College Algebra, Chapter 1, 1.2, Section 1.2, Problem 22

If Jan invests $\$ 4000$ at $4\%$ interest per year, how much additional money must he invest at $\displaystyle 5 \frac{1}{2} \%$ annual interest to ensure that the interest he receives each year is $\displaystyle 4 \frac{1}{2} \%$ of the total amount invested?

Let $T$ be the total amount invested and $x$ be the additional money invested. Also, let $I$ be the total interest


$
\begin{equation}
\begin{aligned}

I =& 4 \frac{1}{2} \% T
&& \text{Model}
\\
\\
4000 (4 \%) + (x) \left( 5 \frac{1}{2} \% \right) =& \left( 4 \frac{1}{2} \% \right) (4000 + x)
&& \text{Substitute the given}
\\
\\
4000(0.04) + (x)(0.055) =& (0.045)(4000 + x)
&& \text{Simplify}
\\
\\
160 + 0.055 x =& 180 + 0.045x
&&
\\
\\
0.055x - 0.045x =& 180 - 160
&&
\\
\\
0.01x =& 20
&& \text{Solve for } x
\\
\\
x =& \$ 2000

\end{aligned}
\end{equation}
$