Lasya invests two different amounts 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 4%?

Introduction / Context:
This question examines your understanding of how simple interest depends on principal, rate, and time. Specifically, it focuses on proportional reasoning when the same time period is used but the rate and principal are different. When the interest amounts are equal, the ratio between principals is inversely proportional to the ratio between the interest rates.

Given Data / Assumptions:

• Investment at higher rate: simple interest rate R1 = 5% per annum.
• Investment at lower rate: simple interest rate R2 = 4% per annum.
• Let principal invested at 5% be P1.
• Let principal invested at 4% be P2.
• Time period is the same for both investments and is denoted by T years.
• Simple interest earned from both investments is equal.

Concept / Approach:
For simple interest, the formula is:
SI = (P * R * T) / 100When the time T is the same for both investments and the interest amounts are equal, we can equate the two simple interest expressions and simplify. This leads to a relationship showing that the ratio of principals is inversely proportional to the ratio of the interest rates, provided time is the same in both cases.

Step-by-Step Solution:
Step 1: Write the simple interest for the first investment.SI1 = (P1 * 5 * T) / 100Step 2: Write the simple interest for the second investment.SI2 = (P2 * 4 * T) / 100Step 3: Since SI1 and SI2 are equal, equate them.(P1 * 5 * T) / 100 = (P2 * 4 * T) / 100Step 4: Cancel common factors T and 100 on both sides.5 * P1 = 4 * P2Step 5: Rearrange to express the ratio P1 : P2.P1 / P2 = 4 / 5So, P1 : P2 = 4 : 5

Verification / Alternative check:
Assume a convenient time, for example T = 1 year, and choose a trial principal that satisfies the ratio 4 : 5, such as P1 = Rs 4,000 and P2 = Rs 5,000. Then SI1 = (4000 * 5 * 1) / 100 = Rs 200, and SI2 = (5000 * 4 * 1) / 100 = Rs 200. Both interests are equal, which confirms that the ratio 4 : 5 is correct.

Why Other Options Are Wrong:

• 5 : 4 would make SI at 5% too large compared to that at 4%, so interests would not be equal.
• 7 : 6 and 6 : 7 are arbitrary ratios that do not satisfy the condition of equal simple interest when substituted into the formula.

Common Pitfalls:
Many learners forget that when interest amounts are equal, principals and rates are inversely related, not directly. Another frequent error is to try substituting random values without using the proportional relationship, which makes the problem unnecessarily long. Always remember that in simple interest problems with equal time and equal interest, the product P * R must be the same for each case, which leads naturally to an inverse ratio of principals and rates.

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