Geometry and Inequalities on the GRE
Harder quant questions combine two different areas of math, and that’s what we’re going to take a look at today.
First, try this problem (© Manhattan Prep) from our Geometry lesson during class 5.
If 2m + 20 > 100, which of the following could be the value of n?
Indicate all such values.(A) 6
(B) 39
(C) 40
(D) 45
(E) 50
(F) 51
Note: there may be more than one answer, and we have to pick all of the correct answers (and none of the incorrect ones) in order to earn any credit on this problem.
Let’s see. They give us an inequality and we also have a diagram. The question asks for the value of angle n.
It’s interesting “ the problem doesn’t mention either the rectangle or the triangle explicitly. Further, angles m and n are both found in the triangle, and we’re told nothing and asked nothing about the rectangle. How strange “ why is the rectangle there?
For a subtle but important reason, it turns out. The rectangle does give us one piece of information: there’s a 90-degree angle, which allows us to deduce that the triangle is a right triangle. How does that help?
If the third, unmarked angle is equal to 90 degrees, then the other two angles, m and n, must also add up to 90 degrees. Okay. What’s so special about that?
It gives us a second equation!
m + n = 90
2m + 20 > 100
Now we’re actually getting somewhere! There is one annoying thing though “ that second equation is actually an inequality. How are we supposed to combine one equation with one inequality?
It turns out that there’s a very neat technique we can use to combine an equation with an inequality. The resulting answer will also be in the form of an inequality “ that is, it will provide us with a range of possible values, not just one value. (Hmm. I think I’m starting to see why this is a choose ALL that apply problem!)
First, simplify the inequality:
2m + 20 > 100
2m > 80
m > 40
Now rewrite that inequality as an equation: the variable m is greater than 40, or m = GT40. Note that the greater than sign is now incorporated into the equation using the abbreviation GT.
Next, plug m = GT40 into the other equation:
GT40 + n = 90
You can solve equations like this in the same way you normally solve all equations, as long as you carry the GT designation through. You also have to watch for occasions when the sign has to flip and become LT (less than) instead. (One example: when multiplying or dividing by a negative.)
Okay, let’s keep going:
n = 90 “ GT40
How do we simplify that? When we subtract a GT, we do the math with the numbers (90 “ 40) and we flip the sign (GT becomes LT):
n = LT50
Confused about why we flipped the sign? Try a real number to understand how this works. We know that m = GT40; let’s say that m is 41. 90 “ 41 = 49. What if m were 42 instead? Then, 90 “ 42 = 48. The biggest possible result is 49, and as m keeps getting bigger, the resulting value keeps getting smaller. That’s why we flip the sign to less than.
Okay, so the variable n is less than 50. Among the answers, the values 6, 39, 40, and 45 are all possible values for n. The final two answers (50 and 51) are not possible values for n.
The correct answer is 6, 39, 40, and 45.
Although this question looks like a geometry question from the start, it is really an algebra question in disguise. Once we figured out that we had an equation and an inequality, we didn’t have to think about geometry any longer “ the diagram was there only so that we would have to take an extra step in figuring out what this problem is really about.
This is a great example of why we shouldn’t dismiss certain categories of questions simply because we think we’re not good at them. Someone who’s great at algebra but really dislikes geometry might give up on this question too quickly, before realizing that it is actually an algebra question in disguise.
Takeaways for decoding disguises:
(1) First, simply be aware that some problems will, at first, look like one thing when they are actually about something else entirely. Keep an eye out for these while studying, and remind yourself to give yourself a little time to get into each problem before you decide whether it’s too hard or a weakness on which you don’t want to spend more time.
(2) There are some pretty neat math techniques that you probably never learned in school, and combining an equation with an inequality is one of them. Practice this technique so that you can confidently handle the trickiest part: knowing when to switch GT to LT or vice versa. Two switch times are: (a) when multiplying or dividing by a negative, and (b) when subtracting the GT or LT term.