## Real World Applications – Algebra I

### Topic

Taking out a Loan to Purchase a Car

### Student Exploration

When making big purchases, such as buying a house or a vehicle, people usually don’t have all of the money to make the purchase. They usually take out loans from a bank to help make the big purchase, and then the person pays the bank back over time.

Suppose you want to take a step into becoming more independent by purchasing a used car. Used cars can run around $20,000. Unfortunately, you only have only saved a certain amount of money at the time. Suppose that two different banks offer a loan. Use different strategies to find out which bank offers the better deal for your needs.

Let’s say the first bank will let us initially put down $2,500 and pay $150 per month. Let’s also say that another bank has convinced us that if we put down more money, the monthly payment would be lower (which is usually the case). This second bank has told us that if we put down $3,000, we can pay $125 a month.

There are a few questions to help us decide which would be the better option for us.

We can first figure out in how many months the two loan payments will be the same. We can let \begin{align*}x\end{align*} represent the number of months that the payments will be the same. Our equation would be:

\begin{align*}2500 + 150x = 3000 + 125x\end{align*}

To solve for \begin{align*}x\end{align*}, we get all the terms with \begin{align*}x\end{align*} on one side of the equation (in this case we’ll decide to put these variables on the left side of the equation so there are no negative \begin{align*}x-\end{align*}terms), and the rest of the terms on the right side of the equation.

Subtract \begin{align*}125x\end{align*} on both sides of the equation, then subtract 2500 on both sides of the equation.

\begin{align*}2500 + 150x - 125x = 3000\end{align*}

Combine like terms, \begin{align*}150x\end{align*} and \begin{align*}-125x\end{align*} because they both have the same variable raised to the same power.

\begin{align*}2500 + 25x &= 3000\\ 2500 - 2500 + 25x &= 3000 - 2500\\ 25x &= 500\end{align*}

Now divide both sides by 25 to isolate the variable.

\begin{align*}X= \frac{500}{25}=20\end{align*}

What does \begin{align*}x = 20\end{align*} mean?

We have to answer our question, which was asking how long it would take for the payments to be the same? So, it would take 20 months for the payments to be the same.

We can also look at which bank loan is “better.” On one hand, the “better” deal might mean paying more every month with a low down payment. On the other hand, the “better” deal might mean paying more as a down payment and then paying less every month. Which do you think is the better deal?

We can use the four-step problem-solving plan to help us answer this question.

Step 1 is to understand the problem. We know what a down payment is, and we know what monthly payments are. We are trying to figure out which is the better deal.

Step 2 is to devise a plan. For this plan, we need to interpret what the “better deal” means to us. What we can do is substitute different numbers in the monthly plan for each loan and interpret the results.

Step 3 is to carry out the plan and solve. Let’s go ahead and substitute some numbers and figure out what it means.

The first loan has a down payment of $2,500 and a monthly payment of $150. The second loan has a down payment of $3,000 and a monthly payment of $125. Let’s find out how much total we would have paid for 15 months.

For the first loan, we can find out the total amount paid by multiplying 150? by 15 and then adding 2,500. \begin{align*}150(15) + 2500 = \$4,750\end{align*}. For the second loan, we would multiply 125 by 15 and then add 3,000. \begin{align*}125(15) + 3000 = \$4,875\end{align*}. Just looking at these results, it looks like we would have paid more of the loan off using the second payment plan.

Let’s substitute another amount for the number of months and see if this is always the case. Let’s find out of this is the same case in 25 months. For the first loan, we have \begin{align*}150(25) + 2500 = $6,250\end{align*} paid off. For the second loan, we have \begin{align*}125(25) + 3000 = 125(25) + 3000 = \$6,125\end{align*}. This time, the first loan looks better. What does this mean?

Step 4 asks us to interpret our results. Looking at month 15, it seems like the first loan is the best option. But, for the \begin{align*}25^{th}\end{align*} month, it seems like the second loan is the better option. Since both of these loans represent a linear relationship, before 20 months, the first loan is the better choice in looking at how much of the loan we’ve paid off. After 20 months, the second loan is better because more money would have been paid off.

If the loan was a total of $5,500, the first loan would be the best choice, since it more would be paid off in 20 months. If the loan is more than $5,500 (which for this car, it is), then the second loan is the better option.

One step further: How could you convince someone that the first option is the best loan choice?

### Extension Investigation

As we grow up to be independent and mature adults, we take responsibilities like making big purchases. Try researching a big purchase, such as a home or a car, and find out how much it costs. Then research two different banks (or the car dealership, since they have payment plans as well). Find out which is your interpretation of a “better” deal. What would this mean to you? Which payment plan would you choose, and why?