1. A sum of $15,000 was invested in three funds for 1 year. At the end of the year Fund A had earned 8% interest, Fund B had
earned 11% interest and Fund C had earned 5% interest. A total of $1,230 was earned in interest. Twice as much money was initially
invested in Fund A as was in Fund B.
a. How much money was initially invested in each account (at the beginning of the year)? Write down your system of
three equations and the corresponding matrix. Fully explain your answers.
A+B+C=15000
.08A+.11B+.05C=1230
A-2B+OC=O
on my 3 by 4 augmented matrix calculator, I get 8000,4000, 3000. Check that
https://matrix.reshish.com/gauss-jordanElimination.php
To solve this problem, we need to set up a system of equations based on the given information.
Let's denote the amount of money initially invested in Fund A as "x" and the amount initially invested in Fund B as "y." Since the amount invested in Fund A is twice as much as in Fund B, we can write the equation:
x = 2y (Equation 1)
We are also given that the total amount invested is $15,000:
x + y = 15,000 (Equation 2)
To find the amount initially invested in Fund C, we can subtract the amounts in Funds A and B from the total amount invested:
Amount invested in Fund C = 15,000 - (x + y)
Now, let's calculate the interest earned in each fund and set up the last equation:
Interest earned in Fund A = x * 8% = 0.08x
Interest earned in Fund B = y * 11% = 0.11y
Interest earned in Fund C = (15,000 - (x + y)) * 5% = 0.05 * (15,000 - (x + y))
The total interest earned is given as $1,230, so we can write:
0.08x + 0.11y + 0.05(15,000 - (x + y)) = 1,230 (Equation 3)
Now we have a system of three equations (Equations 1, 2, and 3) that we can solve simultaneously.
To do this, we can write the system of equations in matrix form:
| 1 -2 0 | | x | | 0 |
| 1 1 0 | x | y | = | 15000 |
| 0 -1 1 | | z | | 1230 |
The first column represents the coefficients of x in each equation, the second column represents the coefficients of y, and the last column represents the coefficients of z (which is the amount invested in Fund C).
To solve this system of equations, we can use matrix operations such as Gaussian elimination or inverse matrix.
Once we solve the system, the values of x, y, and z will represent the amount of money initially invested in each fund.