If J is the simple interest earned on a principal amount K, and K is the simple interest earned on a principal amount L, with the rate of interest and time period being the same in both cases, what is the relation between J, K, and L?

Introduction / Context:
This conceptual question examines how simple interest scales when you change the principal while keeping the rate and time the same. The letters J, K, and L represent interest and principal amounts, and you must identify the correct algebraic relationship among them.

Given Data / Assumptions:

• In the first situation, K is the principal and J is the simple interest earned on K.
• In the second situation, L is the principal and K is the simple interest earned on L.
• The rate of simple interest and the time period are the same in both situations.
• J, K, and L are all positive real numbers representing amounts of money.

Concept / Approach:
For simple interest, SI = (P * R * T) / 100. If R and T are fixed, the simple interest is directly proportional to the principal P. Therefore, when the principal changes from L to K, and the corresponding interests are K and J, you can set up proportional relationships and derive a relation connecting J, K, and L.

Step-by-Step Solution:
Step 1: Let R% be the common rate and T be the common time in years. Step 2: In the first case, J is the interest on principal K, so J = (K * R * T) / 100. Step 3: In the second case, K is the interest on principal L, so K = (L * R * T) / 100. Step 4: From the second equation, K / L = (R * T) / 100. Step 5: From the first equation, J / K = (R * T) / 100. Step 6: Since both J / K and K / L equal (R * T) / 100, they must be equal to each other. Step 7: So J / K = K / L. Cross multiply to get J * L = K * K. Step 8: Therefore, the correct relationship is K x K = J L.

Verification / Alternative check:
You can test the relation with simple numbers. Suppose L = 1000, R = 10%, and T = 1 year. Then K, the interest on L, is (1000 * 10 * 1) / 100 = 100. Now J is the interest on K with the same rate and time, so J = (100 * 10 * 1) / 100 = 10. Then K x K = 100 x 100 = 10000, and J x L = 10 x 1000 = 10000. This confirms that K x K = J L holds.

Why Other Options Are Wrong:
J x J = K L would require J squared to equal K times L, which does not follow from the proportional relationships. L x L = J K would imply that L squared equals J times K, which is not supported by the equations for simple interest. J K L = 1 suggests a unitless product equal to one, which is not meaningful for money amounts and contradicts dimensional consistency.

Common Pitfalls:
Students often confuse which quantity is acting as principal and which is acting as interest in each scenario. Another error is to overcomplicate the algebra instead of using proportional reasoning and simple substitution. Keeping the roles of J, K, and L clear and directly applying the simple interest formula avoids such mistakes.

Final Answer:
The correct relationship among J, K, and L is K x K = J L.