Fractional Canceling Error

I love “this post”:http://scienceblogs.com/builtonfacts/2008/11/testing_123.php from Matt Spring on “Built on Facts”:http://scienceblogs.com/builtonfacts/ the other day. In it he describes a mathematical manipulation on a test that “he think[s] should almost be worth _negative_ points.” In this case it was a student who took the part on the left side, cancelled the _m_ to get the part on the right:

frac{F-mg}{m} rightarrow F-g

(1)

Which is, of course, wrong wrong wrong. There’s a little rule in math that, in order to cancel out a denominator, it must be able to cancel out on both sides of the subtraction sign in the numerator. Here’s what the formula _should_ look like, after applying the rule properly:

frac{F}{m}-frac{mg}{m}

(2)

As you can see, the _m_ simply can’t cancel out from both sides. Just for fun I ran a little check on myself just to make sure _I_ did it right (since I’m a little rusty on my math sometimes). Let’s let _F_ be equal to 26, _m_ to 7, and g to 3. Substitute those into the original equation (1) and solve:

frac{26-7(3)}{7} = frac{26-21}{7} = frac{5}{7}

(3)

Now, substitute those same numbers in for (2) and solve:

frac{26}{7}-frac{7(3)}{7} = frac{26}{7}-frac{21}{7} = frac{5}{7}

(4)

Same answer, so I know I did it correctly.

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