Question

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You’ve studied and now you’re geared up for the ACT math section (whoo!). But are you ready to take on the most challenging math questions the ACT has to offer? Do you want to know exactly why these questions are so hard and how best to go about solving them? If you’ve got your heart set on that perfect score (or you’re just really curious to see what the most difficult questions will be), then this is the guide for you.

We’ve put together what we believe to be the most 21 most difficult questions the ACT has given to students in the past 10 years, with strategies and answer explanations for each. These are all real ACT math questions, so understanding and studying them is one of the best ways to improve your current ACT score and knock it out of the park on test day.



Brief Overview of the ACT Math Section
Like all topic sections on the ACT, the ACT math section is one complete section that you will take all at once. It will always be the second section on the test and you will have 60 minutes to completed 60 questions.

The ACT arranges its questions in order of ascending difficulty. As a general rule of thumb, questions 1-20 will be considered “easy,” questions 21-40 will be considered “medium-difficulty,” and questions 41-60 will be considered “difficult.”

The way the ACT classifies “easy” and “difficult” is by how long it takes the average student to solve a problem as well as the percentage of students who answer the question correctly. The faster and more accurately the average student solves a problem, the “easier” it is. The longer it takes to solve a problem and the fewer people who answer it correctly, the more “difficult” the problem.

(Note: we put the words “easy” and “difficult” in quotes for a reason—everyone has different areas of math strength and weakness, so not everyone will consider an “easy” question easy or a “difficult” question difficult. These categories are averaged across many students for a reason and not every student will fit into this exact mold.)

All that being said, with very few exceptions, the most difficult ACT math problems will be clustered in the far end of the test. Besides just their placement on the test, these questions share a few other commonalities. We'll take a look at example questions and how to solve them and at what these types of questions have in common, in just a moment.



But First: Should You Be Focusing on the Hardest Math Questions Right Now?
If you’re just getting started in your study prep, definitely stop and make some time to take a full practice test to gauge your current score level and percentile. The absolute best way to assess your current level is to simply take the ACT as if it were real, keeping strict timing and working straight through (we know—not the most thrilling way to spend four hours, but it will help tremendously in the long run). So print off one of the free ACT practice tests available online and then sit down to take it all at once.

Once you’ve got a good idea of your current level and percentile ranking, you can set milestones and goals for your ultimate ACT score. If you’re currently scoring in the 0-16 or 17-24 range, your best best is to first check out our guides on using the key math strategies of plugging in numbers and plugging in answers to help get your score up to where you want it to. Only once you've practiced and successfully improved your scores on questions 1-40 should you start in trying to tackle the most difficult math problems on the test.

If, however, you are already scoring a 25 or above and want to test your mettle for the real ACT, then definitely proceed to the rest of this guide. If you’re aiming for perfect (or close to), then you’ll need to know what the most difficult ACT math questions look like and how to solve them. And luckily, that’s exactly what we’re here for.

Answer 1

The z-score for an SAT score of 720 is 1.74, indicating it is relatively high. A math SAT score 1.5 standard deviations above the mean is 697.5. The person who took the ACT math test performed better based on z-scores.

Step-by-step explanation:

The question is asking about the distribution of scores in the math section of the SAT exam. The mean (µ) is 520 and the standard deviation (σ) is 115. To calculate the z-score for an SAT score of 720, we use the formula: z = (x - µ) / σ. Plugging in the given values, we have z = (720 - 520) / 115 = 1.74. This means the SAT score of 720 is 1.74 standard deviations above the mean. In the context of this situation, a score of 720 is relatively high compared to the average.

To find the math SAT score that is 1.5 standard deviations above the mean, we multiply the standard deviation by 1.5 and add it to the mean. So, x = µ + (1.5 * σ) = 520 + (1.5 *115) = 697.5. Therefore, a math SAT score of 697.5 is 1.5 standard deviations above the mean.

To determine who did better on their respective tests, we need to compare the scores to the mean and standard deviation of each test. For the SAT math test, the person scored 700, which is (700 - 514) / 117 = 1.59 standard deviations above the mean. For the ACT math test, the person scored 30, which is (30 - 21) / 5.3 = 1.7 standard deviations above the mean. Since a higher z-score means a higher relative score compared to the mean, the person who took the ACT math test did better in terms of their test score.

Answer 2

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Step-by-step explanation: