Number sense
Many GMAT Quantitative Problems, like the foregoing one, test number sense. What is number sense? Number sense is a good intuition for what happens to different kinds of numbers (positive, negative, fractions, etc.) when you perform various arithmetic operations on them.
Number sense is what allows some folks to "see" shortcuts such as estimation or visual solutions. For example, in the problem above, there's absolutely no need to do any detailed calculations: in fact, folks with number sense can probably do all the math they need to do in their heads.
Examples of a few number sense facts
1. Making the numerator of a fraction bigger makes the whole fraction bigger.
2. Making the denominator of a fraction bigger makes the whole fraction smaller.
3. (big positive) + (small negative) = something positive
4. (small positive) + (big negative) = something positive
5. Multiplying by a positive decimal less than one makes something smaller.
6. Dividing by a positive decimal less than one makes something smaller.
Of course, it would be near impossible to make anything like a complete list. The left-brain reductionist dreams of something like an exhaustive list one could study, but number sense is all about left-brain pattern matching. If you're not familiar with the distinction of left/right hemisphere, see this GRE post which touches on similar issues.
How do you get number sense?
If you don't have it, how do you get it? That's not an easy question. There's no magical shortcut to number sense, but here are some concrete suggestions.
1. Do only mental math. You shouldn't be using a calculator to practice for the GMAT anyway. Try to do simpler math problems without even writing anything down. Furthermore, look for opportunities every day, in every situation, to do some simple math or simple estimation (e.g. there are about 20 cartons of milk on the grocery store's shelf --- about how much would it cost to buy all twenty?)
2. Look for patterns with numbers. Add & subtract & multiply & divide all kinds of numbers --- positive integers, negative integers, positive fractions, negative fractions, and look for patterns.
3. This is a BIG one --- in any GMAT practice problem that seemed (to you) to demand incredibly long calculations, but which had a very elegant solution of which you would have never dreamt ---- that problem & its solution are pure gold. In a journal, write down what insights were used to simplify the problem dramatically. Force yourself to articulate this, and return to this solution and to your notes on it often. Over time, you should develop an array of problems like this, and if you study those solutions, you probably wills start to see patterns.
4. Similar to #3: search the two fora, GMAT Club and Beat the GMAT, for similarly difficult questions, and look for elegant solutions. That's a great place to ask the experts (including yours truly) for more detailed explanations of their choices in the solution.
5. Here's a variant on a game you can play, alone or with others who also want practice. Pick four single digit numbers at random --- some repeats are allowed. You could roll a die four times, and use the results. Now, once you have those four numbers, your job is to use all four of them, and any arithmetic, to generate each number from 1 to 20. By "any arithmetic," I mean any combination of:
a. add, subtract, multiply, divide
b. exponents
c. parentheses & fractions
For example, if the four numbers I picked were {1, 2, 3, 4}, I could get 2 from
(4 – 3) + (2 – 1)
or
or
For any one number, you only need to come up with it in one way. Here, I show three ways just to demonstrate the possibilities. Using similar combination, you have to get every number from 1-20 with these four, or with whatever four you pick. Actually, the set {1, 2, 3, 4} is a very good warm-up set. When you want more of a challenge, use {2,3,3,5}.
Practice problem
Here's a practice problem that demands number sense. If you didn't get anywhere with the practice problem, you may want to study the solution below carefully.
2) http://gmat.magoosh.com/questions/54
Practice problem solution
1) Notice that all three of these are close to fractions that equal 1/3. The fractions that equal 1/3 would be, respectively, 50/150, 110/330, and 300/900. First of all, only the second one has a higher denominator, so the second one is more than 1/3 and the other two are less than 1/3. Therefore, II is the greatest.
Now, from I and III, which is greater. Well, think about it this way. 50/150 = 300/900, because both of those equal 1/3. How much less than one third is each one of these? Well, 147/150 is 3/150 less than 1/3, and 299/900 is 1/900 less than 1/3. Well, clearly, 3/150 > 1/900 (the latter has a smaller numerator and a larger denominator!) Therefore, starting from 1/3, 147/150 goes down further than does 299/900. Therefore, 147/150, dropping down a larger distance, must be the minimum value. Therefore, the correct order is I, III, II. Answer = B
This post was written by Mike McGarry, GMAT expert at Magoosh, and originally posted here.
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