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Manipulating Algebraic Equations and Expressions with Fractions: Expressions

A student writes,

Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better?

This is a surprisingly subtle question. It turns out that the answer is partly “Do what’s mathematically sound,” but mostly “Do what’s useful.” It’ll take me 6 posts to show what that means:

  • Manipulating Algebraic Equations and Expressions with Fractions 1: A Quick Quiz
  • Manipulating Algebraic Equations and Expressions with Fractions 2: Expressions
  • Manipulating Algebraic Equations and Expressions with Fractions 3: Another Example
  • Manipulating Algebraic Equations and Expressions with Fractions 4: Equations
  • Manipulating Algebraic Equations and Expressions with Fractions 5: A Word Problem
  • Manipulating Algebraic Equations and Expressions with Fractions 6: Systems of Equations

Let’s begin with “Do what’s mathematically sound.” When you multiply through by a common denominator, you’re changing the value of every single term (except when the term is equal to zero or the denominator is equal to one). That’s okay when you’re dealing with an equation, so long as you do exactly the same thing to both sides of the equation, but it’s not okay when you’re asked to evaluate an expression.

Consider this example:

maeaewf1_img1a

It may be tempting to multiply through by 10, the least common denominator, but that will yield an answer ten times too great.  Instead, take a look at the answers to see what form they take, and manipulate the expression toward answers of that form. It looks as though we’ll want to get the m and n denominators together. We can accomplish this by giving them a common denominator. In this case, that means multiplying the term m/5 by 2/2, to get {2m}/10.  Notice that this effectively multiplying m/5 by 1. The single most commonly useful way to manipulate an expression is to multiply it by 1.

maeaewf2_img1

Since the two terms now have the same denominator, we can join the numerators.

maeaewf2_img2

If it not clear how to translate that further, then rewrite it as,

maeaewf2_img3

Since 1/10=0.1, you can finally rewrite the expression in the useful form.

0.1(2m+n)

Before we turn to the more interesting cases, let’s be reiterate explicitly the lessons we can draw from this example:

-You can’t multiply through by a common denominator unless you have an equation.

-It’s often useful—especially when you’re asked to evaluate an expression—to see what form the answers take.

– The single most commonly useful way to manipulate an expression is to multiply it by 1.

Next post: another example!

 

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