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:
[tex]x+y=50000_{\text{ }}(1)[/tex]She needs to make $6000 a year from the interest, so:
[tex]I1+I2=6000_{\text{ }}(2)[/tex]Where:
[tex]\begin{gathered} I1=x\cdot r1\cdot t \\ r1=0.15 \\ t=1 \\ ------ \\ I2=y\cdot r2\cdot t \\ r2=0.07 \end{gathered}[/tex]So, we have the following system:
[tex]\begin{gathered} x+y=50000_{\text{ }}(1) \\ 0.15x+0.07y=6000_{\text{ }}(2) \end{gathered}[/tex]Let's solve using substitution:
From (1) solve for x:
[tex]y=50000-x_{\text{ }}(3)[/tex]Replace (3) into (2):
[tex]\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=\frac{2500}{0.08} \\ x=31250 \end{gathered}[/tex]Replace x into (3):
[tex]\begin{gathered} y=50000-31250 \\ y=18750 \end{gathered}[/tex]Answer:
She will be able to invest $18750 in the CD
And she must invest at least 31250 in the 15% bond
