Julie has $8,000 to invest, and hopes to earn $330 in interest in the first year.

Question

Julie has $8,000 to invest, and hopes to earn $330 in interest in the first year. She wants her investment in treasury bills to be $3000 less than her investment into corporate bonds. The expected returns (in simple interest) on each investment are: Treasury Bills: 3% Treasury Bonds: 2% Corporate Bonds: 5% How much should she invest into each one?

Answer 1

Let's start this problem by first noting that we are being asked to calculate simple interest for all of the investments. The formula for simple interest is A = P(1+rt), where A is the amount present after t years, P is the initial investment, and r is the rate expressed in decimal form (not percent).

Next, let's denote P₁ as the initial investment that Julie makes in treasury bills, P₂ as the initial investment that Julie makes in treasury bonds, and P₃ as the initial investment that Julie makes in corporate bonds. We know that she only has $8000 to invest initial, so we can say that P₁ + P₂ + P₃ = $8000. We also know that she wants to invest $3000 less in treasury bills than her investment in corporate bonds. Based on this knowledge, we can say that P₁ = P₃ - $3000.

Now we have 3 variables, and two equations concerning those three variables. In order to solve this problem in a simple manner, we would like to be able to use a system of equations. However, in order for that to happen, we would need 3 linearly independent equations that describe our 3 variables. All we have left to do is produce one more equation or relationship based on the knowledge we've been given.

We know that after t = 1 year, Julie wants her total investments to be worth $8330 ($8000 initial investment plus $330 in growth). The amount after the first year can be modeled as such: A = P(1+r). Now, let's denote A₁ = P₁(1+0.03), where A₁ is the total value of the investment in treasury bills after one year, A₂ = P₂(1+0.02), where A₂ is the total value of the investment in treasury bonds after one year, and A₃ = P₃(1+0.05), where A₃ is the total value of the investment in corporate bonds after one year. Notice that I have substituted the respective rate of growth for each investment as a decimal in the place of r. Based on the knowledge that we have, we can say that A₁ + A₂ + A₃ = $8330, or in other words that the total worth of her investments after one year is $8330. Now, we can substitute in the expressions for A₁, A₂, and A₃ in order to get an equation in terms of P₁, P₂, and P₃. Doing so will produce P₁(1+0.03) + P₂(1+0.02) + P₃(1+0.05) = $8330.

Now we have the following equations in terms of P₁, P₂, and P₃:

P₁ + P₂ + P₃ = $8000

P₁ = P₃ - $3000

P₁(1+0.03) + P₂(1+0.02) + P₃(1+0.05) = $8330

At this point, you can enter these equations into a system of equations calculator and arrive at the correct answer. Alternatively, by hand we can first rearrange these equations.

P₁ + P₂ + P₃ = $8000

P₁ - P₃ = - $3000

P₁(1.03) + P₂(1.02) + P₃(1.05) = $8330

From this form, we can write the system of equations into a coefficients matrix, a variables vector, and a solutions vector, in the form Ax = b. We can then row reduce this matrix to find the solution. From this system of equation, we can determine that:

P₁ = $2000

P₂ = $1000

P₃ = $5000

This tells us that Julie should initial invest $2000 into treasury bills, $1000 into treasury bonds, and $5000 into corporate bonds in order to achieve her investment goals for the first year.

Hope this was helpful!

Answer 2

set x=treasury bills, y=treasury bonds, and z=corporate bonds

0.03x + 0.02y + 0.05z = 330

x+y+z=8000

x=z-3000

0.03(z-3000)+0.02y+0.05z=330

(z-3000)+y+z=8000

0.03z-90+0.02y+0.05z=330

z-3000+y+z=8000

0.08z+0.02y=420

2z+y=11000

y=11000-2z

0.08z+0.02(11000-2z)=420

0.08z+220-0.04z=420

0.04z=200

z=5000

y=11000-2(5000)

y=11000-10000

y=1000

x=z-3000

x=2000

2000=treasury bills, 1000=treasury bonds, and 5000=corporate bonds