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Intuition and Common Sense on Geometry DS Questions

Posted in Test Prep on September 23, 2011 by

“Who cares?”  That should be the operative question on your mind as you tackle Data Sufficiency problems.

Here’s what I mean:  Suppose I ask you the question “What is x?”  I then give you a statement that says “2,346.456 x + 87,234 = 912,347π”.  Is the statement sufficient?

“Yes,” you’d say (hopefully!).

“But do you know what x is?” I’d respond back.

And what would you say?  Yup, you got it:  “Who cares?!”

All we care about is the ability to find x.  We don’t care what x actually is.  Because the statement gives us a simple linear equation with only one variable, we have the ability to find the value of that variable, and that’s all that matters.

This seems a pretty elementary point in and of itself, but it’s one that many students often forget when out in the field tackling tough DS problems.  Specifically on Geometry DS problems, the temptation can be to plug numbers into formulas and tackle the problem as if it were a PS question.  But often times, this is completely unnecessary and a tremendous waste of time.  Many times, you can solve Geometry DS problems intuitively using common sense and simple logic.  But it all hinges on the ability to identify exactly what information you need.

Let’s take a look at the following official DS problem:

A circular tub has a band painted around its circumference, as shown above.  What is the surface area of this painted band?

(1)  x = 0.5

(2)  The height of the tub is 1 meter

Stop!  Don’t write any formulas!  It’s great if you know the formula for the volume of a right circular cylinder, and that might come in handy on PS problems and maybe a more intricate DS problem.  But let’s take a moment to think about what information is really necessary here.  We want to know what the surface area of that band is.  Ask yourself:  What’s keeping us from knowing that?  What’s missing?

Well, we can’t very well know the surface area if we don’t know how wide the cylinder is.  What determines how wide it is?  Radius!  And if we know radius, we also know circumference.  But is that enough?  Nope.  We also don’t know how high x is.  So the two missing pieces of info can be boiled down to:  “x = ?  and r = ?”

This makes perfect sense when you think about it.  How can you know the surface area of something if you don’t know its dimensions?  In this case, the two dimensions are the circumference around the cylinder (which can be determined by radius) and the height of the band, and we need both to get the surface area.

Now that we’ve figured out intuitively what information we need, let’s look at the statements:

Statement (1) gives us the value of x.  Great…nothing about the radius, though.  Insufficient.

Statement (2) gives us the height of the entire cylinder.  Great…nothing about either the radius or the value of x.  Insufficient.

Statements (1) and (2) together give us the height of the entire cylinder and the value of x.  Awesome….where’s the value of the radius?  Still nowhere to be found.  Answer:  E.

We didn’t write down a single equation, and yet we still got out of the problem quickly and with the correct answer.

Now, try your intuitive skills on this other official DS geometry problem.  Remember, try to do it without equations!  Use your common sense!  And post your step-by-step intuitive solutions in the comments!

The inside of a rectangular carton is 48 centimeters long, 32 centimeters wide, and 15 centimeters high.  The carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns, as indicated in the figure above.  If the cans are 15 centimeters high what is the value of k?

(1) Each of the cans has a radius of 4 centimeters.

(2) Six of the cans fit exactly along the length of the carton.