It seems obvious that each ACT / SAT question must have exactly one answer, and this answer must be clearly and objectively correct. However, as we’ll explain below, that simple fact alone disqualifies vast swaths of reasonable questions that you otherwise see in everyday tests at school. This obvious fact also leads to a little-known secret that, when applied correctly, guarantees the cracking of each question. You don’t have to trust me -- read through this guide and ask any perfect or near-perfect scorer. They’ll agree on the secret.
The premise is simple. Each ACT / SAT question must have exactly one right answer. This seems obvious enough -- after all doesn’t every multiple choice question have one right answer?
It turns out that most multiple choice questions you’ve seen in life have relatively lax standards. Suppose your science teacher, Mr. Smith, gave you a multiple-choice quiz last week. Those multiple-choice questions are the same as those of the SAT / ACT right? Absolutely not! Mr. Smith is allowed to write imperfect questions. If there are two right answers, Mr. Smith will give some credit back. In the worst case, some students will have inaccurate scores. What if choice B is only a little more correct than choice C, and you put down C? Tough luck -- this question only counts for 2% of your grade anyway. A few bad questions a month is part of life. In short, the multiple-choice questions you are used to in school have much wider tolerances for error and fuzziness because they matter less in your academic career.
Why the SAT / ACT Can't Tolerate Any Question Mistakes
The ACT / SAT is a totally different ball game, a totally different league. The writers have to make a test that contains hundreds of questions, yet they can't make a single mistake. Not a single one of their questions can have two right answers or no right answers. Each question must have one right answer that stands objectively and clearly above all the other answers. This means that, if you put each question in front of 100 experts, all of must answer it exactly the same way, without any doubt.
What happens if the SAT or ACT makes a mistake? The consequences would be tremendous. Many students on the margin would lose their scholarships. Students would literally lose their deserved acceptances to their dream colleges -- a product that the average family pays five to six digits for. Colleges who obsess about assembling the dream class would be going off flawed data. The truth is, the stakes are super high with the SAT / ACT, so there is no margin for error.
Of course, the pain isn’t just to students and colleges -- it gets transmitted to the test makers as well. Even a few mistakes a year results in scandals (see the June 2015 SAT blowup over “just” 5 minutes timing difference). Both students and colleges will stop using the error-prone test. And, to kick it off, these mistakes have resulted in lawsuits that have cost the ACT / SAT hundreds of thousands of dollars to battle. Therefore, neither test can tolerate any chance of two right answers, no right answer, or any other question mistake.
Why Each SAT / ACT Question Must Have One Very Clear Answer
The ACT / SAT is also not allowed to have unclear answers or answers that rely on fuzzy reasoning. Suppose an ACT science question asked: How many planets are there in the solar system? A) 8; B) 9 … It seems that the answer is pretty clear -- most scientists would say A) 8. But this level of clarity is not good enough for the ACT. Very recently we had nine planets, so some educators might argue that students who put down nine are answering as they’ve been taught and should be given credit; these educators would have a good argument. Other fringe scientists may not accept the consensus and argue that nine is still right. Also, the ACT runs the risk of a new planet being discovered between publication and test date.
If two answers are close to each other in how good they are, this creates headaches for the test makers. First, the test makers might make a mistaken judgment call and claim the slightly worse answer is the right one -- this leads to the horror show above. Another scenario is more insidious: it affects students who put down the “less good” answer but are at the cusp of a big prize. Maybe the student is right at the cutoff of a sports recruitment or a huge scholarship. She would be hugely incentivized to get the test to accept her answer as correct. In fact, many students do cause an administrative or legal headache for the ACT / SAT by making a fuss.
With that environment in mind, you hopefully have a better understanding as to why the ACT / SAT can afford zero mistakes on the test. You should also understand why their multiple-choice questions can’t have a best answer that is just 20% better than the next choice. Now we’ll discuss how we can use this information to your best advantage.
Every SAT/ACT question has exactly one correct answer, and, once you learn this method, that correct answer will look very different from all the other options.
Three Ways to Think About Having One Very Clear Answer
OK, the SAT / ACT has to have one very clear answer -- that’s a little theoretical. How can you think about the degree of clarity in a way that will help you on the test? For my students, I’ve come up with three rules that illustrate what “very clear” means. These rules get at the same single, central idea from three different directions. You should make note of these three rules to remind yourself on the test what clarity means.
Rule 1: The 10x Rule of Clarity
It turns out that the clarity of the right answer is so important that the best answer is not just 20% or 2x better than the next best answer, but in fact 10x better. That’s right, you might think D and E are close answers, but, to an infinitely knowledgeable test-taker, it turns out that E is actually ten times as good as D.
Rule 2: Panel of 100 Experts Agree
Another way to think about how clear the right answer must be is to realize that, if there were a panel of 100 experts, all of them would have to agree on what the right answer is. If even one or two of them disagree, suddenly the question is no longer objective -- it’s subjective and up for debate -- the test maker's worst nightmare. Because questions must be objective, a panel of carefully thinking experts must agree on the correct answer.
Rule 3: Provably Correct
One final useful way to think about how clear the right answer needs to be is to realize that it must be provably correct. If given a long enough time, you could write almost a math-style proof on why the answer is correct and the other choices are wrong. If you couldn’t write a math-style proof, then some part of the logic process has to be based on a “hunch." Hunches are neither clear nor objective, and therefore, the ACT / SAT cannot rely on these. Again, the ACT / SAT must have questions that can be solved using precise, analytical logic.
How do these rules help? When you’re stuck on a question and two answers are looking very close to each other, you’ll realize this can’t be how the question is meant to be answered by the 10x Rule of Clarity. If you ever find yourself in a situation where you have to rely on a subjective judgment, where you catch yourself saying “my opinion is this” or “it seems likely that the answer is this," then the Panel of Experts rule will tell you that you can do more to answer your question.
The most powerful rule, the one rule to rule them all, is the Provably Correct rule. This rule tells you that you never need to rely on fuzzy reasoning or a feeling to answer a question. If you have enough background information, enough time, and enough logic, you can prove that every answer you choose is correct.
Thinking about the Provably Correct rule in the negative is also helpful. It means that, for all the wrong answers, you must be able to identify a fatal flaw that disqualifies them.
These rules also mean that, if you are getting stuck solving the hardest problems for you, the solution is not to “get better intuition” or “get subtler at fuzzy thinking” -- but rather to learn how to penetrate the analytical, logical core of each question.
How do you apply these rules? One of the best ways to learn is to try them on real questions. For the rest of this article, we will demonstrate the rules on a math question, a grammar question, and finally a reading question. The rule will be most obvious in math, but most insightful in reading.
For each of these three example problems, we will write math-style proofs to show beyond any doubt that the answer we choose is the right one. By proofs, I just mean breaking the problem down into very small but clear steps. No background on proofs in a math class is needed. I use the word much more loosely in this article, usually to emphasize that an explanation is crystal clear instead of fuzzy.
The easiest place to start demonstrating these concepts is math. Math is the subject where it is most obvious that each question has one very clear, objective, provably correct answer. Since it is so obvious to everyone that math answers are objective, the following demonstration is less subtle than in reading, but it’s still useful to go through this example to learn.
One of the most difficult ACT Math questions is as follows:
Consider all pairs of positive integers w and z whose sum is five.
For how many values of w does there exist a positive integer x that satisfies both 2^w = x and x^z = 64. (Statement 2)
- Infinitely Many
You can get at the single very clear answer by completing a proof, as I'll show you:
- First, we'll start with the information given to us. The two integers w and z must add up to five. That gives us four options for integer pairs for (w,z): (1,4), (2,3), (3,2) and (4,1).
- Let’s call the second sentence of the question above statement (2). Now we can prove for all four pairs above whether statement (2) holds:
- In each case, since z is a positive integer 2^w = x is a positive integer, we can ignore the restriction that x is a positive integer.
- For (1,4), statement (2) gives that 2^1=x=2. And x^z = 2^4 = 16 =/= 64. Thus, for the first pair statement (2) is false.
- For (2,3), statement (2) gives that 2^2=x=4. And x^z = 4^3 = 64 indeed. Thus, for the second pair statement (2) is true.
- For (3,2), statement (2) gives that 2^3=x=8. And x^z = 8^2 = 64 indeed. Thus, for the third pair statement (2) is true.
- For (3,2), statement (2) gives that 2^4=x=16. And x^z = 16^1 =/= 64. Thus, for the fourth pair statement (2) is false.
- Therefore, the above is a mathematical proof in the most original sense, that there are two pairs that satisfy the answer. The answer is two. This corresponds to B.
Note that, as a mathematical proof, the above explanation is watertight. (I would know -- I've taken dozens of courses in theoretical math and spent countless hours writing proofs.)
This proof passes the 10x rule of clarity (no other answer would be even 1/10th as correct). It would pass the panel of 100 experts rule -- in fact, I bet every single professional mathematician in the world would agree with the proof above. Finally, the proof is fully analytical -- it breaks the entire solution into small but obvious pieces. Proofs in reading, science, or writing won’t be nearly as perfect, but the above serves as a guideline for later in the article.
First, all proofs depend on a set of indisputable, underlying facts (in rigorous proofs, these are called axioms). Here, the two underlying facts I cited were that: 1) a positive integer taken to a positive integer power is positive, and 2) there are exactly four unique pairs of positive integers (w,z) such that w+z=5. Understanding these facts is assumed to be part of being an expert in math, and if you find yourself missing these facts when constructing proofs, then you know the problem is an underlying content problem.
Most of the underlying facts in math and grammar, and some in reading, need to be memorized beforehand. If you lack these facts, no amount of logic and no amount of time will let you solve the problems.
Second, the proof method is best used as training wheels, as an illustrative tool. On the real test, they are too time-consuming to use on more than a few rare occasions. On a real test, proofs are most useful in reading, then grammar, and least of all in math. After all, in math, the fact that there is a single, clear, objective, right answer is usually obvious.
When are proofs useful then? Proofs are useful when you are stuck on the hardest 1-3 problems in each section and have extra time. Proofs are also useful when you are practicing the SAT / ACT. Whenever you feel a question has “two right answers," you can do a proof exercise to convince yourself that’s not the case. Also, I put the math proof first because it’s the simplest to understand; in fact, it’ll be the reading proof at the end that you’ll find most helpful on SAT / ACT training.
How to Use the Proof Method on Math Problems
Here are some general guidelines to follow when you start to solve math problems using this method:
First, read through the question and break down the information it gives you. Then, identify the axioms, or indisputable math facts, you'll need to apply in order to solve this problem. This is where having strong mathematical knowledge comes in handy. If, for example, the question is about triangles, you should be able to quickly come up with all the triangle rules and information you know. After you've done this, you can start the proof. Work through the problem, making a new line for each new statement, until you've solved it and figured out your answer.
The following question is from an ACT English section, and it's similar to a grammar question you might find on SAT Reading. Grammar is a great area to illustrate the Provably Correct concept because it’s an area where many students use fuzzy thinking. Many students, especially native English speakers, are used to “sounding phrases out” and choosing the one that “feels best.” However, it’s also obvious that grammar follows hard, explicit logical rules like math does. And those hard logical rules, not your ear, are the only method guaranteed to get you every question right.
Consider now the following question:
Choose the best replacement for the underlined portion.
A musician balancing a cello case, two Buddhist monks in saffron robes, and a group of stockbrokers in crisp, charcoal gray suits get on the subway at the Wall Street station.
- No Change.
- charcoal gray suits,
- charcoal, gray suits
- charcoal gray, suits
Like math questions, you can follow a set of steps to solve English questions using proofs.
Every question on the English section will relate to at least one grammar fact. Your first step is identifying which grammar fact they are referring to. This requires a strong knowledge of English grammar, but if you study enough, you'll be able to easily identify the particular grammar rule you need. For this example, the sentence has multiple phrases with the same grammatical structure; therefore, the grammar fact you need to use is parallel construction. Go through the answer choices, applying the grammar rule to each of them, until you have clearly identified one correct answer and three incorrect answers.
This one fact is particularly important for this question:
Grammar fact (parallel construction): When there are multiple phrases that have the same grammatical structure, these phrases are to be separated by a comma. Conversely, separation by a comma strongly suggests phrases are parallel. E.g. The US flag is red, white, and blue. The words “red”, “white”, “and blue” are parallel construction and separated by a comma.
Now, let’s examine the answers. Note that the only difference is in the placement of the comma (if it exists at all). We will prove the right answer by deconstructing all versions and showing that all but one is nonsensical or ungrammatical.
Choice A: No Change. The sentence is talking about “charcoal gray suits”. The word “charcoal” modifies gray (it’s a type of gray), and the phrase “charcoal gray” modifies suit. This makes sense. Also, the commas imply parallelism between the three nouns in the sentence: the group of stockbrokers, the Buddhist monks, and the musician. This is also correct.
Choice B: “charcoal gray suits,” This option puts a comma at the end of the phrase. This separates the sentence into four suggested parallel phrases:
A musician balancing a cello case,
two Buddhist monks in saffron robes,
and a group of stockbrokers in crisp, charcoal gray suits,
get on the subway at the Wall Street station.
The first three are noun phrases and contain subjects (musicians, monks, and stockbrokers, respectively). The fourth phrase, however; doesn't include a subject and is instead a verb phrase which violates suggested parallel construction This means that placing the comma after the word "suits" would not be signifying parallel construction.
To be rigorous, you must be aware that, in addition to parallel construction, commas can only be used to set off nonessential clauses, along with a few other minor cases. The verb clause is an essential part of this sentence; without it the sentence would not make sense, and it wouldn't be grammatically correct. The placement of the comma for option B is therefore inappropriate. This disproves B.
Choice C: “charcoal, gray suits” By our first Grammar Fact, this suggests that charcoal and gray are parallel. This means both are modifying the word “suits." The suits are both gray (makes sense) and charcoal (doesn’t make sense). The suits are not literally made of the same charcoal that you barbecue with! This parallelism gives the sentence the wrong meaning and thus can be provably disqualified.
Choice D: “charcoal gray, suits” By our first Grammar Fact again, the commas here would strongly suggest that the phrases “crisp," “charcoal gray” and “suits” are parallel. However, the first two are adjectives, and the final word is a noun, again violating parallelism and disproving this option as the correct answer.
And there we have it, we have “proven” above that the right answer must be A.
(To be even more rigorous, we would want to list all valid uses for commas and eliminate these cases in each of the answers above. This gets truly arduous, but it will advance this proof from “10x correct” territory to “100x correct” territory. This again is a demonstration between the trade offs between full rigor and time spent.)
Who Is This Proof Most Useful For?
The proof is best used for a student who is stuck between two answers which both look right. In this case, many students have complained that they can’t tell whether A or C is correct after looking at the question long and hard. They both “sound” correct.
A proof allows you to show that one answer must be very right while the others are very wrong. In the case above, we relied on the role of the comma in parallelism. You'll want to use this method practically, and only if you have substantial time to eliminate all ambiguity. You can use it 1) on a real test if you have extra time left 2) if you are studying and want to conquer the most difficult questions 3) if you’re working on improving content and don’t mind spending extra time demonstrating to yourself why one answer is exactly right.
Proofs aren’t infinitely powerful. After all, you have to know the underlying Grammatical Fact put out at the beginning. A proof doesn’t give you an answer if you don’t know the subject! Second, proofs take much too long to implement on all questions on a live test. In a live test, you absolutely want to eliminate some choices “by ear” when they sound egregious, and you absolutely want to take timing shortcuts that give you 90% of the accuracy in 10% of the time.
However, even if you don’t do an actual proof on the test, just knowing that a proof must exist is incredibly empowering. Even when you are using intuition or fuzzy feelings, you then know that the intuition or feeling must be overlaying a cold, hard fact. If you are going by intuition, you know that the final word in the answer cannot possibly be just a feeling. Provably Correct is something that should totally change your perspective on an ACT / SAT questions.
ACT / SAT Reading is my favorite area to apply our rule to! This is because reading seems so touchy-feely, so subjective, that it’s tempting for students to think of the section as uncertain, subjective, and intuition-based. In fact, reading questions are exactly the opposite: they are certain, objective, and analytical.
Reading is the opposite of math in that proofs are the least obvious but the most helpful tool to improve your score. Let’s get to the question:
Consider the following paragraph:
"We plan makers are accustomed to things turning out not quite as good as we had in mind. Our world view includes the “diminished excellence” component. Diminished excellence is a condition of the world and therefore never an occasion for sorrow, whereas flawed competence comes out of character and therefore is frequently the reason for the bowed head, the furrowed brow."
In the last paragraph, a comparison is made between "diminished excellence" and "flawed competence." From the narrator's point of view, the conditions are different because the one is:
- A source of sorrow while the other is a source of pride.
- Based in the family while the other is based in the self.
- Inherent in the environment while the other is inherent in the individual.
- A sign that the individual can improve the world while the other is a sign that the individual can't.
If you want to really learn the proof method, I strongly encourage you to work through this problem. Give yourself 10 or even 20 minutes if you need. Write out your logic and compare it against the rigor below. If you are confused, introspect about your confusion. In just a few moments you’ll see an explanation that will prove beyond a doubt that one of the answers is clearly 10x correct.
What Not to Do
First, let’s go over what a student using “fuzzy thinking” might do. Frank the fuzzy thinker might look at F and think, “The paragraph does mention one of them being sorrow, so this looks fine.” He may then go onto G and go, “Well, there was no discussion of family in this paragraph, so that’s clearly out.” For H he thinks, “Yes, one of them is about the world while the other isn’t, so let’s keep H.” Finally, he goes onto J and thinks, “Well, yes, one of them is improvable, while the other isn’t”; so he keeps J.
Frank has eliminated F because it feels a bit off to him, and he eliminated G because of a "feeling" he had. However, both H and J sound good. Frank would estimate that H sounds about twice as good as F, but J sounds the best of any of the answer choices, beating H by maybe 10-20%. Frank thinks the answer really depends on how you see the question-- it's subjective anyway, so he chooses J. Unfortunately for Frank, he chose the wrong answer. Even worse, the way he solved the problem demonstrates the worst of fuzzy logic!
Rules That Frank the Fuzzy Thinker Broke
Note that his final reasoning broke every one of our three “clear answer” rules. First, he thought that the best answer was only 10-20% better than the next, and at most 2-3x better than the third best answer, violating the 10x clear rule. Second, he thought the answer was subjective and broke the “consensus of 100 experts” rule. Finally, his reasoning lacked substantial analytical rigor. He relied on how he felt about the answers and used simple “word matching," breaking the Provably Correct rule.
Breaking the Provably Correct rule on reading questions invariably shows some patterns. Frank illustrates some of them:
- Associative thinking: Frank saw the word "sorrow" in the paragraph and thought that, since answer F contains that word, it has a high chance of being right. Likewise, he ruled out G based on only the single word “family." While it is tempting to use word-matching to choose answers, this is the lowest form of non-analytical, fuzzy thinking. Reading questions are more subtle than hunting for the right word.
- Drawing inferences from the outside: To Frank, whether something is “inherent in the environment” (from the source paragraph) is the same as “a sign that the individual can’t improve the world” (answer J). However, this latter statement is actually not stated in the paragraph at all!
- Dropping or adding words to force things to fit: Frank keeps F even though the word “pride” isn’t anywhere in the paragraph. F otherwise seems like a good answer, so Frank ignores the minor inconvenience that an entire word is out of place.
How to Solve This Problem Analytically With a Proof
Now, let’s see why the above question is really not a subjective, “two good answers” situation. We’ll do this by bringing out our usual tool of analytical rigor, the proof. First, read the paragraph word-by-word slowly and carefully. Think about what each sentence means after reading it. Then, re-read the entire paragraph.
I will start the proof by restating a large majority of the paragraph in my own words. The following statements are logically contained within the original paragraph:
We are plan makers. We are used to things turning out less than our plan. The way we see the world includes a part called “diminished excellence." Diminished excellence is a condition of the world. Because of this, “diminished excellence” is not an occasion for sorrow. However, “flawed competence” comes out of character. Because of this, “flawed competence” is often why there is the bowed head, the furrowed brow.
Each statement is a rigorous transformation of the original and totally implied by the original. We will use both the original and the implied transformation to prove the answers.
Choice F: A source of sorrow while the other is a source of pride.
The first part of this answer is true. It's true that one is implicitly a cause for sorrow. The paragraph states that “diminished excellence” isn’t a cause for sorrow, but the conjunction “however” implies strongly that “flawed competence” indeed causes sorrow.
However, for all F to be true, the second part must be true as well; we must have a source for pride. Since “flawed competence” is taken by sorrow already, if F were true, then “diminished excellence” must be a cause for pride. Intuitively, “diminished excellence” doesn’t seem like a good cause for “pride”, but let’s prove it.
The paragraph just says diminished excellence is a condition of the world and not a cause for sorrow. Nowhere do we have it explicitly said or strongly implied that “diminished excellence” is a cause for pride. This means F cannot be wholly true. Therefore F is wrong.
Choice G: One is based in the family while the other is based in the self.
Reading all the logical statements in the original paragraph, it is easy to see that no statement says anything about a family, nor anything that would imply a family (such as a group, relative, etc). Therefore, we can deduce that “based in the family” cannot possibly be a conclusion of the paragraph. Thus G is ruled out as an answer.
(If you’re looking for extra credit, it is indeed true the paragraph says that “flawed competence” is based out of character, which is strongly associated with the self, so the second half of statement G is true. However half-true is just not good enough!)
Choice H: One is based in the environment while the other is inherent in the individual.
The paragraph clearly does state that diminished excellence is a condition of the world and thus the environment. This proves the first part of the statement as true. Now, the paragraph says the other term, “flawed competence," comes out from the character, which almost definitionally is inherent to the individual. This proves the second part of the statement as true.
As a bonus, further reinforcing this proof is the fact that the entire paragraph is a parallelism between a concept with roots in the world, versus another with roots in character. This world vs. character contrast is exactly targeted by this answer which compares environment vs. individual. Thus H has very strong support, a proof in fact.
Choice J: A sign that the individual can improve the world while the other is a sign that the individual can't.
With fuzzy thinking, J looks similar to H. However, let us be precise. J says that one is a sign that the individual can improve the world. This first concept J refers to cannot possibly be “diminished excellence” since the paragraph does not say anything at all about the individual with respect to “diminished excellence”.
Thus, if J were true, the first part of J must refer to “flawed competence”. It is true that the paragraph says “flawed competence” comes from the character and thus the individual. However, the paragraph says nothing about flawed competence being changeable. Also, even if “flawed competence” implied any potential for change, nothing in the paragraph speaks about the ability of this change to “improve the world”. Thus, J is disproven.
As a bonus, you can also point out multiple other logically-rigorous, fatal flaws in J. For example, the paragraph says nothing about if “diminished excellence” modulates whether an individual can impact the world.
The fuzzy thinking here is that the paragraph talks about two concepts that come from the world versus the individual, while choice J is instead talking about the impact of the individual on the world. Same concepts, but totally unrelated. It would be as unrelated as if the paragraph talked about the weather of New York compared to that of Los Angeles, and the answer talked about flights between New York and Los Angeles. Thus J is rejected too.
We have written relatively objective, logical, and analytical proofs that show that H must be the right answer, and all the others must be wrong answers. Hopefully, if you thought the question was vague, subtle, and subjective before, the proof shows that the question is, in fact, analytical with a very clear, objective, and logical answer.
For Reading questions, you'll want to first start by reading the passage, then transforming it into your own words, while keeping its entire original meaning intact, like we did in the example. Then, go through each of the answer options and compare them to both the original passage and your rewording to see if they are true. Remember, each part of the answer must be true. If only half the answer is true, then it is not the right answer.
Takeaways From the Proof
I hope that the proof above gives you confidence that, with enough time and application of logic, you can clearly show only one answer is true. By transforming what seems like a fuzzy question into cold hard logic, hopefully, the above demonstrated that, on the ACT / SAT, all questions are in fact analytical and provable.
But if all questions are analytical, logical, and provable, then why doesn’t the ACT / SAT just directly test these skills in straightforward ways? The reason is that most of reading / English in academia is indeed subjective and often fuzzy. Who’s a better writer, James Joyce or Vladimir Nabokov? Subjective. What social actions did Orwell want to encourage in his readers by writing his bestseller 1984? Again, subjective. Many interesting and difficult academic subjects are inherently subtle and subjective. I can tell you this from firsthand experience, from writing countless college essays on topics like film studies and the Jewish Bible.
Since much of real academia has a feeling of intuition, subtlety, and subjectiveness, the ACT / SAT wants to mimic these factors. They dress their questions up to look as subtle and fuzzy as possible. But we know from the beginning of the article that the ACT / SAT cannot afford truly fuzzy or subjective questions. Therefore, the fuzziness is a ruse. It becomes a set of trap answers for the student. The core of any ACT / SAT question is a hard, analytic question, and if you only see a fuzzy question, this means you haven’t seen the core of the question yet!
Now that you know that each SAT / ACT answer is provable, you’ll no longer approach difficult questions the wrong way. It is so common for students to think that they need to develop a finer sense of intuition or better fuzzy thinking to get the hard questions. In fact, what you should be working on is a fast analytic breakdown of every question.
You can definitely do this yourself, and the above steps offer pretty good guide-by-examples of how to do it. For those interested, we also should mention that PrepScholar trains you in this method as well. Our program will detect when non-rigorous thinking is the major source of your errors. We see this mostly in high-performing students trying to nail those final questions, but we also see it to some degree in all students. Our program will provide you with lessons that teach you how to be analytic and give you practice problems for you to apply your new skills. If you liked our lesson here, give our program a free try:
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Fred is co-founder of PrepScholar. He scored a perfect score on the SAT and is passionate about sharing information with aspiring students. Fred graduated from Harvard University with a Bachelor's in Mathematics and a PhD in Economics.