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: $\\ I=\frac{P\times R\times T}{100}$

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,

$\\ I=\frac{x\times 15\times 1}{100}\\ =\frac{15x}{100}$

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,

$\\ I_{2}=\frac{(70000-x)\times 7\times 1}{100}\\ \\ =\frac{490000-7x}{100}$

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

$\\ I=I_{1}+I_{2}\\ \Rightarrow 9000=\frac{15x}{100}+\frac{490000-7x}{100}\\ 900000=15x-7x+490000=8x+490000\\ 410000=8x\\ x=\frac{410000}{8}=51250$

⇒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.