Common Mistakes: Incorrectly Combining Or Splitting Up Fractions

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This is part of a series of posts about mistakes that I’ve seen multiple students make while working through GMAT Math questions.

Intro

Lots of questions deal with fractions. Fractions often create confusion because we don’t have a good intuitive feel for them, compared to integers.

Here is a list of things that we can do with fractions:

1. Divide the top and bottom by the same amount: 12/24 = (12÷2)/(24÷2) = 6/12 … this is how we reduce fractions
2. Multiply the top and bottom by the same amount: 3/7 = (3*2)/(7*2) = 6/14 … this is often what we do before we add or subtract fractions
• Or when we want to eliminate square roots in denominators, as in 4/√2 = 4/√2 * √2/√2 = (4√2)/(√2√2) = (4√2)/(2) = 2√2
3. Combine two fractions with the same denominator into one fraction (addition or subtraction): 3/x+ 4/x = 7/x (NOT 7/2x)
4. Split up a fraction with two or more terms in the numerator into two or more fractions: (2 + x)/x = 2/x+ x/x = 2/x+ 1

The Mistake

A few things you CANNOT do with fractions:

1. Split up a fraction with two or more terms in the DENOMINATOR into two or more fractions: 12/(2 + 6) = 12/8 = 3/2 … NOT 12/(2 + 6) = 12/2 + 12/6 = 6+ 2 = 8
• This mistake is more common with algebra, when it is not as clear that this move is illegal: 2x/(x + 2) cannot be simplified, but I have seen students try to make it 2x/(x + 2) = 2x/x + 2x/2 = 2 + x. ILLEGAL.
2. Flip both sides of an equation that has more than one fraction on one side: 1/4 + 1/4 = 1/2. It is NOT true that 4 + 4 = 2
• Again, this is a mistake made mostly with variables, when it is not clear that it is ILLEGAL: 1/x+ 1/y = 1/6 is NOT the same as x+ y = 6
• What you can do it combine the left side: 1/x+ 1/y= y/xy + x/xy = (x+ y)/xy. Then, you can flip both sides: (x+ y)/xy = 1/6 can be rewritten as xy/(x+ y) = 6

Conclusion

Be very careful when you’re simplifying fractions that contain variables. If you’re not sure if the assumption you are making is true, test out with a numerical example like the ones above before applying the technique to the algebra.

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