SAT Math Guides SAT Practice Quizzes SAT Subject Math Guides SAT Subject Physics Guides ACT Math Guides Follow me @ErikTheRedTutor The Fine Print All the HTML and PDF content on Erik The Red by Erik Jacobsen is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. Please feel free to improve, share, and make copies of this content, as long as you use the same, similar, or compatible license. In any case, please leave in an attribution to me (Erik Jacobsen) and to erikthered.com/tutor. The TeX source code is available by request under the same license. I'm a private math and physics tutor in the Basking Ridge, NJ area. I'm currently teaching at Newark Academy in Livingston, NJ. I specialize in ACT math, pre-calculus and calculus, and physics. If you need extra help, or you would like to improve your test scores, or you have comments or suggestions, you can find me @ErikTheRedTutor on Twitter. Or, you can contact me here: tutor < at > erikthered.com (replace the "< at >" with the usual at-sign). SAT and ACT are registered trademarks of the College Entrance Examination Board and ACT, Inc., respectively. Neither company sponsors nor endorses this web site. Not that such sponsorship wouldn't be nice to have. Yet, I toil on, producing all this free stuff. Does anyone actually read this tiny text? :) This page was last updated on: 2020-02-08. |
## SAT Facts and FAQsThis web site was developed and designed for the version of the SAT that was administered from March, 2005, to March, 2016. The primary changes to the math section of the SAT made in 2016 included a switch from multiple-choice questions with five answers to ones with four answers, along with the introduction of a no-calculator section and a different topic emphasis. Although the ACT test and SAT subject tests have essentially remained the same, the SAT has not, and many of the SAT resources to be found here are out of date. However, the calendar is still current, the SAT history page is still a labor of love, and many of the math problems found here remain useful for practice. Just be aware that some math topics represented in the resources may no longer be tested, and that some FAQs may no longer be relevant. Thanks for visiting! SAT FAQs What is the SAT, and why is it important? What do I need to know about the new SAT? When and where can I take the SAT? When do I get my SAT score back? What does my score percentile mean? Is the SAT different every time it is administered? What are the best study resources / materials / books for the SAT? What math do I have to know for the SAT? Is it advanced math? Why does the "Blue Book" give score ranges instead of exact scores? Do I have to report all of my scores, or can I choose which scores to report? When do most people take the SAT? Which is the best month, or easiest month, to take the SAT? Can I use mechanical pencils on the SAT? Are calculators allowed on the SAT? Can I take my cell phone into the test center? What is my SAT score if I got 3 wrong? How about 5 wrong? 10 wrong? 20 wrong? I am not in high school. Can I still take the SAT test? My mom scored X on the SAT and I scored Y. Who did better? SAT Subject FAQs Can I take subject tests different from the ones I originally signed up for on the test day? How is the math on the subject tests different from the regular SAT math? What is my SAT Subject score if I got 3 wrong? How about 5 wrong? 10 wrong? 20 wrong? General SAT FAQs What is the SAT, and why is it important? The SAT is a test administered to high school students in order to assess their ability to succeed in college, independently of high school grades and other indicators. The SAT is a paper-based test, consisting of multiple-choice questions (five possible answers) along with an essay and ten free-response math questions. For many years, the SAT has been a requirement for admission to most four-year colleges and universities in the U.S., and the test has a history spanning more than a century. (For more information, such as what the letters "SAT" and "ACT" stand for, see the (Mostly) Brief History Of The SAT And ACT Tests.) Many changes to the test have been made in the last 80 years to continue to make it relevant and valuable for college admissions. More recently, however, the importance of the SAT itself has diminished: all four-year colleges and universities now allow the use of the ACT as a replacement for the SAT, and some colleges have made both the SAT and ACT optional. Questions remain as to the usefulness of these tests in predicting future success in college. Nonetheless, about 2.5 million students take the SAT each school year. This PDF file has charts showing the number and percent of seniors taking the SAT and ACT tests over the last 20 years. This U.S. map shows the relative popularity of the SAT and ACT tests for each state. What do I need to know about the new SAT? The "new" SAT (soon to be called just the SAT) will be administered for the first time in March, 2016. For the test overall, the major changes are: - There is no guessing penalty (previously, 0.25 points were subtracted for incorrect answers). Your raw score is the number of correct answers.
- The essay portion of the test is now optional and scored separately, making the maximum score 1600 rather than 2400
- For multiple-choice questions, there are four answers instead of five
For math, the major changes are: - On some sections, calculators will not be allowed
- Questions involving trigonometry will appear on the SAT
- Fewer topic areas will be tested; in particular, geometry will become a minor portion of the content (from 25% to below 10%)
In many ways, the new SAT looks to be quite similar to the old SAT; on the other hand, many of the changes also make the SAT more like the competing ACT test. Which test is right for you (old SAT, new SAT, or ACT) is best determined by taking official practice tests (or using tests released by the College Board or ACT, Inc. for practice). For now, the only official practice tests for the new SAT are available here: - Practice Test 1 (Answers/Scoring and Explanations)
- Practice Test 2 (Answers/Scoring and Explanations)
- Practice Test 3 (Answers/Scoring and Explanations)
- Practice Test 4 (Answers/Scoring and Explanations)
If you are currently (2015-2016 school year) a senior in high school, you should not be taking the new SAT. If you are currently a sophomore in high school or younger, you will almost certainly be taking the new SAT. If you are a junior, then your situation is complicated. The PSAT to be administered in the fall of 2015 will be based on the new SAT. However, for many juniors, taking the old SAT before the test changes in March may be a good option: test preparation materials and expertise for the old test are generally easy to find, compared to those for the new test. As well, your particular skill set may be better suited to the old SAT than the new. When and where can I take the SAT? The SAT is administered in October, November, December, January, March, May, and June. The SAT is administered nationally on the same dates for every state in the U.S. For the actual test dates for the current test year, see the SAT & ACT Calendar. The test is given on a Saturday, and under special circumstances you can take it instead on the following Sunday. If you need to, you can figure out what date the SAT will be on well into the future, as follows: For November, December, May, and June, the test is given on the first Saturday of the month. For October and March, the test is given on the second Saturday of the month. Finally, in January, the test is given on the fourth Saturday of the month. (Exceptions: the October SAT is one week earlier if Labor Day falls before September 5; the March SAT is one week earlier if Easter falls before March 30.) The dates of all past regular SAT administrations can be found in this PDF file. The SAT is administered in many locations in the U.S. and countries around the world; typically, these test centers are in schools. To find a test center appropriate for your location, go to this College Board web page and select "Test Centers" in the "Search By" drop-down box. For U.S. locations, specify the state without specifying a city to obtain a list of all the test centers in that state. How long is the SAT? The total test time for the SAT is 3 hours and 45 minutes. You should get three five-minute breaks during the test, so the total elapsed time for the SAT is about 4 hours. The breakdown of the test is: 70 minutes for critical reading, 60 minutes for writing (including the essay), and 70 minutes for math. There is also a non-graded section which takes an additional 25 minutes. The test will begin sometime between 8 a.m. and 9 a.m. local time, depending on the particular testing site, and end by 1 p.m. Generally, you should plan to arrive at the testing location a comfortable amount of time before 8 a.m. Shown below is the time breakdown of a 10-section SAT administration. (Note that students at some test centers will receive a 9-section SAT which is 25 minutes shorter than the 10-section SAT. See After The Test (SAT Test Forms) for more details.) **Section 1.**25 minute section: Essay**Section 2.**25 minute section: Math, Reading, or Writing*5-minute break***Section 3.**25 minute section: Math, Reading, or Writing**Section 4.**25 minute section: Math, Reading, or Writing*5-minute break***Section 5.**25 minute section: Math, Reading, or Writing**Section 6.**25 minute section: Math, Reading, or Writing*5-minute break***Section 7.**25 minute section: Math, Reading, or Writing**Section 8.**20 minute section: Math or Reading**Section 9.**20 minute section: Math or Reading**Section 10.**10 minute section: Writing
How is the SAT scored? Each section (math, critical reading, and writing) is scored on a scale of 200 to 800, so the total score is from 600 to 2400. In detail, the method is this: For each section, your correct answers are added. Next, 0.25 points are subtracted for each incorrect multiple-choice answer, resulting in your "raw" score. Finally, a curve is applied to convert your raw score into a number from 200 to 800 for math and reading, and from 20 to 80 for writing. The curve corrects for the difficulty of the test compared to other SATs. For the writing section, your score on the multiple-choice questions (20 to 80) is combined with the score (2 to 12) on your essay to produce a final writing score (200 to 800). You can see how the most recent high school senior class (2014) scored on the SAT using the SAT composite score percentile ranks table. The average SAT scores for the senior class years since the writing section was introduced are:
(*In 2006, only tests taken by seniors through March of their senior year were included. For 2007 and later, tests taken through June of the senior year were included.) For all of the yearly average SAT scores from 1952 to the present, see this PDF file for a plot and this PDF file for the data. When do I get my SAT score back? Scores are first released via the Internet about 3 weeks after the exam is administered. Generally, the scores will appear on the College Board web site starting at roughly 5 A.M. Eastern Time (9 A.M. or 10 A.M. UT depending on whether or not U.S. daylight saving time is in effect). Not everyone's scores will appear immediately. On the score release day, you will get the scores for math, critical reading, and writing, as well as the writing multiple choice subscore and the essay score. About a week later, the web site will have your "full" score report, which will detail the number of questions answered correctly, incorrectly, or omitted, broken down by category and difficulty of question. For the actual score release dates for the current test year, see the SAT & ACT Calendar. What does my SAT score percentile mean?
After you have taken an SAT, your scores are compared to those of other
people who have previously taken an SAT. (Your score is The comparison group of test takers used to generate your percentile rank is the latest graduating senior class (including international students), and this comparison group remains fixed for all SAT tests administered in the following school year. For example, for the SAT tests administered in the 2014-2015 school year (August to June), the comparison group is the senior class of 2014. The latest scores available for each person in that group are used. The percentile ranks for the class of 2014 are available here. Is the SAT different every time it is administered? Generally, yes. However, individual questions are typically reused and will appear on multiple tests, along with other questions that are new. Questions are "retired" when they appear in a test administered in either January, May, or October, since these tests are made public. For example, a particular math question may first appear on the March 2006 test, again on the November 2008 test, and finally on the October 2010 exam. On occasion, a test may even be reused in large part or in its entirety. For example, the December 2005 SAT was reused as the January 2007 SAT, and a large portion of the November 2005 SAT was reused in the November 2010 SAT. Even on a particular SAT test date, the test is administered using multiple arrangements of the sections; these are called "forms", and multiple forms are given on the same day (even in the same testing location). For each form, the test sections may be arranged in a different order, the essay question may be different, and the non-graded section may be different or not there at all. There may be as many as five different forms of the test on a given test date (including up to four different essay questions). The College Board used to publicize the way the various forms for that test were arranged shortly after each test was administered, but this is no longer done. See After The Test (SAT Test Forms) for more details about how the SAT is arranged on test day and for the arrangement of actual test forms on two test dates. What are the best study resources / materials / books for the SAT? There are many opinions as to which materials are the "best", and the right answer will depend on you as well: not just your particular skill level, but also how you study, how much time you have to study before you take the test, etc. That having been said, there are study materials that can help just about everyone. First and foremost is the "Blue Book" (the real title is: "The Official SAT Study Guide"; see the resources), which contains a lot of questions that have appeared on actual SATs. A second "must-have" resource is the yearly practice test. Each year, the College Board releases an "Official SAT Practice Test", which is a previously administered SAT. Currently, each practice test is one of three SAT exams. You can download them, with answers, directly from the College Board in PDF format, using the following links: - SAT January 2006
(Same as Official SAT Practice Tests 2009-10, 2011-12, and 2013-14. Answers here.) - SAT October 2005
(Same as Official SAT Practice Tests 2006-07, 2008-09, 2010-11, and 2012-13. Answers here.) - SAT March 2005
(Same as Official SAT Practice Test 2007-08. Answers included.)
You could, for example, use a test in the beginning of your studies as a benchmark (or base) score, and then another near the end of your studies. However you use them, try to do so under test-like conditions: use a countdown timer (or have someone time you) in an area free of distractions, as you do each section (you don't have to do an entire test all in one sitting, however). It may be useful to use older (pre-2005) released SAT exams, particularly if you need extensive practice with particular types of questions (for example, hard geometry questions or easy algebra questions). Be aware, however, that some of the topics on the current SAT may not appear in the older tests, and vice versa. These SATs can be found in books, all published by the College Board, such as "10 Real SATs"; a complete list of the SAT tests released in book form by the College Board since 1983 is available here. What math do I have to know for the SAT? Is it advanced math?
The math concepts that you will be tested on when you take the SAT are
those that you have learned over the years, from integer math, fractions,
and percents to algebra, geometry, and functions. **(20% -- 25%) Numbers and Operations**: Integers, Fractions, Ratios, Percents ...**(35% -- 40%) Algebra and Functions**: Solving Expressions, Function Graphs ...**(25% -- 30%) Geometry**: Lines, Slopes, Circles, Triangles, Rectangles, Solids ...**(10% -- 15%) Data, Stats, and Probability**: Averages, Medians, Charts, Tables ...
For a complete list of the topics and question types that have appeared on recent released SAT tests, see the SAT Question Index. Why does the "Blue Book" give score ranges instead of exact scores? The "Blue Book" (see the resources) contains ten practice tests. The first three practice tests are actual SATs previously administered in October 2006, January 2007, and May 2007, and you can get an exact score when you take one of these three tests. The remaining seven practice tests are made up of problems that appeared on previous SATs from 1995 to 2004. However, each of these seven tests was never given as an actual SAT, so it isn't possible to come up with an exact score. You may see exact scores from practice tests in other test prep books; however, those scores are just educated guesses as to what you will score on the real thing. Note: the first edition of the Blue Book contained eight practice tests, none of which were real SATs, so score ranges were given for all eight. Seven of these reappear in the second edition of the book. The version of the Blue Book released in 2012 with a DVD contains the same tests as the second edition, but also includes the January 2008 SAT on the DVD. Do I have to report all of my scores, or can I choose which scores to report? The answer depends on the school(s) to which you want to apply. The College Board will allow you to report any or all test scores. For example, suppose you take the regular SAT twice, and the Math Level 1, Physics, and U.S. History subject tests. You could report only your best SAT and only the Physics subject test scores, if that is what you wanted to do. However, each college and university can set its own policy for scores. College A may leave the choice up to you as to which scores to send, whereas University B may want to see all of your scores. For a list of the score policies for various colleges and universities, see this College Board PDF file. (When you are ready to report scores, you should also check directly with the schools for their score policy information.) For details of the College Board score report policy, see this College Board web page. When do most people take the SAT? The 2013 SAT national test dates in order from most popular to least popular were: October, June, May, March, January, November, and December. Such data are not released by the College Board, but it is possible to estimate a test date's popularity by counting unique visitor accesses to this web site: this graph shows the relative number of SAT downloads from erikthered.com for last year. Many students take the SAT for the first time in May or June as juniors. Some students will take the SAT for the first time in October as seniors, along with many other seniors taking it for the second time. Which is the best month, or easiest month, to take the SAT? A common misconception is that SATs given in a particular month (say, October) are easier (or harder) than average. While it is true that the difficulty of the SAT varies somewhat from one month to the next, the data show that no month is consistently harder or easier than average. The following table shows how the difficulty of the SAT math section has varied in January, May, and October in recent years:
Also shown in the table is the variation in your math score on the
different tests, given that you had gotten two questions wrong. (I chose to
list scores corresponding to two wrong in order to show the maximum
variation in scores due to variations in test difficulty. Scores closer to
the national averages will vary less than those shown in the table.)
The SAT curve will adjust your
score up for harder tests and down for easier tests. In this way, there is
no benefit to having an easier test, and on the other hand, there is no
downside to having a harder test. So, the answer to the question
is: What is the SAT curve? The SAT "curve" generates a scaled score (200-800) from your raw score (the number of questions right minus a quarter point for each multiple-choice question wrong, rounded to the nearest integer). This curve is designed to correct for minor variations in the difficulty of the test. In this way, there is no advantage to getting an easier test and no disadvantage to getting a harder test. For example, suppose that on all the math sections, you get only 3 multiple-choice questions incorrect, and all the others correct. You will receive a raw math score of 50 (the calculation is: 54 - 3 - 0.75 = 50.25, which is rounded to 50). On the January, 2010 SAT you would have received a math score of 710; on the May, 2011 SAT you would have received a 720; and on the January, 2006 SAT you would have received a 740. The January 2010 test was a little easier than average, the May 2011 test was close to average, and the January 2006 test was harder than average. Another way to look at it: in order to score a 710 in math, you would need a raw score of 50 on the easier test (January 2010), 49 on the average test (May 2011), and only 47 on the harder test (January 2006).
A common myth about the SAT curve is that the average test taker should
avoid a particular test month if a large group of strong students will be
taking the SAT that month, and instead take the test when a large group of
weaker students will take the test. The (incorrect) assumption here is that
the curve will push down the average student's score in the first situation
(large group of strong students) and pull it up in the second situation
(large group of weak students). The reality is that the curve only reflects
the difficulty of that particular SAT, not the quality of students taking
the test. For example, suppose that in a particular month, a large group of
strong students take the test. Even if they all get perfect 2400s, your
score will be the same as it was had they not taken the test. In the same
way, a large group of weaker students taking the test will not affect your
score. For those The curves from previously released SAT tests (those given in January, May, and October) are available in this PDF file (6 pages). The math curves from previously released ACT tests (those given in April, June, and December) are available in this PDF file (2 pages). Can I get my test back? In general, no. When you first get your scores via the web, you will only get the three scores for math, reading, and writing along with the percentiles for each score. About a week later, you will get a more detailed report on the web giving the number of correct, incorrect, and omitted answers in several categories, broken down by difficulty of question (easy, medium, and difficult). In math, you will see four categories of questions: numbers and operations, algebra and functions, geometry and measurement, and data analysis. You will not get your test booklet back, and you can not see which questions you got right or wrong. However, if you take the SAT in January, May, or October, you can get (for an extra fee) the test booklet back so that you can see exactly what the questions and answers were along with your answers. Note that you get a new (unused) test booklet, not the one that you actually used. See the College Board SAT Question and Answer Service web page for more details. Also note that the three months listed above are available only for Saturday test takers in the U.S. or Canada; otherwise (Sunday tests and/or outside the U.S. or Canada), only the May test is available. In the other months (March, June, November, and December), you can get (for an extra fee) a list of the question types and difficulties. Also listed for each question is whether you omitted the question, answered correctly, or answered incorrectly. Note that you do not get the questions, the correct answers, or even your actual answers. See the College Board SAT Student Answer Service web page for more details. Can I use mechanical pencils on the SAT?
No. (I could go on about how annoying this is for me since I love
mechanical pencils, but I've already said too much.) The official reason is
that: "Mechanical pencils might punch through the answer sheet and the
marks they make cannot be guaranteed to scan properly." I know of people
who have taken the test using a wide range of lead types (#1, #2, #3, etc.)
in mechanical pencils without any scanning issue. Also, a regular pencil
can have a sharper point than a mechanical pencil. The restriction may more
likely be due to a concern that mechanical pencils could help students to
cheat. (Mechanical pencils are disallowed on the ACT test as well.) You
could probably get away with using a mechanical pencil, especially one that
is made to look like a regular pencil, but the risk of getting your scores
cancelled is probably not worth it. Bring at least two standard #2 (also
known as "HB") wood pencils. (Pencils and calculators are Are calculators allowed on the SAT?
Yes, except for cell phone calculators, or calculators with a full QWERTY
keyboard. Although no questions on the SAT Reasoning Test
will If I had more time, I would put a calculator review here. But for now, I would say: If you have a TI-83 or TI-84 and are already comfortable with it, of course go ahead and bring it. Otherwise, something like a TI-30 XS Multiview (about $14) is a good choice. Can I take my cell phone into the test center?
Officially, no, but in practice, yes. Although the College Board will
strongly suggest that you not bring a phone with you at all, the real
prohibition is against the What is my SAT score if I got 3 wrong? How about 5 wrong? 10 wrong? 20 wrong? The table below shows what you would score (on average) on math, critical reading, or writing for a given number of incorrect multiple choice answers:
The table above gives the scores that you would get for a given number of
incorrect multiple choice answers on a I am not in high school. Can I still take the SAT test? Yes. According to the College Board: "Anyone may take the SAT, regardless of age." However, if you are not yet in ninth grade, your scores will be erased at the end of the testing year unless you specifically request that they be kept. Along with students too young for high school, adults take the SAT as well, for a variety of reasons. Some are entering college later in life, others are tutors or work for tutoring companies. If you are an adult taking the SAT, have at least one form of current, government-issued ID with a picture, such as a driver's license or passport. The name on the ID needs to match the name you registered with the College Board to take the test. The proctor may also pay extra attention to your seating: you may be put in a different room from the high school students, or in the same room but seated adjacent only to other adults. In this way, according to the SAT Standard Testing Room Manual, the adults "can be more closely monitored." My mom scored X on the SAT and I scored Y. Who did better? There have been three major changes to the SAT since the 1970s and 1980s: 1) in 1994, the content of the test was changed and calculators were allowed; 2) in 1995, the scales were "recentered" so that average scores were closer to 500; and 3) in 2005, the content of the test was changed again and a writing section was added. (For the details, see the (Mostly) Brief History Of The SAT And ACT Tests.) This means that a score of 500 on SAT math today is not the same as a score of 500 on SAT math in the 70s or 80s, just like the words "but" and "butt" mean very different things even though they sound the same. Even if the tests were the same today as they were thirty years ago, the common approach to the test by students now (plenty of test preparation, often with access to previously administered SATs) is not the same as then (little or no test prep, with few old SATs available). However, percentile ranks of SAT scores can give us some idea of how to compare scores from now and then. The percentile rank of an SAT score is defined as the percent of students scoring lower than the given score. (See here for a more detailed definition.) So, if the percentile rank of a score today is the same as the percentile rank of a score thirty years ago, then these two scores are reasonably considered the same, even though the tests and student preparation for the tests were quite different. The following table shows the SAT percentiles used during the 2012-13 school year versus those used during the 1982-83 school year. To use it, first look up your score in the first column. Then, find the corresponding percentile column for your year in the subject you're comparing. Next, find the closest match to this percentile in the other year's column, and read off the corresponding score in the first column. For example, suppose that you recently scored a 500 in math. This is a 45 percentile rank. In 1983, that rank (well, a rank of 44) corresponded to a math score of about 450. I.e., you scored about as well as someone who scored a 450 in 1983. If your mom scored more than 450, well, she did better than you did! Another example: Suppose that you recently scored a 500 in the critical reading section. This is a 51 percentile rank. In 1983, that rank (well, a rank of 50) corresponded to a verbal score of about 420. I.e., you scored about as well as someone who scored a 420 on verbal in 1983. This table can be used reasonably well for other nearby years (say, 1979 vs. 2010) since average SAT scores didn't change that much from the mid-70s until the recentering, and haven't changed much since 2005.
Those who took the SAT after the re-centering of the scales in early 1995 but before the changes to the SAT in 2005 can use the College Board's SAT I Score Equivalents table to compare their scores with those from before 1995. SAT Subject FAQs Can I take subject tests different from the ones I originally signed up for on the test day? Yes, as long as you take no more than the number of tests that you signed up for, you can choose to take different subject tests than those you originally chose. Your test booklet will have all the available subject tests in it. All that really matters is that you properly code for the test(s) you are taking on the scoring sheet. How is the math on the subject tests different from the regular SAT math?
The math on both levels of the subject tests is somewhat different than
math on the regular SAT (and level 2 math is more advanced). The regular
SAT math is what you have covered by Algebra II in high school, and a lot
of it is covered in your math classes previous to high school. SAT math is
30% numbers (integers, fractions, the number line, primes, etc.); 40%
algebra (including some functions); 25% geometry (almost all plane
geometry, with a little 3-D geometry); and, 5% graphs, statistics, and
probability. SAT math does The subject tests (formerly called SAT II or IIC) are more "achievement" type tests. In other words, the questions are oriented around: "have you seen this material?". SAT Subject Math Level 1 adds: trigonometry, more functions (symmetry, compound), complex numbers, more solid geometry (spheres, cones, etc.). Subject Math Level 2 adds: series, logarithms, inverse functions, ellipses, hyperbolas, radians, more trigonometry (secant, cosecant, cotangent, and laws of sines and cosines), and more coordinate geometry and functions compared to Math Level 1. What is my SAT Subject score if I got 3 wrong? How about 5 wrong? 10 wrong? 20 wrong? Each subject test has a different "curve" (the table that converts from your raw score to your scaled score). Also, the curve for each test varies a little from one administration to another, to adjust for the difficulty of the test compared to other tests. The table below gives your score for various numbers of questions wrong on a typical subject test.
The table above gives the scores that you would get for a given number of
incorrect multiple choice answers on a |