Divide And Conquer

The SBC GMAT Files

Quant: Divide And Conquer Part 2: A Bottom Up Approach

In the previous article, Master GMAT instructor Roy detailed what I call the “top-down” approach to problem solving – find out what the question is asking, then break that target down to a series of steps, each of which is perfectly manageable on its own.

Today, I’d like to discuss the opposite approach to quantitative problem solving questions: the “bottom – up” approach. This approach is your fallback position when the top-down way is a dead-end, which happens when the end target proves difficult to quantify and break down.
A GMATPREP example:

If M=√4 + ∛4 + ∜4, then the value of M is
A) Less than 3
B) equal to 3
C) Between 3 and 4
D) equal to 4
E) Greater than 4

The top-down approach would begin with breaking the problem down to a series of mini questions: what is the value of √4? of ∛4? of ∜4? And that’s pretty much where this approach ends – I don’t know the value of ∛4 and ∜4! The typical reaction is to focus solely on the obvious difficulty elements of the problem, but no matter how much you wrinkle your head, the correct value of these advanced roots of 4 will not “magically” appear to you in a vision.

At this point, the average test-taker is no longer really trying to solve the question – his mind is too occupied by twin mental processes: lamenting the cruelty of a test that requires him to remember these roots by heart, and the railing against the unjust cosmos which had somehow revealed the need to memorize these to everyone else but him.

Instead of going down that unproductive path to a blind guess, remember this: focusing on the hard part first seldom helps in the GMAT. When the bottom-down approach fails because it includes a step or two that are really insurmountable on their own, bypass the problem – start with the easy part. In other words – forget about what you don’t know, and start with what you KNOW.

What you do know is that √4=2. Replace √4 with 2, and the question now asks whether the value of 2 + ∛4 + ∜4 is within the ranges provided in the answer choices: from “less than 3”, all the way to “greater than 4”. ok, that’s easier to take apart:

If the sum of 2 + ∛4 + ∜4 is “less than 3” or “equal to 3”, then the two advanced roots should equal less than 1 (or equal to 1), and they should both be fractions.
If the sum is between 3 and 4, then the two advanced roots together should equal between 1 and 2 – both are bigger fractions, or one number greater than 1, one small fraction.
On the other extreme, If the sum of 2 + ∛4 + ∜4 is “greater than 4”, then the two advanced roots together should equal more than 2. Again, at least one of them will need to be greater than 1, probably both.

So the real issue of the question is “are ∛4 and ∜4 fractions, or greater than 1?”

This is a much easier question than the intimidating question we originally faced. Since the base is greater than 1, ∛4 and ∜4 cannot be fractions smaller than 1 [ ∛4 raised to the 3rd power should equal 4, but raising a fraction to the 3rd or 4th power will result with an even smaller fraction – not with the base of 4]. So the sum of the three roots, while impossible to calculate precisely without a pocket calculator, must be greater than 4: √4 + ∛4 + ∜4 is greater than 2+1+1=4.

The answer is E.

Notice how the simple action of replacing √4 with 2 can put a difficult question in a much easier context: not “what is the value of ∛4 and ∜4 (impossible to calculate), but “are ∛4 and ∜4 fractions or greater than 1?” (doable).

Main takeaways:
1) Math skills: higher order roots are scary looking, but at the end of the day, a root is just a number. Specifically, a number that, when raised to the exponent, will yield the base of the root.”¨2) Don’t be intimidated by a seemingly unsolvable portion of the problem. Start where it’s easy – start with what you know.

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