Q. 367 - Intermediate Algebra

Question

When she graduates college, Linda will owe \($43,000$ in student loans. The interest rate on the federal loans is $4.5\%$ and the rate on the private bank loans is $2\%$. The total interest she owes for one year was \)$1,585$.

What is the amount of each loan?

Step-by-Step Solution

Answer

The amount of federal loan at the rate of $4.5\%$ is about $$29,000$.

The amount of private bank loan at the rate of $2\%$ is about $$14,000$.

Step 1. Assumption of variables.

Let the amount of federal loan at $4.5\%$ be $$x$ and the amount of private bank loan at $2\%$ be $$y$.

The total amount of loan is $\$43,0000$

That means, $x+y=43,000$.

Step 2. Assumption of a linear equation.

The total interest charged is $\$1,585$.

The equation is,

$$\begin{gathered} 4.5\%x+2\%y=1585 \\ 0.045x+0.02y=1585 \end{gathered}$$

Step 3. The linear equations.

The combination of linear equations is,

$$\begin{gathered} x+y=43000...........\left(1\right) \\ 0.045x+0.02y=1585.........\left(2\right) \end{gathered}$$

Step 4. Finding the value of y from equation (1).

Solving equation (1),

$$\begin{gathered} x+y=43000, \\ y=43000-x \end{gathered}$$

Step 5. Finding the value of x .

Substituting $y=43000-x$, in the equation

$0.045x+0.02y=1585$,

$$\begin{gathered} 0.045x+0.02(43000-x)=1585 \\ 0.045x+860-0.02x=1585 \\ 0.025x=725 \\ x=29000 \end{gathered}$$

Step 6. Finding the value of y .

Substituting $x=29,000$ in the equation

$y=43000-x$,

$$\begin{gathered} y=43,000-29,000 \\ y=14,000 \end{gathered}$$

Step 7. Checking the solution for the equation x + y = 43000 .

Substituting $$\begin{gathered} x=29000 \\ y=14000 \end{gathered}$$ in the equation

$x+y=43000$,

$$\begin{gathered} 29000+14000=43000 \\ 43000=43000 \end{gathered}$$

This is true.

Step 8. Checking the solution of the equation 0 . 045 x + 0 . 02 y = 1585 .

Substituting $$\begin{gathered} x=29,000 \\ y=14,000 \end{gathered}$$ in the equation

$0.045x+0.02y=1585$,

$$\begin{gathered} 0.045\left(29000\right)+0.02\left(14000\right)=1585 \\ 1305+280=1585 \\ 1585=1585 \end{gathered}$$

This is true.