SAT Hardest Questions | Download PDF

Download SAT Math Study Guide PDF

 This page brings together some of the SAT Hardest Questions in Math and Reading & Writing, along with clear, step-by-step explanations that show not just the right answer, but why wrong answers are tempting.

You’ll learn:

• What makes certain SAT questions harder than others

• How high-scoring students approach difficult problems

• Where students typically make mistakes under time pressure

• How to practice these questions in a way that actually raises your score

Whether you’re aiming to move from the 600s to the 750s, or you’re already a strong scorer looking to improve accuracy in harder modules, this guide is built to reflect how the Digital SAT really works – and how top students prepare for it.

Instructor note: These examples are based on real SAT practice tests and common patterns we see while working with U.S. and NRI students preparing for the Digital SAT.

Download SAT Hardest Math Questions PDF

If you’re targeting a 700+ SAT Math score, practicing only average-level questions isn’t enough. This PDF compiles some of the hardest SAT-style math questions to help you build confidence, reduce careless errors, and improve performance on the most challenging problems.

Download SAT Math Hardest Questions PDF 

SAT Math Test Prep Toughest Practice Questions With Answers

Question 1:

Ans. Choice A is correct. It’s given that Tilly earns p dollars for every w hours of
work. This can be represented by the proportion pw . The amount of money, x , Tilly
earns for 39w hours of work can be found by setting up the proportion 39
p x w w = .This can be rewritten as 39pw xw = . Dividing both sides by w results in x p =39 .

Question 2:

Ans. Choice D is correct. It′s given that Juan rides his bike at an average rate of
5.7 minutes per mile. The number of minutes it will take Juan to ride x miles can
be identifyd by multiplying his average rate by the number of miles, x , which
yields 5.7x . Therefore, the function mx x ( )=5.7 models the number of minutes it
will take Juan to ride x miles.

Question 3:

Ans. Choice B is correct. Adding the second equation in the given system to the first
equation in the given system yields 3 3 12 6 x xy +- + = +- ( ) ( ), which is equivalent
to y =6.

Question 4:

Ans. Choice D is correct. In the given equation, s is the speed, in miles per hour, of a certain car t seconds after it began to accelerate. Therefore, the speed of the
car, in miles per hour, 5 seconds after it began to accelerate can be found by
substituting 5 for t in the given equation, which yields s = + 40 3 5( ), or s =55.
Thus, the speed of the car 5 seconds after it began to accelerate is 55 miles per
hour.

Question 5:

Ans. Choice D is correct. By the Pythagorean theorem, if a right triangle has a
hypotenuse with length c and legs with lengths a and b, then 222 cab = + . In
the right triangle shown, the hypotenuse has length c and the legs have lengths
a and b. It’s given that a = 4 and b = 5. Substituting 4 for a and 5 for b in the
Pythagorean theorem yields 2 22 c = + 4 5 . Taking the square root of both sides of this equation yields 2 2 c =± + 4 5 . Since the length of a side of a triangle must be positive, the value of c is 2 2 4 5 + .

Question 6:

Ans. The correct answer is 40. Subtracting 5 from both sides of the given equation
yields 4 160 x = . Dividing both sides of this equation by 4 yields x = 40. Therefore, the solution to the given equation is 40.

Question 7:

Ans. The correct answer is 7. It’s given that the x-intercept of the graph shown is ( ) x,0 . The graph passes through the point ( ) 7,0 . Therefore, the value of x is 7.

Question 8:

Ans. Choice B is correct. The y-intercept of the graph of a function in the xy-plane is
the point on the graph where x =0. It′s given that ( ) 110 fx x = -2. Substituting 0

for x in this equation yields ( ) ( ) 110 f 0 02 = – , or f( ) 0 2 =- . Since it′s given that
y fx = ( ), it follows that y =-2 when x =0. Therefore, the y-intercept of the
graph of y fx = ( ) in the xy-plane is ( ) 0, 2 – .

Question 9:

Ans. Choice D is correct. If the graph of y gx = ( ) is the result of shifting the graph
of y fx = ( ) down k units in the xy-plane, the function g can be defined by an
equation of the form gx fx k ( ) ( ) = – . It’s given that ( ) 3 fx x =7 and the graph of
y gx = ( ) is the result of shifting the graph of y fx = ( ) down 2 units. Substituting

 Question 10:

Ans. Choice A is correct. The solution to a system of equations is the ordered pair
( ) x y, that satisfies all equations in the system. It’s given by the first equation in
the system that x + =7 10. Substituting 10 for x +7 into the second equation
yields 2 10 = y, or y =100. The x-coordinate of the solution to the system
of equations can be found by subtracting 7 from both sides of the equation
x + =7 10, which yields x =3. Therefore, the ordered pair ( ) 3, 100 is a solution to
the given system of equations.

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Instructor Insight:
We see this type of question frequently in harder second modules. Students usually miss it by rushing or choosing an answer that “sounds right” instead of checking the text.

Why These SAT Questions Feel So Difficult

From an instructor’s point of view, the hardest SAT questions don’t rely on advanced math or obscure vocabulary. They’re difficult because they test how well students think under pressure.

On the Digital SAT, especially in the second module, questions are designed to look familiar at first glance. Students often recognize the topic and assume they know the next step – and that’s where mistakes happen. Rushing the setup, skipping a key detail, or trusting an answer that “feels right” can cost valuable points.

What we consistently see is this: students who slow down slightly, re-read what the question is actually asking, and eliminate answer choices deliberately perform far better than students who try to move quickly through hard questions.

Practicing questions like the ones above – and reviewing why tempting wrong answers fail –  is one of the fastest ways to improve accuracy and raise your SAT score.

SAT English Reading & Writing Toughest Practice Questions With Answers

Q1. For painter Jacob Lawrence, being _______ was an important part of the artistic process. Because he paid close attention to all the details of his Harlem
neighborhood, Lawrence’s artwork captured nuances in the beauty and vitality of the Black experience during the Harlem Renaissance and the Great Migration.

Which choice completes the text with the most logical and precise word or phrase?
A) skeptical
B) observant
C) critical
D) confident

Ans. Choice B is the best answer because it most logically completes the text’s
discussion of Jacob Lawrence’s artistic process. In this context, “observant”
means watchful and perceptive. The text emphasizes that the “close attention”
Lawrence paid to “all the details” of his neighborhood allowed him to reflect subtle
elements of “the beauty and vitality of the Black experience” in his artwork. This
context indicates that being observant of his surroundings was an important part
of Lawrence’s work as an artist.

Q2. Mônica Lopes-Ferreira and others at Brazil’s Butantan Institute are studying the freshwater stingray species Potamotrygon rex to identify whether biological characteristics such as the rays’ age and sex have _______ effect on the toxicity of their venom—that is, to see if differences in these traits are associated with considerable variations in venom potency.

Which choice completes the text with the most logical and precise word or phrase?
A) a disconcerting
B) an acceptable
C) an imperceptible
D) a substantial

Ans. Choice D is the best answer because it most logically completes the text’s
discussion of the research that Lopes-Ferreira and her colleagues are conducting
on the stingray species Potamotrygon rex. As used in this context, “a substantial”
effect means an effect that is sizable or noteworthy. The text indicates that the
researchers are seeking to identify whether there are “considerable variations”
in the potency of stingray venom that are associated with variation in the
stingrays’ age and sex. This context suggests that the researchers want to find
out whether stingray age and sex have a substantial effect on venom toxicity.

Q3. Researchers have struggled to pinpoint specific causes for hiccups, which happen when a person’s diaphragm contracts _______ . However, neuroscientist Kimberley Whitehead has found that these uncontrollable contractions may play an important role in helping infants regulate their breathing. Which choice completes the text with the most logical and precise word or phrase?
A) involuntarily
B) beneficially
C) strenuously
D) smoothly

Ans. Choice A is the best answer because it most logically completes the text’s
discussion of diaphragm contractions and hiccups. In this context, “involuntarily”
means done without any control, or by reflex. The text explains that when a
person’s diaphragm repeatedly contracts and results in hiccups (which may be
beneficial for infants), those muscle contractions are “uncontrollable.” This context
indicates that the diaphragm contractions occur without the person’s control.

Q4. Critics have asserted that fine art and fashion rarely _______ in a world where artists create timeless works for exhibition and designers periodically
produce new styles for the public to buy. Luiseño/ Shoshone-Bannock beadwork artist and designer Jamie Okuma challenges this view: her work can be
seen in the Metropolitan Museum of Art and purchased through her online boutique.

Which choice completes the text with the most logical and precise word or phrase?
A) prevail
B) succumb
C) diverge
D) intersect

Ans. Choice D is the best answer because it most logically completes the text’s
discussion about the relationship between fine art and fashion. As used in this
context, “intersect” means to connect or overlap. The text indicates that Jamie
Okuma challenges the position held by critics because her work can be seen
at an art museum and can be bought by the public from her online boutique.
The text also presents the critics’ view as being influenced by a perception that
fine artists create works that are “timeless” and meant for exhibition, whereas
fashion designers periodically produce new styles that are meant for purchase.
This context suggests that the critics believe that fine art and fashion tend not to
overlap—in other words, that they rarely intersect.

Q5. Scholarly discussions of gender in Shakespeare’s comedies often celebrate the rebellion of the playwright’s characters against the rigid expectations
by Elizabethan society. Most of the comedies end in marriage, with characters returning to their socially dictated gender roles after previously defying them, but there are some notable exceptions. Which choice completes the text with the most logical and precise word or phrase?
A) interjected
B) committed
C) illustrated
D) prescribed

Ans. Choice D is the best answer because it most logically completes the text’s
discussion of gender roles in Shakespeare’s comedies. As used in this context,
“prescribed” would mean laid down as rules. The text indicates that the characters
in the comedies often defy gender roles that are “socially dictated” (even if most
characters do return to those roles eventually) and that scholars have been
very interested in these acts of defiance. This context indicates that what the
characters are rebelling against are standards of behavior prescribed by the
society of the time.

Q6. In studying the use of external stimuli to reduce the itching sensation caused by an allergic histamine response, Louise Ward and colleagues found that
while harmless applications of vibration or warming can provide a temporary distraction, such stimuli actually offer less relief than a stimulus that seems less benign, like a mild electric shock. Which choice completes the text with the most
logical and precise word or phrase?
A) deceptive
B) innocuous
C) novel
D) impractical

Ans. Choice B is the best answer because it most logically completes the text’s
discussion of Ward and colleagues’ findings. As used in this context, “innocuous”
means mild or unharmful. The text describes the vibration and warming that Ward
and colleagues used to alleviate itching as “harmless applications” and goes on to
contrast these applications with another stimulus that actually offers more relief
even though it seems to be stronger and “less benign.” This context conveys the
idea that vibration and warming were innocuous stimuli.

Q7. The province of Xoconochco was situated on the Pacific coast, hundreds of kilometers southeast of Tenochtitlan, the capital of the Aztec Empire.
Because Xoconochco’s location within the empire was so _______ , cacao and other trade goods produced there could reach the capital only after a
long overland journey. Which choice completes the text with the most
logical and precise word or phrase?
A) unobtrusive
B) concealed
C) approximate
D) peripheral

Choice D is the best answer because it most logically completes the text’s
discussion of the location of the province of Xoconochco within the Aztec Empire.
As used in this context, “peripheral” means situated toward the outer bounds
rather than the center. The text indicates that Xoconochco was located on a
coast, hundreds of kilometers away from the capital of the Aztec Empire. The
text also states that trade between the province and the capital required “a long
overland journey.” This context suggests that Xoconochco was situated toward
an edge of the empire’s territory rather than near its center.

Q8. The following text is from Charlotte Brontë’s 1847 novel Jane Eyre. Jane works as a governess at Thornfield Hall. I went on with my day’s business tranquilly; but ever and anon vague suggestions kept wandering across my brain of reasons why I should quit Thornfield; and I kept involuntarily framing advertisements and pondering conjectures about new situations: these thoughts I did not think to check; they might germinate and bear fruit if they could.
Which choice best states the main purpose of the text?
A) To convey a contrast between Jane’s outward
calmness and internal restlessness
B) To emphasize Jane’s loyalty to the people she
works for at Thornfield Hall
C) To demonstrate that Jane finds her situation
both challenging and deeply fulfilling
D) To describe Jane’s determination to secure
employment outside of Thornfield Hall

Ans. Choice A is the best answer because it most accurately describes the main
purpose of the text, which is to show that while Jane calmly goes about her daily
tasks, she is experiencing internal agitation about possibly seeking a new job.
At the start of the text, Jane says, “I went on with my day’s business tranquilly,”
indicating that she is outwardly calm. This outward calmness is then contrasted
with her intense internal restlessness, as Jane says that thoughts of leaving her job
keep running through her mind, that she is “involuntarily framing advertisements”
(meaning that she can’t stop herself from thinking up potential listings for jobs),
and that she often wonders what new “situations” (or jobs) would be like.

Q9. Text 1
Most animals can regenerate some parts of their bodies, such as skin. But when a three-banded panther worm is cut into three pieces, each piece grows into a new worm. Researchers are investigating this feat partly to learn more about
humans’ comparatively limited abilities to regenerate, and they’re making exciting progress. An especially promising discovery is that both humans
and panther worms have a gene for early growth response (EGR) linked to regeneration.

Text 2
When Mansi Srivastava and her team reported that panther worms, like humans, possess a gene for EGR, it caused excitement. However, as the team pointed
out, the gene likely functions very differently in humans than it does in panther worms. Srivastava has likened EGR to a switch that activates other genes involved in regeneration in panther worms, but how this switch operates in humans remains unclear.

Based on the texts, what would the author of Text 2 most likely say about Text 1’s characterization of the discovery involving EGR?
A) It is reasonable given that Srivastava and her
team have identified how EGR functions in both
humans and panther worms.
B) It is overly optimistic given additional
observations from Srivastava and her team.
C) It is unexpected given that Srivastava and her
team’s findings were generally met with
enthusiasm.
D) It is unfairly dismissive given the progress that
Srivastava and her team have reported.

Ans. Choice B is the best answer because it reflects how the author of Text 2
would most likely respond to Text 1 based on the information provided.

Text 1

discusses the discovery of a regeneration-linked gene, EGR, in both three-
banded panther worms (which are capable of full regeneration) and humans

(whohave relatively limited regeneration abilities). Text 1 characterizes this discovery
as “especially promising” and a sign of “exciting progress” in understanding human regeneration. The author of Text 2, on the other hand, focuses on the fact that the team that reported the EGR finding pointed out that while EGR’s function in humans isn’t yet known, it’s likely very different from its function in panther worms. Therefore, the author of Text 2 would most likely say that Text 1’s enthusiasm about the EGR discovery is overly optimistic given Srivastava’s team’s observations about EGR in humans.

Q10. The following text is adapted from William Shakespeare’s 1609 poem “Sonnet 27.” The poem is addressed to a close friend as if he were physically
present. Weary with toil, I [hurry] to my bed, The dear repose for limbs with travel tired; But then begins a journey in my head To work my mind, when body’s work’s expired: For then my thoughts—from far where I abide— [Begin] a zealous pilgrimage to thee, And keep my drooping eyelids open wide, What is the main idea of the text?
A) The speaker is asleep and dreaming about
traveling to see the friend.
B) The speaker is planning an upcoming trip to the
friend’s house.
C) The speaker is too fatigued to continue a
discussion with the friend.
D) The speaker is thinking about the friend instead
of immediately falling asleep.

Ans. Choice D is the best answer because it most accurately states the main idea of
the text. The speaker describes the experience of being “weary” and “tired” and
going to bed to seek “dear repose” (that is, sleep), but instead of sleeping, the
speaker is kept awake (“keep my drooping eyelids open wide”) by thoughts of a
friend (“my thoughts… [Begin] a zealous pilgrimage to thee”).

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Testprepkart Student Experience: Why These Questions Feel Hard

One of our Indian NRI students from California came to us after scoring 670 in SAT Math on a Digital SAT practice test. When we reviewed his performance, the issue wasn’t weak concepts-he understood linear equations, ratios, and systems well.

The problem was how he approached harder questions under time pressure.

On questions like the ones above, he often rushed through the setup, assumed he recognized the pattern, and chose an answer that looked right instead of fully checking the logic. In the adaptive second module, that habit cost him multiple points.

Once we slowed him down and had him apply a consistent process-rewriting the equation, identifying what the question was actually asking, and eliminating choices one by one-his accuracy on hard questions improved noticeably. Within three weeks, his Math score moved into the 720–740 range.

We see this pattern repeatedly with U.S. students preparing for the Digital SAT. These questions aren’t hard because they use advanced math.They are difficult because, particularly in the more difficult modules, they put patience, accuracy, and astute judgment to the test. One of the best strategies to raise your score is to practice problems like these and learn why incorrect responses seem fascinating.

Why the SAT Toughest Questions Challenge Students

Don’t just read the solutions.

To get real value from these hardest-level questions:

1. Try each question under timed conditions
   
2. Write down why you chose your answer
   
3. Compare your reasoning to the explanation
   
4. Identify where your decision-making broke down
   

That review process – not the question itself – is what leads to real score improvement.

How Top – Scoring Students Approach Hard SAT Questions

Students who score well on the hardest SAT questions follow a simple but disciplined approach:

• They pause before solving and restate the question in their own words
  
• They predict what a reasonable answer should look like
  
• They eliminate wrong choices instead of hunting for the right one
  
• They check units, signs, and logic before moving on
  

This approach matters even more on the Digital SAT, where adaptive modules reward consistent decision-making over speed.

Digital SAT Math Test Prep Topics Breakdown

Download Math Topice wise Practice Questions

Digital SAT Reading & Writing Topics Breakdown

Download English Topice wise Practice Questions

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