GMAT Math : Symmetry

Many GMAT Math questions can be easily solved by using the principle of symmetry

Let’s illustrate this with an example:

If P = (1+x+x2) and R = (1+y+y2), what is the value of PxR?

A. x2y2 + x2y + xy2 + x2 + xy + y2 + x + xy + 1

B. x2y2 + x2y + xy2 + x2 + xy + y2 + x + y + 1

C. x2y2 + x2y + xy2 + x2 + xy + y2 + y2x + 1

D. x2y2 + x2y + xy2 + x2 + xy + y2 + xy + xy + x + 1

E. x2y2 + x2y + xy2 + x2 + xy + y2 + x + xy + x + 1

Wow, lots of xs and ys! Readers are encouraged to take a crack 🙂

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Yes, we know this is a straightforward GMAT question. But why spend the 20-30 seconds multiplying P with R and then double checking your solution for another 20-30 seconds– when there is a way to solve this is less than 10 seconds 🙂

Trick: Look for symmetry. What this means is if two or more terms are symmetrical in the question, the correct answer may often need to preserve the symmetry

P = (1+x+x2) and R = (1+y+y2) are super symmetrical. If you replace x with y in P you will get R and if you replace y with x in R you will get P.

So, PxR i.e (1+x+x2)(1+y+y2) should also preserve symmetry and this is true – just replace x with y and y with x and you are still left with (1+x+x2)(1+y+y2).

This means the answer equation should have similar terms for x as it does for y – there should not be an ‘imbalance’.

Option (A) has an ‘x’ term but no ‘y’ term

Option (C) has 2 of y2x terms but only 1 x2y term.

Option (D) has an ‘x’ term but no ‘y’ term

Option (E) has two ‘x’ terms but no ‘y’ term.

…so we go with Option (B) which is perfectly balanced. No pen was needed to solve this question, just a sharp eye for symmetry 🙂