A person has $50,000 to invest. As the person's financial consultant, you recommend that the money be invested in Treasury bills that yield 4%. Treasury bonds that yield 8%, and corporate bonds that yield 12%. The person wants to have an annual income of $3960, and the amount invested in corporate bonds must be half that invested in Treasury bills. Find the amount of each investment.

Question

A person has $50,000 to invest. As the person's financial consultant, you recommend that the money be invested in Treasury bills that yield 4%. Treasury bonds that yield 8%, and corporate bonds that yield 12%. The person wants to have an annual income of $3960, and the amount invested in corporate bonds must be half that invested in Treasury bills. Find the amount of each investment.

Answer 1

To find the amount invested in each type of investment, let's break down the problem step by step.

Let's denote:
- x as the amount invested in Treasury bills,
- y as the amount invested in Treasury bonds, and
- z as the amount invested in corporate bonds.

Since the person wants an annual income of $3960, we can set up the equation:

0.04x + 0.08y + 0.12z = 3960 (equation 1)

We also know that the amount invested in corporate bonds must be half that invested in Treasury bills. So, we can set up another equation:

z = 0.5x (equation 2)

Now we have two equations with two variables, x and y. We can solve for x and y by substituting equation 2 into equation 1 to eliminate z:

0.04x + 0.08y + 0.12(0.5x) = 3960
0.04x + 0.08y + 0.06x = 3960
0.1x + 0.08y = 3960 (equation 3)

Now we have two equations with two variables, equation 2 and equation 3. We can solve for x and y using any method such as substitution or elimination.

Let's solve the system of equations by substitution:
From equation 2, we can rewrite it as:
x = 2z (equation 4)

Substituting equation 4 into equation 3:
0.1(2z) + 0.08y = 3960
0.2z + 0.08y = 3960 (equation 5)

Now we have a system of two equations with two unknowns: equations 2 and 5. We can solve this system to find the values of z and y.

To solve the system, we can multiply equation 5 by 100 to eliminate decimals:
20z + 8y = 396000 (equation 6)

Now, we can solve the system of equations 2 and 6:
2z = 20z + 8y (equation 2)
20z + 8y = 396000 (equation 6)

Rearranging equation 2:
20z = 8y
z = (8/20)y
z = (2/5)y (equation 7)

Substituting equation 7 into equation 6:
20(2/5)y + 8y = 396000
8y + 8y = 396000
16y = 396000
y = 396000 / 16
y = 24750

Now that we have found the value of y, we can substitute it back into equation 7 to find the value of z:
z = (2/5) * 24750
z = 9900

Finally, we can substitute the values of y and z back into equation 2 to find the value of x:
x = 2z
x = 2 * 9900
x = 19800

Therefore, the amount invested in each type of investment is:
- $19,800 in Treasury bills,
- $24,750 in Treasury bonds, and
- $9,900 in corporate bonds.