Free CLT Practice Test 2026 (Including PDF With Answers)

Welcome to your 2026 Free CLT Practice Test!

On this page, you’ll find 30 sample questions that mirror the actual test’s format and difficulty level and cover the Math, Verbal Reasoning, and Grammar/Writing sections.

The questions were created by math and English teachers and include thorough explanations to help you understand the reasoning behind the answers and learn from your mistakes.

You can choose from three versions of this free practice test, plus a complete practice package with 430 questions:

Timed interactive quiz with a score report
Printable PDF for offline use
Self-paced practice questions further down this page

Ready? Let’s get started!

What to Expect on the Classic Learning Test (CLT)? A Quick Overview Before You Start

The CLT is an alternative to the SAT/ACT, measuring college-readiness through three concise multiple-choice sections focused on reading, writing, and math.

Tests are taken online, from home or at school, in just two hours, with no unnecessary extras. Each section balances analytical thinking and subject knowledge, all without penalties for guessing.

Remote Takers:

The entire process is remotely proctored with video and screen recording, optional 1‑minute break transitions, and the CLT scores are released within about 3 weeks.

Section	Time Limit	No. of Questions	Details
Verbal Reasoning	40 min	40	Comprehension & critical reading
Grammar & Writing	35 min	40	Grammar rules, style, and clarity
Quantitative Reasoning	45 min	40	Algebra, geometry, logic
Total Testing Time	120 min	120	Plus ~20 min pre-test setup
Optional Essay	30 min	–	Unscored – writing sample only

Free CLT Practice Test (Self-Paced) & PDF

The following sample test includes 30 practice questions, taken from our complete practice package, that cover Verbal Reasoning, Grammar/Writing, and Quantitative Reasoning, with 10 questions for each section.

Each question is followed by a detailed and clear explanation.

To simulate the real exam conditions, please avoid using a calculator and outside tools such as Google and ChatGPT.

(On the actual Classic Learning Test, it’s forbidden to use those.)

If you’d like to print these questions, download our free CLT PDF practice test here.

Good Luck!

CLT Verbal Reasoning Practice Questions

Read the passage and carefully answer the corresponding questions.

The final two questions for each passage are analogy-based. Use your understanding of the passage and the relationships demonstrated within it to arrive at your answers.
Answers are based on the evidence given in a passage. No outside knowledge is required.

Science

This passage is adapted from The Rising Prevalence of Hypertension in Young Adults: Causes, Consequences, and Prevention Strategies by Dr. Steven M. Klein, published in 2023.

¶1 – Over the past two decades, the incidence of hypertension in young adults has increased dramatically, puzzling medical professionals. Dr. Hannah Morgan, a cardiologist at the University of Washington, explains that high blood pressure, once thought to be a condition primarily affecting older adults, is now being increasingly diagnosed in individuals as young as 18. “Hypertension is no longer a disease of middle age,” Morgan states. “Young people, especially those in their 20s and 30s, are now at higher risk.”

¶2 – Hypertension, often called the “silent killer,” is a condition in which blood pressure is consistently too high. The force of blood against the walls of the arteries is elevated, which can cause damage over time. The condition is typically asymptomatic, making it difficult to detect in its early stages. While the immediate effects of hypertension may be absent, its long-term consequences are severe, including heart disease, stroke, and kidney failure. Recent data from the National Heart Institute shows that nearly 25% of adults aged 18 to 34 have elevated blood pressure, but only half of them are aware of their condition.

¶3 – The rise of hypertension in younger populations can be traced to several lifestyle factors. Dr. Morgan points to increased stress levels, poor eating habits, and a lack of physical activity as major contributors. Additionally, the growing consumption of processed foods and sugary beverages is making hypertension more common in this age group. Dr. Morgan notes that many young adults are less likely to visit a doctor regularly, particularly once they transition out of pediatric care. “Without routine check-ups, hypertension often goes unnoticed until it’s too late,” she warns.

¶4 – The challenge of early detection is further compounded by the general perception among young people that they are invincible and too young to experience serious health issues. As a result, many young adults do not think about preventive care or managing health conditions like high blood pressure. According to research conducted at the Health Sciences Institute, only 12% of individuals aged 18 to 30 with hypertension follow recommended treatment protocols, and fewer still adhere to lifestyle changes such as regular exercise and reducing salt intake. “Convincing young people to take their health seriously is one of the biggest hurdles we face,” says Dr. Jonathan Stevens, a public health researcher at the institute.

¶5 – Public health efforts are beginning to address this issue through screening programs and awareness campaigns. Many urban areas have started offering free screenings to help young adults identify and manage their blood pressure. However, experts argue that these efforts must be expanded to reach a larger portion of the population. “We need a broader public health approach,” says Dr. Stevens. “More needs to be done to raise awareness and ensure young adults have access to health services.”

¶6 – Dr. Morgan also advocates for policy-level changes to improve access to healthcare, especially for young adults in lower-income communities where healthcare resources are limited. “Hypertension is a public health crisis,” she says. “We need to address the root causes of the problem, such as food deserts and lack of affordable healthcare.”

¶7 – Experts agree that one of the main reasons young people neglect their health is the absence of obvious symptoms. Unlike conditions like chest pain or fatigue, hypertension often presents no immediate warning signs. This makes it difficult for young adults to recognize the importance of managing the condition. Dr. Morgan notes that many young patients don’t feel sick and, therefore, don’t perceive the need for medical intervention. “The psychological barrier is significant,” she explains. “They feel healthy, so they don’t understand the importance of taking medication or making lifestyle changes.”

¶8 – In light of these challenges, public health campaigns are becoming more focused on educating young people about the dangers of hypertension and the importance of early detection. These campaigns include community screenings, online resources, and educational outreach. Researchers believe that increasing awareness and access to care will be critical in reducing the rising prevalence of hypertension in young adults.

¶9 – Ultimately, addressing the issue of hypertension in young adults requires a multifaceted approach. It is not only a medical issue but also a social and policy challenge. According to Dr. Morgan, the healthcare system must evolve to better engage younger populations and provide the necessary tools for prevention and treatment. “This is not just a personal health issue,” she concludes. “It’s a public health crisis that requires immediate action.”

The Rising Prevalence of Hypertension in Young Adults: Causes, Consequences, and Prevention Strategies, IRE Journals, 2023.

Table: Blood Pressure Status and Treatment Among Adults Aged 18–34

Category	Number of Individuals	Percentage of Total Sample
Normal Blood Pressure	1,200	48%
Diagnosed with Hypertension	600	24%
Undiagnosed Hypertension	500	20%
Receiving Treatment	150	6%
Hypertension Under Control	50	2%
No Screening in Past Year	1,000	40%
Total Sample Size	2,500	100%

Question 1

Based on the passage, the author believes one of the main reasons hypertension is untreated in young adults is that they

Answer & Explanation:

Correct Answer: C) Are unaware of the seriousness of the condition.

Explanation:

The passage emphasizes that hypertension often develops without symptoms, making young adults less likely to take it seriously.

Experts like Dr. Morgan and Dr. Stevens point out that when people do not feel unwell, they assume they are healthy, even if their blood pressure is elevated. This misperception leads to a lack of urgency in seeking treatment.

Furthermore, the passage mentions that young adults tend to skip regular health checkups, believing they are too young for chronic conditions. Because of this, they often remain unaware that their condition requires medical attention.

The combination of no symptoms and a low sense of risk creates a serious barrier to treatment.

Question 2

The passage primarily focuses on:

Answer & Explanation:

Correct Answer: A) The causes and consequences of hypertension in young adults, and the strategies to address it.

Explanation:

The central aim of the passage is to examine why hypertension is becoming more common among young adults and what can be done about it. It outlines several contributing factors such as unhealthy diets, chronic stress, and a lack of physical activity.

The passage then explores the consequences of untreated hypertension, including serious diseases like heart failure and kidney damage.

In addition to outlining the problem, the passage presents possible solutions. These include raising awareness, providing free health screenings, and improving access to care.

The balanced focus on both the problem and potential responses make option A the most accurate summary.

Question 3

Based on the passage, a primary reason young adults with hypertension often go undiagnosed is:

Answer & Explanation:

Correct Answer: B) They do not receive regular health screenings after childhood.

Explanation:

The passage clearly states in Paragraph 3 that “many young adults are less likely to visit a doctor regularly, particularly once they transition out of pediatric care,” and Dr. Morgan emphasizes that “without routine check-ups, hypertension often goes unnoticed until it’s too late.” This directly supports answer B as the primary reason for undiagnosed hypertension.

The text explicitly notes in Paragraph 2 that hypertension is “typically asymptomatic,” eliminating option A. While the passage mentions healthcare access issues in Paragraph 6, it does not mention cost of equipment (D) or avoidance due to labeling (C), making B the only fully supported answer.

Question 4

In Paragraph 5, Sentence 5, Dr. Stevens’s statement that “More needs to be done to raise awareness and ensure young adults have access to health services” most nearly suggests:

Answer & Explanation:

Correct Answer: C) Current outreach efforts are inadequate for the scale of the hypertension problem.

Explanation:

Dr. Stevens emphasizes the need for a “broader public health approach,” indicating that while some efforts (like screenings and campaigns) have begun, they are not yet reaching enough young adults to address the growing crisis effectively. His statement that “more needs to be done” implies dissatisfaction with the current scale and impact of these interventions.

Choices A and D misrepresent the urgency of his statement, while B contradicts the premise that awareness is still lacking among young adults.

Question 5

The passage implies that to address the rising prevalence of hypertension in young adults, hospitals should:

Answer & Explanation:

Correct Answer: B) Increase the availability of health screenings and provide education about the risks of hypertension

Explanation:

The passage consistently emphasizes prevention and early detection as key strategies for addressing hypertension in young adults. In Paragraph 5, it specifically mentions public health efforts “through screening programs and awareness campaigns,” with Dr. Stevens advocating for “a broader public health approach” to expand these services. Paragraph 8 further reinforces this by describing how current campaigns focus on “educating young people about the dangers of hypertension and the importance of early detection.”

Hypertension : Silent Killer ::

Answer & Explanation:

Correct Answer: D) Smoking: Lung Cancer

Explanation:

This analogy works because both pairs involve a hidden threat that may not show symptoms until serious damage is done. Hypertension is described as a silent killer in the passage because people often do not realize they have it until they suffer a major health event. Smoking and lung cancer have a similar dynamic.

While other options mention related conditions, they do not carry the same idea of a silent progression toward serious illness. Option D best mirrors the idea of a condition that quietly harms the body over time.

Ready for More Passages?

You’ve just tackled 10 Verbal Reasoning questions, giving you a taste of the CLT!

Whether you found them challenging or straightforward, our full program offers more questions drawn from the official Author Bank, plus clear explanations to help you master this section.

CLT Grammar/Writing Practice Questions

Read the passage and answer the corresponding questions.

Questions will ask you to either correct an error or suggest an improvement in the passage.

If no correction or improvement is necessary, select the option “NO CHANGE.”

Note: Most questions correspond to a numbered portion of the passage. For these questions, answer choices represent alternatives that could be substituted for the numbered portion. Select the best answer choice.

Modern/Influential Thinkers

This passage is adapted from The Fire Next Time and other writings by James Baldwin, a 20th-century American novelist, essayist, and civil rights activist. The version below draws from Baldwin’s reflections on race, identity, and the role of truth in social justice. ©1963 by James Baldwin Estate.

I stand before you as a witness to the truth—not the truth [71] recorded in textbooks, or carved into monuments, but the truth as it is lived and endured. For those forced to the margins of society, this truth is not abstract. It lives in the body, in memory, [72] and in the silence between words. And though it has been denied and distorted, it remains, waiting to be spoken.

I do not speak as one who has conquered hatred or mastered hope. I speak as one who has known fear, humiliation, and rage—and yet [73] refuse to surrender to them. My name, my body, and my voice has all been shaped by a country that has too often confused cruelty with order, and silence with peace. But I have not allowed that confusion [74] to stole my right to speak plainly or love deeply. For hatred, no matter how justified it may seem, can never lead to liberation. It only builds new prisons.

Let me be clear: the struggle for justice in America is not the burden of Black people alone. It is the work of all who wish to live in a world [75] that is governed by truth and sustained with dignity. We cannot afford to believe that what happens to one group has no bearing on the rest. The chains worn by your neighbor may not bind your wrists, but the weight of injustice [76] bend the whole house.

The younger generation must not be lulled into thinking the myths they inherited are harmless. Many of these myths—about superiority, innocence, and destiny—[77] is weapons disguised as ideals. To love this country truly is not to excuse its crimes or romanticize its beginnings. It is to demand that it live up to its promises and reject comfort when others are forced to endure cruelty.

Love, as I understand it, is not ease or sentimentality. [78] It is a call to action, a refusal of allowing lies to thrive where truth must live. It requires that we see one another clearly, not through fear or prejudice, but through the clarity of shared humanity. Love tells the truth—not to shame, but to heal and to build.

In each of us, there is a light that oppression cannot extinguish. There is a will to be free that no law can erase. The voices of the past—those who cried from the holds of slave ships, who marched through fire hoses and mobs—still live in our bones. Their courage [79] are our inheritance. Their unfinished work is our responsibility. Their unfinished work is our responsibility. We must carry it forward—not in bitterness, but with [80] resolve, clarity, and love.

Question 1

[71] Which of the following choices best matches the tone of the passage?

I stand before you as a witness to the truth—not the truth recorded in textbooks, or carved into monuments, but the truth as it is lived and endured.

Answer & Explanation:

Correct Answer: A) NO CHANGE

Explanation:

Option A is the most effective choice because it preserves the formal, solemn tone of the passage.

The phrase “I stand before you as a witness to the truth” evokes a sense of moral authority and personal testimony, and the contrast between textbook or monumental truth and lived experience reinforces the gravity of the speaker’s message. This tone aligns with themes of injustice, endurance, and social marginalization.

Option B introduces a casual and conversational tone (“read in history class,” “fancy plaques,” “gritty truth”), which undermines the seriousness of the statement.

Option C, while closer in intent, uses less elevated diction (“present the facts” and “hurt and happen”) that weakens the rhetorical force.

Option D is entirely too informal and slang-heavy (“spill the tea,” “y’all”), clashing starkly with the sober and dignified tone of the passage. Therefore, A is the most appropriate choice.

Question 2

[72] and in the silence between words.

Answer & Explanation:

Correct Answer: A) NO CHANGE

Explanation:

The sentence presents a list with parallel prepositional phrases: “in the body, in memory, and in the silence between words.” Each item in the list is balanced and grammatically consistent, and no additional punctuation is needed within the final phrase.

The phrase “in the silence between words” is a cohesive unit and should not be broken up by commas, as it expresses a single, continuous idea.

Option B incorrectly inserts a comma after “and,” which unnecessarily separates the conjunction from the final item in the list. Option C introduces a comma after “silence,” disrupting the connection between “silence” and “between words,” which together convey the full meaning.

[78] Which of the following choices best improves the sentence’s structure and punctuation while preserving its intended meaning?

a refusal of allowing lies to thrive where truth must live.

Answer & Explanation:

Correct Answer: D) a refusal to allow lies to thrive where truth must live

Explanation:

Option D offers the most structurally sound and properly punctuated version of the sentence. The phrase “a refusal to allow” is grammatically conventional and free of awkward interruptions or misplaced punctuation.

It maintains parallelism with the earlier clause, “a call to action,” and presents a clear, forceful structure in keeping with the rhetorical tone.

Option A uses “of allowing,” which creates an awkward and non-idiomatic phrase. Option B inserts unnecessary commas that interrupt the sentence’s rhythm and clarity.

Option C misuses the em dash as it introduces an abruptness inappropriate for the tone and needlessly separates the phrase from its main clause. Therefore, Option D best preserves the sentence’s structure and flow while adhering to punctuation norms.

Question 9

[79] are

Answer & Explanation:

Correct Answer: B) is

Explanation:

The noun “courage” is a singular, abstract noun and must take a singular verb. The correct verb in this case is “is,” which properly aligns with the subject in both number and tone.

The phrase “Their courage is our inheritance” presents a timeless, dignified statement appropriate for formal or rhetorical writing. The present tense “is” underscores the idea that this courage continues to be a living legacy.

Options like “are,” “were,” and “have been” disrupt the grammatical agreement and the tone. “Are” incorrectly treats “courage” as plural. “Were” unnecessarily shifts the focus to the past, weakening the sense of continuity.

“Have been” introduces the present perfect tense, which implies a process or change, rather than a static truth. Only “is” captures the solemn, enduring nature of the idea being conveyed.

Question 10

[80] The author is thinking of changing the final sentence of the passage:

We must carry it forward—not in bitterness, but with resolve, clarity, and love.

Which of the following choices best matches the tone and message of the passage?

Answer & Explanation:

Correct Answer: D) We must take up the task—not in hate, but with dedication, understanding, and compassion.

Explanation:

Option D preserves the solemn, purposeful tone of the original sentence while offering a slightly varied but still dignified and emotionally resonant set of words: dedication, understanding, and compassion. These reflect the spirit of continuing a noble effort in a constructive and humane way.

Option A (NO CHANGE) is strong and stylistically sound, but D presents an equally suitable version with slightly softer and more inclusive emotional language. Option B uses informal and imprecise language (brains, good vibes, team spirit) that clashes with the serious and reflective tone.

Option C, while closer in tone, lacks the strong rhetorical rhythm and intensity of D and does not flow as powerfully.

Time to Master Grammar & Writing?

You’ve just worked through 10 CLT-like Grammar & Writing questions.

If you’re ready to strengthen your understanding of every rule and nuance with more expert-crafted problems and step-by-step guidance, our comprehensive program is designed to help you succeed.

CLT Quantitative Reasoning Practice Questions

Read and answer each question carefully.
Calculators are NOT permitted during the math portion of this exam.
Figures are NOT necessarily drawn to scale.

You may use any of the following formulas to help you in your calculations:

Area of a circle = πr², where r is the radius of the circle
Circumference of a circle = 2πr, where r is the radius of the circle
There are 360 degrees in a circle.
There are 2π radians in a circle.
Volume of a sphere = $$\frac{4}{3} \pi r^3$$, where r is the radius of the sphere
Surface area of a sphere = 4πr², where r is the radius of the sphere
Area of a rectangle = length × width
Area of a triangle = $ \frac{1}{2} \times (\text{base} \times \text{height}) $
The sum of the measures of the interior angles of a triangle is 180°.
Pythagorean theorem (for a right triangle): If a, b, and c are the side lengths of the triangle, and c is the hypotenuse, then a² + b² = c².

Trigonometry:

$ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} $
$ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} $
$ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} $
$ \csc \theta = \frac{1}{\sin \theta} $
$ \sec \theta = \frac{1}{\cos \theta} $
$ \cot \theta = \frac{1}{\tan \theta} $
$ \tan \theta = \frac{\sin \theta}{\cos \theta} $
sin² θ + cos² θ = 1
30° – 60° – 90° triangles have side lengths in a ratio of $ 1 : \sqrt{3} : 2 $, corresponding to their opposite side.
45° – 45° – 90° triangles have side lengths in a ratio of $ 1 : 1 : \sqrt{2} $, corresponding to their opposite angle.

Question 1

A student draws the following right triangle and calculates that

$ \cos X = \frac{3}{5} $

From this, the student concludes that in any right triangle, the length of the adjacent side is always three-fifths the length of the hypotenuse.

Which of the following choices is a counterexample that disproves the above statement?

Answer & Explanation:

Correct option: B.

A. Triangle with sides 6, 8, 10 gives

$ \cos(x) = \frac{6}{10}$

$\quad \cos(x) = \frac{3}{5} $

It approves the statement.

B. For $ \sin(x) = \frac{4}{5} $, different ratio → may not support

C. $ \cos(x) = \frac{3}{5} $ claims to approve the statement

D. Hypotenuse = 20, adjacent = 12. So,

$ \cos(x) = \frac{3}{5} $

It approves the statement.

So, B is the only option that disapproves of the statement.

Question 2

In a circle with center O, points P and Q lie on the circle such that

OP = 4 cm and chord PQ = 43 cm. What is the length of the minor arc PQ?

Answer & Explanation:

Correct option: C.

OP=OQ= 4 cm

$ PQ = 4\sqrt{3} \, \text{cm} $

Triangle OPQ is isosceles

Use Law of Cosines:

PQ² = OP² + OQ² – 2(OP)(OQ)cos(∠POQ)

$ (4\sqrt{3})^2 = 4^2 + 4^2 – 2(4)(4)\cos(\theta) $

16 (3) = 16 + 16 – 32cos(θ)

48 = 32 – 32cos(θ)

16 = -32cos(θ)

cos(θ) = -0.5

$ \theta = 120^\circ = \frac{2\pi}{3} \text{ radians} $

Now, use the arc length formula:
$\text{Arc length} = r \times \theta = 4 \times \left( \frac{2\pi}{3} \right) = \frac{8\pi}{3} \, \text{cm}$

So, C is the correct option.

Question 3

Three circular fountains are located in a city park. Fountain X has a circumference of approximately 62.8 feet, Fountain Y has a diameter of 18 feet, and Fountain Z has a radius equal to the sum of the radii of Fountains X and Y.

Which of the following correctly ranks the area of the fountains from greatest to least?

Answer & Explanation:

Correct option: D.

Let’s find the area of each fountain.

Fountain X

Circumference ≈ 62.8 ft
Use the formula: C = 2πr

So:
2πr = 62.8
r = 62.8 ÷ (2π) ≈ 62.8 ÷ 6.28 ≈ 10 ft
Area = πr² = π(10)² = 100π ft²

Fountain Y

$ \text{Diameter} = 18\,\text{ft} \Rightarrow \text{Radius} = \frac{D}{2} = \frac{18}{2} = 9\,\text{ft} $

$ \text{Area} = \pi r^2 = \pi(9)^2 = 81\pi\,\text{ft}^2 $

Fountain Z

Radius = radius of X + radius of Y
Radius = 10 + 9 = 19 ft
Area = πr² = π(19)² = 361πft²

Compare areas

Fountain Z > Fountain X > Fountain Y

So, the correct answer is (D).

Question 4

A chemist observes a ring-shaped molecule that forms a perfect circle with a radius of 12 nanometers.

A 90° section of the ring is covered by a chemical compound. What is the length of the part of the ring that is not covered by the compound?

Answer & Explanation:

Correct option: B

Circumference = 2πr = 2 × π × 12 = 24π nm

$ \text{Part of the ring covered} = \frac{90}{360} \times 24\pi = \frac{1}{4} \times 24\pi = 6\pi \, \text{nm} $

Part of the ring NOT covered = 24π – 6π = 18π nm

So, B is the correct option.

Question 5

What is the next term in the following sequence?

$ 1,\ \frac{1}{3},\ \frac{1}{7},\ \frac{1}{13},\ \ ? $

Answer & Explanation:

Correct option: C

Let’s solve the sequence:

$ 1,\ \frac{1}{3},\ \frac{1}{7},\ \frac{1}{13} $

The sequence shown above is not a specific type of sequence, like arithmetic (common difference) or geometric (common ratio).

Write the denominators clearly

$ 1 = \frac{1}{1} $
$ \frac{1}{3} $
$ \frac{1}{7} $
$ \frac{1}{13} $

So, the denominators are:
1, 3, 7, 13

Let’s look at how these change:

3 – 1 = 2
7 – 3 = 4
13 – 7 = 6

So the differences between denominators are increasing by 2:
+2, +4, +6 → next difference = +8

Next denominator = 13 + 8 = 21

So, the next term is: $ \frac{1}{21} $

Question 6

Which of the following is always true for integers a and b if a < b and a is positive?

Answer & Explanation:

The correct answer is (C).

Let’s analyze:

A) a² > b²

If a<b, then their squares are less than each other respectively. So this is a false option.

Let’s check with examples.

False: if a = 2,b = 5 → a² = 4,b² = 25 → 4 > 25? No.

B) a – b > 0

Since a<b, this implies that a-b<0, so this statement is also false.

Let’s check it with an example.

If a = 2, b = 5 → 2 − 5 = −3 < 0.

C) -a> -b

Multiplying both sides by −1 reverses the inequality: a<b is true.

D) b – a < 0

If b-a<0, then b<a, which is false.

Let’s check it with an example.

If a = 2, b = 5 → b – a = 3 → 3 < 0?

No → False

Question 7

How many five-digit numbers between 20000 and 60000 (inclusive) satisfy both of the following conditions?

Statement 1: Each digit is a nonzero multiple of 3.
Statement 2: The sum of the digits is a multiple of 9.

Answer & Explanation:

The correct answer is (B) – 36.

Nonzero multiples of 3 = {3,6,9}
So each digit must be from {3,6,9}

As per the condition that the numbers should be between 20000 and 60000, possible thousands digits from {3,6,9} that are between 2 and 6: only 3 and 6.

So the first digit has 2 choices, which are 3 or 6.
The remaining four digits: each from {3,6,9} also have 3 choices each.

So total such numbers = 2 × 3 × 3 × 3 × 3 = 162

Condition 2:

Each digit in {3,6,9} → possible digit sums range from:
Minimum = 3 + 3 + 3 + 3 + 3 = 15
Maximum = 9 × 5 = 45

Multiples of 9 in this range: 18, 27, 36, 45

We count how many of the 162 five-digit numbers have digit sums equal to 18, 27, 36, or 45.

So, only option B is correct.

Question 8

A cylindrical tank that is 6 feet tall has a diameter of 1 foot. The tank is currently filled with oil to 50% of its capacity. A company wants to add a chemical to the oil that requires 5 ounces of additive for every 20 cubic inches of oil.

How many ounces of additive are needed to treat all the oil currently in the tank?

Answer & Explanation:

Correct option: D

Given:

Height of the cylindrical tank = 6 feet
Diameter = 1 foot → radius = 0.5 feet
Tank is 50% full
Additive needed: 5 ounces per 20 cubic inches of oil

Find the total volume of the tank
Volume of a cylinder:
V = πr²h
r = 0.5 feet
h = 6 feet
V = π(0.5)² × 6
V = π(0.25) × 6 = 1.5π cubic feet

Since the tank is 50% full, so
Volume of oil = 0.5 × 1.5π = 0.75π cubic feet

Convert volume to cubic inches

1 cubic foot = 1728 cubic inches
0.75π cubic feet = 0.75π × 1728 = 1296π cubic inches

Use the proportion to find how many ounces of additive are needed

5 ounces per 20 in³

So:
$ x \text{ ounces} = \left( \frac{5}{20} \right) \times 1296\pi $
$ x = \frac{1}{4} \times 1296\pi = 324\pi \text{ ounces} $

Question 9

If θ is defined such that $ \cos \theta = \frac{1}{2} $, what is the value of tan θ?

Answer & Explanation:

Correct option: A

We have $ \cos \theta = \frac{1}{2} $

From the unit circle:
$ \cos(60^\circ) = \frac{1}{2} $
θ = 60°
$ \tan(60^\circ) = \sqrt{3} $
$ \cos(300^\circ) = \frac{1}{2} $
θ = 300°
$ \tan(300^\circ) = -\sqrt{3} $

So:
$ \tan \theta = \pm \sqrt{3} $
Therefore, A is the correct option.

Question 10

In triangle PQR, PQ=QR, m∠P=m∠R, m∠PQS=20°, and m∠SQR=70°. m∠PSQ and m∠TSR are vertical angles.

What is m∠TSR?

Answer & Explanation:

Correct option: D

Triangle PQR is an isosceles triangle because PQ = QR, so the angles P and R are also equal.

Let’s call angle P as x, then angle Q is also x.

Given that m∠PQS=20°, and m∠SQR=70°.

In triangle PQR, using the triangle angle sum:

sum of the angles in a triangle = 180 °

x+x+20°+70°=180°

2x+90°=180°

2x=180°-90°

2x=90°

x=90°÷2=45°

So m∠P=45°.

Now we will find m∠PSQ.

Using the triangle angle sum in triangle PQS:

sum of the angles in a triangle = 180 °

m∠PSQ+20°+45°=180°

m∠PSQ+65°=180°

m∠PSQ=180°-65°

m∠PSQ=115°

Since m∠PSQ and m∠TSR are vertical angles, they are equal.

So, m∠TSR = m∠PSQ = 115°

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