All practice questions

Let $x$, $y$, and $z$ be integers such that $|x| < |y| < |z|$, $x + y + z = -3$, and $xyz > 0$. What is the smallest possible value of $|x - y| + |y - z| + |z - x|$?

A positive integer is said to have exactly $8$ positive factors. What is the fifth smallest positive integer that satisfies this condition?

What integer represents the midpoint between $-9$ and $7$ on a standard number line?

What is the value of the mathematical expression below?

On a map of a city, the scale is $0.5\text{ inches} = 12\text{ miles}$. If the actual distance between two towns is $96\text{ miles}$, what is the distance between the two towns on the map, in inches?

Let $x$ and $y$ be integers such that $|x| \leq 4$ and $|y| \leq 4$. How many distinct pairs of integers $(x, y)$ satisfy the inequality $||x| - |y|| \geq 2$?

The writer wants to add the following sentence to the paragraph:

'These disparate sheets were often stored loosely in scrolls or portfolios, which made them prone to damage and difficult to consult.'

[1] In 1570, Flemish cartographer Abraham Ortelius published the *Theatrum Orbis Terrarum*, which is widely considered the first modern atlas. [2] Prior to this publication, maps were printed in various sizes and formats, making them difficult to bind or compare. [3] Ortelius resolved this issue by compiling existing maps, engraving them onto uniform copper plates, and printing them as a cohesive book. [4] This standardized format revolutionized cartography, allowing merchants and scholars to view the entire known world in a single, organized volume.

To make the paragraph most logical, where should this sentence be placed?

[1] During a chilly trolley ride in New York City in 1902, Mary Anderson noticed that the motorman had to open the front window to wipe away the accumulating sleet. [2] This forced the driver to endure the freezing wind and rain just to maintain a clear view of the street ahead. [3] Upon returning home to Alabama, she sketched a design for a rubber blade that could be operated from inside the vehicle by turning a hand lever. [4] Many manufacturing firms initially refused to purchase the rights to the design, dismissing it as a dangerous distraction to drivers. [5] This manual wiper, which she patented the following year, successfully kept the window clean without requiring the driver to open the glass.

To make this paragraph most logical and cohesive, where should Sentence 5 be placed?

A high-speed fiber-optic cable transmits data at a rate of $8.0 \times 10^7$ bytes per second. A digital archive containing a total of $3.2 \times 10^{11}$ bytes needs to be transmitted. How many seconds will it take to complete the transmission?

Read the following passage segment:

Although the new commercial drone is capable to fly in turbulent weather, the operators decided to postpone the survey flight until the storm subsided.

Which of the following options represents the correct idiomatic phrasing for the underlined portion of the sentence?

A cleaning mixture is prepared by combining bleach and water in a ratio of $1:8$. If a container holds a total of $36$ cups of this mixture, how many cups of bleach were used?

Passage Excerpt:

In the early nineteenth century, a Cherokee silversmith named Sequoyah became convinced that written language was the key to his people's independence. Over the course of a decade, he painstakingly designed a system of eighty-five characters representing the syllables of the spoken Cherokee language. His invention allowed Cherokee speakers to achieve literacy rapidly, leading to the establishment of the first Native American newspaper, *The Cherokee Phoenix*, in 1828.

Given that all the choices are grammatically correct, which choice most clearly and vividly emphasizes the widespread and immediate adoption of Sequoyah's system by the Cherokee community?

An artist has a rectangular sheet of stained glass that measures $84$ inches by $120$ inches. The artist wants to cut the sheet into congruent square tiles of the largest possible side length, such that there is no glass wasted. What is the total number of square tiles the artist will obtain?

In 1856, eighteen-year-old chemist William Henry Perkin was attempting to synthesize quinine, a treatment for malaria, in his home laboratory. Instead of obtaining the expected clear liquid, he produced a dark, sticky sludge. Rather than discarding the failed experiment, Perkin washed the flask with alcohol and noticed a beautiful purple residue. This accidental discovery resulted in mauveine, the world’s first synthetic organic dye. Before Perkin’s breakthrough, purple textiles were exceedingly rare, as the color could only be extracted from expensive natural sources like marine snails. Consequently, Perkin's new dye became super popular and hit it big in the nineteenth-century fashion industry, transforming the worlds of chemistry and apparel design.

Given that all the choices are grammatically correct, which one best maintains the established tone and style of the passage?

In 1911, explorer Hiram Bingham climbed into the Peruvian Andes and encountered Machu Picchu, an ancient citadel from the past that was built by the Inca.

Which choice best replaces the underlined portion of the passage to eliminate redundancy?

The passage below discusses the ecology of deep-sea hydrothermal vents. Read the passage and determine which precise and clear verb best completes the sentence to describe the ecological relationship between the primary-producing bacteria and the surrounding community. Which word best completes the blank?

Two positive integers, $a$ and $b$, are such that $a < b$. The greatest common divisor of $a$ and $b$ is $12$, and their least common multiple is $720$. If $a$ is a multiple of $5$ but $b$ is not a multiple of $5$, what is the value of $a$?

Point $P$ is located at $-14$ on a standard number line. Point $Q$ is located $8$ units from point $P$ in the positive direction. Point $R$ is located $11$ units from point $Q$ in the negative direction. What integer represents the location of point $R$?

Although lacking a brain or central nervous system, the slime mold *Physarum polycephalum* exhibits remarkable problem-solving capabilities. During laboratory experiments, the single-celled organism can navigate complex mazes to locate food. To achieve this, the slime mold leaves a trail of extracellular slime as it moves, creating a physical record of where it has been. Conversely, this externalized spatial memory prevents the organism from wasting energy re-exploring dead ends.

Which choice best transitions from the preceding sentence to the sentence that follows by establishing a logical cause-and-effect relationship?

Passage Excerpt:

In the summer of 1967, postgraduate student Jocelyn Bell Burnell [1] from a new radio telescope she had helped build. Among the miles of data, she noticed a strange "scruff"—a series of repeating radio pulses that occurred every 1.33 seconds. Although some of her colleagues initially wondered if the signals came from an extraterrestrial civilization, Bell Burnell and her advisor soon realized they had discovered a rapidly spinning neutron star, later named a pulsar.

Which choice most effectively emphasizes the tedious, labor-intensive nature of Bell Burnell's work in analyzing the telescope's output?