Question

A sum becomes five times of itself in 8 years at simple interest. What is the rate of interest per annum?

• A. 37.5%
• B. 25%
• C. 50%
• D. 12.5%

Hint:

Whenever a sum becomes $n$ times itself, the interest earned is always $(n-1)$ times the principal.
For $5$ times, the interest earned is $400%$.
To find the rate, simply divide this interest percentage by the time period:
\[ \text{Rate} = \frac{400%}{8\text{ years}} = 50%\text{ per annum} \]
This simple shortcut avoids any algebraic variable manipulation.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question deals with Simple Interest (SI).
We are told that a principal sum of money grows to five times its original value over a duration of $8\text{ years}$.
This growth is entirely due to the accumulation of simple interest.
We need to calculate the annual rate of interest ($R%$) that causes this specific growth.

Step 2: Key Formula or Approach:

• Let the principal sum be $P$.
  
• If the sum becomes $n$ times itself, the final Amount \( A = nP \).
  
• Simple Interest is calculated as: \( SI = A - P = nP - P = (n - 1)P \).
  
• The standard formula for Simple Interest is: \( SI = \frac{P \times R \times T}{100} \).
  
• Substituting \( SI = (n - 1)P \), we get the rate formula: \( R = \frac{(n - 1) \times 100}{T} \).
  

Step 3: Detailed Explanation:

• Let the principal sum of money invested be denoted as $P$.
  
• The final accumulated amount after $8\text{ years}$ is $5$ times the principal, so:
  \[ A = 5P \]
  
• The simple interest earned ($SI$) is the difference between the final amount and the initial principal:
  \[ SI = A - P = 5P - P = 4P \]
  
• The time period ($T$) for this investment is given as:
  \[ T = 8\text{ years} \]
  
• Now, write down the formula for simple interest:
  \[ SI = \frac{P \times R \times T}{100} \]
  
• Substitute the values of $SI = 4P$ and $T = 8$ into the formula:
  \[ 4P = \frac{P \times R \times 8}{100} \]
  
• Since $P$ is positive and non-zero, we can divide both sides of the equation by $P$ to eliminate it:
  \[ 4 = \frac{8R}{100} \]
  
• Rearrange the equation to solve for the rate of interest $R$:
  \[ 4 \times 100 = 8R \]
  \[ 400 = 8R \]
  \[ R = \frac{400}{8} = 50% \]
  

Step 4: Final Answer:
The rate of interest per annum is $50%$, which corresponds to Option (C).