Let’s look at some applications of using order of operations. Here is an example. Suppose a group of friends orders four pizzas, two servings of garlic bread, and five servings of barbecue wings. Write an expression that can be used to find the amount of change they would receive from eighty-five dollars. Then calculate the change. First we wanna know what is this question asking us to do. It’s asking two things of us: one write an expression and two calculate the change. Next we need to look at the information we were given. We know they ordered four pizzas, two servings of garlic bread, five servings of barbecue wings. We also know that they paid with eighty-five dollars. And then we have the chart with the prices for each menu item.
So now why do we need order of operations for this problem? why is it an application of order of operations? Order of operations tells us how we operate a mathematical expression and the order that we need to do so. To write this expression and then calculate it correctly, we’re going to need to know the order which we should operate — the order which we should solve each piece of the expression. So let’s do that. We’re trying to calculate the amount of change they would receive. The amount of change is equal to the amount paid minus the total cost. In our problem, the amount paid was eighty-five dollars. Next we have to calculate the total cost of the meal. The total cost of the food will be the cost of the pizza plus the garlic bread plus the barbecue wings. We ordered four pizzas and they cost thirteen dollars. So that’s a good start. We also ordered two servings of garlic bread that cost two dollars each, two times two, five servings of barbecue wings for three dollars each. So if we take all the information we’ve gathered and translate it into an expression, this would be the correct expression. The key to the success of this expression is these parentheses. The parentheses here tell us that we need to solve for the total cost of the meal before we subtract that from the eighty-five dollars that the friends paid for their meal.
Let’s see what happens when we do that. We’ll have to use order of operations to make sure that we solve the expression in the correct order. Let’s start with what in the parentheses. With 𝑛 parentheses, we’ll also follow the order of operations. Since there are no exponents in our parentheses, we’re going to multiply and divide from left to right: four times thirteen is fifty-two, two times two equals four, and five times three equals fifteen. Now that we’ve solved all the multiplication and division within our parentheses, we’ll move on to adding and subtracting. But remember this is still all within the parentheses because that’s what we do first. Moving from left to right, the first addition problem we see is fifty-two plus four which equals fifty-six. There is still one more addition problem within the parentheses: fifty-six plus fifteen equals seventy-one.
Now we’re finished within our parentheses and we can look at the rest of the problem. The rest of our problem has no exponents, multiplication, division, or even addition. So we can finally solve the last part — the subtraction. And we see that eighty-five minus seventy-one is fourteen. But we can’t forget the problem. The problem is asking us to calculate the amount of change which would be given in dollars. So the correct final answer would be fourteen dollars. The problem wasn’t just asking for us to calculate the change though. The problem also asked for an expression, which is this one.
Here’s another example. Cinema tickets cost nine dollars and ninety-eight cents for adults and four dollars for children. Write the expression that represents the cost of two adults’ tickets and six children’s tickets. What is the total cost? Just like in the other problem, the first thing I want to do is to find out what this problem is asking of me. It’s asking that I write the expression. It’s also asking what is the total cost. Next we wanna know what information are we given to solve this problem. We know that adults cost nine ninety-eight, that children cost four dollars. We also know the number of adults and children attending. So we want two adults and six children.
Okay, let’s go for it. We’re looking for the total cost. The total cost is going to be the cost of the adults plus the cost of the children. The cost of the adults will be the price of an adult ticket times the number of adults attending. In this case the price of an adult ticket is nine ninety-eight and two adults are attending. For children, we’re gonna do the same thing. We’re going to need to find the price per child and the number of children attending. In this case, the price for a child’s ticket is four dollars and there are six children attending. This is the first step. This is our expression that represents the cost of two adult tickets and six children’s tickets. The next step is asking us what is the total cost, which means we need to solve the expression we’ve just written. And when we solve expressions, we use the order of operations. Starting with the P for parentheses, we do have parentheses in this expression, two sets. So we need to solve what’s in the parentheses first. First we multiplied nine ninety-eight by two, which gives us nineteen dollars and ninety-six cents. Next we multiply four times six to get twenty-four. This step finishes off the parentheses.
We don’t have any exponents. We don’t have multiplication or addition left in the problem. The only thing we have left to do is add and subtract from left to right. When we add nineteen ninety-six and twenty-four, we get forty-three ninety-six for the total cost. But we can’t forget the context of our problem and that means we’re talking about money. So our final answer, the total cost, is forty-three dollars and ninety-six cents.
So why does order of operations matter to these word problems anyway? When we deal with word problems and are creating expressions, we will often deal with more than one operation in the same problem. We’ll need to use order of operations to keep straight which operation must be done first. We use parentheses to group things together that should be done first. After you set up your expression, you use “Please Excuse My Dear Aunt Sally or PEMDAS” to make sure that you solve your expression in the correct way.