All practice questions

Question 1361Question →

If $\frac{x}{2} + \frac{x}{3} = 10$ , what is the value of $x$ ?

25

12

2

6

15

Show answer & explanation

Answer: 12

Answer

The value of

x

is 12.

The option showing 12 is correct because converting the fractions

\frac{x}{2}

and

\frac{x}{3}

to have a common denominator of 6 produces

\frac{3x + 2x}{6} = 10

, or

\frac{5x}{6} = 10

. Multiplying both sides by 6 gives

5x = 60

, and dividing by 5 yields

x = 12

.

Step-by-Step Solution

1

Find a common denominator for the fractional terms on the left side of the equation.

The least common denominator of 2 and 3 is 6. Rewriting the fractions gives

\frac{3x}{6} + \frac{2x}{6} = 10

.

Fractions must have identical denominators before their numerators can be added.

2

Combine the fractions on the left side.

\frac{5x}{6} = 10

Adding the numerators

3x + 2x

yields

5x

over the shared denominator 6.

3

Clear the denominator by multiplying both sides of the equation by 6.

5x = 60

Multiplying both sides by 6 eliminates the fraction while maintaining equality.

4

Isolate

x

by dividing both sides by 5.

x = 12

Dividing 60 by 5 yields the solution for

x

.

Key Concept

Solving linear equations involving fractions by clearing denominators using the least common multiple.

Estimated Time:45s

Question 1362Question →

An agricultural research station recorded the annual grain harvest of a farm over a four-year period. In the second year, the harvest increased by $20\%$ compared to the first year. In the third year, the harvest increased by $25\%$ compared to the second year. In the fourth year, due to unfavorable weather, the harvest decreased by $20\%$ compared to the third year. If the grain harvest in the fourth year was $3,600$ metric tons, what was the grain harvest, in metric tons, in the first year?

2,400

2,880

3,000

3,060

4,500

Show answer & explanation

Answer: 3,000

Answer

3,000 metric tons

To find the initial amount, represent the successive percentage changes as sequential decimal multipliers. A

20\%

increase corresponds to a multiplier of

1.20

, a

25\%

increase corresponds to

1.25

, and a

20\%

decrease corresponds to

0.80

. The net change factor is

1.20 \times 1.25 \times 0.80 = 1.20

. Since the fourth-year harvest equals

1.20

times the first-year harvest, dividing

3,600

by

1.20

yields

3,000

metric tons.

Step-by-Step Solution

1

Define variables and write multiplier expressions for each year's percent change

Let

x

be the harvest in the first year. Year 2 harvest

= 1.20x

; Year 3 harvest

= 1.25 \times (1.20x)

; Year 4 harvest

= 0.80 \times (1.25 \times 1.20x)

.

Percentage increases and decreases must be applied sequentially to the immediately preceding period's value.

2

Calculate the combined percentage multiplier for the four-year period

Combined multiplier

= 1.20 \times 1.25 \times 0.80 = 1.20

.

Multiplying the decimal factors yields the overall factor relating the first year's harvest to the fourth year's harvest.

3

Set up and solve the equation for the first year's harvest (

x

)

1.20x = 3,600 \implies x = \frac{3,600}{1.20} = 3,000

.

Dividing the fourth year's total harvest by the combined multiplier gives the initial first year harvest.

Key Concept

Successive Percentage Changes and Base Values

Question 1363Question →

If $k$ is a positive integer such that $k$ is divisible by $6$ and $k + 1$ is divisible by $5$ , what is the remainder when $k^2 + 5k$ is divided by $30$ ?

0

4

6

14

24

Show answer & explanation

Answer: $6$

Answer

The remainder when

k^2 + 5k

is divided by

30

is

6

.

The correct answer is

6

. Since

k

is a multiple of

6

and leaves a remainder of

4

when divided by

5

, the general form of

k

modulo

30

is

24

. Evaluating

k^2 + 5k

modulo

30

yields

24^2 + 5(24) = 576 + 120 = 696

, which gives a remainder of

6

when divided by

30

.

Step-by-Step Solution

1

Determine the congruence class of

k

modulo 6 and modulo 5.

k \equiv 0 \pmod 6

and

k \equiv 4 \pmod 5

.

Given that

k

is divisible by 6, its remainder modulo 6 is 0. Since

k + 1

is divisible by 5,

k + 1 \equiv 0 \pmod 5

, which implies

k \equiv 4 \pmod 5

.

2

Find the smallest positive integer value of

k

modulo 30 satisfying both conditions.

k \equiv 24 \pmod{30}

.

The multiples of 6 are 0, 6, 12, 18, 24, 30, ... Among these, 24 gives a remainder of 4 when divided by 5. Since 5 and 6 are coprime,

k \equiv 24 \pmod{30}

.

3

Substitute

k \equiv 24 \pmod{30}

into the expression

k^2 + 5k

and compute the remainder modulo 30.

The remainder is

6

.

k^2 + 5k = k(k+5) \equiv 24(24+5) = 24(29) \pmod{30}

. Using modular arithmetic,

24 \equiv -6 \pmod{30}

and

29 \equiv -1 \pmod{30}

, so

(-6)(-1) = 6 \pmod{30}

.

Key Concept

Chinese Remainder Theorem and Modular Arithmetic Properties

Estimated Time:1m 30s

Question 1364Question →

In stark contrast to the previous director’s penchant for grand, speculative hypotheses, Dr. Vance adopted a markedly __________ posture toward the newly unearthed fossil fragments, refusing to attribute them to a novel hominin species until exhaustive anatomical comparison had been completed.

Which two of the following answer choices, when inserted into the sentence, produce completed sentences that are similar in meaning?

Select all that apply

circumspect

guarded

empirical

dogmatic

scholarly

erudite

Show answer & explanation

Answer: circumspect; guarded

Answer

The correct answer choices are 'circumspect' and 'guarded'. Both words mean cautious, prudent, and restrained, which perfectly completes the contrast with the previous director's penchant for speculative hypotheses.

The sentence presents a clear structural contrast introduced by 'In stark contrast to'. The previous director favored 'grand, speculative hypotheses', whereas Dr. Vance refuses to jump to conclusions before completing exhaustive comparisons. Therefore, the blank must describe a cautious, non-committal attitude. The words 'circumspect' (careful to consider all circumstances and consequences; cautious) and 'guarded' (cautious and non-committal) fit the sentence perfectly and produce sentences of equivalent meaning.

Step-by-Step Solution

1

Analyze structural clues and pivot phrases in the sentence stem.

The opening phrase 'In stark contrast to' signals an opposite relationship between the previous director's style ('grand, speculative hypotheses') and Dr. Vance's posture.

Identifying sentence polarity dictates the expected meaning of the blank.

2

Determine the required contextual meaning for the blank.

The elaboration clause ('refusing to attribute them... until exhaustive anatomical comparison had been completed') indicates that the blank requires a word meaning cautious, restrained, or non-committal.

The sentence context requires an attitude characterized by care and hesitation.

3

Evaluate option pairs for semantic equivalence and contextual fit.

'circumspect' and 'guarded' both mean cautious, wary, and restrained in taking a definitive stand. Inserting either word creates sentences with equivalent meaning.

GRE Sentence Equivalence requires selecting two words that produce coherent sentences with identical overall meaning.

Key Concept

Contextual Synonym Pair Selection in Sentence Equivalence

Estimated Time:1m 0s

Question 1365Question →

Let $p = 0.00036 \times 10^{-2}$ and $q = 9.0 \times 10^{-7}$ . Which of the following statements are true? Select all that apply.

Select all that apply

p + q = 4.5 \times 10^{-6}

\frac{p}{q} = 4

\sqrt{p \times q} = 1.8 \times 10^{-6}

p - q = 2.7 \times 10^{-7}

\sqrt{p^2 + q^2} = p + q

Show answer & explanation

Answer: $p + q = 4.5 \times 10^{-6}$ ; $\frac{p}{q} = 4$ ; $\sqrt{p \times q} = 1.8 \times 10^{-6}$

Answer

The correct statements are the ones asserting

p + q = 4.5 \times 10^{-6}

,

\frac{p}{q} = 4

, and

\sqrt{p \times q} = 1.8 \times 10^{-6}

.

Converting

p

to

3.6 \times 10^{-6}

and

q

to

0.9 \times 10^{-6}

shows that their sum is

4.5 \times 10^{-6}

, their ratio is

4

, and the square root of their product

\sqrt{3.24 \times 10^{-12}}

is

1.8 \times 10^{-6}

.

Step-by-Step Solution

1

Convert

p

into standard scientific notation.

p = 0.00036 \times 10^{-2} = 3.6 \times 10^{-4} \times 10^{-2} = 3.6 \times 10^{-6}

.

Expressing numbers in consistent scientific notation enables direct arithmetic operations.

2

Align

q

to the same power of ten for comparison.

q = 9.0 \times 10^{-7} = 0.9 \times 10^{-6}

.

Writing terms with matching exponents allows straightforward addition and subtraction.

3

Test each statement using the simplified values.

p + q = (3.6 + 0.9) \times 10^{-6} = 4.5 \times 10^{-6}

(True);

\frac{p}{q} = \frac{3.6 \times 10^{-6}}{0.9 \times 10^{-6}} = 4

(True);

\sqrt{p \times q} = \sqrt{3.24 \times 10^{-12}} = 1.8 \times 10^{-6}

(True);

p - q = 2.7 \times 10^{-6} \neq 2.7 \times 10^{-7}

(False);

\sqrt{p^2 + q^2} \neq p + q

(False).

Evaluating each given equation determines all correct options.

Key Concept

Decimal arithmetic and scientific notation require aligning powers of ten for addition/subtraction and proper application of radical properties.

Question 1366Question →

While early twentieth-century historians portrayed the nineteenth-century industrial revolution as an unmitigated triumph of human ingenuity that fundamentally liberated society from natural constraints, recent ecological histories offer a far more __________ perspective, demonstrating how industrial expansion merely exchanged localized vulnerabilities for systemic climatic risks.

Which of the following words best completes the blank in the passage?

chastening

celebratory

sanguine

peripheral

exultant

Show answer & explanation

Answer: chastening

Answer

The correct completion is 'chastening', as it captures the sobering and humbling reversal required by the sentence structure.

The sentence structure uses the subordinating conjunction 'While' to establish a contrast between early twentieth-century views and recent ecological histories. Early views saw industrialization as an 'unmitigated triumph', so recent histories must present an opposing, critical view. 'Chastening' means subdued, humbled, or made sober, which perfectly matches the description of revealing 'systemic climatic risks'.

Step-by-Step Solution

1

Identify the structural contrast signal in the sentence.

The sentence begins with the concessionary pivot 'While', indicating that the second half of the sentence will present a contrast to the first half.

Pivot words dictate the logical direction of the blank relative to the surrounding context.

2

Analyze the tone and meaning of the first clause.

The first clause describes earlier views of industrialization in highly positive terms ('unmitigated triumph', 'liberated society').

Establishing the initial tone allows determination of the required opposite valence for the target blank.

3

Determine the required semantic tone for the blank and select the matching word.

The blank must describe a perspective that counters 'unmitigated triumph' by highlighting risks and vulnerabilities. 'Chastening' (sobering/subduing) accurately completes this contrast.

The word must reflect the humbling nature of recognizing new systemic risks.

Key Concept

Contrast and Reversal Clues

Question 1367Question →

If $k = 4.5 \times 10^{-8}$ , what is the value of $\frac{(0.000009)^2 \times 400,000}{k}$ expressed in scientific notation?

7.2 \times 10^2

7.2 \times 10^4

7.2 \times 10^{-14}

7.2 \times 10^{10}

7.2 \times 10^{-2}

Show answer & explanation

Answer: $7.2 \times 10^2$

Answer

7.2 \times 10^2

Converting

0.000009

to

9 \times 10^{-6}

and squaring yields

81 \times 10^{-12}

. Multiplying this by

400,000 = 4 \times 10^5

produces

324 \times 10^{-7}

. Dividing by

4.5 \times 10^{-8}

gives

72 \times 10^1

, which in proper scientific notation (

a \times 10^n

with

1 \le a < 10

) is

7.2 \times 10^2

.

Step-by-Step Solution

1

Convert the components of the numerator into scientific notation

0.000009 = 9 \times 10^{-6}

and

400,000 = 4 \times 10^5

Converting all numbers to scientific notation simplifies subsequent exponent operations.

2

Square the first decimal term

(9 \times 10^{-6})^2 = 9^2 \times (10^{-6})^2 = 81 \times 10^{-12}

Applying the exponent to both the coefficient and the power of 10 gives

81 \times 10^{-12}

.

3

Multiply the terms in the numerator

(81 \times 10^{-12}) \times (4 \times 10^5) = (81 \times 4) \times 10^{-12 + 5} = 324 \times 10^{-7}

Multiply coefficients and add powers of 10.

4

Divide the numerator by the denominator

k = 4.5 \times 10^{-8}

\frac{324 \times 10^{-7}}{4.5 \times 10^{-8}} = \left(\frac{324}{4.5}\right) \times 10^{-7 - (-8)} = 72 \times 10^1

Divide coefficients and subtract the denominator exponent from the numerator exponent.

5

Convert the final result to standard scientific notation format

a \times 10^n

where

1 \le a < 10

72 \times 10^1 = 7.2 \times 10^2

Shift the decimal point one place to the left and increase the exponent of 10 by 1.

Key Concept

Decimals and Scientific Notation Operations

Estimated Time:2m 0s

Question 1368Question →

Complete the passage by filling in the blank with the word that best fits the context based on structural clues and transition signals.

Fill in the blanks below

In early computational linguistics, researchers operated under the central premise that exponentially increasing training corpus volume would automatically rectify semantic misinterpretations in machine translation. Far from being an assumption, however, this foundational belief obscured how deeply polysemous terms destabilize purely frequency-driven algorithms. Indeed, where pioneer models treated syntactic transition markers as mere auxiliary noise, modern transformer architectures explicitly rely on structural contrast and continuation signals to disambiguate context-bound meanings, demonstrating that raw data volume without structural context fails to eliminate lexical ambiguity.

Show answer & explanation

Answer

The word 'innocuous' (or synonymous terms such as 'benign' or 'harmless') correctly completes the blank.

The structural frame 'Far from being an [blank] assumption, however...' sets up an explicit reversal. The clause immediately following reveals that the belief had detrimental effects, as it 'obscured' critical flaws and led to 'destabilize[d]' algorithms. To maintain structural coherence with the pivot marker, the word in the blank must express the opposite of harmful or destabilizing—specifically, that the assumption appeared entirely harmless, benign, or innocuous.

Step-by-Step Solution

1

Identify structural transition signals in the target sentence.

The sentence employs the concession/reversal frame 'Far from being an [blank] assumption, however...'

The structural marker 'Far from being... however' indicates that the descriptor in the blank stands in direct contrast to the actual outcome described in the main clause.

2

Analyze the context following the contrast signal.

The main clause states that the belief 'obscured how deeply polysemous terms destabilize' processing models.

Because the actual effect of the assumption was negative, misleading, and destabilizing, the contrasted expectation in the blank must describe an assumption that appeared completely harmless or innocent.

3

Select a vocabulary word matching the required context and tone.

The term 'innocuous' (meaning producing no injury or offense; harmless) fits the contrast structure precisely.

Inserting 'innocuous' completes the structural logic: what seemed like a harmless assumption actually masked severe destabilizing flaws.

Key Concept

Deciphering word meaning using contrast/reversal signals ('Far from being', 'however') paired with elaboration of cause-and-effect outcomes.

Question 1369Question →

Despite the lead architect's insistence that the historic facade be preserved in its original state, the municipal council deemed the proposed structural additions __________, arguing that modern interventions would irreparably compromise the building's aesthetic integrity.

Which of the following two words, when inserted into the blank, produce sentences that are equivalent in meaning?

Select all that apply

incongruous

harmless

discordant

innocuous

impracticable

ostentatious

Show answer & explanation

Answer: incongruous; discordant

Answer

The two words that produce equivalent sentences are 'incongruous' and 'discordant'.

Both 'incongruous' and 'discordant' accurately capture the idea that modern additions would conflict with or clash against the original historic facade, directly completing the sentence to express that the additions would compromise aesthetic integrity.

Step-by-Step Solution

1

Analyze structural clues and sentence polarity

The opening word 'Despite' establishes a contrast, while the phrase 'irreparably compromise the building's aesthetic integrity' indicates that the council viewed the additions negatively as clashing or out of harmony with the historic facade.

Determining the required tone ensures target words convey incompatibility or visual conflict.

2

Identify word pairs among the choices

Two distinct synonym pairs exist: 'incongruous' / 'discordant' (meaning out of place or incompatible) and 'innocuous' / 'harmless' (meaning benign or safe).

Sentence equivalence requires selecting two options that form a coherent near-synonym pair.

3

Evaluate contextual fit of the pairs

'Incongruous' and 'discordant' both describe additions that clash with the building's character. In contrast, 'innocuous' and 'harmless' contradict the council's belief that damage would occur.

Matching contextual fit eliminates valid dictionary synonym pairs that violate the logical polarity of the sentence.

Key Concept

Identifying Synonym and Near-Synonym Pairs

Question 1370Question →

In her reevaluation of early modern agrarian societies, historian Ellen Wood challenges the long-held assumption that peasant households operated entirely outside commercial logic. Wood demonstrates that while these communities prioritized subsistence over profit maximization, households would systematically husband their surplus yields to build financial reserves against poor harvests and feudal dues. Rather than signaling an early embrace of capitalist accumulation or market speculation, this careful stewardship functioned as a defensive strategy aimed at preserving domestic autonomy. Consequently, the retention of agricultural surpluses served not to generate venture capital, but to mitigate external vulnerabilities without risking land tenure.

In the context of the passage, the word husband most nearly means

marry or enter into a marital domestic partnership

conserve and manage frugally

invest in speculative market ventures

redistribute altruistically to neighboring households

surrender under economic duress

Show answer & explanation

Answer: conserve and manage frugally

Answer

The term 'husband' in this context means to conserve and manage frugally.

The passage emphasizes that peasant households engaged in 'careful stewardship' by setting aside surpluses to build reserves against crop failure and feudal dues. In this context, the verb 'husband' means to manage or conserve resources frugally and prudently to maintain autonomy.

Step-by-Step Solution

1

Analyze the structural and context clues surrounding the target word in the passage.

The text states that households would 'husband their surplus yields to build financial reserves' and characterizes this action as 'careful stewardship' and a 'defensive strategy.'

Establishing the surrounding context reveals whether the word carries a positive, negative, or functional tone.

2

Determine the specific secondary meaning of the verb 'husband'.

As a verb, 'husband' means to manage, conserve, or use resources prudently and economically.

GRE vocabulary-in-context questions frequently test secondary definitions of common words.

3

Evaluate the answer choices against the contextual meaning.

The choice 'conserve and manage frugally' aligns precisely with the concept of careful stewardship described in the text.

Matching contextual nuance ensures elimination of distractors that rely on primary dictionary definitions or contextual contradictions.

Key Concept

Reading Comprehension: Passage-Based Word in Context Analysis

Question 1371Question →

At the start of 2024, a technology company allocated a total research budget of $B$ dollars between Department X and Department Y, such that Department X received $60\%$ of the total budget and Department Y received the remaining $40\%$ . In 2025, Department X's budget was increased by $25\%$ , while Department Y's budget was decreased by $20\%$ . In 2026, the combined budget of both departments was reduced by a uniform $10\%$ . If the total combined budget of the two departments in 2026 was $\$288,900$ , what was the initial total research budget $B$ at the start of 2024?

\$270,000

\$297,000

\$300,000

\$321,000

\$340,000

Show answer & explanation

Answer: $\$300,000$

Answer

\$300,000

The initial budget

B

is divided into

0.60B

for Department X and

0.40B

for Department Y. In 2025, Department X's budget becomes

0.60B \times 1.25 = 0.75B

and Department Y's budget becomes

0.40B \times 0.80 = 0.32B

, yielding a combined 2025 budget of

0.75B + 0.32B = 1.07B

. In 2026, a

10\%

reduction results in a combined budget of

1.07B \times 0.90 = 0.963B

. Setting

0.963B = 288,900

and dividing gives

B = \$300,000

.

Step-by-Step Solution

1

Express the 2024 budget allocations for each department in terms of

B

.

Department X received

0.60B

and Department Y received

0.40B

.

Department X was allocated

60\%

of the total initial budget

B

, leaving

40\%

for Department Y.

2

Calculate the budget of each department in 2025 after their respective percentage changes.

Department X:

0.60B \times (1 + 0.25) = 0.75B

; Department Y:

0.40B \times (1 - 0.20) = 0.32B

.

Department X increased by

25\%

(

1.25

multiplier) and Department Y decreased by

20\%

(

0.80

multiplier).

3

Determine the combined budget for 2025 and apply the 2026 reduction.

Combined 2025 budget =

0.75B + 0.32B = 1.07B

. Combined 2026 budget =

1.07B \times (1 - 0.10) = 0.963B

.

The two department budgets sum to

1.07B

, which then undergoes a uniform

10\%

reduction (

0.90

multiplier).

4

Solve for the initial budget

B

using the given 2026 total of

\$288,900

.

0.963B = 288,900 \implies B = \frac{288,900}{0.963} = 300,000

.

Dividing the 2026 total by the net multiplier

0.963

gives the original budget

B

.

Key Concept

Successive percentage changes with initial weighted allocations

Estimated Time:2m 0s

Question 1372Question →

A high-precision instrument measures the mass of a single micro-particle $P$ as $4.8 \times 10^{-8}$ grams and a single micro-particle $Q$ as $8.0 \times 10^{-9}$ grams. A sample consists of a combination of $P$ and $Q$ particles in a ratio of $3$ particles of $P$ for every $5$ particles of $Q$ . If the total mass of the sample is $9.2 \times 10^{-5}$ grams, what is the total number of particles in the sample?

Show answer & explanation

Answer: 4000

Answer

The total number of particles in the sample is 4,000.

Aligning powers of 10 shows that one P particle weighs

4.8 \times 10^{-8}

g and one Q particle weighs

0.8 \times 10^{-8}

g. A combined unit of 3 P particles and 5 Q particles has a mass of

3(4.8 \times 10^{-8}) + 5(0.8 \times 10^{-8}) = 18.4 \times 10^{-8} = 1.84 \times 10^{-7}

g. Dividing the total sample mass of

9.2 \times 10^{-5}

g by

1.84 \times 10^{-7}

g yields 500 units. Since each unit contains 8 particles (3 + 5), the total number of particles is

500 \times 8 = 4,000

.

Step-by-Step Solution

1

Convert the mass of particle Q so that it shares the same exponent (

10^{-8}

) as particle P.

Mass of single particle Q =

0.8 \times 10^{-8}

grams.

Aligning powers of 10 is necessary before performing addition of masses.

2

Find the combined mass of a fundamental ratio group consisting of 3 particles of P and 5 particles of Q.

Mass of one ratio group =

3(4.8 \times 10^{-8}) + 5(0.8 \times 10^{-8}) = 14.4 \times 10^{-8} + 4.0 \times 10^{-8} = 18.4 \times 10^{-8} = 1.84 \times 10^{-7}

grams.

Determines the mass contributed by each set of 8 particles.

3

Divide the total mass of the sample by the mass of a single ratio group.

Number of groups =

\frac{9.2 \times 10^{-5}}{1.84 \times 10^{-7}} = \frac{9.2}{1.84} \times 10^2 = 5 \times 10^2 = 500

groups.

Determines how many full ratio sets of particles make up the sample.

4

Multiply the number of groups by the total number of particles contained in each group (

3 + 5 = 8

).

Total particles =

500 \times 8 = 4000

.

Yields the total count of individual particles in the sample.

Key Concept

Operations with scientific notation, decimal place value alignment, and weighted proportional sums.

Question 1373Question →

In his analysis of early modern political thought, the historian notes that while the philosopher initially presents sovereign authority as absolute, he subsequently moves to qualify this assertion by outlining specific scenarios where a subject's duty of obedience yields to the right of self-preservation.

In the context of the passage, which of the following words most nearly means "qualify"?

certify

limit

invalidate

endorse

elucidate

Show answer & explanation

Answer: limit

Answer

The word 'limit' best expresses the contextual meaning of 'qualify' as used in the passage.

In formal and academic context, the verb 'qualify' frequently means to limit, moderate, or restrict a previous statement by adding conditions or exceptions. The passage describes a shift from viewing authority as 'absolute' to recognizing specific exceptions, indicating that the philosopher is limiting the scope of his initial claim.

Step-by-Step Solution

1

Examine the surrounding contextual clues in the sentence.

The text contrasts an initial position of 'absolute' authority with later 'specific scenarios where... obedience yields'.

This structural contrast signals that the author is placing boundaries or exceptions on the original premise.

2

Determine the secondary meaning of 'qualify' that fits this context.

In academic and analytical writing, 'to qualify' means to modify, limit, or restrict the scope of a statement.

'Limit' directly matches this definition of setting conditions on a broad claim.

Key Concept

Secondary Meanings of High-Frequency Words

Estimated Time:1m 0s

Question 1374Question →

Fill in each blank in the passage below to complete the text in a logically coherent manner.

Fill in the blanks below

While early architectural historians dismissed the metropolis's subterranean channels as mere conduits built solely for drainage, recent structural re-evaluations indicate that these passages were to the broader water-distribution network, demonstrating that civic utility and monumentality were far more than previously recognized.

Show answer & explanation

Answer

Blank 1: peripheral; Blank 2: integral; Blank 3: intertwined

The opening concession ('While early architectural historians dismissed...') establishes a contrast between the past view of the conduits as minor ('peripheral') and the modern finding that they were fundamental ('integral') to the water network. This shift logically leads to the synthesis that civic utility and architectural monumentality were closely linked ('intertwined').

Step-by-Step Solution

1

Analyze the opening contrast signal 'While early... dismissed... as mere [Blank 1]' against 'recent structural re-evaluations indicate'.

Blank 1 must denote something minor, unimportant, or secondary (e.g., 'peripheral').

The concession phrase 'While early... dismissed' requires Blank 1 to express a minimized role that contrasts with modern findings.

2

Track the structural dependency between Blank 1 and Blank 2.

Blank 2 must express a central or vital role (e.g., 'integral').

The shift introduced by 'recent... re-evaluations indicate that these passages were [Blank 2]' directly opposes their initial dismissal as minor conduits.

3

Evaluate the concluding synthesis clause 'demonstrating that civic utility and monumentality were far more [Blank 3]'.

Blank 3 must describe a close connection or union between utility and design (e.g., 'intertwined').

The resolution of the passage relies on showing that practical function (utility) and grand design (monumentality) worked in tandem rather than separately.

Key Concept

Multi-Blank Dependency Tracking

Question 1375Question →

During an operational quality review, a manufacturing plant evaluated its annual defect rate over a three-year period. In Year 1, the defect rate was $d\%$ , where $d > 0$ . In Year 2, the defect rate decreased by $20\%$ from Year 1. In Year 3, the defect rate increased by $25\%$ from Year 2. Which of the following statements must be true? Select all that apply.

Select all that apply

The defect rate in Year 3 is equal to the defect rate in Year 1.

The defect rate in Year 2 is equal to

80\%

of the defect rate in Year 1.

The net percentage change in the defect rate from Year 1 to Year 3 is an increase of

5\%

.

The defect rate in Year 1 is

20\%

greater than the defect rate in Year 2.

The ratio of the defect rate in Year 3 to the defect rate in Year 2 is

4 : 5

.

Show answer & explanation

Answer: The defect rate in Year 3 is equal to the defect rate in Year 1.; The defect rate in Year 2 is equal to $80\%$ of the defect rate in Year 1.

Answer

The statements confirming that the defect rate in Year 3 equals the defect rate in Year 1, and that the defect rate in Year 2 is 80% of Year 1, are correct.

The statement asserting that Year 3 defect rate equals Year 1 defect rate is true because

0.80 \times 1.25 = 1.00

. The statement asserting that Year 2 defect rate is 80% of Year 1 is true because a 20% reduction leaves 80% of the initial value.

Step-by-Step Solution

1

Express the Year 2 defect rate in terms of Year 1.

d_2 = d_1 \times (1 - 0.20) = 0.80 d_1

A

20\%

decrease from Year 1 corresponds to multiplying the base value

d_1

by

0.80

.

2

Express the Year 3 defect rate in terms of Year 2 and Year 1.

d_3 = d_2 \times (1 + 0.25) = (0.80 d_1) \times 1.25 = 1.00 d_1

A

25\%

increase over Year 2 multiplies

d_2

by

1.25

. Substituting

d_2 = 0.80 d_1

shows

d_3 = d_1

.

3

Evaluate each given choice against the derived relationships.

Year 3 equals Year 1 (True). Year 2 is

80\%

of Year 1 (True). Net change is

0\%

not

5\%

(False). Year 1 is

25\%

greater than Year 2, not

20\%

(False). Ratio of Year 3 to Year 2 is

5:4

, not

4:5

(False).

Only the first two statements hold true under rigorous percentage calculations.

Key Concept

Successive Percent Change and Base Value Shifts

Estimated Time:1m 30s

Question 1376Question →

Match each formal academic vocabulary term on the left to its corresponding functional near-synonym on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Pusillanimous matches with Craven; Truculent matches with Belligerent; Mercurial matches with Volatile; Perfidious matches with Treacherous.

Each vocabulary term on the left is matched precisely with its primary semantic near-synonym on the right.

Step-by-Step Solution

1

Analyze the core definition of Pusillanimous

Identified meaning as lacking courage or timid, matching Craven.

Both words describe a cowardly disposition.

2

Analyze the core definition of Truculent

Identified meaning as aggressively defiant or combative, matching Belligerent.

Both words describe hostile and aggressive behavior.

3

Analyze the core definition of Mercurial

Identified meaning as subject to sudden or unpredictable changes, matching Volatile.

Both words describe rapid, erratic shifts in temperament or condition.

4

Analyze the core definition of Perfidious

Identified meaning as deceitful and untrustworthy, matching Treacherous.

Both words refer to betrayal of trust or allegiance.

Key Concept

Identifying Synonym and Near-Synonym Pairs

Question 1377Question →

An agricultural research station monitored the water volume in a reservoir over a three-month period. In the first month, the reservoir's water volume decreased by $x\%$ . In the second month, the remaining water volume decreased by $(x + 10)\%$ . In the third month, heavy rainfall increased the water volume remaining at the end of the second month by $50\%$ . If the final water volume at the end of the third month was exactly $84\%$ of the initial water volume prior to the first month, what is the value of $x$ ?

10

15

20

25

30

Show answer & explanation

Answer: $20$

Answer

The value of

x

is

20

.

The correct answer is

20

. Applying successive percent change multipliers to the initial volume

V_0

gives

V_3 = 1.50 \times (1 - \frac{x+10}{100}) \times (1 - \frac{x}{100}) V_0

. Setting this equal to

0.84 V_0

yields the quadratic equation

k^2 - 1.90k + 0.34 = 0

where

k = \frac{x}{100}

. Solving gives

k = 0.20

, which means

x = 20

.

Step-by-Step Solution

1

Express the successive volume changes as percentage multipliers relative to the initial volume

V_0

.

After month 1:

V_1 = V_0 \left(1 - \frac{x}{100}\right)

. After month 2:

V_2 = V_1 \left(1 - \frac{x+10}{100}\right) = V_0 \left(1 - \frac{x}{100}\right)\left(0.90 - \frac{x}{100}\right)

. After month 3:

V_3 = V_2 \times 1.50 = 1.50 V_0 \left(1 - \frac{x}{100}\right)\left(0.90 - \frac{x}{100}\right)

.

Successive percentage changes must be applied sequentially to the updated volume at each stage.

2

Set the expression for final volume

V_3

equal to

0.84 V_0

and simplify the equation.

1.50 \left(1 - \frac{x}{100}\right)\left(0.90 - \frac{x}{100}\right) = 0.84 \implies \left(1 - \frac{x}{100}\right)\left(0.90 - \frac{x}{100}\right) = \frac{0.84}{1.50} = 0.56

.

The problem states that the final volume is

84\%

of the initial volume

V_0

.

3

Substitute

k = \frac{x}{100}

and formulate a quadratic equation.

(1 - k)(0.90 - k) = 0.56 \implies k^2 - 1.90k + 0.90 = 0.56 \implies k^2 - 1.90k + 0.34 = 0

. Multiply by

100

:

100k^2 - 190k + 34 = 0 \implies 50k^2 - 95k + 17 = 0

.

Converting to a standard quadratic form

ak^2 + bk + c = 0

allows solving for

k

.

4

Solve the quadratic equation for

k

.

k = \frac{95 \pm \sqrt{(-95)^2 - 4(50)(17)}}{2(50)} = \frac{95 \pm \sqrt{9025 - 3400}}{100} = \frac{95 \pm \sqrt{5625}}{100} = \frac{95 \pm 75}{100}

. Thus

k = 0.20

or

k = 1.70

.

The quadratic formula provides the roots for

k

.

5

Select the valid physical root and determine

x

.

Since a percentage decrease in water volume cannot exceed

100\%

(

k \le 1

), we discard

k = 1.70

. Therefore

k = 0.20

, which corresponds to

x = 20

.

A initial decrease of

170\%

is physically impossible in this context.

Key Concept

Multi-step percentage change and successive base updates leading to non-linear equations

Estimated Time:3m 0s

Question 1378Question →

At the beginning of a dry season, a municipal water reservoir contains an initial volume of $V$ gallons of water. During the first month, $\frac{1}{4}$ of the initial volume is released for agricultural irrigation. During the second month, $\frac{1}{5}$ of the remaining volume evaporates, after which $30,000$ gallons of water are pumped into the reservoir from an underground aquifer. If the final volume of water in the reservoir is equal to $\frac{3}{4}$ of the initial volume $V$ , what was the initial volume $V$ , in gallons?

75,000

150,000

200,000

250,000

600,000

Show answer & explanation

Answer: 200,000

Answer

200,000 gallons

Subtracting the first month's release of

\frac{1}{4}V

leaves

\frac{3}{4}V

. Evaporating

\frac{1}{5}

of this remaining amount leaves

\frac{4}{5} \times \frac{3}{4}V = \frac{3}{5}V

. Adding

30,000

gallons gives a final volume of

\frac{3}{5}V + 30,000

. Setting this equal to

\frac{3}{4}V

results in

\frac{3}{4}V - \frac{3}{5}V = \frac{3}{20}V = 30,000

, which solves to

V = 200,000

.

Step-by-Step Solution

1

Determine the volume remaining after the first month's irrigation release.

The volume remaining is

V - \frac{1}{4}V = \frac{3}{4}V

.

Releasing

\frac{1}{4}

of the initial volume leaves

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

of the initial volume.

2

Calculate the volume remaining after the second month's evaporation.

The volume remaining after evaporation is

\left(1 - \frac{1}{5}\right) \times \frac{3}{4}V = \frac{4}{5} \times \frac{3}{4}V = \frac{3}{5}V

.

Evaporating

\frac{1}{5}

of the remaining water leaves

\frac{4}{5}

of that remaining amount.

3

Formulate the linear equation including the pumped volume and solve for

V

.

\frac{3}{5}V + 30,000 = \frac{3}{4}V \implies 30,000 = \frac{3}{4}V - \frac{3}{5}V = \frac{15 - 12}{20}V = \frac{3}{20}V \implies V = \frac{30,000 \times 20}{3} = 200,000$.

Equating the expression for the final water volume to

\frac{3}{4}V

yields a single linear equation in

V

.

Key Concept

Sequential fractional reduction of remaining amounts and solving linear equations involving rational numbers.

Question 1379Question →

If $x$ is a real number that satisfies the equation $x - \sqrt{x + 2} = 4$ , what is the value of $(x + 2)^{\frac{3}{2}}$ ?

27

8

7\sqrt{7} + 2\sqrt{2}

-27

81

Show answer & explanation

Answer: $27$

Answer

27

Isolating the radical gives

x - 4 = \sqrt{x + 2}

. Squaring both sides yields

(x - 4)^2 = x + 2

, which expands to

x^2 - 8x + 16 = x + 2

, or

x^2 - 9x + 14 = 0

. Factoring yields

(x - 7)(x - 2) = 0

, giving solutions

x = 7

and

x = 2

. Testing

x = 7

in the original equation gives

7 - \sqrt{9} = 4

, which is valid. Testing

x = 2

gives

2 - \sqrt{4} = 0 \neq 4

, which is extraneous. Substituting the valid root

x = 7

into

(x + 2)^{\frac{3}{2}}

gives

(7 + 2)^{\frac{3}{2}} = 9^{\frac{3}{2}} = (\sqrt{9})^3 = 3^3 = 27

.

Step-by-Step Solution

1

Isolate the radical term in the given equation.

x - 4 = \sqrt{x + 2}

Isolating the radical allows both sides to be squared cleanly.

2

Square both sides to eliminate the square root and form a quadratic equation.

(x - 4)^2 = x + 2 \implies x^2 - 8x + 16 = x + 2 \implies x^2 - 9x + 14 = 0

Squaring eliminates the radical and allows standard quadratic solving techniques.

3

Factor the quadratic equation to find candidate solutions.

(x - 7)(x - 2) = 0 \implies x = 7 \text{ or } x = 2

Factoring provides potential real roots.

4

Check candidate solutions in the original equation to eliminate extraneous roots.

For

x = 7

:

7 - \sqrt{7 + 2} = 7 - 3 = 4

(valid).
For

x = 2

:

2 - \sqrt{2 + 2} = 2 - 2 = 0 \neq 4

(extraneous).

Squaring both sides can introduce false solutions that fail the original radical equation.

5

Evaluate the target expression

(x + 2)^{\frac{3}{2}}

using

x = 7

.

(7 + 2)^{\frac{3}{2}} = 9^{\frac{3}{2}} = (9^{\frac{1}{2}})^3 = 3^3 = 27

Applying exponent rules (

a^{\frac{m}{n}} = (\sqrt[n]{a})^m

) gives the exact required value.

Key Concept

Solving radical equations requires isolating the radical, squaring both sides, checking for extraneous solutions introduced by squaring, and evaluating fractional exponents via roots and integer powers.

Estimated Time:2m 0s

Question 1380Question →

What is the value of $\sqrt{4^5 + 4^5 + 4^5 + 4^5}$ ?

16

64

128

256

1024

Show answer & explanation

Answer: 64

Answer

64

Combining the four terms inside the radical gives

4 \times 4^5 = 4^6

. Taking the square root yields

\sqrt{4^6} = 4^3

, which evaluates to 64.

Step-by-Step Solution

1

Rewrite the sum inside the radical as multiplication

4^5 + 4^5 + 4^5 + 4^5 = 4 \times 4^5

Adding four identical terms is equivalent to multiplying the term by 4.

2

Apply the product rule for exponents

4^1 \times 4^5 = 4^{1+5} = 4^6

When multiplying exponential expressions with the same base, add their exponents.

3

Simplify the radical expression

\sqrt{4^6} = (4^6)^{1/2} = 4^3 = 64

Taking the square root of a power is equivalent to dividing the exponent by 2.

Key Concept

Exponent Addition and Radical Laws

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept