Six SHSAT Questions

3 English and 3 math

Here are 6 random questions to try. Takers earning below 75% are unlikely to receive an offer. You have 1 minute 48 seconds per question on the real exam.

Question 1 of 6

2024 Test A English, question 40
Passage 5 Reasons Physical Books Might Be Better Than E-Books

5 Reasons Physical Books Might Be Better Than E-Books

by Shaunacy Ferro

Though e-book1 readers have become a more common sight around town, traditional books still have their evangelists.2 According to The New York Times, e-book sales have been falling in 2015. Print definitely isn’t dead. In fact, according to some research, it may actually be a better choice for some readers. While scientists are still trying to tease out exactly how digital reading affects us differently, here are five ways e-books might be inferior to their dead-tree cousins.

E-books can reduce reading comprehension.

In a study of middle schoolers, West Chester University researchers found that students who read on iPads3 had lower reading comprehension than when they read traditional printed books. They discovered that the kids sometimes skipped text in favor of interactive features in the e-books, suggesting that certain multimedia in children’s e-books can be detrimental to the practice of reading itself. However, the researchers noted that some interactive features in e-books are designed to enhance comprehension, and that those might be more helpful than game-type interactive graphics.

Young kids can get distracted by e-books.

Similar results were found by a small study by the Joan Ganz Cooney Center that consisted of 32 kids reading e-books and print books with their parents. It found that “enhanced” e-books might be distracting. Kids who read enhanced e-books—ones with interactive, multimedia experiences—were more engaged with them physically, but in the end they remembered fewer narrative details than those who read print books or basic e-books.

You remember less about a book’s timeline.

Another study of adults also found that e-books can be hard to absorb. The researchers asked 25 people to read a 28-page story on a Kindle4 and 25 to read the story in paperback, then asked the readers to put 14 events from the story in chronological order. Those who read the story on a Kindle performed worse on the chronology test than the book readers, though they performed about the same as print readers in other tests. Earlier research by the same scholars, from Stavanger University in Norway, found that Norwegian 10th graders also remembered more about texts if they read them in print rather than on a computer screen.

They’re not great as textbooks.

While e-book textbooks are often cheaper (and easier to carry) than traditional door-stop5 textbooks, college students often don’t prefer them. In some surveys of college kids, the majority of students have reported preferring print books. However, a 2012 study from the UK’s National Literacy Trust of kids ages 8 to 16 found that more than 50 percent of children reported preferring screen reading.

They’re tiring.

Staring at a lit screen can be tiring for the eyes and the brain. A 2005 study from Sweden found that reading digitally required a higher cognitive workload than reading on paper. Furthermore, staring at LED6 screens at night can disrupt sleep patterns. A 2014 Harvard study found that people who used e-readers with LED screens at night slept worse and were more tired the next day. So, if you’re going to go for an e-book, go for one without the backlight.

The take-away message? If you’re really trying to absorb material, you might want to go for a physical book. And if you’re going to be up all night studying, turn off the backlight.

However, all this may not mean that reading on a Kindle is really going to melt your brain. For instance, reading an e-book on a computer is a much different experience than reading on a Kindle, which is specifically designed for consuming books. So, too, is playing with an interactive e-book on an iPad, compared to using a simpler e-book device that only presents the text, with no opportunities to click away into digital distractions.

And some studies have found that part of the difference between the way people absorb information from e-books versus paper might be due to approaching e-books differently—in one test, participants didn’t regulate their study time with digital books like they did with paper texts, leading to worse performances. It’s possible that our expectations of e-book reading—as well as the different designs of the digital reading experience on a computer or iPad or Kindle—might affect how we approach the text and how much effort we put into studying them. As generations of e-book readers evolve, and people become more accustomed to the idea of sitting down with a digital textbook, these factors could change—for better or for worse.

“5 Reasons Physical Books Might Be Better Than E-Books” by Shaunacy Ferro from MENTAL FLOSS, October 2, 2015. Copyright © 2015 Mental Floss, Inc.

1e-book: an electronic book, a publication made available digitally for reading on a computer or other device

2evangelists: advocates or supporters

3iPads: tablet computers

4Kindle: device for reading e-books

5door-stop: a reference to the idea that some textbooks are large and heavy enough to hold a door open

6LED: Light-emitting diode; LEDs are often used as a backlight for electronic devices such as smart phones and tablets.

Question

The studies described in paragraph 6 are important to the author’s observations in paragraphs 7 and 8 because the findings

  • Ashow that the author has personal opinions to share beyond the ideas included in the studies.
  • Bgive the author the opportunity to reach practical conclusions about reading that are based on scientific evidence.
  • Chighlight the author’s statement that digital reading experiences vary based on the individual and the device.
  • Dsupport the author’s inference that more studies on more types of devices are needed.
Show the answer

Correct answer: B

The question asks how the findings from the studies described in paragraph 6 are important to the author’s observations in paragraphs 7 and 8.

A. Incorrect. The evidence from the studies mentioned in paragraph 6 is not substantially supported by the personal opinions the author shares in paragraph 7 (“And if you’re going to be up all night studying, turn off the backlight”) or in paragraph 8 (“melt your brain” and “digital distractions”).

B. CORRECT. In paragraph 6, the author cites studies from 2005 and 2014 that found that LED screens can “disrupt sleep patterns” and that “people who used e-readers with LED screens at night slept worse and were more tired the next day.” Then, in paragraph 7, the author suggests that “if you’re going to be up all night studying, turn off the backlight,” and in paragraph 8, the author explains some differences between reading e-books on various devices. This information is not based on a personal opinion; instead, it is based on practical conclusions drawn from scientific evidence.

C. Incorrect. In paragraph 6, the author cites a 2005 study in which researchers found that “reading digitally required a higher cognitive workload than reading on paper.” Then, in paragraph 8, the author concludes that “reading an e-book on a computer is a much different experience than reading on a Kindle, which is specifically designed for consuming books.” Although the findings in paragraph 6 are important, they do not highlight the information presented by the author that digital reading experiences vary based on the reader and the device.

D. Incorrect. The author states in paragraph 8 that “reading an e-book on a computer is a much different experience than reading on a Kindle” and mentions how the same thing is true for “an interactive e-book on an iPad” compared with an e-book on “a simpler e-book device.” Readers may conclude that this statement is meant to indicate the author’s belief that more studies on more types of devices are needed. However, the author does not state or imply this belief in the passage.

Question analysis

This question asks why the studies on screen fatigue and sleep in paragraph 6 of an article about e-books matter to the observations the author makes in the next two paragraphs. Students must trace the connection between the research and the later paragraphs, and each choice describes that link in abstract terms that are easy to confuse.

NYS standards
  • 7R5 structure and genre features
  • 7R8 evaluating arguments and claims
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Tracing how evidence leads to an author's conclusions is grade 7 argument work, but this item spans three paragraphs, and the four explanations of the link are abstract and close in wording, more demanding than typical grade 7 work.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Evidence
Answered from several paragraphs or stanzas
Reading level
Passage: about grade 10, 829 words; question and choices: about grade 11, 80 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every choice describes a link between evidence and commentary, so the student has to know both what paragraph 6 found and what paragraphs 7 and 8 do with it. The tempting wrong answer borrows paragraph 8's real point that devices differ and wrongly credits the sleep and workload studies with it. Seeing that the backlight advice comes straight from the Harvard study is what makes the general wording of the right choice convincing.

  1. How, not what major Working it out

    The question asks how paragraph 6's findings function for the paragraphs after them, a relationship between evidence and conclusion, so knowing what the sleep and workload studies found is only the first step. (Anderson & Krathwohl, 2001)

  2. Evidence in several places moderate Working it out

    The links run across three paragraphs: the Harvard backlight finding becomes “turn off the backlight” in paragraph 7, and the tired brain returns as “melt your brain” in paragraph 8. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

  3. The right answer is reworded minor Choosing the answer

    The right choice names no study, screen or device, speaking only of “practical conclusions” and “scientific evidence”, while two wrong choices echo paragraph 8's talk of devices. (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)

Strategy: Note what the paragraph 6 studies found, then find the sentences in paragraphs 7 and 8 that draw on those findings and describe what the author does with them.

Traps

  • Half right: One half of a two-part choice is right and the other is not.
  • Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.
  • Wrong purpose or structure: Names a purpose, structure or effect that the part does not serve, or describes what a part says when the question asks what it does.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Understand the link asked about

    This asks how evidence connects to the author’s comments. First see what the studies in paragraph 6 found. Then see what the author does with those findings in paragraphs 7 and 8.

  2. Read the studies in paragraph 6

    A 2005 study found that “reading digitally required a higher cognitive workload.” Cognitive workload means how hard your brain has to work. A 2014 study found that people who read on lit screens at night “slept worse and were more tired the next day.”

  3. See how the author uses them

    In paragraph 7 the author advises, “if you’re going to be up all night studying, turn off the backlight.” That tip comes straight from the sleep study. Paragraph 8 then warns you not to take the findings too far. It says they “may not mean that reading on a Kindle is really going to melt your brain.” Both are practical, real-life conclusions drawn from science.

  4. Check each choice

    • B fits: the studies let the author reach “practical conclusions” based on “scientific evidence.”
    • A is wrong because the advice in paragraphs 7 and 8 grows out of the studies, not out of opinions beyond them.
    • C is wrong because nothing says reading changes from person to person, and the studies compare screens with paper.
    • D is wrong because the author never says more studies are needed.

    Watch out: C is half right. Paragraph 8 does say devices differ, but the part about each person is never stated.

  5. Choose the answer

    The findings in paragraph 6 lead straight to the author’s practical advice. So the answer is B.

Why this answer and not the closest wrong one

C comes closest, because paragraph 8 does say a computer and a Kindle feel different. But the studies in paragraph 6 compare screens with paper, and no one says reading changes from person to person. B fits the clear link: a sleep study turned into the advice to turn off the backlight.

The original DOE question
Question 40 as printed on page 61 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 61, question 40. Open page 61 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).
The passage “5 Reasons Physical Books Might Be Better Than E-Books” as printed (pages 58–59)The passage “5 Reasons Physical Books Might Be Better Than E-Books” as printed on page 58 of the DOE handbook.The passage “5 Reasons Physical Books Might Be Better Than E-Books” as printed on page 59 of the DOE handbook.

Question 2 of 6

2026 Practice Test A English, question 37
Passage Massachusetts: Lowell National Historical Park

Massachusetts: Lowell National Historical Park

During the first half of the 19th century, Lowell, Massachusetts, quickly transformed itself from a farm town to a bustling industrial city. In time, Lowell became a model of industry, gaining global recognition for its state-of-the-art technology, innovative canal and dam system, mill architecture, boardinghouses, churches, and ethnic neighborhoods. Young Yankee women, immigrant families, and European tourists all flocked to Lowell to find work at one of the many textile mills, or visit the industrious city that was becoming a popular tourist destination. As one Scottish traveler observed during his visit to America, “Niagara and Lowell are the two objects I will longest remember in my American journey, the one the glory of American scenery, the other of American industry.” Today, Lowell National Historical Park welcomes visitors to enjoy the sights of Lowell and learn about the history of one of America’s most significant industrial cities.

The Boston merchants who founded Lowell in 1821 and named it after Francis Cabot Lowell chose to locate the town along Massachusetts’s Merrimack River to take advantage of the kinetic energy offered by the Pawtucket waterfalls. Over six miles of canals powered the waterwheels of Lowell’s mills, whose massive five- and six-story brick buildings dominated the city’s landscape. . . . The most recognized of these buildings are the Lowell Manufacturing Company chartered in 1821, the Suffolk or Wannalancit Mill completed around the 1880s, the Boott Mill Company established in 1835, and the Boott Mill Boardinghouse that opened in 1838. By the 1850s, 40 textile mills employing over 10,000 workers stretched for about a mile along the river. . . .

The city’s female workforce was significant in the history of Lowell. From the early to mid-1800s, women left the constricted lifestyle of small rural towns and rural areas for independent industrial city life. Most were young single Yankee girls, who were tired of the limited opportunities offered by their domestic work. Women found that Lowell’s mills offered monthly wages for their services and provided them room and board. Although these women gained economic independence in Lowell, the mill boardinghouse keepers constantly supervised their social activities, for which they hardly had any time, considering their daily 12- to 14-hour work schedules. At the end of the day, the factory bell signaled the “mill girls” to return to their boardinghouses. They were expected to adhere to the strict code of conduct respecting curfew and attending church.

Yankee “mill girls” continued to dominate the Lowell workforce until the 1840s, when the city began to find it difficult to compete with the growing industrial development in other New England communities. As profits fell, the mill industry cut wages. These wage cuts, deteriorating working conditions, and long workdays led the “mill girls” to protest and organize strikes. When their demands went unheard, the women left Lowell, and immigrant groups replaced them in the workforce. Despite the low wages and unhealthy work conditions, immigrants were eager to find work.

The immigrants replacing the Yankee “mill girls” during the 1840s were predominantly Irish Catholics, who traveled to America during the Great Potato Famine. Although Lowell received an influx of Irish families during this time, the Irish were a part of the city’s history from its birth, and before the “mill girls” arrived, they built Lowell’s historic canals, mills, and boardinghouses. Initially, Lowell’s Protestant community was slow to welcome Irish immigrants, but the hostility between Yankee Protestants and Irish Catholics eventually disappeared. Irish immigrants dominated the industrial scene until the Civil War, when other immigrant groups began to work in the city mills.

Like the Irish, the French-Canadians, Greeks, Poles, Portuguese, Russian Jews, and Armenians who came to work in Lowell’s mills faced long work hours, low wages, and poor living conditions in the city’s crowded tenements. By the time Lowell’s industry declined, the city had become an ethnic melting pot, where each group claimed its own distinct neighborhood, like the Irish immigrants’ “New Dublin” or “Acre,” and the French-Canadians’ “Little Canada.” The city officially began to close down its mills in the 1920s and ’30s after Lowell’s outdated mills could no longer compete against the state-of-the-art cotton mills in other communities and working conditions continued to decline as Lowell’s companies stopped reinvesting in their mills. . . . Despite a brief resurgence during World War II, the city shut down its last surviving mill by the mid-1950s.

From “Massachusetts: Lowell National Historical Park”—Public Domain/National Park Service

Yankee: native to New England
Niagara: a town in northwestern New York State well known as the location of Niagara Falls, a series of waterfalls on the Canadian border
domestic work: household duties like cooking and cleaning

Question

A central idea that Lowell was “one of America’s most significant industrial cities” (paragraph 1) is conveyed in the passage primarily through a description of the

  • Acanals, mills, and boardinghouses that were built by immigrants.
  • Bmill girls and immigrants who comprised Lowell’s workforce.
  • Cdevelopment of the mills and the workforce established to support them.
  • Dcultural diversity of the people who lived in the area.
Show the answer

Correct answer: C

The question asks which description conveys the central idea that Lowell was “one of America’s most significant industrial cities” (paragraph 1).

A. Incorrect. While the passage does state that the city’s canals, mills, and boardinghouses were built by early immigrants from Ireland, these details do not contribute to the development of the overall idea that Lowell was a significant industrial city.

B. Incorrect. The details about the women and immigrants who worked in the mills are important to the passage, but these details alone do not show that Lowell was a significant industrial city.

C. CORRECT. The details throughout the passage about the development of the mills and the people who worked in them convey the significance of Lowell in early American industry. Paragraph 2 shares details about the businesses that started in the 1800s and contributed to the development of industry in the region (“The most recognized of these buildings are the Lowell Manufacturing Company chartered in 1821, the Suffolk or Wannalancit Mill completed around the 1880s, the Boott Mill Company established in 1835, and the Boott Mill Boardinghouse that opened in 1838”). Then paragraphs 3–6 provide specific details about the people who worked in the mills at different points in time.

D. Incorrect. Although the passage references Lowell’s culturally diverse community (“Young Yankee women, immigrant families, and European tourists all flocked to Lowell to find work at one of the many textile mills” [paragraph 1]), these groups alone are not what marked Lowell as a significant industrial city.

Question analysis

This central idea question asks which kind of description in an article about the Massachusetts mill city of Lowell does most to show that it was one of America's most significant industrial cities. Students need a map of all six paragraphs, because every choice names something the passage really describes, and the question turns on which description does the main work.

NYS standards
  • 7R2 theme or central idea
  • 7R5 structure and genre features
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Grade 7 students trace how a central idea is developed, usually through a few supporting details. Here the strands to weigh run through six dense paragraphs of dates, mills and worker groups, and the stem is long and passive, so it asks more than typical.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Evidence
Answered from the whole text
Reading level
Passage: about grade 15, 764 words; question and choices: about grade 10, 63 words.
Corrected from the DOE handbook
  • A grammatical error in the explanation was corrected.
  • An inaccurate detail in the explanation was corrected.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every choice names something the passage really describes, so truth rules nothing out; the question turns on which description carries the claim that Lowell mattered as an industrial city. Because four of the six paragraphs follow the workers, the workforce alone looks like the passage's main thread, and students who forget paragraph 2's mills take the half for the whole. Diversity and the immigrant builders are real themes too, but neither says anything about Lowell's industrial standing.

  1. A very close wrong answer major Choosing the answer

    The workforce choice is the key with one half removed: both describe the people who worked the mills, but only the key also names the development of the mills, which paragraph 2 builds with dates and numbers. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  2. Evidence in several places moderate Working it out

    The industrial picture is built in two stages, the canals and mills of paragraph 2 and the successive workforces of paragraphs 3 to 6, so each choice must be measured against the whole passage. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

  3. A lot to read minor Understanding the question

    Six paragraphs pack in mill names, dates from 1821 to the 1950s and a string of worker groups, from Yankee mill girls to Irish, French-Canadian and Greek immigrants. (Kintsch, 1998; Gorin & Embretson, 2006; Freedle & Kostin, 1993)

  4. Complex sentences minor Understanding the question

    The stem is a long passive sentence that quotes paragraph 1 inside it and holds the real question, “primarily through a description of the,” until its last words. (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)

Strategy: Note in a phrase what each of the six paragraphs is about, then test each choice against the stem's claim: would that description make a reader see Lowell as a major industrial city?

Traps

  • Too narrow: A single detail offered as the central idea or a summary, or a summary that keeps minor details and leaves out an essential part.
  • True but not the answer: A choice says something accurate about the text that does not answer this question.

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: The handbook’s explanation of Option B says “The details about the women and immigrants who worked in the mills is important to the passage.” The subject is “details,” so the site now says “are important.”
  • Explanation: The handbook’s explanation of Option C says paragraph 2 describes “the businesses that started in the early 1800s,” but the sentence it then quotes includes “the Suffolk or Wannalancit Mill completed around the 1880s,” late in the century. The site now says “the businesses that started in the 1800s.”

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Figure out what the question asks

    This is a central idea question. A central idea is a main point that the whole passage builds. The point here is that Lowell was “one of America’s most significant industrial cities.” Significant means important. You need the description that does the most to show this.

  2. Map the six paragraphs

    • Paragraph 1: Lowell grows from a farm town into a famous industrial city.
    • Paragraph 2: The mills: canals, huge brick buildings, mills started across the 1800s, and by the 1850s, “40 textile mills employing over 10,000 workers.”
    • Paragraphs 3 and 4: The Yankee “mill girls” who worked in the mills.
    • Paragraphs 5 and 6: The Irish and other immigrants who took their place.
  3. See how the picture is built

    The passage shows Lowell’s importance in two parts. First it describes the mills: how they were built and how big they grew. Then it follows the workers who ran them, group after group. Together, the mills and the workforce show a major industrial city. Workforce means all the people who do the work. Either part alone tells only half the story.

  4. Check each choice

    • A is wrong. Paragraph 5 does say the Irish built the canals, mills and boardinghouses. But that is one fact in one sentence, not the main description.
    • B is wrong. The mill girls and immigrants fill paragraphs 3 to 6. But this choice leaves out the mills themselves, the heart of paragraph 2.
    • C is right. It covers both parts: how the mills grew and the workers who kept them running.
    • D is wrong. Lowell did become an “ethnic melting pot.” But a mix of cultures doesn’t show that a city was an industrial leader.

    Watch out: Four of the six paragraphs are about workers, so B looks like the main thread. It is only half of it.

  5. Choose the answer

    The central idea is carried by the growth of the mills plus the workforce gathered to run them. So the answer is C.

Why this answer and not the closest wrong one

B is the closest wrong answer. It matches the second half of C, the people who worked in the mills. But it drops the mills, with their six miles of canals and, by the 1850s, 40 mills. Workers alone don’t make a city an industrial giant. C keeps both halves, so it fits the whole passage.

The original DOE question
Question 37 as shown on the DOE’s digital test, page 52 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test A English, page 52, question 37. This is the DOE’s version, before the corrections listed in the analysis.
The passage “Massachusetts: Lowell National Historical Park” as printed (page 50)The passage “Massachusetts: Lowell National Historical Park” as printed on page 50 of the DOE handbook.

Question 3 of 6

2026 Practice Test C English, question 11
Passage Excerpt from “Niagara Falls”

Excerpt from “Niagara Falls”

by Rupert Brooke

The real secret of the beauty and terror of the Falls is not their height or width, but the feeling of colossal power and of unintelligible disaster caused by the plunge of that vast body of water. If that were taken away, there would be little visible change, but the heart would be gone.

The American Falls do not inspire this feeling in the same way as the Canadian. It is because they are less in volume, and because the water does not fall so much into one place. By comparison their beauty is almost delicate and fragile. They are extraordinarily level, one long curtain of lacework and woven foam. Seen from opposite, when the sun is on them, they are blindingly white, and the clouds of spray show dark against them. With both Falls the colour of the water is the ever-altering wonder. Greens and blues, purples and whites, melt into one another, fade, and come again, and change with the changing sun. Sometimes they are as richly diaphanous1^1 as a precious stone, and glow from within with a deep, inexplicable light. Sometimes the white intricacies of dropping foam become opaque and creamy. And always there are the rainbows. If you come suddenly upon the Falls from above, a great double rainbow, very vivid, spanning the extent of spray from top to bottom, is the first thing you see. If you wander along the cliff opposite, a bow springs into being in the American Falls, accompanies you courteously on your walk, dwindles and dies as the mist ends, and awakens again as you reach the Canadian tumult. And the bold traveller who attempts the trip under the American Falls sees, when he dare open his eyes to anything, tiny baby rainbows, some four or five yards in span, leaping from rock to rock among the foam, and gambolling beside him, barely out of hand’s reach, as he goes. One I saw in that place was a complete circle, such as I have never seen before, and so near that I could put my foot on it. It is a terrifying journey, beneath and behind the Falls. The senses are battered and bewildered by the thunder of the water and the assault of wind and spray; or rather, the sound is not of falling water, but merely of falling; a noise of unspecified ruin. So, if you are close behind the endless clamour, the sight cannot recognise liquid in the masses that hurl past. You are dimly and pitifully aware that sheets of light and darkness are falling in great curves in front of you. Dull omnipresent foam washes the face. Farther away, in the roar and hissing, clouds of spray seem literally to slide down some invisible plane of air.

Beyond the foot of the Falls the river is like a slipping floor of marble, green with veins of dirty white, made by the scum that was foam. It slides very quietly and slowly down for a mile or two, sullenly exhausted. Then it turns to a dull sage green, and hurries more swiftly, smooth and ominous. As the walls of the ravine close in, trouble stirs, and the waters boil and eddy. These are the lower rapids, a sight more terrifying than the Falls, because less intelligible. Close in its bands of rock the river surges tumultuously forward, writhing and leaping as if inspired by a demon. It is pressed by the straits into a visibly convex form. Great planes of water slide past. Sometimes it is thrown up into a pinnacle of foam higher than a house, or leaps with incredible speed from the crest of one vast wave to another, along the shining curve between, like the spring of a wild beast. Its motion continually suggests muscular action. The power manifest in these rapids moves one with a different sense of awe and terror from that of the Falls. Here the inhuman life and strength are spontaneous, active, almost resolute. . . . A place of fear.

One is drawn back, strangely, to a contemplation of the Falls, at every hour, and especially by night, when the cloud of spray becomes an immense visible ghost, straining and wavering high above the river, white and pathetic and translucent. The Victorian lies very close below the surface in every man. There one can sit and let great cloudy thoughts of destiny and the passage of empires drift through the mind; for such dreams are at home by Niagara. I could not get out of my mind the thought of a friend, who said that the rainbows over the Falls were like the arts and beauty and goodness, with regard to the stream of life—caused by it, thrown upon its spray, but unable to stay or direct or affect it, and ceasing when it ceased. In all comparisons that rise in the heart, the river, with its multitudinous waves and its single current, likens itself to a life, whether of an individual or of a community. A man’s life is of many flashing moments, and yet one stream; a nation’s flows through all its citizens, and yet is more than they. In such places, one is aware, with an almost insupportable and yet comforting certitude, that both men and nations are hurried onwards to their ruin or ending as inevitably as this dark flood. Some go down to it unreluctant, and meet it, like the river, not without nobility. And as incessant, as inevitable, and as unavailing as the spray that hangs over the Falls, is the white cloud of human crying. . . . With some such thoughts does the platitudinous2^2 heart win from the confusion and thunder of a Niagara peace that the quietest plains or most stable hills can never give.

From LETTERS FROM AMERICA by Rupert Brooke—Public Domain

1^1diaphanous: sheer, translucent

2^2platitudinous: clichéd, common

Question

In paragraph 4, the idea that human life and history ultimately end in the same way is illustrated mainly through

  • Athe discussion of how the rainbows visible in the Falls are like the art and beauty created by humankind.
  • Bthe comparison between the movement of water in the Falls and the human experience.
  • Cthe inclusion of details that show that every observer’s experience with the Falls is different.
  • Dthe acknowledgment that contemplating the Falls at night sparks an awareness of humankind’s destiny.
Show the answer

Correct answer: B

The question asks how paragraph 4 of the excerpt mainly illustrates the idea that human life and history ultimately end in the same way.

A. Incorrect. In paragraph 4, the author compares the rainbows visible in the Falls to “the arts and beauty,” which share the rainbows’ dependence on “the stream of life”—that is, they are “caused by it, thrown upon its spray, but unable to stay or direct or affect it, and ceasing when it ceased.” This comparison does not illustrate the idea that human life and history ultimately end in the same way but rather that art and beauty, like human beings, are unable to “stay or direct or affect” their ultimate fate.

B. CORRECT. The author concludes in paragraph 4 that the movement of the Falls can be compared to people and history when he writes, “both men and nations are hurried onwards to their ruin or ending as inevitably as this dark flood.” This comparison illustrates the idea that human life and history ultimately end in the same way.

C. Incorrect. Although in paragraph 4 the author states that “One is drawn back, strangely, to a contemplation of the Falls” and provides an account of “a friend, who said that the rainbows over the Falls were like the arts and beauty and goodness, with regard to the stream of life,” the author does not provide details to show that every observer’s experience with the Falls is different. Rather, the author explains that “the Victorian lies very close below the surface in every man” (paragraph 4).

D. Incorrect. While the setting at night does provide a place of contemplation, the author notes the “dark flood” as the overall illustration of the human experience.

Question analysis

In this question on the closing paragraph of Rupert Brooke’s “Niagara Falls,” students identify the main way the author illustrates the idea that human life and history ultimately end in the same way. It is a lightly reworded version of a 2018 Test B question. The paragraph is long and reflective, mixing several images with abstract thoughts, so students must decide which of them actually illustrates this particular idea.

NYS standards
  • 7R1 textual evidence and inferences
  • 7L5 figurative language and word relationships
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Most grade 7 work asks how an idea is shown in plain modern prose. This paragraph is a century-old reflection in long sentences with words like “certitude” and “unavailing,” and several of its images could each seem to carry the idea.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Evidence
Answered from one paragraph or stanza
Reading level
Passage: about grade 9, 987 words; question and choices: about grade 11, 81 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Paragraph 4 offers three pictures, the spray as a night ghost, the friend’s rainbows and the river as a life, and only one of them shows people and nations meeting an end. The 2026 stem’s “ultimately end” makes the rainbows’ “ceasing when it ceased” more tempting, and the night-time “thoughts of destiny” sound like the idea itself. Students who ask what the paragraph uses to show the idea, rather than where it mentions endings or destiny, land on the dark flood.

  1. Complex sentences major Understanding the question

    The idea sits inside a hedged sentence, “one is aware, with an almost insupportable and yet comforting certitude, that both men and nations are hurried onwards to their ruin or ending,” and the comparison comes last: “as inevitably as this dark flood.” (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)

  2. The right answer is reworded moderate Choosing the answer

    The key sums up the paragraph’s river-and-life likeness in its own words, “movement of water” and “human experience,” while the other choices borrow the paragraph’s vocabulary: “rainbows,” “art and beauty,” “night” and “destiny.” (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)

  3. Hard words minor Understanding the question

    “Multitudinous,” “certitude,” “insupportable” and “unavailing” all sit in the lines that carry the idea, and the footnoted “platitudinous” closes the paragraph, so the comparison has to be read through them. (Perfetti, 2007; Freedle & Kostin, 1993)

Strategy: Find the sentence in paragraph 4 that actually states the idea about life and history ending, then ask how that very sentence makes its point and match a choice to it.

Traps

  • Supports only part of the claim: In a question about evidence: a line that backs one half of the claim but not the other, or a related point rather than the claim itself.
  • Outside knowledge: A choice is plausible in the real world but not supported by this text.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Figure out what the question asks

    This asks how an idea is shown in paragraph 4. The idea is that people’s lives and whole nations end in the same way: they all come to an end. You need the picture the author mainly uses to show that idea. Paragraph 4 has several pictures, so find the one tied to this idea.

  2. Find the sentence with the idea

    Paragraph 4 says “both men and nations are hurried onwards to their ruin or ending as inevitably as this dark flood.” Men means people. Nations means countries and their history. Inevitably means in a way that can’t be stopped. The “dark flood” is the river’s dark water rushing over the Falls.

  3. See how the comparison works

    Just before, the author says “the river, with its multitudinous waves and its single current, likens itself to a life.” Multitudinous means very many. A current is water flowing one way. Likens itself to means seems like. A life has many moments but moves one way, like the river. And just as the water must rush on, people and nations rush toward their end. So the idea is shown by comparing moving water with human life.

  4. Check each choice

    • A does not fit. A friend compares the rainbows to “the arts and beauty and goodness.” That picture is about art, not about lives and nations ending.
    • B fits. The river’s movement stands for human lives and history.
    • C does not fit. The paragraph never says that each visitor sees the Falls differently.
    • D does not fit. The night view and the “thoughts of destiny” tell when and where the author thinks. They do not show how lives end.

    Watch out: D is tempting because destiny, or what happens in the end, sounds like the idea itself. But the question asks what illustrates the idea, and that is the river.

  5. Choose the answer

    The author pictures people and nations rushing to their end like the river’s water. So the answer is B, “the comparison between the movement of water in the Falls and the human experience.”

Why this answer and not the closest wrong one

D is the closest wrong choice. Paragraph 4 really does say the Falls at night stir “great cloudy thoughts of destiny.” But that only tells you the author was thinking about such things. The picture that shows lives and nations ending is the river. They are “hurried onwards to their ruin or ending as inevitably as this dark flood.”

The original DOE question
Question 11 as shown on the DOE’s digital test, page 19 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test C English, page 19, question 11.
The passage “Excerpt from “Niagara Falls”” as printed (pages 134–135)The passage “Excerpt from “Niagara Falls”” as printed on page 134 of the DOE handbook.The passage “Excerpt from “Niagara Falls”” as printed on page 135 of the DOE handbook.

Question 4 of 6

2018 Test B Math, question 53

Question

At North High School, a survey asked two questions, Question A and Question B. For each question, students could answer either “yes” or “no.” Of the 800 students who responded to the survey, 720 answered “yes” to Question A, and 640 answered “yes” to Question B. What is the least possible number of these students who could have answered “yes” to both questions?

  • A80
  • B160
  • C560
  • D640
Show the answer

Correct answer: C

None of the 80 students (800 − 720) who answered “no” to Question A could have answered “yes” to both questions.

Therefore, the least possible number of students who could have answered “yes” to both questions can be found by subtracting the 80 who answered “no” to Question A from the 640 who answered “yes” to Question B or 640 − 80 = 560.

Question analysis

In a survey of 800 students at North High School, 720 said yes to the first question and 640 said yes to the second. Students must find the smallest possible number who said yes to both. It tests reasoning about overlapping groups, and asking for the least possible overlap rather than an exact count makes it unusual.

NYS standards
  • 7.SP.8b sample spaces for compound events
  • 7.EE.4 equations and inequalities from problems
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students work with Venn diagrams and two-way tables when all counts are given. Finding the smallest possible overlap requires reasoning about extremes rather than computing, which is rare in grade 7 and new to most students.
Estimated difficulty
Among the hardest on the SHSAT: a non-obvious approach, heavy reading, or a well-hidden trap.
Solution
2 steps
Reading level
Question and choices: about grade 8, 60 words.
Corrected from the DOE handbook
  • A repeated phrase and a stray comma in the explanation were removed.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Most students have never been asked for a smallest overlap, so they fall back on a picture of nested groups and give the largest one. The key move is to push the two yes groups apart as far as possible: only the 80 students who said no to Question A can absorb some of Question B's yes answers. Once that is seen, the arithmetic is one subtraction, so the question rewards reasoning far more than computation.

  1. No obvious method major Working it out

    Nothing in the wording points to the method: the smallest overlap appears only when all 80 students who said no to Question A are imagined saying yes to Question B. (Schoenfeld, 1985)

  2. Turning words into math moderate Understanding the question

    "The least possible number of these students who could have answered “yes” to both questions" asks for an extreme case, which has to be turned into an arrangement of the 800 students. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  3. A very close wrong answer minor Choosing the answer

    640 is only 80 above the right 560, so a rough sense of size cannot choose between them; only asking whether the overlap is the least or the greatest separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Add the two yes counts and compare the sum with the number of students; any amount above the total must come from students counted in both groups.

Traps

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: The handbook prints “(800 − 720)” twice in its first sentence, after “80 students” and again after “Question A,” and puts a comma between subject and verb in “the least possible number of students who could have answered “yes” to both questions, can be found.” The site keeps the first “(800 − 720)” and drops the comma.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Least overlap of two groups
least overlap=a+b−t\text{least overlap} = a + b - t
a is the size of the first group and b is the size of the second. t is the total number of people. If a + b is more than t, some people must be in both groups. The extra amount is the smallest number that can be in both.
  1. Sort out what you know

    There were 800 students. Of them, 720 said yes to Question A and 640 said yes to Question B. You need the least number who could have said yes to both. The overlap is the group in both yes groups. You want the smallest overlap.

  2. Find who said no to A

    Students who said no to Question A cannot be in the overlap.

    800−720=80800 - 720 = 80

    So 80 students said no to Question A.

    Watch out: 80 is not the answer. It is only a step, and it is choice A.

  3. Keep the overlap small

    To make the overlap as small as you can, put as many B yes answers outside it as possible. The only students outside it are the 80 who said no to A. So pretend all 80 of them said yes to B.

  4. Count the rest of the yes group

    The other yes answers to B must come from students who also said yes to A.

    640−80=560640 - 80 = 560

    So at least 560 students said yes to both.

    Watch out: 640 is the most that could say yes to both, not the least. That is choice D.

  5. Give the answer

    The fewest students who could say yes to both questions is 560. So the answer is C, 560.

Check your answer

Use the overlap formula: 720+640−800=560720 + 640 - 800 = 560. There are 1,360 yes answers but only 800 students, so 560 students must be counted twice. You can also check the groups. 560 said yes to both, 160 to A only and 80 to B only. That makes 560+160+80=800560 + 160 + 80 = 800.

The original DOE question
Question 110 as printed on page 163 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 163, question 110. Open page 163 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D). This is the DOE’s version, before the corrections listed in the analysis.

Question 5 of 6

2018 Test A Math, question 54

Question

Number line with points J, K, L and M marked from left to right. Only K has a value: 3/8.

On the number line above, JK=312JK=3\frac{1}{2} , JM=934JM=9\frac{3}{4} , and LM=118LM=1\frac{1}{8} . What is the position of point LL?

  • A5185\frac{1}{8}
  • B5145\frac{1}{4}
  • C5125\frac{1}{2}
  • D6146\frac{1}{4}
Show the answer

Correct answer: C

First, use JK=312JK=3\frac{1}{2} to find the location of JJ:

38−J=312\frac{3}{8}-J=3\frac{1}{2}

J=38−312=−318J=\frac{3}{8}-3\frac{1}{2}=-3\frac{1}{8}

Now, use JM=934JM=9\frac{3}{4} to find the location of MM:

M−(−318)=934M-\left(-3\frac{1}{8}\right)=9\frac{3}{4}

M+318=934M+3\frac{1}{8}=9\frac{3}{4}

M=934−318=658M=9\frac{3}{4}-3\frac{1}{8}=6\frac{5}{8}

Finally, use LM=118LM=1\frac{1}{8} to find the location of LL:

658−L=1186\frac{5}{8}-L=1\frac{1}{8}

L=658−118=548=512L=6\frac{5}{8}-1\frac{1}{8}=5\frac{4}{8}=5\frac{1}{2}

Question analysis

This number line question gives the lengths JK, JM and LM as mixed numbers and asks for the position of point L. Students must chain three distance steps, moving left from K to J, right to M and back to L, with fractions in eighths and a negative position. Losing track of where K sits is the main danger.

NYS standards
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students add and subtract rational numbers and find distances on a number line one at a time. This item chains three such steps with mixed numbers and a negative position, more than typical grade 7 work demands.
Estimated difficulty
Among the hardest on the SHSAT: a non-obvious approach, heavy reading, or a well-hidden trap.
Solution
4 steps
Reading level
Question and choices: about grade 9, 20 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every number is a mixed fraction, one position is negative, and the figure's even spacing hides how long KL really is. The distance route, KL = 9 3/4 − 3 1/2 − 1 1/8, avoids the negative entirely but ends one step short of a position unless K's 3/8 is added back. That last step is the one most students skip, which is why the question sits at the top of the scale.

  1. A very close wrong answer major Choosing the answer

    The choices 5 1/8, 5 1/4 and 5 1/2 lie within 3/8 of one another, so no estimate separates them; only exact eighths and adding K's 3/8 pick out the right one. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  2. Error-prone arithmetic moderate Working it out

    Every step mixes halves, quarters and eighths, and placing J means 3/8 − 3 1/2 = −3 1/8, a subtraction that crosses zero. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  3. Reading a figure, table or graph moderate Understanding the question

    Only K is labeled, and the points are drawn evenly spaced although KL is over four times LM, so the picture gives no help with sizes. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  4. Many steps minor Working it out

    Three linked moves, K to J, J to M and M back to L, each depend on the last, so an early slip carries through. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Start from K's labeled position, subtract JK to place J, add JM to place M, then subtract LM, keeping every fraction in eighths.

Traps

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distance on a number line
d=b−ad = b - a
d is the distance between two points. b is the point on the right and a is the point on the left. Subtracting left from right always gives a positive distance. It works even when a is negative.
Equal fractions
ab=a×kb×k\frac{a}{b} = \frac{a \times k}{b \times k}
a is the top number (numerator) and b is the bottom number (denominator). Multiplying both by the same number k gives an equal fraction. It is like cutting each piece into k smaller pieces. Use it to give fractions the same bottom before you add or subtract.
  1. Sort out what you know

    Only K has a value: 38\frac{3}{8}. JK means the distance from J to K. You know JK = 3123\frac{1}{2}, JM = 9349\frac{3}{4} and LM = 1181\frac{1}{8}.

    You need the position of L, the number where L sits. The points are drawn evenly spaced, but the distances are not equal. So use the numbers, not the picture.

  2. Write the fractions in eighths

    The fractions use halves, fourths and eighths. Give them all a bottom of 8:

    312=348934=9683\frac{1}{2} = 3\frac{4}{8} \qquad 9\frac{3}{4} = 9\frac{6}{8}

    Watch out: 34\frac{3}{4} is 68\frac{6}{8}, not 48\frac{4}{8}. That slip leads to choice B.

  3. Find the distance from K to M

    JM is JK and KM put together. So take JK away from JM:

    968−348=6289\frac{6}{8} - 3\frac{4}{8} = 6\frac{2}{8}

    KM is 6286\frac{2}{8}, or 6146\frac{1}{4}. Watch out: this is a distance, not where L sits. Choice D stops here.

  4. Find the distance from K to L

    KM is KL and LM put together. So take LM away:

    628−118=5186\frac{2}{8} - 1\frac{1}{8} = 5\frac{1}{8}

    KL is 5185\frac{1}{8}. Watch out: this is still a distance. Choice A stops here.

  5. Find the position of L

    L is 5185\frac{1}{8} to the right of K, and K is at 38\frac{3}{8}. Add:

    38+518=548=512\frac{3}{8} + 5\frac{1}{8} = 5\frac{4}{8} = 5\frac{1}{2}

    So the answer is C, 5125\frac{1}{2}.

Check your answer

Place every point. J is 3483\frac{4}{8} left of K, at −318-3\frac{1}{8}. M is 9689\frac{6}{8} right of J, at 6586\frac{5}{8}. L is 1181\frac{1}{8} left of M, at 5485\frac{4}{8}, which is 5125\frac{1}{2}. Same answer.

The original DOE question
Question 111 as printed on page 91 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 91, question 111. Open page 91 of the handbook (PDF)

Question 6 of 6

2024 Test B Math, question 41

Question

A bag contains 2 green, 4 blue, and 3 yellow disks. Alana selects 4 disks from the bag at random, one at a time, without replacing them. What is the probability that she selects all 4 blue disks?

  • A82,187\frac{8}{2{,}187}
  • B1126\frac{1}{126}
  • C2566,561\frac{256}{6{,}561}
  • D169\frac{16}{9}
Show the answer

Correct answer: B

If Alana picks 4 blue disks in a row, the number of blue disks and the total number of disks will both decrease by 1 after each selection.

For the first selection, there are 4 blue disks out of 9 total disks, so the probability of selecting the first blue disk is 49\frac{4}{9}.

The probabilities of selecting blue for the next three selections are 38\frac{3}{8}, 27\frac{2}{7}, and 16\frac{1}{6}.

49×38×27×16=243,024=1126\frac{4}{9}\times\frac{3}{8}\times\frac{2}{7}\times\frac{1}{6}=\frac{24}{3{,}024}=\frac{1}{126}

Question analysis

A bag holds green, blue and yellow disks, and Alana draws four of them one at a time without putting any back. Students find the probability that all four disks are blue. It tests the probability of dependent events, and the difficulty is keeping track of how the bag changes after each draw.

NYS standards
  • 7.SP.8a compound event as fraction of outcomes
  • 7.SP.8b sample spaces for compound events
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 compound probability is usually found with lists or tree diagrams for two stages. Four dependent draws, each changing the counts, are too many to list and need the multiplication rule, which grade 7 does not formally teach.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 5, 37 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Students who know the multiplication rule for successive draws still have two adjustments to make on every draw, and each wrong choice skips at least one of them. Leaving the bag at 9 disks, or putting each disk back, both produce fractions over 6,561 that look as exact as the right one. Writing the four fractions in a row before multiplying, and cancelling as you go, keeps the arithmetic small.

  1. Something you must already know moderate Working it out

    The rule for successive draws without replacement is assumed: each draw's chance is multiplied in, with both the blue count and the total lowered by one. (Anderson & Krathwohl, 2001)

  2. Error-prone arithmetic moderate Working it out

    The product 4/9 × 3/8 × 2/7 × 1/6 = 24/3,024 has to be reduced to 1/126, and two wrong choices hide behind 9 to the fourth power, 6,561. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Write the chance of blue for each of the four draws in turn, lowering both the blue count and the total by one each time, and multiply.

Traps

  • Without replacement: In a second pick there is one fewer item, which changes the probability.
  • Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Probability
P=number of wanted outcomestotal number of outcomesP = \frac{\text{number of wanted outcomes}}{\text{total number of outcomes}}
P is the probability, the chance that something happens. It is a number from 0 to 1. The top counts the results you want. The bottom counts all the results that can happen. It works when every result is equally likely.
Chance of events in a row
P(A then B)=P(A)×P(B after A)P(A \text{ then } B) = P(A) \times P(B \text{ after } A)
A and B are two events that happen one after the other. P(A) is the chance of the first. P(B after A) is the chance of the second, counting only what is left after the first. You multiply because the second must also happen inside the share of times the first did.
  1. Sort out the question

    The bag holds 2 green, 4 blue and 3 yellow disks. That is 9 disks in all. Alana takes 4 disks, one at a time.

    “Without replacing them” means a disk she takes does not go back in the bag. You need the chance that all 4 disks are blue.

  2. Find the chance for each pick

    For all 4 to be blue, each pick must be blue. After each blue pick, one blue disk is gone. One disk is gone from the total, too. So the top and the bottom each go down by 1.

    PickBlue disks leftDisks leftChance of blue
    1st4949\frac{4}{9}
    2nd3838\frac{3}{8}
    3rd2727\frac{2}{7}
    4th1616\frac{1}{6}
  3. Multiply the four chances

    All four picks must go right, one after another. Use the rule for events in a row and multiply.

    49×38×27×16=4×3×2×19×8×7×6=243,024\frac{4}{9} \times \frac{3}{8} \times \frac{2}{7} \times \frac{1}{6} = \frac{4 \times 3 \times 2 \times 1}{9 \times 8 \times 7 \times 6} = \frac{24}{3{,}024}

  4. Simplify the fraction

    Divide the top and the bottom by 24. That keeps the value the same.

    24÷243,024÷24=1126\frac{24 \div 24}{3{,}024 \div 24} = \frac{1}{126}

    Watch out: Using 49\frac{4}{9} for every pick gives 2566,561\frac{256}{6{,}561}, choice C. That is right only if each disk goes back in. Keeping 9 on the bottom each time gives 82,187\frac{8}{2{,}187}, choice A. Choice D is more than 1, and no chance can be more than 1.

    So the answer is B, 1126\frac{1}{126}.

Check your answer

Cancel before you multiply. On top, 4×2=84 \times 2 = 8 cancels the 8 on the bottom. That leaves 39×7×6=3378\frac{3}{9 \times 7 \times 6} = \frac{3}{378}. Divide the top and bottom by 3 to get 1126\frac{1}{126} again.

The original DOE question
Question 98 as printed on page 211 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 211, question 98. Open page 211 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).

Sample #964470