Breaking Down A GMATPrep Geometry Problem
Recently, we talked about quant problems where it seems that the GMAT test writers are more interested in our ability to set up the problem than in our ability to solve down to a specific number. This seems to be one focus for them going forward, and we looked at a probability problem (click here to view) that illustrated this principle. Ive got another one for you today.
Set your timer for 2 minutes. and GO!
*A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r?(A) [pmath] pi r^2 [/pmath]
(B) [pmath] pi r^2 + 10 [/pmath]
(C) [pmath] pi r^2 + {1}/{4} pi^2 r^2 [/pmath]
(D)[pmath] pi r^2 + (40-2pi r)^2[/pmath]
(E)[pmath] pi r^2 + (10-{1/2}pi r)^2 [/pmath]
Bonus question: figure out how someone would mistakenly arrive at any (or all!) of the four wrong answer choices.
Okay, so we have a 40-meter-long wire and its going to get cut once, but we dont know exactly where (and we never will its a theory problem, not a real-number problem). Theres one key variable in play: the radius, r. The answer choices all have this variable, so one option is to plug in a real number somewhere and turn this thing into an arithmetic problem. Another option is to start drawing and keep track of the things algebraically as we go.
The other key question to answer before we start: what does the question ask? Lets see, it wants the total area of the circle and the square we need to calculate each and add them up. We know formulas for the area of a circle and the area of a square; write them down.
Area of a circle = [pmath] pi r^2[/pmath](where r is the radius)
Area of a square = [pmath]s^2[/pmath](where s is the length of one side)
So were looking for: [pmath] pi r^2+s^2[/pmath]
Glance at the answers. If you had to choose right now, what would you NOT pick?
Okay, lets try a real number (in an actual testing situation, either youd try this or youd try algebra well try algebra next). Draw a line on your paper and indicate that the length is 40. Cut it somewhere and pick two lengths (that add up to 40, of course). I first picked 15 and 25, until I realized that I would rather have even numbers (think about why), so I changed my numbers to 16 and 24.
I decided that Id make the circle out of the 16-meter piece. Draw that what does the 16 actually represent in terms of the circle?
Right! The circumference and we have a formula for that!
[pmath] C = 2 pi r = 16[/pmath]
Solve for r:
[pmath] r = {16}/{2pi} = {8}/{pi} [/pmath]
Okay, now we have a value for the radius (though its ugly). We want the area for a circle with this radius, so plug in:
[pmath]A=pi r^2=pi(8/pi)^2=64/pi[/pmath]
What about the square? Well, we have a 24-meter piece left to make that. All four sides of a square are the same length, so divide by 4: each side of the square is 6 meters long. We get to the area simply by squaring the length of one side, so the area of the square is 36.
Great! The total area of the two shapes is [pmath]64/pi+36[/pmath].
Now check the answers to find a match, using [pmath]r=8/pi[/pmath]. All five answers start with [pmath]pi{r}^2[/pmath], so we dont even need to check that half of it we know that is the area of the circle. We only need to check the area-of-the-square portion of each answer.
(A) [pmath]pi r^2[/pmath] This one doesnt include the area of the square at all. Eliminate.
(B) [pmath]pi r^2+10[/pmath] This says the square is 10, but I want it to be 36. Eliminate.
(C) [pmath]{1/4}pi^2 r^2={1/4}pi^2(8/pi)^2=2[/pmath]. Nope.
(D) [pmath](40-2pi r)^2 = [40-2pi(8/pi)]^2=(40-16)^2=24^2[/pmath] Im not sure what that does equal, but it doesnt equal 36. Eliminate.
(E) [pmath](10-{1/2}pi r)^2=[10-{1/2}pi(8/pi)]^2=(10-4)^2=6^2=36[/pmath] Bingo!
The correct answer is E.
Thats one way. What about an algebraic approach? In this case, algebra can be a much faster approach, but you have to feel confident that you know what youre doing. Otherwise, itll be easy to make a mistake and get one of those wrong answers instead.
Were going to start out with the same general idea: were looking for [pmath]pi r^2+s^2[/pmath]. Look at the answers notice anything? The first portion is already in all 5 answers! We only need to deal with the second portion: how do we write [pmath]s^2[/pmath] in terms of r (that is, using the variable r)?
When we cut the wire in two parts, the part that were going to use to make the circle does not represent the radius of the circle; it represents the circumference of the circle. So the circle portion of the wire has a length of [pmath]2pi r[/pmath]. The total length of the wire is 40, so the amount we have left over to make the square has a length of [pmath]40-2pi r[/pmath].
That length represents the perimeter of the square (always draw things out to make sure); the formula for perimeter is 4s. In order to calculate the area of a square, we need to find the length of one side, so we need to divide the perimeter by 4:
[pmath]{40-2pi r}/4=10-{1/2}pi r[/pmath]
Thats now the length of one side of the square. What do we do to that to get the area? Right! We square it! Go look at your answers. Does anything match the square portion (that is, the second half of each answer)? Answer E it is!
Bonus question: how are the four wrong answers constructed?
Answer A reflects the area of the circle but omits the area of the square.
Im not 100% sure how they built answer B. One way might have something to do with that 10 that shows up in the right answer. Maybe someone does part of the work but forgets about or omits the messy half and is left just with the 10 for the square.
And now that Im looking at choice C, I think I mightve been right about choice B. C reflects the other bit of the right answer for the area of the square: we solve down to the [pmath]{1/2}pi r[/pmath] piece (but forget or omit the 10) and then square that to get [pmath]{1/4}pi^2 r^2[/pmath].
Answer D is the closest to the real thing. Whats the difference? The square portion wasnt divided by 4! [pmath]40-2pi r[/pmath]represents the length of the entire piece used to make the square rather than the length of just one side of the square.
Key Takeaways for Set-up Geometry Problems:
(1) Look at your answer choices! Use the construction of the choices to help you figure out what steps to take. When they want a set-up, as opposed to a more normal solution (such as a value), you know that you need to work your way through that set-up very carefully. (And you also know you have the time to do so, because you arent going to have to solve for a value!)
(2) Make the situation as real and concrete as you can. Draw and label everything. When they mention something for which there is a formula (area, perimeter, circumference, ), write that formula down. Visualize cutting the wire and making the shapes. How would it happen physically? And what formula corresponds to that?
(3) If youre not sure how to do the problem algebraically, step through with real numbers instead. It might take longer, but thats okay if it prevents you from making a mistake (and if you can still get it done in 2 minutes).
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.