A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct ?
The correct answer is (A) R = S.
Explanation:
Let us set up the mathematical equations for the two compounding scenarios over 1 year: Scenario 1 (Compounded half-yearly at R% annual rate): Time = 1 year. Since it is compounded half-yearly, the number of compounding periods is n = 2, and the half-yearly rate is R/2 %. The amount Q is given by: Q = P × (1 + R / 200)^2. Scenario 2 (Compounded annually at S% annual rate): Time = 1 year. The number of compounding periods is n = 1, and the annual rate is S%. The amount Q is given by: Q = P × (1 + S / 100)^1. Since both scenarios yield the same amount Q from the same principal P in 1 year: P × (1 + R / 200)^2 = P × (1 + S / 100) (1 + R / 200)^2 = 1 + S / 100 Expanding the left side: 1 + (R / 100) + (R^2 / 40000) = 1 + (S / 100) Subtracting 1 from both sides: (R / 100) + (R^2 / 40000) = S / 100 Multiplying the entire equation by 100: R + (R^2 / 400) = S Since R is a positive interest rate, R^2 / 400 is a positive value. Therefore, S = R + (R^2 / 400), which means S > R, or equivalently, R < S. Thus, option (c) is correct.
हिंदी में प्रश्न एवं आदर्श उत्तर
कोई मूलधन P, अर्धवार्षिक रूप से संयोजित R% वार्षिक चक्रवृद्धि ब्याज दर से 1 वर्ष में Q हो जाता है । यदि वही मूलधन P, वार्षिक रूप से संयोजित S% वार्षिक चक्रवृद्धि ब्याज दर से 1 वर्ष में Q हो जाता है, तो निम्नलिखित में से कौन-सा एक सही है ?
सही उत्तर (A) R = S है।
व्याख्या:
Let us set up the mathematical equations for the two compounding scenarios over 1 year: Scenario 1 (Compounded half-yearly at R% annual rate): Time = 1 year. Since it is compounded half-yearly, the number of compounding periods is n = 2, and the half-yearly rate is R/2 %. The amount Q is given by: Q = P × (1 + R / 200)^2. Scenario 2 (Compounded annually at S% annual rate): Time = 1 year. The number of compounding periods is n = 1, and the annual rate is S%. The amount Q is given by: Q = P × (1 + S / 100)^1. Since both scenarios yield the same amount Q from the same principal P in 1 year: P × (1 + R / 200)^2 = P × (1 + S / 100) (1 + R / 200)^2 = 1 + S / 100 Expanding the left side: 1 + (R / 100) + (R^2 / 40000) = 1 + (S / 100) Subtracting 1 from both sides: (R / 100) + (R^2 / 40000) = S / 100 Multiplying the entire equation by 100: R + (R^2 / 400) = S Since R is a positive interest rate, R^2 / 400 is a positive value. Therefore, S = R + (R^2 / 400), which means S > R, or equivalently, R < S. Thus, option (c) is correct.