Final answer:
To find the equivalent simple interest rate to a 10.1% compounded quarterly rate, first calculate the compound amount after a year, then use it to find the required simple interest rate. After calculating, the equivalent simple interest rate is approximately 10.449%.
Step-by-step explanation:
To find the equivalent simple interest rate that would yield the same amount as a 10.1% interest rate, compounded quarterly after one year, we need to calculate the amount of money that would be accrued through compound interest and then determine the simple interest rate that would give us the same final amount.
First, let's calculate the amount of money after one year with compound interest. The formula for compound interest is A = P(1 + r/n)^(nt) where:
- P = principal amount (initial investment)
- r = annual interest rate (decimal)
- n = number of times interest is compounded per year
- t = number of years
- A = amount of money accumulated after n years, including interest
If we assume the principal amount P is $1 for simplicity, the calculation for compounded quarterly would be:
A = 1(1 + 0.101/4)^(4*1) = 1(1.02525)^4 ≈ 1.104487.
This means that after one year, we will have approximately $1.104487 with a 10.1% interest rate compounded quarterly.
Using the simple interest formula I = PRT where I is the interest, P is the principal amount, R is the rate of interest per year, and T is the time period in years, we want to find R such that the interest I plus the principal P equals approximately $1.104487.
So, I + P = 1.104487.
We know that I = PRT, so:
1.104487 = 1 + (1 * R * 1)
R = 1.104487 - 1 = 0.104487
To express R as a percentage, we multiply by 100:
R * 100 = 10.4487%.
Therefore, the equivalent simple interest rate is approximately 10.449%