Final answer:
Josie owes a total of $65,000 in student loans; by setting up a system of linear equations based on the total loan amount and the yearly interest, we found that the federal loan amount is $20,000 and the private bank loan amount is $45,000.
Step-by-step explanation:
The student Josie wants to determine the amount of each loan she took out: one with a federal interest rate of 2.4% and the other with a private bank interest rate of 5%, given that the total interest she owes for one year is $2,730.00. To solve this, we can use a system of linear equations where:
- x represents the amount of the federal loan.
- y represents the amount of the private bank loan.
The total amount of loans taken is $65,000, so:
x + y = 65,000 (1)
The total interest for one year is $2,730, so the second equation using the interest rates for the loans would be:
0.024x + 0.05y = 2,730 (2)
We can solve this system of equations using substitution or elimination. For simplicity, we will use the elimination method:
Multiply equation (1) by 0.024:
0.024x + 0.024y = 1,560 (3)
Now, subtract equation (3) from equation (2):
(0.024x + 0.05y) - (0.024x + 0.024y) = 2,730 - 1,560
0.026y = 1,170
y = 1,170 / 0.026
y = 45,000
Now plug the value of y back into equation (1):
x + 45,000 = 65,000
x = 65,000 - 45,000
x = 20,000
Hence, the amount of the federal loan is $20,000 and the amount of the private bank loan is $45,000.