Hey guys,
This problem essentially comes down to whether or not you recognize the ability to use one of the greatest tools at your disposal for this test:
The Difference of Squares Rule
x^2 - y^2 = (x+y)(x-y)
The Difference of Squares rule is the ultimate GMAT "decoder" - the Rosetta Stone of GMAT algebra. It can transform a pretty ugly piece of math into something that's much easier to calculate.
In this case, the problem can be transformed from:
(1001^2 - 999^2) / (101^2 - 99^2)
To:
(1001 + 999)(1001 - 999) / (101 + 99)(101 - 99)
In this form, the first term in both numerator and denominator will add to have zeroes in the units digit, which makes for easy division, and the second term in both cases becomes 2, so that directly factors out:
2000 * 2 / 200 * 2
= 2000/200
=10
Difference of Squares saves the day again!
Regarding Difference of Squares (don't worry about knowing the name, but it's an important enough concept that you should at least keep it at the top of your mind somehow):
1) Make sure that you can recognize the chance to use it whenever you see two squared terms with some subtraction
2) Also be able to note that you can take a term in the other form (two parentheticals multiplied, like (x+y)(x-y)) and move it to the x^2 - y^2 form
3) Know that Difference of Squares is an important tool when you're stuck with a square root in the denominator of a fraction:
1/(1 + sqrt 2)
will not be a GMAT answer choice. You'd need to use Difference of squares to get rid of the radical terms.
If you multiply both numerator and denominator by the denominator (1 + sqrt 2), you'll end up with:
(1 + sqrt2 )/ (1+sqrt2)^2
When you square that term in the denominator, you'll end up with: 1 + 2sqrt2 + 2
You won't be able to get rid of the radicals that way. Instead, you need to use Difference of Squares:
1/(1 + sqrt 2) * (1-sqrt2)/(1-sqrt2)
= (1-sqrt 2)/(1^2 - sqrt2^2)
=(1-sqrt 2)/(1-2)
= (1-sqrt 2)/(-1)
= sqrt 2 - 1 (multiply by -1 to get the negative out of the denominator)
Difference of Squares is one of the few formulas that you really have to know to be successful on the GMAT, and if you know it and keep your eyes open for chances to use it, you'll be able to tackle some pretty ugly algebra with relative ease. Make sure you practice this one![/u]
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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