Sam two student loans totaled $31,000. One of his loans was at 2.8% simple interest and the other at 4.5%. After one year, Sam owed $1024.40 in interest. What was the amount of each loan?
x = loan at 2.8%, y = loan at 4.5%
0.028x + 0.045y = 1024.40
x + y = 31000
solve these two equations together for your answer
post back if you need help
To find the amount of each loan, we can use a system of equations.
Let's assume the amount of the loan at 2.8% interest is x, and the amount of the loan at 4.5% interest is y.
According to the given information, the total amount of the loans is $31,000. Therefore, our first equation is:
x + y = 31,000 -- Equation (1)
Now, let's calculate the interest for each loan.
The interest on the first loan at 2.8% can be calculated using the formula: I = P * R * T, where I is the interest, P is the principal (loan amount), R is the rate, and T is the time (1 year in this case). Plugging in the values, we get:
I1 = x * 0.028 * 1 -- Equation (2)
Similarly, the interest on the second loan at 4.5% can be calculated:
I2 = y * 0.045 * 1 -- Equation (3)
Given that the total interest after one year is $1024.40, our third equation is:
I1 + I2 = 1024.40 -- Equation (4)
Now, we have a system of three equations:
x + y = 31,000 -- Equation (1)
I1 = x * 0.028 * 1 -- Equation (2)
I2 = y * 0.045 * 1 -- Equation (3)
I1 + I2 = 1024.40 -- Equation (4)
To solve this system of equations, we can substitute equations (2) and (3) into equation (4) and simplify:
x * 0.028 + y * 0.045 = 1024.40 -- Equation (5)
To eliminate the decimals, we can multiply this equation by 100:
2.8x + 4.5y = 102440 -- Equation (6)
Now, we can solve the system of equations (1) and (6) using the elimination method.
1) Multiply equation (1) by 2.8 to cancel out the x term:
2.8x + 2.8y = 86800 -- Equation (7)
2) Subtract equation (7) from equation (6) to eliminate the x term:
(2.8x + 4.5y) - (2.8x + 2.8y) = 102440 - 86800
1.7y = 15640
3) Solve for y:
y = 15640 / 1.7
y ≈ 9200
Now, substitute the value of y back into equation (1) to find x:
x + 9200 = 31000
x ≈ 21800
Therefore, Sam's loans were approximately $21,800 and $9,200.