A sum becomes 14 times of itself in 15 years at the rate of simple interest per annum. In how many years will the sum becomes 92 times of itself?

Options

A. 91 years
B. 195 years
C. 105 years
D. 210 years

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the Problem
We need to determine how many years it will take for a sum of money to become 92 times itself, given that it becomes 14 times itself in 15 years.

### Step 2: Define the Variables
Let's assume the principal amount (P) is 1 unit. According to the problem:
- After 15 years, the amount (A) becomes 14 times the principal.
- So, A = 14P = 14 units.

### Step 3: Calculate the Interest Earned
The interest earned (SI) can be calculated as:
\[ \text{SI} = A - P = 14 - 1 = 13 \text{ units} \]

### Step 4: Use the Simple Interest Formula
The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal
- R = Rate of interest per annum
- T = Time in years

From the previous steps:
- SI = 13 units
- P = 1 unit
- T = 15 years

Substituting these values into the formula:
\[ 13 = \frac{1 \times R \times 15}{100} \]

### Step 5: Solve for Rate (R)
Rearranging the equation to find R:
\[ R = \frac{13 \times 100}{15} \]
\[ R = \frac{1300}{15} = 86.67 \text{ (approximately)} \]

### Step 6: Calculate Interest for 92 Times the Principal
Now we need to find out how long it will take for the amount to become 92 times the principal:
- If the principal is 1 unit, then the amount will be 92 units.
- The interest earned in this case will be:
\[ \text{SI} = A - P = 92 - 1 = 91 \text{ units} \]

### Step 7: Use the Simple Interest Formula Again
Now we will use the same formula to find the time (T) required to earn 91 units of interest:
\[ 91 = \frac{1 \times R \times T}{100} \]

Substituting R from Step 5:
\[ 91 = \frac{86.67 \times T}{100} \]

### Step 8: Solve for Time (T)
Rearranging the equation to find T:
\[ T = \frac{91 \times 100}{86.67} \]
\[ T = \frac{9100}{86.67} \approx 105 \text{ years} \]

### Final Answer
Thus, it will take approximately **105 years** for the sum to become 92 times itself.

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