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What is the largest two-digit positive integer $n$ such that when $n$ is divided by $4$ , the remainder is $3$ , and when $n$ is divided by $5$ , the remainder is $2$ ?

Show answer & explanation

Answer: 87

Answer

The largest two-digit positive integer satisfying both remainder conditions is 87.

Any integer satisfying both remainder requirements must be of the form

n = 20m + 7

for an integer

m

. Testing values for

m

shows that

m = 4

yields

n = 87

, which is the largest two-digit integer fitting this rule.

Step-by-Step Solution

Set up modular arithmetic equations for the given remainder conditions.

n \equiv 3 \pmod 4

and

n \equiv 2 \pmod 5

Translate the verbal description of remainders into mathematical congruence relations.

Combine the congruence relations to find the general form of

n

n = 20m + 7

for non-negative integers

m

The least common multiple of

4

and

5

20

, meaning solutions repeat every

20

units.

Find the maximum integer

m

that produces a two-digit integer.

For

m = 4

n = 87

. For

m = 5

n = 107

Two-digit integers are strictly less than

100

Key Concept

Simultaneous Remainders and Divisibility Cycles

Estimated Time:1m 30s

Question 1282Question →

Passage:
In 1930, following the discovery of clay tablets at Ras Shamra in northern Syria, German scholar Hans Bauer initiated the decipherment of the previously unknown Ugaritic cuneiform script. Recognizing that the script contained only thirty distinct signs, Bauer deduced that it was alphabetic rather than logosyllabic. Operating on the hypothesis that the underlying language belonged to the Semitic family, he identified the Ugaritic word for 'king' by scanning for frequent three-letter combinations corresponding to known Semitic triconsonantal roots. Independently, French epigrapher Édouard Dhorme utilized a distinct methodology by focusing on word dividers and contextual occurrences of single-letter prepositions. While Bauer correctly determined the phonetic values of nineteen signs within weeks, several of his initial consonant assignments proved erroneous due to misidentified root patterns. Dhorme subsequently refined Bauer's chart by correcting these specific phonetic assignments and confirming that Ugaritic possessed a consonantal alphabet with three distinct aleph signs denoting different following vowels, a feature absent in standard West Semitic scripts.

Consider each of the choices separately and select all that apply.
Which of the following statements regarding the Ugaritic script or its decipherment is explicitly supported by the passage?

Select all that apply

Hans Bauer inferred that the script was alphabetic after taking note of its small inventory of unique signs.

Édouard Dhorme's primary decipherment technique involved searching for triconsonantal root patterns in the tablets.

The script incorporated multiple aleph characters that corresponded to distinct following vowel sounds.

Show answer & explanation

Answer: Hans Bauer inferred that the script was alphabetic after taking note of its small inventory of unique signs.; The script incorporated multiple aleph characters that corresponded to distinct following vowel sounds.

Answer

The explicitly supported statements are that Hans Bauer inferred the script was alphabetic due to its limited sign count, and that the Ugaritic script contained distinct aleph characters corresponding to different following vowels.

The passage directly supports two details: that Hans Bauer deduced the script was alphabetic because it contained only thirty distinct signs, and that Ugaritic included three distinct aleph signs denoting different following vowels. The statement attributing triconsonantal root searching to Édouard Dhorme is incorrect because the text explicitly assigns that method to Bauer while describing Dhorme's method as focusing on word dividers and single-letter prepositions.

Step-by-Step Solution

Evaluate the statement regarding Hans Bauer's inference about the alphabetic nature of the script.

The text explicitly states: 'Recognizing that the script contained only thirty distinct signs, Bauer deduced that it was alphabetic rather than logosyllabic.' This directly supports the statement.

Direct textual facts must confirm explicit detail claims.

Evaluate the statement concerning Édouard Dhorme's decipherment technique.

The passage notes that searching for triconsonantal root patterns was Bauer's method, while Dhorme 'utilized a distinct methodology by focusing on word dividers and contextual occurrences of single-letter prepositions.' Therefore, this statement misreads the details.

Accurate detail retrieval requires ensuring actions are attributed to the correct subject in the passage.

Evaluate the statement regarding the aleph signs in the Ugaritic script.

The passage explicitly states that Dhorme confirmed 'that Ugaritic possessed a consonantal alphabet with three distinct aleph signs denoting different following vowels.' This directly supports the statement.

Confirming stated orthographical details against passage facts.

Key Concept

Explicit Detail Retrieval in GRE Reading Comprehension

Question 1283Question →

If $n$ is a negative odd integer, which of the following expressions MUST be a positive even integer?

n^2 + 1

-n^2 + 1

(n + 1)^2

2n - 1

n^3 + 1

Show answer & explanation

Answer: $n^2 + 1$

Answer

The expression

n^2 + 1

must be a positive even integer.

Squaring any negative odd integer

n

produces a positive odd integer (

n^2 \ge 1

). Adding

1

to a positive odd integer always yields a positive even integer (

n^2 + 1 \ge 2

Step-by-Step Solution

Analyze the parity and sign of

n^2

Since

n

is negative and odd, any even power of

n

(such as

n^2

) produces a positive odd integer.

Negative times negative yields positive, and odd times odd yields odd.

Analyze the effect of adding

1

n^2

n^2 + 1

is the sum of a positive odd integer and

1

, which results in a positive even integer.

Adding

1

to any odd integer results in an even integer, and adding

1

to a positive integer maintains positivity.

Key Concept

Parity and sign rules for exponents and addition

Question 1284Question →

Read the sentence below concerning observational astrophysics. Based on the explicit causal structural signal, fill in the blank with the word that best completes the sentence.

Fill in the blanks below

The astrophysicist noted that the massive star's sudden envelope expansion did not merely disrupt its immediate orbital neighborhood; consequently, the intense radiation pressure the collapse of the adjacent interstellar cloud, significantly shortening the timeline for protostellar formation.

Show answer & explanation

Answer

The blank is completed by verbs such as 'precipitated', 'hastened', 'accelerated', or 'triggered', which mean to cause an event to happen suddenly or sooner than expected.

The structural marker 'consequently' establishes a direct cause-and-effect relationship between the star's expansion and the surrounding nebular behavior. Combined with the clue 'significantly shortening the timeline for protostellar formation,' the blank requires a verb meaning to bring about, trigger, or speed up an outcome. Words such as 'precipitated', 'hastened', 'accelerated', or 'triggered' fulfill this causal continuation perfectly.

Step-by-Step Solution

Identify the structural transition signal in the sentence.

The conjunctive adverb 'consequently' establishes a clear cause-and-effect relationship, indicating that the second clause expresses a direct result of the preceding stellar event.

Recognizing transition signals clarifies the logical relationship between clauses.

Analyze surrounding context clues to determine the required meaning for the blank.

The phrase 'significantly shortening the timeline for protostellar formation' indicates that the radiation pressure caused the gas collapse to occur prematurely or more rapidly.

The modifier at the end of the sentence defines the nature of the outcome.

Infer a word that satisfies the causal direction and contextual meaning.

A suitable word is 'precipitated' (or 'hastened'/'accelerated'), matching the requirement for a verb meaning to bring about suddenly or prematurely.

'Precipitate' precisely fits the context of initiating or accelerating a physical process.

Key Concept

Interpreting Continuation and Causal Structural Signals

Question 1285Question →

Which terms correctly fill in each blank in the passage below to complete the text in a logically coherent manner?

Fill in the blanks below

Far from proving that the newly unearthed tablet was an ancient forgery, the presence of anomalous grammatical constructions actually served to its authenticity; because such idiosyncratic syntax could not have been devised by a modern hoaxer lacking knowledge of the regional dialect's archaic evolution, the linguistic irregularities were eventually recognized as evidence of the artifact's genuine antiquity.

Show answer & explanation

Answer

Blank 1 requires a verb meaning to confirm or validate (such as 'substantiate' or 'corroborate'), and Blank 2 requires an adjective meaning decisive or irrefutable (such as 'compelling' or 'conclusive').

The structural pivot 'Far from proving... a forgery' establishes a reversal, indicating that the grammatical anomalies served to validate or substantiate authenticity rather than disprove it. Subsequently, the causal explanation that a hoaxer could not have known such archaic syntax dictates that these features provided compelling or decisive evidence of genuine antiquity.

Step-by-Step Solution

Analyze the contrast clue in the first clause.

The opening structural shift 'Far from proving that... was an ancient forgery' indicates that the presence of anomalous grammatical constructions supported, rather than undermined, the tablet's genuineness.

The structural pivot 'far from' reverses the negative expectation associated with a forgery, requiring Blank 1 to express confirmation or validation.

Analyze the causal clue for the second blank.

The second clause explains that a modern hoaxer could not have devised such archaic syntax, demonstrating that the irregularities provided convincing and definitive proof.

The inability of a counterfeiter to know the archaic dialect establishes that the evidence is decisive, compelling, or conclusive for Blank 2.

Track inter-blank dependency to ensure semantic alignment.

Both blanks work in tandem: confirming authenticity in the first slot logically aligns with treating the linguistic evidence as decisive proof in the second slot.

Choosing a term meaning to discredit for the first blank would directly contradict the positive assertion made in the second clause, disrupting cross-blank coherence.

Key Concept

Multi-Blank Dependency Tracking

Question 1286Question →

A water storage tank is initially filled to $\frac{4}{5}$ of its total capacity with water. First, $\frac{1}{4}$ of the water in the tank is drained for irrigation. Next, an amount of water equal to $\frac{1}{3}$ of the remaining water in the tank is added back. Finally, $\frac{3}{8}$ of the water currently in the tank is removed for domestic use. Which of the following statements about the volume of water in the tank must be true? Select all such statements.

Select all that apply

After the second operation, the volume of water in the tank is equal to the initial volume of water before any operations.

The final volume of water in the tank is equal to

\frac{5}{8}

of the initial volume of water.

The total volume of water removed from the tank during the first and third operations combined is equal to

\frac{1}{2}

of the total capacity of the tank.

The volume of water removed in the first step is equal to

\frac{1}{4}

of the total capacity of the tank.

The final volume of water in the tank is equal to

\frac{2}{5}

of the total capacity of the tank.

Show answer & explanation

Answer: After the second operation, the volume of water in the tank is equal to the initial volume of water before any operations.; The final volume of water in the tank is equal to $\frac{5}{8}$ of the initial volume of water.; The total volume of water removed from the tank during the first and third operations combined is equal to $\frac{1}{2}$ of the total capacity of the tank.

Answer

The correct statements are: (1) After the second operation, the volume of water in the tank is equal to the initial volume of water before any operations; (2) The final volume of water in the tank is equal to 5/8 of the initial volume of water; and (3) The total volume of water removed from the tank during the first and third operations combined is equal to 1/2 of the total capacity of the tank.

The statement regarding the volume after the second operation is correct because adding 1/3 of the remaining 3/5 capacity adds 1/5 capacity, returning the total volume to 4/5 capacity. The statement comparing the final volume to the initial volume is correct because (1/2) divided by (4/5) equals 5/8. The statement regarding total water removed in the first and third operations is correct because 1/5 capacity plus 3/10 capacity equals 1/2 capacity.

Step-by-Step Solution

Define total tank capacity as

T

and find initial water volume.

Initial volume

V_0 = \frac{4}{5}T

The problem states the tank is initially

\frac{4}{5}

full.

Calculate water remaining and removed after Operation 1.

Removed =

\frac{1}{4} \times \frac{4}{5}T = \frac{1}{5}T

. Remaining

V_1 = \frac{4}{5}T - \frac{1}{5}T = \frac{3}{5}T

Draining

\frac{1}{4}

of existing water leaves

\frac{3}{4}

of the existing water.

Calculate water remaining after Operation 2.

Added =

\frac{1}{3} \times \frac{3}{5}T = \frac{1}{5}T

. New volume

V_2 = \frac{3}{5}T + \frac{1}{5}T = \frac{4}{5}T

Adding

\frac{1}{3}

of the remaining volume increases it by a factor of

1 + \frac{1}{3} = \frac{4}{3}

Calculate water remaining and removed after Operation 3.

Removed =

\frac{3}{8} \times \frac{4}{5}T = \frac{3}{10}T

. Final volume

V_3 = \frac{4}{5}T \times \left(1 - \frac{3}{8}\right) = \frac{4}{5}T \times \frac{5}{8} = \frac{1}{2}T

Removing

\frac{3}{8}

of current water leaves

\frac{5}{8}

of that volume.

Evaluate each given statement against calculated values.

V_2 = V_0 = \frac{4}{5}T

(True).

\frac{V_3}{V_0} = \frac{\frac{1}{2}T}{\frac{4}{5}T} = \frac{5}{8}

(True). Total removed =

\frac{1}{5}T + \frac{3}{10}T = \frac{1}{2}T

(True).

Direct comparison with calculated step outcomes confirms these three statements are valid.

Key Concept

Sequential Fraction Multiplication and Part-to-Whole Relationships

Estimated Time:1m 45s

Question 1287Question →

Read the passage below:

While early nineteenth-century commentators dismissed the philosopher’s final treatise as an act of intellectual (i)________, claiming its convoluted prose signaled a mind past its peak, modern scholars view the text through a dramatically different lens. Rather than serving as evidence of conceptual (ii)________, the work's intricate, non-linear structure represents a calculated attempt to dismantle traditional metaphysics. Far from stumbling into ambiguity, the philosopher sought to (iii)________ the dogmatic assumptions of contemporary epistemology by forcing readers to re-examine the very foundations of rational thought.

Which of the following combinations of terms best completes the passage to maintain cross-sentence contextual coherence throughout?

(i) senescence, (ii) exhaustion, (iii) subvert

(i) erudition, (ii) exhaustion, (iii) subvert

(i) senescence, (ii) vigor, (iii) subvert

(i) senescence, (ii) exhaustion, (iii) affirm

(i) fervor, (ii) neutrality, (iii) bypass

Show answer & explanation

Answer: (i) senescence, (ii) exhaustion, (iii) subvert

Answer

The combination '(i) senescence, (ii) exhaustion, (iii) subvert' correctly completes all three blanks to maintain contextual coherence across the passage.

The combination of 'senescence', 'exhaustion', and 'subvert' establishes complete thematic and structural harmony across all three sentences. 'Senescence' matches the descriptive clue 'mind past its peak' in sentence 1. 'Exhaustion' maintains the contrastive frame introduced by 'Rather than serving as evidence of...' in sentence 2, where modern scholars reject the notion that the work represents depleted intellect. Finally, 'subvert' complements 'dismantle traditional metaphysics' in sentence 3, completing the author's intended goal of challenging dogmatic assumptions.

Step-by-Step Solution

Analyze Blank (i) using clues in the first sentence.

The clause 'claiming its convoluted prose signaled a mind past its peak' provides a direct restatement clue. The word in blank (i) must mean mental decline or deterioration.

'Senescence' means the condition or process of deterioration with age, perfectly matching 'past its peak'.

Analyze Blank (ii) using cross-sentence contrast clues between sentence 1 and sentence 2.

Sentence 2 opens with a contrast ('dramatically different lens', 'Rather than serving as evidence of...'). Blank (ii) restates the early critics' view of decline.

'Exhaustion' (state of being depleted of strength/resources) reflects the early critics' negative assessment that modern scholars are refuting.

Analyze Blank (iii) using continuation clues from sentence 2 to sentence 3.

Sentence 2 states the structure was a 'calculated attempt to dismantle traditional metaphysics'. Sentence 3 elaborates on this intent ('forcing readers to re-examine...').

'Subvert' (undermine or overturn) logically aligns with 'dismantle' and challenging 'dogmatic assumptions'.

Key Concept

Cross-Sentence Contextual Coherence in Multi-Blank Text Completion

Estimated Time:2m 0s

Question 1288Question →

Read the following passage:

"Although early twentieth-century historians routinely dismissed folk remedies as mere superstitious relics of pre-scientific culture, recent ethnobotanical studies have forced a reevaluation of these traditional pharmacopeias. Far from being entirely devoid of empirical efficacy, many herbal treatments were carefully refined over centuries of trial and error. Consequently, modern pharmacologists no longer automatically discount the therapeutic potential of indigenous plant-based therapies, choosing instead to systematically isolate their active biochemical compounds; paradoxically, this rigorous scientific scrutiny has validated several remedies that orthodox medicine had long rejected."

Based on the structural clues and transition signals in the passage, which of the following best captures the meaning of the word "discount" as it is used in the text?

dismiss or disregard

reduce the financial price of

overestimate the value of

confirm through laboratory testing

misinterpret the origin of

Show answer & explanation

Answer: dismiss or disregard

Answer

The word 'discount' in this passage means 'dismiss or disregard'.

The passage establishes a direct parallel between early historians who 'routinely dismissed' herbal remedies and modern pharmacologists who 'no longer automatically discount' their potential. The structural continuation signal 'Consequently' coupled with the phrase 'no longer automatically' indicates a shift away from earlier skepticism, demonstrating that 'discount' is used in its secondary sense meaning to dismiss, minimize, or write off as unimportant.

Step-by-Step Solution

Locate the target word and identify surrounding structural transition signals.

The target word 'discount' occurs in the clause: 'Consequently, modern pharmacologists no longer automatically discount the therapeutic potential...'

Structural transitions such as 'Consequently' and 'no longer' dictate the semantic relationship between this clause and preceding claims.

Analyze parallel structures and contrast relationships across the text.

The text contrasts early historians who 'routinely dismissed' remedies with modern pharmacologists who 'no longer automatically discount' them.

The phrase 'no longer automatically [discount]' reverses the prior action of 'dismissing' relics as superstitious.

Select the contextual definition that maintains sentence coherence.

The meaning 'dismiss or disregard' seamlessly fits the logical structure of stopping an unwarranted rejection of empirical evidence.

This secondary meaning of 'discount' aligns perfectly with the passage's argument concerning scientific reevaluation.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Question 1289Question →

An investment fund had an initial value of $\$4{}000$ . During the first year, the value of the fund increased by $15\%$ . During the second year, the new value of the fund increased by $10\%$ . What was the total value of the fund at the end of the second year?

\$5{}060

\$5{}000

\$4{}600

\$4{}840

\$5{}290

Show answer & explanation

Answer: $\$5{}060$

Answer

The total value of the fund at the end of the second year was

\$5{}060

To find the final value after successive percent increases, compute the growth sequentially. After Year 1, the fund grows to

\$4{}000 \times 1.15 = \$4{}600

. In Year 2, the

10\%

increase applies to the new amount of

\$4{}600

, resulting in

\$4{}600 \times 1.10 = \$5{}060

. Thus,

\$5{}060

is the correct final value.

Step-by-Step Solution

Calculate the fund's value at the end of the first year.

\$4{}000 \times (1 + 0.15) = \$4{}000 \times 1.15 = \$4{}600

15\%

increase on the initial

\$4{}000

base increases the value to

115\%

of its original amount.

Calculate the fund's value at the end of the second year using the updated base value.

\$4{}600 \times (1 + 0.10) = \$4{}600 \times 1.10 = \$5{}060

The second year's

10\%

increase applies to the new base value of

\$4{}600

, not the initial

\$4{}000

Key Concept

Successive Percent Changes and Base Shift

Question 1290Question →

Although evolutionary biologists long suspected that the orchid's intricate floral architecture evolved to attract specialized pollinators, recent genetic mapping suggests that these structural adaptations were surprisingly ______ , arising instead as unintended side effects of metabolic shifts during rapid speciation. Select the two options that produce completed sentences identical in meaning.

Select all that apply

fortuitous

adventitious

advantageous

beneficial

ornamental

morphological

Show answer & explanation

Answer: fortuitous; adventitious

Answer

The two words that correctly complete the sentence with equivalent meanings are 'fortuitous' and 'adventitious'.

The sentence uses the contrast word 'Although' and the clarifying phrase 'arising instead as unintended side effects' to show that the orchid's features were not deliberately evolved adaptations, but rather accidental occurrences. The words 'fortuitous' and 'adventitious' both mean happening by chance or unintentionally, yielding two completed sentences with identical meaning.

Step-by-Step Solution

Analyze structural clues and sentence direction

The sentence begins with the concession signal 'Although', setting up a contrast between the long-held belief (that floral architecture evolved purposefully for attraction) and recent genetic findings. The phrase 'arising instead as unintended side effects' indicates that the blank must describe something accidental, unplanned, or occurring by chance.

Establishing the target meaning and tone is necessary before evaluating vocabulary options.

Identify candidate words fitting the context

'Fortuitous' (occurring by chance) and 'adventitious' (accidental/unplanned) both match the context of an unintended side effect.

Sentence Equivalence requires selecting words that maintain logical sentence coherence.

Evaluate remaining options for false synonym pairs and topical traps

'Advantageous' and 'beneficial' form a synonym pair but fit the initial hypothesis rather than the contrasting finding. 'Ornamental' and 'morphological' relate topically to floral traits and biology but lack synonym pairs and do not convey the idea of chance.

Eliminating invalid synonym pairs and unpaired topical distractors guarantees semantic equivalence.

Key Concept

Eliminating False Synonyms and Topically Related Distractors

Question 1291Question →

While early nineteenth-century economic theorists dismissed the merchant's detailed journal as mere anecdotal clutter, modern historians regard the text as __________, recognizing that its granular records of daily transactions offer invaluable insights into regional trade networks. Which two of the following answer choices, when inserted into the sentence, produce completed sentences that are similar in meaning?

Select all that apply

illuminating

revealing

incandescent

luminescent

exhaustive

inconsequential

Show answer & explanation

Answer: illuminating; revealing

Answer

The sentence is completed with equivalent meaning by selecting 'illuminating' and 'revealing'.

The words 'illuminating' and 'revealing' both mean providing valuable insight or making significant details clear. When inserted into the blank, both words emphasize how modern historians view the detailed merchant journal as a source of rich understanding, directly contrasting with earlier dismissals of the text.

Step-by-Step Solution

Analyze structural pivot signals and sentence polarity.

The initial clause uses 'While' to establish a contrast between early economic theorists who 'dismissed' the journal as 'mere anecdotal clutter' and modern historians who place positive value on it ('invaluable insights').

Identifying contrast signals establishes that the target blank requires a positive adjective signifying informative value.

Evaluate candidate words for contextual fit.

'Illuminating', 'revealing', and 'exhaustive' fit the context of providing deep historical insight.

The missing word must describe a historical text that provides detailed, useful understanding.

Identify synonymous choice pairs that yield equivalent sentence meaning.

'Illuminating' and 'revealing' are contextual synonyms that both denote providing significant insight or clarity, completing the sentence with identical semantic meaning.

GRE Sentence Equivalence requires choosing two options that produce equivalent overall sentence meaning, eliminating unpaired words like 'exhaustive' and non-contextual pairs like 'incandescent' / 'luminescent'.

Key Concept

Sentence Equivalence: Contextual Synonym Pair Selection

Question 1292Question →

In a manufacturing facility, three automated machines—Machine A, Machine B, and Machine C—produce components at constant individual rates. Machine A working alone completes $\frac{3}{10}$ of a standard daily order in $3$ hours. Machine B working alone completes $\frac{2}{5}$ of the order in $6$ hours. Machine C working alone completes $\frac{1}{4}$ of the order in $5$ hours.

Initially, all three machines work together for $3$ hours. At the end of $3$ hours, Machine A malfunctions and stops, while Machines B and C continue working together without interruption until the daily order is finished. How many total hours does it take from the start to complete the entire daily order?

Show answer & explanation

Answer: 6

Answer

The total time required from the start to complete the entire daily order is 6 hours.

Each machine's rate per hour is found by dividing the given fraction by hours: Machine A is

\frac{1}{10}

, B is

\frac{1}{15}

, and C is

\frac{1}{20}

. Summing these gives an initial combined rate of

\frac{13}{60}

per hour. In the first 3 hours, the three machines complete

3 \times \frac{13}{60} = \frac{13}{20}

of the order, leaving

1 - \frac{13}{20} = \frac{7}{20}

of the order remaining. When Machine A stops, the remaining combined rate of B and C is

\frac{1}{15} + \frac{1}{20} = \frac{7}{60}

per hour. Dividing the remaining

\frac{7}{20}

\frac{7}{60}

yields 3 additional hours. Adding the initial 3 hours gives a total duration of 6 hours.

Step-by-Step Solution

Determine individual hourly work rates for each machine.

Machine A rate =

\frac{1}{10}

order/hr, Machine B rate =

\frac{1}{15}

order/hr, Machine C rate =

\frac{1}{20}

order/hr.

Divide the fraction of the job completed by the duration in hours to find the unit rate for each machine.

Calculate the combined rate of all three machines and the fraction completed in the first 3 hours.

Combined rate =

\frac{13}{60}

order/hr; Work completed in 3 hours =

\frac{13}{20}

, leaving

\frac{7}{20}

of the order unfinished.

Sum the three rates using a common denominator of 60, then multiply by 3 hours to find the total work done during the first stage.

Calculate the combined rate of Machines B and C, and find the additional time required to complete the remaining fraction.

Combined rate of B and C =

\frac{7}{60}

order/hr; Additional time needed = 3 hours.

Divide the remaining fraction

\frac{7}{20}

by the combined rate

\frac{7}{60}

to determine the remaining hours needed.

Add the initial duration and additional duration to obtain the total time.

Total time =

3 + 3 = 6

hours.

The question asks for the total elapsed time from the start of the job.

Key Concept

Adding rational fractions with different denominators to solve multi-stage work and rate problems.

Question 1293Question →

The evolutionary biologist observed that urban wildlife populations had not merely adapted to noise pollution; indeed, sustained anthropogenic disturbance had ________ behavioral shifts across generations, effectively accelerating the speciation process. Select the two answer choices that, when used to complete the sentence, fit the meaning of the sentence as a whole and produce completed sentences that are alike in meaning.

Select all that apply

catalyzed

stymied

engendered

mitigated

observed

documented

Show answer & explanation

Answer: catalyzed; engendered

Answer

The correct options are the pair meaning to cause or bring about ('catalyzed' and 'engendered').

The sentence relies on the continuation transition 'indeed' to build upon the statement that wildlife 'had not merely adapted'. To complete the causal structure leading to accelerated speciation, the blank requires verbs meaning to produce or bring about. Both 'catalyzed' and 'engendered' fit this requirement precisely, creating coherent sentences with identical meanings.

Step-by-Step Solution

Analyze structural signals and overall sentence polarity.

The clause following the semicolon contains the explicit continuation signal 'indeed', which serves to amplify the preceding assertion ('not merely adapted'). The concluding phrase 'effectively accelerating the speciation process' further reinforces that the blank requires a verb meaning to cause, instigate, or promote.

Recognizing continuation and causal structural signals dictates the necessary definition and tone of the target word pair.

Identify vocabulary pairs that fulfill the sentence logic.

'Catalyzed' (to cause or accelerate an action) and 'engendered' (to bring into existence or cause) both logically complete the sentence and yield identical meanings.

Sentence equivalence items require selecting two choices that produce sentences with identical overall semantic meaning.

Evaluate and eliminate distractors.

'Stymied' and 'mitigated' express negative or reductive forces, directly opposing the continuation signal. 'Observed' and 'documented' fit the general thematic domain of scientific research but fail to express the causal force required by the clause.

Distractor analysis eliminates options that reverse structural signals or present non-causal topical words.

Key Concept

Interpreting continuation structural signals ('indeed') and causal mechanisms to select contextually synonymous word pairs.

Question 1294Question →

A technology company allocates its quarterly research budget among three key projects: Project Alpha, Project Beta, and Project Gamma. Initially, Project Alpha receives $\frac{2}{5}$ of the total budget. Of the remaining budget, Project Beta is allocated $\frac{4}{7}$ , and Project Gamma receives the rest. During a mid-quarter review, $\frac{1}{6}$ of Project Alpha's allocated budget and $\frac{1}{4}$ of Project Beta's allocated budget are transferred to Project Gamma. After these transfers, Project Gamma's final budget is what fraction of the total initial quarterly research budget?

\frac{5}{21}

\frac{13}{50}

\frac{43}{105}

\frac{83}{165}

\frac{16}{105}

Show answer & explanation

Answer: $\frac{43}{105}$

Answer

\frac{43}{105}

Project Alpha receives

\frac{2}{5}

of the total budget, leaving

\frac{3}{5}

. Project Beta gets

\frac{4}{7}

\frac{3}{5}

, which is

\frac{12}{35}

. Project Gamma's initial share is

\frac{3}{5} - \frac{12}{35} = \frac{9}{35}

. Transferring

\frac{1}{6}

of Alpha's share contributes

\frac{1}{15}

, and transferring

\frac{1}{4}

of Beta's share contributes

\frac{3}{35}

. Adding these to Gamma's initial share yields

\frac{9}{35} + \frac{3}{35} + \frac{1}{15} = \frac{12}{35} + \frac{1}{15} = \frac{36 + 7}{105} = \frac{43}{105}

Step-by-Step Solution

Determine Project Alpha's initial allocation and the remaining budget fraction.

Project Alpha gets

\frac{2}{5}

of the total budget

B

. The remaining fraction is

1 - \frac{2}{5} = \frac{3}{5} B

Subtractions from the whole yield the remaining unallocated portion.

Calculate the initial allocations for Project Beta and Project Gamma.

Project Beta receives

\frac{4}{7} \times \frac{3}{5} B = \frac{12}{35} B

. Project Gamma receives the rest of the remaining budget:

\frac{3}{5} B - \frac{12}{35} B = \frac{21}{35} B - \frac{12}{35} B = \frac{9}{35} B

Project Beta's share is a fraction of the remaining budget, not the total budget.

Calculate the transferred amounts from Project Alpha and Project Beta to Project Gamma.

Transfer from Alpha =

\frac{1}{6} \times \frac{2}{5} B = \frac{1}{15} B

. Transfer from Beta =

\frac{1}{4} \times \frac{12}{35} B = \frac{3}{35} B

Transfers are fractional parts of each project's individual initial allocation.

Sum Project Gamma's initial share and the two transferred amounts.

Gamma's final fraction =

\frac{9}{35} + \frac{1}{15} + \frac{3}{35} = \frac{12}{35} + \frac{1}{15} = \frac{36}{105} + \frac{7}{105} = \frac{43}{105}

Finding the least common denominator (

105

) allows exact addition of the rational numbers.

Key Concept

Multi-step sequential operations with fractions and rational numbers

Question 1295Question →

Read the following passage:

In eighteenth-century macroeconomic theory, mercantilist scholars insisted that bullion accumulation was the primary engine of sovereign wealth. However, David Hume’s price-specie-flow mechanism demonstrated that unchecked trade surpluses would inevitably induce domestic inflation, which would in turn check the export competitiveness of the nation. Indeed, rather than sustaining perpetual economic dominance, an unbridled influx of precious metals served to reverse trade balances through automatic price adjustments.

In the context in which it appears, the word "check" most nearly means

inspect for technical accuracy

curb or restrain

verify the legal validity of

stimulate or bolster

log or record systematically

Show answer & explanation

Answer: curb or restrain

Answer

In this context, "check" most nearly means "curb or restrain".

The passage establishes a direct structural contrast using the pivot word 'However' and the elaboration 'rather than sustaining... served to reverse.' The text notes that 'unchecked' surpluses cause inflation, which in turn acts upon export competitiveness in a negative, dampening manner. Therefore, the word 'check' is used in its secondary sense meaning to curb, restrain, or halt.

Step-by-Step Solution

Identify the target vocabulary word and analyze the immediate syntactic context.

The word "check" acts as a verb governing the direct object "the export competitiveness of the nation."

Determining the grammatical role isolates how the target word interacts with neighboring clauses.

Locate structural signals, pivot markers, and contextual contrast indicators.

The sentence begins with the pivot marker "However" and features internal opposition between "unchecked trade surpluses" and the resulting action on "export competitiveness." Furthermore, the subsequent sentence uses the contrast frame "rather than sustaining perpetual economic dominance, [it] served to reverse trade balances."

Structural signals establish that domestic inflation acts as a opposing force that counteracts or diminishes export competitiveness.

Evaluate option definitions against the required contextual valence.

The definition "curb or restrain" accurately captures the halting/reducing effect implied by the structural contrast between an unbridled surplus and a reversed trade balance.

Secondary definitions of polysemous words must be selected based on structural alignment rather than default frequency.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Estimated Time:2m 0s

Question 1296Question →

Pair each formal vocabulary term in Column A with its most accurate semantic near-synonym in Column B.

Click a left item, then click its matching right item

Show answer & explanation

Answer

'Penurious' matches with 'Parsimonious', 'Lugubrious' matches with 'Doleful', 'Fastidious' matches with 'Punctilious', and 'Apocryphal' matches with 'Spurious'.

Each pair represents words that share core semantic definitions and equivalent formal register: penurious/parsimonious (excessively frugal), lugubrious/doleful (mournful), fastidious/punctilious (overly meticulous), and apocryphal/spurious (dubious authenticity).

Step-by-Step Solution

Analyze the core definition and tone of each word in Column A.

'Penurious' relates to stinginess, 'Lugubrious' relates to sorrow, 'Fastidious' relates to meticulousness, and 'Apocryphal' relates to doubtful authenticity.

Establishing primary meanings allows clear identification of matching nuances in Column B.

Evaluate Column B candidates for exact or functional near-synonyms.

'Parsimonious' aligns with 'Penurious'; 'Doleful' aligns with 'Lugubrious'; 'Punctilious' aligns with 'Fastidious'; 'Spurious' aligns with 'Apocryphal'.

Functional near-synonyms share both primary definitions and formal connotation in academic contexts.

Key Concept

Identifying Synonym and Near-Synonym Pairs

Question 1297Question →

A technology company's annual revenue increased by $40\%$ from 2021 to 2022, and then decreased by $15\%$ from 2022 to 2023. If the company's annual revenue in 2023 was $\$476,000$ , what was its annual revenue in 2021?

\$336,000

\$380,800

\$400,000

\$404,600

\$560,000

Show answer & explanation

Answer: $\$400,000$

Answer

\$400,000

The correct answer is

\$400,000

. Expressing the multi-step percentage change relative to the initial 2021 revenue

R

gives an intermediate 2022 value of

1.40R

and a final 2023 value of

1.40R \times 0.85 = 1.19R

. Solving

1.19R = 476,000

yields

R = 400,000

Step-by-Step Solution

Define the unknown variable and express successive percentage changes algebraically.

Let

R

be the 2021 annual revenue. The 2022 revenue is

R \times (1 + 0.40) = 1.40R

. The 2023 revenue is

1.40R \times (1 - 0.15) = 1.40R \times 0.85

Sequential percentage adjustments apply to the newly resulting base values rather than the initial value.

Calculate the net multiplier for the two-year change.

1.40 \times 0.85 = 1.19

. Thus, the 2023 revenue equals

1.19R

Multiplying the growth factor (

1.40

) by the decay factor (

0.85

) gives the overall percentage relationship between 2021 and 2023.

Solve for the initial revenue

R

using the given 2023 revenue.

1.19R = 476,000 \implies R = \frac{476,000}{1.19} = 400,000

Dividing the final amount by the combined multiplier determines the original starting revenue.

Key Concept

Successive Percent Change and Base Shift

Question 1298Question →

Read the passage below:

"Far from viewing the early cartographer’s coastal charts as mere (i)________ creations, modern historians recognize that these seemingly crude maps were (ii)________ to the kingdom’s naval supremacy; the simple rendering of shorelines was not born of technical ineptitude, but rather reflected a deliberate prioritization of strategic utility over geographic embellishment."

Which set of terms, when inserted into the blanks, best completes the passage in a logically coherent manner?

(i) ornamental, (ii) integral

(i) vital, (ii) peripheral

(i) redundant, (ii) irrelevant

(i) ornamental, (ii) detrimental

(i) indispensable, (ii) ancillary

Show answer & explanation

Answer: (i) ornamental, (ii) integral

Answer

The passage is best completed by choosing '(i) ornamental' and '(ii) integral'.

The passage establishes a clear contrast between an outdated view of early charts and their actual strategic value. The opening pivot 'Far from viewing... as mere...' requires Blank (i) to represent a diminished, decorative status ('ornamental'), which aligns with the phrase 'geographic embellishment' at the end. Consequently, modern historians recognize these charts as central ('integral') to naval supremacy because of their 'strategic utility.'

Step-by-Step Solution

Analyze structural clues and transition signals in the passage.

The opening phrase 'Far from viewing... as mere (i)________' indicates that Blank (i) describes a trivialized or superficial view that modern historians reject. The final clause ('prioritization of strategic utility over geographic embellishment') clarifies that the maps favored functional utility over decorative detail.

Tracking cross-clause clues isolates the required polarity for Blank (i).

Determine the required meaning for Blank (i).

Blank (i) must mean decorative or non-essential, matching 'geographic embellishment.' The term 'ornamental' fits precisely.

The word 'mere' combined with 'geographic embellishment' requires a term denoting superficial decoration.

Determine the required meaning for Blank (ii) based on inter-blank dependency.

Because modern historians reject the view that the maps were merely ornamental, they must view them as central or essential to naval power. The term 'integral' fits this role.

The contrast between the rejected view in clause 1 and the modern perspective requires a positive, high-importance term for Blank (ii).

Key Concept

Multi-Blank Dependency Tracking

Question 1299Question →

Let $n$ be a positive integer that is a factor of $180$ . If $n$ is divisible by $6$ but is not divisible by $4$ , what is the maximum possible number of positive divisors of $n$ ?

Show answer & explanation

Answer: 12

Answer

The prime factorization of

180

2^2 \times 3^2 \times 5^1

. Any factor

n

180

is of the form

2^a \times 3^b \times 5^c

. For

n

to be divisible by

6

, we must have

a \ge 1

and

b \ge 1

. For

n

not to be divisible by

4

, we must have

a < 2

, which forces

a = 1

. To maximize the total number of divisors

(a+1)(b+1)(c+1)

, we select the largest possible values for

b

and

c

, which are

b = 2

and

c = 1

. This yields

n = 2^1 \times 3^2 \times 5^1 = 90

, giving

(1+1)(2+1)(1+1) = 12

positive divisors.

Step-by-Step Solution

Find the prime factorization of 180.

180 = 2^2 \times 3^2 \times 5^1

Any positive factor

n

180

must have the form

n = 2^a \times 3^b \times 5^c

, where

0 \le a \le 2

0 \le b \le 2

, and

0 \le c \le 1

Apply the divisibility conditions to determine the possible values of the exponent

a

a = 1

Since

n

is divisible by

6 = 2 \times 3

a \ge 1

and

b \ge 1

. Since

n

is not divisible by

4 = 2^2

a < 2

. Thus,

a

must equal

1

Maximize the divisor count formula

(a+1)(b+1)(c+1)

using the available ranges for

b

and

c

b = 2

and

c = 1

To maximize the number of positive divisors, choose the maximum allowable values for

b

(

b=2

) and

c

(

c=1

Calculate the maximum number of positive divisors.

(1+1)(2+1)(1+1) = 2 \times 3 \times 2 = 12

For

n = 2^1 \times 3^2 \times 5^1 = 90

, the total number of positive divisors is

12

Key Concept

Divisor count formula and prime factor constraints

Estimated Time:1m 30s

Question 1300Question →

A charity foundation allocates its annual grant budget among three sectors: Healthcare, Education, and Environmental Conservation. First, $\frac{2}{5}$ of the total annual budget is allocated to Healthcare. Next, $\frac{5}{9}$ of the remaining budget is allocated to Education. The final remainder of $\$120,000$ is allocated to Environmental Conservation. What was the foundation's total annual grant budget?

\$32,000

\$240,000

\$360,000

\$450,000

\$2,700,000

Show answer & explanation

Answer: $\$450,000$

Answer

\$450,000

The option specifying

\$450,000

is correct because allocating

\frac{2}{5}

leaves

\frac{3}{5}

of the budget. Taking

\frac{5}{9}

\frac{3}{5}

gives

\frac{1}{3}

of the total. The combined allocations leave

1 - (\frac{2}{5} + \frac{1}{3}) = \frac{4}{15}

of the total budget, which equals

\$120,000

. Dividing

\$120,000

\frac{4}{15}

yields

\$450,000

Step-by-Step Solution

Determine the fraction of the total budget remaining after the Healthcare allocation.

Since Healthcare receives

\frac{2}{5}

of the total budget, the remaining fraction is

1 - \frac{2}{5} = \frac{3}{5}

Sequential fraction operations require updating the remaining whole after each step.

Calculate the fraction of the total budget allocated to Education.

Education receives

\frac{5}{9}

of the remaining

\frac{3}{5}

, which is

\frac{5}{9} \times \frac{3}{5} = \frac{15}{45} = \frac{1}{3}

of the total budget.

To find a fraction of a remaining fraction, multiply the two fractions together.

Find the final remaining fraction allocated to Environmental Conservation.

Subtract both allocations from the whole:

1 - \left(\frac{2}{5} + \frac{1}{3}\right) = 1 - \frac{11}{15} = \frac{4}{15}

The final dollar amount corresponds to the fraction of the total budget left over.

Solve for the total annual grant budget.

\frac{4}{15} \times \text{Total} = \$120,000 \implies \text{Total} = \$120,000 \times \frac{15}{4} = \$450,000

Dividing the part by its corresponding fractional proportion yields the total whole.

Key Concept

Sequential Fraction Operations and Finding the Whole from a Fractional Part

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