---
title: The Hardest ACT Math Question Types · PrepScholar
description: Want to challenge yourself with really hard ACT math problems? Here are the 21 most difficult math questions we've seen on the ACT, ever.
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# [The Hardest ACT Math Question Types](https://blog.prepscholar.com/21-hardest-act-math-problems)

![author image](http://cdn2.hubspot.net/hubfs/360031/author_courtney.png)

Posted by [Courtney Montgomery](https://blog.prepscholar.com/author/courtney-montgomery) | Updated on August 29, 2024

[ACT Math](https://blog.prepscholar.com/topic/act-math)

 

![Feature_chess-1](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/Feature_chess-1.jpg?width=304&name=Feature_chess-1.jpg)

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?

If you're up for the challenge, preparation starts by understanding the toughest question types you'll face on the ACT. So 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 most difficult math question types the ACT has given to students in the past 10 years, complete 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.

Let's dive in!

 

## 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](https://blog.prepscholar.com/complete-official-act-practice-tests-free-links) 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](https://blog.prepscholar.com/realistic-act-practice-testing-with-parent-proctoring), 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](https://blog.prepscholar.com/complete-official-act-practice-tests-free-links) 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](https://blog.prepscholar.com/how-to-improve-low-act-math-scores-tips-from-a-perfect-scorer), your best best is to first check out our guides on using the key math strategies of [plugging in numbers](https://blog.prepscholar.com/plugging-in-numbers-sat-act-math-strategy) and [plugging in answers](https://blog.prepscholar.com/plugging-in-answers-a-critical-sat-math-act-math-strategy) 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)](https://blog.prepscholar.com/how-to-get-36-on-act-math-8-strategies-by-a-perfect-scorer), 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.

 

![body_green_light](https://360031.fs1.hubspotusercontent-na1.net/hubfs/360031/body_green_light.jpg)

 

*Ready, set...*

 

## Hardest Types of ACT Math Questions

Now that you're positive that you should be trying out these difficult math questions, let’s get right to it! We've broken hard ACT questions into sections based on type. Additionally, the answers to these questions are in a separate section below, so you can go through them all at once without getting spoiled.

 

### Type #1: Graph Interpretation

Knowing how to read and interpret graphs can be tricky, especially since that information can also require you to use *other* math skills, like finding points on a line or calculating angles.

To unpack a graph question, first ask yourself—what is the prompt *actually* asking me to do? Figuring out the core mathematical concept behind the graphed information will make it easier to answer the question.

Check out the two graph interpretation questions below. Notice that both are asking you to interpret the graph data by using other mathematical concepts! With that in mind, try solving them on your own to get practice solving these tough questions. (Remember: we've included the answers below!)

 

#### Graph Interpretation Questions

**QUESTION #1:**

In the figure below, line $q$ in the standard $(x,y)$ coordinate plane has equation $-2x+y=1$ and intersects line $r$, which is distinct from line $q$, at a point on the $x$-axis. The angles $∠a$ and $∠b$ formed by these lines and the $x$-axis are congruent. What is the slope of line $r$?

![body-graph-1-1](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-graph-1-1.png?width=200&height=193&name=body-graph-1-1.png)

F. $-2$

G. $-{1}/2$

H. $1/2$

J. $2$

K. Cannot be determined

 

ANSWER #1:

First, turn our given equation for line q into proper [slope-intercept form](https://blog.prepscholar.com/lines-slopes-coordinate-geometry-act-math-review-practice).

$−2x+y=1$

$y=2x+1$

Now, we are told that the angles the lines form are congruent. This means that the slopes of the lines will be opposites of one another \[Note: perpendicular lines have opposite *reciprocal* slopes, so do NOT get these concepts confused!\].

Since we have already established that the slope of line $q$ is 2, line $r$ must have a slope of -2.

**Our final answer is F**, -2

 

**QUESTION #2:**

The graph of the equation $h=-at^2 + bt + c$, which describes how the height, $h$, of a hit baseball changes over time, $t$, is shown below.

![body-graph-2](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-graph-2.png?width=250&height=149&name=body-graph-2.png)

If you alter only this equation's $c$ term, which gives the height at time $t=0$, the alteration has an effect on which of the following?

I. The $h$-intercept  
II. The maximum value of $h$  
III. The $t$-intercept

F. I only

G. II only

H. III only

J. I and III only

K. I, II, and III

 

ANSWER #2:

The equation we are given ($−at^2+bt+c$) is a parabola and we are told to describe what happens when we change $c$ (the $y$-intercept).

From what we know about functions and function translations, we know that changing the value of $c$ will shift the entire parabola upwards or downwards, which will change not only the $y$-intercept (in this case called the "$h$-intercept"), but also the maximum height of the parabola as well as its $x$-intercept (in this case called the $t$-intercept). You can see this in action when we raise the value of the $y$-intercept of our parabola.

![body-question-2-graphs](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-question-2-graphs.png?width=192&height=400&name=body-question-2-graphs.png)

Options I, II, and III are all correct.

Our final answer is K, I, II, and III

 

### Type #2: Trigonometry Questions

Trigonometry questions are tough for a number of reasons. First, they require you to memorize formulas and values in order to properly interpret questions. There's also a lot of conversion involved: you have to know how (and when!) to convert radians to degrees and visa versa.

And don't forget functions! Trigonometry uses non-linear functions, which is tough because its graph isn't a line or part of a line. Because trigonometry combines so many advanced math concepts, they can trip you up.

Test your trig skills on the advanced questions below. Try to work the, without peeking at the answers!

 

#### Trigonometry Questions

**QUESTION #1:**

The equations of the two graphs shown below are $y\_1(t) = a\_1\\sin (b\_1t)$ and $y\_2(t)=a\_2\\cos(b\_2t)$, where the constants $b\_1$ and $b\_2$ are both positive real numbers.

![body-graph-3](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-graph-3.png?width=363&height=277&name=body-graph-3.png)

Which of the following statements is true of the constants $a\_1$ and $a\_2$?

A. $0 < a\_1 < a\_2$

B. $0 < a\_2 < a\_1$

C. $a\_1 < 0 < a\_2$

D. $a\_1 < a\_2 < 0$

E. $a\_2 < a\_1 < 0$

 

**ANSWER #1:**

The position of the a values (in front of the sine and cosine) means that they determine the [amplitude](https://blog.prepscholar.com/act-trigonometry-the-complete-guide) (height) of the graphs. The larger the a value, the taller the amplitude.

Since each graph has a height larger than 0, we can eliminate answer choices C, D, and E.

Because $y\_1$ is taller than $y\_2$, it means that $y\_1$ will have the larger amplitude. The $y\_1$ graph has an amplitude of $a\_1$ and the $y\_2$ graph has an amplitude of $a\_2$, which means that $a\_1$ will be larger than $a\_2$.

**Our final answer is B**, $0 < a\_2 < a\_1$.

 

 

**QUESTION #2:**

For x such that $0 < x <π/2$, the expression ${√{1 – \\cos^2 x}/{\\sin x} + {√{1 – \\sin^2 x}/{\\cos x}$ is equivalent to:

F. 0

G. 1

H. 2

J. $-\\tan{x}$

K. $\\sin{2x}$

 

**ANSWER #2:**

If you remember your [trigonometry shortcuts](https://blog.prepscholar.com/act-trigonometry-the-complete-guide), you know that $1−{\\cos^2}x+{\\cos^2}x=1$. This means, then, that ${\\sin^2}x=1−{\\cos^2}x$ (and that ${\\cos^2}x=1−{\\sin^2}x$).

So we can replace our $1−{\\cos^2}x$ in our first numerator with ${\\sin^2}x$. We can also replace our $1−{\\sin^2}x$ in our second numerator with ${\\cos^2}x$. Now our expression will look like this:

${√{\\sin^2 x}/{\\sin x}+{√{\\cos^2 x}/{\\cos x }$

We also know that the square root of a value squared will cancel out to be the original value alone (for example,$√{2^2}=2$), so our expression will end up as:

$={\\sin{x}}/{\\sin{x}}+{\\cos{x}}/{\\cos{x}}$

Or, in other words:

$=1+1$

$=2$

**Our final answer is H**, 2.

 

### Type #3: Logarithms

Logarithms can be hard because they require you to differentiate between exponential and logarithmic questions and understand logarithmic notation. If you can't read the question, you can't answer it, after all!

You'll also need to understand exponents to work with logarithms since the two concepts are intertwined. Working through the following equation can help you learn how to manage these tough problems.

 

#### Logarithm Questions

**QUESTION #1:**

What is the real value of $x$ in the equation $\\log\_2{24} – \\log\_2{3} = \\log\_5{x}$?

F. 3

G. 21

H. 72

J. 125

K. 243

 

**ANSWER #1:** If you’ve brushed up on your log basics, you know that $\\log\_b(m/n)=\\log\_b(m)−\\log\_b(n)$. This means that we can work this backwards and convert our first expression into:

$\\log\_2(24)-\\log\_2(3)=\\log\_2(24/3)$

$=\\log\_2(8)$

We also know that a log is essentially asking: "To what power does the base need to raised in order to achieve this certain value?" In this particular case, we are asking: "To which power must 2 be raised to equal 8?" The answer to this question is 3, since $(2^3=8)$, so $\\log\_2(8)=3$

Now this expression is equal to $\\log\_5(x)$, which means that we must *also* raise our 5 to the power of 3 in order to achieve $x$. So:

$3=\\log\_5(x)$

$5^3=x$

$125=x$

**Our final answer is J**, 125.

 

### Type #4: Functions

Functions are one of those question types that are tough for some students but easier for others. That's because as functions get more complex, so does finding the correct answer. If you understand how functions work, you can break down more complex problems into simpler steps.;

Try your hand at the hard function question below to test your knowledge.

 

#### Function Questions

**QUESTION #1:**

The functions $y = \\sin x \\and y = \\sin(x + a) + b$, for constants $a$ and $b$, are graphed in the standard $(x,y)$ coordinate plane below. The functions have the same maximum value. One of the following statements about the values of $a$ and $b$ is true. Which statement is it?

![body-graph-4](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-graph-4.png?width=300&height=125&name=body-graph-4.png)

A. $a < 0$ and $b = 0$

B. $a < 0$ and $b > 0$

C. $a = 0$ and $b > 0$

D. $a > 0$ and $b< 0$

E. $a > 0$ and $b> 0$

 

ANSWER #1:

The only difference between our [function graphs](https://blog.prepscholar.com/act-trigonometry-the-complete-guide) is a [horizontal shift](https://blog.prepscholar.com/reflections-rotations-and-translations-act-geometry-strategies-and-practice), which means that our b value (which would determine the vertical shift of a sine graph) must be 0.

Just by using this information, we can eliminate every answer choice but A, as that is the only answer with $b=0$. For expediency's sake, we can stop here.

**Our final answer is A**, $a<0$ and $b=0$

 

Advanced ACT Math note: An important word in ACT Math questions is "must," as in "something *must* be true." If a question **doesn't** have this word, then the answer only has to be true for a particular instance (that is, it *could* be true.)

In this case, the majority of the time, for a graph to shift horizontally to the left requires $a>0$. However, because $\\sin(x)$ is a periodic graph, $\\sin(x+a)$ would shift horizontally to the left if $a=-π/2$, which means that for at least one value of the constant $a$ where $a<0$, answer **A** is true. In contrast, there are no circumstances under which the graphs could have the same maximum value (as stated in the question text) but have the constant $b≠0$.

As we state above, though, on the real ACT, once you reach the conclusion that $b=0$ and note that only one answer choice has that as part of it, you should stop there. Don't get distracted into wasting more time on this question by the bait of $a<0$!

 

![body_finish-1](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body_finish-1.jpg?width=227&name=body_finish-1.jpg)

*Whoo! You made it to the finish line—go you!*

 

## What Do the Hardest ACT Math Questions Have in Common?

Now, lastly, before we get to the questions themselves, it is important to understand what makes these hard questions “hard.” By doing so, you will be able to both understand and solve similar questions when you see them on test day, as well as have a better strategy for identifying and correcting your previous ACT math errors.

**In this section, we will look at what these questions have in common** and give examples for each type. **In the next section, we will give you all 21 of the most difficult questions as well as answer explanations for each question,** including the ones we use as examples here.

Some of the reasons why the hardest math questions *are* the hardest math questions are because the questions do the following:

 

### #1: Test Several Mathematical Concepts at Once

EXAMPLE QUESTION:

Consider the functions $f(x) = √{x}$ and $g(x) = 7x + b$. In the standard $(x,y)$ coordinate plane, $y = f(g(x))$ passes through $(4,6)$. What is the value of $b$?

A. $8$

B. $-8$

C. $-25$

D. $-26$

E. $4 – {7√6}$

 

As you can see, this question deals with a combination of functions and coordinate geometry points.

 

### #2: Require Multiple Steps

Many of the most difficult ACT Math questions primarily test just one basic mathematical concept. What makes them difficult is that you have to work through multiple steps in order to solve the problem. (Remember: the more steps you need to take, the easier it is to mess up somewhere along the line!)

EXAMPLE QUESTION:

An integer from 100 through 999, inclusive, is to be chosen at random. What is the probability that the number chosen will have 0 as at least 1 digit?

A. $19/900$

B. $81/900$

C. $90/900$

D. $171/900$

E. $271/{1{,}000}$

Though it may sound like a simple probability question, you must run through a long list of numbers with 0 as a digit. This leaves room for calculation errors along the way.

 

### #3: Use Concepts You're Less Familiar With

Another reason the questions we picked are so difficult for many students is that they focus on subjects you likely have limited familiarity with. For example, many students are less familiar with algebraic and/or trigonometric functions than they are with fractions and percentages, so most function questions are considered “high difficulty” problems.

EXAMPLE QUESTION:

The functions $y =$ sin$x$ and $y =$ sin$(x + a) + b$, for constants $a$ and $b$, are graphed in the standard $(x,y)$ coordinate plane below. The functions have the same maximum value. One of the following statements about the values of $a$ and $b$ is true. Which statement is it?

![body-graph-5](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-graph-5.png?width=313&height=138&name=body-graph-5.png)

A. $a < 0$ and $b = 0$

B. $a < 0$ and $b > 0$

C. $a = 0$ and $b > 0$

D. $a > 0$ and $b < 0$

E. $a > 0$ and $b > 0$

Many students get intimidated with [function](https://blog.prepscholar.com/algebra-functions-on-act-math-strategies-and-practice-questions) problems because they lack familiarity with these types of questions.

 

### #4: Give You Convoluted or Wordy Scenarios to Work Through

Some of the most difficult ACT questions are not so much mathematically difficult as they are simply tough to decode. Especially as you near the end of the math section, it can be easy to get tired and misread or misunderstand exactly what the question is even asking you to find.

EXAMPLE QUESTION:

In the complex plane, the horizontal axis is called the *real axis* and the vertical axis is called the *imaginary axis.* The complex number $a + bi$ graphed in the complex plane is comparable to the point $(a,b)$ graphed in the standard $(x,y)$ coordinate plane.

The modulus of the complex number $a + bi$ is given by ${√{a^2+b^2}$ . Which of the complex numbers $z\_1$ , $z\_2$ , $z\_3$ , $z\_4$ , and $z\_5$ below has the greatest modulus?

![body-graph-6](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-graph-6.png?width=250&height=230&name=body-graph-6.png)

F. $z\_1$

G. $z\_2$

H. $z\_3$

J. $z\_4$

K. $z\_5$

This question presents students with a completely foreign mathematical concept and can eat up the limited available time.

 

### #5: Appear Deceptively Easy

Remember—if a question is located at the very end of the math section, it means that a lot of students will likely make mistakes on it. Look out for these questions, which may give a false appearance of being easy in order to lure you into falling for bait answers. Be careful!

EXAMPLE QUESTION:

Which of the following number line graphs shows the solution set to the inequality $|x–5|<1$ ?

![body-graph-7](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body-graph-7.png?width=245&height=314&name=body-graph-7.png)

 

This question may seem easy, but, because of how it is presented, many students will fall for one of the bait answers.

 

### #6: Involve Multiple Variables or Hypotheticals

The more difficult ACT Math questions tend to use many different variables—both in the question and in the answer choices—or present hypotheticals. (Note: The best way to solve these types of questions—questions that use multiple integers in both the problem and in the answer choices—is to use the strategy of [plugging in numbers](https://blog.prepscholar.com/plugging-in-numbers-sat-act-math-strategy).)

EXAMPLE QUESTION:

If $x$ and $y$ are real numbers such that $x > 1$ and $y < -1$, then which of the following inequalities *must* be true?

**A.** ${x/y} > 1$

**B.** ${|x|^2} > |y|$

**C.** ${x/3} – 5 > {y/3} – 5$

**D.** ${x^2} + 1 > {y^2} + 1$

**E.** ${x^{-2}} > {y^{-2}}$

Working with hypothetical scenarios and variables is almost always more challenging than working with numbers.

 

![body_bakery](https://360031.fs1.hubspotusercontent-na1.net/hub/360031/hubfs/body_bakery.jpg?width=513&name=body_bakery.jpg)

*Now picture something delicious and sooth your mind as a reward for all that hard work.*

 

## The Take-Aways

Taking the ACT is a long journey; the more you get acclimated to it ahead of time, the better you'll feel on test day. And knowing how to handle the hardest questions the test-makers have ever given will make taking your ACT seem a lot less daunting.

**If you felt that these questions were easy**, make sure not underestimate the effect of adrenaline and fatigue on your ability to solve your math problems. As you study, try to follow the timing guidelines (an average of one minute per ACT math question) and try to take full tests whenever possible. This is the best way to recreate the actual testing environment so that you can prepare for the real deal.

**If you felt these questions were challenging**, be sure to strengthen your math knowledge by checking out our [individual math topic guides for the ACT](https://blog.prepscholar.com/topic/act-math). There, you'll see more detailed explanations of the topics in question as well as more detailed answer breakdowns.

 

## What’s Next?

**Felt that these questions were harder than you were expecting?** Take a look at [all the topics covered on the ACT math section](https://blog.prepscholar.com/whats-tested-on-act-math-concepts-subjects-and-skills) and then note which sections you had particular difficulty in. Next, take a look at our [individual math guides](https://blog.prepscholar.com/topic/act-math) to help you strengthen any of those weak areas.

**Running out of time on the ACT math section?** [Our guide to helping you beat the clock](https://blog.prepscholar.com/how-to-stop-running-out-of-time-on-act-math) will help you finish those math questions on time.

**Aiming for a perfect score?** Check out our [guide on how to get a perfect 36 on the ACT math section](https://blog.prepscholar.com/how-to-get-36-on-act-math-8-strategies-by-a-perfect-scorer), written by a perfect-scorer.

 

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 Courtney Montgomery

 Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She has years of tutoring experience and writes creative works in her free time.

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