A certain sum of money becomes three times its original value in 5 years 4 months at simple interest. For this simple interest investment, what is the yearly rate of interest, expressed in percent per annum, that causes the amount to become three times the principal in that time?

Introduction / Context:
This question extends the common idea of a sum doubling under simple interest to the case where a sum becomes three times its original value. Given the time required for tripling, we must determine the annual simple interest rate. This kind of problem appears frequently in banking and competitive exams and tests whether you understand the linear growth nature of simple interest with respect to time and rate.

Given Data / Assumptions:

• Initial principal is P (some positive amount).
• Final amount after the given time is 3P (the sum becomes three times).
• Interest is at simple interest, not compound interest.
• Time t = 5 years 4 months.
• We need to find r, the annual rate of simple interest in percent.

Concept / Approach:
Under simple interest, the amount A after time t years at rate r percent is:
A = P * (1 + r * t / 100).
Here, A is given to be 3P. Thus, 3P = P * (1 + r * t / 100). Cancelling P leads to a simple equation in r and t. We must convert 5 years 4 months into an exact number of years as a fraction, then isolate r. Because simple interest grows linearly, the total interest fraction is r * t / 100, which in this case must equal 2 (since 3P = P + 2P).

Step-by-Step Solution:
Step 1: Convert 5 years 4 months to years. Four months is 4/12 years = 1/3 years. Step 2: Therefore, t = 5 + 1/3 = (15/3) + (1/3) = 16/3 years. Step 3: Use the simple interest amount formula: A = P * (1 + r * t / 100). Step 4: Since the amount becomes 3P, we have 3P = P * (1 + r * t / 100). Step 5: Cancel P from both sides to get 3 = 1 + r * t / 100. Step 6: Subtract 1 to obtain r * t / 100 = 2. Step 7: Therefore, r = (2 * 100) / t = 200 / (16/3) = 200 * (3/16) = 600 / 16. Step 8: Simplify 600 / 16 = 37.5, so r = 37.5 percent per annum. Step 9: Hence, the yearly simple interest rate is 37.5 percent.

Verification / Alternative check:
To verify, compute the total interest fraction using r = 37.5 and t = 16/3 years. The fraction r * t / 100 = (37.5 * 16/3) / 100. First compute 37.5 * 16 = 600, so the numerator is 600/3 = 200. Then 200 / 100 = 2. This means the interest is 2P, so the final amount is P + 2P = 3P, exactly as the question states. This confirms that 37.5 percent is indeed the correct annual rate for the given time and tripling effect.

Why Other Options Are Wrong:
Any rate less than 37.5 percent, such as 18.75 %, 25 %, or 27.5 %, results in r * t / 100 being less than 2, so the amount would be less than 3P. A larger rate like 42.25 % would make r * t / 100 exceed 2, leading to an amount greater than 3P. Only 37.5 % gives exactly the required threefold increase in the principal over 5 years 4 months.

Common Pitfalls:
Common mistakes include misreading the phrase "three times its original value" as meaning the interest equals the principal instead of twice the principal. Another frequent error is incorrectly converting 4 months into years, sometimes treating 5 years 4 months as 5.4 years, which is incorrect. Some students also forget to subtract 1 when going from 3 = 1 + r * t / 100, leading to r * t / 100 = 3 instead of 2. Being precise with conversions and algebraic steps avoids these issues.

Final Answer:
The yearly rate of simple interest that makes a sum become three times its original value in 5 years 4 months is 37.5 % per annum.