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Manipulating Algebraic Equations and Expressions with Fractions: Another Example

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:

Be sure that you’ve read the first two posts in this series before tackling this post, since this offers an opportunity to practice the skill we discussed there

AAA equation fix


Because is an expression rather than equation, we don’t have the option of multiplying through by a common denominator. What could we do? Well, as we saw in our last post, the single most commonly useful way to manipulate an expression is to multiply it by 1. But wait! Isn’t there a standard way to manipulate a compound fraction? Should we use that instead?

Yep, and as we’ll see, that standard way to manipulate a compound fraction is just one way to multiply the expression by 1.

You may remember that the standard way to divide by a fraction is to multiply by the reciprocal of that fraction; lots of American students learn this as “invert (the denominator) and multiply (by that inverse).” That mnemonic works well enough, but it may obscure the underlying logic.

Let’s go ahead and manipulate for now, and then double back and tackle the underlying logic. Let’s rewrite the problem so that we’re multiplying the numerator (x/5) by the reciprocal of the denominator:

{x/5}/{10/y} becomes {x/5} x {y/10} which in turn becomes xy/50.

As with the problem from my last post, we’d better take a look at the answer choices to see what form our expression should take. We’re going to want it in the form of answers C, D, and E: a decimal fraction multiplied by the expression .

If you can’t translate automatically, do it in stages:






Remember that I wrote above that multiplying by the reciprocal (or “inverting and multiplying”) is just a special way of multiplying by 1? Let’s see why.

Consider the general form {a/b}/{c/d}. We could simplify this fraction by multiplying by a special form of 1, {d/c}/{d/c}:


Of course, that’s the very same result that you’d get by going directly to “invert and multiply.” Fortunately, if you’re well-practiced with that maneuver, you don’t need to worry about the underlying logic during the test.


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