Intermediate Algebra, Chapter 2, Cumulative Exercises, Section Cumulative Exercises, Problem 32

Shamil Mamedov invested some money at $5\%$ simple interest and $\$2,000$ more than that amount at $6\%$. The interest for the year totaled $\$670$. How much was invested at each rate?

Step 1: Read the problem, we are asked to find the amount invested on each rate.
Step 2 : Assign the variable. Then organize the information in the table.
Let $x = $ amount invested at $5\%$ interest rate.
Then, $x + 2000 = $ amount invested at $6\%$ interest rate

$
\begin{array}{|c|c|c|c|c|c|}
\hline
& \rm{Principal} & \cdot & \text{Interest Rate} & = & \rm{Interest} \\
\hline
5 \% & x & \cdot & 0.05 & = & 0.05x \\
\hline
6 \% & x + 2000 & \cdot & 0.06 & = & 0.06(x + 2000) \\
\hline
\end{array}
$


The total interest earned is equal to the sum of the interests at each rate.

Step 3: Write an equation from the last column of the table
$0.05x + 0.06(x + 2,000) = 670$

Step 4: Solve

$
\begin{equation}
\begin{aligned}
0.05x + 0.06x + 120 &+ 670\\
\\
0.11x &= 670 - 120\\
\\
0.11x &= 550\\
\\
x &= 5000
\end{aligned}
\end{equation}
$

Then by substitution,
$x + 2,000 = 5,000 + 2,000 = 7,000$

Step 5: State the answer
In other words, Shamil invested $5,000$ and $7,000$ at $5\%$ and $6\%$ interest rates respectively.

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