Chapter 10.2, Problem 37E

To determine

To calculate: Quantity that she invest in each type to satisfy the given conditions.

Answer to Problem 37E

She has to invest $\$4000$ in long term bonds.

Explanation of Solution

Given information:

Short term bond pays: $4\%$

Long term bond: $5\%$

Total annual income: $5.1\%$

Calculation:

Lets assume,

  $S=$ amount invested in short term.

  $m=$ amount invested in medium term.

  $l=$ amount invested in long term.

So,

  $\begin{array}{l}s+m+l=10000\\ s=m\end{array}$

  $.04s=$ annual return on short terms.

  $.05m=$ annual return on medium terms.

  $.06l=$ annual return on long terms.

The problem is looking for an overall return of $5.1\%$ on these $3$ investment, so in words

(sum of the returns)/ (total investment) $=5.1\%$

  $\frac{.04s+.05m+.06l}{10000}=.05l$

Multiply both sides by $10000$

  $.04s+.05m+.06l=510$

Multiply both sides by $100$

  $\begin{array}{l}4s+5m+6l=51000\\ s+m+l=10000\end{array}$

Multiply both sides $2$ by $6$

  $6s+6m+6l=60000$

Subtract $1$ from $2$

  $2s+m=9000$

Since,

  $\begin{array}{l}s=m\\ 2m+m=9000\\ 3m=9000\\ m=3000\\ s=3000\\ s+m+l=10000\\ 3000+3000+1=10000\\ 1=10000-6000\\ 1=4000\end{array}$

He has to invest $\$4000$ in long term bonds,

  $\begin{array}{l}\frac{.04s+.05m+.06l}{10000}=.051\\ \frac{.04\left(3000\right)+.05\left(3000\right)+.06\left(4000\right)}{10000}=.051\\ \frac{120+150+240}{10000}=.051\\ \frac{510}{10000}=.051\\ .051=.051\end{array}$