A mortgage broker purchased two trust deeds for a total of $290,000. One trust deed earns 9% simple annual interest, and the second one earns 12% simple annual interest. If the total annual interest from the two trust deeds is $31,500, what was the purchase price of each trust deed?
To solve this problem, we can set up a system of equations:
Let x be the amount of money invested in the trust deed that earns 9% interest.
Let y be the amount of money invested in the trust deed that earns 12% interest.
From the problem, we know that:
x + y = $290,000 -- Equation 1 (total amount invested)
We also know that the total annual interest from the two trust deeds is $31,500. The interest earned can be calculated as follows:
Interest from trust deed 1 (at 9% interest) = (x) * 0.09 -- Equation 2
Interest from trust deed 2 (at 12% interest) = (y) * 0.12 -- Equation 3
The total annual interest from the two trust deeds is given as $31,500, so we have:
(x) * 0.09 + (y) * 0.12 = $31,500 -- Equation 4
Now we have the system of equations:
x + y = $290,000 -- Equation 1
(x) * 0.09 + (y) * 0.12 = $31,500 -- Equation 4
To solve this system, we can use the substitution method or the elimination method. I will use the elimination method.
Multiplying Equation 1 by 0.09 will allow us to eliminate one variable:
0.09x + 0.09y = $26,100
Now we can subtract Equation 4 from the modified Equation 1:
0.09x + 0.09y - (0.09x + 0.12y) = $26,100 - $31,500
0.09x + 0.09y - 0.09x - 0.12y = -$5,400
0.09y - 0.12y = -$5,400
-0.03y = -$5,400
To solve for y, we can divide both sides of the equation by -0.03:
y = -$5,400 / -0.03
y = $180,000
Now that we have the value for y, we can substitute it back into Equation 1 to find the value of x:
x + $180,000 = $290,000
x = $290,000 - $180,000
x = $110,000
Therefore, the purchase price of each trust deed is $110,000 for the one earning 9% interest and $180,000 for the one earning 12% interest.
Let's assume the purchase price of the trust deed with 9% interest is $x, and the purchase price of the trust deed with 12% interest is $y.
According to the given information, the total purchase price of the two trust deeds is $290,000:
x + y = $290,000 --- Equation (1)
The annual interest earned from the trust deed with 9% interest can be calculated as follows:
0.09x
Similarly, the annual interest earned from the trust deed with 12% interest is:
0.12y
The total annual interest from the two trust deeds is given as $31,500:
0.09x + 0.12y = $31,500 --- Equation (2)
Now we have a system of two equations (Equation 1 and Equation 2):
x + y = $290,000
0.09x + 0.12y = $31,500
We can solve this system of equations to find the values of x and y.
$X @ 9%, $Y @ 12%.
Eq1: X + Y = $290,000.
Eq2: 0.09x + 0.12y = $31,500.
Multiply both sides of Eq1 by 0.12 and subtract Eq2 from Eq1.