Question

After four years in college, Josie owes \($65$, $800$in student loans. The interest rate on the federal loans is $4.5$% and the rate on the private bank loans is $2$%. The total interest she owes for one year was \)$2878.50$. What is the amount of each loan?

Short answer

The amount of federal loan and private bank loan is$\$62,500$and$\$3,300$

Step-by-step solution

Step 1. Given Information  

We are given that Josie owes $$65,800$in student loans and the total interest she owes for one year was $$2878.50$.

Step 2. Assumptions and Formation of equations

Let $x$and $y$be the amount of federal loan and private bank loan respectively.

Josie owes $$65,800$in student loans, so

$x+y=65800-\left(1\right)$

The total interest she owes for one year was $$2878.50$, so the other equation will be,

$$\begin{gathered} 4.5\%x+2\%y=2878.5 \\ 0.045x+0.02y=2878.5-\left(2\right) \end{gathered}$$

Step 3. Solving the equations.

The first equation can be written as,

$x=65800-y-\left(3\right)$

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

role="math" $$\begin{gathered} 0.045x+0.02y=2878.5 \\ 0.045(65800-y)+0.02y=2878.5 \\ 2961-0.045y+0.02y=2878.5 \\ -0.025y=-82.5 \end{gathered}$$

Dividing by $-0.025$on both sides, we get

$y=3,300$

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

$$\begin{gathered} x=68500-3300 \\ x=62,500 \end{gathered}$$

Hence the amount of federal loan and private bank loan is $\$62,500$and $\$3,300$.

Step 4. Checking the solution

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

$$\begin{gathered} x+y=65800 \\ 62500+3300=65800 \\ 65800=65800 \\ 0.045x+0.02y=2878.5 \\ 0.045\left(62500\right)+0.02\left(3300\right)=2878.5 \\ 2812.5+66=2878.5 \\ 2878.5=2878.5 \end{gathered}$$

This is true, hence the solution is correct.