A certain sum of money becomes three times its original value in 5 years at simple interest. At the same constant rate of simple interest, in how many years will this sum become five times its original value?

Introduction / Context:
This simple interest problem considers a sum that grows from its original value to three times in 5 years, and then asks for the time needed to reach five times at the same rate. Because simple interest produces linear growth over time, we can use the triple condition to find the rate and then extend the time to reach five times the principal.

Given Data / Assumptions:

• Principal P becomes 3P in 5 years at simple interest.
• Rate of simple interest is R% per annum (constant).
• We want the time T needed so that the amount becomes 5P.
• Simple interest formula: A = P * (1 + R * T / 100).

Concept / Approach:
First, we find R using the condition that the amount becomes three times the principal in 5 years. Then we plug the same R into the amount formula with A = 5P and solve for T. Linear growth under simple interest makes this two-step approach straightforward and intuitive.

Step-by-Step Solution:
For tripling: 3P = P * (1 + R * 5 / 100). Divide both sides by P: 3 = 1 + (R * 5 / 100). So R * 5 / 100 = 2, which means R = 2 * 100 / 5 = 40% per annum. Now we want A = 5P: 5P = P * (1 + R * T / 100). Divide both sides by P: 5 = 1 + (40 * T / 100). So 40 * T / 100 = 4, giving T = 4 * 100 / 40 = 10 years. Therefore, it will take 10 years to become five times the principal.
Verification / Alternative check:
In 5 years the amount becomes 3P, so the total interest gained is 2P. This means an interest of 2P in 5 years, or 0.4P per year, consistent with a 40% rate. To reach 5P from P, we need interest of 4P. Since 1 year gives 0.4P, time required is 4P / 0.4P per year = 10 years, confirming our result.

Why Other Options Are Wrong:
5 years would only triple the sum, not make it five times. 8, 12, and 15 years correspond to different multiples of P under a 40% simple interest rate and do not yield exactly 5P. Using those times in the formula would lead to amounts that are either less than or greater than five times the principal.

Common Pitfalls:
Some learners confuse three times the principal with three times the interest. Others mistakenly think the time to go from 3P to 5P is the same as the first 5 years. It is important to calculate the actual rate first, then apply it consistently to reach the required multiple. Mixing simple interest and compound interest ideas is another frequent error.

Final Answer:
The sum will become five times its original value in 10 years.