Lasya invests two different sums of money at simple interest, one at 5% per annum and the other at 4% per annum; if the simple interest earned from both investments is equal, what is the ratio of Lasya's investments at 5% and at 4%?

Introduction / Context:
This question checks understanding of how simple interest depends on principal, rate, and time. Lasya invests two different amounts at two different annual simple interest rates, yet the interest earned from each investment is the same. The problem asks for the ratio of the two principal amounts, which is a standard concept used in aptitude exams on simple interest and proportional reasoning.

Given Data / Assumptions:
- Investment 1 earns simple interest at 5 percent per annum. - Investment 2 earns simple interest at 4 percent per annum. - The simple interest earned from both investments is equal. - The time period for both investments is assumed to be the same. - Required ratio = amount invested at 5 percent : amount invested at 4 percent.

Concept / Approach:
For simple interest, SI = (P * R * T) / 100. When the time T is the same for both investments and the interests are equal, the products P1 * R1 and P2 * R2 must be equal. Therefore P1 * R1 = P2 * R2. From this, the ratio of the principals is P1 / P2 = R2 / R1. This means the principal is inversely proportional to the rate when interest and time are the same.

Step-by-Step Solution:
Step 1: Let the amount invested at 5 percent be P1. Step 2: Let the amount invested at 4 percent be P2. Step 3: Simple interest for P1 is SI1 = (P1 * 5 * T) / 100. Step 4: Simple interest for P2 is SI2 = (P2 * 4 * T) / 100. Step 5: Given that SI1 = SI2, so (P1 * 5 * T) / 100 = (P2 * 4 * T) / 100. Step 6: Cancel common factors 100 and T (T is nonzero), giving 5 * P1 = 4 * P2. Step 7: Rearranging, P1 / P2 = 4 / 5. Step 8: Hence, the ratio of investments at 5 percent and 4 percent is 4 : 5.

Verification / Alternative check:
Choose simple values that satisfy the ratio 4 : 5. For example, let P1 = Rs 4,000 and P2 = Rs 5,000 and take a common time of 1 year. Then SI1 = (4,000 * 5 * 1) / 100 = Rs 200 and SI2 = (5,000 * 4 * 1) / 100 = Rs 200. The interests are equal, confirming that the ratio 4 : 5 is correct.

Why Other Options Are Wrong:
A ratio of 5 : 4 would imply P1 / P2 = 5 / 4, which would make the interest at 5 percent larger than at 4 percent, contradicting the condition of equal interest. Ratios 7 : 6 and 6 : 7 are arbitrary and do not satisfy the necessary inverse proportionality between principal and rate. Only the ratio 4 : 5 correctly balances the higher rate with a smaller principal.

Common Pitfalls:
A typical mistake is to assume that the ratio of principals is the same as the ratio of rates, rather than the inverse. Another error is forgetting that the time is the same for both investments and therefore not cancelling it properly. Remember that for equal simple interest over equal time, principal is inversely proportional to the rate of interest.

Final Answer:
The ratio of Lasya's investments at 5 percent and 4 percent is 4 : 5.