Jane needs a short-term loan to buy a new washing machine. She needs to borrow $1500 at 20% compounded annually and plans to have it paid off in 1 year. Jane writes the formula 1500(1.2)t and finds out that this loan will cost her $1800.
Which equation shows how Jane can rewrite the formula to find the annual percentage rate that would cost her the same amount if it compounded semi-annually?
A≈1500(1.095)2t
A≈1500(1.095)12t
A≈1500(1.2)12t
A≈1500(1.2)2t
To answer this question, we need to understand how to adjust the formula for a loan with different compounding periods.
In this case, Jane wants to find an equation that represents the same amount of money when compounded semi-annually.
The formula for calculating the future value (A) of an investment with compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = Future Value
P = Principal amount (initial investment)
r = Annual interest rate (as a decimal)
n = Number of times that interest is compounded per year
t = Number of years
In Jane's case, she initially used the formula 1500(1.2)t, assuming that the interest is compounded annually. Now, she wants to find the equivalent interest rate if it were compounded semi-annually.
To adjust the formula for semi-annual compounding, we need to divide the annual interest rate by the number of compounding periods per year (2 in this case) and multiply the number of years by the number of compounding periods. This gives us the modified equation:
A ≈ P(1 + r/n)^(nt)
Plugging the values for Jane's case, we have:
A ≈ 1500(1 + 0.2/2)^(2*1)
Simplifying further:
A ≈ 1500(1.1)^2
Therefore, the correct equation is:
A ≈ 1500(1.1)^2t
So, the correct answer is A≈1500(1.1)2t.