Question

A retired couple has $170,000 to invest to obtain annual income. they want some of it invested in a safe certificates of deposit yielding 6%. the rest they want to invest in AA bonds yielding 12% per year. how much should they invest in each to realized exactly $18,000 in interest per year

Answer 1

$40,000 for 6% and $130,000 for 12%

EXPLANATION :

The formula for interest is :

`I=P* r* t`

Where I is the interest

r is the rate of interest

t is the time in years

Their total investment is $170,000, and they want to divide it into two parts, let's say P1 and P2.

P1 + P2 = 170,000

One investment has 6% interest, and the other has 12%.

So we have r1 = 6% or 0.06

and r2 = 12% or 0.12

Solve for the interest in t = 1 year

`\begin{gathered} I_1=P_1r_1t \\ I_1=P_1(0.06)(1) \\ I_1=0.06P_1_{} \end{gathered}`
`\begin{gathered} I_2=P_2r_2t \\ I_2=P_2(0.12)(1) \\ I_2=0.12P_2 \end{gathered}`

They want to have an interest of $18,000 per year.

So I1 + I2 = 18,000

This will be :

`\begin{gathered} I_1+I_2=18000 \\ 0.06P_1+0.12P_2=18000 \end{gathered}`

From the first equation we have :

P1 + P2 = 170,000

Express P1 in terms of P2 :

P1 = 170,000 - P2

Substitute this P1 to the equation above.

`\begin{gathered} 0.06P_1+0.12P_2=18000 \\ 0.06(170000-P_2)+0.12P_2=18000 \\ 10200-0.06P_2+0.12P_2=18000 \\ -0.06P_2+0.12P_2=18000-10200 \\ 0.06P_2=7800 \\ P_2=(7800)/(0.06)=130,000 \end{gathered}`

Now we have P2, solve for P1

`\begin{gathered} P_1=170000-P_2 \\ P_1=170000-130000 \\ P_1=40000 \end{gathered}`

Therefore, the investment will be $40,000 and $130,000