Question

A business has two loans totaling \($85,000$. One loan has a rate of $6$% and the other has a rate of $4.5$% This year, the business

expects to pay \)$4,650$in interest on the two loans. How much is each loan?

Short answer

The amount of loan with $6\%$interest rate is $\$55,000$and with $4.5\%$interest rate is $\$30,000$.

Step-by-step solution

Step 1. Given Information

We are given that a business has total loan equal to $\$85,000$and the business expects to pay $$4,650$in interest on the two loans.

Step 2. Assumptions and Formation of equations

Let $x$and $y$be the loan amount with $6\%$and $4.5\%$interest rates.

The total loan is $85,000$, so

$x+y=85000-\left(1\right)$

The other equation representing the interest will be,

$$\begin{gathered} 6\%x+4.5\%y=4650 \\ 0.06x+0.045y=4650-\left(2\right) \end{gathered}$$

Step 3. Solving the equations.

The first equation can be written as,

$x=85000-y-\left(3\right)$

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

$$\begin{gathered} 0.06x+0.045y=4650 \\ 0.06(85000-y)+0.045y=4650 \\ 5100-0.06y+0.045y=4650 \\ -0.015y=-450 \end{gathered}$$

Dividing by $-0.015$,

$$\begin{gathered} y=\frac{-450}{-0.015} \\ y=30,000 \end{gathered}$$

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

$$\begin{gathered} x=85000-30000 \\ x=55,000 \end{gathered}$$

Hence the amount of loan with $6\%$interest rate is $\$55,000$and with $4.5\%$interest rate is $\$30,000$.

Step 4. Checking the solution 

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

$$\begin{gathered} x+y=85000 \\ 55000+30000=85000 \\ 85000=85000 \\ 0.06x+0.045y=4650 \\ 0.06\left(55000\right)+0.045\left(30000\right)=4650 \\ 3300+1350=4650 \\ 4650=4650 \end{gathered}$$

This is true, hence the solution is correct.