Q. 155 - Intermediate Algebra

Question

Mark wants to invest \($10,000$ to pay for his daughter’s wedding next year. He will invest some of the money in a short term CD that

pays $12$% interest and the rest in a money market savings account that pays $5$% interest. How much should he invest at each rate if he wants to earn \)$1095$ in interest in one year?

Step-by-Step Solution

Answer

Mark should invest $\$8,500$  in CD and $\$1,500$ in marketing saving account.

Step 1. Given Information

We are given that Mark wants to invest $$10,000$ in short term CD at $12\%$ interest and in market savings account at $5\%$ interest.

Step 2. Assumptions and Formation of equations

Let $x$ and $y$ be the amount of invested in CD and market savings account respectively.

Mark invest $\$10,000$, so

$x+y=10000 -\left(1\right)$

He wants to earn $$1095$in interest in one year, so

$$\begin{gathered} 12\%x+5\%y=1095 \\ 0.12x+0.05y=1095 -\left(2\right) \end{gathered}$$

Step 3. Solving the equations.

The first equation can be written as,

$x=10000-y -\left(3\right)$

Now, putting the value of $x$ in second equation, we get

$$\begin{gathered} 0.12x+0.05y=1095 \\ 0.12(10000-y)+0.05y=1095 \\ 1200-0.12y+0.05y=1095 \\ -0.07y=-105 \end{gathered}$$

Dividing by $-0.07$, we get

$$\begin{gathered} y=\frac{-105}{-0.07} \\ y=1,500 \end{gathered}$$

Now, putting the value of $y$ in third equation, we get

$$\begin{gathered} x=10000-1500 \\ x=8,500 \end{gathered}$$

Hence he should invest $\$8,500$ in short term CD and $\$1,500$ in marketing saving account.

Step 4. Checking the solution

Checking the solution by putting the value of $x,y$ in the equations, we get

$$\begin{gathered} x+y=10000 \\ 8500+1500=10000 \\ 10000=10000 \\ 0.12x+0.05y=1095 \\ 0.12\left(8500\right)+0.05\left(1500\right)=1095 \\ 1020+75=1095 \\ 1095=1095 \end{gathered}$$

This is true, hence the solution is correct.