Introduction / Context:This problem combines two simple interest calculations with different principals, different effective time periods, and a relationship between their rates. It is a good example of how to set up equations when multiple loans and rates are involved. The key is to carefully track time and ensure that each loan's interest is computed for its correct duration.Given Data / Assumptions:
- First loan: Rs. 725 lent for 1 full year at rate r% per annum
- Second loan: Rs. 362.50 lent after 8 months, so it earns interest for 4 months
- Rate for second loan = 2r% per annum
- Total interest from both loans after 1 year = Rs. 33.50
- We must find r
Concept / Approach:Simple interest for each loan is computed separately using:
SI = (P * R * T) / 100For the first loan, T is 1 year. For the second loan, time is 4 months, which is 4 / 12 = 1 / 3 of a year. We express both interests in terms of r and set their sum equal to 33.50 to solve for r.
Step-by-Step Solution:First loan interest = (725 * r * 1) / 100 = 7.25rSecond loan interest = (362.50 * 2r * (4 / 12)) / 100Simplify 2 * (4 / 12) = 8 / 12 = 2 / 3Second loan interest = (362.50 * (2 / 3) * r) / 100362.50 * (2 / 3) = 241.666..., so interest = (241.666... * r) / 100A cleaner exact method: 362.50 * 2 = 725; so second loan interest = (725 * r) / (3 * 100) = (29r) / 12Total interest = 7.25r + 29r / 12Write 7.25r as 87r / 12, so total = (87r + 29r) / 12 = 116r / 12 = 29r / 3Given total interest = 33.50, so 29r / 3 = 33.5029r = 100.50r = 100.50 / 29 = 3.46 approximatelyTherefore, the original rate is about 3.46% per annumVerification / Alternative check:Let r = 3.46%. Interest on first loan: 725 * 3.46 / 100 ≈ 25.085. Interest on second loan for 4 months at 2r = 6.92% per annum is:
SI2 ≈ (362.50 * 6.92 * (1 / 3)) / 100 ≈ 8.415Total interest ≈ 25.085 + 8.415 = 33.50, matching the given total, so the rate is consistent.
Why Other Options Are Wrong:At 4.5%, 5% or 6%, the calculated total interest would be much higher than Rs. 33.50. These rates do not satisfy the equation 29r / 3 = 33.50 and therefore cannot be correct.
Common Pitfalls:Common mistakes include treating both loans as if they were lent for the full year, forgetting that the second loan earns interest only for 4 months, or not doubling the rate for the second loan. Algebraic manipulations with fractions must also be done carefully to avoid arithmetic errors.
Final Answer:The original annual rate of interest is approximately
3.46% per annum.
