Dalajit lends Rs. 10,800 to Jaabir for 3 years and Rs. 7,500 to Kabir for 2 years at the same rate of simple interest, and receives a total of Rs. 1422 as interest from both loans together. What is the common rate of interest per annum (in percent)?

Introduction / Context: In this question, one lender, Dalajit, gives two loans to two different people for different time periods but at the same unknown simple interest rate. The total interest he receives from both loans is known. We must use this combined information to find the common annual rate of simple interest. Given Data / Assumptions:

• Loan 1: Principal P1 = Rs. 10,800 lent to Jaabir for T1 = 3 years.
• Loan 2: Principal P2 = Rs. 7,500 lent to Kabir for T2 = 2 years.
• Common rate R% per annum for both loans.
• Total simple interest from both loans: SI_total = Rs. 1422.
• Simple interest formula: SI = (P * R * T) / 100.

Concept / Approach: We write separate simple interest expressions for each loan using the same rate R, then add them and set the sum equal to the given total interest of Rs. 1422. This gives a single equation in one unknown R, which we can solve straightforwardly to obtain the rate per annum. Step-by-Step Solution: Interest from first loan: SI1 = (P1 * R * T1) / 100 = (10800 * R * 3) / 100. Simplify SI1 = 10800 * 3 * R / 100 = 32400R / 100. Interest from second loan: SI2 = (P2 * R * T2) / 100 = (7500 * R * 2) / 100. Simplify SI2 = 7500 * 2 * R / 100 = 15000R / 100. Total interest: SI_total = SI1 + SI2 = (32400R + 15000R) / 100 = 47400R / 100. We are given SI_total = 1422, so 47400R / 100 = 1422. Thus, 474R = 1422, giving R = 1422 / 474 = 3% per annum. Verification / Alternative check: Compute SI1 at 3%: SI1 = 10800 * 3 * 3 / 100 = 10800 * 0.09 = 972. Compute SI2 at 3%: SI2 = 7500 * 3 * 2 / 100 = 7500 * 0.06 = 450. Total = 972 + 450 = 1422, which matches the given total interest, confirming that R = 3% per annum is correct. Why Other Options Are Wrong: Rates of 3.5 percent, 4 percent, 4.5 percent, or 2.5 percent lead to interest totals that differ from Rs. 1422 when substituted into the formula. For example, a higher rate like 4 percent would produce a larger total interest, while 2.5 percent would produce less. Common Pitfalls: A common error is to try to find separate rates for each loan, even though the question clearly states the rate is the same. Another pitfall is arithmetic mistakes when combining the two simple interest expressions or when solving for R. Keeping all steps aligned with SI = (P * R * T) / 100 ensures accuracy. Final Answer: The common rate of interest per annum is 3 percent.