We tend to see a lot of plane geometry questions on the GMAT involving benches, walkways, or other additions being constructed “around” shapes. We’re usually given a few different parts (length, width, or radius) and then asked to find something like the new total area, the original perimeter, or the new length with the addition.
Here’s how to avoid a couple potential missteps on these types of questions!
No real-world values! Sometimes we’ll have to set up equations with the formulas and look to see how the formulas relate to one another, like in this question:
A rectangular fountain with length L and width W is surrounded by a rectangular bench, which is 3.5 inches wide. What is the area of the bench?
(A) LW – 7L – 7W
(B) 3.5L + 3.5W + 12.25
(C) LW – 3.5W – 3.5L
(D) 7L + 7W + 49
(E) LW + 3.5L + 3.7W + 12.25
Notice how we aren’t given the length or the width, and all we know about this bench is that it’s 3.5 inches wide. One easy way to get past the fact that we aren’t given values is to pick our own numbers. Let’s say the Length was 4 and the width was 2. The fountain’s area would be 8. Adding 3.5 to both sides of the fountain would increase the length to 11 and the width to 9. The new area is 99, and the area of the bench is 99-8 = 91. All we’d have to do is plug in L = 4 and W = 2 into the answer choices and look for the answer that yields 91.
Another way to solve is to just use algebra and not real-world values. The dimensions of the rectangular fountain will be found using the expression L x W.
The dimensions of the total rectangle (the fountain + the bench) is:
(L + 7)(W + 7) = LW + 7L + 7W + 49
Forgetting both sides. We’ll add 7 rather than 3.5 to the length and the width because the bench goes completely around. Make sure you don’t forget to draw a quick sketch to visualize how the length and width will change.
The area of the bench is the area of the total rectangle less the area of the rectangular fountain:
(LW + 7L + 7W + 49) – (LW) = 7L + 7W + 49
The correct answer is (D).
Keep an eye out for these question-types on Test Day, and just like in this practice question, there’s often more than one way to solve. Picking numbers, recalling the common formulas, and carefully drawing diagrams are just a few extra tricks you have up your sleeve!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing advice to help students better prepare for the GMAT and the SAT.