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Math  /  Algebra

QuestionWhich of the two annual interest rates would yield the larger amount in 1 year? 12%12 \% compounded monthly or 1212%12 \frac{1}{2} \% compounded annually
Select the correct choice and fill in the answer box to complete your choice. (Do not round until the final answer. Then round to three decimal places as needed.) A. The effective interest rate for 12%12 \% compounded monthly is □\square %\% which would yield the larger amount in 1 year. B. The effective interest rate for 1212%12 \frac{1}{2} \% compounded annually is □\square \% which would yield the larger amount in 1 year.

Studdy Solution

STEP 1

1. We are comparing two different interest rates to determine which yields a larger amount after one year.
2. The first interest rate is 12%12\% compounded monthly.
3. The second interest rate is 12.5%12.5\% compounded annually.
4. We need to calculate the effective annual interest rate for both options.

STEP 2

1. Calculate the effective annual interest rate for 12%12\% compounded monthly.
2. Calculate the effective annual interest rate for 12.5%12.5\% compounded annually.
3. Compare the two effective annual interest rates to determine which is larger.

STEP 3

To find the effective annual interest rate for 12%12\% compounded monthly, use the formula for compound interest:
A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}
where PP is the principal amount, rr is the annual nominal interest rate (as a decimal), nn is the number of compounding periods per year, and tt is the time in years.
For 12%12\% compounded monthly: - r=0.12r = 0.12 - n=12n = 12 - t=1t = 1
The effective annual rate (EAR) is calculated as:
EAR=(1+0.1212)12−1\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1

STEP 4

Calculate the effective annual rate:
EAR=(1+0.1212)12−1=(1+0.01)12−1\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = \left(1 + 0.01\right)^{12} - 1
EAR=1.126825−1=0.126825\text{EAR} = 1.126825 - 1 = 0.126825
Convert to percentage:
EAR=12.6825%\text{EAR} = 12.6825\%

STEP 5

For 12.5%12.5\% compounded annually, the effective annual interest rate is simply the nominal rate, since it is compounded once per year:
EAR=12.5%\text{EAR} = 12.5\%

STEP 6

Compare the two effective annual rates:
- 12%12\% compounded monthly yields an effective rate of 12.6825%12.6825\%. - 12.5%12.5\% compounded annually yields an effective rate of 12.5%12.5\%.
Since 12.6825%12.6825\% is greater than 12.5%12.5\%, the 12%12\% compounded monthly yields the larger amount in 1 year.
The correct choice is:
A. The effective interest rate for 12%12\% compounded monthly is 12.683%12.683\% which would yield the larger amount in 1 year.

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