Simplifying & Interpreting Expressions

At its core, Algebra describes relationships between values. These relationships come together to create an algebraic equation, which represents some larger concept or meaning. Algebraic equations are scattered across everyday life, waiting to be uncovered.

Let’s look at an example.

If you’ve ever shopped for, well, anything really, you know the following to be true: stuff goes on sale. Clothes, shoes, groceries, books, electronics, etc. They all go on sale. As a consumer, it’s important to be able to calculate sale prices so you know how much you’ll spend on an item. And it turns out, calculating the sale price is an excellent exercise in simplifying and interpreting algebraic expressions.

How Do I Calculate the Sale Price?

Say you walk into your favorite clothing store only to find out that everything is 30% off. This store rarely has sales, so you’re pretty excited. You quickly grab a pair of jeans, two t-shirts, and a sweater and make your way over towards check-out. On the way to the registers, you find a cool hat which you add to your haul. Success!

In order to calculate the sale price of each item, we’ll have to create a general expression* which can be applied to everything you plan to purchase. Following general standards of mathematical nomenclature, let’s use x as our variable, which in this case represents the original price of the article of clothing in question.

If something is 30% off, that means 30% of the original price is taken off of the original price. Let’s write this as an expression with x in place of the original price:

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Now, let’s say you’re less concerned with the sale price of each item individually, but more interested in what your total cost of the purchase will be after the sale. One way to do this is to apply the expression above five separate times to each individual item, and then add those new prices together to calculate one grand total. What would this look like as an expression?

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It would look like the expression above, with the following variable assignments:

  • j = original price of jeans

  • t1 = original price of t-shirt 1

  • t2 = original price of t-shirt 2

  • s = original price of sweater

  • h = original price of hat

I think we can agree that this is not the ideal expression to calculate the total price of your purchase. The expression is way too long and cumbersome to compute, and at first glance it doesn’t look like there’s an easy way to simplify it due to the multitude of variables. So let’s go back to our first expression with x, and see if we can simplify that further.

How Do I Simplify the Expression?

In the first expression, we are taking 30% of x and subtracting that result from itself. To simplify this expression, we just need to perform a basic subtraction of the coefficients.** So what does that leave us with?

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Since 1.0 minus 0.30 equals 0.70, we can simply the first equation to the value shown above. So now the question is, how do we interpret this new expression?

How Do I Interpret the Expression?

If an item of clothing is 30% “off,” that means the remaining 70% is “on.” We write this mathematically as 0.70x. Simply put, with the sale, you will only have to pay 70% of the original price of the item. So, what does this simplification do to our grand total expression?

First, we replace all individual “30% off” expressions with their “70% on” equivalents.

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Then, we combine like terms to finish our simplification.

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The end result is an expression that is easier to calculate and interpret. The expression above shows that your grand total can be calculated by adding up the original price of each individual item and multiplying that total by 70%.

We just saw how calculating the sale price can be a great example of simplifying and interpreting algebraic expressions. If you’re looking for a challenge, think about how our grand total expression might change in different scenarios:

  • How might we represent a sales tax?

  • What if you also had a coupon for $15 off?

  • What if the hat you picked out was already 20% off before the store-wide sale?

The example we used was fairly straightforward, but practicing with more complex scenarios will help you master the art of simplifying and interpreting algebraic expressions.

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*An expression is an equation without an equals sign that represents one final value.

**A coefficient is a numeric multiplier of a variable. If there is no number in front of a variable, the coefficient is one.