Final answer:
The gain from quarterly compounding compared to annual compounding at a rate of 4.5% per annum is derived using the compound interest formula. Without specific principal and time values, the exact gain cannot be determined, but the compound interest formula shows how compounding frequency increases the total amount of interest accumulated over time.
Step-by-step explanation:
When an amount is lent at a nominal rate of 4.5% per annum and compounded quarterly, the gain from compounding more frequently compared to annual compounding can be calculated using the formula for compound interest. To find the gain in rupees, one would calculate the future value using quarterly compounding and subtract the future value using annual compounding from it.
Compound Interest Formula
For quarterly compounding, the formula is A = P(1 + r/n)nt, where:
- A is the future value of the investment/loan,
- P is the principal amount,
- r is the annual interest rate (decimal),
- n is the number of times that interest is compounded per year,
- t is the time the money is invested or borrowed for (in years).
For annual compounding, the formula simplifies as n is equal to 1. To determine the specific gain over compounding annually, a specific principal amount and time period would be needed. Without these values, the correct option from the given choices cannot be determined.
However, as we see in one of the examples provided, compound interest can exceed simple interest even over a short period and with small amounts, as compound interest is a calculation on the principal plus the accumulated interest. This effect intensifies with larger sums and longer periods due to the nature of exponential growth.