GMAT Arithmetic 101

The GMAT Quant section tests a variety of mathematical concepts, including algebra, geometry, and arithmetic. Generally speaking, the test-makers are not looking to trick or confuse you. Most GMAT math problems are rather straightforward. Read on to find out the specific arithmetic topics covered, tips for success, and plenty of practice problems!

What kind of arithmetic is tested on the GMAT?

The nine essential arithmetic concepts you need to look out for on the GMAT are these:
(Click on the concept to go to that section directly)

Now let’s take a quick tour through these various areas. Some of the topics overlap a bit. And you may need to combine ideas to solve more challenging problems, but here are the basics.

Arithmetic Operations

This includes the fundamental operations such as addition, subtraction, multiplication, and division. For example, typical problems may test your mastery of order of operations, or spotting common factors to cancel in complicated fractions. It’s important to have good number sense too.

Number Properties

A good number of GMAT math problems rely on your knowing your number properties. Primarily, we’re talking about properties of integers. So, be prepared to analyze evens and odds, consecutive integers and multiples, prime numbers, positives and negatives, and all the related concepts.

For example, did you know that 1 is not a prime number? On the other hand, 0 is definitely an integer!

Fractions

Fractions are incredibly important and useful. However, they often get a bad rap. But as long as you remember the fundamental rules, then fractions can be your friends!

• Adding or subtracting: Always make sure you have a common denominator.
• Multiplying: Just multiply the tops (numerators) and multiply the bottoms (denominators). This is the easiest operation for fractions!
• Division: Remember that you need to flip the fraction that is being divided. Or, if you have a stack of fractions, take the reciprocal of the fraction on the bottom. Then multiply, and that’s all there is to it!

Ratio and Proportions

Ratios are nothing more than fractions in disguise! But they are often used just to compare the parts of a whole with each other.

For example, if you tell me that the ratio of boys to girls in your classroom is 3 : 5, then I know that 3/8 of the class are boys and 5/8 are girls (note how I just added 3 and 5 to get the denominator of 8).

Proportions are equations that compare two fractions or ratios.

Here is a great resource to fill in the details: GMAT Quantitative: Ratio and Proportions

Percents

Percents represent yet another way of talking about fractional amounts. It literally tells you how many out of a hundred (per cent = per hundred). If you want to find p% of a number A, just multiply p × A. Percent change involves adding or subtracting a percentage of the original amount — for example, 50% more of 40 is equal to 40 + (0.50)(40) = 60.

Here’s more on the topic of percents:

Powers and Roots of Numbers

Knowing your Laws of Exponents and Roots will definitely help you to get through the GMAT Quantitative section. Here are a few links to get you started.

But what about those “impossible” problems with huge tricky exponents, in which they ask for the units digit of the answer? Don’t fall into the trap of trying to work out the number exactly — that will eat up too much precious time. Instead, review these methods: GMAT Quant: Difficult Units Digits Questions.

Statistics

GMAT statistics problems generally ask you to work out the average (arithmetic mean), median, mode, range, and interpret standard deviation. Further topics to be aware of include descriptive statistics and statistical significance.

Often, the problems can be done without much of computation. For example, did you know that if you multiplied all of the data by the same number, $$k$$, then the mean, median, mode, and standard deviation also gets multiplied by that same factor $$k$$? By contrast, adding the same number $$k$$ to all of the data only changes the mean, median, and mode, while leaving the standard deviation the same!

Statistics make the world go ’round! (Image by Wynn Pointaux from Pixabay)

Combinatorics

The term combinatorics is just a fancy word for counting. So it can’t be that hard, right? Well, just be careful. Does the order of items make a difference? Can the same item be counted more than once in a list?

Combinations questions ask how many ways that groups can be made out of a pool of distinct items. For example, how many 4-person committees are possible if chosen from a group of 7 people? Order doesn’t matter.

Permutations questions, on the other hand, are all about order. For instance, in how many ways can 4 people line up against the wall? If you are thinking 24, then you already know your factorials.

Check out GMAT Permutations and Combinations for more!

Discrete Probability

GMAT Probability questions come in many flavors. From problems that involve counting to geometric probability and problems based on dice, you have to be prepared for just about anything.

But there are just a few fundamental concepts to keep in mind. The main idea behind probability is that it’s a fraction of the desired outcomes over the total number of outcomes. To get started, check out this article: GMAT Probability Rules.

Also, you can expect probability to be a favorite topic for GMAT data sufficiency questions — for practice, see GMAT Data Sufficiency Practice Questions on Probability.

GMAT Arithmetic Tricks: 3 Tips for Success

Here are three tips for the GMAT arithmetic section — two fairly general pointers and one specific content area pitfall to avoid.

Tip #1: It’s All About Number Sense

You can avoid a ton of work and headache by simply estimating. A careful approximation may allow you to eliminate answer choices and avoid lengthy computations at the same time. That’s a win-win!

For example, if you have to multiply 98 × 103 for some reason, then just estimate 100 × 100 = 10000.

Tip #2: Look for Patterns

GMAT Arithmetic questions do not require extensive mathematical expertise to solve. In fact, most questions rely on either basic number properties or recognizing a simple pattern. For an example of the latter, consider the following example:

What is the remainder when $$5^{31}$$ is divided by 3?

Clearly, we cannot work out the expression directly. Calculators are not permitted on the Quant section, and this power is way too large to compute by hand. But let’s look for a pattern. First consider small powers.

$$5^1 = 5$$. Remainder: 2

$$5^2 = 25$$. Remainder: 1

$$5^3 = 125$$. Remainder: 2

$$5^4 = 625$$. Remainder: 1

It seems as though every odd power yields a remainder of 2 when you divide by 3, and so since the exponent 31 is odd, we’ll guess (correctly): 2.

Tip #3: Be Careful with Roots and Squares

There’s a standard set of traps that the GMAT test-makers rely on in powers and roots questions.

If $$b > 0$$, then the equation $$x^2 = b$$ always has two solutions, the positive and negative square root of $$b$$.

For example, if one of the two equations in a Data Sufficiency problem is $$(x + 1)^2 = 25$$, then that statement cannot be sufficient by itself (unless further information was given in the setup before the statements). Instead, you have to be aware that both $$x +1 = 5$$ and $$x+1=-5$$, or $$x = 4, -6$$ are the solutions.

GMAT Arithmetic Review: Practice Problems

Ok, now let’s practice! Each of the problems deals with one or more of the GMAT arithmetic topics discussed above.

Happy solving!

Problem Solving Questions

1. A large aquarium contains only two kinds of fish, guppies and swordtails. If $$\dfrac{3}{4}$$ of the number of guppies is equal to $$\dfrac{2}{3}$$ of the number of swordtails, then what fraction of fish in this aquarium are guppies?

2. $$\dfrac{4^6-4^5}{3} =$$

3.

4. $$\sqrt{81+81+81+81+81+81+81+81} =$$

5.

6. If $$k$$ is a positive integer, what is the smallest possible value of $$k$$ such that $$1040k$$ is the square of an integer?

7.

8. If $$k$$ is the greatest positive integer such that $$3^k$$ is a divisor of $$15!$$ then $$k =$$

9.

10. A certain school district has specified 5 different novels and 4 different non-fiction books from which teachers can choose to assemble a summer reading list. Each summer reading list must have exactly 3 novels and 2 non-fiction books. How many different summer reading lists are possible?

Data Sufficiency Questions

For each of the below problems, choose one of the following:

1. Ann and Bob planted trees on Friday. What is the ratio of the number of trees that Bob planted to the number of trees that Ann planted?

(1) Ann planted 20 trees more than Bob planted.

(2) Ann planted 10 percent more trees than Bob planted.

The answer is (B) Statement 2 ALONE is sufficient to answer the question, but statement 1 alone is NOT sufficient.

2.

3. If $$P$$ and $$Q$$ are positive integers, and if $$P > 1$$, does $$P = Q$$?

(1) $$P$$ is a factor of $$Q$$

(2) $$P$$ is a multiple of $$Q$$

The answer is (C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.

4.

5. What is the value of $$x$$?

(1) $$(x-5)^2 = 0$$

(2) $$(x-3)^2 = 4$$

The answer is (A) Statement 1 ALONE is sufficient to answer the question, but statement 2 alone is NOT sufficient.

6.

7. A box contains only red chips, green chips, and blue chips. If a chip is randomly selected from the box, what is the probability that the chip is either red or green?

(1) The probability of selecting a green chip is 1/3.

(2) The probability of selecting a blue chip is 1/7.

The answer is (B) Statement 2 ALONE is sufficient to answer the question, but statement 1 alone is NOT sufficient.

8.

9. A certain aquarium holds three types of fish: angelfish, swordtails, and guppies. What is the ratio of the number of guppies to the number of angelfish?

(1) There are 200 fish in the aquarium

(2) 45 percent of the fish are swordtails, and there are twice as many swordtails as there are angelfish.

The answer is (B) Statement 2 ALONE is sufficient to answer the question, but statement 1 alone is NOT sufficient.

Conclusion

The GMAT Quantitative section covers a lot of material. From word problems to quadratic equations, data sufficiency questions to problem-solving, and multiple-choice questions, there’s just a wide range of things to keep track of on the GMAT exam. However, I hope that after reading this article you will know what to expect in areas of GMAT arithmetic on test day.

Author

• Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!