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Free PSAT Practice Test - Math Arithmetic Questions
This section is divided into two parts: one where calculators are allowed, and one where they are not. It covers algebra, advanced math, problem-solving, data analysis, and some geometry and trigonometry.
Q1: If 2x+3=3x−4, what is the value of 7x?
1. 49
2. -49
3. -7
4. 7
A1: The correct answer is 1
To solve for 7x, we first get all the x-terms on one side of the equation:
2x−3x=−4−3.
Simplifying, we get:
−x=−7
By multiplying both sides by -1, we find:
x=7
Now, multiply both sides by 7 to achieve the value of 7x:
7*x = 7*7 → 7x = 49
Q2: What are the solutions to the quadratic equation x2−4x−5=0?
1. 1 and 5
2. -1 and -5
3. 1 and -5
4. -1 and 5
A2: The correct answer is 4.
This equation can be factored as (x−5)(x+1)=0, which gives us the solutions x=5 and x=−1. Therefore, the correct answer is A) -1 and 5
PSAT Math Tip – Practice to simplify:
In the PSAT/NMSQT math section, you will encounter challenging questions, some of which may be time-consuming and require multiple steps of calculations to complete. Since time is of the essence – you must be able to identify questions that can be simply answered – such as question 2. You could perform a complete quadratic formula which will take much longer. However, by thorough practice, you can identify easily that there's a simpler method – such as factoring the equation.
Which of the following choices represents the same relationship as demonstrated in this table?
1. 
2. 
3. Y is equivalent to the difference of the value X and a constant C, where C equals 1.
4. Y = X *4
A3: The correct answer is 2.
As can be seen from the table, the relationship is Y = X + 2: 2 = 0 + 2 7 = 5 + 2 12 = 10 + 2 17 = 15 + 2 22 = 20 + 2 27 = 25 + 2
Answer (A) is incorrect because in the table when X = 0, Y = 2 and in the graph when X = 0, Y = 1. Answer (C) is incorrect because if Y = X – 1, then when X = 0, Y = -1 instead of 2. Answer (D) is incorrect because if Y = X * 4, then when X = 0, Y = 0 instead of 2. Only answer (B) represents the same relationship: -8 = -10 + 2 -6 = -8 + 2
-4 = -6 + 2 -2 = -4 + 2 0 = -2 + 2 2 = 0 + 2 Therefore, the correct answer is (2).
Q4: In circle O, the length of chord AB is 24 cm, and the radius of the circle is 13 cm. What is the distance from the center of the circle, O, to the midpoint of chord AB?
1. 5 cm
2. 10 cm
3. 12 cm
4. 15 cm
A4: The correct answer is 1.
The radius to the midpoint of a chord bisects the chord at a right angle. Using the Pythagorean theorem in the resulting right triangle, we have:
132=122+d2
Where d is the distance from O to the midpoint of AB. Solving for d, we find:
Q5: Based on a random survey, a study found that the average number of hours spent on social media per day by teenagers is 3 hours, with a margin of error of 0.5 hours. Which of the following conclusions is most accurate?
1. The actual average number of hours spent on social media by all teenagers is likely between 2.5 and 3.5 hours.
2. It is impossible for any teenager to spend less than 2.5 or more than 3.5 hours on social media per day.
3. Every teenager spends between 2.5 and 3.5 hours on social media per day.
4. The average number of hours spent on social media by all teenagers is exactly 3 hours.
A5: The correct answer is 1.
The margin of error in a statistical analysis provides a range around the estimated mean within which the true mean is likely to fall. It does not guarantee that all individual observations (in this case, the hours each teenager spends on social media) will fall within this range, nor does it assert that the estimated mean is the absolute true mean for the entire population.
Option A is the most appropriate conclusion because it correctly interprets the margin of error: the actual average is likely within the range of 2.5 to 3.5 hours. This option understands the concept of a confidence interval in statistical terms, which is essentially what the estimated mean plus or minus the margin of error represents.
Option B incorrectly interprets the margin of error as a hard limit beyond which no values can exist, which is not true. The margin of error only pertains to the estimate of the mean, not to the distribution of individual values.
Option C is incorrect because it assumes that the margin of error applies to each measurement in the population, which is not the case. The margin of error applies to the estimate of the mean, not to individual data points.
Option D is incorrect because it misunderstands the nature of an estimate and the margin of error in statistical analysis. The presence of a margin of error means there is uncertainty around the estimate, so stating the mean is exactly 3 hours ignores this uncertainty.
The PSAT/NMSQT Math section includes 44 questions to be taken in 70 minutes – which are divided into two modules (22 questions in 35 minutes). Since there are less than two minutes for each question – make sure to work smart and efficiently.
Use the tools you have (built-in calculator, formulas sheet) and make sure to not dwell too much on difficult questions – start by recognizing easy questions you can easily answer and save more time for difficult and long questions. That way – you can guarantee more "easy" questions will be answered without time pressure and have more time to answer each "difficult" question.
Understanding the PSAT/NMSQT Score:
Each section of the PSAT/NMSQT has a score ranging between 160-760. The overall score is the addition of both sections – therefore it ranges from as low as 320, to a maximum of 1520. Remember that the higher your score – the higher percentile you are among other students, and you are better prepared for your SATs.
Regarding the score for achieving the National Merit Scholarship – the score is a bit different, as it gives larger emphasis to the Reading and Writing section. Double your Reading and Writing
section score, add the Math section score, and divide the sum by 10. The score can be as high as 228 – and we recommend aiming for at least 210 to be eligible for the scholarship. Therefore – make sure to practice thoroughly, and put your extra time for additional Reading and Writing practice.
We will now continue to the PSAT/NMSQT Reading and Writing section – where your verbal, grammar, and comprehension skills will be measured.
PSAT Practice Test - Reading & Writing Questions
This section includes passages from literature, historical documents, social sciences, and sciences.
In terms of reading, students are required to demonstrate their ability to interpret, synthesize, and use evidence from the passages to answer questions.
In terms of writing, the questions measure grammar, vocabulary in context, and editing skills.
PSAT/NMSQT Reading Questions
Q1: In the heart of the Amazon rainforest, there exists a species of frog whose skin contains a substance with the remarkable ability to repel water. This substance not only keeps the frog dry during the heaviest of rainstorms but also has the potential to revolutionize the way we design rainwear.
What is the main purpose of the passage?
1. To argue for the protection of the Amazon rainforest
2. To describe a unique characteristic of a species of frog
3. To criticize current rainwear designs
4. To promote a new type of rainwear
A1: The correct answer is 2
The passage focuses on introducing a species of frog in the Amazon and a unique feature of its skin. While the text hints at the potential application of this feature in rainwear design, the primary focus is on the frog's characteristics, making option B the correct answer.
While Option A may be implied through the importance of the frog species, it is not mentioned in the text.
Option C is incorrect since there is no regard for the current rainwear designs – it only implies possible methods to improve them.
Option D is incorrect, as there is no "new type of rainwear" – but a potential to change for the better the rainwear design.
Q2: On a cold, starry night, the young astronomer gazed through her telescope, her breath forming clouds in the chilly air. Each star seemed to tell a story, spanning billions of years, and yet appearing as mere flickers in the night sky. The astronomer pondered the vastness of the universe and her place within it. It was a humbling experience, one that filled her with a profound sense of wonder and an insatiable curiosity. She realized that each discovery was not an endpoint but a gateway to more questions, more mysteries to unravel.
In the passage, the phrase "each discovery was not an endpoint but a gateway" suggests that the astronomer views discoveries as…
1. final answers
2. insignificant findings
3. starting points for further inquiry
4. obstacles to understanding
A2: The correct answer is 3.
The phrase implies that discoveries lead to more questions and further exploration, making option C correct.
Option A is incorrect because the passage explicitly states discoveries are not final answers.
Option B is incorrect as it contradicts the passage's message that discoveries are valuable and lead to more exploration.
Option D is incorrect because the passage portrays discoveries positively, as opportunities for more understanding, not as obstacles.
PSAT Practice Test Reading Tip – Read the Question First:
While the texts may not be too long in the PSAT/NMSQT, it is still highly recommended to begin each question with the task required. If the task requires an overall description of the text, continue to read the text, and find a recurring theme, then continue to the answer choices. If the task refers to specific parts of the text – emphasize that part, and you can not only save valuable time but also be more accurate and keener to the answer.
Q3: The following text is from F. Scott Fitzgerald's 1925 novel The Great Gatsby. In this excerpt, Nick Carraway describes his first impression of Jay Gatsby's extravagant parties at his mansion on Long Island. "There was music from my neighbor's house through the summer nights. In his blue gardens men and girls came and went like moths among the whisperings and the champagne and the stars.
At high tide in the afternoon I watched his guests diving from the tower of his raft, or taking the sun on the hot sand of his beach while his two motor-boats slit the waters of the Sound, drawing aquaplanes over cataracts of foam.
On weekends his Rolls-Royce became an omnibus, bearing parties to and from the city between nine in the morning and long past midnight, while his station wagon scampered like a brisk yellow bug to meet all trains. And on Mondays eight servants, including an extra gardener, toiled all day with mops and scrubbing-brushes and hammers and garden-shears, repairing the ravages of the night before."
Which choice best states the main purpose of the text?
1. To detail the process of cleaning up after Gatsby's parties
2. To highlight the opulence and extravagance of Gatsby's lifestyle
3. To introduce the character of Jay Gatsby through the eyes of Nick Carraway
4. To emphasize the non-stop, frenetic pace of life at Gatsby's mansion
A3: The correct answer is 2.
This passage vividly describes the lavishness of Gatsby's parties, from the music and gardens to the luxury cars and boats, culminating in the need for extensive cleanup efforts.
While option 1 touches on an aspect of the text, it misses the broader depiction of luxury and excess that is the focal point.
Option 3, although true in the context of the novel, is not the main purpose of this specific passage, which focuses more on setting a scene of luxury than on character introduction.
Option 4 captures an element of the passage but does not fully encompass the emphasis on Gatsby's wealth and the extravagant lifestyle that the detailed descriptions in the text convey, making option B the most accurate.
Q4: The following text is adapted from Charles Dickens' 1859 novel A Tale of Two Cities. It opens with the famous lines that paint a picture of the contrasting conditions of the time:
"It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we had nothing before us, we were all going direct to Heaven, we were all going direct the other way."
As used in the text, what does the phrase "the worst of times" most nearly mean?
1. The most challenging
2. The least memorable
3. The most prosperous
4. The least complicated
A4: The correct answer is 1.
The phrase "the worst of times" contrasts with "the best of times," indicating a period of significant difficulty or adversity. This contrast is used to highlight the extreme dichotomies of the era Dickens describes.
Option 2, "The least memorable," is incorrect because the phrase implies difficulty rather than a lack of memorability.
Option 3, "The most prosperous," directly contradicts the intended meaning of hardship.
Option 4, "The least complicated," is also incorrect because "worst" conveys negative connotations of difficulty or hardship, not simplicity or lack of complexity. Therefore, the phrase "the worst of times" most closely aligns with the meaning of "the most challenging" times, making option 1 the correct answer.
Q5: Text 1 Economic analyst Laura Wells conducted a comprehensive study on the impact of technology on workforce productivity. Her findings indicate a significant increase in productivity levels in
sectors that have integrated advanced technology into daily operations. However, Wells also notes a paradoxical increase in job dissatisfaction among employees in these sectors, attributing this trend to the reduction in human interaction and the increased pace of work.
Text 2 Sociologist Dr. Henry Tam argues that while technology can enhance operational efficiency, it should not be viewed solely as a tool for maximizing productivity. He emphasizes the importance of maintaining a balance between technological advancement and the preservation of workplace culture and employee well-being. Tam suggests that the definition of productivity should encompass not only output but also job satisfaction and employee engagement.
Based on the texts, how would Dr. Tam (Text 2) most likely respond to the findings presented by Wells (Text 1) regarding the increase in job dissatisfaction?
1. By agreeing that technology improves productivity but at the cost of employee satisfaction
2. By disputing the claim that technology is the sole reason for increased productivity levels
3. By cautioning against over-reliance on technology without considering its impact on employee morale
4. By asserting that the reported job dissatisfaction is unrelated to technological integration
A5: The correct answer is 3.
Answer 3: Dr. Tam's emphasis on balancing technological advancement with the preservation of workplace culture and employee well-being suggests he would caution against over-reliance on technology, especially if it negatively impacts employee morale. This aligns with Wells' observation of increased job dissatisfaction due to reduced human interaction and a faster pace of work.
Answer 1: While Dr. Tam acknowledges the benefits of technology, his focus is on balancing these benefits with employee well-being, not just accepting the trade-off as inevitable.
Answer 2: Dr. Tam does not dispute the benefits of technology on productivity; his concern is more about the holistic definition of productivity that includes employee well-being.
Answer 4: Given Dr. Tam's concern for employee well-being alongside technological advancement, it's unlikely he would dismiss the link between job dissatisfaction and technological integration as presented by Wells.
Before summarizing both subjects as a conjoined section, let's proceed to the PSAT/NMSQT Writing questions.
PSAT/NMSQT Writing Questions
Q1: During a renovation of an ancient library, a manuscript believed to be from the 16th century was discovered behind a wall. The manuscript contained detailed illustrations of celestial bodies and contained annotations in a script like that used by astronomers of the period. In 2022, a group of researchers including historian Dr. Elena Rodriguez, chemist Dr. Li Wei, and paleographer Dr. Marcus Antonius conducted a thorough examination of the manuscript's paper and ink. Their analysis revealed the presence of synthetic ultramarine blue, a pigment not synthesized until the early 18th century. Consequently, the team inferred that _______
Which choice most logically completes the text?
1. the dating of pigments cannot accurately determine the age of a manuscript.
2. the manuscript must have been a 16th-century astronomer's prediction of future pigments.
3. the use of synthetic pigments in the 16th century was more common than previously thought.
4. the illustrations in the manuscript could not have been original 16th-century work.
A1: The correct answer is 4.
Answer 4: The presence of synthetic ultramarine blue, a pigment not available until the early 18th century, indicates that the illustrations could not have been created in the 16th century. This choice logically follows from the evidence provided by the researchers' analysis.
Answer 1: While the dating of pigments is a complex process, the specific finding of a pigment known not to exist until the 18th century provides a clear indication that the manuscript's illustrations were not original to the 16th century. This option undermines the direct evidence provided by the pigment analysis.
Answer 2: Suggesting that a 16th-century astronomer could predict the development of synthetic pigments is highly improbable and not supported by historical evidence or the context of the analysis.
Answer 3: The discovery of synthetic ultramarine blue in the manuscript contradicts the notion that such pigments were used in the 16th century, making this option inconsistent with the findings of the researchers.
Q2: Dr. Elsie Widdowson, a renowned British dietitian and nutritionist, played a pivotal role in the development of wartime rationing during the Second World War. Her extensive research on food composition led to significant advancements in nutritional science, _______ the fortification of bread with calcium and the enrichment of other staple foods to combat dietary deficiencies.
Which choice completes the text so that it conforms to the conventions of Standard English?
1. involves
2. involved
3. involving
4. will involve
A2: The correct answer is 3.
Answer 3: The word "involving" is the correct choice because it correctly modifies the clause that precedes it, indicating that the advancements included specific actions like the fortification of bread. The use of "involving" creates a participial phrase that provides additional information about Dr. Widdowson's research and its outcomes without creating a new independent clause, maintaining grammatical consistency and coherence within the sentence.
Answer 1: "Involves" is present tense, which does not fit the past context of Dr. Widdowson's contributions.
Answer 2: "Involved" would imply a new, independent clause, but without a conjunction to connect it to the previous clause, it would create a comma splice error.
Answer 4: "Will involve" suggests future action, which is incorrect because the sentence discusses historical events and contributions.
PSAT Practice Test Writing Tip – Verbally Speak the Sentence:
This tip may seem weird but trust us- it is one of the most useful tips for writing questions – especially ones that require logical completion. Try to complete the sentence using any of the options provided – and while you may wind up with more than one possible answer, in most cases, you will easily disqualify at least one answer, increasing your chances of answering correctly – especially since there's no negative scoring.
Q3: Renowned architect Zaha Hadid was celebrated for her futuristic building designs. In creating the aquatic center for the 2012 Olympics, _______ Hadid employed a wave-like roof structure that mimicked the fluidity of water. This design choice not only paid homage to the aquatic events but also pushed the boundaries of architectural design.
Which choice completes the text so that it conforms to the conventions of Standard English?
1. (2012) and
2. (2012),
3. (2012), and
4. (2012)
A3: The correct answer is 2.
Answer 2: Option B) (2012), is the correct choice because it properly uses a comma after the year to separate the introductory clause from the main clause. This punctuation ensures clarity and adheres to the conventions of Standard English by correctly separating the parenthetical element (the year) from the rest of the sentence.
Answer 1: Option A) (2012) and is incorrect because the conjunction "and" is unnecessary. The sentence structure does not require a conjunction to connect two independent clauses, making "and" redundant and grammatically incorrect in this context.
Answer 3: Option C) (2012), and is incorrect because the use of the comma and the conjunction "and" together is redundant when there's no independent clause following "and." The sentence flows directly into a dependent clause, making the conjunction unnecessary.
Answer 4: Option D) (2012) is incorrect because it ends the parenthetical statement without a comma, which can lead to confusion and disrupt the flow of the sentence. A comma is needed after the year to clearly separate the introductory element from the main clause.
Q4: In the realm of classical music, Ludwig van Beethoven is often celebrated for his symphonic masterpieces. His Ninth Symphony is particularly renowned for its complexity and emotional depth. _______ Beethoven composed this symphony when he was completely deaf, a testament to his extraordinary talent and perseverance.
Which choice completes the text with the most logical transition?
1. Similarly,
2. Therefore,
3. Interestingly,
4. Consequently,
A4: The correct answer is 3.
Answer 3: "Interestingly," is the correct choice because it introduces a surprising or noteworthy piece of information that adds depth to the previous statement. It signals to the reader that a fascinating fact related to Beethoven's achievements is about to be shared, which in this case is his ability to compose despite his deafness.
Answer 1: "Similarly," is incorrect because it implies that the following information will parallel the preceding statement in a way that shows likeness. However, Beethoven's deafness does not resemble his achievements; instead, it contrasts them, making this transition illogical.
Answer 2: "Therefore," suggests a cause-and-effect relationship between the two clauses. In this context, Beethoven's renown for his symphonic masterpieces does not directly cause him to compose while deaf. The use of "therefore" would mislead the reader into expecting a consequential relationship that does not exist.
Answer 4: "Consequently," like "therefore," implies a direct result or outcome, which is not the relationship between Beethoven's fame and his composing under the condition of deafness. The fact that he composed while deaf is not a consequence of his renown but an independent, remarkable fact.
Q5: The advent of electric cars has led to a significant reduction in greenhouse gas emissions. This shift towards sustainable transportation, _______ adopting renewable energy sources for charging stations, can further decrease the environmental impact of vehicles.
Which choice completes the text so that it conforms to the conventions of Standard English?
1. coupled with
2. and coupled to
3. coupling with
4. couples with
A5: The correct answer is 1.
Answer 1: "coupled with" is the correct choice because it is a standard phrase used to indicate that two factors are being considered together or combined to achieve a particular result. In this context, it suggests that both the advent of electric cars and the adoption of renewable energy sources work together to reduce environmental impact.
Answer 2: "and coupled to" is incorrect because "coupled to" is not the standard phrasing used in this context, and the addition of "and" is unnecessary and makes the sentence awkward. The phrase "coupled with" is sufficient to convey the idea of two elements being linked.
Answer 3: "coupling with" is grammatically incorrect in this context because "coupling" is a gerund, and using "with" after a gerund in this manner is not standard usage. The phrase "coupled with" is the correct idiomatic expression.
Answer 4: "couples with" is incorrect because it changes the verb tense and suggests an ongoing action, which does not fit the context of a completed action implied by the rest of the sentence. "Coupled with" is the correct phrase to indicate that two completed actions or states are being considered together.
The PSAT/NMSQT Reading and Writing section includes 54 questions to be taken in 64 minutes – which are divided into two modules (27 questions in 32 minutes). With just over a minute for each question – you must make your time count. Remember that the module is built so that most of the "easy" questions are at the beginning, and the Reading and Writing parts are mostly separated. Therefore – you should practice a personalized working method to be as efficient as possible and get a high score.
How Do I Practice for the PSAT?
To effectively prepare and practice for the PSAT, consider the following steps:
· Familiarize yourself with the Test Format: Know the sections (Math, Reading, Writing) and types of questions.
· Use Official PSAT Practice Tests: The College Board offers sample questions and full PSAT practice tests.
· Review Key Concepts: Focus on algebra, data analysis for Math; grammar rules for Writing; and critical reading skills.
· Take Timed Tests: When taking PSAT Practice tests, you simulate actual test conditions, which in turn improves your speed and accuracy.
· Analyze Mistakes: Review wrong answers to understand and learn from them
Is The PSAT Very Hard?
The PSAT, designed as a precursor and practice for the SAT, is generally considered to be slightly easier and less comprehensive. It assesses similar skills in reading, writing, and math but does not delve as deeply into content, making the SAT more challenging.
Preparation levels and familiarity with the test format significantly influence perceptions of difficulty for both tests. The PSAT also serves as a qualifier for the National Merit Scholarships, offering an added incentive for high performance.
What is A Good PSAT Score?
A standout PSAT score is typically in the top 10% of test-takers, around the 90th percentile, equating to scores near 1210-1220 for 11th graders.
The PSAT's scoring system ranges from 320 to 1520, divided between the Math and Evidence-Based Reading and Writing sections, with each section scored from 160 to 760. High scores not only indicate college readiness but can also lead to scholarship opportunities.
