In simple interest, let J be the simple interest earned on a principal amount K, and let K be the simple interest earned on a principal amount L. If the rate of interest and time period are the same in both cases, what is the correct relation between J, K, and L?

Introduction:
This question tests algebraic manipulation using the simple interest formula when the same rate and time apply to different principals. The twist is that one interest value becomes the principal in the next statement. By writing both statements using SI = (P * r * t) / 100, we can eliminate the common factor (r*t/100) and derive the required relationship among J, K, and L.

Given Data / Assumptions:

• J is SI on principal K at rate r% for time t
• K is SI on principal L at the same rate r% for the same time t
• Simple interest: SI = (P * r * t) / 100
• Let m = (r * t) / 100 (a common multiplier)

Concept / Approach:
Express both interests using the same multiplier m. Then substitute one equation into the other, and simplify to get a clean identity involving only J, K, and L. The common multiplier cancels, which is the main simplification trick.

Step-by-Step Solution:
Let m = (r * t) / 100 J = K * m K = L * m Substitute K = L * m into J = K * m: J = (L * m) * m = L * m^2 Now compute K^2: K^2 = (L * m)^2 = L^2 * m^2 Compute J * L: J * L = (L * m^2) * L = L^2 * m^2 Hence K^2 = J * L

Verification / Alternative check:
Pick a sample m = 0.2 and L = 100. Then K = 20, J = 4. Check: K^2 = 400 and J*L = 4*100 = 400. Relationship holds, confirming the derivation.

Why Other Options Are Wrong:
J*J = K*L would imply J proportional to sqrt(K*L), which does not match the chain definition. L*L = J*K is not consistent with the multiplier cancellation. J*K*L = 1 is dimensionally meaningless here. J = K + L is unrelated to simple interest structure.

Common Pitfalls:
Forgetting that K is used as a principal in the first statement but is an interest in the second, or not factoring out the common term (r*t/100).

Final Answer:
The correct relation is K * K = J * L.