Carletta has 7000 dollars to invest

Question

As her financial consultant, you recommend that she invest in Treasury Bills that yield 5%, Treasury Bonds that yield 6%, and corporate bonds that yield 7%. Carletta wants to have an annual income of 405 dollars, and the amount invested in corporate bonds must be half that invested in Treasury Bills. Find the amount she should place in each investment

Answer 1

Well, this is definitely a system of linear equations, so let's set us up some linear equations!  There are three things she can invest in, so we need three variables:  x for the amount invested in treasury bills, y for the amount invested in treasury bonds, and z for the amount invested in corporate bonds.

 

She only has 7000 to split up among the three, so we know the sum of them:

x + y + z = 7000

Her income from these investments will be the sum of the return on each.  That is, the sum of the 5% she gets on her treasury bills investments, the 6% on the treasury bonds, and the 7% on the corporate bonds, or

.05x + .06y + .07z = 405

And since she wants half as much invested in corporate bonds (z) as in treasury bills (x), we know that

z = .5x

Three independent equations is enough to solve for three variables, so we can solve, first by using the last equation to drop z out of the first two:

 

x + y + (.5x) = 7000

.05x + .06y + .07(.5x) = 405

 

1.5x + y = 7000

.085x + .06y = 405

 

Rearrange the first equation to get y = 7000 - 1.5x, and we can plug it into the last

 

.085x + .06(7000-1.5x) = 405

.085x + 420 - .09x = 405

-.005x = -15

.005x = 15

x = 3000

 

Go back and plug this into 1.5x + y = 7000:

1.5(3000) + y = 7000

4500 + y = 7000

y = 2500

 

And we know z = .5x = .5*3000 = 1500

 

Check our answer:

3000 + 2500 + 1500 = 7000, so the total amount checks out

.05*3000 + .06*2500 + .07*1500 = 150 + 150 + 105 = 405, so the income checks out.

And 1500 is half of 3000, so the relative investment amounts check out, and we have the answer we want.