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Question
The line shown models the recommended combinations of the number of guppies and angelfish that can survive in a 50-gallon aquarium, based on the gallons of water that each type of fish needs. Based on this model, how many gallons of water does each angelfish need?
Hint :
The point (0, 5) represents 0 guppies and 5 angelfish in the aquarium.
Explanation
Each (x, y) point on the graph represents a number of guppies (x) and a corresponding number of angelfish (y) that can survive in a 50-gallon aquarium filled with water.
To determine the gallons of water for each angelfish, notice that the graph intersects the y-axis at (0, 5). This point represents 0 guppies and 5 angelfish, so it is possible for 5 angelfish to survive in 50 gallons of water.
Therefore, the total gallons of water (50) is equal to the number of angelfish (5) multiplied by the gallons of water for each angelfish. Let a represent the gallons of water for each angelfish.
Now solve the equation for a.
| 50 = 5a | |
| 10 = a | Divide by 5 on both sides |
Each angelfish needs 10 gallons of water.
Things to remember:
An amount per item is equal to the total amount divided by the number of items.
Passage:
Existing solutions for preventing food waste are often considered ______ because a high percentage of food that is potentially edible continues to be thrown away. But Susanne Lee wants to help consumers in Maine reduce food waste with an app called Food Rescue US, which connects local grocers, restaurants, and caterers with volunteers who deliver it to soup kitchens and shelters.
Question
Which choice completes the text with the most logical and precise word or phrase?
| A. utilitarian | |
| B. insufficient | |
| C. beneficial | |
| D. flexible |
Explanation
Identify words in the text that provide clues to the best answer.
The sentence with the blank states that despite efforts to prevent food waste, a high percentage of food "continues to be thrown away," which indicates that attempts to reduce waste have been largely ineffective.
"Insufficient" means not enough to meet a goal—here, of preventing food waste. Existing solutions have not met that goal and would therefore be considered insufficient.
(Choice A) "Utilitarian" means useful for achieving a purpose. Solutions that fail to reduce the high percentage of food waste are not useful for achieving their intended purpose.
(Choice C) "Beneficial" means producing good or helpful results. However, the efforts to prevent food waste haven't helped much, given the high percentage that is thrown away.
(Choice D) "Flexible" means capable of adapting to new or changing requirements. Solutions that are considered flexible would not be explained by a high percentage of food waste. The text doesn't discuss whether attempts to reduce food waste have been changed or adapted.
Things to remember:
Select the word that supports the meaning found in the surrounding context.
Question
The rational function g is defined by an equation in the form g(x) = a / (x - b), where a and b are constants. The partial graph of y = g(x) is shown. If h(x) = g(x - 5), which equation could define function h ?
| A. h(x)=2/x | |
| B. h(x)=2/x−5 | |
| C. h(x)=2/x−10 | |
| D. h(x)=2(x−5)/x−5 |
Hint :
The function h(x) = g(x − 5) is defined in terms of g(x), so h represents a transformation to the function g.
The graph of a rational function has a vertical asymptote at any value of x where the denominator is equal to 0 and the numerator is not equal to 0.
The function h(x) = g(x − 5) is defined in terms of g(x), so h represents a transformation to the function g.
For any function f(x), the graph of y = f(x − h) + k is the graph of y = f(x) transformed by a horizontal shift of h units and a vertical shift of k units.
In h(x) = g(x − 5), the number 5 is subtracted from the input x so it represents a horizontal shift of 5 units to the right.
Notice that the graph of the rational function g(x) approaches x = 5 but never intersects x = 5, so g(x) has a vertical asymptote at x = 5.
After a shift 5 units to the right, the graph of h(x) must have a vertical asymptote at x = 10.
The graph of a rational function has a vertical asymptote at any value of x where the rational expression is undefined because the denominator is equal to 0 (and the numerator is not 0).
Set the denominator of the equation in each choice equal to 0 and solve for x to see which could have a vertical asymptote at x = 10.
Only the equation in Choice C could represent a rational function with a vertical asymptote at x = 10.
Note: The equation in Choice D is undefined when x = 5. However, its graph does not have a vertical asymptote because both the numerator and denominator are 0 when x = 5. Therefore, this graph has a hole at x = 5.
Things to remember:
-
For any function f(x), the graph of y = f(x − h) + k is the graph of y = f(x) transformed by a horizontal shift of h units and a vertical shift of k units.
-
The graph of a rational function has a vertical asymptote at any value of x where the denominator is equal to 0 and the numerator is not equal to 0.
-
A vertical asymptote is a vertical line that a graph of a function approaches but never intersects.
Alternate Method :
The graph of a rational function has a vertical asymptote at any value of x where the rational expression is undefined because the denominator is equal to 0 (and the numerator is not 0).
Identify the denominator of the given function g(x).
Set the denominator x − b equal to 0 and solve for x to see that the g(x) has a vertical asymptote at x = b. To find the value of b, identify the vertical asymptote of g(x) from the given graph.
The graph of g(x) approaches x = 5 but never intersects x = 5, so g(x) has a vertical asymptote at x = 5. Therefore, b must equal 5. Plug b = 5 into the given equation for g(x).
The question asks for an equation of h(x). The function h(x) = g(x − 5) is defined in terms of g(x). The function notation g(x − 5) represents the output (or value) of g(x) when the input is x − 5.
To get an equation for h(x), substitute x − 5 for each instance of x in g(x) = a / (x − 5).
| g(x) = a / (x − 5) | Given function g(x) |
| g(x − 5) = a / x − 5 − 5 | Substitute x − 5 for each instance of x |
| g(x − 5) = a / (x − 10) | Simplify |
Therefore, an equation for h(x) is h(x) = a / (x − 10). The only equation from the choices that could represent h(x) is h(x) = 2 / (x − 10).
Note: It is possible to plug the coordinates of a point from the given graph into g(x) = a / (x − 5) and solve for a to confirm that a = 2.
Things to remember:
-
The graph of a rational function has a vertical asymptote at any value of x where the denominator is equal to 0 and the numerator is not equal to 0.
-
A vertical asymptote is a vertical line that a graph of a function approaches but never intersects.
Question
Data sets X and Y are shown in the graphs above. Each data set consists of 18 integers. Which of the following statements must be true?
| A. Data sets X and Y have the same mean, but the standard deviation of data set X is greater than the standard deviation of data set Y. | |
| B. Data sets X and Y have the same mean, but the standard deviation of data set Y is greater than the standard deviation of data set X. | |
| C. Data sets X and Y have the same standard deviation, but the mean of data set X is greater than the mean of data set Y. | |
| D. Data sets X and Y have the same standard deviation, but the mean of data set Y is greater than the mean of data set X. |
Hint :
The standard deviation of a data set is a measure of how spread out the data points are from the mean.
Explanation
Analyze each bar graph to determine how the mean and standard deviation of each data set compare. The horizontal axis gives the data values, and the vertical axis gives the number of those values in that set.
Notice that both graphs are symmetric. In a symmetric distribution, the mean and the median (or middle value) are equal. Each graph is symmetric about the same middle value (7), so the mean of each data set is equal to 7.
Eliminate Choices C and D because the data sets do not have different means.
The standard deviation (SD) of a data set is a measure of how spread out the data points are from the mean. The more the values are spread out from the mean, the greater the standard deviation.
Each mean is 7, so analyze how spread out the data points are from 7 in each graph.
The data points of set X are further from the mean than the data points of set Y, so the standard deviation of data set X is greater than the standard deviation of data set Y. Eliminate Choice B.
Therefore, the statement in Choice A must be true: Data sets X and Y have the same mean, but the standard deviation of data set X is greater than the standard deviation of data set Y.
Note: It also is possible to use the mean formula to calculate the mean of each data set.
Things to remember:
- The standard deviation of a data set is a measure of how spread out the data points are from the mean. The more the values are spread out, the greater the standard deviation.
- In a symmetric distribution, the mean and the median are equal.
Passage
A staple of modern history books, timelines were first developed by Jacques Barbeu-Dubourg in the 1750s. His early timelines and those made by others generally charted the lives of individuals on a chronological grid, reflecting the Enlightenment assumption that history could be measured against an absolute scale of time, moving inexorably onward from zero. Drawn from calendars and chronologies, ______ plotted the lives of two thousand men between 1200 B.C.E. and 1750 C.E.
Question
Which choice completes the text so that it conforms to the conventions of Standard English?
| A. Chart of Biography, a timeline created in 1765 by Joseph Priestley, | |
| B. Joseph Priestley, creator of the 1765 Chart of Biography, | |
| C. there was a creator of timelines, Joseph Priestley, who in the 1765 Chart of Biography | |
| D. the timeline creator who in 1765 published Chart of Biography, Joseph Priestley |
Explanation
Rule: The first noun after an introductory participle phrase should do the action described by that phrase.
Evaluate what the opening phrase refers to and choose the answer that begins with the noun it describes.
"Drawn from calendars and chronologies" is a participle phrase referring to something that is based off of calendars and chronologies. Because a chart can be based off of calendars and chronologies, Chart of Biography, a timeline created in 1765 by Joseph Priestley, is correct.
(Choice B) A person isn't created from calendars and chronologies, so the participle phrase cannot refer to the noun "Joseph Priestley," meaning that this noun shouldn't immediately follow the participle phrase.
(Choices C and D) The participle phrase should be followed by the noun it describes, but in these answers the closest noun—"creator"—refers to a person. The participle phrase refers to something that is based on calendars and chronologies, which a person would not be.
Things to remember:
Select the answer with a noun that does the action described by the participle phrase.
Question
In triangle PQR, the measure of ∠P is 102° and the measure of ∠Q is 47°. In triangle STU, the measure of ∠S is 102° and the measure of ∠T is 47°. Which additional piece of information is needed to determine whether triangles PQR and STU are similar?
| A. The measure of angles R and U | |
| B. The measure of angle R | |
| C. The measure of angle U | |
| D. No additional information is needed |
Hint :
Angles that are equal in measure are congruent.
Explanation
The Angle-Angle similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Angles are congruent if they are equal in measure.
It is given that angles P and S measure 102° and that angles Q and T measure 47°. Therefore, triangles PQR and STU are similar because they have two pairs of congruent angles.
The given information is sufficient to determine that triangles PQR and STU are similar. No additional information is needed.
Note: Interior angle measures of a triangle sum to 180°, so it is possible to subtract 102° and 47° from 180° to determine that ∠R and ∠U both measure 31°. However, this is not necessary to determine similarity.
(Choices A, B, and C) These choices may result from the misconception that all three pairs (instead of any two pairs) of angle measures must be known to determine whether triangles are similar.
Things to remember:
The Angle-Angle similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Passage:
Existing solutions for preventing food waste are often considered ______ because a high percentage of food that is potentially edible continues to be thrown away. But Susanne Lee wants to help consumers in Maine reduce food waste with an app called Food Rescue US, which connects local grocers, restaurants, and caterers with volunteers who deliver it to soup kitchens and shelters.
Question
Which choice completes the text with the most logical and precise word or phrase?
| A. utilitarian | |
| B. insufficient | |
| C. beneficial | |
| D. flexible |
Explanation
Identify words in the text that provide clues to the best answer.
The sentence with the blank states that despite efforts to prevent food waste, a high percentage of food "continues to be thrown away," which indicates that attempts to reduce waste have been largely ineffective.
"Insufficient" means not enough to meet a goal—here, of preventing food waste. Existing solutions have not met that goal and would therefore be considered insufficient.
(Choice A) "Utilitarian" means useful for achieving a purpose. Solutions that fail to reduce the high percentage of food waste are not useful for achieving their intended purpose.
(Choice C) "Beneficial" means producing good or helpful results. However, the efforts to prevent food waste haven't helped much, given the high percentage that is thrown away.
(Choice D) "Flexible" means capable of adapting to new or changing requirements. Solutions that are considered flexible would not be explained by a high percentage of food waste. The text doesn't discuss whether attempts to reduce food waste have been changed or adapted.
Things to remember:
Select the word that supports the meaning found in the surrounding context.
Passage:
"The Tuft of Flowers" is an early 1900s poem by Robert Frost. In the poem, the speaker discusses a sense of solitude followed by a personal reflection: ______
Question
Which quotation from "The Tuft of Flowers" most effectively illustrates the claim?
| A. "I went to turn the grass once after one / Who mowed it in the dew before the sun. / The dew was gone that made his blade so keen / Before I came to view the leveled scene." | |
| B. "A leaping tongue of bloom the scythe had spared / Beside a reedy brook the scythe had bared. / I left my place to know them by their name, / Finding them butterfly weed when I came." | |
| C. "And dreaming, as it were, held brotherly speech / With one whose thought I had not hoped to reach. / 'Men work together,' I told him from the heart, / 'Whether they work together or apart.'" | |
| D. "But he had gone his way, the grass all mown, / And I must be, as he had been,—alone, / 'As all must be,' I said within my heart, / 'Whether they work together or apart.'" |
Explanation
Find the quote from the poem that matches the claim: the speaker discusses a sense of solitude followed by personal reflection.
| "But he had gone his way, the grass all mown, | But the other worker had left after mowing the grass, |
| And I must be, as he had been,—alone, | And I was left by myself like he had been, |
| 'As all must be,' I said within my heart, | And I thought to myself, "All people are alone," |
| 'Whether they work together or apart.'" | "No matter whether they work alongside someone or by themselves." |
The speaker mentions being alone, which shows a sense of solitude (aloneness; isolation). His personal reflection is his thought to himself that everybody is alone even when working with others. Because it contains the claim's idea that a sense of solitude is followed by a personal reflection, the right answer is Choice D.
(Choices A, B, and C) The speaker offers personal reflections in all of these lines, but despite the fact that he is the only person there, he doesn't express feelings of solitude in any of them. Therefore, none of these answers support the claim.
-
Choice A: These lines indicate the speaker's realization that the mower had been there earlier before he arrived.
-
Choice B: These lines indicate that the speaker notices some flowers that the mower left, and he identifies them as butterfly weed.
-
Choice C: These lines show the speaker imagining what he would say to the mower about brotherhood.
Things to remember:
Paraphrase each set of lines to determine which ones contain the ideas referred to by the question's claim.
Passage:
Despite the initial theory that tattooing originated in ancient Egypt, tattooing practices, which appear to have been used for cultural, medicinal, and religious reasons, have been traced back to a variety of ancient inhabitants who lived throughout Europe and North America. Archaeological findings from Greece, Scotland, Italy, and the United States all reveal that tattooing has held cultural significance for millennia and can be linked to widely scattered civilizations as far back as 3500 B.C.E. These archaeological findings therefore imply that ______
Question
Which choice most logically completes the text?
| A. it is more important to determine where the art of tattooing originated than to understand its cultural significance. | |
| B. the cultural group responsible for inventing the art of tattooing is unlikely to be identified. | |
| C. many cultural practices unified ancient societies as well as characterized unique ethnic groups. | |
| D. the origins of most cultural traditions are difficult to pinpoint using standard archaeological dating methods. |
Explanation
Summarize the text to determine the topic being discussed. Then, select the answer that is specific to that topic and can be inferred based on the information provided.
This text begins by stating that tattooing is no longer thought to have originated in ancient Egypt because of findings in Europe and North America. It also notes that these findings go back thousands of years and can be traced to "widely scattered civilizations."
Because this evidence reveals that tattooing occurred in many places over a long period of time, it can be inferred that it would be difficult to identify the first group to practice tattooing. Therefore, the choice that logically completes the sentence is as follows: the cultural group responsible for inventing the art of tattooing is unlikely to be identified.
(Choice A) The text states that tattooing appeared to be culturally significant for many ancient cultures, but determining whether discovering the origin of tattooing is more important than its cultural significance is not a logical conclusion based on the preceding sentences.
(Choice C) This answer choice generally references "many cultural practices" rather than the specific topic of the text—ancient tattooing practices. The concluding statement in a paragraph should be specific to the topic of that paragraph.
(Choice D) This answer choice makes a general reference to "most cultural traditions" and "archaeological dating methods," but the only cultural traditions mentioned relate specifically to tattooing and no specific archaeological dating methods are discussed.
Things to remember:
A concluding statement will be based on the topic presented in the text.
Passage:
During the Golden Age of Mexican Cinema, between 1930 and 1960, the Mexican film industry all but dominated Latin American cinema due to the industry's high production levels, quality, and economic success when people like Fernando de Fuentes, Emilio Fernández Romo, Joaquín Pardavé, and Dolores del Río participated in the creation of movies in various ways. However, the number of films these individuals participated in could have extended well beyond this time period as most of them began working before 1930 and continued to work well after 1960; it's feasible, therefore, that ______
Question
Which choice most effectively uses data from the table to complete the text?
| A. Dolores del Río acted in significantly fewer films than did Emilio Fernández Romo, who is credited with 33 acting performances. | |
| B. Joaquín Pardavé's 28 writing credits include only film scripts he wrote between 1942 and 1955. | |
| C. Dolores del Río acted in well over 27 films and Fernando de Fuentes produced more than 37. | |
| D. Fernando de Fuentes actually directed only 8 films and wrote scripts for 37. |
Explanation
Determine what the text claims and choose an answer with information that logically completes the claim.
The text claims that the total number of film credits for each person listed in the table could have been higher than shown because their careers included years outside the Golden Age.
Therefore, it is feasible (reasonable; likely) that these individuals could have participated in more films than those listed in the table. The answer that supports this idea with data from the table states that it's feasible Dolores del Río acted in well over 27 films and Fernando de Fuentes produced more than 37.
(Choice A) This answer compares how many films Dolores del Río and Emilio Fernández Romo acted in over the course of their lives. However, because the table only indicates how many films each acted in during the Golden Age, it's not possible to know whether Dolores acted in significantly fewer than Emilio.
(Choice B) This answer focuses on the writing of Joaquín Pardavé from 1942 to 1955, but the text's claim deals with years beyond the period from 1930 to 1960.
(Choice D) This answer inaccurately cites data from the table, as Fernando de Fuentes directed 37 and wrote 42 films during the period, not 8 and 37 films, respectively.
Things to remember:
Although some of the choices might contain accurate information from the table, only the correct answer will support the text's claim.
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