A certain sum of money becomes double of itself in 5 years at a rate of simple interset. In how many years will it become 16 times of itself at the same rate of simple interset ?

Options

A. 20
B. 60
C. 75
D. No option is correct.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know that a certain sum of money (let's denote it as \( P \)) becomes double in 5 years at a certain rate of simple interest. This means:
- Initial amount (Principal) = \( P \)
- Amount after 5 years = \( 2P \)
- Time = 5 years

### Step 2: Calculate the interest earned in 5 years
The formula for the amount in simple interest is:
\[ A = P + SI \]
Where \( SI \) is the simple interest earned. Since the amount after 5 years is \( 2P \), we can write:
\[ 2P = P + SI \]
This simplifies to:
\[ SI = 2P - P = P \]
Thus, the simple interest earned in 5 years is equal to the principal \( P \).

### Step 3: Calculate the annual interest
To find the annual interest, we divide the total interest earned over 5 years by the number of years:
\[ SI_{1 \text{ year}} = \frac{SI}{5} = \frac{P}{5} \]

### Step 4: Determine the time to become 16 times the principal
Now we want to find out how long it will take for the principal \( P \) to become \( 16P \). Using the same formula for the amount:
\[ A = P + SI \]
We need to find the time \( t \) when:
\[ 16P = P + SI \]
This simplifies to:
\[ SI = 16P - P = 15P \]

### Step 5: Relate the interest to time
Since we know the interest earned in one year is \( \frac{P}{5} \), we can express the total interest \( SI \) in terms of time:
\[ SI = \text{Annual Interest} \times t = \frac{P}{5} \times t \]
Setting this equal to the interest needed to reach \( 16P \):
\[ \frac{P}{5} \times t = 15P \]
Now, we can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{t}{5} = 15 \]

### Step 6: Solve for \( t \)
Multiplying both sides by 5 gives:
\[ t = 15 \times 5 = 75 \]

### Final Answer
Thus, it will take **75 years** for the sum of money to become 16 times itself at the same rate of simple interest.

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