A certain sum of money doubles itself in 7 years 8 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 twice the original principal in that time?

Introduction / Context:
This question checks your understanding of how to determine the rate of simple interest when the time required for a sum to become a multiple of itself is given. When a sum doubles, triples, or reaches some other multiple under simple interest, there is a very direct relationship between the rate, time, and the multiple. Here, the sum doubles in 7 years 8 months, and we must find the annual rate of simple interest in percent.

Given Data / Assumptions:

• Initial principal is P (some positive amount).
• Final amount after 7 years 8 months is 2P (the sum doubles).
• Interest is simple interest at a constant rate r percent per annum.
• Time t is 7 years 8 months, which we must convert to a single value in years.

Concept / Approach:
Under simple interest, the amount A after time t years at rate r percent on principal P is given by:
A = P * (1 + r * t / 100).
Here, the amount becomes 2P, so we set 2P = P * (1 + r * t / 100). Cancelling P, we get a simple equation in terms of r and t. The time must be expressed entirely in years, so we convert 7 years 8 months to a fractional form. Solving for r gives the annual simple interest rate in percent.

Step-by-Step Solution:
Step 1: Convert 7 years 8 months to years. Eight months is 8/12 years = 2/3 years. Step 2: Therefore, t = 7 + 2/3 = (21/3) + (2/3) = 23/3 years. Step 3: Use the simple interest amount formula: A = P * (1 + r * t / 100). Step 4: Since the sum doubles, A = 2P. So 2P = P * (1 + r * t / 100). Step 5: Cancel P from both sides to get 2 = 1 + (r * t / 100). Step 6: Subtract 1 to obtain r * t / 100 = 1. Step 7: Therefore, r = 100 / t = 100 / (23/3) = 100 * (3/23) = 300/23. Step 8: Express 300/23 as a mixed fraction: 23 * 13 = 299, remainder 1, so r = 13 1/23 percent. Step 9: Hence, the annual simple interest rate is 13 1/23 percent per annum.

Verification / Alternative check:
We can check by computing the simple interest using r = 300/23 percent over t = 23/3 years. The total interest fraction is (r * t / 100) = (300/23) * (23/3) / 100 = (300/3) / 100 = 100/100 = 1. This means the interest equals the principal P, so the final amount equals P + P = 2P, which agrees with the given condition that the sum doubles. This confirms the correctness of the rate 13 1/23 percent.

Why Other Options Are Wrong:
18 3/4 %, 26 2/23 %, 30 %, and 12 1/2 % correspond to different multiples or different times when substituted into the formula A = P * (1 + r * t / 100). None of them give exactly a doubling for t = 23/3 years. For example, 12 1/2 % (which is 12.5 %) over 23/3 years yields a total interest fraction less than 1, so the amount would be less than 2P. Only 13 1/23 % makes the sum exactly double in the specified time.

Common Pitfalls:
Students often mishandle the conversion of months into years, using 7.8 years instead of 7 years 8 months, or simplifying 8/12 incorrectly. Another common mistake is to forget to divide by 100 when rearranging the formula. Some may also try to use compound interest logic here, but the question explicitly states simple interest. Keeping track of fractions carefully and following the algebraic steps avoids these errors.

Final Answer:
The yearly rate of simple interest that causes the sum to double in 7 years 8 months is 13 1/23 % per annum.