Q. 366 - Intermediate Algebra

Question

Jack has $$12,000$ to invest and wants to earn $7.5\%$ interest per year. He will put some of the money into a savings account that earns $4\%$ per year and the rest into CD account that earns $9\%$ per year. How much money should he put into each account?

Step-by-Step Solution

Answer

The amount she has put into the savings account is $\$3,600$ and into the CD account is $\$8,400$.

Step 1. Assumption of values

Let the amount to be invested in the savings account be $\$x$ and in the CD account be $\$y$.

The total amount of investment is $\$12,000$.

That means, $x+y=12000$

Step 2. Formation of equation.

The interest earned on a savings account is $4\%$ and on CD account is $9\%$.

The total interest to be earned is $7.5\%$per year.

This means, 

$$\begin{gathered} 4\%x+9\%y=7.5\%(12,000) \\ 0.04x+0.09y=0.075\left(12000\right) \\ 0.04x+0.09y=900 \end{gathered}$$

Step 3. The linear equations

The combination of linear equations is,

$$\begin{gathered} x+y=12,000............\left(1\right) \\ 0.04x+0.09y=900.....\left(2\right) \end{gathered}$$

Step 4. Finding the value of y.

Solving equation (1),

$$\begin{gathered} x+y=12000, \\ y=12000-x \end{gathered}$$

Step 5. Finding the value of x .

Substituting $y=12000-x$ in the equation

$$\begin{gathered} 0.04x+0.09y=900, \\ 0.04x+0.09(12000-x)=900 \\ 0.04x+1080-0.09x=900 \\ -0.05x=-180 \\ x=3600 \end{gathered}$$

Step 6. Finding the value of y

Substituting $x=3600$ in equation

$$\begin{gathered} y=12000-x, \\ y=12000-3600 \\ y=8400 \end{gathered}$$

Step 7. Checking the solution for the equation x + y = 12000

Substituting $$\begin{gathered} x=3600 \\ y=8400 \end{gathered}$$ in the equation

$$\begin{gathered} x+y=12000, \\ 3600+8400=12000 \\ 12000=12000 \end{gathered}$$

This is true.

Step 8. Checking the solution for the equation 0 . 04 x + 0 . 09 y = 900 .

Substituting $$\begin{gathered} x=3600 \\ y=8400 \end{gathered}$$ in the equation,

$$\begin{gathered} 0.04x+0.09y=900 \\ 0.04\left(3600\right)+0.09\left(8400\right)=900 \\ 144+756=900 \\ 900=900 \end{gathered}$$

This is true.