An investor has $50,000 to invest in two types of bonds. Bond A is a conservative investment that pays
a 5% return, bond B is a riskier investment that pays a 10% return. How much should be invested at
each rate to get $3,750 total return on investment? Set up a system of equations to represent this
problem and write it as a matrix equation.
Where did you get all that money!??
A+B = 50000
.05A + .10B = 3750
[A,B] = [25000,25000]
You are correct. My apologies for the mistake in the equations.
The correct system of equations is:
A + B = 50000 ----(1)
0.05A + 0.10B = 3750 ----(2)
Rewriting equation (1) with the values you provided, we get:
A + B = 50000
Rewriting equation (2), multiplying both sides by 100 to eliminate the decimals, we get:
5A + 10B = 375000
The matrix equation representing this system of equations is:
[A B] * [x; y] = [B'],
where
[A B] = [[1 1]; [5 10]],
[x; y] = [A; B],
[B'] = [[50000]; [375000]].
Substituting the values you provided, we have:
[1 1] * [A; B] = [25000]
[5 10]
So, $25,000 should be invested in bond A and $25,000 should be invested in bond B to get a total return of $3,750.
Let's assume that the amount invested in Bond A is x dollars and the amount invested in Bond B is y dollars.
According to the problem, the total amount invested is $50,000. Therefore, the first equation would be:
x + y = 50000
The returns on investment for Bond A and Bond B are 5% and 10% respectively. We can calculate the total return by multiplying the amount invested by the corresponding return rate:
0.05x + 0.10y = 3750
To represent this problem as a matrix equation, we can rewrite the equations in matrix form:
⎡1 1⎤ ⎡x⎤ ⎡50000⎤
⎢ ⎥ * ⎢ ⎥ = ⎢ ⎥
⎣0.05 0.10⎦ ⎣y⎦ ⎣ 3750 ⎦
Let x be the amount (in dollars) invested in bond A and y be the amount (in dollars) invested in bond B.
According to the given information, the investor has to invest a total of $50,000, so we have the equation:
x + y = 50000 ----(1)
The return on investment for bond A is 5%, which can be expressed as 0.05x. Similarly, the return on investment for bond B is 10%, which can be expressed as 0.10y.
The total return on investment is $3,750, so we have the equation:
0.05x + 0.10y = 3750 ----(2)
Rewriting equation (2) by multiplying both sides by 100 to eliminate the decimals, we get:
5x + 10y = 375000 ----(3)
The system of equations to represent this problem can be written as:
x + y = 50000
5x + 10y = 375000
Writing the system of equations as a matrix equation, we have:
[A] * [x; y] = [B], where
[A] = [[1 1]; [5 10]],
[x; y] = [[x]; [y]],
[B] = [[50000]; [375000]].