A person lends Rs. 1,540 for 5 years and Rs. 1,800 for 4 years, both at the same rate of simple interest. If the total interest received from the two loans together is Rs. 1,788, what is the annual rate of interest (in percent per annum)?

Introduction / Context:
This question combines two simple interest loans with the same rate but different principals and time periods. The total interest from both loans is known, and we are asked to find the common annual rate of interest. It tests the ability to form and solve a single equation involving multiple simple interest terms.

Given Data / Assumptions:

• First loan: Principal P1 = Rs. 1,540, time T1 = 5 years
• Second loan: Principal P2 = Rs. 1,800, time T2 = 4 years
• Rate of interest for both loans = R% per annum
• Total simple interest from both loans = Rs. 1,788

Concept / Approach:
Simple interest for each loan is given by:
SI = (P * R * T) / 100We compute the simple interest expressions for both loans in terms of R, sum them, and set the result equal to 1,788. Solving this equation gives us the rate R.

Step-by-Step Solution:
Interest from first loan: SI1 = (1540 * R * 5) / 100Interest from second loan: SI2 = (1800 * R * 4) / 100Total interest = SI1 + SI2 = 1788SI1 = (1540 * 5 * R) / 100 = (7700R) / 100SI2 = (1800 * 4 * R) / 100 = (7200R) / 100Total interest = (7700R + 7200R) / 100 = (14900R) / 100So (14900R) / 100 = 1788149R = 1788R = 1788 / 149R = 12Therefore, the annual rate of simple interest is 12% per annum
Verification / Alternative check:
Check total interest using R = 12%. For the first loan: SI1 = (1540 * 12 * 5) / 100 = 1540 * 0.6 = Rs. 924. For the second loan: SI2 = (1800 * 12 * 4) / 100 = 1800 * 0.48 = Rs. 864. Total interest = 924 + 864 = Rs. 1,788, which matches the given total. This confirms that 12% is correct.

Why Other Options Are Wrong:
At 14.5%, 11% or 10.5%, the calculated total interest would not be Rs. 1,788. Each would give a different sum either higher or lower than 1,788. Only 12% yields exactly the required amount.

Common Pitfalls:
Students may forget to divide by 100 in the simple interest formula or incorrectly add the two interest expressions. Another mistake is to substitute numbers too early and lose track of the common factor R. Keeping R symbolic until the final step makes the algebra straightforward and avoids arithmetic errors.

Final Answer:
The annual rate of simple interest is 12% per annum.