500 New SAT Math Problems
Just 19.99 on Amazon

Hi everyone! The latest edition of 500 New SAT Math Problems is now available in paperback from Amazon. This edition just has been modified from the previous edition to account for the changes on the Digital SAT.

The paperback is now on sale on Amazon for only $19.99. Note that once the sale ends (by the end of today), the price of this book will go up to $42.99. 

The promotion has ended. Thanks to everyone who participated. The book is now available at its regular price here: 500 New SAT Math Problems

If you have any questions, feel free to contact me at steve@SATPrepGet800.com 

Thank you all for your continued support!

A Trick For Free Two Day Shipping

I would like to finish this post with a little trick you can use to get free 2 day shipping on any of the books you decide to purchase without making any additional purchases. If you have never used Amazon Prime you can sign up for a free month using the following link.

Sign Up For Amazon Prime For Free

If you have already had a free trial of Amazon Prime you can simply open up a new Amazon account to get a new free trial. It just takes a few minutes! You will need to use a different email address than the one you usually use.

This next part is very important! After you finish your transaction, go to your Account, select “Manage my prime membership,” and turn off the recurring billing. This way in a month’s time Amazon will not start charging you for the service.

After shutting off the recurring billing you will still continue to receive the benefit of free 2 day shipping for one month. This means that as long as you use this new Amazon account for your purchases you can do all of your shopping on Amazon for the next month without having to worry about placing minimum orders to get free shipping.

Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.

Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:

Sign Up For Amazon Prime For Free

If you think your friends might be interested in this special offer, please share it with them on Facebook:

Thank you all for your continued support!

500 New SAT Math Problems
Just 19.99 on Amazon

Hi everyone! The latest edition of 500 New SAT Math Problems is now available in paperback from Amazon. This edition just has been modified from the previous edition to account for the changes on the Digital SAT.

The paperback is now on sale on Amazon for only $19.99. Note that once the sale ends (by the end of today), the price of this book will go up to $42.99. 

The promotion has ended. Thanks to everyone who participated. The book is now available at its regular price here: 500 New SAT Math Problems

If you have any questions, feel free to contact me at steve@SATPrepGet800.com 

Thank you all for your continued support!

A Trick For Free Two Day Shipping

I would like to finish this post with a little trick you can use to get free 2 day shipping on any of the books you decide to purchase without making any additional purchases. If you have never used Amazon Prime you can sign up for a free month using the following link.

Sign Up For Amazon Prime For Free

If you have already had a free trial of Amazon Prime you can simply open up a new Amazon account to get a new free trial. It just takes a few minutes! You will need to use a different email address than the one you usually use.

This next part is very important! After you finish your transaction, go to your Account, select “Manage my prime membership,” and turn off the recurring billing. This way in a month’s time Amazon will not start charging you for the service.

After shutting off the recurring billing you will still continue to receive the benefit of free 2 day shipping for one month. This means that as long as you use this new Amazon account for your purchases you can do all of your shopping on Amazon for the next month without having to worry about placing minimum orders to get free shipping.

Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.

Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:

Sign Up For Amazon Prime For Free

If you think your friends might be interested in this special offer, please share it with them on Facebook:

Thank you all for your continued support!

1000 New SAT Math Problems
Just 29.99 on Amazon

Hi everyone! The latest edition of 1000 New SAT Math Problems is now available in paperback from Amazon. This edition just has been modified from the previous edition to account for the changes on the Digital SAT.

The paperback is now on sale on Amazon for only $29.99. Note that once the sale ends (by the end of today), the price of this book will go up to $64.99.

The promotion has ended. Thanks to everyone who participated. The book is now available at its regular price here: 1000 New SAT Math Problems

If you have any questions, feel free to contact me at steve@SATPrepGet800.com 

Thank you all for your continued support!

A Trick For Free Two Day Shipping

I would like to finish this post with a little trick you can use to get free 2 day shipping on any of the books you decide to purchase without making any additional purchases. If you have never used Amazon Prime you can sign up for a free month using the following link.

Sign Up For Amazon Prime For Free

If you have already had a free trial of Amazon Prime you can simply open up a new Amazon account to get a new free trial. It just takes a few minutes! You will need to use a different email address than the one you usually use.

This next part is very important! After you finish your transaction, go to your Account, select “Manage my prime membership,” and turn off the recurring billing. This way in a month’s time Amazon will not start charging you for the service.

After shutting off the recurring billing you will still continue to receive the benefit of free 2 day shipping for one month. This means that as long as you use this new Amazon account for your purchases you can do all of your shopping on Amazon for the next month without having to worry about placing minimum orders to get free shipping.

Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.

Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:

Sign Up For Amazon Prime For Free

If you think your friends might be interested in this special offer, please share it with them on Facebook:

Thank you all for your continued support!

Hi all,

A few months back I explained how each of my books has been designed for a specific reason. In this post I would like to give a brief description about each of the books to help you decide which one (or ones) is right for you.

SAT Prep Official Study Guide Math Companion – This contains my solutions to all problems in the Blue Book tests (1-10 and DVD). Most problems are solved in several different ways using methods that are much simpler and more efficient than the College Board’s solutions.

28 SAT Math Lessons To Improve Your Score In One Month (Advanced Course) – This is a one month course designed for students currently scoring over 600 in SAT math. If used properly by these students, a perfect 800 or near perfect score can be achieved. Please note that this book contains well over 300 problems, most of which are level 4 and 5. Some of these problems are extremely rare and quirky in nature, but everything has been included for the student that is going for a perfect score.

28 SAT Math Lessons To Improve Your Score In One Month (Intermediate Course) – This is a one month course designed for students scoring between 500 and 600 in SAT math. It contains all the most important strategies, and problem types designed to take students’ scores closer to a 700. This course takes all the guesswork out of what to study.

28 SAT Math Lessons To Improve Your Score In One Month (Beginner Course) – This is a one month course designed for students scoring below 500 in SAT math. This is a very low intensity course designed to get unmotivated or frustrated students with a weak mathematical background to begin solving SAT math problems. It contains only the most basic strategies and problem types designed to take students’ scores closer to a 600. This course takes all the guesswork out of what to study.

320 SAT Math Problems Arranged By Topic And Difficulty Level – As the title suggests, this book has problems arranged by topic and difficulty level. Note that the problems in this book are uniformly distributed among all levels and topics. If you are looking for a book that focuses on only difficult SAT math problems, then this is NOT the right book for you. The very advanced student will get much more value out of the Advanced Course 28 Lessons book.

The 32 Most Effective SAT Math Strategies – In here you will find all the most important and significant tips, tricks and tactics to solve problems you will face in SAT math. Internalize these strategies and you will be able to answer more SAT math problems in less time.

I have actually updated all of the books’ product pages with this information, so I hope this make things a bit clearer for those of you that are unsure which of my 6 books might be best for you.

Common SAT Math Strategies

Welcome back to the GET 800 blog – The only place you need to visit for SAT math hints, tips and tactics!

I have been asked some excellent questions on my Facebook page lately. And today I will address one of these questions that had been asked only a few days ago. If you have any SAT math related query that you want answered, simply post it up there and I will come back with an answer ASAP.

There are three very basic SAT math strategies that every high school student should know: Plugging In Answer Choices, Taking Guesses, and Picking Numbers. Take a look at these links before you continue with this message, as the information below will make more sense after a review of these strategies.

Sometimes students get confused about when to use which of these strategies. Although there are no definite rules I can give you that will work 100 percent of the time, today I will provide you with some general guidelines.

Plugging In

Let’s begin with the strategy of plugging in answer choices:

This strategy can only be used for multiple choice questions. You simply plug each answer choice into the question until you find the one works. It is always best to start with choice (C) as your first attempt, unless there is a specific reason not to. An example of such a reason would be that the word “least” appears in the problem. In this case start with the smallest answer choice (which will usually be choice (A) or (E)). Similarly, if the word greatest appears in the problem, start with the largest answer choice.

Plugging in can be used on problems from every topic and difficulty level. It’s often a great way to avoid having to perform messy algebra.

This strategy is most useful when the last part of the question says “What is_____” where the blank contains a single quantity.

Here are some examples:

(1) What is the value of x?
(2) What is the length of the original rectangle?
(3) What is the second number in the list?
(4) Which of the following is a perfect cube?
(5) What is the least such value of x?

For the first four of these examples you would start with choice (C). For the fifth example you should start with the smallest answer choice.

This strategy is generally NOT useful when you are asked to find a more complicated expression. For example, do not try to plug in answer choices if the question ends with “What is the value of x + y?” In this case you will probably want to use the strategy of “Taking a guess,” or possibly “Picking numbers.”

During your SAT math practice sessions, you should try to apply this strategy on every multiple choice question. The more you attempt to use it, the easier it will be to detect when it can be applied.

Here is a straightforward example of a Level 1 Number Theory problem where “starting with choice (C)” is a useful strategy.

Three consecutive integers are listed in increasing order. If their sum is 138, what is the second integer in the list?

(A) 45
(B) 46
(C) 47
(D) 48
(E) 49

I suggest you try to solve this problem in 3 ways:

• by starting with choice (C)
• algebraically
• using the fact that in a list of consecutive integers, the arithmetic mean is equal to the median.

And here is a Level 4 Geometry problem where this strategy is effective:

The sum of the areas of two squares is 85. If the sides of both squares have integer lengths, what is the least possible value for the length of a side of the smaller square?

(A) 1
(B) 2
(C) 6
(D) 7
(E) 9

You should solve this problem by starting with choice (A).

There are more examples in my original blog post titled Starting With Choice (C) – A Basic SAT Math Strategy.

Taking Guesses

Next let’s talk about the strategy of taking a guess:

This strategy is similar to plugging in, except with this one you are not using the answer choices. The reason you are not using the answer choices is either because it is a grid in problem (so there are no answer choices), or the question is asking for a more complicated expression such as “the value of x + y,” or “the perimeter of a geometric figure. In the first case you may want to take a guess for x or y, and in the second case you may want to take a guess for the width of a side of the figure. For examples of applying this strategy see my original blog post titled Basic SAT Math Strategy – Take A Guess. For a more difficult example, see my blog post titled Taking a Guess – A Further Example Of How To Use This Technique.

Picking Numbers

And finally, let’s talk about the strategy of picking numbers:

This strategy is applied by choosing specific values for the unknown quantities in the problem. A new problem is formed that is easier to solve. After solving this easier problem, you must then plug the specific values you have picked into every answer choice, and eliminate any answer choices that do not come out correct. If more than one answer choice has not been eliminated, you can try picking new numbers to eliminate more choices.

Here are some occasions when the strategy of picking numbers will usually work:

• When one or more variables appear in the answer choices.
• When the word “percent” is in the problem (pick 100).
• When there is a variable in the problem, and it is implied that the answer does not depend on the value of the variable.

Here is an Algebra problem where “Picking numbers” can be used.

Which of the following is equal to (x + 66)/22 ?

(A) (x+33)/11
(B) x+3
(C) 3x
(D) x/22+3
(E) (x+3)/11

Note that you usually want to avoid picking numbers that are too simple. For example, if you choose x = 0, then the answer to the question becomes 3. Now, if you plug a 0 in for x into every answer choice you get the following:

(A) 3
(B) 3
(C) 0
(D) 3
(E) 3/11

Note that (A), (B), and (D) are all correct.

See if you can solve this problem by picking a better number. You may also want to try to solve it algebraically.

For examples of applying this strategy see my original blog post on Picking Numbers.

Picking Numbers In Percent Problems

If the word “percent” appears in a problem it’s usually a great idea to choose the number 100. This often works even when there is no variable in the problem.

For examples of picking numbers in percent problems, see my blog post titled Picking A Number To Solve Percent Problems.

Summary

To summarize, here are some general guidelines for choosing which strategy to use:

• If the answer choices are all numbers, and you are being asked for a simple quantity, try plugging in first.
• If the answer choices are all numbers, and you are being asked for a more complicated quantity, try taking a guess.
• If the answer choices contain variables, try picking a number.
• Any question that mentions percents, pick the number 100.
• In grid in questions with variables, try taking a guess.

There is a lot of information in this blog post with links to many of my other posts, so take your time to read and digest the information. As always, the best way to internalize the concepts above is by practice. So get your Blue Book and get practicing! If you would like additional practice, check out my 28 SAT Math Lessons Series. Click on the picture below for more information about these books.

If you liked this article, please share it with your Facebook friends:

And if you have any questions on this topic, then please post them on my Facebook wall:

I will answer your questions right away.

Hello everyone.

If you have been reading this blog, then you are probably familiar with what the SAT is. In case you are not, here is a link to one of my articles that gives detailed information on this: Overview of the Math Sections of the SAT

One thing that I have not yet talked about is the PSAT.

First, did you know that the “P” in PSAT stands for preliminary? Most people incorrectly think that it stands for practice. 

Also, almost twice the number of test takers take the PSAT compared to the SAT. This is because the PSAT determines eligibility and qualification for the National Merit Scholarship Program.

In addition to College Board Blue Book tests, almost all of my students take the PSAT to gauge how well they are going to do on the SAT. There are a few major differences between the PSAT and the SAT and this post will shed some light on these differences.

Test Length and Structure

The PSAT consists of 5 sections. There are two 25 minute reading sections, two 25 minute math sections, and one 30 minute writing section. This gives a total testing time of 2(25) + 2(25) + 30 = 130 minutes, or 2 hours and 10 minutes. If we add in the 5 minute break after section 2 and the 1 minute break after section 4 we get a total time of 2 hours and 16 minutes from the beginning to the end of the test.

The SAT is much, much, much longer than the PSAT. It consists of 10 sections – that’s right, double the amount that is on the PSAT. There is one 25 minute essay, one 25 minute writing section, one 10 minute writing section, two 25 minute reading sections, one 20 minute reading section, two 25 minute math sections, one 20 minute math section, and one 25 minute experimental section which may be reading or math. This gives a total testing time of 7(25) + 2(20) + 10 = 225 minutes, or 3 hours and 45 minutes. If we add in the three 5 minute breaks, we get a total time of 4 hours from the beginning to the end of the test.

So although the PSAT is supposed to prepare you for the SAT, one thing that it definitely does NOT prepare you for is the amount of endurance necessary to get through the SAT. The SAT is the first, and maybe only test that you will ever take that requires this kind of time commitment. Fatigue alone can be responsible for causing a decrease in score from your PSAT to your SAT.

Scoring

The PSAT is graded out of 240 points with each of reading, writing and math consisting of a total of 80 points.

The SAT is graded out of 2400 points with each of reading, writing and math consisting of a total of 800 points.

So you can essentially convert your PSAT score to the corresponding SAT score by tacking on an extra zero at the end.

This is not entirely accurate however.

In other words, suppose a student were to score a 50 in math on their PSAT. Does that mean that if they had taken the SAT instead that day that they would have scored a 500? The answer in general is no – they most likely would have scored a bit lower than that. A rough estimate is about 5% lower.

So, in other words, after taking your PSAT, to get your approximate SAT score tack on a zero to the end, and take away 5 percent. This is equivalent to multiplying your PSAT score by 9.5.

For example, a student that scored 170 on their PSAT would have received approximately 170(9.5) = 1615 on their SAT. Of course 1615 is not actually a possible SAT score, but again, this is just a rough estimate.

Note that there is no statistical data to support the accuracy of this computation. After all, nobody takes a PSAT and an SAT on the same day. This is just my own personal way of estimating what I believe is happening.

Other than “the missing zero” all other aspects of scoring the PSAT and the SAT are the same. Multiple choice questions and grid ins are handled the same way, and the raw score is computed in the same fashion. For more details on how this is done, see this article titled Overview of the Math Sections of the SAT.

Just one comment on the scaled score – although for most students the scaled score is handled almost identically on both the PSAT and SAT, the strongest students may be penalized a bit more heavily on the PSAT. For example just one wrong question on the math section of the PSAT could drop your scaled score down 5 points. This could really hurt if you are applying for a National Merit Scholarship.

Material Covered

Almost all of the topics covered on the PSAT and SAT are identical. The only exception is a few math concepts that are left out of the PSAT. These concepts are not put on the PSAT because a large percentage of students have not yet covered these topics in school by October of their junior year (which is when most students take the PSAT).

The topics that are left off of the PSAT mostly pertain to functions and their graphs. These are topics that are generally introduced in Algebra 2.

The SAT also has one type of section that the PSAT does not – the essay. Students do NOT have to write an essay on the PSAT, but on the SAT this is one of the 10 sections.

Final Note

As mentioned in the section on scoring above, if you decide to take your SAT without any preparation, then it is more likely that your score will go down about 5% from your PSAT than go up. So whether you use a tutor, take a prep course, or prepare on your own, please prepare. With good preparation you can show a significant increase in score from your PSAT to your SAT.

If you do decide to study on your own, I suggest you take a look at my “28 SAT Math Lessons” series. Each book in this series is targeted for students in a specific score range. Get the book for your score range and you will learn all the strategies you need to get to the next score level. In addition you will be provided with hundreds of practice problems all targeted specifically for you. Click on the picture below for more information about these books.

If you liked this article, please share it with your Facebook friends:

I’ll see you next week…

Why Can’t We Divide By Zero?

Two weeks ago, I wrote a post on divisibility tricks – how to quickly recognize if a number is divisible by another number without the use of a calculator. I have had a few questions regarding one of the statements I made concerning division involving zero. The answers to these questions are quite a bit more difficult than anything you will need for the SAT. So instead of adding this information to the previous post, I thought it would be better for me to create a new blog post on this subject.

Whenever I ask a student why we can’t divide by zero I always get strange answers. Most students just restate the question in another way. Some typical answers are “because division by 0 is impossible,” “because my teacher told me I can’t,” or “because my calculator gives an error when I try to do it.”

But none of these “answers” explain anything.

We will clear up this mystery by the end of this post.

Some of the questions we will answer in this post are the following:

1. What does it mean for an integer to be divisible by another integer?
2. Is zero divisible by every integer?
3. Are there any integers that are divisible by zero?

Note that this is a pretty technical article with some fairly sophisticated mathematics. You should only read this post if you are genuinely interested in why division by zero is impossible.

So take a deep breath, and let’s begin…

We say that 42 is divisible by 7 because 42 = 7·6. The number “6” itself isn’t particularly important here, but what is important is that 6 is an integer. In other words, 42 is divisible by 7 because there is an integer k such that 42 = 7k.

In general, an integer n is divisible by an integer d if there is another integer k such that n = dk.

In practice we can check if n is divisible by d simply by dividing n by d in our calculator (or sometimes by using the divisibility tricks I mentioned earlier). If the answer is an integer, then n is divisible by d. If the answer is not an integer (it contains digits after the decimal point), then n is not divisible by d.

For example, when we divide 42 by 7 we get 6, an integer. So 42 is divisible by 7.

As another example, every integer n is divisible by 1. Indeed, n = 1·n, and n is an integer.

What about the integer 0? Is 0 divisible by 7?

Yes it is because 0 = 7·0, and 0 is an integer.

In fact, if m is any nonzero integer, then 0 is divisible by m. Indeed, 0 = m·0.

This fact is worth emphasizing since students often get confused by it.

Zero is divisible by every nonzero integer!

Note in the bold statement above the word “nonzero.” Isn’t 0 also divisible by 0 according to that definition? After all, if we set m = 0 in the equation “0 = m·0,” we then get 0 = 0·0 which is a true statement! We will talk about this more in just a bit.

Let’s now check if there are any integers n which are divisible by 0. If n is divisible by 0, then there is an integer k such that n = 0·k. But 0·k = 0. So in other words, if n is divisible by 0, it follows that n must be equal to 0.

In other words, there are no nonzero integers that are divisible by 0!

Note that we have not shown that 0 is divisible by 0. We only proved the conditional statement “if an integer is divisible by 0, then that integer must be 0.” We still need to check if 0 actually is divisible by 0.

So let’s assume for a moment that 0 is divisible by 0. This means that there is an integer k such that 0 = 0·k. But every single integer satisfies this equation. Just to pick two specific values, 0 and 1 both satisfy this equation. This means that 0/0 = 0 and 0/0 = 1. It follows that 0 = 1. But 0 and 1 are different integers. It follows that 0 cannot be divisible by 0. Cool, huh?

Maybe your head is spinning a little after this, so take a deep breath, take some time away from this post and then come back to re-read it – if you are interested in why we can’t divide by zero!

As mentioned, this info is not relevant for SAT math, but (for me at least) it is very interesting and can help raise your level of mathematical maturity. The question of why division by zero is not possible is one of the big holes students have in their understanding, so I hope that after reading this post you are now a little wiser!

Take care! I’ll be back with more SAT math relevant material next week…

Hi all,

This is just a short message to let you know that parts of SATPrepGet800 website will be down at times throughout September.

We are actually updating the site so it looks more modern and is easier to navigate.

So I’ll keep you updated on the progress.

All the best!

A while back I created a thread that explained how SAT math problems involving remainders can be solved quickly and easily. Quickly take a look at these posts to refresh your memories on what I wrote. Click on these links for part 1 (calculator algorithms) , 2 (cyclical nature of remainders) and 3 (recognizing remainder problems) for the posts in this thread.

I realized that I can share with you certain ‘tricks’ that you can employ to quickly and easily divide numbers without using long division or a calculator. Some of these tricks will actually take less time than typing the numbers into your calculator. With practice these tricks can become second nature to you which of course means that you can save precious time on the day of the SAT.

Some of these tricks will be obvious to you, but that is only because you have practiced the trick for that particular type of number so often that it has already become second nature to you.

Here are some easy tricks that you may already know:

• An integer is divisible by 2 precisely when the last digit is 0, 2, 4, 6 or 8.

For example, 326 is divisible by 2, whereas 327 is not.

• An integer is divisible by 3 precisely when the sum of its digits is divisible by 3.

For example 477 is divisible by 3 because 4 + 7 + 7 = 18, and 18 is divisible by 3.

What about 5215? Well, 5 + 2 + 1 + 5 = 13. Since 13 is not divisible by 3, neither is 5215

• An integer is divisible by 4 precisely when the number formed by taking just the last two digits of the integer is divisible by 4.

For example, 2,136,416 is divisible by 4 simply because 16 is divisible by 4.

What about 532,211. Well 11 is not divisible by 4. So neither is 532,211.

• An integer is divisible by 5 precisely when the last digit is 0 or 5.

So 6,235 is divisible by 5, but 2,112,321 is not divisible by 5.

• An integer is divisible by 6 if it is divisible by 2 and by 3.

Let’s check if 2,132,478 is divisible by 6. Well this number ends in 8, so it is divisible by 2. Also, 2 + 1 + 3 + 2 + 4 + 7 + 8 = 27. Since 27 is divisible by 3, so is 2,132,478. Since this number is divisible by both 2 and 3, it is divisible by 6.

• An integer is divisible by 9 precisely when the sum of its digits is divisible by 9.

For example, 477 is divisible by 9 because 4 + 7 + 7 = 18, and 18 is divisible by 9.

What about 5214? Well, 5 + 2 + 1 + 4 = 12. Since 12 is not divisible by 9, neither is 5214. (Note however, that 5214 is divisible by 3).

• An integer is divisible by 10 if it ends in a 0.

So 5960 is divisible by 10, and 7655 is not.

Here are a few more – these are advanced divisibility tricks:

Please take note that these last few tricks are not standard at all, and almost nobody knows them. I am including them more for fun, and because it makes the post more complete (otherwise the only numbers “left out” are 7 and 8). But most students can skip them.

• Here is a nice little algorithm for testing if an integer is divisible by 7: Double the last digit of the integer and subtract this value from the original integer with the last digit removed. If this new number is divisible by 7, so is the original number. Otherwise it is not.

Let’s check if 5362 is divisible by 7:

536 – 2(2) = 532

53 – 2(2) = 49

Since 49 is divisible by 7, so is 532. Since 532 is divisible by 7, so is 5362.

• An integer is divisible by 8 precisely when the number formed by taking just the last three digits of the integer is divisible by 8.

For example, let’s check if 2,136,416 is divisible by 8. Well when we divide 416 by 8 we get 54. So 416 is divisible by 8, and therefore so is 2,136,416.

• Is there a way to test for divisibility by 11? Yes there is. Add up every other digit beginning with the second digit from the left, and then subtract the remaining digits. If this new number is divisible by 11, so is the original number. Otherwise it is not.

Let’s check if 28,765 is divisible by 11. Well (8 + 6) – (2 + 7 + 5) = 0. Since 0 is divisible by 11, so is 28,765.

As mentioned, I do not recommend worrying about these last few for the SAT. I just included them for those of you that might be interested in learning some more advanced tricks. Note that there are many, many more advanced divisibility tricks out there. If I were to include them all, this post would be ten pages long, so I needed to stop here. But feel free to post your favorites in the comments and we can discuss them. In any case, with practice you will be using divisibility tricks without even realizing it!

Here is a video containing this information:

 

You can practice using problems found in my books.

If you have any questions or comments let me know… If you found the post helpful, why not use the share button below to let your friends know of this post too? A share would be much appreciated.

Speak next week!

Combinations On The SAT

Continuing on from last week, today I would like to continue discussing more advanced methods for solving counting problems.

Two weeks ago we discussed some more basic methods for solving counting problems on the SAT. These methods can be viewed by clicking the following link: Listing and the Counting Principle

And last week we talked about permutations. This article can be viewed here: Permutations 

Today I want to focus on another more advanced counting techniques – Combinations.

Combinations

First a definition: A combination is just a subset containing a specific number of the elements of a particular set. The number of combinations of n things taken r at a time is

For example, the number of combinations of {1, 2, 3} taken 2 at a time is

These combinations are 12, 13, and 23.

Note that 21 is the same combination as 12 (but 12 and 21 are different permutations). More on this below…

Just like the permutation formula, you do not need to know the combination formula for the SAT. You can do this computation very quickly on your TI-84 graphing calculator. To compute ₃C₂, type 3 into your calculator, then in the Math menu scroll over to Prb and select nCr (or press 3). Then type 2 and press Enter. You will get an answer of 3.

Difference Between Permutations And Combinations

Let me quickly tell you the difference between permutations and combinations: “Permutation” is just a fancy word for arrangement. Whenever you are counting the number of ways to arrange things you can use the permutation formula that I shared with you last week (or better yet the nPr feature in your calculator). So if you want to “order” things, “stack” things, “rearrange” things, choose specific tasks for people, etc, you would use permutations.

Example 2:

Here is our first SAT math problem where combinations can be used:

2. A chemist is testing 5 different liquids. For each test, the chemist chooses 3 of the liquids and mixes them together. What is the least number of tests that must be done so that every possible combination of liquids is tested?

First note that we can solve this problem by simply listing all of the possibilities. But be careful! This is just a bit tricky.

In the following list a * means we are choosing that liquid, and an O means we are not:

***OO      *OO**
**O*O      O***O
**OO*      O**O*
*O**O      O*O**
*O*O*      OO***

We see that there are 10 combinations.

Alternatively, we can obtain the solution quicker by using what we have learned about combinations: We are counting the number of ways to choose 3 of the 5 liquids. This is ₅C₃ = 10.

Example 3: 

And here is a more difficult SAT math problem that can be solved quickly using combinations:

3. Any 2 points determine a line. If there are 18 points in a plane, no 3 of which lie on the same line, how many lines are determined by pairs of these 18 points?

We need to count the number of ways to choose 2 points from 18. This is the combination ₁₈C₂ = 153.

Let me end this post by saying that on the SAT I have not seen a problem where you absolutely NEED to use permutations or combinations. You can always do counting problems on the SAT either by forming a careful list, or by using the counting principle. But every now and then using permutations or combinations will get you the answer much more quickly.

And that’s all you need to know about counting problems for the SAT, but if you want more practice problems I suggest you take a look at my 28 SAT Math Lessons Series. Click on the picture below for more information about these books.

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