# GMAT Quantitative Problems: Defining the Negative Space

Take a look at the picture with this blog. It’s an iconic optical illusion. Stare at it—what do you see?  The picture is called the Great Wave off Kanagawa, painted by Katsushika Hokusai, a Japanese artist famed for his brilliant compositions. This drawing is of a wave, of course, but do you see the other wave, the reverse wave in the sky?

This image utilizes negative space. You take the whole frame, the great big rectangle, you block out that actual image—and what remains is, in its own right, an interesting picture.

You’ve seen this on the GMAT, of course. Images like this occur frequently on the Quantitative section:

To find the area of the shaded region, we need to subtract the area of the smaller inner circle from the large outer circle—the difference is the area of the ring.

But the concept extends beyond simple pictures and geometry questions. Probability problems sometimes operate on a similar principle, subtracting an easy-to-find probability of failure from 1, the total of all probabilities, the “whole frame,” as it were. Once you’ve subtracted all the failures, then whatever remains, the  “negative space,” must be the chances of success!

The GMAT is a test of critical thinking. It tests your ability to find the most effective path to the solution. Sometimes, you’ll pick numbers, sometimes you’ll do the math directly, sometimes you’ll guess strategically. And sometimes, you’ll define the negative space around the answer, and solve that way. Today’s problem of the day is best solved by that principle. You can’t figure out how many arrangements follow the rule in the question stem. But you can find out how many arrangements don’t follow the rule, and subtract it from the total. Good luck!

Question

Six children, Arya, Betsy, Chen, Daniel, Emily, and Franco, are to be seated in

a single row of six chairs. If Betsy cannot sit next to Emily, how many different

arrangements of the six children are possible?

(A) 240

(B) 480

(C) 540

(D) 720

(E) 840

Solution:

Step 1: Analyze the Question

We have to arrange six children in six chairs, but two of

the children can’t sit together. We’re asked to calculate the

number of different arrangements of children.

We’ll calculate the number of possible arrangements of the

children. Then, we’ll subtract the number of ways Betsy can

sit next to Emily.

Step 3: Approach Strategically

The possible number of arrangements of six elements is

6! = 6 x 5 x 4 x 3 x 2 x 1= 720.

Now we’ll have to calculate the number of ways that would

violate the question stem by putting Betsy next to Emily.

If we number our seats from left to right, there are 5 ways

they can sit together if Betsy is on the left and Emily is on

the right:

Seats 1 & 2

Seats 2 & 3

Seats 3 & 4

Seats 4 & 5

Seats 5 & 6

And there are 5 more ways if Emily is on the left and Betsy

is on the right, for a total of 10. Now, for any one of those

10 ways, the four remaining children can be seated in 4!

ways: 4! = 4 x 3 x 2 x 1 = 24. So we need to subtract

24 x 10 = 240 ways that have Betsy and Emily sitting

together from our original total of 720: 720 – 240 = 480.

Eli Meyer has been a Kaplan teacher since 2003. He has spent the past four years focused almost exclusively on the GMAT, and also has prior experience helping students ranging from middle-schoolers taking the ISEE to professors retaking the GRE for their second PhD. During his Kaplan career, Elis has also written and revised Kaplan course materials and acted as a community liaison on several popular GMAT message boards, all the while helping his students succeed both in and out of the classroom.

• Hsdhszy

Awesome, but seems lengthy…