Never fear! In this post, we’ll tell you everything you need to know about fractions and decimals for the GMAT. We’ll give you a refresher on all the relevant rules and formulas, tips and tricks for every question you’ll see on them on the GMAT, and some example questions with thorough explanations so you can see these strategies in action.

GMAT Fractions: Rules to Know

Below are all the rules that you need to know about fractions for the GMAT.

Definition of a Fraction

A fraction is a visual representation of a number divided by another number. The top number of a fraction is called the numerator, and it’s the number being divided. The bottom number is called the denominator, and it’s the number that the top number is divided by. 

In the fraction $n/d$, $n$ is the numerator (the top number) and $d$ is the denominator (the bottom number). The fraction 1/2, for example, is 1 divided by 2 or one-half.

0 can’t be the denominator in a fraction, because dividing by 0 is undefined.

Two fractions are equivalent when they represent the same number. For example: 2/8 and 4/16 are equivalent, because they both equal 0.25.

When both the numerator and the denominator can be be divided evenly by the same number, the fraction can be simplified into its lowest terms (the smallest equivalent fraction). The largest number that both the numerator and the denominator can be divided by is called the greatest common factor (GCF) or greatest common divisor (GCD). Dividing both by the GCD simplifies the fraction into its lowest form.

For example, 2 is the greatest common factor of both 2 and 8. For the fraction 2/8, when you divide the numerator and the denominator by 2, you get 1/4—the lowest or most simplified form of the fraction. When dealing with fractions in equations, you almost always want them to be in their simplest forms, so that they’re easier to do calculations with.

Multiplying and Dividing With Fractions

Multiplying with fractions is easy: you just multiply the numerators and multiply the denominators.

For example:

$7/10 × 4/9 = 28/90$ or  $14/45$

To divide with fractions, “flip” the fraction after the division sign (called the divisor) so that the denominator becomes the numerator and vice versa, and then multiply with that number.

Example:

$${7/10} ÷ {4/9} = 7/10 × 9/4 = 63/40$$

This “flipped” version of a fraction is called its reciprocal or inversion. The reciprocal or inversion of any fraction $n$/$d$ is $d$/$n$ (where $n$ and $d$ ≠ 0).

Adding and Subtracting With Fractions

Two fractions with the same denominator can be added or subtracted easily. You simply add or subtract the numerators, and leave the denominators the same.

$$3/8 – 2/8 = 1/8$$
$$5/9 – 1/9 = 4/9$$

If you need to add or subtract with fractions that don’t have the same denominator, then you can do the opposite of simplifying and express them as equivalent fractions with the same denominator. As long as you multiply or divide the numerator and the denominator of a fraction by the same number, it will remain equivalent:

$$3/8 × 9/9 = 27/72$$
$$2/8 = 27/72$$

This gives us an always true rule, which is helpful in algebraic expressions:

$${x + y}/z = x/z + y/z$$

When adding or subtracting fractions with different denominators, multiplying the fractions so that the denominators represent the least common multiple (the lowest number that both denominators factor into) is usually the simplest way to go and makes doing calculations easier than working with larger numbers.

Example:

$$1/3 + 3/4$$
$$LCM = 12$$
$$1/3 × 4/4 = 4/12$$
$$3/4 × 3/3 = 9/12$$

$$1/3 + 3/4 = 4/12 + 9/12 = 13/12=1 1/12$$

By multiplying 3 and 4, we see that the LCM is 12. We then convert both fractions so that they both have a denominator of 12. Then it’s easy to add them together!

Mixed Numbers

A number made up of a whole number and a fraction (like 1 and 1/12 above) is called a mixed number. To change a mixed number into a fraction, multiply the whole number by the denominator and then add the result to the numerator. This then becomes the new numerator.

$$6 4/9 = {(6 × 9)+ 4}/9 = {54 + 4}/9 = 58/9$$

GMAT Decimals: Rules to Know

Below are all the rules you need to know about decimals for the GMAT.

Definition of a Decimal

Decimals and fractions are both ways of representing number values in between integers or whole numbers.

In the decimal system, the distance from the decimal point represents the place value of each number. For example, the number 412.735, has 4 in the “hundreds” place, 1 in the “tens” place, and 2 in the “ones” or “units” place; and then after the decimal, 7 in the “tenths” place, 2 in the “hundredths” place, and then five in the “thousandths” place. Here’s a table illustrating this information:

The Zero Rule

After you pass the decimal point, you can add an infinite number of zeros to the end of a number:

$$1.435 = 1.4350 = 1.4350000000000000000 = 1.43500000000000000000000000000000000000000$$

This rule only applies to after the end of the number after the decimal point:

$$1435 ≠ 14350$$
$$1435 = 1435.0 = 1435.0000000000000000000000000$$

Adding and Subtracting With Decimals

To add or subtract two decimals, the decimal places of each need to line up. You can use the zeros rule above if one number has fewer digits to the after decimal place than the other:

7.872 + 6.30285 =

    7.87200
+  6.30285
= 14.17485

Multiplying and Dividing With Decimals

When multiplying decimals, do not line up the decimal point: the decimal gets inserted afterward. Instead, multiply the two numbers as if they were whole numbers. Once you have the product, it’s time to put the decimal back in.

But how do you figure out where the decimal place goes? The rule is that you add up the amount of numbers after the decimal of each number you multiplied, and that sum is the number of decimal places that should be in the product: 

1.56 (two numbers after the decimal)
× 2.3 (one number after the decimal)
= 3.588 (three decimals—sum of one and two above)

To divide any number (a dividend) by a decimal (the divisor) using long division, move the decimal point of the divisor to the right however many places it takes to get to a whole number, and then move the decimal point in the dividend over by that many places as well. If there’s still a decimal left in the dividend after this, make sure you place it directly above the dividend in the answer.

Finally, do the division as you normally would. For example,

90.625 ÷ 12.5 becomes 906.25 ÷ 125

Then you do the long division with 906.25 as the dividend and 125 as the divisor, making sure to place the decimal in the answer directly above its place in the dividend.

Converting Decimals to Fractions

Every decimal can be expressed as a fraction with these steps:

1. Move the decimal point over however many places to the right until it becomes a whole number
2. Use that as the numerator
3. Place in the denominator the power of 10 that corresponds to however many places you moved the decimal over:

$$0.5 = 5/10$$
$$0.05 = 5/100$$
0.005 = 5/1000 or  1/200

Another way to think of it is that the number of places you move the decimal to the right to make the numerator a whole number is the number of 0’s you’ll add after 1 in the denominator.

Numbers less than -1 or greater than +1 with decimals can be expressed as fractions using the above rule in combination with the mixed number rule:

$$7.5 = 7 5/10$$
$$= {(7 × 10) + 5}/10$$
$$= [70 + 5]/10 = 75/10$$

And this can be simplified:

$$75/10 = 15/2$$
$$7.5 = 15/2$$

Converting Fractions to Decimals

When you plug in a fraction as a division problem into a calculator, it automatically gives you the decimal equivalent. Unfortunately, we don’t have access to a calculator on the GMAT Quant section, but the manual conversion isn’t too hard.

You can always find the decimal equivalent of a fraction with long division, by using the numerator as the dividend and the denominator as the divisor. But there’s an alternative method that can be handy as well.

First, find a number you can multiply the denominator of the fraction by to make it 10, or 100, or 1000, or any 1 followed by 0s. Next, multiply both numerator and denominator by that number to get its equivalent expression. Finally, write down just the top number, putting the decimal point in the corresponding place: one space from the right hand side for every zero in the bottom number.

Here’s an example using the fraction 3/4:

$$3/4 = ?/100$$
$$4 × 25 = 100$$
$${3 × 25}/{4 × 25} = 75/100$$
$$= 0.75$$

Scientific Notation of Decimals

“Moving over” decimal places with powers of 10 is a useful concept. Sometimes, numbers are expressed as the product of a number multiplied by 10 to a certain power. The power represents how many places you need to “move” the decimal point to get to its decimal expression. The sign of the exponent indicates which direction: a positive exponent moves the decimal over to the right, and a negative exponent moves it to the left.

Examples:

$$0.0489 = 4.89 × 10^{-2}$$
$$60235 = 6.0235 × 10^4$$
$$540 = 5.4 × 10^2$$
$$29 = 2.9 × 10^1 = 2.9 × 10$$

Terminating and Recurring Decimals

Terminating decimal GMAT questions sound scary if you don’t know what a terminating decimal is, but it’s actually deceptively simple.

All of the decimals in the examples above have an end. They are called terminating decimals because there aren’t an infinite amount of numbers after the decimal point. Any terminating decimal is can be represented as a fraction with a power of ten in the denominator. For example, 0.0462 = 462/10000 = 231/5000.

It’s possible to have an infinite amount of numbers after the decimal point. 1/3 is an example of a recurring decimal, as we can see when we convert it with long division:

$$1/3 = 0.333333333… = 0.\ov 3$$

The above are equivalent expressions: Both the ellipses and the line above the three indicate that the threes after the decimal point go on forever.

Recurring decimals are tough to work with. Knowing which fractions have infinite decimal expressions, like 1/3 and 1/9, helps significantly in deciding whether to convert it into a decimal or leave it as a fraction in solving problems.

The Key Rule for Fractions That Are Terminating Decimals

If the prime factorization of the denominator of a fraction has only 2 and/or 5, then it can be written as something over a power of ten, which means its decimal expression terminates.

If the denominator doesn’t have only 2 and/or 5 as factors, then the decimal expression is recurring. Here are some examples:

1/24 is recurring (24 = 2³ × 3, so 24 has a prime factor of 3 in addition to 2)

1/25 is terminating (25 = 5²)

1/28 is recurring (28 = 2² x 7, so there’s a prime factor of 7 in addition to 2)

1/40 is terminating (40 = 2³ x 5)

1/64 is terminating (64 = 2⁶)

Importantly, this rule only applies for fractions in their simplest forms. For example, 9/12 terminates, even though 12 has 3 as a prime factor, because 9/12 is really just 3/4, which is 3/2²

One key way to express this rule is that that the denominator must be some value equivalent to 2^m5ⁿ, where $m$ and $n$ are integers. So any fraction that can be expressed as $x$/2^m5ⁿ will terminate, and any other fraction won’t.

Note that the number 1 as a denominator satisfies those requirements, as any number to a power of 0 equals 1, and 0 is an integer, so it can be the value of $m$ and $n$:

1 = 2⁰5⁰

If you need a refresher on what prime factorization is, head to our guide to integer properties for the GMAT, which includes an entire section devoted to explaining prime factorization.

GMAT Fractions Questions

Below are the key kinds of GMAT fraction questions. Note that fractions as a concept overlap with some of the other types of questions, such as rate questions and average questions. The line is often blurry between fractions and decimals as well, and sometimes actually converting the given value to decimals from fractions or vice-versa can make the problem clearer. We’ll see an example of that below.

Example GMAT Fractions Question 1: Problem Solving and Averages

Here’s a GMAT averages problem involving fractions:

This is a fraction question baked into an averages question with algebra. As you may know, the formula for averages is simply to add all the numbers together and then divide by the total number of numbers, which is 4 in this case. This gives us the below fraction:

$${(n+2) + (2n-3) + (4n+1) + (7n+4)}/4$$

We also know from the question that the average is 15, so that equation is equal to 15:

$${(n+2) + (2n-3) + (4n+1) + (7n+4)}/4 = 15$$

To simplify this equation, let’s get rid of the fraction by multiplying both sides by 4 (the denominator):

$${{(n+2) + (2n-3) + (4n+1) + (7n+4)}/4} × 4 = 15 × 4$$
$$(n+2) + (2n-3) + (4n+1) + (7n+4) = 60$$

Since the right side of this equation is all addition and subtraction now, we don’t need the parentheses. Let’s simplify and solve:

$$14n + 4 = 60$$
$$14n = 56$$
$$n = 56/14$$
$$n = 4$$

(B) is the answer.

Example GMAT Fractions Question 2: Problem Solving and Rates

Here is a fraction problem in the context of a GMAT rate problem:

First, let’s make sure we understand what the numerator and the denominator represent in these fractions. The rate is per hour, so we’re talking about tanks (the numerator) per hour (the denominator).

So the rate of the small pump is 1/2 tank/hour, and the rate of the larger pump is 2 tank/hour, or 2/1 (in fraction expression). Together, the combined rate of the two pumps is:

$$1/2 + 2/1$$

You probably know this off the top of your head, but just to illustrate the addition of fractions, I’ll show you how to do it out. We need the lowest common multiple of the denominators so that we can render them both as expressions with the same denominator.

The LCM is 2, so:

$$2/1 + 2/1 = 1/2 + 4/2 = 5/2 \tanks \per \hour$$

To get to the time it takes to fill the tank, we need to divide the job (filling 1 tank) over their collective rate (5/2 tanks per hour).

Hence together they will fill the tank in $1/(5/2)$. Let’s use the rule about fraction division—that it’s simply multiplication with the numerator and denominator flipped—to simplify this:
$$1/(5/2) = 1 × 2/5$$
$$= 2/5 \hours$$

The answer is (C).

Example GMAT Fractions Question 3: Problem Solving and Probability

A basic knowledge of fractions is required for GMAT probability problems as well. Here’s an example:

As with many questions on the GMAT, this problem is simpler than the lengthy wording makes it sound.

We can think of the first 2 cards as cards that have already been turned “face up” and are therefore out of the pile. So the probability of picking a stock card goes from 8/48 to 8/46. Let’s simplify:

$$8/48 = 4/23$$

(E) is the answer.

Example GMAT Fractions Question 4: Problem Solving With Algebra

Sometimes it will be useful to come up with your own algebraic equation to solve a GMAT fractions question. Here’s an example:

Let the price of basic computer be $c$, and the price of the printer be $p$.

What do we know? We know that $c$+$p$=2500. We also know that the price of the enhanced computer will be $c$+500, since the question stem tells us that it’s 500 dollars more than the basic computer. So the total price of the enhanced computer and the printer is 500 dollars more than 2500, or 3000 dollars.

Now, we are told that the price of the printer is 1/5 of that new total $3000 price. Let’s figure that out:

$$p = 1/5 × $3000$$
$$= $3000/5$$
$$= $600$$

Now that we know how much $p$ (the printer) is, we can plug this value in the first equation to solve for $c$ (the basic computer):

$$c + $600 = $2500$$
$$c = $2500 – $600$$
$$c = $1900$$

The answer is (D).

Example GMAT Fractions Question 5: Data Sufficiency

Here is a relatively simple data sufficiency fraction problem:

So all we know from the prompt is that 4 servings of Malik’s dish require 1 and 1/2 or in decimal expression 1.5 cups of pasta.

Statement 1 is insufficient because it just says: “Malik will make half as many servings as he did the last time he prepared the dish.” However, we have no idea how many servings Malik prepared last time. Since we don’t know the servings, we can’t find how much pasta is required. Hence, insufficient. Eliminate (A) and (D).

Statement 2 says Malik used 6 cups of pasta the last time he prepared this dish. Just looking at this statement by itself (without statement 1), it doesn’t really indicate anything: if 6 cups of pasta was the last time, we clearly can’t say how many cups of pasta Malik will use the next time. Hence, insufficient. Eliminate (B).

Now, let’s combine statements 1 and 2. We know that Malik used 6 cups of pasta the last time and that he will make half as many servings as he did the last time. That being the case, Malik will clearly require 3 cups of pasta next time (1/2 of 6 = 3). Sufficient.

Thus, (C) is the answer.

Example GMAT Fractions Question 6: Data Sufficiency With Algebra

Here’s a slightly more advanced data sufficiency question with fractions, involving algebra:

This question is basically asking us if $x$ is a fraction/decimal between 0 and 1. Let’s work methodically through the statements.

Statement 1 is insufficient because there are many values not between 0 and 1 that satisfy the condition of being between -1/2 and 3/2. If that’s not obvious, you might want to convert the statement into decimals. In decimal form, all statement 1 is telling us is that $x$ is between -0.5 and 1.5. So if $x$ was 1.1, 1.2, 1.3, -0.4, etc., it would be between -0.5 and 1.5 but not between 0 and 1. Hence, statement 1 is insufficient. Eliminate (A) and (D).

Now let’s test statement 2. Statement 2 is just an overcomplicated way of saying that:

$$x = 3/4 – 1/4$$

So we solve for $x$ very easily:

$$x = 3/4 – 1/4 = 1/2$$

1/2 is between 0 and 1, so statement 2 is sufficient and the answer is (B).

GMAT Decimal Questions

Below are the key kinds of GMAT decimal questions. Like fractions, decimals questions overlap with other kinds of questions, and often come with a fraction aspect as well. The GMAT particularly loves to test you on the concept of terminating versus recurring decimals, so we’ve included several examples of that below.

Example GMAT Decimal Question 1: Problem Solving With Terminating and Recurring Decimals

This question tests you on both fractions and decimals. One key rule to remember for this question is that a fraction in its simplest form with a denominator that has only 2 and/or 5 its prime factors will convert to a terminating decimal:

x/2^m5ⁿ = terminating decimal

Head back to the section on terminating and recurring decimals above if you need more of a refresher.

Now, onto the question.

First, let’s rephrase it algebraically. Let $p$ = the prime integer that’s greater than 100, which = the sum of all the deposits up to and including the day. Let $d$ be the number of days, up to and including the chosen one ($d$ = 1 would be June 1, $d$ = 30 would be June 30).

The average daily deposit up to and including the chosen day will be the sum of the deposit divided by the number of days, or $p$/$d$.

So the question becomes: What is the probability that $p$/$d$ will have less than 5 decimal places?

Now that we know what we’re being asked, the next step is to hone in on only the days that would have a terminating decimal, since those that yield a recurring decimal will by definition have more than 5 decimal places.

As stated above, to be a terminating decimal, $p$/$d$ must = x/2^m5ⁿ, so $d$ must = 2^m5ⁿ. And luckily, because the numerator $p$ is a prime number, all the possible values of $p$/$d$ will be in their simplest forms, so we can test the denominator without worrying that the terminating decimal fraction rule might not apply. .

The days in June (values for $d$) that can be expressed as 2^m5ⁿ and are thus not recurring are day 1, 2, 4, 5, 8, 10, 16, 20, and 25. You can figure this out by doing a prime factorization of each of the 30 days in June, but as long as you’re still know your multiplication tables, you should be able to look at a number between 1 and 30 and realize almost right away if it has a prime factor other than 2 and/or 5.

So now, of days 1, 2, 4, 5, 8, 10, 16, 20, and 25, we have to check if any of them have more than 5 decimal places, which is possible even though they do terminate. We can do this using the rule for converting fractions to decimals. 5 decimals is the ten thousandths place, so to have 5 decimal places or less, $p$/$d$ × 10,000 must yield an integer:

p/d × 10000 = an integer (nothing after the decimal point)

For this to work, $d$ will have to be a factor of 10,000. As it happens, all of these numbers go into 10,000 (10,000 is divisible by 1, 2, 4, 5, 8, 10, 16, 20, and 25), so for all 9 of these $d$’s, $p$/$d$ = a number with less than 5 decimals.

Thus, out of all of the days of June, there are 9 values for $d$ for which $p$/$d$ has less than 5 decimal places, so the probability is 9/30 = 3/10.

(D) is the answer.

Example GMAT Decimal Question 2: Problem Solving and Scientific Notation Format

Here’s an example of a GMAT terminating decimals question in which knowledge of scientific notation comes in handy.

First, let’s multiply both the numerator and the denominator by 2⁴, so that we can get the exponents of both the base numbers in the denominator to be the same:

$$1/{2^3×5^7} × 2^4/2^4 = 2^4/{2^7 × 5^7}$$

Now we can multiply the bases together, and put the exponent 7 with the result:

$$2^4/{2^7 × 5^7} = 2^4/10^7 = 16/10^7$$

16/10⁷ is the same as 16 × 10^-7

16 × 10^-7 is just the scientific notation for the decimal 0.0000016 (you move the decimal to the left 7 times). Thus, $d$ will have two non-zero digits, 16, when expressed as a decimal. The answer is (B).

Example GMAT Decimal Question 3: Problem Solving and Estimation

This is a great example of a GMAT decimal question in which you should use your powers of estimation instead of solving it:

We don’t want those plus signs in this fraction—let’s do the additions and see what the resulting fraction looks like:

$$1+0.0001/{0.04+10} = 1.0001/10.04$$

Now we can see that these tiny little decimals are negligible: basically, the numerator is 1 and the denominator is 10. So this fraction is virtually 1/10, which equals 0.1. (C) is the answer.

Example GMAT Decimal Question 4: Data Sufficiency and Terminating Decimals

Lots of GMAT data sufficiency decimal questions will ask you if a certain equation or variable is a terminating decimal. Here’s an example:

Statement 1 indicates that $x$, the numerator, is a multiple of 2, which has nothing to do with the terminating or recurring property of decimals—that’s based on the denominator.

We can test this by plugging in multiple-of-2 values for $x$: 2/4 is a terminating decimal, but 4/6 is a recurring decimal. So, statement 1 is not sufficient. Eliminate (A) and (D).

Statement 2 says that y is a multiple of 3. You might be tempted to say that this violates the denominator = 2^m5ⁿ rule, but be careful! Statement 2 gives no information about whether or not $x$ and $y$ have common factors. For instance, 12 is a multiple of 3, but 9/12 is terminating, since it simplifies into 3/4. But 8/12 is recurring, as it simplifies 2/3. into So statement 2 is also not sufficient. Eliminate (B).

Now, let’s plug in numbers to test statement 1 and 2 together. 4/9 satisfies both the statements and it’s recurring, but 18/24 also satisfies both requirements and it’s terminating. So even together the statements are not sufficient and the answer is (E).

Tips for GMAT Fractions and Decimals Questions

Below are the key tips for mastering fraction and decimal questions on the GMAT.

#1: Memorize the Decimal Conversion for All Single-Digit Fractions

When it comes to fractions, being able to convert them to and from decimals with ease will help you get the correct answer faster on many different kinds of GMAT questions. Just because a question is ostensibly asking about medians or areas or probability doesn’t mean that you won’t need to work with fractions at some point to solve the question.

Here’s the conversion for 1/2 through 1/9:

1/2: 0.500
1/3: 0.333
1/4: 0.250
1/5: 0.200
1/6: 0.167 (half of 1/3)
1/7: 0.143 (just need to know this one)
1/8: 0.125 (half of 1/4)
1/9: 0.111

#2: Memorize the Terminating Decimal Rule for Fractions

In addition to the basic conversion, take some time to memorize the rules and properties sections above—especially the x/2^m5ⁿ rule for terminating decimals.

#3: Convert Freely Between Fractions and Decimals as Needed

By the time you take the GMAT, you should be able to fluently convert fractions to decimals or vice versa, depending on what will make a given problem easier. As you do more and more practice questions, you’ll become better at detecting which expression will be the easiest to use to solve the question. The GMAT will often give you the format that is harder to work with to start, as they are testing both your fluency with fractions and decimals and your ability to come up with the best route for solving the problem on your own.

For example, if you’re trying to determine where a variable x falls on the number line, it’s probably easier to work in decimals than in fractions. On the other hand, if you’re given a number like 0.111111111111111 and you have to do algebra with it, it’s probably easier to use 1/9—especially since the answer options will likely be spaced far enough apart that .000000001 of a difference isn’t going to leave you stuck between options.

Remember, you don’t have a calculator for the Quant section, so if you encounter a question on the GMAT that seems impossible to solve without one, there’s almost always a rule, property, or different form of expression that you can use to make it easier. Keep an eye out for strange wording that’s obscuring a very basic principle, and try converting the given fractions to decimals or vice versa if you’re stuck.

#4: Look at the Answer Options Before Solving

In the terminating decimal GMAT question about what the fraction with those tiny decimal points was closest in value to, you may have been tempted to solve for the exact value. But a quick glance at the answer options, which are all spaced out by a power of ten, tells you that all you have to do is get the location of the decimal point correct—not the exact value. That’s a pretty wide margin.

You should always look at the answer choices before even beginning to solve a problem—they’ll clue you in to the right approach.

Whenever you see tricky-looking decimals with widely varying answer choices, as in the terminating decimal GMAT question above, that best approach might be simply to estimate.

#5: Don’t Solve Further Than You Need To

Speaking of approaches, as with the first terminating decimal GMAT question, in the example of Malik’s serving dish you may have been immediately tempted to find how many cups of pasta are needed per serving. 4 servings of the dish required 1.5 cups, so the amount of cups per serving is 1.5/4, or 15/40 when we multiply both numerator and denominator by ten to get rid of the decimal in the fraction. This simplifies to 3/8 a cup of pasta per serving.

But this information is actually useless for finding sufficiency, as the two statements gave us everything we needed to solve the problem of how many cups of pasta Malik will use the next time he makes the dish.

Bottom line: Glance at the answer choices first, and then focus on solving only what you need to solve to get to one answer choice.

What’s Next?

Another key topic to become familiar with for the GMAT Quant section is integers.

For more general advice, check out our 10 tips to master the Quant section.

If you’re looking for more tips, here’s our list of the best tips and shortcuts for doing well on the Quant section.

Happy studying!

The post GMAT Fractions and Decimals: Everything You Need to Know appeared first on Online GMAT Prep Blog by PrepScholar.

If Hayley takes the GMAT three times and scores 750, 770, and 800, respectively, what is her mean score? You probably won’t get a statistics question that’s quite that easy on the GMAT, but the good news about GMAT statistics questions is that they test mostly simple concepts. You don’t need to know any complicated formulas or equations to master statistics for GMAT quant questions, but you do need to develop a solid understanding of descriptive statistics principles.

In this article, I’ll give you an overview of what’s tested in GMAT statistics questions, define the four key definitions you need to know, and give you tips for mastering statistics questions. I’ll also walk you through two GMAT statistics sample questions. And, as an added bonus, I’ll offer up a range of terrible statistics puns. (Get it?) By the end of this article, you’ll have mastered statistics for GMAT!

What’s Tested in GMAT Statistics Questions?

Many test takers are intimidated by statistics questions on the quant section, particularly the notorious GMAT standard deviation problems. However, GMAT statistics questions are much simpler than they’re often believed to be.

There are typically between four and eight statistics questions on the GMAT quant section. But because questions on the GMAT quant section often combine more than one type of math, you may see basic statistics principles (like mean) pop up in other types of questions, too.

Statistics for GMAT are mainly concerned with finding the center and the spread of a set of values.

Mean and median are the terms that we use to find the center of a set of values. The center of a set of values helps us understand the middle and the average of that set of values.

Range and standard deviation measure the spread of a set of values. Just like the term implies, the spread of a set of values is all about how far the values are spread out from each other. Range is a very simple way for determining spread. Standard deviation is a much more complicated measure of spread. I’ll talk about what all of these terms mean in the next section.

The 5 Key Definitions for GMAT Statistics Questions

There are five key definitions you need to know in order to master GMAT statistics problems.

Mean

You find the mean, or average, of a set of values by dividing the sum of the values in a data set by the number of values.

How do we find the mean?

Well, consider the data set 5, 10, and 15. To find the mean of this data set, first you would find the sum of all values, which is 30 (5 + 10 + 15 = 30). Next, you would divide 30 (the sum of all the values) by 3, which is the number of values in the data set.

The mean of that data set is 10.

Median

The median is the middle number in a set of values. Here’s how you find the median of a data set.

First, consider the data set: 5, 3, 4, 1, 2.

Order the numbers from smallest to largest.

1, 2, 3, 4, 5.

3 is the middle number, which means the median is 3.

What about if you have an even set of values, such as 5, 4, 1, 2?

Order the numbers from smallest to largest.

1, 2, 4, 5.

Find the sum of the two middle numbers (4+2 = 6).

Divide the sum of the two middle numbers by 2 (6/2 = 3).

3 is the median of that data set.

Mode

The mode is the value in a data set that appears most frequently.

Consider the data set 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8.

The number 4 appears three times in the data set. It appears the most.

The number 4 is the mode of that data set.

Range

The range of a set of values is the difference between the maximum and minimum values of that set.

For instance, in the set 100, 150, and 200, you would find the range by subtracting the minimum value (100) from the maximum value (200).

The range of that set of values is 100, then.

Standard Deviation

Standard deviation GMAT questions often trip up test takers. But GMAT standard deviation questions actually test a relatively simple concept.

Standard deviation measures how far the values in a data set are away from the mean (in other words, the deviation from the mean).

Let’s break down what that means.

In order to find standard deviation, you need to first calculate the mean. Let’s start with the data sets {0, 2, 4} and {1, 2, 3}.

We find the mean of each of these data sets by finding the sum of the data sets and dividing by the number of values in each data set. Both data sets have a mean of 3.

If we look at the first data set, {0, 2, 4} we can see that the numbers are more widely spread away from the mean (3) than the numbers in the data set {1, 2, 3}. That means that the standard deviation for {0, 2, 4} is greater than the standard deviation for {1, 2, 3}.

There’s a four step process for finding standard deviation. I’ll talk about that later when I give you tips for mastering GMAT statistics questions. Until then, make sure you solidly understand what the concept of standard deviation is.

Tips for Mastering GMAT Statistics Questions

GMAT statistics questions can seem intimidating. But don’t worry! These four tips will help you get a better mean score on your GMAT quant practice.

#1: Learn These 4 Rules for Standard Deviation GMAT Questions

Standard deviation GMAT questions often seem incredibly tricky. To make matters worse, standard deviation often pops up in data sufficiency questions, which are tricky enough on their own. However, there are several rules that you can memorize to help boost your understanding of standard deviation.

If every value in a data set is equal, they all equal the mean. That means the standard deviation is zero. This is the lowest possible standard deviation you can have. You can’t have a negative standard deviation.

If you change the values of the numbers in your data set by adding or subtracting the same number to each value, the standard deviation will stay the same. Basically, if you have a set of values such as {1, 2, 3} and add 5 to each value so that you get the set {6, 7, 8}, your standard deviation will be exactly the same for both sets.

Multiplying always changes the standard deviation of a set of numbers, unless you’re multiplying by 1 or -1. If you’re raising the numbers by a power, that always changes the standard deviation as well.

Adding new numbers to a set changes the mean, and thus the standard deviation. There can be exceptions to this rule (if the numbers line up just right), but it will generally be the case.

#2: Understand the Standard Deviation Formula

As I mentioned before, it’s not necessary to know the standard deviation formula in order to solve GMAT statistics questions. However, developing an understanding of how the standard deviation formula works will help you quickly solve standard deviation GMAT questions by giving you a clearer sense of how the different values involved (standard deviation, mean, and variance) relate to each other.

I won’t go through the full standard deviation formula, but it’s helpful to understand the basic steps:

1. Find the mean of a set of values.
2. Find the differences between each value and the mean.
3. Square all the differences and take the averages of the differences, which gives you the variance.
4. Find the square root of the variance.

Because it’s the square root of the variance, standard deviation is often a weird number. Remember, the GMAT won’t ask you to calculate anything you can do easily by hand, so you won’t need to use this particular four-step process. Rather, familiarize yourself with the concept behind it (finding the mean and the dispersion from the mean) to increase your understanding of standard deviation.

#3: Understand the Effect of Changing Numbers on Value Sets

Adding numbers to your data set doesn’t just affect the standard deviation. It can also affect other statistical measurements, such as the mean, mode, median, and range.

If you add another number to your data set, your mean will almost always be affected. The key exception is if the value added is equal to the mean — in that case the mean will stay the same.

If you add another number to your data set, but it’s greater than the minimum and less than the maximum value, your range will not be affected.

Adding numbers may also affect your median and mode, but it occurs on a case-by-case basis.

Make sure you recalculate each of these values if you’re solving a multi-part question where you introduce a new value to a certain set.

#4: Remember That The Numbers Will Work Out

As I mentioned before, the writers of the GMAT know that you’re not allowed to use a calculator on the quant section. That means that you’ll be able to solve every question using your mastery of fundamental math concepts, a pencil, and scratch paper. If you find yourself calculating an extremely long decimal, you’re probably going down the wrong path. Remember that you’ll be looking mainly for whole number answers, or simple fractions or decimals. If it’s anything more than that, it’s wrong.

GMAT Statistics Questions Examples

In this section, I’ll walk you through how to solve a sample problem solving and a sample data sufficiency statistics questions.

Problem Solving Statistics Question

This question is a prime example of why it’s important to master your statistics fundamentals. For this question, we need to focus on how adding numbers affects the spread and the center.

Let’s look at mean first. Adding 5 to each score would increase every single score, which means that the mean will definitely change. Knowing that, we can automatically rule out answers B and C, which don’t include the mean.

Now let’s look at the median. If we add 5 to each score, we will also change the median, because the set of scores will be completely different.

That means that our correct answer is D. The median and the mean will both change.

But before we move to the next question, let’s talk about why the standard deviation won’t change.

Standard deviation has to do with the spread of numbers. If we add 5 to each score, every number is increasing by the same amount (5). That means that the spread will be exactly the same, since every score has increased by 5.

Data Sufficiency Statistics Sample Question

Let’s start by understanding what the question is asking us. The information tells us that the original 23 paragraphs have a total of 2,600 words. It also asks if the average number of words per paragraph is less than 120, once the two ancillary paragraphs are added.

If the word count is less than 120 per paragraph, then the total number of words in the essay would be less than 25 (number of paragraphs) times 120 (average number of words per paragraph), or 3,000 words.

If the original 23 paragraphs had a total of 2,600 words, that means that the two new paragraphs can only have a total of 3,000 – 2,600 words, or 400 total words, or an average of less than 200 words per paragraph.

Now, since this is a data sufficiency questions, let’s evaluate each statement by itself first

The information provided in statement (1) implies that the total number of words in the 2 added paragraphs is more than (2)(100) = 200. Therefore the number of words could be 201, or the number of words could be 500. This statement is not sufficient, because, our total number of words could be way greater than 400. This statement is not sufficient.

The information implies that the total number of words in the 2 added paragraphs is less than (150)(2) = 300, which is in turn less than 400. The statement is sufficient. The correct answer is B.

Review: What You Need to Know for GMAT Statistics Questions

GMAT statistics questions focus on five major topics: mean, median, mode, range, and standard deviation.

Mean and median are concerned with the “center” of a set of numbers.

Range and standard deviation are concerned with the “spread” of a set of numbers.

The best way to approach GMAT statistics questions is to deeply understand these fundamental principles and how to apply them to complicated scenarios.

What’s Next?

We have the guides to cover every type of content that you’ll see on the GMAT quant section. Our comprehensive guides give you overviews of each section, as well as provide you with sample questions that show you how to apply the tips we’ve discussed. Check out our guides to GMAT geometry and GMAT percentages to start.

The GMAT, unlike other standardized tests, doesn’t allow you to bring a list of formulas or rules to test day. GMAT idioms guide to start boosting your understanding of the essential math concepts you need to memorize before test day. Our downloadable PDF will aid your memorization.

Want to switch gears? We have in-depth guides to help you master the GMAT verbal section, as well. Get an in-depth guide to idiomatic language in our GMAT idioms guide, or learn the tips and tricks you need to ace the verbal passages in our guide to mastering the GMAT verbal.

The post GMAT Statistics and Standard Deviation Questions: 4 Key Tips appeared first on Online GMAT Prep Blog by PrepScholar.

Percents are one of the most common math concepts: in fact, you probably use them every day, whether calculating a tip or determining how much a discount really is. It’s not surprising, then, that GMAT percentage questions pop up throughout the quant section. You may see percents tested by themselves, or they might be part of another question type, such as a geometry question.

Unfortunately, GMAT percent problems can be quite tricky even though the concepts they test are fairly simple. But don’t worry! In this guide, I’ll explain exactly what you need to know to master percentage questions on the GMAT and walk you through sample problems using percents.

What’s Tested in GMAT Percentage Questions?

For many test takers, GMAT percentage questions seem very straightforward and easy. After all, we deal with percentages often in day-to-day life, too calculate taxes, interest rates, tips, and more. However, while GMAT percentages may appear easier than say, standard deviation questions, they can actually be quite tricky. Before we get into some tips for how to solve GMAT percentage questions, let’s talk a bit more about what this question type actually tests.

The word percent can be broken down into “per cent,” which means “through 100.” A percent is a part of 100. The formula for finding a percent is: $\part/\whole * 100$.

For the GMAT percent problems, you’ll mainly be concerned with calculating percent change. A percent change asks you to find the difference in percent between a starting and ending number.

The formula for finding percent change is:

$$\percent \change = (\amount \of \change)/(\starting \amount) * 100$$

Let’s look at that in action.

If the price of an item changes from 10 dollars to 15 dollars and we want to find the percent increase, we would plug values into our equation. The amount of change from 10 to 15 is 5 (15 – 10 = 5) and the starting amount is 10. Thus our equation would be:

$$\percent \change = (5 / 10) * 100$$

$$\percent \change = 50%$$

Percentages will often pop up in geometry and arithmetic questions. For geometry questions, you may be asked to determine by what percentage the area of a square increases if the length of its sides is increased by 20%. Or, you may be asked to determine how a decrease in the area of a triangle corresponds to its sides.

You’ll see all kinds of applications in arithmetic questions. You may be asked to calculate the percent change of a savings account, or the profits of a company. You may be asked to calculate percent change of the output of cookies from a popular bakery. Whatever you’re asked to calculate, make sure that you follow the tips suggested in the following section.

5 Key Tips for Solving GMAT Percentages

As I mentioned before, GMAT percent problems can be deceptively tricky. In this section, I’ll outline the most important tips to keep in mind as you’re attacking GMAT percentages.

#1: Language Matters

One of the biggest challenges with percentage questions is tricky wording. You need to read percentage question doubly carefully to make sure that you’re answering the correct question.

Consider the following example:

If you’re purchasing a bicycle that costs 500 dollars, and its price suddenly doubles, the cost of the bicycle is now 1,000 dollars. 1,000 dollars represents an increase of 100% from the original price of the bike, but 200% of the original cost of the bike.

See how the wording there is tricky? Depending on whether you’re searching for the percent change (increase) or trying to determine the proportion of the original cost of the bike, you’ll be looking for two different numbers. It can be easy to fly through a question like this and accidentally pick the wrong right number, if you’re not reading carefully.

#2: Mind the Increase-Decrease Trap (or the Decrease-Increase Trap)

Consider the following example:

The price of watermelon increased by 30%, and then decreased by 30%. The final price is what percent of the original price?

This type of question is very common on the GMAT and is designed to trick you. Inexperienced test-takers may see the increase of 30% and the decrease of 30% and think that 100% will be the correct answer because, in theory, the price of the good has increased and decreased by the same amount.

Not so.

When you go up by a percent and then down by the same percent, you don’t end up with the amount that you started with. For instance, if you are increasing the price of a watermelon by 30%, you’ll use that new price as the starting value to solve the next part of the question. Having different starting values means that you’ll have different answers. In order to find the actual answer, you’ll have to multiply the two multipliers. I’ll explain what that means more in the next tip.

#3: In a Series of Percentage Changes, Multiply the Multipliers

In order to find the percent change in a series of percentage changes, you must multiply the multipliers. Let’s consider the example in the previous question.

The price of the watermelon initially increased by 30%. We can represent that increase as 1.3. That’s the multiplier we’ll use for the increase.

The price of the watermelon then fell by 30%. We can represent that decrease as 0.7.

In order to find the total change, we need to multiply the multipliers.

$$(1.3)(0.7) = 0.91$$

The watermelon’s final price is 91% of the initial price, which means it actually decreased by 9 percent.

Whenever you have more than one multiplier, multiply the multipliers to find the actual percent change.

#4: Starting Value Matters

Make sure you know which value is the starting value when you’re trying to determine percent change.

Remember, we calculate percent change using both the amount of change and the starting value. If we have either of these numbers incorrect, the answer will be wrong.

Take a look at this example:

The price of a good is now 400 dollars, having decreased from 500 dollars.

In this example, the starting value is 500 dollars. The language in this question is slightly confusing and 500 dollars appears second in the statement. Only by reading carefully can you actually find out which value is the starting value.

If you use the incorrect starting value, you won’t be able to get to the right answer. Make sure you’re using the correct starting value as you’re working through a question.

#5: Know Your Multipliers

There are three main kinds of multipliers that you’ll see on the GMAT. Knowing which one to use when is key to finding the correct value on percent problems.

X% of a number

You can find X% of a number by multiplying the number by the percent expressed as a decimal. What does that mean?

Say you’re trying to understand what 35% of 500 is. You’ll multiply 500 by 35 percent to find the correct answer. You would express this in equation form as:

$$(0.35)(500) = x$$

We can say that the formula for finding X% of a number is:

(percent as a decimal)(original value)

X% increase

An increase multiplier is expressed as:

multiplier = 1 + (percent as a decimal)

If you were to use an increase multiplier to find the increase in a value, you would multiply the increase multiplier by the original value. Let’s look at what that means.

Say that the money in your savings account increases 15% over the year from its original value of $500.

To find the percent increase you would use the following equation:

$$(1.15)(500) = X$$

I used the number 1.15 because to find the multiplier of a percent increase problem, I add 1 to the percent expressed as a decimal, e.g. (1 + .15 = 1.15).

X% decrease

A decrease multiplier is expressed as:

multiplier = 1 – (percent as a decimal)

If you were to use a decrease multiplier to find the decrease in a value, you would multiply the decrease multiplier by the original value.

Say the money in your savings account decreased 15% over the year from its original value of $500.

To find the percent decrease, you would use the following equation:

$$(0.85)(500) = x$$

I used 0.85 because, to find a decrease multiplier, I subtract the percent decrease from 1. (1 – .15 = 0.85)

These multipliers often trip up test takers. Many test takers struggle with using the correct formulas to find percent increase and percent decrease. Make sure you remember to add or subtract the percent increase or decrease from 1 in order to get the correct multiplier.

GMAT Percent Problems Examples

Check out these sample GMAT percent problems to see how I solve a variety of GMAT percent problems using the tips I described above.

Problem Solving GMAT Percentages Sample Question

Because y is a positive integer, we can express y percent as y/100. Knowing that, let’s express the information in the question as equations.

We know that x is 50 percent of 50 percent of x, which is a positive integer. We can express that equation as:

$y = (0.50)(0.50x)$

We also know that y percent of x = 100. Remember, we can express y percent as y/100, since we know that y is a positive number. We can always express a percent as the value/100 when the value is a positive integer.

We can express the second equation as

$y/100x = 100$

Now, let’s simplify that first equation.

$y = (0.50)(0.50x)$

$y = 0.25x$

Now, let’s simplify the second equation by multiplying each side by 100.

$yx = 10,000$

Now we can plug our simplified value of y (y = 0.25x) into the second equation in order to isolate x.

$(0.25x)(x) = 10,000$

or

$0.25x^2 = 10,000$

Let’s continue to isolate x by dividing each side by 0.25

$x^2 = 40,000$

Now let’s find the square root of each side to find the value of x.

$x = 200$

The correct answer is C.

Data Sufficiency GMAT Percentages Sample Question

With data sufficiency questions, we always want to evaluate each statement individually first. We’re trying to determine the percent of companies that required neither computer nor writing skills.

From the information in the questions, we can say that the companies surveyed could be placed into one of the four following categories:

1. Requiring computer skills and requiring writing skills
2. Requiring computer skills but not requiring writing skills
3. Not requiring computer skills but requiring writing skills
4. Not requiring computer or writing skills

It’s given that 20 percent of the surveyed companies fell into category 1. It’s necessary to determine what percent of the survey companies fell into category 4.

Let’s look at statement 1. This helps identify the percentage in category 2. Since ½ the companies requires computer skills also required writing skills, then the other ½ of the companies that required computer skills did not require writing skills. However, this information only establishes that 20 percent required computer skills, but not writing skills. Statement 1 is not sufficient.

Let’s look at statement 2. While this category establishes category 3, that is that, 45 percent required writing skills but not computer skills, no further information is available. Statement 2 is not sufficient.

Now let’s look at them both together.

Taking (1) and (2) together, the first three categories add up to 85 percent (20 + 20 + 45 = 85). Therefore, category 4 would be equal to 100 – 85 = 15 percent of the surveyed companies required neither skill.

The correct answer is C.

What’s Next?

Looking for more in-depth GMAT quant guides? We’ve got you covered. Our in-depth guides to GMAT geometry, probability, and rate problems offer comprehensive tips and tricks for each of these GMAT content areas. You’ll learn about the types of questions you’ll see on the GMAT, as well as the important formulas and strategies you need to know to succeed.

Our guides to GMAT problem solving and GMAT data sufficiency examine the two GMAT quant question types in greater detail. Each guide focuses on the content areas covered by the question type and offers specific strategies for succeeding at each. It’s important to be equally comfortable solving data sufficiency and problem-solving questions. Use our guides to help you on the path to test day.

If you’re looking for more overall strategy, our guide to the 10 most important tips for beating the GMAT, will help you increase the effectiveness of your GMAT preparedness before and during test day. Learn the most important tips and strategies you should be employing to achieve your goal score.

The post GMAT Percentages: 5 Key Tips for Percent Problems appeared first on Online GMAT Prep Blog by PrepScholar.

If you’re like me, you probably spent a lot of time in high school memorizing the difference between sine and cosine and sighing over long, multi-step proofs, only to forget all of this hard-earned knowledge the second that classes dismissed for break.

If you’ve forgotten a lot of your high school geometry rules or are just in need of a refresher before taking the GMAT, then you’ve found the right article. In this article, I’ll be giving you a comprehensive overview of GMAT geometry.

First, I’ll talk about what and how much geometry is actually on the GMAT. Next, I’ll give you an overview of the most important GMAT geometry formulas and rules you need to know. Then, I’ll show you four geometry sample questions and explain how to solve them. Finally, I’ll talk about how to study for the geometry you’ll encounter on the GMAT and give you tips for acing test day.

GMAT Geometry: What to Expect

If you feel like you’ve forgotten a lot of the geometry that you learned in high school, don’t worry. The GMAT only covers a fraction of the geometry that you probably studied in high school. In the next section, I’ll talk about the geometry concepts that you’ll actually find on the GMAT.

You’ll find geometry concepts in both data sufficiency and problem-solving questions. Geometry questions make up just under a quarter of all questions on the GMAT quant section. As with all GMAT quant questions, you won’t just need to know how to apply geometry principles in isolation. You’ll need to know how to combine your geometry knowledge with knowledge of other concepts (like number properties, for instance) to get at the correct answer. I’ll talk more about what this actually means when I go over some geometry sample questions.

As I mentioned before, the GMAT only covers a fraction of the geometry that you learned in high school. As with the rest of the content on the GMAT Quant section, you’ll only need to know how to apply high school geometry concepts, which may be a relief to some test-takers.

Unfortunately, unlike some other standardized tests (like the SAT), the GMAT doesn’t provide any formulas for you. You’ll have to memorize all the formulas and rules you’ll need to know for the test.

In the next section, I’ll talk to you about the most important rules and formulas you’ll need to know to answer geometry problem solving and data sufficiency questions.

The Most Important GMAT Geometry Formulas and Rules to Know

The good news about GMAT geometry is that you don’t need to brush up on a whole bunch of topics in order to do well. The bad news about GMAT geometry is that you’ll have to memorize all the rules and formulas you need to know for the test, because none will be provided to you on test day. You also can’t bring in any aids to help you with the exam.

In this section, I’ll talk about the major GMAT geometry formulas and rules that you should study and memorize as you’re preparing for the exam.

Lines and Angles

• A line is a one-dimensional abstraction that goes on forever.

• For any two points, there is one straight line (only one!) that passes through them.

• A line section is a segment of a straight line that has two endpoints. The midpoint is the point that divides the line segment into two equal parts.

• Two lines are parallel if they lie in the same plane and never intersect. Two lines are perpendicular if they intersect at a 90° angle.

• An angle is made when two lines intersect at a point. This point is called the vertex of the angle.

• Angles are measured in degrees (°).

• An acute angle is an angle whose degree measure is less then 90°.

• A right angle’s degree measure is exactly 90°.

• An obtuse angle’s degree measure greater than 90°.

• A straight angle’s degree measure is 180°.

• The sum of the measures of angles on a straight line is 180°.

• The sum of the measures of the angles around a point (which make a circle) is 360°.

• Two angles are supplementary if their sums make a straight angle.

• Two angles are complementary if their sums make a right angle.

• Vertical angles are opposite angles formed by two intersecting line segments.

• A line or a segment bisects an angle if it splits the angle into two, smaller equal angles.

• Vertical angles are a pair of opposite angles formed by intersecting line angles. The two angles in a pair of vertical angles have the same degree measure.

Triangles

• A triangle is a closed figure with three angles and three straight sides.

• The sum of the interior angles of a triangle is 180°.

• Each interior angle is supplementary to an adjacent exterior angle. Together, they equal 180°.

• The formula for finding the area of a triangle is $½bh$.
  • $b$ = base
  • $h$ = height

• An isosceles triangle has two sides of equal length.

• An equilateral triangle has three equal sides and three angles of 60°.

• There are two kinds of special right triangles:
  • Isosceles right triangles have a side relationship of 1:1:$√2$.
  • 30°60°90° triangles have a side relationship of 1:$√3$:2.

• A right triangle has one 90° interior angle. The side opposite the right angle is called the hypotenuse and it’s the longest side of the triangle.

• Pythagorean Theorem for finding side lengths of a right triangle: $a^2 + b^2 = c^2$

• Two triangles are similar if their corresponding angles have the same degree measure.

• Two triangles are congruent if corresponding angles have the same measure and corresponding sides have the same length.

Circles

• The diameter of a circle is a line segment that connects two points on the circle and passes through the center of the circle.

• The radius is a line segment from the center of the circle to any point on it.

• A circle’s central angle is formed by two radii.

• The distance around the circle is called circumference:
  • $C=πd$
  • $C = 2πr$

• An arc is a part of the circumference of a circle.
  • $\Length = (n/360°)C$, where $n$ is the measurement of the central angle of the circle portion in degrees.

• The area of a circle is found with the formula $A = πr^2$.

Polygons

• A polygon is a closed figure that has straight line segments as its sides.

• The perimeter of a polygon is the distance around the polygon (the sum of the length of all its sides).

• The sum of the four interior angles of a quadrilateral is 360°.

• Area of a square: $s^2$

• Area of a rectangle: $l$$w$

• Area of a parallelogram: $b$$h$

• Area of a trapezoid: $1/2(a + b)h$

Solids

• A cylinder is a solid whose horizontal cross section is a circle.

• Volume of a cylinder: $Bh$, where $B$ is the area of the base.

• Area of the base of a cylinder: ?r₂ (because, remember, a cylinder has a circular cross section)

• A cube is a rectangular solid where all the faces are squares.
  • Volume of a cube: $Bh$, where $B$ is the area of the base.

• A rectangular solid is a solid with six rectangular faces.
  • Volume of a rectangular solid: $lwh$

Coordinate Geometry

• The slope of a line tells you how steeply that line goes up or down the coordinate plane.
  • $slope$ = $rise$/$run$
  • $slope = change in $y$ / change in $x$

• The rise is the difference between the $y$-coordinate values of two points on the line; the run is the difference between the x-coordinate values of two points on the line.

• You can also find the slope of a line using the slope-intercept equation, which is $y = mx + b$, where the slope is $m$ and the $b$ is the value of the $y$-intercept.

• Perpendicular lines have slopes that are negative reciprocals of one another.

• To determine the distance between any two points on a coordinate plane, you can use the Pythagorean theorem.

4 Tips for GMAT Geometry Questions

Even the most prepared test-takers can feel a lot of anxiety on test day. Follow these tips to boost your score and help you work your way through tricky GMAT geometry questions.

#1: Use What You Know

For all GMAT geometry questions, start by identifying what you know and what you need to find out. Use the information in the question and in any diagrams to build up your understanding of a figure. For instance, if you know that the measure of two different angles in a triangle are 60 degrees and 80 degrees, respectively, you can use what you know to figure out the measure of the third angle. The more information you have, the more you’ll be able to figure out.

#2: Look for Connections on Multiple Figure Questions

If there is more than one recognizable shape in a diagram, there is a connection between them. Look for what one of the figures tells you about the other. Perhaps the diagonal of a square is the same as the radius of a circle. Or the height of one triangle is the hypotenuse of another. Whatever the connection, it’s probably the key to answering the question.

#3: Don’t Assume That Drawings Are To Scale

You can’t assume that diagrams on the GMAT are to-scale. If you’re assuming a shape is a square and it’s actually a rectangle, you can make big mistakes in your calculations. Only use the information given to you on the diagram or in the question itself. Don’t ever assume anything that you can’t reason out with cold, hard math.

#4: Make Your Own Diagram

If you’re solving a question that involves a shape, but the test doesn’t give you a diagram, make your own. Making your own diagram will help you better visualize a question. You can also re-draw a diagram on your scrap paper even if the test provides you with a diagram to view. Sometimes, re-drawing a diagram will help you get a better understanding of the figure so that you can more easily solve the problem.

GMAT Geometry Practice Questions

One of the most important parts of preparing for the GMAT is to practice solving real GMAT questions. Solving real GMAT geometry questions helps you prepare for the content that you’ll actually see on the test. In this section, I’ll walk you through four real GMAT sample questions that use geometry concepts: two problem-solving questions and two data sufficiency questions.

Problem Solving Sample Question 1

To start, since this problem doesn’t provide a diagram, we want to draw our own on scrap paper. Drawing your own diagram helps you better visualize the problem. So, draw a rectangle and label the sides “8 m” and “10 m,” since we know that from the problem.

Next, let’s take a step back and think about what the question is asking us. It’s asking to figure out the cost of covering a floor in carpet squares. When you’re covering a floor in carpet squares, you want to cover the entire area of the floor. So, our next step is to find the area.

We know that the formula for area of a rectangle is $a = lw$. Let’s solve that using the information we have. $A = (8)(10)$. The area of this rectangle is 80 $m^2$.

Now, we need to figure out how much area each carpet square covers. The formula for finding the area of a square is also $lw$, so let’s go ahead and do that. $Area = (2m)(2m)$. The area covered by each carpet square is 4$m^2$.

To find the number of carpet squares needed to cover the floor, we’ll divide the total area of the floor by the area of each individual carpet square. $80 m^2/ 4 m^2 = 20$ total carpet squares needed to cover the floor.

The cost of each carpet square is 12, so for our final step, we’ll multiply 20 (number of carpet squares needed) by 12 (cost per carpet square) to get a total of $240.

The correct answer is B.

Problem Solving Sample Question 2

As always, let’s start by figuring out what this question’s asking us. It’s asking us to compare the distance Mary walked to the distance Ted walked. In order to do that, we need to first figure out how far they actually walked.

It’s pretty easy to figure out how far Mary walked. We can just add 8 + 6. Mary walked 14 miles.

It’s a little trickier to figure out how far Ted walked. Notice that the diagram is in the shape of a right triangle. That tells us that we can use the Pythagorean theorem to find the length of Ted’s walk, which is really just the missing side of this triangle. Since Ted’s side is across from the right angle, we know that it’s the hypotenuse. Therefore, we can plug in our sides pretty easily. $8^2$ +$6^2$ = $PR^2$ or $64$ + $36$ = $PR^2$, or $100 = $PR^2$. We can then find the square root of 100, which is 10. So, $PR = 10mi$.

Now, we know that Mary walked 14 miles and Ted walked 10 miles. Therefore, the distance Mary walked exceeded the distance Ted walked by 4 miles ($14 – 10 =  4$). 4 is 40% of 10, so the correct answer is B. Mary exceeded the distance Ted walked by 40%.

Data Sufficiency Sample Question 1

This question’s asking us to determine measure of an interior angle of a triangle. For data sufficiency questions, we always want to address each statement separately FIRST. Let’s begin with statement (1).

Statement (1) states that angle BDC measures 60 degrees. Since we know that $\angle ∠ {BDC}$ is on a straight line, we know that the angle adjacent to it ($\angle ∠ {BDA}$) can be added to $\angle ∠ {BDC}$to equal 180°. So, we can find the measure of angle BDA by using the equation: $180 – 60$ = $\angle ∠ {BDA}$. Therefore, we know the measure of $\angle ∠ {BDA}$ is 120°.

Next, we know that all the angles inside a triangle add up to 180°. Since we now know the measure of angle BDA (120) and the measure of $\angle ∠ {ABD}$ (20), we can find the third angle in that triangle by using the equation 180 – (20 + 120) = $\angle ∠ {BAC}$. So, statement (1) is sufficient. We now can eliminate answer B and answer E.

Now, let’s move on to statement (2). We want to forget everything we know about statement (1) at first and address statement (2) by itself.

The statement tells us that the degree measure of $\angle ∠ {BAC}$ is less than the degree measure of $\angle ∠ {BCD}$. However, we don’t have enough information to figure out what the measure of either angle actually is. So, statement (2) is not sufficient.

The correct answer then is A; statement (1) alone is sufficient.

Data Sufficiency Sample Question 2

Remember, when solving data sufficiency questions, you want to take each statement by itself first. Also keep in mind that you can’t assume that any diagrams given are to scale. You might be tempted to say that the triangle pictured is an isosceles right triangle, but you can’t assume that. Keeping all this in mind, let’s look at statement (1).

Statement (1) says that $x = y = 1$. That means that both $x$ and $y$ = 1. Can we use that to find the value of z?

Well, we know that the value of z is equal to 1 + the value of the base of the right triangle. There’s no information in the problem to tell us what the value of the base of the right triangle is. So, the value of the base can vary, so the value of $z$ can vary.

That means that statement (1) isn’t sufficient by itself.

Now, let’s look at statement (2) by itself first. Statement (2) says $w = 2$. However, even though we know that $w = 2$, we don’t know anything about the rest of the sides. That means all the other sides can vary, so z can vary as well. Statement (2) isn’t sufficient by itself either.

Now, let’s look at the two statements together.

Taking (1) and (2) together, we know that $z = y + (z – y)$ [the base of the triangle]. Or, we can say that $z = 1y + (z – 1)$.

The value of $z – 1$ can be determined by applying the Pythagorean theorem to the triangle. We know that the hypotenuse is 2 (from statement (2): $w = 2$) and we know that one side = 1 (from $x = 1$) and one side equals $z – 1$.

We can then write the equation $1^2 + (z – 1)^2 = 2^2$. Since we only have one variable in the equation, we can solve through for z.

You don’t need to solve a data sufficiency question. You only need to know that you can! So since we know we can solve the question using both statements, the correct answer is C. Both statements together are sufficient.

How to Study for Geometry on the GMAT

Studying for the GMAT may seem overwhelming, because there’s a lot of content to review. The good news is that executing a well-thought-out study plan will help you achieve your goals. Here are some tips geometry for the GMAT.

#1: Use High Quality Practice Materials

The best way to prepare for the GMAT is by using real GMAT geometry questions in your prep. Real GMAT geometry questions will simulate the GMAT’s style and content. For instance, you’ll have to use more than one skill in the question, or you’ll get practice using your geometry skills on data sufficiency questions, which are unique to the GMAT. Using resources like GMATPrep or the GMAT Official Guide will give you access to real, retired GMAT questions.

As you might’ve noticed from our practice questions, you’ll rarely see a straightforward question on the GMAT that just asks you to use your geometry skills. You’ll likely have to combine your knowledge of geometry with your knowledge of arithmetic or number properties or ratios… or all of the above! Practicing GMAT-style questions (real, retired GMAT questions if you can get them) will give you practice at using multiple skills in one question.

#2: Memorize Important Formulas

As I mentioned before, you won’t get to use a formula cheat sheet on the GMAT. You’ll have memorize all the formulas you expect to need on test day. Using flashcards is a great way to build your knowledge so that you can quickly recall and use important formulas on test day.

What’s Next?

You’ve read all about the formulas you need to know for GMAT geometry. Are you ready to master them? Using flashcards can be a great way to boost your memory. Before you get started with flashcards, check out our total guide to GMAT flashcards to learn about the best GMAT flashcards out there and the best way to study with flashcards.

Feel like you’ve mastered GMAT geometry? Looking for a new challenge on your quest to total GMAT quant domination? Check out our guide to GMAT probability to conquer a new type of math of the GMAT.

Are you totally confused by the data sufficiency practice questions? If so, don’t worry. Data sufficiency questions may seem strange, but our total guide to data sufficiency on the GMAT will break down everything you need to know to master this question type.

The post Every GMAT Geometry Formula You Need to Know appeared first on Online GMAT Prep Blog by PrepScholar.

Integers are one of the key recurring elements on the GMAT Quant section, so if you’ve started studying, you probably have some questions. What is an integer? Is zero an integer? What does it mean if integers are consecutive?

The good news is, you’ve almost certainly learned everything you need to know about working with integers in middle and early high school math. The bad news is that these rules and properties have probably been relegated to dusty, moss-ridden corner of your brain—and even if they haven’t, you’re going to have to apply them in novel ways on the exam.

Luckily, you’ve come to the right place! In this post, we’ll tell you everything you need to know about integers for the GMAT. We’ll give you a refresher on all the relevant rules and properties of integers, tips and tricks for every kind of integer question you’ll see on the GMAT, and some example questions with thorough explanations so you can see these strategies in action.

What Is an Integer? Is 0 an Integer?

Integers are all multiples of 1. They are all the positive whole numbers and their negative opposites, as well as zero. They do not include fractions, percentages, or numbers with decimals (which rules out figures like pi).

Here is the set of integers around and including zero:

{ … -3, -2, -1, 0, 1, 2, 3 … }

The ellipses indicate that the set of all integers goes on to infinity in either direction, because every single whole number and its corresponding negative is an integer. Note that 0 is an integer too (though it does have some special properties, which we’ll get into below.

Yep, that’s it! The integer as a concept is pleasantly neat and simple. However, it gets a little trickier when we start doing math with them, as they behave in very specific ways.

Key Properties of Integers for the GMAT

Now that you understand the concept, let’s go over the types, rules, and properties of integers in mathematical equations. Note that 0 and 1 are integers with special properties: they have their own subsection devoted to them below.

Above all, you should think of the following rules as just shortcuts. Of course, you already know how to multiply, divide, add, and subtract whole numbers. These properties just make it easier to do so in your head: they help you get to the right answer quickly by ruling on what can and can’t work, given how certain kinds of integers always behave. Doing this kind of efficient mental math is the key to mastering the GMAT Quant section as a whole, and it’s absolutely essential for properties of integers GMAT questions.

Even and Odd Integers: Definitions

At this point in your life, you’re almost certainly familiar with even and odd numbers. Still, you’ll need this baseline info to understand subsequent properties, so make sure you know all of the following definitions before moving on.

Even:

• Any number that results in an integer when divided by 2 is an even integer.
• Even integers end in 0, 2, 4, 6, or 8.

Odd:

• Any integer that isn’t divisible by 2 (as in, doesn’t result in an integer when divided by 2) is an odd integer. Note that I didn’t say “any number”— 4.5 is not divisible by 2, but it’s also not an integer, so it’s not odd.
• Odd integers end in 1, 3, 5, 7, or 9.

Both:

• Non-integers can’t be even or odd. Only integers can be even or odd, because decimal places automatically rule out divisibility by 1 or by 2. So if you see a question on the GMAT that specifies that a certain number is even or odd, you know it must be an integer.

Adding and Subtracting With Integers

The rules for adding and subtracting with integers are fairly intuitive as well, but just make sure that you’ve reviewed all of them before moving on.

Any integer plus or minus another integer results in an integer.

$$4 + 5 = 9$$

$$4 – 5 = -1$$

Negative and positive

Negative and positive numbers have particular rules when it comes to addition and subtraction:

• Adding a negative integer is the same as subtracting the positive.

$$4 + (-5) = 4 – 5 = -1$$

$$(-4) + (-5) = (-4) – 5 = -9$$

• Subtracting a negative integer is the same as adding the positive (double negatives cancel out).

$$4 – (-5) = 4 + 5 = 9$$

$$(-3) – (-4) = (-3) + 4 = 1$$

Odds and Evens

Adding or subtracting two of the same kind results in an even integer; adding or subtracting odd and even results in an odd integer. So:

• Odd + or – odd = even

$$5 + 7 = 12$$

• Even + or – even = even

$$12 – 8 = 4$$

• Odd + or – even = odd

$$8 + 3 = 11$$

$$8 – 3 = 5$$

Multiplying and Dividing With Integers

Multiplying and dividing with integers is right where things start to get a little more complex. It’s essential that you understand all of these moving parts before hitting the examples below.

First, the basics: multiplying integers with other integers always yields a result that is also an integer. However, dividing with integers isn’t so straightforward. 3 / 5, for example, would yield a fraction or a decimal, which is by definition not an integer. Same with 5 / 3.

Negatives and Positives

When multiplying or dividing numbers with the same sign, the result is always positive. When multiplying or dividing numbers with a different sign, the result is always negative. So:

• Negative × or / negative = positive

$$(-5) × (-5) = 25$$

• Negative × or / positive = negative

$$(-5) × 5 = (-25)$$

• Positive × or / negative = negative

$$5 × (-5) = (-25)$$

• Positive × or / positive = positive

$$5 × 5 = 25$$

Odds and Evens

Any integer multiplied by an even number is even; otherwise it’s odd:

• Odd × odd = odd
• Even × even = even
• Odd × even = even

Multiples and Factors

A multiple is the product of an integer and another integer. In other words, any integer that is perfectly divisible by another integer (with nothing left over) is called a multiple of the latter integer. For example, 20 is a multiple of 4, because 20 is divisible by 4.

Factors, also called divisors, are the positive integers that can be multiplied together to create a multiple. For example, 4, 5, 2, 10, 1, and 20 are all the factors of 20. 3 is not a factor of 20, because you can’t multiply 3 by any other integer to get to 20.

Together, we can simplify these rules into an always-true equation:

$$\Multiple/\Factor=\Integer$$

The greatest common factor, or greatest common divisor (GCF or GCD), is the largest factor shared by two numbers. For example, the greatest common factor of 20 and 30 is 10, since 10 is the largest integer that goes into both.

A cousin of the greatest common factor is the least common multiple (LCM). The LCM is the lowest multiple that two factors share. So the LCM of 20 and 30 is 60, as both 20 and 30 factor into 60 (but no number lower than 60).

If $x$ and $y$ are integers and $y = xn$ for some integer $n$, then $x$ is a factor (or divisor) of $y$, and $y$ is a multiple of $x$.

Quotient and Remainders

The remainder is what is left over in a division problem. For factors and their multiples, the remainder will always be 0, because factors go into their multiples perfectly, without anything left over. The quotient is how many times an integer can fit into another integer, regardless of what’s left over.

For example: 20 divided by 3 is 6 with 2 left over, because $3 × 6 = 18$. So 20 divided by 3 has a quotient of 6 and a remainder of 2.

When you divide an integer by another integer on the calculator, if the latter integer isn’t a factor of the former, you’ll automatically get decimal places instead of remainders. Using remainders and quotients is a way to do this math in your head—which is why the GMAT Quant section likes to bring them up, as you don’t have a calculator to use.

If $x$ and $y$ are positive integers, the quotient and remainder ($q$ and $r$, respectively) can be represented with these formulae:

$$y = xq + r,\and 0 ≤ r ≤ x$$

For example, if $y$ is 20 and $x$ is 3:

$$20 = (3)q + r$$
$$20 = (3)(6) + 2$$
$$20 = 18 + 2$$
$$0 ≤ 2 ≤ 3$$
$$q = 6, r = 2$$

So $y$ is only evenly divisible by $x$ if the remainder $r = 0$.

Properties of the Integer 0

As indicated above, zero is an integer. But it does have its own special properties:

• 0 is an even integer (because 2 goes into it 0 times)!
• 0 is is the only integer that is neither positive nor negative. So if you are asked about “negative numbers,” this doesn’t include zero. But if you are asked about “non-positive” or “non-negative” numbers, this does include zero.
• Any number plus or minus 0 is that number:

$$n + 0 = n; n – 0 = n$$

• Any number multiplied by 0 is 0:

$$n × 0 = 0$$

• You cannot divide by 0:

$$n/0 = \undefined$$

Properties of the Integer 1

1 has some special properties as well.

• 1 is an odd integer.
• Any number multiplied or divided by 1 is itself:

$$n × 1 = n, n / 1 = n$$

• Any number other than zero divided by itself is 1:

$$n/n = 1, \if n ≠ 0$$

• The reciprocal of any number is 1 over that number. Any number other than zero multiplied by its reciprocal equals 1:

$$n × 1/n = 1, \if n ≠ 0$$

• Multiplying a number by -1 changes the sign but not the absolute value:

$$5 × 1 = 5; 5 × (-1) = -5$$

Prime Numbers

Prime numbers come up again and again on the GMAT, so they’re a key integer concept to review.

As you likely already know, prime numbers are positive integers that can only be divided by itself and 1. In other words, a prime number only has two factors: 1 and itself. 1 is not considered prime, because its only factor is itself—it would have to have two factors to be considered prime.

There is a predefined set of prime numbers. The first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

2 is the only even prime number, because every even number larger than 2 is divisible by 2 (has 2 as a factor in addition to 1 and itself), and thus can’t be prime.

Prime Factorization

Every integer greater than 1 is either prime or can be broken down into prime factors. The expression of this is called a prime factorization. When you think about it, it makes intuitive sense: when you’re listing out all the factors of a number, any factor that isn’t prime has its own factors other than 1 and itself, and so it can be broken down into those factors, until there’s nothing left but prime (aka indivisible) integers.

As an example, here’s the prime factorization of 30:

$$30 = 5 × 6 = 2 × 3 × 5$$

Any multiple, by definition, will include in its own prime factorization the prime factorization of its factors. For example, the prime factorization of 210, which is a multiple of 30, looks like this:

$$210 = 2 × 3 × 5 × 7$$

210 has to have (2 × 3 × 5) in its prime factorization because, as a multiple of 30, it must include 30’s entire prime factorization. Here’s a slightly more sophisticated example with 9 and 36, a multiple of 9:

$$9 = 3 × 3 = 3^2$$
$$36 = 2 × 2 × 3 × 3 = 2^2 × 3^2$$

36 has prime factors other than 9’s prime factors, but 9 can’t have any prime factors that 36 doesn’t have, because that would mean a number goes into 9 that doesn’t go into 36—which makes no sense.

Exponents and Square Roots

For integer problems, there isn’t too much you need to know about square roots. Just make sure you understand the definition below.

If the square root of a given number is an integer, that means that that the number has to be an integer too and is thus a perfect square. The perfect squares are 1 (1²), 4 (2²), 9 (3²), 16 (4²), and so on.

Consecutive Integers on the GMAT

Consecutive integers GMAT problems come up again and again: they’re a quintessential question type, as they have very specific rules with a lot of moving parts, which is exactly the kind of thing that the GMAT loves to quiz you on.

On the exam, you will see sets (defined lists) of certain kinds of integers, marked by the squiggly brackets called braces. You may also see the word “distinct,” as in, a set of distinct integers. Distinct just means different, so no repeated integers. You may also see the word “inclusive,” meaning that the set is inclusive of the first and last numbers (“all integers from 1 to $n$, inclusive” means that this set includes 1 and $n$).

{ -10, -9, -8, -7 }

The above is a set of consecutive integers, or integers that follow each other in order. Consecutive integers can be represented algebraically:

$$n + 1, n + 2, n + 3 … , \where n \is \an \integer$$

Consecutive Even Integers

{ 2, 4, 6, 8 } is a set of consecutive even integers. Consecutive even integers can be represented algebraically as well:

$$2n, 2n + 2, 2n + 4, 2n + 6 … ,\where n \is \an \integer$$

Consecutive Odd Integers

{ 1, 3, 5, 9, } is a set of consecutive odd integers. Here’s the algebraic representation of consecutive odd integers:

$$2n + 1, 2n + 3, 2n + 5, 2n + 7 … ,\where n \is \an \integer$$

Sum of consecutive integers and divisibility

If the number of integers in a consecutive set is odd: the sum of all the integers is always divisible by that number.

Let’s say $c$ is the number of consecutive integers. In the set { 2, 3, 4, 5, 6 } we have $c = 5$ consecutive integers. So the sum of 2 + 3 + 4 + 5 + 6 should be divisible by 5:

$$2 + 3 + 4 + 5 + 6 = 20$$
$$20/5 = 4$$

If the number of integers in a consecutive set is even, the sum of the integers is never divisible by that number.

Let’s again say that $c$ is the number of consecutive integers. In the set { 2, 3, 4, 5 } we have $c = 4$ consecutive integers. The sum of 2 + 3 + 4 + 5 should not be divisible by 4:

$$2 + 3 + 4 + 5 = 14$$

14  is indeed not divisible by 4.

Factorials

Factorials have some properties that intersect with consecutive integer properties, and questions involving factorials also come up fairly frequently on the GMAT. Below are all the rules you need to know.

If $n$ is an integer greater than 1, then $n$ factorial, represented by the symbol $n!$, is the product of all of the integers from 1 to $n$. Examples:

$$3! = 1 × 2 × 3 = 6$$
$$5! = 1 × 2 × 3 × 4 × 5 = 120$$

$$0! \is \the \same \as 1! = 1$$

Factorials and Permutations

Factorials are useful for questions that asked about the number of ways an object could be ordered. Let’s say you have a couch, a sofa, and a table, and you want to see how many ways you could order them consecutively in a room (couch first and then table second, or sofa first and then couch, etc). There are three different items, so the number of ways is 3!, or 6.

These ordering possibilities are also called permutations. There are 6 different permutations in the example above.

You may see permutations in contexts other than factorials as well. For example, the GMAT could have a question giving you 8 kinds of pasta, 3 kinds of sauce, and 2 kinds of vegetables mix-ins and asking you how many types of pasta you could make by combining one of each of these ingredients. In this case, the answer is simple multiplication: $8 × 3 × 2 = 48$

Factorials and consecutive integers

The product of $n$ consecutive integers is always divisible by $n!$

For example, let’s say we are given this set: { 3, 4, 5, 6 }. That set contains 4 consecutive integers, or $n = 4$, so the product of $3 × 4 × 5 × 6$ is divisible by 4!.

$$3 × 4 × 5 × 6 = 360$$
$$4 × 3 × 2 × 1 = 24$$
$$360/24 = 15$$

Other Sets of Integers

You may also see integer sets in other types of patterns. Part of your task will be to come up with algebraic representations for these sets, or to understand what the given algebraic representation means. For example, you might supply something like this for “a set of multiples of 5”:

$$5n, 5n + 5, 5n + 10 …, \where n \is \an \integer$$

Summary of All Integer Rules and Properties

When you’re sure you comprehend the concepts, check out PrepScholar’s Rules and Properties of Integers for the GMAT, a PDF summary of all the above properties without the lengthy explanations. It’s a nice resource for memorizing the rules you’ll need for properties of integers GMAT questions.

Properties of Integers GMAT Question Examples

Now that you’ve gone over all the rules and properties of integers that you’ll need to know for the GMAT, it’s time to see them in action on real practice questions. All of these are official GMAT questions taken from the GMAC’s free GMATPrep Software.

Data Sufficiency Integer Questions

Data sufficiency questions have their own unique format, and thus their own way of testing you on integers. Below are the key types of data sufficiency properties of integers GMAT questions that you’ll encounter on the exam.

#1: Data Sufficiency with Sets of Integers; Odds and Evens

The question, statement 1, and statement 2 all indicate by themselves that we’re working with integers. The question itself gives us a defined set of only integers, plus non-integers can neither be even nor a multiple of another integer.

As with all data sufficiency questions, we’re being asking us whether statement 1 and statement 2 are enough either by themselves or together to narrow down $n$ to only one possible value. The five different answer options (A through E) represent all the different permutations of the sufficiency of the statements: either 1 is enough by itself but 2 isn’t, 2 is enough but 1 isn’t, both taken together are enough but they’re not enough by themselves, etc.

Like all data sufficiency questions, we can attack this one methodically: First test statement 1, eliminate answer options, then test statement 2, eliminate answer options, and then (if needed) combine them to find the right answer.

So first, let’s use what we know about integers to test Statement 1. Statement 1 says that $n$ is even, so we can rule out 33, 39, and 41, but we’re still left with a bunch of different options of what $n$ could be. Eliminate A and D.

Now onto statement 2. Statement 2 says that $n$ is a multiple of 3. 33, 36, 39, and 42 are all multiples of 3, so that statement by itself will not be sufficient. Eliminate B. Now we only have C and E left.

Finally, let’s see if combining the two statements works to narrow down $n$ to only one possible value. When we put the two statements together—$n$ has to be even, and $n$ has to be a multiple of 3—we’re left with not just one but two potential values for $n$: 36 and 42. Therefore, both statements together are not sufficient to find $n$. Answer E is correct.

#2: Data Sufficiency with Factorials

We know right off the bat that we’re working with integers, because they’ve told us. We are basically being asked to rule out the possibility that the integer $k$ is prime, because it needs to have a factor between 1 and itself. First, let’s use what we know about factorials to simplify the equations given in statement 1:

1. $k > 4!$ means that $k > 4 × 3 × 2 × 1$, which = 24. So $k > 24$.

Now let’s look at the first statement: if $k$ is greater than 24, then $k$ certainly can have a factor that is greater than 1 but less than itself, unless $k$ is prime. If $k$ is 27, for example, then it has only 27 and 1 as factors. Because of this exception of prime numbers, statement 1 is insufficient. Eliminate A and D.

We are now left with B, C and E. Let’s check statement 2. Statement 2 is indicating that there are twelve possibilities for $k$: 13! + 2, 13! + 3, 13! + 4, and so on through 13! + 13. We don’t actually have to do the math for 13!—we can use the knowledge of properties as a shortcut.

13! by definition has 2 as a factor, as it equals $13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1$, and so adding 2 more would still keep 2 as a factor. Same for all the rest: 13! has 3 as a factor, and adding another 3 to that result still maintains 3 as a factor, and so on. Therefore, for each of the 12 possibilities, there is a factor in between 1 and $k$. Statement 2 alone is sufficient to rule out the possibility of a prime number, and the correct answer is B.

#3: Data Sufficiency with Negatives and Positives; Exponents

This time, we don’t know that we’re working with integers until we take the question and the two statements altogether. Statement 1 tells us that $x = 6$, but that doesn’t mean that $y$ is also an integer—in fact, it doesn’t narrow down the possibilities for $y$ at all, integer or not. Eliminate A and D right away.

Statement 2 tells us that $x^2 = y^2$. By itself, that wouldn’t narrow down $x$ or $y$, so eliminate B. But taken together with statement 1, we know that $x^2 = 6^2 = 6 × 6 = 36$. So $y^2 = 36$ too. You might at first think: ahah! $y$ has to be 6. Both statements together are sufficient.

But that’s incorrect. Remember the rules about multiplying negatives and positives: a negative times a negative is a positive, just as much as a positive times a positive is a positive. Therefore, $y$ could be 6 or (-6). We have two possibilities left for $y$, so, even taken together, the statements are still insufficient. The correct answer is E.

#4: Data Sufficiency with multiples and factors; primes

Here we have to determine whether a given equation would result in an integer. First, let’s think about what would have to be true for this to work out: $s$ would have to be a factor of $r$—or, put another way, $r$ would have to be a multiple of $s$. So we can reword the question to simplify it: “Is $s$ a factor of $r$?”

Now, let’s look at statement 1. If every factor of $s$ is also a factor of $r$, that means $s$ must be a factor of $r$ because $s$ is its own factor ($s × 1 = s$). Statement 1 is sufficient, so eliminate B, C and E.

Next, let’s tackle statement 2 with prime factorization. Importantly, $r$ can have more prime factors than $s$, but $s$ can’t have additional prime factors from $r$. Let’s plug in some very simple numbers: $r$ could be 18, which can be broken down into prime factors 2 and 3. And $s$ can be 8, whose only prime factor is 2, and this condition is met. But 18/8 does not yield an integer. Statement 2 is insufficient, so the correct answer is A.

Problem Solving Integer Questions

Problem solving properties of integers GMAT questions are formatted differently than data sufficiency ones, but they still test many of the same skills.

#1: Problem-Solving with Odds and Evens

We know we’re dealing with integers here, a) because of the condition of “odd” and b) because we can’t split a ball into segments in this case—we’re working with whole balls as units.

We can think of this as a consecutive set of integers from 1 through 100, and so we can use our integer properties to answer this question. The probability of selecting an odd ball at random is 50/100 (ball 1, ball 3, ball 5, ball 7, etc.), or ½—the same as the probability for selecting an even ball (ball 2, ball 4, ball 6, etc.).

From our integer addition properties, we know that for the sum of the three numbers on the selected balls (or selected integers) to be odd, either

a. all the numbers have to be odd, or
b. one of the numbers have to be odd the two other numbers have to be even.

Here’s the probability of each permutation of selecting either all odd or just one odd:

$$\Probability \of \odd, \odd, \odd = ½ × ½ × ½ = ⅛$$
$$\Probability \of \odd, \even, \even = ½ × ½ × ½ = ⅛$$
$$\Probability \of \even, \odd, \even = ½ × ½ × ½ = ⅛$$
$$\Probability \of \even, \even, \odd = ½ × ½ × ½ = ⅛$$

All together, that’s 4 permutations of ⅛:

$$4 × ⅛ = ½$$

The correct answer is C.

#2: Problem-Solving with Consecutive Integers; Factorials and Prime Factorization

First, try and recognize the properties at play here. “The product of all the integers from 1 to $n$, inclusive” is simply $n!$, so we have a factorial question. We are told that 990 is a factor of $n!$, so $n!$ is a multiple of 990. Given this fact, we are being asked to find the lowest possible value for $n$.

Thus, this is also a factorization question, and to solve it, we need the prime factorization rule: the prime factorization of any multiple, by definition, will include its factors’ prime factorizations. Refer back to the examples in the “prime factorizations” section above if this is still confusing.

Back to the question: we know that $n!$, as a multiple of 990, is going to have at least all the same prime factors as 990 buried within its prime factorization. Let’s do out the prime factorization:

$$990 = 2 × 5 × 11 × 3 × 3$$

Now, let’s look at what we’re being asked: we want to find the lowest possible value for $n$, given that $n!$ is a multiple of 990. So we need the smallest product that has buried within it the above factorization.

The largest prime factor in that bunch is 11, so $n!$ would have to at least include a multiple of 11 in order to “hold” this prime factorization. And remember, $n! = n × (n-1) × (n – 2) × (n – 3) × (n – 4)$ …and so on. So given that we’re counting down and we NEED to make sure 11 appears in there, 11 is the lowest possible value of $n$, and choice B is correct.

We absolutely shouldn’t waste time doing double-checking this by doing it out algebraically on the GMAT, but for the sake of explanation, here it is:

$$11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 2 × 5 × 11 × 3 × 3 × 2 × 2 × 2 × 7 × 3 × 2 × 5 × 2 × 2 × 2 × 1$$

$2 × 5 × 11 × 3 × 3$ shows up in 11!, so 11! contains 990 as a factor.

#3: Problem-Solving with Exponents

The units digit of a must be either 3 or 7, because only numbers ending in 3 or 7 would result in a number with a units digit of 9 when squared $(3^2 = 9, 7^2 = 49)$.

Plug in 3:

$$(3 + 1)^2 = 4^2 = 16$$

The units digit is 6, which doesn’t work. So plug in 7:

$$(7 + 1)^2 = 8^2 = 64$$

The units digit is indeed 4, and that condition is met. Now we can solve:

$$(a + 2)^2 = (7+2)^2 = 9^2 = 81$$

And the units digit is 1. A is correct.

5 Tips for Integer Questions on the GMAT

Now that you’re familiar with integer properties and have seen them in action in real examples, let’s go over the five best strategies for nailing properties of integers GMAT questions.

#1: Memorize Key Properties

The hard part of the GMAT isn’t the math itself: it’s the reasoning and analysis required to take the basic math and get to the right answer. Remember, all these “properties” of integers are really just shortcuts: they narrow the scope of answers and make mental math easier to do. You want to save as much time and energy for the reasoning part as possible, so memorize all these shortcuts to the point where they become instinctual. To do your best on integer questions, you should know all the above properties and the set of prime numbers (up to 47) cold.

If you’re given a figure or equation that seems too hard to figure out without a calculator, chances are there is a property shortcut that you’re supposed to be using instead. Consider the example with 13! above: the point was to use integer properties to solve it, not to calculate the numerical value of 13!, which likely would’ve required writing it out on your scratchboard and taken up a ton of time. Ultimately, the GMAT will reward you for making quick and accurate work of integer problems—or at least their first steps—by applying these properties.

Along those lines, the answers to data sufficiency problems are always the same, and always in the same order. Memorize them to save time.

#2: Use Properties to Recognize Concepts in Disguise

The GMAT loves to test you on rules and properties in disguise. They won’t actually name those rules and properties in the question; it’s up to you to decode the question and supply the correct property yourself.

For example, in the “problem solving with factorials and factorization” question above, we were given this wording in the question: “the product of all the integers from 1 to $n$, inclusive, is a multiple of 990.” The product of all integers from 1 to $n$ inclusive is simply $n!$, which you needed to know to solve the problem quickly and efficiently. It’s vital that you understand integer properties well enough to recognize masked concepts and definitions like this one, because they’ll come up again and again.

#3: Use Properties to Eliminate Wrong Answers

Just like in the examples above, you can eliminate wrong answer options—sometimes right off the bat—by getting rid of the ones that defy the rules about the kinds of integers you’re given. For example, if you know you’re looking for an even number because you have to multiply two odds together, before you do the math, you can already eliminate all the answer choices that give odd numbers. If you only know that the answer must be an integer itself, you can get rid of any options with fractions or decimals.

Eliminating wrong answer choices also helps with guessing strategically, which you’ll need to do if you’re stuck on a question or pressed for time.

#4: Don’t Assume a Number is an Integer Unless Told

Unless they explicitly say so or give you a set or condition that can only apply to integers, never assume an unknown number is an integer. The only time you can assume you are working with integers without being explicitly told is if the “unit” given cannot be split into less than whole numbers. For example, if you’re finding the number of employees hired by a company, it’s not going to be something like 3.5, because you can’t have half of an employee.

If you’re not told, use the list of properties of integers for the GMAT that you have memorized to deduce whether you’re working with integers—but never, ever presume so.

#5: Practice, Practice, Practice!

There’s no substitute for this part: practice integer questions until you’ve got them down pat. Check out our posts on the best 28 GMAT practice tests along with 2000 GMAT practice questions for every question type for a comprehensive list of all the best sources. As a first step, you should download the official free GMATPrep software from mba.com, which contains two full-length practice GMATs and 30 extra Quant practice problems with explanations—but note that you can’t select only integer problems from those 30 (they’re randomized). After that, there are several great options available for purchase detailed in the guide linked above.

To test your memorization, ask yourself after you’ve done an integer problem—even if you got it right—if there was any way to get to the correct answer faster. Check your answer against the explanation every time.

If you’re struggling with a particular problem, chances are that plenty of other students have tackled that same exact problem. Take advantage of this: you should google a question after you’ve tried to solve it to find online boards in which professional GMAT instructors and other Quant whizzes explain the best way(s) to go about answering it. GMAT Club, Manhattan Prep’s GMAT Forum, and Beat the GMAT are all great forums to use for answer explanations; just be sure that the person posting the answer is a reliable enough source (like a verified GMAT instructor or someone who’s been “upvoted” several times).

What’s Next?

Integers are only one of several key concepts covered on the challenging GMAT Quant section, so be sure to check out our complete guide to all the math that is on the GMAT and how to review it before creating your study plan. You should also make sure you understand how scoring works for the Quant section and the test as a whole so you can set goals for yourself accordingly.

When you’re ready to start memorizing integer properties, you can always refer back to this article or to PrepScholar’s Rules and Properties of Integers for the GMAT (a PDF of just the rules, without the lengthy explanations).

Happy studying!

The post The Ultimate GMAT Guide to Integer Properties appeared first on Online GMAT Prep Blog by PrepScholar.

If Hayley studies for five hours a day for 30 days leading up to the GMAT, what’s the probability that she’ll meet her goal GMAT score?

While you won’t see a question exactly like that on the GMAT, you’ll encounter a number of probability questions on the quantitative section of the GMAT. Many GMAT test-takers are intimidated by probability questions, but if that’s you, don’t worry! In this guide, I’ll teach you everything you need to know to conquer probability on the GMAT.

First, I’ll walk you through the four essential GMAT probability rules you’ll need to answer every probability question. Then, I’ll offer several other tips for acing probability questions. Next, I’ll walk you through four sample GMAT probability questions with explanations. Finally, I’ll give you some tips on practicing for probability questions. What’s the probability that you’ll ace  probability GMAT questions after reading this guide? Pretty high, I’d say.

The 4 GMAT Probability Rules You Need to Know

For many test-takers, probability seems incredibly complicated. How can you possibly predict how likely an event is to occur?

However, probability (even on the GMAT) is actually pretty simple. As you may know from other reading you’ve done on the GMAT quantitative section, the GMAT quantitative section only tests high school math concepts. That means that you only need to know basic, high school level probability rules to do well on the test.

In this section, I’ll walk you through the four basic GMAT probability rules you need to do well on the GMAT quant section.

#1: Probability Equals the Number of Desired Outcomes Divided by the Number of Possible Outcomes

What does probability really mean? Well, it’s basically a way of figuring out how likely something is to happen.

You can figure out the probability of an event occurring (such as getting heads when you flip a coin) by dividing the number of desired outcomes (in this case, heads), by the number of possible outcomes (in this case, two: heads or tails).

So, the probability of getting heads when you flip a coin is $1/2$.

Obviously, not all probability questions will be the simple. But developing a good understanding of what probability actually means will help you figure out the answer to a question. If you’re trying to find the probability of something happening, remember that you first need to figure out how many desired outcomes there are and how many possible outcomes there are.

#2: The Probability of Two Discrete Events Occurring Is the Product of the Two Individual Probabilities

You can find the probability of two discrete events happening by finding the product of the two individual probabilities. What does that mean?

Well, a discrete event is basically an event that occurs which doesn’t have an effect on another event. So, if you flip a coin twice, each coin flip is its own event. They don’t affect each other.

So, to find the probability of two discrete events happening, you need to first find the individual probabilities. We already know that, when you try to get heads while flipping a coin, the probability is $1/2$. So, to find the probability of flipping a coin twice and getting two heads is $1/2 x 1/2$.

#3: The Probability of Getting One or Another Result Is the Sum of the Two Probabilities

The probability of getting one result or another means that you’re looking for one singular event to happen. So, if you’re looking to find the probability of flipping a coin and getting heads or tails, you will find the sum of the two probabilities.

The probability of flipping a coin and getting heads is $1/2$; the probability of flipping a coin and getting tails is $1/2$. So, the probability of flipping a coin and getting heads or tails is $1/2 + 1/2$ (or, 1).

#4: The Probability of Something Not Happening Is One Minus the Probability That It Will Happen

If you’re trying to find the probability of something not happening, first you need to find the probability that it will happen. Sound backwards? Trust me; it’s simple!

If you’re trying to find the probability of flipping a coin and not getting heads, first find the probability of flipping a coin and getting heads. Remember, that probability is $1/2$.

Next, subtract that probability from 1.

So, the probability of flipping a coin and not getting heads is $1 – 1/2$. What’s $1 – 1/2$? Well, it’s $1/2$, of course! And if you remember, $1/2$ is also the probability of flipping a coin and getting tails… or, in other words, the probability of flipping a coin and getting an outcome that’s not heads.

That’s it! Those are the four simple probability rules you need to ace the GMAT probability questions you see. Don’t worry if you’re still a bit nervous. In the next section of the article, I’ll talk about how to apply those four rules by showing you four sample probability GMAT questions.

Sample GMAT Probability Questions

In this section, I’ll walk you through how to solve different sample probability GMAT questions. First, I’ll show you two examples of problem solving questions that use probability. Next, I’ll show you two examples of data sufficiency questions that use probability.

Problem Solving Sample Question 1

For probability questions, we first want to figure out what we’re trying to find out. Or, in other words, what event are we determining the probability for?

We’re trying to figure out the probability of reaching into a box one time and pulling out a number that has a hundreds digit of 2.

Remember, probability is number of desired outcomes / number of total outcomes.

Let’s look for total outcomes first. Since the numbers in the box span from 101 to 350, there are 250 total outcomes. Don’t get tricked here! If you subtract 100 from 350, you’ll get 249. But you need to remember to count the number 350. So there are 250 total outcomes.

Next, let’s look for desired outcomes. Our desired outcome is any number with a hundreds digit of 2. Since the numbers go from $101 – 350$, there will be 100 numbers with a hundreds digit of 2.

Probability is desired outcomes over total outcomes. So, probability of finding a number with a hundreds digit of 2 is $100/250$, or $2/5$.

The correct answer is A.

Problem Solving Sample Question 2

This question is the type of GMAT probability question that often confuses test-takers. However, we can use our four simple rules to easily solve it.

We’re trying to find the probability that either event M or event R will occur. Remember, when you’re trying to find the probability of an either or question, you will add the two probabilities. So, to figure out the probability of either event M or event R occurring, we need to add the probabilities of event M and event R occurring.

Now, let’s find out what the probability of events M and R occurring are.

Let’s start with event M. The probability of event M not occurring is 0.8. That means we can work backwards to figure out the probability of event M occurring. You figure out the probability of an event not occurring by subtracting it from 1. That means we can reverse that process and subtract 0.8 from 1 to find the probability of event M occurring. $1 – 0.8 = 0.2$. 0.2 is the probability of event M occurring.

We can do the same thing for event R. The probability that event R will not occur is 0.6. So, the probability that event R will occur is $1 – 0.6 = 0.4$.

Now, we add our two probabilities together. $0.4 + 0.2 = 0.6$. There is a 0.6 probability that event M or event R will occur.

Last but not least, we need to convert our decimal into a fraction. 0.6 = 3/5, so our correct answer is C.

Data Sufficiency Sample Question 1

For data sufficiency questions, we need to evaluate each statement alone before evaluating the statements together. Let’s look at statement (1) first.

We’re trying to figure out what the probability is that a marble will be blue when selected at random from a jar. That means we need to figure out the number of desired outcomes and the number of total outcomes.

Statement (1) tells us that there are 24 marbles in the jar. That means that we know the number of total outcomes: 24.

However, statement (1) doesn’t give us any information about how many blue marbles there are in the jar. We know that there are eight red marbles, but we have no way of using that information to figure out how many blue marbles there are. That means we don’t know how many desired outcomes there are.

Let’s look at statement (2) by itself now.

Statement (2) doesn’t tell us how many marbles there are in the jar. That means we don’t know the total outcomes. It also doesn’t tell us the desired outcomes, because we have no idea how many blue marbles are in the jar. So, statement (2) alone is not sufficient.

Now, let’s take the two statements together. We know that there are 24 total outcomes. We know that there are 8 red marbles. (We know both of these things from statement [1]).

If we add in our information from statement (2), we can figure out that there are 12 white marbles. Since the probability of picking a white marble is $1/2$, we can find that $½(24) = 12$.

We know now that there are 8 red marbles and 12 white marbles in the jar. From there we can figure out that there are 4 blue marbles in the jar ($24 – 8 – 12$ = blue marbles).

4 is therefore our desired outcome. So, our total probability of picking a blue marble is $4/24$, or $1/6$.

Therefore, both statements together are sufficient, so the correct answer is C.

Data Sufficiency Sample Question 2

Let’s evaluate each statement by itself first.

From statement (1), we know that the probability that Jill will get a job from neither company is 0.3. However, we don’t have any information to figure out what the probability of her getting either job actually is. So, statement (1) by itself is not sufficient.

Let’s look at statement (2). From statement (2), we know that the probability that Jill will receive a job from one of the companies is 0.5. However, we don’t have any information to figure out the probability for the other job, or to figure out the probability of the jobs together. So, statement (2) is not sufficient.

Now, let’s look at the statements together.

We know that the probability of Jill getting one job is 0.5. We also know that the probability of her not getting a job at either company is 0.3.

If we subtract those probabilities from 1, we can find the probability of getting a job from one company. Remember, so far, we have the probability of her getting neither job (0.3) and getting one job (0.5). All we have left is to figure out the probability of getting both!

$1 – (0.5) – (0.3) = 0.2.$ Taken together, we can figure out that Jill has a 0.2 probability of getting both jobs. So, we can use both statements together to correctly answer the question. That means that answer C is correct.

4 Other Tips For GMAT Probability

Here are a few other tips to keep in mind as you’re preparing for GMAT probability questions.

#1: Memorize GMAT Probability Rules

The biggest tip that I can give you is to memorize those four GMAT probability rules I discussed earlier in the article. If you know exactly what a question is asking you, you’ll be able to figure out what information you’re looking for and answer the question quickly and easily.

Using flashcards can be a great way to improve your fluency with different concepts. You can build your own flashcards with probability problems and practice quickly recognizing which of those four rules you’ll need to answer the question. If you’re looking for more ways to effectively use flashcards in your prep, check out our ultimate guide to GMAT flashcards.

#2: Know the GMAT Probability Clue Words

Looking for clue words in your GMAT probability questions will help you know which skill you need to use.

If the question uses the word “or,” it’s asking you to find the sum of two probabilities. Remember our sample from the previous section: heads or tails.

If the question uses the word “and,” it’s asking you to find the product of two probabilities. Remember our sample from the previous section: to get heads twice (or, the probability of getting heads and heads), you need to multiply.

Memorizing those clue words and what they ask you to do will help you work quickly and confidently through your GMAT probability questions.

#3: Use High-Quality Practice Materials

The best way to prepare for the GMAT is by using real GMAT practice questions in your prep. Real GMAT practice questions will simulate the GMAT’s style and content. Using resources like GMATPrep or the GMAT Official Guide will give you access to real, retired GMAT questions.

#4: Practice GMAT-Style Probability Questions

As you might’ve noticed from our practice questions, you’ll rarely see a straightforward question on the GMAT that just asks you to use your probability skills. You’ll likely have to combine your knowledge of probability with your knowledge of arithmetic or number properties or ratios… or all of the above! Practicing GMAT-style questions (real, retired GMAT questions if you can get them) will give you practice at using multiple skills in one question.

Review: How to Attack GMAT Probability Questions

While probability for the GMAT might seem difficult, it’s actually quite simple.

There are four simple rules you should memorize in order to solve every single probability question you’ll encounter.

If you memorize those four rules, pay attention to your probability clue words, and practice real GMAT questions, you’ll be well on your way to acing your GMAT probability questions.

What’s Next?

Confused as to why every data sufficiency question has the same five answers? Don’t worry, you’re not alone! And our expert guide to GMAT data sufficiency is here to help. Check it out to learn all about these unique questions.

If Hayley answers four GMAT probability questions at a rate of one per minute, how many GMAT probability questions will she answer over four hours? In other words: looking to learn more about rate questions on the GMAT? Check out our guide to GMAT rate questions.

Do you want to take a step back and learn more about the GMAT quant section as a whole? We have just the guide for you! Our guide to the GMAT quant section will take you through every question type, give you some sample questions and explanations, and break down the types of content you’ll see on the exam.

The post 4 Key Rules for GMAT Probability Questions appeared first on Online GMAT Prep Blog by PrepScholar.