## Real World Applications – Algebra I

### Topic

Selling Candy

### Student Exploration

One of the most common ways that teenagers start to make money is by selling candy. If you buy a case of candy, it can cost you about $26 for a box of 30 pieces of big candy bars, which means that if you sell each candy bar for $1, you can get a $4 profit. If you and your friend Jay are both selling candy, we can represent this relationship by using and proving the distributive property.

You and Jay sold a lot of candy bars. Let’s say you sold 15 boxes in one week and she sold 8 boxes.

We can represent the total amount that the two of you earned in two different ways.

The first way to represent how much money you both made total is by adding the total number of boxes sold and then multiplying this by 4. So, we would get the sum, which is \begin{align*}15 + 8\end{align*} and then multiplying that by 4, which is \begin{align*}4(15 + 8)\end{align*}.

Another way to represent this relationship is if you figured out how much you made first, before Jay. Since you knew that you sold 15 bars, you multiplied your earnings by 4, so \begin{align*}4 \times 15\end{align*} or $60, and then added Jay’s earnings. Then, your total earnings would be \begin{align*}(4 \times 15) + (4 \times 8)\end{align*}. These two methods would yield the same answer, right? Do the calculations to check. Since these are two different ways of representing the total profit, what does this mean?

These two ways actually help prove the distributive property, and why the distributive property holds true. If we were to distribute the 4 to 15 and 8, we would get the second expression above.

But, let’s say that you and Jay both decided that $4 is not a very good profit, and wanted to see what kinds of profit you can make by changing how much you want your profit to be. We can also represent this using the distributive property!

Let’s say that “\begin{align*}x\end{align*}” will represent the number of dollars you want to add to your $4 current profit. We can use the expression \begin{align*}(x + 4)(15 + 8)\end{align*} to represent the total amount of profit between you and your friend. We can also represent this as the sum of both of your profits individually. Your profit would be \begin{align*}(x + 4)(15)\end{align*}, which represents the amount added to your current profit and then the sum multiplied by 15, or the number of boxes of candy you sold. Jay’s profit would be \begin{align*}(x + 4)(8)\end{align*}, which represents the amount added to her current profit and then the sum multiplied by 8, or the number of boxes of candy she sold. To find the total profit between the two of you, you’d take the sum of the two quantities, or adding \begin{align*}15(x + 4) + 8(x + 4)\end{align*}. Since a common factor in this expression is \begin{align*}(x + 4)\end{align*}, we can also rewrite this as \begin{align*}(x + 4)(15 + 8)\end{align*}. We just factored this expression by finding the common factor, or by grouping!

### Extension Investigation

Check out other ways that you can represent the distributive property and factoring in your world. When would this be useful to you?

If we went back to the example above about selling candy, could you figure out a way to sell something and find a way to make the maximum profit? If so, how? What type of relationship would this represent? (Linear? Quadratic?) If it’s quadratic, what would the point of that would yield the maximum profit be called?