GMAT Tech


GMAT blog, GMAT prep, GMAT tech, computer adaptive testThe GMAT is in some ways a technological marvel. Thanks to the wonders of the internet, thousands of locations across the globe are instantly reporting scores on the same test. The computer-adaptive test adapts to your skill level, adjusting difficulty on a question by question basis. Every center is equipped with a state-of-the-art scanner that records examinees’ handprints as a security measure.

Unsurprisingly, technology can also help you prepare for this test. Every GMAT student knows that paper-based quizzes can’t produce a test-like experience. Full-length practice Computer Adaptive Tests, like those offered by Kaplan and from www.mba.com, are key to success. But you can take the online prep a step further; most GMAT prep books, like Kaplan’s or the Official Guide, are also available as PDFs. Learning your lessons from a tablet or computer screen get your eyes used to reading on a monitor, and forces you to take your notes on separate paper and not directly on the questions themselves. The more test-like your practice, the better!

When it comes to study schedules, technology can be a great help too. Sharing your schedule with your fellow students or with your instructor via an online calendar can help them help you keep on pace. And setting up automatic email reminders for your study sessions can make sure you don’t lose track of your GMAT prep on a busy day.

Finally, modern technology is great for finding likeminded students. Having a tough day studying? Looking for study-buddies in your area? Try posting on a GMAT forum or facebook page for encouragement, resources, and fellow GMAT students.



GMAT Prep: Be Assertive to Make Time


GMAT blog, GMAT prep, GMAT study planA few weeks ago, a group of break-dancers started dancing outside my GMAT classroom at a local university.

Now, a part of me thought this was very fun. I like to pretend I’m still cool to college students. So, I was smiling and trying not to bop my head to the music when I went out and asked them to turn down the music. They were pretty nice about it, too, and turned down their music. For about fifteen minutes. The second time I asked them to turn it down, I was a little less nice—and they were a little less happy to comply.

The third time, I didn’t ask. I called the Campus Police and had them rousted.

I felt bad about it. I was becoming “The Man.” I was an authority figure.  I was stern.  I wasn’t a “cool guy” anymore.  But I got over my guilt quickly, by reminding myself that I was a GMAT teacher, and my students were GMAT students. We had goals to meet, and hours to work, and we couldn’t do that with techno blaring in through the closed door.

The lesson here is that being assertive is an important part of the GMAT preparation process. Of course, “being assertive” is not code for “being a jerk.” The last thing you want to do is alienate the friends and family who will support you if things get tough! But I’ve heard so many stories of students who didn’t stand up for themselves. One got stuck repeatedly answering the door to his apartment during a CAT,  because his roommates had scheduled a delivery when they were out. Another had weekly family picnics; she couldn’t bring herself to tell her extended family that she needed a week or two off!

Preparing for the GMAT takes 120 to 150 hours, and it’s up to you to find that time. Spend time with your friends and family, but don’t let them become an obstacle. Tell them when you need time for a CAT. They’ll understand! A picnic, or a birthday party, or even an impromptu breakdancing session, is one day. The GMAT, your B-school, and your MBA can change the rest of your life.



Tough GMAT Probability Questions


GMAT blog, flipping a coin, probability of heads or tailsTackling some of the tougher GMAT probability questions efficiently relies on both steady practice and your ability to make two key decisions well. First, you will need to quickly and accurately assess the total number of possible outcomes (the denominator of your probability equation). Second, within a multitude of possible approaches, you will need to determine the most efficient route to calculate the number of desired outcomes (the numerator of your probability equation).

With the clock ticking away on your GMAT CAT, figuring out the total number of possibilities can be time-consuming and fraught with room for error. For instance, if a question asks about the probability of getting at least 2 heads on 5 coin tosses, you could sit there all day writing out possibilities:

HHTTT

HTHTT

HTTHT

So forth and so on. I know I got dizzy with the possibilities just writing those three out.  There is a better and more efficient way. For every coin that you toss there are 2 possibilities. You can think of the total possibilities like a permutation problem.

__2__    __2__   __2__   __2__  __2_

1st             2nd          3rd         4th         5th

Just like in a GMAT permutations question when we are trying to determine the total number of codes possible or 4-digit numbers, we would multiply these individual probabilities together. Therefore, there are 2x2x2x2x2 = 2^5 = 32 total possibilities.

Next, we need to look at the numerator  (desired outcomes). We want to find the all of the possibilities that have at least 2 heads, which means that we could have 2 heads, 3 heads, 4 heads, or 5 heads.  To do so, we would need to count all of the different ways that these possibilities could be arranged. Again, we find ourselves in a situation that will be time-consuming and fraught with error. Instead of going down this path, remember that the sum of the probabilities of a complete set of mutually exclusive possible outcomes is 1.  Thus, as is often the case on “at least” probability questions, we can look for those options that are restricted. Then we only have to count the options that have 1 or 0 heads.

TTTTT

HTTTT

THTTT

TTHTT

TTTHT

TTTTH

There are only 6 of those, instead of the 26 possibilities the other way.

Finally, we can either subtract 6/32 from 1 in order to remove all of the restricted possibilities from 1 or we can subtract 6 from 32 and use the result as the desired possibilities. Either way, the answer is 26/32, which you can reduce down to 13/16.

Let’s look at another to make sure we have this down.

Question

A test has 5 multiple-choice questions. Each question has 4 answer options (A,B,C,D). What is the probability that a student will choose “B” for at least four questions if she leaves no questions blank?  Pause a moment and try it for yourself first.

Step 1: Total number of possibilities

There are 5 questions and each has 4 possibilities, so our total possibilities would be 4x4x4x4x4 = 4^5 = 1024

Step 2: Approach Desired Strategically

Here there are far more possibilities for 0, 1, 2, or 3 “B’s,” so let’s get a total for 4 or 5 “B’s”.

All B’s — B,B,B,B,B

Four B’s — A,B,B,B,B – B,A,B,B,B – B,B,A,B,B – B,B,B,A,B — B,B,B,B,A

C,B,B,B,B – B,C,B,B,B – B,B,C,B,B – B,B,B,C,B – B,B,B,B,C

D,B,B,B,B – B,D,B,B,B – B,B,D,B,B – B,B,B,D,B – B,B,B,B,D

3 x 5 = 15 because we can repeat the same pattern for each letter other than B

We can also calculate the total possibilities of 4 B’s by calculating the possibilities for each “no-B” position.

No-B,B, B, B, B = 3x1x1x1x1 = 3

B, No-B, B, B, B = 1x3x1x1x1 = 3

B, B, No-B, B, B = 1x1x3x1x1 = 3

B, B, B, No-B, B = 1x1x1x3x1 = 3

B, B, B, B, No-B = 1x1x1x1x3 =3

A total of 15 possibilities with 4 B’s in the mix

 

That gives a total of 16 different ways that a student can choose at least 4 B’s here.

16/1024 = 1/64 as our final probability.

 

Keep these two decisions in mind each time that you approach a tough probability question on the GMAT quantitive section. You don’t have to write out all of the possible outcomes in order to tackle these on test day!

The language of probability can take a while to learn, especially if you are unfamiliar or out of practice with it to start.  Post your questions below, and we can help you get on track.



Three GMAT Challenges


GMAT blog, GMAT study plan, GMAT goal, MBA applicationsPiecing together the time to study for the GMAT can be challenging.  In today’s blog, I’m going to talk about three students (whose names I’m changing to protect their identities).  Each had a major obstacle to studying, and each overcame it in a different way. I hope these students’ examples can help some of you reach your GMAT and MBA goals.

 

Case Study 1: Vincent, the Entrepreneur

The Challenge: Vincent was a busy man when I was tutoring him. His schedule was very flexible—his main source of income was a business that he started and ran himself—but he was distracted at all hours by emails and phone calls related to his work.

The Solution: Vincent needed a time and place where he could study in peace.

Because of his flexible work schedule, it was easier for Vincent to find time than it is for some other students. He dedicated a daily block of time to studying, and had the discipline to stick with it—though as his tutor, I was standing by ready to make sure he stuck with it if he got distracted! 

Vincent had a harder time finding the space he needed to study. Local coffee shops were noisy, and didn’t have reliable internet connections for his CATs. But fortunately, there was a quiet study space regularly available in the local public library. Not only did that let him work in peace, but it also forced him to turn off his cell phone and disconnect from the world.

Ultimately, Vincent got a 700—though he didn’t quite reach his goal, he significantly improved his score, posting a result that combined with his entrepreneurial experience to make a top-tier-worthy application.

 

Case Study 2: Brandon, the Financier

The Challenge: Brandon had a lot of things to cope with. He was a long time out of college, so his writing and grammar skills were rusty (especially since he was a non-native, though fluent, speaker of English). Moreover, though he worked with numbers quite a bit in his job at a bank,the GMAT quantitative section proved challenging since he seldom had to do algebra, let alone geometry or probability. 

Brandon had a relatively easy work schedule and a strong work ethic, and he was able to make consistent, steady progress across the board. But after 4 weeks and 60 points of improvement, he was exhausted and burnt out. 

The Solution: Brandon and I sat down to start working on his applications. 

This was something that had to get done, so it was a good use of time—but for Brandon, it was also a welcome relief from the constant effort of GMAT studying, especially when rusty fundamentals meant nothing was coming easily Working on the applications boosted his confidence, since seeing awesome application essays reminded him that he was a strong candidate already, and his test score was just the final piece of the puzzle. And finally, writing application essays with questions like “What are your goals at business school?” restored his focus on why he was studying for the test in the first place! 

After spending a few days writing and revising application material, Brandon was ready and energized to get back to GMAT studying—and his practice test scores kept rising.

 

Case Study 3: Sally, the Management Consultant

The Challenge: Sally was working as a consultant while taking my class. She worked 70-80 hours/week during her busy periods and 50 hours/week at slower times. She spent most of her work week away from home. And perhaps most frustrating of all, Sally’s subordinates were studying for the GMAT on every train ride to and from their work site. She wanted to study with them, but didn’t think she’d be able maintain the respect necessary to manage them—especially since some of them were outscoring her on practice tests! 

The result: Sally decided not to take the test.

I realize this might not seem like an inspiring outcome, but it’s actually quite brilliant. The GMAT is not something that fits everyone’s schedule at any given time—it’s a major commitment. Forcing yourself to take a test you’re not ready for is just going to put a mediocre score on your record for the next five years. And more importantly, there is more than one path to success. Sally’s hard work has earned her a raise to a pay grade normally reserved for MBA’s! She’s hoping that with a few more years of such progress, she’ll be able to achieve her long-term career goals through an executive MBA program, which will be a better fit for her busy, hardworking lifestyle. I look forward to helping her again when that time comes.



GMAT Critical Reasoning Inference Questions, Part 2


GMAT Blog, GMAT Critical reasoning inference questionListen to a politician speaking, and you’ll hear a lot of platitudes and vague statements. Occasionally, a senator or congressman will make a statement about a specific number or an exact proposal; rarely, those statements will even be correct. But mostly, you’ll hear things like, “the hidden costs will total billions,” or “this program will have far-reaching negative impacts,” or “some have suggested that this proposed law will do nothing but enrich corporations.”

When you think about it, these claims make perfect sense. With a claim as vague as the ones above, it’s hard to be proven wrong or caught in a lie. For instance, “hidden costs” could refer to net costs, but it also could refer to gross costs even if the proposal actually netted a profit. “Billions” could refer to two billion, or it could refer to two hundred billion!

In other words, the vaguer the claim, the more likely it is to be true. And on GMAT Critical Reasoning Inference questions, which ask you to identify what must be true on the basis of a short statement, the vaguest answer is most likely to be correct.

It may seem contradictory that strong words like “must” are rarely the answer to a question that asks what “must be true.” But when you think about it, it makes sense. It’s very, very easy to conclude something is possible. It’s much harder to prove something is certain. For instance, it can be easy to prove that a type of thing might have a certain quality—for instance, you can prove that some swans are white by pointing to a single white swan. But proving that all those things have that quality requires you to rule out every possible exception. Even if you show me 999 white swans, the 1,000th swan might turn out to be black!

You should always spend a few moments trying to predict the answer to an Inference question, even though it’s not always possible. And if the answer is too hard to predict, your next step should be to carefully check the answers for one that must be true. But sometimes, the computer adaptive test will give you a very high-difficulty problem, or you’ll be stuck between two answer choices, or you just won’t have time and need to guess strategically on a problem or two to beat the clock. And in those cases, picking the vaguest answer is one of the most reliable guessing strategies on the GMAT.

Consider the following GMAT practice problem. Without reading the text, can you figure out which answer is most likely correct? Then, go through the whole problem properly, and see why it’s a good fit. Good luck!

 

Problem

Randall: Many of the productions of my plays by
amateur theater groups are poorly done, and such
interpretations do not provide a true measure of my
skills as a dramatist.

Which one of the following can be properly inferred
from Randall’s statement?

(A) Some amateur theater groups’ productions of
Randall’s plays provide a true measure of his
skills as a dramatist.
(B) All amateur theater group productions of
Randall’s plays that are not poorly done provide
a true measure of his skills as a dramatist.
(C) All of the productions of Randall’s plays by
amateur theater groups that do not provide a
true measure of his skills as a dramatist are
poorly done.
(D) If a production of a dramatist’s play is well done,
then it provides a true measure of his or her
skills as a dramatist.
(E) At least some amateur theatrical groups’
productions of Randall’s plays fail to provide a
true measure of his skills as a dramatist.

 

Solution:

Step 1: Identify the Question Type
The keywords “properly inferred” in
the question stem are a sure sign of an
Inference question.

Step 2: Untangle the Stimulus
Randall’s comments can be reduced
to an if/then statement: If productions
of his plays are poorly done, then they
don’t provide a true measure of his
skills as a dramatist. And many amateur
theater groups perform his plays poorly.

Step 3: Predict the Answer
While we may not be able to predict
what the correct answer choice will
infer, we can be certain that it is a
statement that must be true if we accept
Randall’s statement as true.

Step 4: Evaluate the Choices
Since many amateur productions
are poorly done, and no poorly done
production provides a true measure of
Randall’s skills, it must be true that at
least some amateur groups’ productions
do not provide a true measure of his
skills, so (E) is correct. Don’t be afraid
of (E) because it seems “obvious.” This
is not a test maker trick—an “obvious”
answer is one that must be true, so it
works as a valid Inference. (A) seriously
distorts Randall’s statement. Just
because some amateur productions
don’t do him justice doesn’t mean that
there are other productions that do. If
the GMAT tells you that some marbles
are red, you can’t automatically infer
that some are not red. (B) is another
sort of distortion. Randall’s statement
about certain poorly done productions
in no way guarantees anything about
productions that aren’t poorly done.
(C) is far too extreme. Randall does
establish a correlation between poor
production quality and failure to provide
a true measure of his skills, but that
correlation has only been established
for a certain set of productions and
can’t be extended to all productions.
(D) attempts to extract a broad principle
from Randall’s statement, but his
statement is too particular to allow this
kind of extrapolation.

The answer is (E).

 

 

 



GMAT Practice Accuracy


GMAT practice test accuracy, calibration with official GMATOne of the most common questions I field from students posting on MBA forums, is whether Kaplan CATs are accurate.

Some people see official scores so far above or below their expectation that they assume a math error is the only explanation. Others hear that a batch of Kaplan tests from around 2007 had some scoring irregularities, and assume (incorrectly) that we haven’t fixed things in the past five years. And still others just haven’t practiced enough to understand the ins and outs of the GMAT’s adaptive testing. But the question is always the same: are Kaplan tests mathematically representative of the real GMAT?

The answer is “yes.” Kaplan uses the Official GMAT tests to normalize our scores; students who take a Kaplan test and the Official GMAT in the same weekend usually get scores no further apart than the test’s statistical margin of error, 29 points.

But that’s not the whole story. As I mentioned, many students do see scores on test day that surprise them. Test scores on practice tests can vary wildly from exam to exam. Assuming an “accurate” test, this seems impossible; the GMAT produces very consistent scores.

The key lies in the fact that practice tests are practice. The GMAT isn’t purely a test of grammar, logic and math. It’s a mental game, testing your endurance and focus. Students who force tests into a busy schedule will find their late-night scores plummeting. Conversely, students who were nervous going into the real test but relaxed under the low pressure of a diagnostic may find their practice scores leagues higher than their official results.

So when you take a Kaplan test, you can be confident that it’s an accurate mathematical representation of your score. But you can’t be sure it’s an accurate real-life approximation. Instead, you need to ask yourself: how did I feel when I took the mock test? How will I feel on Test Day? If you realize there’s a discrepancy, take that into account when you look at your score. And try to minimize the factors that could disrupt your score; you can reduce study-stress by planning out a study schedule, and use stress-reduction techniques on the day of the GMAT to make sure your head stays in the game.



Kaplan’s GMAT Course Undergoes Continuous Improvement


GMAT BlogOne thing I like about Google is that they are constantly churning out both new products and improvements to additional products.  Google knows that in order to stay relevant and lead the market, innovation is fundamental.  Kaplan does, too.

For more than 70 years, Kaplan has been training ambitious individuals to reach and exceed their goals on standardized tests so they can reach and exceed their goals professionally.  We have been teaching the GMAT to prospective business students almost since its inception in 1954.  In short, Kaplan Test Prep is a product leader and, like Google, we have multiple teams devoted to continuous product improvement and innovation.

Instead of letting all this hard work and commitment go unnoticed, I want our students to know what is going on behind the scenes.  Not only do the smart people behind these projects deserve some recognition, but it is also important that everyone is up-to-date with everything we have to offer.  After all, the worst resource is the one that goes unused!

Recent Product Updates:

  • All nine of our computer adaptive practice tests contain an Integrated Reasoning section.  We offer the most full-length practice tests in the business and we were the first to develop IR lessons and realistic practice sections.  By the way, practice tests taken at a Pearson VUE Testing Center will include the IR section, as well, and that Test Day Experience is only available to Kaplan GMAT students.
  • We offer eight practice test timing options.  It is important to provide our students with a realistic testing experience during their preparation with us.  Some individuals require special accommodations to sit for the GMAT due to various physical and cognitive challenges.  In order to maintain an ultra-realistic testing experience for all of our students, we have fully fleshed out our special ADA timing options within our program to meet every need of every student.
  • Students can flag quiz questions and create quizzes comprised of flagged items.  Our GMAT Quiz Bank tool houses the highest number of GMAT practice questions in the business.  Students can create hundreds of quizzes across multiple layers of granularity.  A quiz parameter example might be: a timed quiz (timing based on q-type) focusing on arithmetic and specifically set properties offering medium and high difficulty problem solving and data sufficiency questions pulled from both used and unused items.  Now, we have added a new functionality: flagging.  Students can flag questions and create quizzes from those flagged questions for further review.
  • Question statistics and enhanced explanation display.  Our answer explanations are extremely robust and always reflect the most efficient and strategic way to solve a GMAT question.  We know that studying can be an arduous process.  Just like the need for a comfortable chair, the ergonomics (i.e., visual appeal) of the explanation display makes a difference.  No detail is too small!  Furthermore, in order to help you identify wrong answer pathologies and to self-diagnose performance, we now offer question statistics.  We tell you the percent of test takers that chose each answer choice option for each question.
  • The Lessons On Demand and Workshops list now include run times.  Sounds simple, I know, but it makes planning study sessions easier.  Before, a student would not know how long one of our professionally produced videos would last until they clicked on it.  A simple but helpful solution of adding the total run time next to the video title lets you make the most of your study session.
  • We adjust our 1-8 Integrated Reasoning scoring scale in tandem with GMAC.  As more and more test takers sit for the New GMAT with the IR section, more and more data will help GMAC communicate what a given score within the scoring scale means in relation to a percentile distribution.  As GMAC collects this data and updates their percentile distribution chart, so will we.  For more on the Integrated Reasoning scoring scale, click here.
  • Improved answer choice selection functionality.  Again, this may seem like a nominal improvement, but the results justify such attention to detail.  In the past, when a test taker selected an answer choice, the cursor of the mouse needed to hover over the bubble preceding the choice.  Now, test takers can click anywhere on the answer choice to select it.  So what?  Well, that small improvement could mean an aggregated decrease of 1 minute of time spent just placing the cursor in the right spot on the screen.  One minute can easily equal one question!

From now on, I am going to track our course and product improvements for our students, teachers, and everyone else interested in learning with Kaplan.  Check back often for updates.  Also, please reply with any and all suggestions you may have to make our GMAT products even better.  Thanks!



New GMAT Transition Coincides with Venus Transit


GMAT Blog

NASA’s SDO observes Venus prior to solar transit

June 5, 2012 has finally come and gone.  To those of us within the gravitational pull of the GMAT, this date was no less than a celestial event.  June 5th not only marked the transit of Venus across the sun, but also the launch of the New GMAT.

What has changed?  A new section called Integrated Reasoning (IR) has replaced the Analysis of an Issue essay and taken its time allotment.  Hence, the GMAT is still the same total length.  That is, you write a 30-minute Analysis of an Argument essay, then take the new 30-minute Integrated Reasoning section, then take the 75-minute Quantitative section, and finally complete the 75-minute Verbal section (note: you get two 8-minute breaks; one between IR and Quant, and then another between Quant and Verbal).

Integrated Reasoning questions appear in four different formats and across twelve questions total in the 30-minute time frame.  The formats are: Graphics Interpretation, Two-Part Analysis, Table Analysis, and Multi-Source Reasoning.  A given prompt, or question setup, may have multiple questions and, like the rest of the GMAT, IR is computer adaptive at the question level.  Thus, once a question has been answered, you cannot return and change the answer.  It is also interesting to note that test takers have access to a very basic on-screen calculator during this section only (i.e., still no calculators on the Quantitative section).

I have written at length about the New GMAT in previous posts and invite you to read through them to learn more (here’s a dozen: one, two, three, four, five, six, seven, eight, nine, ten, eleven, and twelve).  However, I am much more interested in your actual experiences over my anticipated ones.  We want to hear from the pioneers out there—those of you who have been among the first to take the New GMAT.

What was it like? 

How did you prepare? 

Was it challenging? 

Did you really feel like the questions were forcing you to integrate reasoning? 

What surprised you?

While Venus will not traverse the sun again until 2117, brave explorers destined for b-school greatness will take the New GMAT just about every day from here on in. 

Tell us about your experience blazing the trail that others will soon follow.  If you have taken it, we want to hear about it!  Boast, warn, and teach – whatever you think the experience calls for…



GMAT Fences


GMAT BlogIn my years of teaching, I’ve seen all kinds of clever solutions to GMAT math problems. I’ve also seen all kinds of errors. Some are utterly bizarre—and fortunately, seldom repeated, because the students who make those mistakes usually face-palm when they review their tests and go on to learn from their missteps. But some errors are so common and so often repeated that they earned their own names. One such example is the “fencepost error.”

Here’s a simple example: Say we are setting up a straight fence that’s exactly 100 ft long, with posts every 10 feet. How many posts do we need?

Did you say 10? Tempting, but that’s the right answer to the wrong question. There are 10 sections of fence, each 10 feet long.  But there are actually 11 fenceposts, because you start with a fencepost, at 0 feet!

GMAT Blog

 

 

See?

This error can trap the unwary GMAT student in a few different ways. The most common is in finding sums of series consecutive integers.  The formula for the total of an evenly spaced list is to multiply the average value of that list by the number of items. So, you need to know exactly how many numbers you’re adding. So if a question asks for the sum of numbers between, say, 37 and 59, some students might just say there are 59 – 37 = 22 numbers on that list—but that doesn’t count 37, the first number. To ensure you solve for the correct value, you need to add the first “fencepost” and count all 23 numbers (write them out and count if you want to confirm!)

Also, rather oddly, this error shows up on the verbal section in sentence correction too. Clauses are the “fenceposts” in sentences, in a sense, and connecting words (therefore, however, so) are the fence sections. You can tell a sentence is improperly constructed if you don’t have exactly one more clause than you have connecting words.

Keep this error in mind as you solve your practice problems—and today’s question of the day. It’s a common mistake, but also an easy one to avoid once you’re aware of it. Practice will make sure you’re not fooled on Test Day.

Question of the Day:

In a new housing development, trees are to be planted along the sidewalk of a certain street. Each tree takes up one square foot of sidewalk space, and there are to be 14 feet between each tree. How many trees can be planted if the road is 166 feet long?

A) 8

(B) 9

(C) 10

(D) 11

(E) 12

Solution:

Step 1: Analyze the Question

Though this is not a Geometry question, a quick sketch of the situation will help illustrate how to solve it. GMAT Fencepost illustration

 

 

 

So we know that the unit of one tree and one space is 1 foot + 14 feet = 15 feet.

Step 2: State the Task

To find how many trees can be planted, determine the feet required for a tree and the space between trees. Divide the total length of the street by the unit of one tree and the space between trees.

Step 3: Approach Strategically

Each tree takes up 1 foot, and each space takes up 14 feet. Together they take up 15 feet. Now find how many times 15 goes into the total number of feet on one side of the street:

166 / 15 = 11, with a remainder of 1 foot.

We can plant one last tree in the remaining foot, bringing the total number of trees to 12. This means along the street, we can plant 12 trees with 11 spaces between them, as long as we start and end with a tree. (E) is correct.

Step 4: Confirm Your Answer

Make sure your answer makes sense in the context of the question. Did you take into account the remainder of the division? Will an entire tree fit in the remaining space? You can use these questions to confirm your work.