Question

A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.If we let x be the amount the woman invests in the 15% bond, how much will she be able to invest in the CD? AnswerSet up and solve the equation for how much the woman should invest in each option to sustain a $6,000 annual return.She must invest at least $Answer in the 15% bond.

Answer 1

Let:

x = Amount the woman invest in the 15% bond

y = Amount the woman invest in the CD

The woman has $50000 to invest, so:

`x+y=50000_{\text{ }}(1)`

She needs to make $6000 a year from the interest, so:

`I1+I2=6000_{\text{ }}(2)`

Where:

`\begin{gathered} I1=x\cdot r1\cdot t \\ r1=0.15 \\ t=1 \\ ------ \\ I2=y\cdot r2\cdot t \\ r2=0.07 \end{gathered}`

So, we have the following system:

`\begin{gathered} x+y=50000_{\text{ }}(1) \\ 0.15x+0.07y=6000_{\text{ }}(2) \end{gathered}`

Let's solve using substitution:

From (1) solve for x:

`y=50000-x_{\text{ }}(3)`

Replace (3) into (2):

`\begin{gathered} 0.15x+0.07(50000-x)=6000 \\ 0.15x+3500-0.07x=6000 \\ 0.08x+3500=6000 \\ 0.08x=6000-3500 \\ 0.08x=2500 \\ x=(2500)/(0.08) \\ x=31250 \end{gathered}`

Replace x into (3):

`\begin{gathered} y=50000-31250 \\ y=18750 \end{gathered}`

Answer:

She will be able to invest $18750 in the CD

And she must invest at least 31250 in the 15% bond