Question

A retired woman has $70,000 to invest but needs to make $9,000 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%. Set up and solve the equation for how much the woman should invest in each option to sustain exactly a $9,000 annual return.

Answer 1

Let, x be the amount she invest in the bond which pays 15% of interest.

Then, $70000-x would be the amount she would be investing in a CD that pays 7% of interest.

We know the simple interest formula is:
`\\ I=(P* R* T)/(100)`

Case I: Let
`I_(1)` be the interest she gets on investing $x amount in bond which gives 15% of interest.

Thus,
`P_(1)=x, R_(1) =15\%,  T_(1) =1`

Hence we have now,

`\\ I=(x* 15* 1)/(100)\\ =(15x)/(100)`

Case II: Let
`I_(2)` be the interest she gets on investing $(70000-x) amount in CD which gives 7% of interest.

Thus,
`P_(2)=70000-x, R_(2) =7\%,  T_(2) =1`

Hence we have now,

`\\ I_(2)=((70000-x)* 7* 1)/(100)\\ \\ =(490000-7x)/(100)`

Since, the woman needs to make a total interest of $9000 each year,

`\\ I=I_(1)+I_(2)\\ \Rightarrow 9000=(15x)/(100)+(490000-7x)/(100)\\ 900000=15x-7x+490000=8x+490000\\ 410000=8x\\ x=(410000)/(8)=51250`

⇒The amount she invest in bond = x= $51,250

Thus, the amount she invest in CD = $70,000 - x= $70,000 - $51,250= $18,750

Therefore, the amount that women should invest in bond is $51,250 and in CD is $18,750 so that she can get a total interest of $9000 every year.