This question was found on page 57 of the Manhattan GMAT Formulas & Strategy Booklet

If the length of the side of a cube decreases by two thirds, by what percentage will the volume of the cube decrease?

So I was reviewing this question and the explanation seemed wrong to me.

Here's what they said: Plug in a number for a side and chose 9. The volume would be 9x9x9 = 729

Since the length of the side of a cube decreases by two thirds, its new length is now 3 units.

It's new volume is 3 x 3 x 3 = 27

So change/original = (729-27)/729 = 96.3%

The problem never said the length of every side decreased. It said the length of THE SIDE of a cube decreased which means one side to me.











Change/Original = (729 - 297)/729 =59.3% decrease

Any thoughts or explanations as to why we're supposed to assume that all sides of the cube are to shrink by two thirds if this wasn't specified????

Cool pictures! Read the problem again carefully. The question asks "by what percentage will the volume of the CUBE decrease" - that is, it is still a cube, even after the length change. And a cube's sides are all the same length, by definition.

Also, you will get problems wrong on the test if you try to think this hard about them - I agree that some of the OG / GMAT questions are not so nice, but they wouldn't do this to you! The above is standard wording - because a cube's sides all have the same length, you only have to talk about "the side" because "the side" is the same everywhere on a cube.

Thanks for answering that. I'm not sure what happened, but somewhere along the way I became a GMAT detail oriented psycho.

yeah. well, i must say that the pictures rather made my day, too.

a couple of comments:

1) the pyramid you've drawn is impossible. if you look at the front or back triangular face, you'll notice that the base of those 'triangles' is 9 units long, yet somehow the higher segments drawn parallel to the base - which must be shorter - are also 9 units long. if both of those are 9, then those faces can't be triangles; they must be infinitely long rectangular stripes, of width 9 everywhere, of which the sides will never meet at the top to finish off the triangle.
to see that this is the case, look back at your first diagram, which is admirably drawn, and notice that the yellow sides are parallel to each other. that most certainly can't be part of a pyramid, as it's impossible for any of the sides of a pyramid to be parallel (else they couldn't come together in a point at the top).
2) since you're being very literal, i'll go ahead and note that your diagram doesn't actually accomplish what you set out to accomplish anyway: your diagram actually decreases the lengths of two parallel sides of the cube.

in any case, all nitpicking aside, make sure you learn the important lesson here, which is that gmat problems don't tend to deal with particularly nasty shapes. if you look through the geometry problems in the og - even the rather difficult ones - you'll notice that the vast majority of them still employ simple shapes such as circles, rectangles, squares, cubes, circular cylinders, and the like. they just find exceptionally creative ways to ask tricky questions about them, that's all.