Question

grandma has $250000 to invest. she divides her money into two accounts. one account is in ultra-safe treasury bills paying 4% interest, and the other is in riskier corporate bonds paying 6%. if she needs $12,000 per year in income from her investments, how much should she invest in each account

Answer 1

for the ultra-safe treasury bills:

$\begin{gathered} I_1=PV\cdot r\cdot t \\ I_1=x\cdot0.04\cdot1 \\ I_1=0.04x \\ \text{where:} \\ x=\text{amount 1} \end{gathered}$

For riskier corporate bonds:

$\begin{gathered} I_2=PV\cdot r\cdot t \\ I_2=y\cdot0.06\cdot1 \\ I_2=0.06y \\ \text{Where:} \\ y=\text{amount 2} \end{gathered}$

she needs $12,000 per year, so:

$\begin{gathered} I_1+I_2=12000 \\ 0.04x+0.06y=12000 \end{gathered}$

grandma has $250000 to invest, therefore:

$x+y=250000$

Let:

$\begin{gathered} x+y=250000\text{ (1)} \\ 0.04x+0.06y=12000\text{ (2)} \\ \text{From (1) solve for x:} \\ x=250000-y\text{ (3)} \\ \text{ Replace (3) into (2)} \\ 0.04(250000-y)+0.06y=12000 \\ 10000-0.04y+0.06y=12000 \\ 0.02y=12000-10000 \\ 0.02y=2000 \\ y=\frac{2000}{0.02} \\ y=100000 \end{gathered}$

Replace y into (3):

$\begin{gathered} x=250000-100000 \\ x=150000 \end{gathered}$

Therefore, grandma needs to invest $150000 in ultra-safe treasury bills, and