Paula borrowed $5000 from her brother Mario. She agreed to repay the money at the end of 5 years, giving Mario the same amount of interest that he would have received if the money had been invested at 2.25% compound quarterly.
How much money did Paula repay her brother?
We can use the formula for compound interest to calculate the amount that Mario would have at the end of 5 years if he invested $5000 at 2.25% compound quarterly:
A = P(1 + r/n)^(nt)
where A is the amount at the end of the investment period, P is the principal (initial amount invested), r is the annual interest rate (expressed as a decimal), n is the number of times per year that interest is compounded, and t is the number of years.
In this case, P = $5000, r = 0.0225 (2.25% expressed as a decimal), n = 4 (quarterly compounding), and t = 5.
A = 5000(1 + 0.0225/4)^(4*5)
A = 5000(1.005625)^20
A = $5863.27
So if Mario had invested $5000 at 2.25% compound quarterly, he would have had $5863.27 at the end of 5 years. Therefore, Paula should repay Mario $5863.27 to fulfill their agreement.
To calculate the amount Paula repaid her brother, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (repayment amount)
P = principal amount (borrowed amount)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case:
P = $5000
r = 2.25% = 0.0225 (since it's in decimal form)
n = 4 (quarterly compounding)
t = 5 years
Plugging in the values into the formula:
A = 5000(1 + 0.0225/4)^(4*5)
A = 5000(1 + 0.005625)^(20)
Now, let's calculate step by step:
Step 1: Calculate the term inside the parentheses:
1 + 0.005625 = 1.005625
Step 2: Calculate the exponent:
(1.005625)^20 ≈ 1.116155628
Step 3: Multiply the principal amount by the calculated exponent:
A = 5000 * 1.116155628
Step 4: Calculate the final amount:
A ≈ $5580.78
Therefore, Paula repaid her brother approximately $5580.78.