To buy both a new car and a new house, Tina sought two loans totalling $260,394. The simple interest rate on the first loan was 3.7%, while the simple interest rate on the second loan was 8.8%. At the end of the first year, Tina paid a combined interest payment of $22,611.73. What were the amounts of the two loans?

Question

To buy both a new car and a new house, Tina sought two loans totalling $260,394. The simple interest rate on the first loan was 3.7%, while the simple interest rate on the second loan was 8.8%. At the end of the first year, Tina paid a combined interest payment of $22,611.73. What were the amounts of the two loans?

Answer 1

x = = amt of car loan at 0.037

y = amt house loan at 0.088

0.037x + 0.088y = 22611.73
x + y = 260394

Solve these two equations simultaneously

Answer 2

To find the amounts of the two loans, we can set up a system of equations:

Let's assume the first loan amount as x and the second loan amount as y.

From the given information, we know that the total loan amount is $260,394. So, our first equation is:

x + y = 260,394 ----(Equation 1)

Now, let's calculate the interest paid on each loan and add them to get the total interest paid.

For the first loan, the interest paid can be calculated using the formula: Interest = Principal * Rate * Time
So, the interest paid on the first loan is: x * 0.037 * 1 = 0.037x

For the second loan, the interest paid can be calculated using the same formula: Interest = Principal * Rate * Time
So, the interest paid on the second loan is: y * 0.088 * 1 = 0.088y

The combined interest paid at the end of the first year is given as $22,611.73. So, our second equation is:

0.037x + 0.088y = 22,611.73 ----(Equation 2)

Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of x and y.

Using any method of solving systems of equations (substitution, elimination, or matrices), we can find the values of x and y.

Let's solve the system of equations using the elimination method:

Multiply Equation 1 by 0.037 and Equation 2 by 100 to eliminate the decimals:

0.037x + 0.037y = 9636.078
0.037x + 8.8y = 2261173

Subtract the first equation from the second equation:

(0.037x + 8.8y) - (0.037x + 0.037y) = 2261173 - 9636.078
8.8y - 0.037y = 2251536.922
8.763y = 2251536.922
y = 2251536.922 / 8.763
y ≈ 256950.69

Now, substitute the value of y back into Equation 1 to find x:

x + 256950.69 = 260394
x = 260394 - 256950.69
x ≈ 3443.31

Therefore, the amounts of the two loans are approximately $3,443.31 and $256,950.69.