If you’re like me, you probably haven’t thought about remainders in over ten years, when you first learned about long division in elementary school. Remember those lessons on what we call the number that’s left over in a division problem? Yeah, that number’s called a remainder.

And guess what? Remainders show up a lot on the GMAT. Of course, the remainder problems you’ll encounter on the GMAT are much different than the ones you worked on when you were ten years old. GMAT remainder questions can often be quite tricky, but don’t worry! In this guide, I’ll give you a comprehensive overview of GMAT remainder problems so that you feel ready to solve them when you see them on test day.

First, I’ll walk you through a brief refresher course on remainders. Next, I’ll offer some tips for solving GMAT remainder questions. Finally, I’ll provide some sample GMAT remainder problems and explanations for you to study.

What Is A Remainder?

If you’re a bit fuzzy on what a remainder actually is, don’t sweat it. Many GMAT test takers haven’t worked with remainders in many years. Let’s start by going over the basics of what a remainder is and how you’ll see them tested on the GMAT.

Think back to that fourth grade math class you took a long time ago. Let’s review the specific terminology we use to talk about division:

• When you divide $x$ by $y$, it’s the same as the fraction “$x/y$.”
• If we were dividing 6 by 3 (or, $6/3$), 6, the term we’re dividing by something else, would be the “dividend.”
• 3, the number that’s doing the dividing, is the “divisor.”
• In the case of the simple division problem $6/3$, 2 is our answer, or our “quotient.”

But, as I’m sure you know, not all division problems are as nice and neat as $6/3$. That’s where remainders come in.

Simply put, the remainder is the fraction part (or what “remains”) when you divide two numbers that don’t result in a whole number quotient.

For instance, when you divide $8/3$, the remainder is 2. When you divide 8 by 3, you have two sets of 3, with 2 leftover.

An easy way to think about remainders is to think about them as mixed numbers. For instance, the fraction $8/3$ is the same thing as the mixed number $2 2/3$. $2/3$ represents our remainder. We express it as $2/3$ because we have 2 parts left out of the 3 parts we need to make a whole number. The denominator will always be the same as the divisor.

Now that we’ve reviewed the basics, let’s move on to some more complicated, GMAT-style remainder math. Let’s say that we’re dividing our dividend $a$ by our divisor $b$ to yield our whole number quotient $c$ and our remainder $d$. That translates into the following equation:

$$a/b=c+d/b$$

For instance, in the simple problem we worked through before:

• $a$: 8 (dividend)
• $b$: 3 (divisor)
• $c$: 2 (quotient)
• $d$: 2 (remainder)

It’s worth memorizing this basic remainder equation – it’ll come up fairly often on the GMAT.

Remember, remainders can also be expressed as decimals. For instance, the remainder $2/3$ could be represented as .66 (repeating). That’s what you’ll likely see when you’re using a calculator (e.g., $8/3=2.66666$).

GMAT remainder problems are obviously more complex than what we’ve gone over in this section. For the GMAT, you’ll be asked to apply your knowledge of the basic relationship between dividend, divisor, quotient, and remainder to solve moderate to advanced algebraic equations.

As with all GMAT quant questions, we’ll have to rely on more than just our basic remainders knowledge to get the right answer. In the next section, I’ll give you some tips on solving GMAT remainder questions and then walk you through solving four sample questions.

Tips for Solving GMAT Remainder Questions

While GMAT remainder problems can be quite complicated, there are some things you can to do to more easily solve the problems you encounter. Keep in mind these tips as you’re working on GMAT remainder problems.

#1: Memorize the Remainder Relationship Formula

In the previous section, I gave you the formula for finding the remainder of a division equation:

$a/b=c+d/b$, where $a$ = dividend; $b$ = divisor, $c$ = quotient, and $d$ = remainder.

Knowing this equation is the key to answering remainder questions on the GMAT. You should also be comfortable moving around the variables in the equation, so that you understand the other relationships this equation yields such as:

$$a=cb+d$$

Being able to quickly recall the remainder equation and manipulate it in different ways by moving the variables on either side of the equal sign will really help you on the GMAT. You’ll be able to correctly plug-in the different numbers and formulas the GMAT throws at you so that you can figure out exactly what you need to solve.

#2: Plug-In Numbers for Variables

A great way to work out tricky remainder questions is to plug-in numbers for variables into your GMAT remainders equation. While this strategy won’t work all the time, plugging in numbers can be helpful if you’re stuck and unsure how to get to an answer. Let’s look at a sample question to see how this works:

What is the remainder when $x$ is divided by $3$, if the sum of the digits of $x$ is 5?

E. 6

That means we can plug in a number for $x$ that has digits that add together to equal five. For instance, we can use the number 14, because $1 + 4 = 5$. If we divide $14/3$, we find out that the remainder is 2. Let’s keep testing this theory. If we plug in the number 50 for $x$ (because $5 + 0 = 5$), we get the equation $50/3$, which equals 16 remainder 2. We can test this one more time with $23$ ($2 + 3 = 5$). If we divide $23/3$, we get 7 remainder 2. That tells us that the answer is A: 2.

Plugging in numbers doesn’t always work, particularly if you’ve got a number of different unknown variables. However, if you’ve got a relatively small set of potential numbers (e.g., numbers whose digits add up to five), you can plug in numbers to test for the correct answer.

#3: Learn Remainder Shortcuts

There are several nifty remainder “shortcuts” that you should keep in mind when you’re working on GMAT remainder questions. These shortcuts will save you time so you don’t have to completely write out calculations.

The possible remainders when a number is divided by a divisor $b$ can range from 0 to one less than $b$. For example, if $b$ = 5, the possible remainders are then from 0 – 4 (which is one less than 5). If $b$ = 10, the possible remainders range from 0 – 9 (which is one less than 10).

If a number is divided by 10, its remainder is the last digit of that number. If you divide it by 100, its remainder is the last two digits of that number, and so on. For, example 49 divided by 10 equals 4 with a remainder of 9.

You can take the decimal portion of the quotient and multiply it by the divisor to get the remainder. For example, if we know $9/5=1.8$ we can multiply .8 by the divisor 5, which gives us the remainder, 4. Keep in mind that you won’t have a calculator on the GMAT, though, so this tip may or may not save you time.

GMAT Remainder Problem Examples

Now that we’ve learned some tips for solving remainder questions, let’s see them in action. In these GMAT remainder problems, I’ll walk you through how to solve each question using the equations, tips, and tricks we’ve discussed earlier in the article.

Problem Solving Remainder Sample Question

Let’s start with this question by plugging what we know into our remainder equation. In this case, we know the following:

$x$/$y$ = $q$ + 9, where $q$ is the quotient (which is unknown), $x$ is the dividend (unknown), and $y$ is the divisor (also unknown).

From the question, we also know that $x/y = 96.12$. Remember how we talked about decimals earlier? When we’re giving an answer to a division problem that has a decimal in it, the whole number (in this case, 96) is the quotient and the decimal is the remainder. So, in this case, we express our answer as 96 + 0.12, with 96 as the quotient and 0.12 is the remainder.

We can also express our remainder as $\remainder/\divisor$, which gives us the equation:

$$x/y = q + 9/y$$

Since we now have two values for $x/y$, we can set them equal to each other, which yields:

$$96 + 9/y = 96.12$$

We can solve this equation through by subtracting 96 from the left side of the equation so we get:

$$9/y = .12$$

That leaves us with $9/y = 0.12$, or $y = 75$.

Data Sufficiency Remainder Sample Question

When we approach data sufficiency questions, we always want to solve each statement alone before looking at them together. Keeping that in mind, let’s look at statement (1) first.

Remember, we’re trying to figure out if we can determine the tens digit of a positive integer.

Statement (1) tells us that, when our positive integer $x$ is divided by 100, it has a remainder of 30.

This statement is a good example of something that you can plug a number into. For this, I’m going to try dividing different numbers by 100. I’ll start with 100. When you divide 100 by 100, you don’t get a remainder. When you divide 110 by 100, you get a remainder of 10. When you divide 120 by 100, you get a remainder of 20. When you divide 130 by 100, you get a remainder of 30.

You can also use the remainder shortcut that, if any number is divided by 100, its remainder is the last two digits of that number. That tells us that tens digit of our number has to be 3. This statement is sufficient.

Now let’s try statement (2). Remember, we want to look at each statement by itself first.

We can try plugging in numbers for statement (2). Using the same method of plugging in every number from 100 to 300 counting by tens, I find that if I divide 140 by 110, my quotient is 1 remainder 30. However, if I divide 250 by 110, I get a quotient of 2 remainder 30.

In this case, I have two different numbers that both yield me a remainder of 30. That means statement (2) is not sufficient.

I can’t use statement (1) and statement (2) together because they are mutually exclusive. Therefore, my answer is A.

Review: GMAT Remainders

While remainders may seem like elementary math you haven’t touched in years and don’t need to remember, remainder questions will often appear on the GMAT.

The most important step in solving GMAT remainder problems is to memorize the remainders relationships equations. However, plugging-in numbers and recognizing patterns can also help you solve these tricky questions.

What’s Next?

Are you confident in solving remainders questions now? Looking to move onto a new GMAT quant challenge? We have in-depth guides on many of the math concepts you’ll see on the GMAT. Check out our guides to GMAT geometry and GMAT rate problems to boost your knowledge on two other commonly tested GMAT concepts.

If you’re looking for a more generalized overview of the GMAT quant section, our GMAT quant guide will give you a solid overview of the content of the GMAT quant section, while our GMAT quant practice guide will help you better understand how and what to practice to ace the quant section.

Looking to completely change it up? If you want to focus on verbal instead, our in-depth guide to the GMAT verbal section will give you a great overview of the GMAT verbal section, while also suggesting resources you can use to practice.

The post GMAT Remainder Problems: 3 Key Tips appeared first on Online GMAT Prep Blog by PrepScholar.

For many GMAT test takers, Data Sufficiency questions are the most difficult questions on the GMAT. But what do the hardest GMAT Data Sufficiency questions look like? What skills and concepts do they test? What do they have in common? What Data Sufficiency strategies can we use to get these challenging GMAT Quantitative questions right?

In this article, I’ll go over the five hardest GMAT Data Sufficiency questions, what you’ll need to know to solve them, how to approach them on test day, and what we can learn from hard GMAT Quant questions about mastering Data Sufficiency.

How We Found These GMAT Data Sufficiency Questions

To gather the hardest GMAT questions, our GMAT experts took advantage of the computer adaptive algorithm used on the test. Over the course of the test, the difficulty levels of questions change based on how well you performed on previous questions. Get a few questions right, move up a difficulty level. Get a few questions wrong, move down a difficulty level. By the end of the test, every test taker should be presented with questions that perfectly match their ability.

Our GMAT experts took the practice tests on the GMATPrep software multiple times without missing a single question on the Quantitative section. We collected the questions they received into a master list of the hardest GMAT Quantitative questions. We then looked at activity on various online forums to determine which of these hard GMAT math questions test takers struggled with the most from each question type. This left us with the five hardest GMAT Data Sufficiency questions out there, ready for you to study!

GMAT Data Sufficiency Question 1

This particular problem gives us four different numbers on a number line ($A$, $B$, $C$, and $D$) and tells us the distance between two sets of points ($A$ ↔ $B$ and $C$ ↔ $D$). We should also note that these points are not necessarily in alphabetical order. Whenever we have GMAT Quant questions dealing with shapes, graphs, number lines, etc., it’s a really good call to draw out examples — this is the visual equivalent of plugging in numbers.) Applying this trick here, and remembering that the distance between $A$ and $B$ is longer than between $C$ and $D$, we see that our line could look like this:
this:
this:
and so on.

We need to find the distance between $B$ and $D$. This means that we need to gather information

1. about the order of the points
2. about how our first set of points ($A$ and $B$) relate to our second set of points ($C$ and $D$)

Statement 1

If $A$ and $B$ are two different points and are both the same distance from $C$, this means that the distance between $A$ and $C$ must also be 18 and that $C$ must be directly between the two points like so:
We also know that $D$ is only 8 away from $C$, so it is closer to $C$ than either $A$ or $B$. However, we still don’t know where $D$ is compared to these points. It could be between points $A$ and $C$, making it 26 away from $B$:
or between points $C$ and $B$, making it 10 away from $B$:
Since we don’t know whether the distance between $B$ and $D$ is 26 or 10, Statement 1 is insufficient.

Statement 2

Statement 2 tells us that $A$ is to the left of $D$. Well, $A$ is to the left of $D$ in both of the number lines above, and the distance between $B$ and $D$ is not the same in either. So this doesn’t tell us much. If the statement told us that $A$ was directly to the left of $D$, this might be a little more helpful … but it didn’t and it isn’t. Statement 2 is insufficient.

BOTH

Well, we already established that $A$ is to the left of $D$ (fulfilling Statement 2) in both of the number lines we created to fulfilling Statement 1, so even with the information from both statements, we don’t know whether the distance between $B$ and $D$ is 26 or 10. Since we still can’t solve for a single solution, the correct answer is E: Statements 1 and 2 TOGETHER are NOT sufficient to answer the question.

GMAT Data Sufficiency Question 2

Whenever we have a word problem, like this one, we want to translate the words into math. Scanning over the problem, we see the phrases “36 or fewer” and “more than 36” — these are classic signs that we’re dealing with inequalities. This particular problem gives us two scenarios for calculating how much Bob is paid based on how many total items he produces in a given week (one for 36 or fewer items, one for more than 36 items), so we want to create two equations: one for each scenario. Letting $i$ = the number of items Bob makes in a given week, we can translate our first scenario as

$\If i ≤ 36\, \then \total \pay=x×i$

Our second sentence is a little more complicated. If Bob produces more than Bob is paid $x$ for the first 36 items (or $36x$). Then for all of the items after 36 (or $i-36$), he is paid $1.5x$ (or $1.5x×(i-36)$). Putting that together,

$\If i > 36\, \then \total \pay=x×36 + 1.5x×(i-36)$

So we have two equations, each with three variables ($i$, $x$, and $\total \pay$) … which means we need a bunch of information to figure out an answer. To figure out a value for $i$, we need information about

• which of the two equations to use
• the value of $x$
• the total pay

Statement 1

This statement tells us how much Bob was paid last week, but it doesn’t tell us anything about the specific value of $x$ or which of the two equations we should use. So we could have:

$i=1 \and x=480 → 480=480×1$

or

$i=32 \and x=15 → 480=15×32$

or

$i=76\ \and x=5 → 480=5×36 + 1.5(5)×(40)$

and so on. Statement 1 is insufficient.

Statement 2

This one tells us how much Bob was paid this week, and it compares the number of items he produced this week to the number he produced last week. Well, we don’t know anything about how many items Bob produced last week, so the last piece of information doesn’t tell us much about $x$ — he could have produced 1 item last week and 3 this week or 100 items last week and 102 this week. And, like in Statement 1, we don’t know whether or not $i$ is greater than 36, so we don’t know which statement to use. So we could have:

$i=4 \and x=145 → 580=145×4$

or

$i=29 \and x=20 → 580=20×29$

or

$i=41\, \and x=13{1/3} → 580=13{1/3}×36 + 1.5(13{1/3})×(5)$

and so on. Statement 2 is insufficient.

BOTH

What if we put the two statements together? Well, now we know something: the additional two items Bob produced this week earned him \$30 more than he earned last week. This means that Bob earned an extra /$15 per item. But we’re still missing a key piece of information: which scenario are we dealing with?

1. Did Bob produce 36 or fewer items this week? If so, then both items were produced at a rate of $x$, so that $x=15$.
2. Did Bob produce at least 38 items this week? If so, then both items were produced at a rate of $1.5x$, so that $1.5x=15$ → $x=10$?
3. OR did Bob produce exactly 35 items last week and 37 items this week? If so, then the first item was produced at a rate of $x$ and the second item was produced at a rate of $1.5x$, so that $x+1.5x=30$ → $2.5x=30$ → $x=12$.

We’ve got a few options here, so let’s try each individually. Remember, we want to solve for the number of items Bob produced last week, so we’ll use that equation:

1. $x=15$, $480=15i$ → $i=32$
2. $x=10$, $480=36(10)+1.5(10)(36-i)$ → $480=360+15(36-i)$ → $120=15(i-36)$ → $8=i-36$ → $i=44$

We already have two possible solutions, so we don’t need to look at our third, more complicated option. We cannot determine whether Bob made 32 or 44 items last week, so we cannot solve the problem with both statements. The correct answer is E: Statements 1 and 2 TOGETHER are NOT sufficient to answer the question.

GMAT Data Sufficiency Question 3

Reading through the question, we see that we’re dealing with a group of houses where some have a swimming pool and some have a patio. Scanning over the statements, we see that some houses have only a pool, some houses have only a patio, some have neither, and some have both. Almost anytime we see the word “both” in GMAT Quant questions, we’re dealing with an overlapping sets problem — we are looking at two criteria (here, having a pool and having a patio) and where they overlap (here, having “both” a pool and a patio).

Overlapping sets problems have a lot of information, so it’s really easy to get lost in them. A good trick is to use a visual representation to keep track of what you know:

• For two overlapping criteria, use a table, where each axis represents one criterion.
• For three overlapping criteria, use a venn diagram, where each circle represents a criterion.

Here, we have two overlapping sets, so we’re going to use a table. We’ll go ahead and fill in only what was stated directly in the question. We want to find the total number of houses that have a Pool, so we’ll represent that in our table as $x$:

Because of the way we’ve set the table up, the two numbers in each row should add up to the total at the end of the row and the two numbers in each columns should add up to the total at the bottom of the column. This means that if we have at least two of the three values in each row or column, we should be able to solve for the third. Looking at our table, we see that our total row along the bottom has two values. If there are 75 houses in total and 48 of those houses have patios, 75 – 48 = 27 of those houses must not have patios. We can go ahead and fill that information in our table:

Doesn’t seem like we can get much more out of our table at this point, so we’ll move on to our Statements.

Statement 1

To start we’ll fill in the information directly given in the statement:

We see that our first column has two values, so we should be able to solve for the third. If there are 58 houses with patios and 38 of those houses do not have pools, 48 – 38 = 10 of those houses must have pools:

Looking at the row and the column that contain $x$, we see that we only have one number value for each, meaning that we can’t solve for $x$. Statement 1 is insufficient.

Statement 2

This statement doesn’t give us any concrete numbers to work with, but it does tell us that two of our values (houses with both pools and patios and houses with neither pools nor patios) are equal to each other. When we know that the same number shows up in two places, but we don’t know what that number is, it’s a good idea to represent that number with a variable — if we represent both values as, say, $n$, we know that they are the same number and can combine or eliminate them down the line:

Now we’re getting somewhere! We don’t have two number values in any row or column, but we can use both the top row and the second column to represent No Patio/Pool with variables: if there are $x$ total houses with pools and $n$ of those houses have patios, $x-n$ must not have patios, and if there are 27 total houses that do not have patios, and $n$ of those houses do not have pools, $27-n$ must have pools:

Since the number of houses with no patio and a pool equals both $x-n$ and $27-n$, we can set the two equal to each other to solve for $x$:

$x-n=27-n$

$x=27$

We were able to determine that 27 houses have pools, which means that Statement 2 is sufficient. The correct answer is B: Statement 2 alone is sufficient to answer the question.

GMAT Data Sufficiency Question 4

Right away, the word “ratio” tips us off that we’re dealing with ratios in this problem, and the word “greater” indicates that we’re dealing with inequalities. However, as we read through the rest of the problem, things start to get a little more confusing: one company, two divisions, full-time and part-time employees … this is a lot to process.

We do see the words “either” and “both” though, which should get some overlapping sets wheels turning in our minds. We see that, like the problem above, we have two criteria: employees can belong to Division X or Division Y and can be full-time or part-time. Since this problem doesn’t have any concrete numbers, it isn’t strictly necessary to make a table like we did in the problem above. However, it can still be helpful to define the relationships between our sets and build equations:

We know that the two numbers in each row should add up to the total at the end of the row and the two numbers in each columns should add up to the total at the bottom of the column. So we can now build 6 different equations:

1. $\Full\-\Time \@ \X + \Part\-\Time \@ \X = \Employees \@ \X$
2. $\Full\-\Time \@ \Y + \Part\-\Time \@ \Y = \Employees \@ \Y$
3. $\Full\-\Time \@ \Z + \Part\-\Time \@ \Z = \Employees \@ \Z$
4. $\Full\-\Time \@ \X + \Full\-\Time \@ \Y = \Full\-\Time \@ \Z$
5. $\Part\-\Time \@ \X + \Part\-\Time \@ \Y = \Part\-\Time \@ \Z$
6. $\Employees \@ \X + \Employees \@ \Y = \Employees \@ \Z$

Now that we have this set up, let’s figure out what the question is asking for. Like with all word problems, we want to translate words into math. Whenever we’re dealing with ratios, we should remember that ratios can (and should) be expressed as fractions:

Is ${\full\-\time \@ \X}/{\part\-\time \@ \X} > {\full\-\time \@ \Z}/{\part\-\time \@ \Z}$?

or in other words, are there more full-time employees for every part-time employee at Division X than at the entire company?

Statement 1

This Statement gives us information about the ratio of full-time employees to part-time employees at Division Y compared to Company Z:

${\full\-\time \@ \Y}/{\part\-\time \@ \Y} < {\full\-\time \@ \Z}/{\part\-\time \@ \Z}$

Now, before we rule this statement out because it doesn’t tell us anything about Company X, let’s see how we can use our equations to substitute X back into the inequality. Looking at equations 4 and 5, we see that we can rearrange the equations to give:

4. $\Full\-\Time \@ \Y = \Full\-\Time \@ \Z – \Full\-\Time \@ \X$
5. $\Part\-\Time \@ \Y = \Part\-\Time \@ \Z – \Part\-\Time \@ \X$

Subbing those into our inequality gives us:

${\full\-\time \@ \Z – \full\-\time \@ \X}/{\part\-\time \@ \Z – \part\-\time \@ \X} < {\full\-\time \@ \Z}/{\part\-\time \@ \Z}$

Let’s think about what we know about fractions. To make a fraction smaller, we need to either

1. decrease the numerator relative to the denominator
2. increase the denominator relative to the numerator

We know that we are decreasing both the numerator and denominator, so we must be decreasing the numerator by a greater percentage than we are decreasing the denominator. This means that the number of full-time employees at Division X is larger relative to the number of part-time employees at Division X than the number of full-time employees at Company Z to the number of part-time employees at Company Z. In other words, the ratio of the number of full-time employees to the number of part-time employees is greater for Division X than for Company Z. Statement 1 is sufficient.

Statement 2

Like with Statement 1, let’s translate this into math:

$\full\-\time \@ \X > {1/2}\full\-\time \@ \Z$

$\part\-\time \@ \Y > {1/2}\part\-\time \@ \Z$

Given equation 5, the second half of our statement also tells us that

$\part\-\time \@ \X < {1/2}\part\-\time \@ \Z$

This means we can write the ratio of full-time employees at Division X as

${>{1/2}\full\-\time \@ \Z}/{<{1/2}\part\-\time \@ \Z}$

or, cancelling the {1/2} in both the numerator and denominator,

${>\full\-\time \@ \Z}/{<\part\-\time \@ \Z}$

To make a fraction larger, we need to either:

1. increase the numerator relative to the denominator
2. decrease the denominator relative to the numerator

Here, we’re doing both: full-time employees at Division X is greater than full-time employees at Company Z and part-time employees at Division X is less than part-time employees at Company Z. This means that

${\full\-\time \@ \X}/{\part\-\time \@ \X} > {\full\-\time \@ \Z}/{\part\-\time \@ \Z}$

which is exactly what we’re trying to solve for. Statement 2 is sufficient.

Since both statements are sufficient to solve the problem individually, the correct answer is D.

GMAT Data Sufficiency Question 5

The word “remainder” tells us that we’re dealing with, what else, a remainder problem. Remainder problems scare a lot of students because they don’t involve an easy to use/memorize formula. However, this means that we have a great opportunity to plug in numbers.

Even though this isn’t technically a “word problem”, we still need to translate the words into math to build an equation:

${(n-1)(n+1)}/24 = \? \| \R\: r$

Let’s make a note that $n$ must be a positive integer and move on to our statements.

Statement I

This statement tells us that $n$ is not divisible by two — in other words, it’s telling us that $n$ is odd. Let’s try plugging in numbers. When we select numbers to plug in, our goal is to prove that the statement is insufficient: in other words, we want to pick numbers that will give us different results. We also want to pick numbers that are easy to work with to save time.

We see that one of the values in our numerator is (n-1), which means that picking 1 will give us a zero in our numerator. That seems like it’ll give us an interesting result, so we’ll give it a shot:

${(1-1)(1+1)}/24$

${(0)(2)}/24$

$0/24$

$0 | \R\: 0$

So when $n=1$, $r=0$. Let’s try our next odd number up, $3$ — based on the size of the denominator, it seems like our numerator will be smaller than the denominator, giving a solution of 0 with positive remainder:

${(3-1)(3+1)}/24$

${(2)(4)}/24$

$8/24$

$0 | \R\: 8$

So when $n=3$, $r=8$. This means that $r$ can be either 0 or 8 given Statement 1. Since we can’t find a single value for $r$, Statement 1 is insufficient.

Statement II

This statement tells us that $n$ is not divisible by three. That knocks $n=3$ out of the running. $n=1$ still works, however, so we know that $r=0$ is still a possibility given Statement 2.

Since we tried only odd numbers last time, let’s try an even number this time to see if that changes things up: we’ll do 2 to keep our numbers easy to work with:

${(2-1)(2+1)}/24$

${(1)(3)}/24$

$3/24$

$0 | \R\: 3$

So when $n=2$, $r=3$. This means that $r$ can be either 0 or 3 given Statement 2. Like before, since we can’t find a single value for $r$, Statement 2 is insufficient.

BOTH

Putting these two statements together, we know that $n$ must be odd and cannot be divisible by 3: so we have 1, 5, 7, 11, etc. These numbers are going to get pretty big pretty fast, so let’s try them from smallest to greatest. We already know that $r=0$ when $n=1$, so we want to find a positive value for $r$ to prove that both statements are insufficient:

${(5-1)(5+1)}/24$

${(4)(6)}/24$

$24/24$

$1 | \R\: 0$

So when $n=5$, $r=0$. That’s the same as when $n=1$. Let’s try the next number up, 7:

${(7-1)(7+1)}/24$

${(6)(8)}/24$

$48/24$

$2 | \R\: 0$

So when $n=7$, $r=0$. We’re starting to see the hints of a pattern here. Let’s try one more, 11, to be sure:

${(11-1)(11+1)}/24$

${(10)(12)}/24$

$120/24$

$5 | \R\: 0$

So when $n=7$, $r=0$. Once we’ve tried at least 4 numbers in a series and confirmed that we’ve done a reasonable job picking numbers that would give us different results, we can usually determine that we have a pattern. Here, we can say confidently that given Statement 1 and Statement 2, $r$ will always be 0. This means that the correct answer is C: BOTH statements together are sufficient.

Key Takeaways: Learning From The Hardest Data Sufficiency Questions

So what can the hardest GMAT Quantitative questions teach us about GMAT Data Sufficiency questions in general?

1. Visuals — drawings, tables, Venn diagrams, graphs, what have you — are our friends, and not only on Geometry questions. On the GMAT, advanced quant questions are hard to conceptualize, and drawing things out keeps us from having to keep track of a lot of complicated relationships in our heads.
2. Whenever we have words, we need to translate them into math. Like visuals, building equations helps us take hard GMAT math questions and distill them into something we can work with. Use math-y keywords, like “greater than”, “equal to”, “divided by”, etc. to break sentences down into their component parts.
3. The hardest GMAT Data Sufficiency questions often involve more logic than simple math, especially around number sense concepts. Being comfortable making inferences based on what we know can save us a lot of time compared to slogging through a bunch of proofs.
4. That said, picking numbers to plug in is a great Data Sufficiency strategy that can help us avoid overthinking a problem or confirm our logic. Always pick numbers that you think will yield two different solutions, making the statement insufficient.

What’s Next?

What are the math concepts tested on the GMAT? The best GMAT math tricks and shortcuts? The most important Data Sufficiency tips? These articles expand on the concepts used in these five problems, explaining what you need to know about GMAT Data Sufficiency before test day.

Looking to improve your Quant score? This article explains what exactly a good GMAT Quantitative score is.

If you’d like similar analyses of the hardest questions from other GMAT question types, check out our post on the five hardest Sentence Correction questions.

The post The 5 Hardest GMAT Data Sufficiency Questions appeared first on Online GMAT Prep Blog by PrepScholar.

Luckily, we’ve done the hard work of distilling everything you need to know about GMAT exponents! In this post, we’ll cover all the relevant rules, properties, formulas, and shortcuts. We’ll also walk you through an example of every main kind of exponent question that you’ll encounter on the test, so you can see these formulas in action. By the end, you really will be an exponent expert!

What Are Exponents? Definitions and Terminology

Below are the basic definitions and terms that you need to know for GMAT exponents. Most of this is probably very familiar to you, but make sure you understand everything before moving on.

Definition of Exponents

Let’s start with the basics. An exponent indicates how many times a given number should be multiplied by itself. The number itself is called the base number, and for any base number $k$, the exponent $n$ says how many times to use that number in a multiplication. The exponent is written in superscript to the right of the base number:

$$k^n$$

We’re most familiar with the “square,” which is what it’s called when the exponent is 2.

$$k^2 = k × k$$

But an exponent can be any number. Here are some basic examples:

$$2^2 = 2 × 2 = 4$$
$$3^2 = 3 × 3 = 9$$
$$4^2 = 4 × 4 = 16$$
$$5^2 = 5 × 5 = 25$$

$$2^3 = 2 × 2 × 2 = 8$$
$$3^3 = 3 × 3 × 3 = 27$$
$$4^3 = 4 × 4 × 4 = 64$$

$$2^4 = 2 × 2 × 2 × 2 = 16$$
$$3^4 = 3 × 3 × 3 × 3 = 81$$
$$4^4 = 4 × 4 × 4 × 4 = 256$$
$$5^4 = 5 × 5 × 5 × 5 = 625$$

Another common term for exponents is “powers.” We can refer to $k^n$ as “the $n$th power of $k$,” or “$k$ to the power of $n$.” Raising any number to the power of 2 is called squaring that number, and raising the number to the power of 3 is called cubing the number.

Definition of Roots

The square root of any number $y$ is a number that, when squared, equals $y$. (If $y = k^2$, then $k$ is the square root of $y$.)

The same goes for the cube root: the cube root of any number $y$ is a number that, when cubed, equals $y$.

And so on for every exponent.

The square root of any number is represented by this symbol, called a “radical”:

$$√4 = 2$$
$$√25 = 5$$

Any root beyond a square root (cube roots and up) is represented by a little number in superscript to the left of the radical that indicates the power. Below are examples of cube roots:

$$√^3{8} = 2$$
$$√^3{125} = 5$$

Properties of Exponents

Now that we’ve reviewed the definitions, let’s get into the properties of exponents that you need to know for the GMAT.

Property of the Exponent 1

Any number to the power of 1 is just itself, which is why you don’t really see numbers with exponents of 1 next to them, as it just isn’t necessary.

$$2^1 = 2$$
$$8^1 = 8$$
$$5093^1 = 5093$$

Property of the Exponent 0

Any number to the power of 0 is 1:

$$2^0 = 1$$
$$5^0 = 1$$
$$217^0 = 1$$
$$509395934234^0 = 1$$

Squaring a Number Between 0 and 1

As you can see from the examples above, when you square (or raise to any higher power than 2) a base number that is greater than 1, the result will be a larger number.

When you square (or raise to any higher power than 2) a base number that is between 0 and 1, the result will be a smaller number. Here’s an example:

$$(1/2)^2 =1/4$$
$$1/4 < 1/2$$

Positive and Negative Powers

Positive and negative powers and their roots have particular properties.

Every Positive Number Has Two Square Roots

Every positive number has two square roots: the positive square root, and the negative square root. This is because a negative multiplied by a negative is a positive. Here’s an example:

$$√4 = 2  (-2)$$
$$2 × 2 = 4$$
$$(-2) × (-2) = 4$$

Cubes and Other Odd Powers Have Only One Real Root

Cubes, on the other hand, have only one real root, because a negative times a negative times a negative equals a negative, and a positive times a positive times a positive equals a positive.

$$√^3{8} = 2$$
$$2^3 = 8$$
$$√^3{-8} = -2$$
$$(-2)^3 = -8$$

Odd powers will always have the same sign as their roots.

Negative Squares

The square root of a negative number is not a real number.

$√(-9)$ = not a real number

This applies to all even powers ($x^2$, $x^4$, $x^6$…and so on)

Negative Exponents

Any base number to a negative power equals 1 divided by the base number to the positive version of the exponent:

x^-r = 1/x^r  , where $r$ is any positive integer and $x$ is any positive number.

$$5^(-2) = 1/(5^2) = 1/25$$

Fractional Exponents

Fractional exponents look tricky at first. Here are all the equivalencies ($x$ is any positive number and $r$ and $s$ are any positive integers):

$$x^{r/s} = (x^{1/s})^r = (x^r)^{1/s} = √^s{r}$$

The two middle steps here simply illustrate the ‘exponent to the power of another exponent’ rule; the most important thing to remember is that x^r/s = ^s√(x^r). Let’s see this in action:

$$27^{2/3} = (27^{1/3})^2 = (27^2)^{1/3} =√^3{27^2} = √^3{729} = 9$$

$$8^{1/3} =√^3{8} = 2$$

$$9^{1/2} = √9 = 3$$

So any number to the power of $(1/2)$ is just the square root of that number, and any number to the power of $1/3$ is just the cube root, and so on.

The Cyclicity Property

Successive powers have what’s called a “cyclicity” that manifests in the units digit. Take 3, for example:

$$3^1 = 3$$
$$3^2 = 9$$
$$3^3 = 27$$
$$3^4 = 81$$
$$3^5 = 243$$
$$3^6 = 729$$

Pay attention to the units digits. Notice a pattern? The last digit repeats itself after a cycle of 4, and the cycle is 3, 9, 7, 1.

This repetition of numbers after a particular stage is called the cyclicity of numbers. This comes in handy when you need to find the unit digit of a given number to a certain power on a GMAT question, as you just need to find the number on which the cycle stops.

Rules for Manipulating Exponents

Below are the rules for doing algebra and arithmetic with exponents that you’ll need to know for GMAT exponent questions.

Multiplying and Dividing With Exponents

Below are the rules for multiplying and dividing with exponents. Note: You can’t combine bases or exponents when adding or subtracting terms—the algebraic equivalents are much more complex. Which is why we’re jumping first into multiplication and division!

Multiplying and Dividing with Same Base, Different Exponents

If you have to multiply two values together that have the same base number but different exponents, you can simply add the exponents together. Here’s an example:

$$3^2 × 3^5 = 3^(2+5) = 3^7 = 2187$$

Algebraically, this rule is represented as follows:

$$x^r × x^s = x^(r+s)$$

where $x$ is any positive number and $r$ and $s$ are any real numbers.

This division rule is the opposite of the rule above. If you have to divide two values together that have the same base number but different exponents, you can simply subtract the exponent of the denominator from the exponent of the numerator:

$${x^r}/{x^s} = x^(r-s)$$

where $x$ is any positive number and $r$ and $s$ are any real numbers. Here’s an example:

$$4^5/4^2 = 4^(5-2) = 4^3 = 64$$

Multiplying and Dividing with Different Bases, Same Exponent

If you have to multiply numbers with the different base numbers but the same exponent, you simply multiply the base numbers and keep the exponent as it is:

$$x^r × y^r = (xy)^r$$

where $x$ and $y$ are any positive numbers and $r$ is any real number. Here’s an example

$$3^2 × 5^2 = (3 × 5)^2 = 15^2 = 225$$

For division, the opposite rule applies. A fraction to any positive power equals the numerator to that power divided by the denominator to that power:

$$(x/y)^r = {x^r}/{y^r}$$

where $x$ and $y$ are any positive numbers and $r$ is a real number.

$$(3/4)^2 = {3^2}/{4^2} = 9/16$$

Exponents Raised to Another Power

Exponents can be layered, like in this example:

$$(4^2)^3$$

When you have a value with an exponent raised to the power of another exponent like this, you simply multiply the two exponents:

$$(4^2)^3 = 4^(2×3) = 4^6 = 4096$$

In general terms, we can write the rule as such:

$$(x^r)^s = x^{rs}$$

where $x$ is a positive number and $r$ and $s$ are real numbers.

Simplifying Exponents and Base Numbers With Factors

Base numbers that are already squares, cubes, etc of other numbers can be simplified. To use the square root as a base number, you multiply the exponent by 2 to get an equivalent expression. Here’s an example:

$$4^14 = (2^2)^14 = 2^28$$

For the cube root, you’d multiply the exponent by 3:

$$8^7 = 2^21$$

And so on for all the powers.

GMAT Exponent Practice Problems

Below are examples of the key kinds of GMAT exponent questions you will encounter on the exam, including data sufficiency and problem solving varieties.

GMAT Exponent Question 1: Exponents and Cyclicity

This question is simpler than it looks! We just need to find the cyclicity of powers of 4 and powers of 9. One thing you can do is plug in number values for $a$ and $b$, starting with 1 for each. Let’s do $a$ first:

If $a$ = 1, $x$ = 4¹ = 4
If $a$ = 2, $x$ = 4² = 16
If $a$ = 3, $x$ = 4³ = 64
If $a$ = 4, $x$ = 4⁴ = 256

…Notice a pattern yet? There is a cycle of 2: the units digits for powers of 4 is either 6 or 4.

Now let’s do $b$:

If $b$ = 1, $y$ = 9¹ = 9
If $b$ = 2, $y$ = 9² = 81
If $b$ = 3, $y$ = 9³ = 729
If $b$ = 4, $y$ = 9⁴ = 6561

The units digits for powers of 9 is either 9 or 1, so there’s also a cyclicity of 2.

Now, let’s multiply the possibilities of units digits for $x$ with the possibility of units digits for $y$. A cycle of 2 possibilities times a cycle of 2 other possibilities = 4 possible combinations: (4×9), (4×1), (6×9), and (6×1)

$$4 × 9 = 36$$
$$4 × 1 = 4$$
$$6 × 9 = 54$$
$$6 × 1 = 6$$

So we’re looking for either 6 or 4. (B) is the answer.

GMAT Exponent Question 2: Exponents, Cyclicity and Remainders

Cyclicity is a concept that comes up a lot on the more challenging GMAT exponent questions, so we’ve included another example. Unlike the last question, this one doesn’t ask directly about units digits, which makes it even tougher!

We’re given that $n$ is a positive integer, so the cyclicity rule applies here. First, let’s find the cyclicity of 3, so we can determine the units digit of 3⁽⁸ⁿ⁺³⁾ :

3¹ → the units digit is 3
3² → the units digit is 9
3³ → the units digit is 7
3⁴ → the units digit is 1
3⁵ → the units digit is 3 again

… and so on

So, the units digit of 3 in positive integer power has a cyclicity of 4 for the units digit (3, 9, 7, 1).

3⁽⁸ⁿ⁺³⁾ will have the same units digit as 3³, which is 7, which we can see when we plug in values for $n$ in 3⁽⁸ⁿ⁺³⁾:

$n$ = 1 → 3¹¹, units digit is 7
$n$ = 2 → 3¹⁹, units digit is 7
$n$ = 3 → 3²⁷, units digit is 7

Thus, the units digit of 3⁽⁸ⁿ⁺³⁾ + 2 = 7+2 = 9.

Lastly, we have to divide by 5. Any positive integer with the units digit of 9 divided by 5 gives the remainder of 4. The answer is (E).

You could also just plug in an easy number for $n$ from the start and do some heavy multiplication and long division, but this will take much longer, since the calculations are very unwieldy.

GMAT Exponent Question 3: Factors and Simplifying

First, let’s simplify 4¹⁷ to get to the same base number:

4¹⁷ = 2³⁴, so we have 2³⁴ – 2²⁸

Remember, this is a subtraction equation with exponents, which means we can’t do the same simple swapping that we can with division: 2³⁴ – 2²⁸ does NOT equal 2⁶. Instead, let’s do a factoring and utilize the distributive property. A common factor of 2³⁴ and 2²⁸ is actually 2²⁸:

$$2^34 – 2^28 = 2^28 × (2^6 – 1) = 2^28 × (64 – 1) = 2^28 × 63$$

Suddenly, this problem is a lot easier. 2²⁸ is made up of all only 2’s, so the greatest prime factor is just going to be 2. What about the prime factors of 63?

$$63 = 9 × 7 = 3 × 3 × 7$$

So the prime factors of 2²⁸ × 63 are 2, 3, and 7. Hence, the largest prime factor is 7, and (D) is the answer.

GMAT Exponent Question 4: Negative Exponents

Remember: odd powers have the same sign as their roots, so given that $-37 = x^5$, then $x$ must also be negative.

The next thing to acknowledge is that, since we don’t have a calculator, we’re not going to be able to find the 5th root of -37. And the GMAT wouldn’t want us to: they want us to use our understanding of exponent rules and properties to get to the right answer.

So first, let’s try to find a known number whose value when raised to the fifth is in the ballpark of -37, so we can establish a baseline. $x = -2$ fits the bill:

$$(-2)^5 = -32$$

That’s pretty close. What happens when we try $x = -3$?

$$(-3)^5 = -243$$

That’s way off. So $x$ must be a teeny tiny bit less -2 (remember, we’re in the negatives here, so less means a tiny bit closer to -3 on the number line). Something around (-2.1). Even though that sounds vague, that’s all we need to know to estimate our way to the right answer.

We can easily eliminate (B), since we’ve already shown that $x$ must be less than -2.

We can also eliminate (C), since $(-2)^2 = 4$, and $x$ is actually a little more like (-2.1), so it’s definitely not going to be less than 4.

We can also get rid of (A). $√-x$ is about equal to $√-(-2.1)$, and the double negative cancels out, leaving $√2.1$. There’s no way the $√2.1$ is greater than 2.

$x^4$ would be $(-2.1)^4$, or 16.something, so it’s definitely not greater than 32.

That leaves us with (D). Let’s check it:

$$x^3 ≈ (−2.1)^3 ≈ −8.something.$$

-8.something, no matter what that “something” is, is definitely less than −8.nothing, so option (D) must be true.

GMAT Exponent Question 5: Exponents and Consecutive Integers

Some GMAT questions combine exponents with the concept of consecutive integers. Here’s an example:

Let’s solve this one by picking numbers.

Let $r$ = 1, $s$ = 2, and $t$ = 3:

$$3^r + 3^s + 3^t = 3 + 3^2 + 3^3 = 3 + 9 + 27 = 39$$

The greatest prime factor of 39 is 13 (13 × 3 = 39).

Does this work for every consecutive integer set? Let’s pick another and see. Let $r$ = 2, $s$ = 3, and $t$ = 4:

$$3^r + 3^s + 3^t = 3^2 + 3^3 + 3^4 = 9 + 27 + 81 = 117$$

The greatest prime factor of 117 is 13 (13 × 3 × 3 = 117).

We can now see that this will work for every consecutive integer set. (E) is the answer.

GMAT Exponent Question 6: Data Sufficiency and Exponents

Here is a typical data sufficiency exponents question:

Rather than jumping right to the statements, let’s first simplify the question stem. The rule for dividing with the same base, different exponents tells us that $x^(r-s) = x^r/x^s$. So:

$$5^(−k) < 5^(1−2k)$$
$$= 5^(−k) < 5^1/5^2k$$
$$= 5^(k) < 5$$

Now, onto statement 1. Let’s rearrange it slightly:

$$2 < 1 – k$$
$$2+k < 1$$
$$k < -1$$

If $k$ is less than -1, then $5^k$ will always be less than 5, as $5^k$ would equal $1/5^(+ version of k)$. So statement 1 is sufficient.

Onto statement 2.

$$2k<3$$
$$= k < 3/2$$

If $k$ is less than 3/2 or 1.5, $k$ could still be some value like 1.4, and 5^1.4 is greater than 5. So statement 2 is insufficient.

The answer is B.

4 Key Tips for GMAT Exponents

GMAT exponent practice problems can be challenging. Here are our top tips for nailing them.

#1: Memorize the Exponent Rules and Properties Above

The key to GMAT exponent questions is to know the above exponent rules and properties cold. Many exponent questions will not explicitly say so, but they’ll require you to spot the applicability of one of the rules for multiplying, dividing, cyclicity, etc. Implementing these rules with ease will help you simplify equations that seem more complex than they really are.

#2: Memorize Squares of 2-15 and Units Digits Cycles Through 12

Just like you drilled your multiplication tables as a kid, you should drill all the squares of 2 – 15. Also, for those pesky digit and remainder questions, it’s helpful to memorize the units digits exponent cycles for 2-12.

For the lower numbers (like 2 – 5), you may even want to memorize up to the fifth power.

This instantaneous knowledge comes in very handy, like on GMAT Exponent Question 4 above, in which it was extremely helpful to know offhand that 2⁵ = 32.

Here are all the squares of 2-15:

2² =4
3² =9
4² =16
5² =25
6² = 36
7² =49
8² = 64
9² = 81
10² =100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225

For cyclicity questions, here is an ordered list of units digit cycles for 2-12 powers:

2 = 2, 4, 8, 6
3 = 3, 9, 7, 1
4 = 4, 6
5 = 5
6 = 6
7 = 7, 9, 3, 1
8 = 8, 4, 2, 6
9 = 9, 1
10 = 0
11 = 1
12 = 2, 4, 8, 6

#3: Pick Numbers Strategically

Many GMAT exponent practice problems don’t require you to solve all of the equations within them. Sometimes picking a simple number and substituting it for the unknown variable works even better—and makes the problem simpler and easier—than actually solving a complex algebraic equation. Just like in example question 5 above, picking a value to stand in for the unknown can save time and make it much easier to visualize and solve the problem.

#4: Estimate When Given Irrational Roots

Some GMAT questions will give you values they know you can’t find without a calculator, like the fifth root of 37 in one of the examples above. In these kinds of questions, you really are supposed to estimate! Use the value that gets you the closest to approximate, and eliminate answer options that can’t possibly work. By that point, there will be only one left.

What’s Next?

Looking for more GMAT exponents practice? We’ve got you covered.

Alternatively, maybe you’d like to move on to everything you need to know about GMAT integers.

We also have some more general GMAT Quant tips and tricks to help you develop strategies to master the Quant section.

Happy studying!

The post GMAT Exponents: Rules, Tips, and Practice 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.

Luckily, this means that there are several GMAT math tricks, tips, and shortcuts that you can use to improve your performance. In this post, we’ll give you all the major GMAT quant tricks, including tips and shortcuts for each of the two question types as well as some that apply to both. With these GMAT math tricks in your arsenal—plus the boatloads of studying you’re surely doing—you’ll be well prepared to nail the Quant section on test day.

GMAT Math Tricks: What Can They Help You With?

The makers of the GMAT will tell you that there are no such things as “tips” or “tricks” for doing well on the Quant section. Unfortunately, there is some truth to this: while GMAT math tricks can help you a little bit, the only real way to ace the GMAT Quant Section is to invest lots of time in focused, targeted preparation. Yep, that means study, study, study.

That said, each of the GMAT quant tricks below are extremely useful. Some help you execute basic calculations (like multiplication and division with unwieldy numbers) more quickly and efficiently; some help you get to the right answer without even having to solve the equation. Additionally, many of these GMAT Quant tricks are particularly helpful for guessing strategically on questions you’re stuck on—so when all else fails, you can feel like you have a solid plan and a fighting chance to get the right answer.

General GMAT Math Tricks

Below are some overall GMAT Quant tricks—as in, tips and shortcuts that apply to both question types.

#1: Simplify Calculations with Multiples of 10

Working with multiples of 10 is easy in addition, subtraction, multiplication and division. Even when you’re given a less pretty number, you can still use multiples of 10 to solve it painlessly.

Addition and Subtraction with 10

To add two and three digit numbers that aren’t already a multiple of 10 or 100, round the number to the nearest 10s or 100s digit, do the addition, and then add or subtract the result by the number you rounded off. Do the opposite when subtracting.

$$525 + 311$$
$$= 500 + 311 + 25$$
$$= 811 + 25$$
$$= 836$$

Multiplying and Dividing with 10 via the Distributive Property

To multiply by an awkward number, such as 16, you can multiply first by 10, and then multiply by 6, and then add the two products together:

$$n × 16 = (n × 10) + (n × 6)$$

You may remember that the rule that applies to this calculation is called the distributive property, and it works as follows:

$$a (b+c) = (a × b) + (a × c)$$

This works for subtraction as well:

$$a (b – c) = (a × b) – (a × c)$$

Here’s a full example:

$$5 × 37 = 5 × (40 – 3)$$
$$= (5 × 40) – (5 × 3)$$
$$= (200) – (15)$$
$$= 185$$

The distributive property is one of the most helpful tools in your GMAT toolkit because it simplifies unwieldy calculations.

Multiplying and Dividing between 11 and 19 via 10

There’s a slightly different trick for multiplying any two numbers between 11 and 19. Here, you can add the ones digit of one number to the other number, multiply that result by 10, and then add the product of just the ones digits.

Here’s an example:

$$11 × 17 = (18 × 10) + (1 × 7) = 187$$

Squaring between 11 and 19 via 10

To square any number $n$ between 11 and 99, find the nearest multiple of 10, and then find out how much you would have to add or subtract to get there. We’ll call the value that you’d have to add or subtract to get to a multiple of 10 $d$, for “difference.”

Next, do the opposite function with $d$ and the original number $n$ (add it if you had to initially subtract it to get to a multiple of 10, subtract if you had to add) to get two numbers that average out to $n$ ($n$ + $d$ and $n$ – $d$).

Finally, multiply those two numbers and add the square of $d$.

Here’s an example:

$$57^2 = (60 × 54) + 3^2$$
$$57^2 = (60 × 54) + 9$$
$$57^2 = (10 × 6 × 54 ) + 9$$
$$57^2 = 3240 + 9 = 3249$$

Little GMAT math tricks like the above make mental math faster and easier, which is key to success on the Quant section, as you don’t have access to a calculator.

#2: Plug in Numbers, but With Care

Many GMAT Quant questions don’t require you to solve all of the many equations embedded within them. Sometimes picking a simple number and substituting it for the unknown variable works even better—and makes the problem simpler and easier—than actually solving the complex algebraic equation.

For problem solving questions—especially when you’re looking for a rate, ratio, fraction, or percentage of an unknown whole—picking a value to stand in for the unknown can save time and make it much easier to visualize and solve the problem.

Here’s an example problem solving question that shows this strategy in action:

To practice law in their state, the third year law students at Western University have to pass the bar examination. If ⅓ of the class opted not to take the bar examination and ¼ of those who did take the test, did so and failed. What percent of the 3Ls will be able to practice law in their state?

This question provides a perfect use case for plugging in numbers. Since we’re dealing with ⅓ and ¼, you should choose a number that both 3 and 4 factor into neatly. So let’s go with 12, the lowest common multiple.

If the class has 12 people in it and ⅓ don’t take the test, that means 4 don’t take it and 8 do. Of those 8 who did take the test, 2 fail, so 6 in total are able to practice. The answer is asking for the percentage, which is now easy: 6/12 is ½, or 50%.

For data sufficiency questions, however, plugging in numbers is only really helpful for proving that a statement is insufficient. The reverse is much more time consuming, and so it doesn’t make sense over solving the problem, so only resort to it if you strongly suspect that a given statement isn’t sufficient and you don’t know another way to proceed.

Here’s an example of this strategy in action in a data-sufficiency-style question (with just one statement, for the purposes of illustrating number-picking):

If integer $n$ is greater than 1, is 2^$n$ – 1 prime?
1) $n$ is even

For this first statement, let’s plug in some even values for $n$:

Try $n$ = 2. We get 2^$n$ – 1 = 2² – 1 = 3, and 3 is prime.

So far so good, but now try $n$ = 4. We get 2^$n$ – 1 = 2⁴ – 1 = 15, and 15 is not prime. So statement 1 is not sufficient.

4 Tips for Picking Numbers

Here are four tips for picking numbers effectively and avoiding common traps in both question types:

1. Make sure that the number you pick meets all of the conditions in the question.

Note that some of the conditions may not be stated in their “official” terms, so you’ll have to read into the given information to recognize the rules at play. For example, a question stem might tell you that a given number y “has only two factors.” This means that y has to be a prime number. This is yet another example of how the math itself isn’t all that hard on the GMAT—what’s hard is uncovering the buried information that the question stem is masking by putting it in unusual words.

2. Be careful to avoid making assumptions beyond the given conditions. 

For example, if your question states that $a$, $b$, and $c$ are consecutive numbers, you can’t then assume that $a$<$b$<$c$ or that $a$>$b$>$c$. All you know is that they are consecutive—you don’t know the exact order in which they each occur.

Another example is if the question states that $x$ > 5. Many would assume that the number has to be 6 or higher. But unless it is stated that $x$ must be an integer or a whole number, then we can’t make this assumption, as there are an infinite number of decimal values between 5 and 6.

Assuming that the answer is a whole number without being told is a mistake people make all the time on the GMAT.

3. Avoid a number that represents a possible exception to the general rules of a condition.

For example, 2 is the only even prime number and can lead to some confounding results when worked with in an equation, so you may not want to choose it as your “plug-able” number in a prime numbers question.

4. Plug in numbers that are easy to work with.

Don’t use a crazy number like 367—the whole point is to make the problem simpler! As long as they meet all the rules of the conditions given (and don’t have their own confounding special properties), simple numbers like 3, 4, 5, etc. should be fine.

GMAT Data Sufficiency Tricks

Data sufficiency questions are tough for everyone at first, since they’re stylistically different from the math problems you’re used to doing. Once you get used to them, however, you can discern some tricks and shortcuts that are baked into the unique format of these peculiar questions. Below are the best GMAT data sufficiency tricks.

#1: Work Methodically Through the Choices

With their unchanging list of answer options, data sufficiency questions lend themselves perfectly to a special kind of process of elimination: You should always work through the answer choices in the same order.

We’ve pasted the choices below for your review. As soon as possible, you should memorize these answer choices until you know them cold—this will save you a good deal of time on the test. Note that they won’t come with A-E lettering on the real thing (we’ve put that in to make referring to them easier); instead, they’ll each have a bubble to the left that you’ll click on to indicate the answer.

First: test statement 1. If it isn’t sufficient to find one and only one answer, then eliminate (A) and (D). If it is sufficient, eliminate (B), (C), and (E).

Next, test statement 2. If it isn’t sufficient and statement 1 was sufficient, then (A) is the answer. If it is sufficient and statement 1 was also sufficient, then (D) is the answer. If it is sufficient and statement 1 wasn’t sufficient, then (B) is the answer.

If it isn’t sufficient and statement 1 also wasn’t sufficient, then either (E) or (C) is the answer.

You should only put the statements together if, after testing each statement for sufficiency by itself and going through the process of elimination above, both statements are insufficient and you’re left with (E) and (C). At this point, there are only two options: either they’re sufficient when taken together, or they’re not. If putting them together gets to only one answer, then (C) is the answer. If not, then (E) is the answer.

#2: You Don’t Always Have to Solve Questions All the Way

Very often, you don’t necessarily have to determine the value of the expression in the data sufficiency question prompt or in the statements: Your task is simply to determine if the information provided is each statement is enough to do so.

For example, let’s say you get the DS question stem: “What is the value of $x$?” (This is a fairly common type of DS stem.) And let’s say you’re given the following in statement 1:

Statement 1: 22$x$ + 251 = 550

Being the studious person that you are, you might be tempted to solve for $x$ in statement 1 by subtracting 251 from both sides, and then solving for $x$ by dividing by 22 (which is tricky, since it doesn’t result in a whole number). But you don’t actually need to solve for $x$: Just by looking at the equation, you know that statement 1 can lead to only one possible value for $x$, so it is sufficient to determine the value of $x$, as asked.

Already, you can eliminate (B), (C), and (E)—and you didn’t even need to “do” any math!

#3: Use the “$n$ Variables, $n$ Equations” Rule

Data sufficiency questions that ask you to solve for one variable often feature two variables (usually $x$ and $y$) in the statements. Remember the “$n$ variables, $n$ equations” rule of linear equations for these questions: you need $n$ distinct equations to solve for $n$ variables; thus to solve for $x$ and $y$, you need two distinct equations that include both $x$ and $y$.

This means that statement 1 alone is usually not sufficient, so you can eliminate (A) and (D) after a quick glance to make sure that’s true. However, don’t just pick (C) and move on: You must simplify each equation to double-check that one isn’t the same as the other.

Here’s a brief example:

What is the value of $x$?
Statement 1: 8$y$ = 12 – 4$x$
Statement 2: 2$x$ + 4$y$ = 6

Both of these equations are the same, so you can’t solve for one variable by plugging in its equivalent expression of the other:

$x$ + 2$y$ = 3
$x$ = (3 – 2$y$)
2(3 – 2$y$) + 4$y$ = 6
6 – 4$y$ + 4$y$ = 6
6 = 6
$y$ = ?

As long as there are two distinct equations with $x$ and $y$ (and there are no squared variables, which we’ll get into below), then both statements together should be sufficient

#4: Avoid the Square Root Trap

If $x$² is any positive number, then $x$ could be a positive number or a negative number, as a negative times a negative results in a positive as well. This means that there are two possible values for $x$, one negative and one positive. Assuming that “$x$² = [any positive number]” provides sufficiency to get to one value of $x$ is a very common mistake, but it’s easy to avoid by simply remembering this rule!

GMAT Problem Solving Tricks

GMAT problem solving questions are often thornier than they appear. Below are the best GMAT math trick, tips and shortcuts to help you strategically approach even the toughest problem solving questions.

#1: Look at All the Answer Choices Before Solving

This is generally a better strategy than to solve the problem right away and then look for a choice that matches your solution, as the choices themselves can provide clues to how to solve the problem—especially if there’s a property or shortcut that can help you do so. As always, the GMAT almost never requires you to do extremely laborious equations out by hand—they want to see that you can get to the right answer efficiently (as an excellent business person would)!

#2: Estimate to Cross off Wildly Wrong Answers

On a related note, many problem-solving questions test your ability to approximate reasonably, rather than precisely solving a complicated equation.

For example, let’s say you have to multiply a given number by a strange fraction, such a 11/53. This is fairly close to ⅕, or .2. The GMAT wants you to get to the right answer efficiently: they don’t want you to do all the work of dividing the given number by 53 and then multiplying by 11 on your scratchpad. It’s highly likely that the wrong answer choices will be far away from about ⅕ of the number, with only one choice that’s even in the ballpark.

Alternatively, you may get a question that appears to ask you to multiply many large numbers together, but the answer choices are all in exponent form and are all an order of magnitude away. In this case, you might be able to just estimate and find the closest answer as well.

Here’s an example of a good problem solving question to use estimation on:

The first thing to note is that 84% is an unwieldy number. When you see figures like that, it’s a sign that you may want to look for an estimating shortcut.

So what can we eyeball? Since there are 20% more girls than boys, we know that the weighted average will be closer to the girls’ percent than the boys’ percent. So we should look at the answer choices to see what we can already get rid of.  We can easily eliminate A, since 75% is going to be too low to be the weighted average. We can also cross out D and E, since they will both be too high (and are essentially equal).

79.5%, as the unweighted average of 75% and 84%, is the low extreme—the right answer will be slightly higher than that when adjusted for the total number of boys and girls, but just by a little bit, since there’s not a drastic difference in number between the two groups. So C, at 80%, looks right just by estimating. C is in fact the correct answer.

#3: Backsolve

Rather than plugging in numbers of your own choosing, some problem solving questions can be solved by working backward: plug the answer choices in, do the equation(s) with them, and cross off the choices that don’t balance. Usually there’s a faster way to get to the right answer, but this method can be a lifesaver when you really just don’t know how else to solve a given question.

The best way to approach backsolving is to start with C: the value that’s in the middle of all the choices. This way, even if it doesn’t balance the equation, you can determine whether the number that will work will be higher or lower (and rule out the higher or lower answer choices accordingly).

Here’s an example of a problem solving question that lends itself to backsolving:

As always when backsolving, start with C, $x$ = 4.

$$4^3 + 5(4) = 64 + 20 = 84$$

This is just over 80, so C could be the answer and, even if not, we can already eliminate D and E. Our powers of estimation tell us that 2 (A) is definitely going to yield a result far below 80, so let’s just check that 3 (B) doesn’t work:

$$3^3 + 5(3) = 27 + 15 = 42$$

That’s below 80, so $x$ = 4 is the smallest possible integer that satisfies the condition and C is the answer.

In Conclusion: Summary of GMAT Math Tricks

Simplify unwieldy calculations with the distributive property and other other tricks that allow you to do basic arithmetic with easy numbers like 10. Plugging in numbers is a great strategy for when you can’t think of another way to solve the equation, but be careful with the numbers that you choose. Plugging in numbers works best to prove insufficiency for data sufficiency questions, but it may make more sense to backsolve for problem solving questions.

You can “game” data sufficiency questions by working methodically through the answer choices, and remember that you don’t always have to solve or “do out” a complex equation to prove sufficiency. And if you encounter two variables on a data sufficiency problem, you will need two equations to solve for both of them.

Avoid traps by remember that positive squares can have positive or negative square roots, and by not presuming a number is an integer unless told.

For problem solving questions, look at the answer choices before solving the problem. Once you’ve identified what the question is asking of you, estimate to cross off answers that can’t possibly work. Sometimes estimating will get you all the way to the right answer. Finally, don’t be afraid to backsolve—but always start with C, or the middle value, when doing so.

What’s Next?

Check out our guide to 10 tips to master the quant section.

When you’re ready to conquer the subject areas tested, check out our GMAT-specific guides to integer properties, geometry formulas, and rate problems.

If you need help getting started and developing a study plan, we’ve got you covered there too.

Happy prepping!

The post GMAT Math Tricks: The 9 Best Tips and Shortcuts appeared first on Online GMAT Prep Blog by PrepScholar.

Let’s start with the obvious: the data sufficiency questions on the GMAT are really weird. They look nothing like any question you’ve seen in math class. For one thing, every single data sufficiency question has the same five answers. For another, data sufficiency questions aren’t just testing your math skills. They’re also testing your ability to analyze a set of data and decide what information you need to answer the question.

In this guide, I’ll walk you through the GMAT data sufficiency section, give you several strategies for solving GMAT data sufficiency questions, and offer tips for preparing for these unique questions. I’ll also take you through solving several retired sample GMAT data sufficiency questions.

GMAT Data Sufficiency Overview

Data sufficiency questions are one of two types of questions on the GMAT quant section. While there is no set number of data sufficiency questions on the quant section, GMAT Club reports that data sufficiency questions generally make up just under half of questions on the quant section. Since there are 31 questions in Quantitative Reasoning, about 15 of them will be data sufficiency questions.

A GMAT data sufficiency question is made up of a question and two statements, labeled (1) and (2). Your job is to decide whether the data given in each of the statements is sufficient (enough) to answer the given question. You’ll need to use the data in the statements, basic knowledge of high-school-level mathematics, and everyday facts (like the number of days in a given month) to answer the question.

There are five possible answer choices for every data sufficiency question:

You’ll have to use high school level math skills in arithmetic, algebra, geometry, and word problems to solve data sufficiency questions. For instance, you may see questions on properties of integers, sets, and counting methods. GMAT data sufficiency questions often require you to use more than one skill. You might be determining the average of a set of numbers for the first statement, and solving for a variable in the second statement. For more information on the math tested on the GMAT quant section, check out our complete guide to GMAT math topics. 

Sample GMAT Data Sufficiency Questions

In this section, I’ll walk you through solving several sample GMAT data sufficiency questions.

Sample Question #1

Correct Answer: C

sample question #1 Explanation

The best way to approach data sufficiency questions is to take each statement individually first, before having to consider them together. To that end, let’s start with statement 1.

Statement 1: The question asks us to determine how much money will be donated to the Red Cross based on the number of cars sold at the dealership. With data sufficiency questions, we always want to start with what we know.

We know that 500 cars are expected to be sold, as it tell us that in statement 1. Now, we need to decide if we can figure out how much money will be donated.

The question tells us that x dollars will be donated for each car sold, so the equation 500x represents the total amount of the expected donation.

However, we don’t know the value of x, and we have no way of determining it from the information given. So, we cannot solve the equation 500x, meaning that statement 1 is NOT sufficient for us to solve this problem.

Statement 2: Just as we took statement 1 by itself, let’s take statement 2 by itself first.

Statement 2 tells us that 60 more cars were sold than expected. If we know that x represents the amount of money donated to the Red Cross for each car, then we know that 60x represents the amount donated beyond the expected amount, because 60 cars were sold and x dollars were donated for each car.

If the total amount of the donation was $28,000, then the total amount that was expected can be found using the equation $28,000 – 60x, with 60x representing the unexpected amount we found before. Since we don’t know what x represents, we can’t find the total amount of the expected donation using Statement 2 alone.

Now that we’ve evaluated both statements individually, it’s time to evaluate them together. The first thing I notice when I look at both statements is that both statements have x in them. That means that I can combine the statements and solve for x.

Combining the two statements yields me the equation 500x = 28000 – 60x. From there, I can determine the total amount of the expected donation since I can combine like terms and solve for x.

Notice that I don’t actually have to solve this equation. All I need to do is know that I can solve it. Since I can solve with the statements together, but not alone, my correct answer is C.

Sample Question #2

Correct Answer: D

The first step in this question is to figure out what you’re trying to solve for. The question asks you how many pine trees the lot contains. Let’s use p as our variable to represent the number of pine trees the lot contains. You’re trying to solve for p in this equation.

Statement 1: Remember, we always want to start out by evaluating each statement individually. Statement 1 says that the ratio of oak trees to pine trees is 8 to 5. The ratio 8 to 5 can also be represented as 8/5. We can also say that the number of oak trees to pine trees is 56 to p, or 56/p, based on the information in the question.

Now, we can set the equations equal to each other because they both represent the same thing (ratio of oak to pine trees). Setting the equations equal to each other yields the equation 8/5 = 56/p. Because there is only one variable in this equation, I will be able to solve for p with no extra information. Statement 1 is therefore sufficient to answer the question.

Statement 2: Even though we already know that Statement 1 is sufficient, we’re still going to solve evaluate Statement 2 by itself first. Statement 2 says that the number of oak trees increased by 4. The question tells us that the original number of oak trees was 56, so 56 + 4 = 60. 60 is the new number of oak trees.

Next, the statement tells us that the ratio of oak trees to pine trees is now 12 to 7. We can also write the ratio of 12 to 7 as 12/7. We can also say that the number of oak trees to pine trees is 60 to p or 60/p. Just as we did with Statement 1, we can set the equations equal to each other, yielding the equation 60/p = 12/7.

Remember, we don’t need to solve for p, we just need to know that we can. Based on the information in Statement 2, we can also solve for p.

Statements 1 and 2 both contain enough information for us to answer the question, so the correct answer is D.

Sample Question #3

Correct Answer: B

Let’s start off by examining the question. We always want to make sure we understand what the question is asking us. We also want to make sure that we simplify the question, if possible, because simplifying the question will give us easier and clear equations to use as we solve the problem. In this case, we can simplify the question. The question “Does 2m – 3n = 0” is equivalent to the simpler question “Does 2m = 3n?”

Statement 1: If you’ve read the explanations for the previous two questions, I probably sound like a broken record by now, but I’ll repeat myself again. Remember, we always want to evaluate each statement individually, before looking at them together.

Let’s look at Statement 1 by itself. Statement 1 says that m doesn’t equal 0. That doesn’t give us a lot of information. Let’s go back to the original equation and see what we can learn there.

In the original, we see that 2m = 3n. In that equation, we also don’t have a lot of information. Statement 1 leaves an infinite range of possible values for m, and, since neither Statement 1 nor the original equation address possible values for n, we have no way to figure out the relationship between m and n. Therefore, Statement 1 is not sufficient.

Statement 2: Even though we know Statement 1 isn’t sufficient, we’re going to try to figure out Statement 2 by itself first. Statement 2 says that 6m = 9n. Right away, I notice that both 6 and 9 are multiples of 3, so the equation can be simplified by dividing each term by 3.

When I divide each term by 3, I get 6m/3 = 9n/3. If I simplify that, I get 2m = 3n. Remember, 2m = 3n is the original equation I’m looking for, so Statement 2 is sufficient and the correct answer is B.

Sample Question #4

Correct Answer: E

Let’s start out by understanding what the question’s asking us. It’s asking us to determine the value of n, which is a member of the set {33, 36, 38, 39, 41, 42}. So, for our statements to be sufficient, they need to help us decide which of those six numbers n is.

Statement 1: Look at Statement 1 alone first. Statement 1 says that n is even. That implies that n can be either 36, 38, or 42, because those are the even numbers of that six number set. However, there’s no other information in the statement that can help us narrow down which one of those three numbers n is. So, Statement 1 is not sufficient.

Statement 2: Statement 2 says that n is a multiple of 3. This implies that n could be 33, 36, 39, or 42. However, there’s no further distinction in Statement 2 to determine which of those four numbers n is. So, Statement 2 is not sufficient.

If we combine Statements 1 and 2 together, n is even and a multiple of three. That leaves us with n equaling either 36 or 42, with no way to determine which of those n is. Therefore, the statements are insufficient alone and together, so the correct answer is E.

4 Key Data Sufficiency GMAT Tips

Follow these data sufficiency GMAT tips to help solve tricky questions.

#1: Evaluate the Statements Individually First

Evaluating each statement individually will help you answer data sufficiency questions quickly and more easily.

Evaluate statement 1 first, then evaluate statement 2 by itself. When you do evaluate statement 2, you’ll need to forget everything you did for statement 1. Pretend that they’re two different questions. Once you’ve determined whether each statement is sufficient on its own, you’ll be able to put them together. This strategy will also save you time. For instance, if neither statement is sufficient on its own, you’ll be able to eliminate answers A, B, and E. If both statements are sufficient on their own, you’ll be able to eliminate A, B, C, and E right away.

Solve the simplest statement first. You’ll notice that the GMAT tends to have one statement that’s extremely simple, and one that’s extremely long and convoluted. You don’t have to evaluate the statements in order. Go with whatever’s easier first.

#2: Decide What Type of Question It Is

There are two basic kinds of data sufficiency questions: value questions and yes/no questions.

Value questions ask you to find a numerical value (e.g., what’s the value of 5x?). For value questions, if you’re able to find a specific value using the information in either statement, then that statement is sufficient.

Yes/no questions ask you whether or not something is true (e.g., is y an even number?). For yes/no questions, a definitive yes or a definitive no answer are both considered sufficient. An answer that is sometimes yes or sometimes no is not sufficient.

Remember: a definitive answer is always sufficient. An answer that may or may not be correct is not sufficient.

Before you start on a question, ask yourself if it is a value question or a yes/no question. This step will help you determine what kind of answer you’re looking for.

#3: Plug in Smart Numbers

You can solve many GMAT data sufficiency questions by plugging in real numbers for the variables in equations. Look for questions that have algebraic answers, or questions that ask for the values of algebraic expressions instead of just the values of variables when plugging in numbers.

When plugging in numbers, try to use easy, whole integers that match the constraints of the question. If the question asks you to use a specific type of number (e.g., a multiple of 3), make sure you’re using a multiple (e.g, use 6 instead of 54) that’ll be easy to do basic calculations with. The writers of the GMAT know that people generally pick positive, whole numbers to plug into their equations. Don’t forget about negative integers, positive and negative fractions, positive and negative decimals, etc., when analyzing a data sufficiency question. You may also need to try to plug in more than one number to test, depending on the question.

#4: Don’t Try to Fully Solve a Problem If You Don’t Have To

GMAT data sufficiency questions aren’t asking for specific answers. They’re asking if the statements contain enough information to find a specific answer. To that end, don’t do the work of fully solving a problem if you don’t have to.

For instance, if the question asks you “What is the value of y” and one of the statements is 24y + 34y = 1200, recognize that you can solve for y without taking the trouble of going through the entire process to solve that problem. You’re evaluating sufficiency of data, not looking for specific answers.

How to Prepare for GMAT Data Sufficiency Questions

While GMAT data sufficiency questions may seem daunting, these data sufficiency GMAT tips can help you more easily master your prep.

#1: Memorize the Five Answer Choices Using the 12TEN Mnemonic

Every data sufficiency question has the same 5 possible answers:

The 12TEN mnemonic can help you remember those answers and save you time:

1: only statement 1

2: only statement 2

T: both statements together

E: either statement

N: neither statement

Using this mnemonic will help you save time on test day because you’ll already know what every question is asking you, and you’ll be able to systematically test each of the answers. You’ll also have a clear idea of what each answer choice means.

#2: Review the Fundamentals

The questions on the GMAT quant section only test high school math concepts. That means that you’ll have likely seen every skill you need to master the test. The key, then, lies in mastering these fundamentals. You won’t be able to use a calculator on the GMAT, so you’ll have to be fast with your basic calculations. Practice multiplying and dividing decimals and fractions. Memorize the exponent rules. Memorize common roots and higher powers. These simple tips will save you time and build your confidence on test day.

#3: Use High-Quality Practice Materials

Setting yourself up with solid practice tools and creating a practice schedule you can stick to will help you prepare for the data sufficiency questions. It’s important to understand what makes good GMAT quant practice, so you don’t waste your studying time.

GMAT quant practice questions should be the same format as the real GMAT. Practicing questions using the same format helps you familiarize yourself with the mechanics of the test.

GMATPrep is an online software that gives you access to two full-length practice CATs, with the option of purchasing more. The software also gives you access to 45 GMAT quant practice questions and lets you sort the questions into the sets so that you can practice specific skills (so, you can make a set that consists only of data sufficiency questions).

Check out our full guide to official and unofficial GMAT quant practice materials for more specific ideas of what to use for your prep.

What’s Next?

Learn more about the types of math skills tested on the data sufficiency questions with our GMAT math overview, or switch over to the verbal section and learn how to master the three verbal question types.

Looking for more practice resources? Check out our list of the best GMAT practice tests.

Finally, read our guide on what a good GMAT quant score is to get a better sense of how data sufficiency questions affect your score.

The post 7 Expert GMAT Data Sufficiency Tips + Examples 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.

If you couldn’t already tell, we’re going to focus on rate problems in this article!

There are a number of different types of rate problems that you’ll see on the GMAT. (Don’t worry – they’ll all make much more sense than the silly examples I provided above.) In this article, I’ll explain what rate problems are, give you the formulas and equations you need to solve the three most common kinds of rate problems on the GMAT, and give you tips on how to practice for GMAT rate problems.

By the end of this article, you’ll have everything you need to start tackling your GMAT questions at a faster and more accurate rate!

Get it?

What Are GMAT Rate Questions?

There are three basic types of rate problems that you’ll see on the GMAT: questions that ask about the rate that work is completed, questions that ask you about the rate of distance traveled, and questions that ask you about interest rates.

A typical work rate question may tell you about two different people that are working at a particular rate. The question may ask you how long it’ll take for the two people to finish a job as they’re both working. For instance, a question may ask how long it takes two students, working together at different rates, to generate a certain product.

A typical distance rate question may tell you that a car travels along a road at a certain speed for one length of time, then continues traveling at a different speed for another length of time. The question may ask you to determine the average rate of travel for the entire trip. For instance, the question may ask you to determine how long it’ll take a car to get from Point A to Point B, if it’s traveling at two different constant rates of speed.

A typical interest rate question may ask you how much interest you’ll earn per year if you’re earning interest at a set rate for a certain amount of money. For instance, the question may ask you to determine how much money Person A earns in one year if they invest $x$ amount of money at $y$ interest rate.

You’ll see all three types of questions throughout the quantitative section of the GMAT. You’ll see rate questions for both data sufficiency and problem solving questions.

In the next section, I’ll talk about the formulas and equations you need to know for each type of rate problem and give you sample questions for each as well.

GMAT Work Rate Problems

As I mentioned before, GMAT work rate problems ask you about the rate of completion of different jobs. While these questions seem pretty simple, GMAT rate problems often trip up test takers.

The basic equation you need to know for GMAT work rate problems is:

$$\amount = \rate*\time$$

What does that mean? Basically, it means that the amount of goods produced is equal to the rate at which the goods are produced multiplied by the time spent producing the goods.

So, for instance, if I were to produce Grumpy Cat stuffed animals at a rate of 2 animals per hour, and I spent two hours producing stuffed animals, my equation would look like this:

$$\amount = 2 \animals/\hour * 2 \hours$$

I create four stuffed animals in two hours.

Obviously, GMAT work rate problems are more complicated than that example. It’s important to be comfortable with the basic equation $amount = rate*time$, though, as well as understand its implications. For instance, if the rate at which I produce stuffed animals doubles, but the time remains the same, the amount of stuffed animals I produce also doubles.

Let’s take a look at using this formula in action in a couple sample questions.

PROBLEM SOLVING WORK RATE SAMPLE QUESTION

Let’s start by assigning variables for the portion of the job that each machine completes alone. $R$ = the rate at which $R$ makes products; $S$ = the rate at which $S$ makes products; $T$ = the rate at which $T$ makes products.

Now, let’s plug those variables into an equation. We know that $R$, $S$, and $T$ working together for 4 hours make one product. We know that $S$ and $T$ working together for 5 hours make one product. That knowledge yields us these equations:

• $4R+4S+4T=1$
• $5S+5T=1$

We’ll isolate the variables from constants in each of these equations, meaning that we end up with:

• $R+S+T = 1/4$
• $S+T = 1/5$

Now, we can substitute $1/5$ for $S + T$ in our original equation. That yields us the equation $R + 1/5 = 1/4$. Let’s isolate $R$, then: $R = (1/4) – (1/5)$.

$R$, therefore, equals $1/20$.

Remember, that $1/20$ is in the form of ($amount$/$time$). 1, therefore is the completed job, and 20 is the time it takes to complete the job.

The answer, then, is $E$. It takes $R$ 20 hours to complete the job.

DATA SUFFICIENCY WORK RATE SAMPLE QUESTION

Let’s start by defining what we know. Let’s say that $K$, $M$, and $P$ are the number of minutes it takes machines $K$, $M$, and $P$, respectively, to complete the task. The amount of the task that $K$ can do in one minute is then $1/K$; the amount $M$ can do in one minute is $1/M$; the amount P can do is $1/P$.

If it takes all three machines working together 24 minutes to do the task, they can complete $1/24$ of the task, together, in one minute.

So, it holds that $1/K + 1/M + 1/P = 1/24$.

Now, let’s look at each statement by itself. Remember, for data sufficiency questions, we always want to assess each statement alone first.

Statement (1) says that $M$ and $P$ together can complete the task in 36 minutes. That means that they can complete $1/36$ of the task in 1 minute. In other words, $1/M + 1/P = 1/36$. We can use that information to then solve for a unique value of K by plugging into our original equation:

$1/K + (1/36) = 1/24$. (Note: you don’t actually need to solve data sufficiency questions; you just need to know that you can. I’m showing the completed equation here to show that it can be solved.)

Statement (1), therefore, is sufficient to solve for $K$.

Now let’s forget everything we learned in statement (1) and look at statement (2). Statement (2) tells us that $K$ and $P$ can do the task together in 48 minutes. So $1/K + 1/P = 1/48$, or they can do $1/48$ of the task in one minute.

However, we don’t have any information here to figure out the value of $K$. Since we have no value for $P$, we can have a number of values for $K$. We can’t solve the equation using this information.

Therefore, statement (2) is not sufficient.

The correct answer, then is $A$.

GMAT Distance Rate Problems

GMAT distance rate questions often ask you to determine how far someone or something is traveling at a certain speed. You may be asked to calculate average speed or average distance or the time spent traveling. There’s one basic equation that you can use to complete these calculations.

The basic equation for distance is as follows:

$$\distance = \rate*\time$$

PROBLEM SOLVING DISTANCE RATE SAMPLE QUESTION

Let’s start with what we know. We know that so far, Bob has been running for a distance of 3.25 miles. We also know that he is running at a constant rate of 8 minutes per mile. Normally, we express speed in miles per hours, with distance over rate, so this constant rate is expressed differently than the normal, We can use this knowledge to figure out how long Bob has been running for by plugging those numbers into our distance equation.

$(3.25 miles)*(8 minutes$/$mile) = 26 minutes$. Bob has been running for 26 minutes.

Let’s then say that $X$ is the number of additional minutes Bob can run south before turning around. The number of minutes that he’ll be running north, then, will be $X + 26$.

Since Bob will be running a total of 50 minutes after his initial 26 minutes of running, we can say that his remaining total time running (50) is equal to the rest of his time running south ($X$) and his time running north ($X + 26$). This knowledge gives us the equation: $X + (X + 26) = 50$.

That means that $X = 12$, or, Bob can run for 12 more minutes before turning around. his total remaining time running (50) is equal to the rest of his time running south (x) + his time running north (x+26)

Now, we can plug 12 minutes into our distance rate equation to figure out how many more miles Bob can run.

$(12 minutes)$/$(8 minutes$/$mile$) = $miles$ Bob can run before turning around.

So, Bob can run 1.5 more miles before turning around. The correct answer is A.

DATA SUFFICIENCY DISTANCE RATE SAMPLE QUESTION

For data sufficiency questions, we always evaluate each statement alone first. Let’s look at statement (1) first.

Statement (1) tells us that $2x + 3y = 280$. We know that $X$ and $Y$ are our variables for distance. We also know that Marta traveled for a total of 5 hours.

Marta’s average speed, then is $(2X + 3Y/5)$ miles per hour, as we know that she’s traveling for $X$ miles per hour for 2 hours and $Y miles$/$per hour$ 3 hours and dividing by 5 will help us, then, find the average speed over the entire length of the trip (5 hours).

Since $(2X + 3Y) = 280$, it also follows that $(2X + 3Y)/5 = 280/5$. Statement (1) is sufficient.

Now let’s look at statement (2). Statement (2) tells us that $Y = X + 10$. Let’s plug that back into our initial equation for speed.

$(2X + 3Y)/5 = R$, where $R$ equals speed. Let’s plug in our equation for $Y$.

$[2X + 3(X +10)]/5 = R$, or $X + 6 = R$. However, we don’t have enough information to solve for speed.

Therefore, statement (2) is not sufficient.

The correct answer, then, is A.

GMAT Interest Rate Problems

GMAT interest rate problems typically ask you to determine how much money a person earns if they place a certain amount of money into account at a particular interest rate.

There are a number of formulas that you should memorize to master these questions.

The simple formula for interest is:

Interest = $P x R x T$

• $P$ = starting principle (or, amount in the account)
• $R$ = annual interest rate (usually expressed as a percent)
• $T$ = number of years

GMAT interest rate questions typically ask about more complicated interest questions, as well as simple interest questions. Here are some other formulas you should know for GMAT interest rate questions:

The formula for annual compound interest is:

Annual Compound Interest = $P(1+R)^T$

• $P$ = starting principle
• $R$ = annual interest rate
• $T$ = number of years

The formula for compound interest is:

Compound Interest = $P(1 + R/X)^xt$

• $P$ = starting principle
• $R$ = annual interest rate
• $T$ = number of years
• $X$ = numbers of times the interest compounds over the year

PROBLEM SOLVING INTEREST RATE SAMPLE QUESTION

Let’s start by understanding what the question tells us. It tells us that John receives his interest compounded quarterly. That means he receives his one quarter (1/4) of his compound interest every three months. Since John’s interest rate is 4 percent, that means he earns 1 percent of his interest every three months.

Every three months, 1/4 of the compound interest is added to John’s account. This new total then accrues interest for the next quarter.

The question asks us how much money is in John’s account after six months. Six months means that John’s account has accrued interest twice.

The amount of money in John’s account will then be found with the equation: $amount = ($10,000)(1.01)(1.01)$.

So, John will have $10,201 in his account after six months. Lucky John!

The correct answer is D.

DATA SUFFICIENCY INTEREST RATE SAMPLE QUESTION

For data sufficiency questions, we always want to assess each statement by itself first.

First, let’s say that $H$ = the amount of money Hannah invested at $Y$ percent. We can use that variable for all of our equations.

We can express Hannah’s earned interest rates with two equations. First, $(X/100)($5,000) = the amount of interest$ Hannah earned from her first account. $(Y/100)(H)$ = $the amount of interest$ Hannah earned from her second account.

Statement (1) says that Hannah earned $900 from both accounts. So, $([X/100][$5,000] + [Y/100][A]) = $900$.

We have no way of further isolating the variables however, so we don’t have enough information to solve for $H$. Statement (1) is not sufficient.

Let’s look at statement (2). Statement (2) tells us that $X = 6$, because the statement tells us that the annual interest rate is 6 percent. However, telling us that $X$ is 6 still doesn’t give us enough information to solve for A. Statement (2) is also not sufficient.

Statements (1) and (2) together are also not sufficient because there is more than one possible value for $H$. The correct answer, therefore, is E: both statements together are not sufficient.

How to Practice for GMAT Rate Problems

While GMAT rate problems may seem tricky, practicing them will significantly increase your ability to solve them on test day. Keep in mind these tips as you’re studying to boost your performance.

#1: Memorize the Formulas and Equations

The best way to feel confident when tackling GMAT rate questions is to know your GMAT rate formulas and equations backwards and forwards… literally! For instance, memorize that $distance = rate*time$ and $time = distance/rate$. Spend time memorizing the other equations and formulas that I suggested in the earlier sections of this article. If you spend your time memorizing equations and formulas, you’ll be able to easily understand what a particular question is asking you and what to do with the different variables in the question.

A great way to memorize formulas is by using flashcards. To learn more about the best ways to use flashcards, check out our total guide to the best GMAT flashcards.

#2: Remember That Rates Are Like Ratios

While rates may seem pretty complicated, they’re really actually ratios! Or, in another form, fractions. For instantly, fuel efficiency (miles per gallon) is really expressed as a fraction: $\miles/\gallon$. Thinking about rates in terms of fractions with numerators and denominators may help them seem easier to understand.

Remember, because rates are really ratios, we can solve GMAT rate questions by thinking about proportions, just like we would do with normal fractions and ratios. This tip can help if you’re struggling with how to set up your rate equations. Setting up rates as a ratio/fraction will help you think about how the different variables relate to each other.

#3: Add or Subtract the Rates

Most GMAT work rate problems will ask you to analyze the work rate of two or more machines or people. For these questions, you’ll likely have to compare individual production with their combined production.

When you’re tackling these problems, remember this fact: you can add or subtract the rates on these questions. You can’t add or subtract times. So, when you’re looking to find a combined amount of goods produced, make sure you’re combining the rates, not the times. For instance, if two machines are creating a certain amount of goods in a set time, you can add or subtract the amount of goods, but the time it takes to make the goods will remain the same.

#4: 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.

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 rate problems skills. You’ll likely have to combine your knowledge of rate problems 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: Solving GMAT Rate Problems

There are three basic types of GMAT rate problems: questions that ask about the rate of work completion, questions that ask about interest rate, and questions that ask about distance.

GMAT rate questions are very formula driven. Spend some time memorizing the equations and formulas in this article to be able to quickly and easily solve any rate problem you encounter on exam day.

What’s Next?

Want to set up a steady rate of study for the GMAT? (I promise, the rate puns are over now.) Our detailed GMAT study plan will help you set up an optimal study schedule and give you a solid foundation for improving your GMAT score.

Looking to master another part of the GMAT quantitative section? We have guides to help you build your knowledge in a number of different content areas. Check out our total guides to GMAT geometry and GMAT probability for content-specific guides.

If you’re looking for a more in-depth overview of the quant section, we have our total guide to GMAT quant, as well as a review of the best GMAT quant practice. We also have a complete guide to mastering the three question types on the GMAT verbal section, if you’d like to focus on a different area of the test.

The post GMAT Rate Problems: How to Conquer All 3 Types appeared first on Online GMAT Prep Blog by PrepScholar.