NJGPA Practice Questions (2026) — Free Sample Questions with Answers

14 free NJGPA questions from our 150-question bank, grouped by content domain. Each answer explains why the right option is right — and why every wrong option is wrong.

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Reading: Literary Texts

Question 1

How does the speaker of Poem 2 transform the lines "I was only ever the part of the lesson / that you could see" from Poem 1?

Show answer & explanation

Answer: C

She repeats them to show that she agrees her mother's part in the lesson was small.: She quotes the idea in order to say "You were wrong about one thing."

She uses them to argue that the lake, not her mother, deserves credit for the lesson.: That is Poem 1's own claim; Poem 2 pushes against it.

She disputes them, arguing that her mother's key help was unseen: the voice that went on talking.: Correct. "You were never the part I could see" (stanza 2), and the "trick" was the voice that "kept talking" after the hand was gone (stanza 3).

She mocks them to show that she resented being left alone in the water as a child.: Her tone is grateful ("You said it like a gift, and it was one"), not resentful.

In a reply poem, find the quoted or echoed line and decide whether the new speaker agrees, extends or disputes it.

Question 2

Select the sentence from paragraph 9 of Passage 2 that best states the theme the author develops through Deja's experience.

She thought about her grandmother's story all the way across the street. It had been right about the fire after all, though not in the way she had once believed. At six, she had thought the lesson was about generosity, about being kind to a neighbor in need. Now it seemed to her that the story was about something closer to survival: a flame you keep to yourself can go out, but one you have given away keeps burning somewhere, waiting to be carried home. Gideon's lights were still on.
Click the sentence to select it.
Show answer & explanation

Answer: "Now it seemed to her that the story was about something closer to survival: a flame you keep to yourself can go out, but one you have given away keeps burning somewhere, waiting to be carried home."

  1. Sentence 4 is the key: it states the lesson of Deja's experience. Keeping the starter to herself left the family's only supply open to loss, while the cup given away in 1994 kept "burning" and came back.
  2. Sentence 3 gives the lesson Deja believed at six (generosity), which the passage replaces, so it is not the theme the author develops.
  3. Sentences 1 and 2 introduce her reflection without stating the lesson, and sentence 5 is a closing image.

A theme sentence states what the character learns; skip sentences that only lead up to it or give a view the story moves beyond.

Question 3

How does revealing Lena's envelope in the opening stage directions (paragraph 1) affect the audience's experience of Daniel's speeches in paragraphs 4–8?

Show answer & explanation

Answer: B

It makes the audience suspect that Daniel already knows about the letter and is pretending.: Nothing suggests Daniel knows; his reaction in paragraphs 28–29 confirms he did not.

It creates dramatic irony, since the audience senses his plans may collapse while he stays unaware.: Correct. The audience knows about the envelope, so his excited talk of booths and "Lena and Daniel" carries tension that he cannot feel.

It lets the audience share Augusto's point of view, because he is the only one who notices it.: Augusto notices, but the stage directions show the envelope to the audience directly; the effect is irony, not his point of view.

It leads the audience to see Lena as dishonest for hiding important news from her family.: One step too far: her repeated attempts to speak ("Dad—") show hesitation, not dishonesty.

Dramatic irony occurs when the audience knows something a character does not; ask how that knowledge changes the meaning of the character's lines.

Reading: Informational Texts

Question 4

How does Passage 2 help explain the observation in paragraph 1 of Passage 1 that the sun "looks weak, as through smoked glass"?

Passage 1 is an original work written in the style of an 1816 farm journal. Abel Hutchins is a fictional farmer; the weather he describes follows historical accounts of that year in New England.

Passage 1: From the Journal of Abel Hutchins, Vermont, 1816

1May 20th. Planted the lower field to corn, though the ground is cold to the hand an inch down. Father always said to plant when the oak leaves are the size of a squirrel's ear, but the oaks are sullen this year and will not leaf. A dry, hard spring. The sun rises red and looks weak, as through smoked glass.

2June 7th. Snow. I write it plainly so that I will believe it later. Snow in June, four inches on the meeting-house steps, and the sheep, which we sheared last week, standing pressed against the barn wall for warmth. Lydia brought the lambs into the kitchen. The children went sledding on the hill behind the meeting-house, and I had not the heart to stop them, though it seemed a kind of mockery. The corn that was up is black.

3June 12th. Replanted the lower field. Lydia says we are fools to do it, and she is right, but a man must do something with his hands.

4July 9th. Frost again in the low ground. The Reverend says the cold is a judgment upon us; Deacon Ferris says it is the spots upon the sun, which he has seen through a smoked glass and which he claims are large enough to swallow the earth. I do not know which to believe. I know only that the corn does not care for either explanation. The swallows have left the barn early, and the brook runs as cold as I have known it in October.

5August 22nd. A hard frost in the night, and the corn is ruined past saving. We will buy what hay we can for the winter, though the price is already twice what it was. Lydia counts the potatoes in the cellar each evening, as if counting could make more of them. Neighbor Kimball is selling his farm and going west to Ohio, where he says the winters are kinder.

6September 30th. The harvest, such as it is, is in. Lydia says perhaps the weather has always been this cruel somewhere in the world, and we have only been lucky until now. I think often of the red sun in May. Something has come between us and the sun, I am sure of it, though I cannot say what. I suppose no one will ever know.

Passage 2: The Volcano Behind the Year Without a Summer — by Isadora Grantham

1In April 1815, Mount Tambora, a volcano on the island of Sumbawa in what is now Indonesia, produced the largest volcanic eruption in recorded history. Its explosions were heard more than a thousand miles away, and its ash buried farms and villages across the island and its neighbors.

2But Tambora's most far-reaching effect was invisible. Along with ash and rock, the eruption blasted enormous amounts of sulfur dioxide gas into the stratosphere, the layer of the atmosphere that begins roughly six to ten miles above the ground. There, the gas reacted with water to form a fine haze of tiny sulfate droplets, called aerosols. Because the stratosphere has no rain to wash them out, such droplets can linger for a year or more, spreading around the globe on high-altitude winds. They reflect part of the incoming sunlight back into space, and less sunlight reaching the surface means a cooler planet.

3The result was 1816, remembered in New England as "the year without a summer." Snow fell in parts of New England in early June. Frosts struck in July and August, killing crops across the region. In western Europe, cold rains ruined harvests, and food prices soared. In New England, some families gave up their farms altogether and moved west, hoping for better land and kinder weather. Scientists estimate that average global temperatures fell by less than one degree Celsius. That may sound small, but a global average hides large regional swings, and a single killing frost in the growing season can destroy a crop that months of mild weather cannot save.

4People at the time had no way to connect their ruined crops to a volcano on the other side of the world. News traveled slowly, and the science of the atmosphere was in its infancy. Even careful observers could record only what they saw. Some blamed sunspots, which happened to be visible to the naked eye that year; others saw the cold as a religious sign. It was not until the twentieth century that scientists firmly linked major eruptions to periods of global cooling.

5Today the evidence comes from several directions. Ice cores drilled from the Greenland and Antarctic ice sheets contain layers of sulfate that settled out of the atmosphere after major eruptions, and the layer from Tambora is among the largest of the past several centuries. Tree rings from the period show stunted growth. Old weather diaries, kept by farmers, ship captains and clergy, record the cold day by day and help scientists reconstruct where and when it struck. Together, these independent lines of evidence point to the same conclusion.

6In 1991, scientists were able to watch the process happen. When Mount Pinatubo erupted in the Philippines, satellites tracked its sulfur cloud as it spread around the globe. Over the following year or so, average global temperatures dropped by about half a degree Celsius, close to what researchers had predicted. Pinatubo's eruption was smaller than Tambora's, but it confirmed the basic mechanism: what goes up into the stratosphere can, for a time, turn down the sun. Researchers now use such events to test the computer models that simulate Earth's climate.

Show answer & explanation

Answer: A

It suggests that a haze of sulfate droplets from Tambora was reflecting part of the sunlight before it reached the ground.: Correct. Passage 2, paragraph 2, explains that the sulfate aerosols spread around the globe and "reflect part of the incoming sunlight," which matches a dim sun in May 1816.

It shows that the sunspots seen that year were large enough to dim the sun noticeably for observers across New England that spring.: Passage 2 mentions that people "blamed sunspots," presenting this as a mistaken explanation, not as the cause.

It indicates that ash from Tambora fell on Vermont and darkened the sky over the farm.: Confuses ash, which fell near the volcano, with the sulfate haze that traveled worldwide.

It explains that viewing the sun through smoked glass makes it look dimmer than it is.: The farmer's "as through smoked glass" is a comparison; Deacon Ferris's smoked glass is a different detail.

When a scientific text explains a historical one, match the mechanism (reflected sunlight) to the specific observation (a weak sun).

Question 5

How does the treatment of the three-hour delay differ between the two passages?

Passage 1: Historic Bridge Rolls to New Home — by Corbin Lattimore, *The Fairmont Ledger*

1KESTREL FALLS — After nine hours, two broken cables and one very long delay, the Hadley Creek Covered Bridge came to rest Saturday on new stone footings in Ashgrove Park, about 600 feet downstream from the spot where it had stood since 1871.

2The 94-foot wooden bridge, one of the few covered bridges remaining in the county, was moved to protect it from flooding. Engineers determined last year that its original site, at a sharp bend in the creek, had flooded four times since 2011 and that another major flood could wash out its supports. Relocating the bridge to the park, where it will carry pedestrians and cyclists, and building a new concrete bridge for cars at the old site was estimated to cost about $1.2 million less than restoring the covered bridge in place.

3The move was carried out by Gantry & Lowe Structural Movers, which lifted the 62-ton structure onto steel beams and hydraulic dollies on Friday. Work began at 7 a.m. Saturday. Winches pulled the bridge at roughly two feet per minute, and crews paused often to check the load.

4Shortly before 11 a.m., a winch cable snapped, and work stopped for nearly three hours while a replacement was brought from a nearby quarry. No one was injured, and project manager Beatrice Coulter said the bridge was not damaged. "That's why we go slow," Coulter said. "When something lets go, you want it to let go at two feet a minute, not twenty." A second cable failed in the afternoon but was replaced in under thirty minutes.

5An estimated 600 people watched from the creek banks and the park hillside over the course of the day. Volunteers from the Kestrel Falls Historical Society and members of a high school engineering club handed out water and helped keep spectators behind safety barriers. The historical society raised $214,000 of the project's $1.9 million cost through donations; state and county funds covered the rest.

6The bridge reached its footings at 4:12 p.m. and was lowered into place by 6:30 p.m. Crews will spend the next several weeks bolting it down, rebuilding its approaches and installing lighting. Town officials expect it to reopen to foot traffic in the fall.

7"It has been standing for a hundred and fifty years," Mayor Elena Haddad said at a brief ceremony after the bridge was set down. "We wanted to make sure it gets another hundred and fifty."

Passage 2: The Day the Bridge Walked — by Farida Jalloh

1I have lived in Kestrel Falls my whole life, and I had crossed the Hadley Creek bridge maybe a thousand times without once thinking of it as something that could move. It was just there, like the creek and the hill: a red tunnel with a plank floor that thumped under my bike tires. When our engineering club signed up to volunteer on moving day, I expected a long morning of handing out water. I did not expect to be standing on the bank at 10:52 a.m. when the cable went.

2It sounded like the crack of a baseball bat, only bigger, and every head on the hillside turned at once.

3For a second nobody breathed. The bridge didn't fall. It didn't even shake that I could see. It just stopped, the way a person stops when she realizes she has forgotten her keys, and sat there on its steel beams halfway between where it had always been and where it was supposed to go. Then the crew foreman blew a whistle, and the workers walked—walked, not ran—to the winch.

4The nearly three hours after that were the strangest part of the day, and the part I think about most. Hundreds of people had come to watch a bridge move, and now it wasn't moving, and nobody left. Someone's grandfather set up a folding table and handed out slices of a sheet cake that said HAPPY MOVING DAY in blue icing. Kids held a contest to guess the minute the bridge would reach the park. An old woman named Mrs. Bellweather told everyone nearby that she had first held hands with her husband inside that bridge in 1961.

5I spent those hours with Ms. Caldwell, our club advisor, and a sketchbook, trying to work out how the movers would replace the cable. She kept asking questions instead of answering mine. Where would the load shift if one winch stopped pulling? Why move at two feet a minute instead of ten? By the time the new cable arrived, I had filled four pages with diagrams, and I understood, in a way no textbook had managed, why slow can be the safest engineering decision there is.

6When the bridge started moving again, just before two o'clock, the whole hillside stood up. It is hard to describe a building traveling. It doesn't roll so much as glide, so slowly that you never see it move, but every time you look away and back, the world has rearranged itself a little around it. A heron that had stood in the creek all morning finally gave up and flew off.

7A second cable broke later that afternoon, and this time nobody gasped; we had all become experts. At 4:12 p.m., the bridge reached its new footings. I know the exact time because I won the guessing contest—well, tied, with a fourth grader named Leo who had guessed 4:12 for no reason he could explain.

8Mrs. Bellweather has asked to be the first person to walk through the bridge when it reopens. She told me it looks smaller in its new spot. I don't think it does. I think it looks exactly as big as it always was. We just saw how much it took to carry it.

Show answer & explanation

Answer: D

Passage 1 presents the delay as a serious threat to the bridge, while Passage 2 treats it as only a minor annoyance for the crowd.: Passage 1 reports that the bridge was "not damaged"; it does not present a threat.

Passage 1 blames the moving company for the delay, while Passage 2 blames the quarry.: Neither passage assigns blame; the quarry supplied the replacement cable.

Passage 1 describes the delay in detail, while Passage 2 mentions it only in passing.: Reversed: Passage 2 gives the delay two full paragraphs (4–5).

Passage 1 reports the delay's cause and safety meaning, while Passage 2 dwells on how the waiting time was spent.: Correct. Passage 1 gives the cable failure and Coulter's explanation; Passage 2 describes the cake, the contest and the narrator's diagrams.

Comparing accounts means noticing what each one emphasizes, such as facts and causes in a news report versus experience in a personal account.

Question 6

Which statement best expresses a central idea of the speech?

The following speech is an original work written in the style of an early-twentieth-century public address. The speaker, Eliza Hartwell, and the town of Larkspur Mills are fictional.

Address at the Opening of the Larkspur Free Library, 1904 — by Eliza Hartwell

1Friends and neighbors of Larkspur Mills: I have been asked to say a few words before we open these doors, and I will try to keep them few, for I know that most of you have come here from a ten-hour shift, and that the books behind me are better company than any speech.

2Twenty years ago there was a library in this town. Some of you will remember it. It occupied two rooms above the bank, and it belonged to the Larkspur Subscribers' Association, which charged five dollars a year for membership. Five dollars was a week's wages for a spinner in those days. The library was open from ten until four, when every spinner was at her frame. It held, I am told, a very fine set of the classics bound in leather, and I am also told that the bindings were in excellent condition when the association dissolved, because they had hardly been opened.

3I do not tell this story to shame the subscribers. They were generous people, and they believed in books. But they believed in books the way one believes in a beautiful garden behind a high wall—as something to be kept, admired and protected rather than walked in. They built a library for the few who already had the leisure to read, and then they wondered why the town did not grow wiser.

4The building we open today rests on a different idea: that knowledge is not a luxury, like silk or silver, which a town may enjoy once its necessities are met. Knowledge is itself a necessity, like bread and clean water. A town that hoards it starves; a town that shares it thrives. This library is not a garden behind a wall. It is a granary, and its doors open outward.

5Consider what is on these shelves. There are novels, yes, and poems, and the leather classics—for we have bought them back. But there are also manuals of mechanics and chemistry, of bookkeeping and law, of farming and medicine. There is a reading room with the newspapers of four cities. There is a corner for children, with low tables and books they may touch with their own hands. And there are lamps, good lamps, because these doors will stay open until nine o'clock every evening but Sunday, so that the spinner and the carter and the machinist may come when their work is done.

6Some have asked why a mill town needs chemistry and law. I will answer with questions of my own. Who most needs to understand the dyes that color our cloth—the owner who buys them, or the dyer who breathes them all day? Who most needs to read a contract—the man who writes it, or the man who must sign it? It is not the comfortable who need knowledge most. It is those who have the least protection without it.

7Others have warned me that books will make working people discontented. I do not doubt it. I hope they will. I hope that a boy who reads about the great bridges of Europe will grow discontented with the rickety footbridge over our river and become the engineer who replaces it. I hope that a girl who reads about the circulation of the blood will grow discontented with being told that fevers come from night air, and become the doctor this town has never had. Discontent that has learned to read does not merely complain. It builds.

8I have been told, too, that a free library is a charity, and that charity weakens those who receive it. But no one calls a public road a charity, though every one of us walks on it. No one calls a bridge a charity, though the rich man's carriage and the peddler's cart cross it alike. A library is a road. It is the road by which a mind travels from what it knows to what it does not, and like every road, it serves the town best when no toll is collected at the gate.

9When these doors open in a moment, I ask only this of you. Do not treat these books the way the subscribers did. Do not keep them clean. Wear them out. Bend their spines, smudge their pages, carry them home in your coat pockets through the rain and bring them back with the corners turned down. A worn book is a book that has done its work, and in ten years, when our librarian tells me that half the collection must be replaced, I shall consider it the finest report this town could give.

10The doors are open. Go in.

Show answer & explanation

Answer: A

Knowledge is a public necessity that serves a town best when everyone can reach it freely.: Correct. Paragraph 4 calls knowledge "a necessity, like bread and clean water," and paragraph 8 says the library, like a road, serves best "when no toll is collected."

Wealthy citizens have a duty to donate their books to the poor of the town as an act of charity.: Hartwell rejects the idea that the library is "a charity" (paragraph 8).

Working people should read so that they become more content with their lives.: Reversed: she hopes books will make readers "discontented" and lead them to build (paragraph 7).

The earlier subscription library failed because its books were poorly chosen.: Its classics were "very fine"; it failed because of fees and hours, not the books.

In a speech, the central idea is the claim that the examples and answers to objections all support.

Question 7

Which statement best expresses a central idea of the memoir?

Halfway Dog — a memoir by Ingrid Castellanos

1The puppy arrived on a Tuesday in a plastic crate with a laminated card tied to the door. HAZEL, the card said. FEMALE. YELLOW LABRADOR. RETURN DATE: APPROXIMATELY 15 MONTHS. I was fourteen, and I remember thinking that "approximately" was a strange word to put on a living thing.

2My family had signed up to be puppy raisers for the Fairhaven Guide Dog School, a nonprofit that trains dogs to guide people who are blind or have low vision. The arrangement is simple to describe. The school breeds the puppies and, when they are about eight weeks old, places each one with a volunteer family. The family raises the dog for a year or more: house-training it, teaching it basic cues such as sit, down and wait, and above all introducing it to the world—buses, elevators, grocery stores, crowds, sirens, slippery floors, automatic doors. Then the dog returns to the school for months of formal training, and if it succeeds, it is matched with a person who will depend on it.

3What is not simple is the part where you give the dog back. Everyone asked me about it. Teachers asked; my friends asked; a woman in the cereal aisle at Lockwood's Market, seeing Hazel in her green "puppy in training" jacket, asked, "How will you ever let her go?" I had a cheerful answer ready: "She has an important job to do." I said it so many times that it stopped meaning anything, the way a word does when you repeat it to yourself in the mirror.

4For the first few months, though, I did not think much about giving her back, because I was too busy. Hazel had to go outside every two hours. She ate a sock, a pencil and most of a library book about ancient Rome. (We paid for the book.) Every outing was a lesson: Hazel learning to lie quietly under a restaurant table, Hazel learning that the hum of an escalator was nothing to fear, and me learning that a puppy in a jacket attracts strangers the way a dropped ice cream cone attracts ants. The school asked raisers to keep a log, and I filled three notebooks. Most entries looked like this one: Walked through hardware store. Startled at paint shaker. Recovered in 4 seconds. Good girl.

5Our raisers' group met every other Saturday with a supervisor named Lucille Gaines, who had trained guide dogs for thirty years. She could watch a puppy cross a parking lot and know, from the set of its ears, whether it was worried. Once, when I complained that Hazel still pulled on the leash, Lucille said something I copied into my log that night: "You're not raising a perfect dog. You're raising a confident one. The school can teach her to guide. It can't teach her to trust the world. That's your job."

6I did not fully understand that until the day we rode the city bus for the first time. Hazel was nine months old. The bus hissed and lurched, the floor vibrated, and a man two seats away was playing music loudly through his phone. Halfway through the ride I looked down and saw that Hazel had fallen asleep with her chin on my sneaker. Months of grocery stores and escalators and paint shakers had added up to this: a dog who could sleep on a bus. Someday, I realized, someone would need to get on a bus with her and trust that she would not panic. Every entry in my notebooks had been written for that person.

7Hazel went back to Fairhaven when she was sixteen months old. I will not pretend that day was easy. I walked her into the kennel building, unclipped her leash and handed it to a trainer, and Hazel trotted off without looking back, which Lucille later told me was the best sign of all. In the car, my mother passed me a box of tissues without saying a word.

8Not every puppy becomes a guide. Some dogs turn out to be too easily distracted, or too cautious, or simply better suited to another kind of life, and the school finds them other work or adoptive homes. For five months I checked my email every day. Then the message came: Hazel had passed her training. She had been matched with Bernadette Kwan, a music teacher who had lost her sight in her twenties, and we were invited to the graduation.

9At the ceremony, Bernadette held Hazel's harness handle and asked me to tell her everything—what Hazel had been like as a puppy, what she was afraid of, what she loved. I told her about the sock and the library book and the paint shaker. Then I told her about the bus. She laughed. "I take the bus to work every day," she said. "She's never once been nervous."

10People still ask me how I could let her go. I no longer give the cheerful answer. What I tell them now is that I never had her to keep. I had her to get ready. That was the job, and Hazel and I did it together.

Show answer & explanation

Answer: A

Raising a guide dog puppy means preparing it for someone else, and the author finds purpose in letting go.: Correct. The memoir moves from the dreaded "giving back" (paragraph 3) to "I had her to get ready" (paragraph 10).

Guide dog schools rely far more heavily on their volunteer families than on the professional trainers they employ.: Overreach: the memoir describes both roles and never ranks them.

Most puppies raised by volunteers do not go on to become working guide dogs.: The memoir says only "Not every puppy becomes a guide"; it gives no proportion.

Training a puppy is mainly a matter of correcting habits like chewing and pulling.: Too narrow: those details are early episodes; Lucille says the real job is building confidence.

A central idea is supported from beginning to end; check that it covers both the opening worry and the closing reflection.

Algebra

Question 8

The graphs of and intersect at two points. Which values of are the solutions to the equation ?

Show answer & explanation

Answer: C

  1. 1. Set the functions equal: , so .
  2. 2. Factor: , so or .
  3. 3. Check: , and . ✓ The intersection points are and .

and : Factoring sign swap: expands to , not .

and : These are the zeros of alone, where ; they ignore .

and : Correct. , and both values give equal outputs.

and : These are the y-coordinates of the intersection points (−1, 0) and (4, 5), not the x-values.

The x-coordinates of the intersection points of y = f(x) and y = g(x) are exactly the solutions of f(x) = g(x).

Question 9

Yusuf pours a cup of tea and sets it on a counter. The temperature of the tea, in degrees Celsius, minutes later is modeled by . Select all the statements that correctly interpret the model.

Select all that apply.
Show answer & explanation

Answer: B, C, E

  1. 1. At : °C, the starting temperature.
  2. 2. The term is the amount by which the tea is warmer than 22°C; it is multiplied by 0.9 each minute, so it shrinks toward 0 and approaches 22°C.
  3. 3. At : °C. The drop from 90 to 83.2 is 6.8°C, about 7.6%, not 10% of the temperature.

The tea's temperature decreases by 10% each minute.: The 10% decrease applies only to the 68-degree gap above 22°C; the temperature itself falls from 90 to 83.2, about 7.6%.

The tea's temperature when it is poured is 90°C.: Correct: at m = 0 the power equals 1, so T = 22 + 68 = 90.

The value 22 is the temperature that the tea approaches as time passes.: Correct: 68(0.9)^m gets closer to 0 as m increases, so T gets closer to 22°C.

The tea cools by 0.9°C each minute.: 0.9 is a multiplying factor, not a constant amount subtracted each minute; that reading treats the model as linear.

The difference between the tea's temperature and 22°C is multiplied by 0.9 each minute.: Correct: T − 22 = 68(0.9)^m, so each additional minute multiplies that difference by 0.9.

After 1 minute, the tea's temperature is 81°C.: Applies 0.9 to the whole 90°C (90 × 0.9 = 81); the model gives 22 + 61.2 = 83.2°C.

In , the constant is the level approached and is the gap from that level; the factor acts on the gap, not on the whole temperature.

Functions

Question 10

Two new clubs began recruiting on the same date. Which statement about the two clubs during weeks 0 through 4 is true?

Riverbend Running Club members
Week01234
Members3036424854

The number of members of the Westlake Cycling Club weeks after the same start date is modeled by .

Show answer & explanation

Answer: B

  1. 1. Westlake: , , , , .
  2. 2. Riverbend: 30, 36, 42, 48, 54 (adds 6 per week).
  3. 3. Week 3: 40 < 48; week 4: 80 > 54, so Westlake first has more members at week 4.
  4. 4. Average rates of change over weeks 0–4: Riverbend (54 − 30)/4 = 6; Westlake (80 − 5)/4 = 18.75 members per week.

Westlake has more members than Riverbend at week 3.: At week 3 Westlake has 5(2)³ = 40 members and Riverbend has 48.

Westlake first has more members than Riverbend at week 4.: Correct. Westlake trails 40 to 48 at week 3 and leads 80 to 54 at week 4.

Riverbend gains members faster than Westlake from week 0 to week 4.: Riverbend gains 6 per week on average, but Westlake gains (80 − 5)/4 = 18.75 per week.

Riverbend has more members than Westlake at every week from 0 to 4.: True for weeks 0 through 3 only; at week 4 Westlake has 80 members and Riverbend has 54.

To compare functions given in different forms, evaluate both on the same inputs; an exponential function eventually overtakes a linear one.

Statistics and Probability

Question 11

A company compared commute times at its two offices. Select all the statements that are supported by the data.

One-way commute times (minutes) of 9 employees at each office
OfficeCommute times
Mapleton12, 15, 18, 20, 22, 24, 25, 28, 34
Northview10, 14, 19, 21, 22, 23, 30, 38, 62
Select all that apply.
Show answer & explanation

Answer: A, B, D

  1. 1. Medians (5th value): Mapleton 22, Northview 22.
  2. 2. Means: Mapleton 198 ÷ 9 = 22; Northview 239 ÷ 9 ≈ 26.6.
  3. 3. IQR: Mapleton Q1 = 16.5, Q3 = 26.5, IQR = 10; Northview Q1 = 16.5, Q3 = 34, IQR = 17.5.
  4. 4. Without 62, Northview has median (21 + 22)/2 = 21.5 (down 0.5) and mean 177 ÷ 8 ≈ 22.1 (down about 4.4).

The two offices have the same median commute time.: Correct: the middle (5th) value is 22 minutes for both offices.

The mean commute time at Northview is greater than Northview's median.: Correct: the 62-minute outlier pulls the mean up to about 26.6, above the median of 22.

The two offices have the same mean commute time.: Mapleton’s mean is 22 but Northview’s is about 26.6; equal medians do not imply equal means.

Northview's interquartile range is greater than Mapleton's.: Correct: 17.5 minutes versus 10 minutes.

Removing the 62-minute commute changes Northview's median more than its mean.: The median drops only 0.5 minute, while the mean drops about 4.4 minutes.

The range is the best measure for comparing the spreads because of the 62-minute commute.: The range is the measure most distorted by the outlier; the IQR is the more appropriate comparison.

The mean is pulled toward an outlier while the median resists it; use the median and IQR to compare data sets with outliers.

Similarity, Right Triangles, and Trigonometry

Question 12

For each pair of triangles described, select whether the triangles must be similar or are not necessarily similar.

Must be similarNot necessarily similar
Triangle 1 has sides 6, 8 and 10; triangle 2 has sides 9, 12 and 15.
Triangle 1 has sides 3, 4 and 5; triangle 2 has sides 6, 8 and 12.
Triangle 1 has angles of 40° and 75°; triangle 2 has angles of 75° and 65°.
Triangle 1 has sides 4 and 6 with a 50° angle between them; triangle 2 has sides 6 and 9 with a 50° angle between them.
Triangle 1 has sides 5 and 7 with a 30° angle opposite the side of length 5; triangle 2 has sides 10 and 14 with a 30° angle opposite the side of length 10.
Both triangles are right triangles, and each has a 35° angle.
Select one option in each row.
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Answer: Triangle 1 has sides 6, 8 and 10; triangle 2 has sides 9, 12 and 15.: Must be similar; Triangle 1 has sides 3, 4 and 5; triangle 2 has sides 6, 8 and 12.: Not necessarily similar; Triangle 1 has angles of 40° and 75°; triangle 2 has angles of 75° and 65°.: Must be similar; Triangle 1 has sides 4 and 6 with a 50° angle between them; triangle 2 has sides 6 and 9 with a 50° angle between them.: Must be similar; Triangle 1 has sides 5 and 7 with a 30° angle opposite the side of length 5; triangle 2 has sides 10 and 14 with a 30° angle opposite the side of length 10.: Not necessarily similar; Both triangles are right triangles, and each has a 35° angle.: Must be similar

  1. 1. AA: two pairs of congruent angles (find third angles from the 180° sum).
  2. 2. SSS: all three side ratios equal.
  3. 3. SAS: two side ratios equal and the included angles congruent.
  4. 4. An angle that is not between the two proportional sides (SSA) does not guarantee similarity.

AA, SAS (included angle) and SSS establish similarity; SSA and unequal side ratios do not.

Question 13

Soraya stands at the top of a cliff, 45 meters above sea level. She sees two boats directly in line with the base of the cliff, on the same side. The angle of depression to the nearer boat is 38°, and the angle of depression to the farther boat is 24°. To the nearest meter, how far apart are the two boats?

Enter a number. Fractions may be entered as a/b.
Show answer & explanation

Answer: 43 meters

  1. 1. The angle of depression equals the angle of elevation from each boat (alternate interior angles), so in each right triangle the 45 m cliff is opposite the angle.
  2. 2. Nearer boat: , so m.
  3. 3. Farther boat: m.
  4. 4. Distance apart: , which is 43 m to the nearest meter.

Solve each right triangle separately for its horizontal leg, then subtract; using 45 tan θ instead of 45 / tan θ puts the angle at the wrong vertex.

Question 14

In the figure, . Which similarity statement must be true?

ABCDE
Point D is on side AB and point E is on side AC of triangle ABC. The marked angles, ∠ADE and ∠ACB, are congruent. Figure not drawn to scale.
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Answer: C

  1. 1. is shared by both triangles, and is given.
  2. 2. By AA, the triangles are similar with and ; the third vertices correspond, so .
  3. 3. Write the statement in that order: .

: Assumes , which would pair with ; the given angles pair with .

: Pairs with , but is the angle shared by both triangles, so must correspond to .

: Correct. Shared and give AA similarity with , , .

: Pairs with , matching to ; the given congruence is at , not .

Match vertices by the congruent angles, not by where they appear on the figure; DE is not parallel to BC here, so D does not correspond to B.

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