40} \]

4. **Equating the Rates**:
- Since both expressions represent the same rate \( r \), we can set them equal to each other:
\[ \frac{k - 1}{16} = \frac{2k - 1}{40} \]

5. **Cross Multiplying**:
- Cross multiplying gives:
\[ 40(k - 1) = 16(2k - 1) \]
Expanding both sides:
\[ 40k - 40 = 32k - 16 \]

6. **Solving for \( k \)**:
- Rearranging the equation:
\[ 40k - 32k = 40 - 16 \implies 8k = 24 \implies k = 3 \]

7. **Finding the Time for \( 4k \)**:
- Now we know \( k = 3 \). We need to find the time when the amount becomes \( 4kP = 12P \).
- We already know that it takes 16 years to get \( kP \) and 24 years to get another \( kP \) (from \( kP \) to \( 2kP \)).
- Therefore, to get from \( 2kP \) to \( 3kP \) will take another 24 years.
- Finally, to reach \( 4kP \) from \( 3kP \) will take another 24 years.

8. **Total Time Calculation**:
- Total time to reach \( 4kP \):
\[ 40 \text{ years (to reach } 2kP) + 24 \text{ years (to reach } 3kP) + 24 \text{ years (to reach } 4kP) = 88 \text{ years} \]

### Final Answer:
The total time taken for the sum to become \( 4k \) times itself is **88 years**.