All practice questions

Question 1881Question →

A company has a total of $150$ employees assigned to either the Marketing department or the Development department. The number of employees in the Development department is $30$ more than $3$ times the number of employees in the Marketing department. How many employees work in the Marketing department?

25

30

40

45

120

Show answer & explanation

Answer: 30

Answer

The number of employees working in the Marketing department is 30.

Letting

m

represent the Marketing department employees, the Development department has

3m + 30

employees. Summing both yields

m + 3m + 30 = 150

, which simplifies to

4m + 30 = 150

. Subtracting 30 gives

4m = 120

, and dividing by 4 yields

m = 30

.

Step-by-Step Solution

1

Define variables for each department.

Let

m

be the number of employees in Marketing. Then the number of employees in Development is

3m + 30

.

Expressing both quantities in terms of a single variable simplifies setting up a linear equation.

2

Set up the linear equation representing total employees.

m + (3m + 30) = 150

The sum of employees in both departments equals the company total of 150.

3

Combine like terms.

4m + 30 = 150

Adding

m

and

3m

gives

4m

.

4

Isolate the variable term and solve for

m

.

4m = 120 \implies m = 30

Subtract 30 from both sides and then divide by 4.

Key Concept

Linear Equations in One Variable - Word Problem Modeling

Estimated Time:1m 0s

Question 1882Question →

In the $xy$ -plane, line $L$ passes through the points $(-3, 4)$ and $(5, -2)$ . Line $M$ is perpendicular to line $L$ and passes through the point $(1, -1)$ . What is the $y$ -intercept of line $M$ ?

-\frac{7}{3}

-\frac{7}{4}

-\frac{1}{4}

\frac{7}{3}

\frac{1}{3}

Show answer & explanation

Answer: $-\frac{7}{3}$

Answer

The

y

-intercept of line

M

is

-\frac{7}{3}

.

The slope of line

L

is

m_L = \frac{-2 - 4}{5 - (-3)} = -\frac{3}{4}

. The perpendicular line

M

has slope

m_M = \frac{4}{3}

. Substituting point

(1, -1)

into the line equation gives

y - (-1) = \frac{4}{3}(x - 1)

, which simplifies to

y = \frac{4}{3}x - \frac{7}{3}

. Thus, the

y

-intercept is

-\frac{7}{3}

.

Step-by-Step Solution

1

Calculate the slope of line

L

using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

m_L = \frac{-2 - 4}{5 - (-3)} = \frac{-6}{8} = -\frac{3}{4}

.

The slope of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

is given by the vertical change divided by horizontal change.

2

Determine the slope of line

M

using the perpendicular line slope relationship.

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

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

3

Find the equation of line

M

using point-slope form

y - y_1 = m(x - x_1)

with point

(1, -1)

and slope

m_M = \frac{4}{3}

.

y - (-1) = \frac{4}{3}(x - 1) \implies y + 1 = \frac{4}{3}x - \frac{4}{3} \implies y = \frac{4}{3}x - \frac{7}{3}

.

The point-slope form allows writing the linear equation directly given one point and the slope.

4

Identify the

y

-intercept of line

M

.

The

y

-intercept is

(0, -\frac{7}{3})

, or simply

-\frac{7}{3}

.

In slope-intercept form

y = mx + b

, the constant term

b

represents the

y

-intercept.

Key Concept

Perpendicular lines have slopes that are negative reciprocals (

m_1 \cdot m_2 = -1

). The equation of a line can be determined using its slope and a given point.

Estimated Time:1m 30s

Question 1883Question →

Pipes $A$ and $B$ , operating together at their respective constant rates, can fill an empty storage tank in $6$ hours. Pipes $B$ and $C$ , operating together at their constant rates, can fill the same tank in $8$ hours. The rate at which Pipe $A$ fills the tank is $1.5$ times the rate at which Pipe $C$ fills the tank. If all three pipes operate together for $3$ hours to fill the empty tank, after which Pipe $B$ is closed, how many additional hours will it take for Pipes $A$ and $C$ together to finish filling the tank?

\frac{6}{5}

\frac{12}{5}

\frac{18}{5}

\frac{21}{5}

3

Show answer & explanation

Answer: $\frac{6}{5}$

Answer

\frac{6}{5}

hours

The rate equations established from the problem yield individual rates of

\frac{1}{8}

tank/hr for Pipe

A

,

\frac{1}{24}

tank/hr for Pipe

B

, and

\frac{1}{12}

tank/hr for Pipe

C

. Working together for

3

hours at a combined rate of

\frac{1}{4}

tank/hr, all three pipes fill

\frac{3}{4}

of the tank, leaving

\frac{1}{4}

of the capacity to be filled. After Pipe

B

closes, Pipes

A

and

C

work at a combined rate of

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

tank/hr. Dividing the remaining

\frac{1}{4}

tank by

\frac{5}{24}

tank/hr gives

\frac{6}{5}

hours.

Step-by-Step Solution

1

Express given conditions in terms of work rates per hour (

r_A, r_B, r_C

).

r_A + r_B = \frac{1}{6}

,

r_B + r_C = \frac{1}{8}

, and

r_A = 1.5 r_C = \frac{3}{2} r_C

.

Filling a full tank in

H

hours means completing

\frac{1}{H}

of the tank per hour.

2

Solve the system of equations for individual rates

r_A, r_B, r_C

.

Substituting

r_B = \frac{1}{8} - r_C

into

r_A + r_B = \frac{1}{6}

yields

\frac{3}{2} r_C + \frac{1}{8} - r_C = \frac{1}{6} \implies \frac{1}{2} r_C = \frac{1}{24} \implies r_C = \frac{1}{12}

. Consequently,

r_A = \frac{1}{8}

and

r_B = \frac{1}{24}

.

Finding individual unit rates allows calculating any combined work scenario.

3

Calculate the fraction of the tank filled in the first 3 hours with all three pipes open.

Combined rate

r_A + r_B + r_C = \frac{1}{8} + \frac{1}{24} + \frac{1}{12} = \frac{3+1+2}{24} = \frac{6}{24} = \frac{1}{4}

of the tank per hour. In

3

hours,

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

of the tank is filled, leaving

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

of the tank empty.

Work completed equals rate multiplied by time.

4

Determine the additional time needed for Pipes

A

and

C

to fill the remaining

\frac{1}{4}

of the tank.

Combined rate of

A

and

C

is

r_A + r_C = \frac{1}{8} + \frac{1}{12} = \frac{5}{24}

per hour. Additional time

t = \frac{\frac{1}{4}}{\frac{5}{24}} = \frac{1}{4} \times \frac{24}{5} = \frac{6}{5}

hours.

Time required equals remaining work divided by active rate.

Key Concept

Combined Work Rates and Systems of Rate Equations

Question 1884Question →

Based on the structural clues and transition signals in the passage, enter a transition word that best completes the blank.

Fill in the blanks below

In marine bioacoustics, researchers historically regarded deep-sea cetacean vocalizations as purely navigational signals; , recent spectral analyses demonstrate that these acoustic patterns carry intricate social information, effectively reversing long-held assumptions regarding their functional simplicity.

Show answer & explanation

Answer

however (or an equivalent contrast signal such as nevertheless, yet, conversely, or nonetheless)

The sentence contrasts an older view ('purely navigational signals') with a modern finding ('intricate social information'). The concluding participial phrase ('effectively reversing long-held assumptions') explicitly confirms that a contrast transition word such as 'however', 'nevertheless', or 'conversely' is required.

Step-by-Step Solution

1

Analyze the relationship between the two main clauses connected across the semicolon.

The first clause notes that vocalizations were historically viewed as 'purely navigational signals', whereas the second clause presents findings showing they carry 'intricate social information'.

Comparing 'purely navigational' with 'intricate social information' reveals a direct contrast in understanding.

2

Identify structural clues that indicate the direction of the thought.

The phrase 'effectively reversing long-held assumptions' signals a reversal of perspective.

A reversal marker requires an explicit contrast or adversarial transition signal to join the clauses logically.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Question 1885Question →

For all real numbers $a$ and $b$ , which of the following expressions are equivalent to $(a + b)^2 - (a - b)^2$ ? Select all that apply.

Select all that apply

4ab

(2a)(2b)

0

2a^2 + 2b^2

2ab

Show answer & explanation

Answer: $4ab$ ; $(2a)(2b)$

Answer

The expressions

4ab

and

(2a)(2b)

are equivalent to

(a + b)^2 - (a - b)^2

.

Expanding the squared expressions gives

(a+b)^2 = a^2 + 2ab + b^2

and

(a-b)^2 = a^2 - 2ab + b^2

. Subtracting the second expression from the first yields

(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = 4ab

. Since

(2a)(2b)

also equals

4ab

, both

4ab

and

(2a)(2b)

are mathematically equivalent to the original expression.

Step-by-Step Solution

1

Expand

(a + b)^2

using the binomial square formula.

(a + b)^2 = a^2 + 2ab + b^2

Squaring a sum produces a trinomial containing two squared terms and a middle product term.

2

Expand

(a - b)^2

using the binomial square formula.

(a - b)^2 = a^2 - 2ab + b^2

Squaring a difference produces a trinomial with a negative middle product term.

3

Subtract the second expanded trinomial from the first.

(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = a^2 - a^2 + 2ab + 2ab + b^2 - b^2 = 4ab

Distributing the negative sign across

(a^2 - 2ab + b^2)

changes the signs inside the second set of parentheses.

4

Compare the simplified result

4ab

against all given choices.

Both

4ab

and

(2a)(2b) = 4ab

are identical to the calculated result.

Algebraic multiplication shows

(2a)(2b) = 4ab

.

Key Concept

Binomial expansion and combination of like terms

Question 1886Question →

In GRE Sentence Equivalence questions, test-takers must evaluate candidate vocabulary choices against sentence logic while avoiding semantic traps. Match each candidate word scenario on the left with its correct analytical classification on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match True Contextual Synonym Pair to valid equivalent words fitting context; False Synonym Trap Pair to valid dictionary synonyms that contradict sentence clues; Topically Related Distractor to subject-associated words lacking a pair; and Unpaired Contextual Fit to plausible single words without a synonymous option.

The correct pairings accurately distinguish between true equivalent pairs, out-of-context synonym traps, topically biased distractors, and unpaired contextual fits based on GRE Sentence Equivalence rules.

Step-by-Step Solution

1

Analyze the role of True Contextual Synonym Pairs in Sentence Equivalence.

Identify that correct choices must form a synonym pair AND satisfy the sentence's logical direction.

Sentence Equivalence requires two options that create sentences with identical meanings.

2

Examine False Synonym Trap Pairs.

Recognize that dictionary synonyms are incorrect if they violate tone or contrast signals.

Test writers frequently include true synonym pairs that create a coherent but contextually incorrect sentence.

3

Evaluate Topically Related Distractors and Unpaired Contextual Fits.

Match subject-matter triggers and single fitting words to their respective failure mechanisms.

Distractors often exploit domain familiarity or single-word plausibility without fulfilling the requirement for equivalent paired sentences.

Key Concept

Eliminating False Synonyms and Topically Related Distractors in Sentence Equivalence

Question 1887Question →

While the lead archivist acknowledged that the primary documents were somewhat fragmented, she maintained that the chronicler’s narrative was essentially __________, as it contained no demonstrable fabrications or intentional distortions. Which two of the following options, when inserted into the sentence, produce completed sentences that are most similar in meaning?

Select all that apply

veracious

truthful

exhaustive

thorough

compelling

lucid

Show answer & explanation

Answer: veracious; truthful

Answer

The correct selections are the options corresponding to 'veracious' and 'truthful'. Both words mean truthful or accurate, perfectly matching the contextual clue that the account contained no fabrications, and creating sentences with identical meanings.

The sentence specifies that the chronicler's narrative 'contained no demonstrable fabrications or intentional distortions.' The blank therefore requires a word that means truthful, honest, or factual. 'veracious' (meaning truthful) and 'truthful' (meaning conforming to truth) fit this definition precisely and create sentences with identical meaning.

Step-by-Step Solution

1

Analyze sentence context and structural pivots

The sentence uses 'While' to contrast fragmented documents with the chronicler's account, and the explanatory clause 'contained no demonstrable fabrications or intentional distortions' indicates that the blank must mean truthful, factual, or accurate.

The target word must describe an account that is free of falsehoods.

2

Evaluate option definitions and identify synonym pairs

Identified two potential synonym pairs: 'veracious' and 'truthful' (both meaning honest/accurate), and 'exhaustive' and 'thorough' (both meaning complete/detailed). 'Compelling' and 'lucid' stand alone without semantic pairs.

Sentence Equivalence requires selecting two words that both fit contextually and produce equivalent sentences.

3

Test synonym pairs against context

'veracious' and 'truthful' directly align with being free of fabrications. 'exhaustive' and 'thorough' fail context because the passage explicitly states the underlying records were fragmented, so the narrative could not be fully complete.

Distractors may form valid dictionary synonym pairs but fail the specific logical requirements of the blank.

Key Concept

Selecting Contextually Valid Synonym Pairs in Sentence Equivalence

Question 1888Question →

If $x$ satisfies the equation $\frac{3x - 1}{2} - \frac{x + 4}{5} = \frac{x + 7}{10} + 2$ , which of the following statements about $x$ must be true? Indicate all such statements.

Select all that apply

x > 3

3x

is an integer

x

is a solution to the equation

3x - 10 = 0

x

is an integer

x < 3.2

Show answer & explanation

Answer: $x > 3$ ; $3x$ is an integer; $x$ is a solution to the equation $3x - 10 = 0$

Answer

The statements that

x > 3

, that

3x

is an integer, and that

x

is a solution to the equation

3x - 10 = 0

are all correct.

Solving the equation yields

x = \frac{10}{3} \approx 3.33

. The statement

x > 3

is correct since

3.33 > 3

. The statement that

3x

is an integer is correct because

3 \times \frac{10}{3} = 10

. The statement that

x

is a solution to

3x - 10 = 0

is correct because

3\left(\frac{10}{3}\right) - 10 = 0

.

Step-by-Step Solution

1

Clear denominators by multiplying the entire linear equation by the least common multiple, 10.

5(3x - 1) - 2(x + 4) = (x + 7) + 20

Eliminating fractions simplifies the expression into a standard linear form.

2

Distribute terms across parentheses.

15x - 5 - 2x - 8 = x + 7 + 20

Expand both sides while correctly applying sign rules.

3

Combine like terms on each side.

13x - 13 = x + 27

Group variable terms together and constant terms together.

4

Isolate the variable

x

.

12x = 40 \implies x = \frac{40}{12} = \frac{10}{3}

Subtract

x

and add

13

on both sides, then divide by

12

.

5

Evaluate each given statement using

x = \frac{10}{3}

.

The value

\frac{10}{3} \approx 3.33

is greater than

3

;

3\left(\frac{10}{3}\right) = 10

is an integer;

3\left(\frac{10}{3}\right) - 10 = 0

is true;

\frac{10}{3}

is not an integer;

\frac{10}{3}

is not less than

3.2

.

Determine which logical assertions hold true for

x = \frac{10}{3}

.

Key Concept

Solving multi-step linear equations with fractions and evaluating properties of rational solutions.

Question 1889Question →

If $x$ and $y$ are positive integers such that $2^x - 2^y = 1920$ , what is the value of $x + y$ ?

11

15

18

22

25

Show answer & explanation

Answer: $18$

Answer

The correct value is 18.

Factoring

2^y

from

2^x - 2^y

yields

2^y(2^{x-y} - 1)

. Factoring 1920 as

2^7 \times 15

allows us to uniquely match the power-of-two factor

2^y = 2^7

(

y = 7

) and the odd factor

2^{x-y} - 1 = 15

(

x - y = 4 \implies x = 11

). Summing

x

and

y

gives

11 + 7 = 18

.

Step-by-Step Solution

1

Factor out the smaller power of 2 from the expression

2^y(2^{x-y} - 1) = 1920

Since

x

and

y

are positive integers and

2^x - 2^y > 0

, we know

x > y

. Factoring

2^y

separates the even power of 2 component from an odd component.

2

Determine the prime factorization of 1920 into a power of 2 and an odd integer

1920 = 128 \times 15 = 2^7 \times 15

Repeatedly dividing 1920 by 2 gives

1920 = 2^7 \times 15

, where 15 is an odd integer.

3

Equate the even and odd components

2^y = 2^7 \implies y = 7

and

2^{x-y} - 1 = 15 \implies 2^{x-y} = 16 = 2^4 \implies x - y = 4

Because

x - y \ge 1

, the quantity

2^{x-y} - 1

must be an odd integer, forcing it to equal 15 and the power of 2 factor to equal

2^7

.

4

Solve for

x

and calculate the final sum

x + y

x = 7 + 4 = 11

, so

x + y = 11 + 7 = 18

Adding

x = 11

and

y = 7

gives the required value.

Key Concept

Factoring exponential expressions by pulling out the common base power and matching unique prime factorizations.

Question 1890Question →

Line $Q$ passes through the coordinates $(-1, 7)$ and $(7, 1)$ in the Cartesian plane. Line $P$ is perpendicular to line $Q$ and intersects the $x$ -axis at $(-6, 0)$ . What is the $y$ -intercept of line $P$ ?

Show answer & explanation

Answer: 8

Answer

The

y

-intercept of line

P

is 8.

The slope of line

Q

is calculated as

\frac{1 - 7}{7 - (-1)} = -\frac{3}{4}

. Since line

P

is perpendicular to line

Q

, its slope is the negative reciprocal,

\frac{4}{3}

. Using the

x

-intercept point

(-6, 0)

, the line equation is

y - 0 = \frac{4}{3}(x + 6)

, which simplifies to

y = \frac{4}{3}x + 8

. Setting

x = 0

yields the

y

-intercept of

8

.

Step-by-Step Solution

1

Calculate the slope of line

Q

Slope of line

Q

is

-\frac{3}{4}

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{7 - (-1)} = \frac{-6}{8} = -\frac{3}{4}

.

2

Determine the slope of perpendicular line

P

Slope of line

P

is

\frac{4}{3}

Perpendicular lines have slopes that are negative reciprocals of each other:

m_P = -\frac{1}{m_Q} = -\frac{1}{-3/4} = \frac{4}{3}

.

3

Find the equation and

y

-intercept of line

P

The

y

-intercept is

8

Line

P

passes through

(-6, 0)

. Using point-slope form:

y - 0 = \frac{4}{3}(x - (-6)) \implies y = \frac{4}{3}x + 8

. Substituting

x = 0

gives

y = 8

.

Key Concept

Perpendicular lines and line equations in coordinate geometry

Question 1891Question →

Read the passage below:

Far from revealing an evolutionary trajectory that was (i)________, the fossil discoveries at the rift valley demonstrated a remarkable structural continuity, compelling paleontologists to conclude that the morphological shifts observed across epochs were (ii)________ rather than abrupt.

Which of the following pairs of terms best completes the passage so that the text maintains logical coherence and inter-blank dependency?

(i) erratic, (ii) incremental

(i) erratic, (ii) sporadic

(i) immutable, (ii) incremental

(i) deliberate, (ii) anomalous

(i) immutable, (ii) sporadic

Show answer & explanation

Answer: (i) erratic, (ii) incremental

Answer

The pair '(i) erratic, (ii) incremental' correctly completes the passage by establishing proper contrast in the first clause and coherent alignment in the second.

The correct completion is '(i) erratic, (ii) incremental'. The opening structure 'Far from revealing [Blank i]... demonstrated structural continuity' establishes a clear contrast, requiring a word in Blank (i) that means irregular or disjointed ('erratic'). In the second clause, the morphological shifts are described as 'Blank (ii) rather than abrupt' while maintaining 'structural continuity', requiring a word that means gradual or step-by-step ('incremental'). Together, both words satisfy the inter-blank dependency and structural pivots.

Step-by-Step Solution

1

Analyze the structural signals and pivot words in the first sentence clause.

The phrase 'Far from revealing...' indicates a reversal between Blank (i) and 'demonstrated a remarkable structural continuity'.

Blank (i) must describe a trajectory opposite to one characterized by structural continuity.

2

Determine the semantic valence required for Blank (i).

'Erratic' (disjointed, unpredictable, or lacking continuity) serves as the exact opposite of structural continuity.

This satisfies the concession/contrast introduced by 'Far from'.

3

Analyze the structural context surrounding Blank (ii).

Blank (ii) describes 'morphological shifts' that are characterized as 'rather than abrupt' and supported by 'structural continuity'.

Blank (ii) requires a word meaning gradual or continuous to contrast with 'abrupt'.

4

Evaluate candidate words for Blank (ii) and verify inter-blank alignment.

'Incremental' (occurring in small, gradual steps) aligns with structural continuity and directly opposes 'abrupt'.

Pairing '(i) erratic' with '(ii) incremental' creates a fully coherent sentence across both blanks.

Key Concept

Multi-Blank Dependency Tracking

Estimated Time:1m 15s

Question 1892Question →

A delivery truck traveled from Warehouse X to Warehouse Y at a constant speed of $50$ miles per hour. On the return trip along the exact same route, heavy traffic reduced the truck's constant speed to $30$ miles per hour. If the total driving time for the entire round trip was $8$ hours, what was the distance, in miles, between Warehouse X and Warehouse Y?

120

150

160

250

300

Show answer & explanation

Answer: $150$

Answer

The distance between Warehouse X and Warehouse Y is

150

miles.

The correct distance between the two warehouses is

150

miles. Since time equals distance divided by rate, the time taken going to Warehouse Y is

\frac{d}{50}

hours and returning is

\frac{d}{30}

hours. Summing these expressions to equal the total

8

hours gives

\frac{d}{50} + \frac{d}{30} = 8

. Solving for

d

yields

\frac{8d}{150} = 8

, which simplifies to

d = 150

miles.

Step-by-Step Solution

1

Define variables and write time expressions for each leg of the trip

Let

d

be the distance in miles between Warehouse X and Warehouse Y. Time outbound is

t_1 = \frac{d}{50}

hours, and time inbound is

t_2 = \frac{d}{30}

hours.

Distance divided by rate gives the time taken for each individual leg of the journey.

2

Set up the equation using total elapsed time

\frac{d}{50} + \frac{d}{30} = 8

The total driving time for both legs combined is given as

8

hours.

3

Solve the algebraic equation for

d

\frac{3d + 5d}{150} = 8 \implies \frac{8d}{150} = 8 \implies 8d = 1200 \implies d = 150

Finding a common denominator of

150

allows combining the fractions to solve for the unknown distance

d

.

Key Concept

Distance-Rate-Time Relationship & Harmonic Mean Rate Modeling

Question 1893Question →

Read the following passage concerning early modern European cartography:

"In early modern European cartography, mapmakers rarely aimed for literal geographic fidelity; rather, their illustrations were designed to inform the viewer's moral worldview. Indeed, far from functioning merely to report topographical coordinates, these elaborate maps served primarily to shape and animate the audience's cultural values. Consequently, modern analysts who evaluate these maps solely by their empirical precision misapprehend their underlying purpose, mistaking a medium of ideological instruction for an unsuccessful attempt at spatial measurement."

In the context of the passage, the word "inform" most nearly means:

impart form or character to

notify regarding empirical facts

verify the accuracy of

denounce or officially condemn

simplify for ease of navigation

Show answer & explanation

Answer: impart form or character to

Answer

The word 'inform' in this context means to 'impart form or character to' (shape or animate).

The passage employs a contrast signal ('rarely...; rather...') followed by an elaboration signal ('Indeed, far from... served primarily to shape and animate'). These structural clues demonstrate that 'inform' does not mean to convey factual data, but rather to give shape, form, or character to the viewer's worldview.

Step-by-Step Solution

1

Analyze the structural pivot in the first sentence

The semi-colon followed by 'rather' creates a contrast between 'literal geographic fidelity' (empirical accuracy) and what map illustrations were designed to do to the viewer's moral worldview.

Contrast signals indicate that the target word must express a function distinct from literal reporting.

2

Examine the elaboration signal in the second sentence

The phrase 'Indeed, far from functioning merely to report... served primarily to shape and animate' restates and clarifies the clause containing 'inform'.

The continuation signal 'Indeed' directly connects 'inform' with its parallel structural equivalent: 'to shape and animate'.

3

Select the definition matching the contextual structural clues

The secondary definition of 'inform'—meaning to imbue, animate, or give form to—aligns precisely with 'shape and animate'.

Relying on structural context resolves the polysemous word to its secondary scholarly definition rather than its common usage.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Estimated Time:1m 30s

Question 1894Question →

Determine which terms best complete each corresponding blank in the passage to maintain cross-sentence contextual coherence.

Fill in the blanks below

Early attempts to codify maritime law in the seventeenth century were hardly ; rather than establishing universally accepted protocols, they reflected the partisan economic interests of individual seafaring powers. Consequently, modern historians who characterize these early treaties as foundational models of international altruism the pervasive commercial rivalries that dictated their terms. It was only when systemic piracy threatened all merchant fleets alike that sovereign states moved past unilateral maneuvering and approached oceanic governance with genuine .

Show answer & explanation

Answer

Blank 1 requires 'disinterested', Blank 2 requires 'gloss over', and Blank 3 requires 'reciprocity'.

The passage establishes a contrast between initial partisan lawmaking and later cooperative governance. For the first blank, the contrast with 'partisan economic interests' demands 'disinterested' (meaning impartial). For the second blank, the causal transition 'Consequently' indicates that scholars viewing partisan treaties as altruistic are ignoring historical evidence, meaning they 'gloss over' rivalries. For the third blank, moving past 'unilateral maneuvering' when faced with shared threats implies engaging in 'reciprocity' (mutual exchange and cooperation).

Step-by-Step Solution

1

Analyze Blank 1 using internal contrast clues within the first sentence.

The clause 'rather than establishing universally accepted protocols, they reflected the partisan economic interests' indicates that the first blank must mean unbiased or impartial. 'Disinterested' fits this exact definition.

The contrast signal 'rather than' opposes partisan interest with impartiality.

2

Analyze Blank 2 using the causal transition 'Consequently' connecting Sentence 1 and Sentence 2.

Since the treaties were actually driven by partisan self-interest, historians who view them as altruistic are failing to acknowledge the true rivalries. Thus, they 'gloss over' these rivalries.

The shift signal 'Consequently' requires Blank 2 to express a failure to account for the partisan reality established in Sentence 1.

3

Analyze Blank 3 using the condition specified in the final sentence.

The text states that states 'moved past unilateral maneuvering' when threats impacted 'all merchant fleets alike'. This shift toward shared, non-unilateral action requires a word meaning mutual exchange or cooperation, which is 'reciprocity'.

Moving past 'unilateral maneuvering' logically leads to bilateral or multilateral mutual cooperation.

Key Concept

Cross-Sentence Contextual Coherence in Multi-Blank Text Completion

Question 1895Question →

Match each formal vocabulary term on the left with its most precise functional near-synonym on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Fastidious pairs with Punctilious; Prevaricate pairs with Dissemble; Munificent pairs with Bountiful; Taciturn pairs with Reticent.

Each left term aligns with its functional near-synonym based on shared core definitions: 'Fastidious' and 'Punctilious' both indicate extreme care regarding details; 'Prevaricate' and 'Dissemble' both signify misleading evasiveness; 'Munificent' and 'Bountiful' express broad generosity; and 'Taciturn' and 'Reticent' describe quiet restraint.

Step-by-Step Solution

1

Analyze the primary definition and tone of each left item.

'Fastidious' means demanding and attentive to detail. 'Prevaricate' means to lie or mislead. 'Munificent' means remarkably generous. 'Taciturn' means reserved in speech.

Establishing core semantic definitions allows accurate alignment with corresponding synonyms.

2

Match each term to its nearest functional synonym among the right items.

'Punctilious' shares the exact nuance of meticulous detail with 'fastidious'. 'Dissemble' matches the deceitful evasion of 'prevaricate'. 'Bountiful' reflects the lavish generosity of 'munificent'. 'Reticent' mirrors the quiet reserve of 'taciturn'.

Direct semantic correspondence establishes correct synonym pairs without contextual ambiguity.

Key Concept

Identifying Synonym and Near-Synonym Pairs

Estimated Time:1m 0s

Question 1896Question →

An investor allocates a total principal of $\$10,000$ between two investment options, Account X and Account Y. Account X earns simple interest at an annual rate of $8\%$ . Account Y earns interest at an annual rate of $8\%$ compounded semi-annually. Neither account receives additional deposits or withdrawals for a period of $2$ years. Which of the following statements MUST be true?

Select all such statements.

Select all that apply

If the total principal is divided equally between Account X and Account Y, the total interest earned after

2

years is greater than

\$1,640

.

Allocating a greater portion of the total principal to Account Y strictly increases the total interest earned after

2

years.

If the entire

\$10,000

principal is placed in Account X, the total interest earned after

2

years is

\$1,664

.

Show answer & explanation

Answer: If the total principal is divided equally between Account X and Account Y, the total interest earned after $2$ years is greater than $\$1,640$ .; Allocating a greater portion of the total principal to Account Y strictly increases the total interest earned after $2$ years.

Answer

The correct statements are the statement regarding equal division yielding total interest greater than $1,640, and the statement that allocating a greater portion of principal to Account Y increases the total interest earned.

The statement regarding equal principal division is true because

5,000 placed in Account X yields

800 in simple interest and

5,000 placed in Account Y yields

849.29 under semi-annual compounding, giving a sum of

1,649.29 which exceeds

1,640. The statement regarding shifting allocation to Account Y is true because Account Y's effective two-year yield of 16.99% is higher than Account X's two-year simple yield of 16.00%.

Step-by-Step Solution

1

Calculate the effective 2-year return for Account X under simple interest.

Simple interest over 2 years at an annual rate of 8% is

2 \times 8\% = 16\%

. Total return factor is

0.16

.

Simple interest accumulates linearly as

I = P \cdot r \cdot t

.

2

Calculate the effective 2-year return for Account Y under semi-annual compound interest.

Semi-annual rate is

\frac{8\%}{2} = 4\% = 0.04

. The number of compounding periods in 2 years is

2 \times 2 = 4

. Total compound multiplier is

(1.04)^4 = 1.16985856

. Thus, the total interest earned is

16.985856\%

of the principal.

Compound interest uses the period rate raised to the power of total compounding periods:

(1 + r_{period})^n

.

3

Evaluate the statement for equal division of $10,000.

With

5,000 in Account X: interest =

5,000 \times 0.16 =

800. With

5,000 in Account Y: interest =

5,000 \times 0.16985856 \approx

849.29. Total interest =

800 +

849.29 =

1,649.29. Since

1,649.29 > $1,640, this statement is TRUE.

Summing the interest from both accounts confirms the total exceeds $1,640.

4

Evaluate the statement comparing allocation shifts.

Since Account Y earns approximately

16.99\%

over 2 years and Account X earns

16.00\%

, the yield of Y exceeds X by approximately

0.99\%

. Thus, increasing the allocation to Account Y increases overall interest earned. This statement is TRUE.

Reallocating principal to an option with a higher effective yield strictly increases overall earnings.

5

Evaluate the statement for placing all funds in Account X.

If all

10,000 is in Account X, interest =

10,000 \times 0.16 =

1,600. The claim of

1,664 is FALSE.

1,664 results from calculating annual compound interest (

10,000 \times (1.08^2 - 1)$), which does not apply to simple interest.

Key Concept

Simple Interest vs. Compound Interest

Estimated Time:2m 0s

Question 1897Question →

In convex pentagon $ABCDE$ , the measure of interior angle $\angle A$ is $100^\circ$ . The measures of the remaining four interior angles, $\angle B$ , $\angle C$ , $\angle D$ , and $\angle E$ , are in the ratio $2 : 3 : 3 : 3$ . What is the measure, in degrees, of $\angle B$ ?

40^\circ

80^\circ

108^\circ

120^\circ

160^\circ

Show answer & explanation

Answer: $80^\circ$

Answer

The correct answer is

80^\circ

, which corresponds to the measure of

\angle B

.

The sum of interior angles of a 5-sided polygon (pentagon) is

(5-2) \times 180^\circ = 540^\circ

. Subtracting

\angle A = 100^\circ

leaves

440^\circ

for the remaining four angles. The total ratio parts for these four angles is

2 + 3 + 3 + 3 = 11

. Dividing

440^\circ

by 11 gives

40^\circ

per ratio unit. Since

\angle B

corresponds to 2 ratio parts, its measure is

2 \times 40^\circ = 80^\circ

.

Step-by-Step Solution

1

Calculate the sum of all interior angles of the pentagon.

The sum of interior angles for an

n

-sided polygon is given by

(n - 2) \times 180^\circ

. For a pentagon (

n = 5

), the sum is

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

Determining the total interior angle sum is necessary to find the sum of the unknown angles.

2

Subtract the known angle measure

\angle A

from the total interior angle sum.

The combined sum of angles

\angle B + \angle C + \angle D + \angle E = 540^\circ - 100^\circ = 440^\circ

.

Isolating the sum of the remaining four angles allows distribution according to the given ratio.

3

Determine the value of one ratio unit.

The sum of ratio parts is

2 + 3 + 3 + 3 = 11

parts. One part is equal to

440^\circ / 11 = 40^\circ

.

Finding the magnitude of a single ratio unit enables calculation of any individual angle.

4

Multiply the single unit value by the ratio coefficient for

\angle B

.

Since

\angle B

corresponds to 2 parts,

\angle B = 2 \times 40^\circ = 80^\circ

.

This yields the requested measure of angle

\angle B

.

Key Concept

Sum of interior angles of an n-sided polygon:

(n - 2) \times 180^\circ

, combined with proportional partitioning of angle sums.

Alternative Method

Express the angles in terms of a variable

x

. Let the remaining angles be

2x, 3x, 3x, 3x

. Set up the equation

100^\circ + 2x + 3x + 3x + 3x = 540^\circ

, simplify to

100 + 11x = 540

, solve for

x = 40^\circ

, and then evaluate

\angle B = 2x = 80^\circ

.

Estimated Time:1m 15s

Question 1898Question →

A rectangular floor measuring $126\text{ cm}$ by $180\text{ cm}$ is to be completely covered with identical square tiles of the largest possible side length, without cutting any tiles or leaving gaps. What is the total number of square tiles required to cover the floor?

18

35

70

140

306

Show answer & explanation

Answer: 70

Answer

The total number of square tiles required is 70.

To cover the rectangular floor with the largest possible identical square tiles, the side length of the square tile must be the greatest common divisor of

126

and

180

. Factoring both numbers gives

126 = 2 \times 3^2 \times 7

and

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

. The greatest common divisor is

2 \times 3^2 = 18\text{ cm}

. Dividing the dimensions by

18\text{ cm}

gives

7

tiles along the length and

10

tiles along the width. Multiplying

7 \times 10

yields a total of

70

tiles.

Step-by-Step Solution

1

Find the prime factorization of both dimensions of the floor.

126 = 2^1 \times 3^2 \times 7^1

and

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

Prime factorization allows systematic determination of the greatest common divisor.

2

Calculate the Greatest Common Divisor (GCD) of 126 and 180 to determine the largest possible square tile side length.

\gcd(126, 180) = 2^1 \times 3^2 = 18\text{ cm}

The square tiles must fit evenly along both the length and width without cutting.

3

Determine the number of tiles along each dimension.

Along length:

126 \div 18 = 7

; along width:

180 \div 18 = 10

Dividing each total length by the tile side length gives the tile count per side.

4

Calculate the total number of tiles required.

7 \times 10 = 70\text{ tiles}

The floor forms a 7 by 10 grid of tiles.

Key Concept

Greatest Common Divisor (GCD) applications in geometric tiling

Question 1899Question →

Three industrial machines, $X$ , $Y$ , and $Z$ , continuously produce liquid chemical mixtures composed solely of Compounds $A$ and $B$ .

- Machine $X$ produces the mixture at a constant rate of $100$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $3:2$ by volume.
- Machine $Y$ produces the mixture at a constant rate of $150$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $1:4$ by volume.
- Machine $Z$ produces the mixture at a constant rate of $R$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $7:3$ by volume.

If all three machines operate simultaneously for $4$ hours, the resulting combined mixture contains equal total volumes of Compound $A$ and Compound $B$ .

Which of the following statements must be true? Select all that apply.

Select all that apply

The operating rate

R

of Machine

Z

is

175

liters per hour.

Machine

Z

accounts for more than

40\%

of the total volume of the combined mixture produced in

4

hours.

In the 4-hour output, the volume of Compound

B

produced by Machine

Y

is greater than the total volume of Compound

B

produced by Machines

X

and

Z

combined.

The ratio of the volume of Compound

A

produced by Machine

X

to the volume of Compound

A

produced by Machine

Z

is

2:3

.

If Machine

Y

were turned off and only Machines

X

and

Z

operated for

4

hours, the combined output would have Compound

A

and Compound

B

in a ratio of

7:3

.

Show answer & explanation

Answer: The operating rate $R$ of Machine $Z$ is $175$ liters per hour.; Machine $Z$ accounts for more than $40\%$ of the total volume of the combined mixture produced in $4$ hours.; In the 4-hour output, the volume of Compound $B$ produced by Machine $Y$ is greater than the total volume of Compound $B$ produced by Machines $X$ and $Z$ combined.

Answer

The statements asserting that Machine Z's rate is 175 liters per hour, that Machine Z accounts for over 40% of the total output volume, and that Machine Y produces more Compound B than Machines X and Z combined are all true.

The correct options are those stating that Machine Z operates at 175 liters per hour, that Machine Z accounts for over 40% of the 4-hour output, and that Machine Y produces more Compound B than Machines X and Z combined. Machine X produces 240 liters of Compound A and 160 liters of Compound B in 4 hours. Machine Y produces 120 liters of Compound A and 480 liters of Compound B in 4 hours. Machine Z produces 2.8R liters of Compound A and 1.2R liters of Compound B in 4 hours. Equating total Compound A (360 + 2.8R) to total Compound B (640 + 1.2R) gives 1.6R = 280, so R = 175 liters per hour. Machine Z thus contributes 700 liters out of 1700 total liters, which is approximately 41.18% (greater than 40%). Machine Y produces 480 liters of Compound B, which exceeds the 370 liters of Compound B produced by Machines X and Z combined (160 + 210 liters).

Step-by-Step Solution

1

Calculate the volumes of Compound A and Compound B produced by Machines X and Y in 4 hours.

Machine X produces

400

L total (

240

L of A,

160

L of B). Machine Y produces

600

L total (

120

L of A,

480

L of B). Combined from X and Y:

360

L of A and

640

L of B.

Decompose each machine's total 4-hour output into individual compound amounts using part-to-whole fraction multipliers.

2

Formulate Machine Z's 4-hour output expressions and solve for rate R.

Machine Z produces

4R

L total (

2.8R

L of A,

1.2R

L of B). Setting total A equal to total B:

360 + 2.8R = 640 + 1.2R \implies 1.6R = 280 \implies R = 175

L/hr.

The condition of equal total volumes of Compound A and Compound B forms a single linear equation in terms of R.

3

Determine Machine Z's proportion of the total volume produced by all three machines.

Machine Z volume =

4 \times 175 = 700

L. Total volume =

400 + 600 + 700 = 1700

L. Percentage =

\frac{700}{1700} \times 100\% \approx 41.18\% > 40\%

.

Evaluate whether Machine Z contributes more than 40% of the complete 1700-liter mixture.

4

Compare Compound B output from Machine Y against the combined Compound B output from X and Z.

Machine Y produces

480

L of B. Machines X and Z produce

160 + (1.2 \times 175) = 370

L of B.

480 > 370

is true.

Direct numerical comparison verifies the inequality statement.

5

Evaluate the remaining ratio claims.

Compound A ratio X to Z is

240 : 490 = 24 : 49 \neq 2 : 3

. Combined X and Z ratio of A to B is

730 : 370 = 73 : 37 \neq 7 : 3

. Both claims are false.

Check remaining distractors for mathematical accuracy.

Key Concept

Multi-rate and multi-component ratio mixture balancing

Question 1900Question →

While contemporary scholars often praise the treatise for its seemingly progressive assertions, historical context reveals that the author's primary motivation was to ______ traditional orthodoxy rather than dismantle it. Which two options, when inserted into the sentence, produce completed sentences that are most similar in meaning?

Select all that apply

bolster

subvert

buttress

promulgate

undermine

scrutinize

Show answer & explanation

Answer: bolster; buttress

Answer

The correct selections are 'bolster' and 'buttress', as both words mean to strengthen or support, providing the exact logical opposition to 'dismantle' demanded by the structural contrast clue.

The sentence structure hinges on the contrast indicator 'rather than dismantle it', establishing that the author sought to preserve or support orthodoxy. Both 'bolster' and 'buttress' mean to strengthen or reinforce, completing the sentence with equivalent meanings that satisfy this contrast.

Step-by-Step Solution

1

Analyze structural transition and context clues in the stem.

The phrase 'rather than dismantle it' signals a contrast between what scholars praise (progressive assertions) and the author's actual goal (supporting or reinforcing traditional orthodoxy).

Identifying sentence polarity determines the required semantic direction for the blank.

2

Evaluate candidate options for synonym pairs and contextual fit.

Two distinct synonym pairs exist: 'bolster'/'buttress' (meaning to support/strengthen) and 'subvert'/'undermine' (meaning to weaken/overthrow).

Sentence Equivalence requires selecting pair members that produce coherent, identical meanings.

3

Eliminate false synonym traps and unpaired topical distractors.

The 'subvert'/'undermine' pair opposes the needed logical shift. 'Promulgate' and 'scrutinize' are topically related to treatises and analysis but lack synonymous matches and fail contextual fit.

Topical relevance does not equal contextual fit or pair equivalence.

Key Concept

Eliminating False Synonyms and Topically Related Distractors

Estimated Time:1m 15s

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