Articles published in Fractions, Decimals, Percents

The 4 Math Strategies Everyone Must Master, Part 1


We need to know a lot of different facts, rules, formulas, and techniques for the quant portion of the test, but there are 4 math strategies that can be used over and over again, across any type of math—algebra, geometry, word problems, and so on.

Do you know what they are?

Try this GMATPrep® problem and then we’ll talk about the first of the four strategies.

* ” If mv < pv < 0, is v > 0?

“(1) m < p

“(2) m < 0”

All set?

How did you do the problem? Most quant questions have more than one possible approach and this one is no exception—but I want to use this problem to talk about a particular technique called Testing Cases.

This question is called a “theory” question: there are just variables, no real numbers, and the answer depends on some characteristic of a category of numbers, not a specific number or set of numbers. When we have these kinds of questions, we can use theory to solve—but that can get very confusing very quickly. Instead, try testing real numbers to “prove” the theory to yourself.

(Note: I chose a particularly tough question for this exercise; testing cases can also be useful and fast on easier questions!)

This problem gives one inequality:

“mv < pv < 0″

The test writers are hoping that you’ll say, “Oh, let’s just divide by v to get rid of it, so the equation is really m < p < 0.” But that’s a trap! Why?

When you divide an inequality by a negative, you have to flip the signs. But you don’t know whether v is positive or negative, so you don’t know whether to flip the signs! Never divide an inequality by a variable if you don’t know the sign of the variable.

The question itself contains a clue (two, actually!) pointing to this trap. The given inequality asks about “< 0” and the question also asks whether v > 0? Less than zero and greater than zero are code for “I’m testing you on positives and negatives.”
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GMAT Challenge Problem Showdown: October 14, 2013


challenge problem
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!

Here is this week’s problem:

If acdx, and y are positive integers such that ay < x and  is the lowest-terms representation of the fraction , then c is how much greater than d? (If  is an integer, let d = 1.)

(1)  is an odd integer.

(2) a = 4

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GMAT Challenge Problem Showdown: September 30, 2013


challenge problem
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:

For how many different pairs of positive integers (ab) can the fraction  be written as the sum ?

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Ratios: Box ‘Em Up (Or Just Pour A Drink)


On the GMAT, you may see a 3 to 5 ratio expressed in a variety of ways:

3 to 5
x/y = 3/5
5x = 3y (Yes, that’s the same as the other 3. Think about it.)

In the real world, we encounter ratios in drink recipes more often than anywhere else (3 parts vodka, 5 parts cranberry),don't drink and derive perhaps explaining why–after drinks that strong–we forget how to handle them.

Keep in mind: ratios express a “part to part” relationship, whereas fractions and percentages express a “part to whole” relationship. So the fraction of the above drink is 3/8 vodka (or 37.5% of the whole). Either way, hold off on mixing that drink until after this post.

I like to set up ratios using a “ratio box.” The box is a variant on the “Unknown Multiplier” technique from page 65 of our FDPs book, but it’s a nice way to visually manage ratios without resorting to algebra.

Let’s take the beginning of a typical ratio question:

“The ratio of men to women in a class is 3:2…”

Instead of doing anything fancy with variables, I just set up a tracking chart:

Men Women Total
Ratio 3 2 5

From this point alone, I have sufficient information to answer a bunch of questions.

-What fraction of the students are men? (3/5)

-What percent of the students are women? (40%)

-What is the probability of choosing a man? (3/5)


However, I have nowhere near enough information to answer anything about the REAL numbers of students in this class. Suppose the GMAT were to add a little more information:

“The ratio of men to women in a class is 3:2. If there are 35 students in the class…”

Now we can calculate almost everything about the real numbers of people. First, make a bigger box with 3 lines. The unfilled box looks like this:

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Challenge Problem Showdown- April 8, 2013


challenge problem
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:

If ab, and c are integers such that 0 < a < b < c < 10, is the product abc divisible by 3?

(1) If   is expressed as a single fraction reduced to lowest terms, the denominator is 200.

(2) c “ b < b “ a?

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5 Simple Math Tricks for Faster Computations


BenGMAT FranklinFor every five hours of studying combinatorics-type questions, the average GMAT student increases their chances of being able to correctly answer a question type that is found only on the very difficult end of the GMAT spectrum. Meanwhile, the same student will have to compute hundreds of basic computations without the aid of a calculator. For students who know how to quickly do these computations, they are rewarded with extra minutes that can be spent double-checking their work and critically thinking about whether their answers make sense. As BenGMAT Franklin might say- a second saved is a second earned on the GMAT… but it doesn’t matter if those extra seconds come from being faster at doing combinatorics questions or quicker at computations. So check out these five math tricks, learn the ones that you like, and practice them daily to give yourself some extra time to finish off that 37th and final quant question.

Note: like everything else on the GMAT, being able to do something and being able to do something QUICKLY are two different tasks. If you like any of the following tricks, make sure you know it inside and out before you try using it during your test.

1. Add or Subtract 2 or 3 Digit Numbers

To add numbers that aren’t already a multiple of ten or one-hundred, round the number to the nearest tens or hundreds digit, add, and then add or subtract by the number you rounded off. Do the opposite when subtracting.


144 + 48 = 144 + 50 – 2 = 192

1385 – 492 = 1385 – 500 + 8 = 893


This math trick comes down to the order of operations- adding and subtracting occur in the same step and can happen in either order. Like many other computation tricks, this one comes down to replacing one tricky computation with two simpler ones.
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This Fraction Problem Is Harder Than It Looks


I’ve spoken with multiple students lately who received a disappointing (lower than they were expecting) score on the quant section and who all said that the quant felt relatively easy or straightforward. How is that possible?gmat factions

First of all, thinking that a test like the GMAT is easy is actually a warning sign: things probably are not going very well. If the test was going very well, then you’d be seeing some seriously hard “ next to impossible “ problems.

Second, the test writers are phenomenal at writing questions that don’t seem all that complicated but are in fact your worst nightmare. My worst nightmare is not an impossible question “ I know I can’t do it, so I just pick and move on. My worst nightmare is a question that I think I can do, and I spend a decent chunk of time doing it, and then I get it wrong anyway “ even though I’m sure I got it right!

Try this GMATPrep problem and you might see what I mean. Set your timer for 2 minutes. and GO!

*  Of the 3,600 employees of Company X, 1/3 are clerical. If the clerical staff were to be reduced by 1/3, what percent of the total number of the remaining employees would then be clerical?


(A) 25%

(B) 22.2%

(C) 20%

(D) 12.5%

(E) 11.1%

What’s hard about this one? It looks completely straightforward!

Hmm so I have 3,600 employees and 1/3 are clerical. That’s easy: 1,200 are clerical. Then I need to take 1/3 of that, so that’s 400. They want the percent of the total, so 400/3600 = 4/36 = 1/9 = ugh. Let’s see, a little quick long division right, 1/9 is 11.1%. Answer E. Done!

Where’s my mistake? Go find it. (Actually, find both of them.) Also note that my wrong answer was right there in the answer choices!!
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How to Make Weighted Average Problems Easy


Most people don’t like weighted averages, and for good reason. The formula is complicated, and these often come in the form of story problems, which are hard to set up. We’re going to talk today about a couple of great little techniques to make these fast and easy well, easier anyway!

First, try this GMATPrep problem. Set your timer for 2 minutes. and GO!

*  A rabbit on a controlled diet is fed daily 300 grams of a mixture of two foods, food X and food Y. Food X contains 10 percent protein and food Y contains 15 percent protein. If the rabbit’s diet provides exactly 38 grams of protein daily, how many grams of food X are in the mixture?


(A) 100

(B) 140

(C) 150

(D) 160

(E) 200

gmat weighted average rabbitWow. I’m glad I don’t have to feed this rabbit. This sounds annoying. : )
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Breaking Down GMATPrep Weighted Average Problems


employeesThis week, we’re going to tackle a GMATPrep question from the quant side of things. We’ll tackle a medium-level question this week in order to learn how to master weighted average questions in general, and in the next article, we’ll try a very hard one “ just to see whether you learned the concept as well as you thought you did. : )

Before we begin, I want to mention that every weighted average problem I’ve seen on GMATPrep is a Data Sufficiency question. This doesn’t mean that they’ll never give us a Problem Solving weighted average problem, but it does seem to be the case that the test-writers are more concerned with whether we understand how weighted averages work than with whether we can actually do the calculations. So we’re going to work on that conceptual understanding today and then we’ll discuss a neat calculation shortcut next week (built on the same principles!), just in case we do need to solve.

Let’s start with a sample problem. Set your timer for 2 minutes. and GO!
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Recognizing Relative Numbers On The GMAT


Given the statement, the ratio of men to women in the room is 3 to 5, how many men are in the room?

You probably recognize pretty quickly that it is not possible to answer the question above.  Just given a ratio, it is not possible to identify the actual number of men in the room.  At this point we know the number of men in the room must be a multiple of 3, but the actual number could be 3 or 3,000 (although I am not sure I have been in a room that large).

Along with ratios in their traditional form (3 to 5 or 3:5), there are other types of numbers that are ratios, slightly disguised

a) Fractions: The container is 2/3 full.

This statement is expressing that there are 2 full parts for every 3 total parts of the container (a ratio of 2 to 3).

b) Percentages: 33% of company employees have Master’s degrees.

This statement is expressing for every 33 employees with Master’s degrees there are 100 total employees (a ratio of 33 to 100).

c) Percentage or fractional increase: The company’s profits increased 25% (or ¼) from 2010 to 2011.

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