All practice questions

Match each advanced vocabulary word 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

The correct pairings are: obdurate matches obstinate; laconic matches taciturn; abstruse matches recondite; effervescent matches ebullient.

Each left-hand term is paired with its precise semantic equivalent: obdurate means obstinate (stubborn); laconic means taciturn (succinct/uncommunicative); abstruse means recondite (esoteric/obscure); and effervescent means ebullient (exuberant).

Step-by-Step Solution

1

Analyze the meaning of 'obdurate'.

Defines an unyielding or stubborn disposition.

Find a right-hand term with equivalent meaning, which is 'obstinate'.

2

Analyze the meaning of 'laconic'.

Refers to using concise or minimal phrasing in speech.

Match with 'taciturn', which describes being untalkative or concise.

3

Analyze the meaning of 'abstruse'.

Describes ideas that are complex, deep, or hard to comprehend.

Match with 'recondite', which means obscure or little-known.

4

Analyze the meaning of 'effervescent'.

Refers figuratively to vivacious, enthusiastic demeanor.

Match with 'ebullient', which means overflowing with enthusiasm.

Key Concept

Identifying direct semantic equivalence and primary definitions of high-frequency GRE vocabulary.

Question 1842Question →

Read the passage below:

While early musicologists dismissed the composer��s late symphonic sketches as mere (i)________ fragments never intended for performance, recent archival discoveries reveal that these draft scores were in fact (ii)________ to her final masterpiece, as key thematic motifs from the sketches recur directly throughout the completed work.

Which of the following combinations of words logically completes the blanks in the passage?

(i) chimerical, (ii) integral

(i) chimerical, (ii) peripheral

(i) pragmatic, (ii) integral

(i) provisional, (ii) superfluous

(i) indispensable, (ii) peripheral

Show answer & explanation

Answer: (i) chimerical, (ii) integral

Answer

The combination '(i) chimerical, (ii) integral' logically completes both blanks by establishing the required contrast between the initial dismissal of the sketches and their actual importance to the final work.

The correct choice establishing '(i) chimerical' and '(ii) integral' satisfies both local context clues and structural contrast. 'Chimerical' fits the view of the sketches as unrealizable fragments, while 'integral' aligns with the evidence that their core motifs appear throughout the final masterpiece.

Step-by-Step Solution

1

Analyze structural signals and contrast pivots in the sentence.

The opening word 'While' and the phrase 'in fact' indicate a contrast between how early musicologists viewed the sketches ('dismissed... as mere (i)________') and what recent discoveries demonstrate ('were in fact (ii)________').

Tracking inter-blank relationships requires identifying the logical pivot connecting the two clauses.

2

Determine the required meaning for Blank (i).

Because the sketches were 'dismissed' and described as 'never intended for performance', Blank (i) requires a word meaning fanciful, unrealistic, or imaginary, such as 'chimerical'.

The context clue 'mere... fragments never intended for performance' defines the negative characterization in the first clause.

3

Determine the required meaning for Blank (ii) using cross-sentence evidence.

The concluding clause notes that 'key thematic motifs from the sketches recur directly throughout the completed work,' indicating that the draft scores were essential or central, making 'integral' the correct fit.

The final clause provides the evidence confirming the exact degree of importance needed for the second blank.

Key Concept

Multi-Blank Dependency Tracking

Question 1843Question →

A positive integer $n$ yields a remainder of $7$ when divided by $12$ . What is the remainder when the expression $5n + 3$ is divided by $6$ ?

0

1

2

3

5

Show answer & explanation

Answer: $2$

Answer

The remainder when

5n + 3

is divided by

6

is

2

.

Since

n

leaves a remainder of

7

when divided by

12

, we can write

n = 12k + 7

for some integer

k \ge 0

. Substituting this into

5n + 3

yields

5(12k + 7) + 3 = 60k + 38

. Factoring out

6

gives

6(10k + 6) + 2

. Because

6(10k + 6)

is divisible by

6

, the remainder of the expression when divided by

6

is

2

.

Step-by-Step Solution

1

Express

n

in terms of the division algorithm for divisor

12

.

n = 12k + 7

for some non-negative integer

k

An integer that leaves a remainder of

7

when divided by

12

can be represented as

12k + 7

.

2

Substitute the expression for

n

into

5n + 3

and simplify.

5(12k + 7) + 3 = 60k + 35 + 3 = 60k + 38

Algebraic expansion allows us to analyze the entire expression modulo

6

.

3

Determine the remainder of

60k + 38

when divided by

6

.

60k + 38 = 6(10k + 6) + 2

Since

60k + 36

is an exact multiple of

6

, the leftover term

2

is the remainder.

Key Concept

Properties of Integers and Remainder Arithmetic

Estimated Time:1m 30s

Question 1844Question →

Let $m$ and $n$ be positive integers such that $m$ is divisible by $18$ and $n$ is divisible by $15$ . Which of the following integers MUST be a divisor of the product $m \cdot n$ ? Select all such values.

Select all that apply

54

90

135

36

60

Show answer & explanation

Answer: $54$ ; $90$ ; $135$

Answer

The integers

54

,

90

, and

135

must be divisors of the product

m \cdot n

.

Since

m

is a multiple of

18

(

2^1 \times 3^2

) and

n

is a multiple of

15

(

3^1 \times 5^1

), their product

m \cdot n

must be a multiple of

18 \times 15 = 270

. The prime factorization of

270

is

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

. Any integer that divides

270

is guaranteed to divide

m \cdot n

for all valid values of

m

and

n

. The numbers

54

(

2^1 \times 3^3

),

90

(

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

), and

135

(

3^3 \times 5^1

) are all divisors of

270

.

Step-by-Step Solution

1

Express

m

and

n

in terms of their minimal prime factorizations.

m = 18a = 2^1 \times 3^2 \times a

and

n = 15b = 3^1 \times 5^1 \times b

for positive integers

a

and

b

.

Divisibility conditions specify the minimum prime factors that

m

and

n

must contain.

2

Find the minimal guaranteed prime factorization of the product

m \cdot n

.

m \cdot n = (2^1 \times 3^2 \times a) \times (3^1 \times 5^1 \times b) = 2^1 \times 3^3 \times 5^1 \times (ab) = 270 \times ab

.

Multiplying

m

and

n

combines their guaranteed prime factor powers.

3

Determine which choices divide

270 = 2^1 \times 3^3 \times 5^1

without requiring additional factors of

a

or

b

.

54 = 2^1 \times 3^3

,

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

, and

135 = 3^3 \times 5^1

all divide

270

. Numbers requiring

2^2

(

36

and

60

) do not necessarily divide

270

.

A number MUST divide

m \cdot n

if its prime factor powers do not exceed the minimum guaranteed powers in

m \cdot n

.

Key Concept

Divisibility of Integer Products via Prime Factorization

Question 1845Question →

A sequence $a_1, a_2, a_3, \dots, a_n$ is defined by $a_1 = 1$ and $a_n = 2a_{n-1} + 3$ for all integers $n \geq 2$ . What is the value of $a_4$ ?

11

13

25

29

61

Show answer & explanation

Answer: $29$

Answer

The value of

a_4

is

29

.

Using the recursive rule

a_n = 2a_{n-1} + 3

starting with

a_1 = 1

, we compute

a_2 = 2(1) + 3 = 5

,

a_3 = 2(5) + 3 = 13

, and finally

a_4 = 2(13) + 3 = 29

.

Step-by-Step Solution

1

Calculate the second term

a_2

using the recursive formula with

a_1 = 1

.

a_2 = 2(1) + 3 = 5

The recursive rule requires the previous term to compute the next term.

2

Calculate the third term

a_3

using

a_2 = 5

.

a_3 = 2(5) + 3 = 13

Apply the recursive definition

a_n = 2a_{n-1} + 3

for

n = 3

.

3

Calculate the fourth term

a_4

using

a_3 = 13

.

a_4 = 2(13) + 3 = 29

Apply the recursive definition

a_n = 2a_{n-1} + 3

for

n = 4

.

Key Concept

Recursive Sequences

Question 1846Question →

In his investigation of neural network architecture, the neuroscientist argued that advanced cognitive mechanisms do not eradicate primitive reflexes, but rather function to arrest their autonomous execution, thereby allowing higher-order executive control to intervene.

In the context of the sentence, the word "arrest" most nearly means which of the following?

imprison

inhibit

apprehend

amplify

evaluate

Show answer & explanation

Answer: inhibit

Answer

In this passage, 'arrest' is used in its secondary definition meaning to inhibit, check, or bring to a halt, referring to stopping the autonomous execution of primitive reflexes.

In the context of neural function, 'arrest' means to check, halt, or slow down progress. Using 'inhibit' accurately conveys that advanced cognitive mechanisms hold back automatic responses to allow deliberate decision-making.

Step-by-Step Solution

1

Analyze the sentence context and structural clues.

The sentence states that cognitive mechanisms do not 'eradicate' primitive reflexes, but function to affect their 'autonomous execution' so that executive control can intervene.

Understanding the contrast with 'eradicate' and the goal of enabling intervention reveals that reflexes must be paused or suppressed rather than destroyed or legally detained.

2

Evaluate the primary versus secondary meanings of 'arrest'.

'Arrest' primarily refers to legal apprehension or imprisonment, but secondarily means to check, halt, or inhibit a biological or mechanical process.

A biological process like reflex execution cannot be legally detained, so the secondary definition meaning 'to inhibit or stop' must be applied.

3

Select the option that matches the secondary meaning in context.

'Inhibit' accurately matches the meaning of stopping or checking autonomous reflex execution.

Inhibition directly describes preventing an automated response from proceeding.

Key Concept

Secondary Meaning of Polysemous Words in Context

Question 1847Question →

Let $A = 0.000032 \times 10^{n}$ and $B = 8.0 \times 10^{n-3}$ , where $n$ is an integer. If $\frac{A^2}{B} = 1.28 \times 10^{-4}$ , what is the value of $n$ ?

-2

1

3

4

9

Show answer & explanation

Answer: $3$

Answer

The value of

n

is

3

.

Rewriting

0.000032

as

3.2 \times 10^{-5}

allows

A

to be expressed as

3.2 \times 10^{n-5}

. Squaring

A

yields

10.24 \times 10^{2n-10}

. Dividing by

B = 8.0 \times 10^{n-3}

gives

1.28 \times 10^{n-7}

. Equating

n - 7 = -4

directly leads to

n = 3

.

Step-by-Step Solution

1

Convert

A

to standard scientific notation in terms of

n

A = 3.2 \times 10^{-5} \times 10^n = 3.2 \times 10^{n-5}

The decimal

0.000032

equals

3.2 \times 10^{-5}

because the decimal point is shifted

5

places to the right.

2

Calculate

A^2

A^2 = (3.2 \times 10^{n-5})^2 = (3.2)^2 \times 10^{2(n-5)} = 10.24 \times 10^{2n-10}

Apply the power of a product rule

(ab)^k = a^k b^k

and exponent power rule

(10^p)^q = 10^{pq}

.

3

Divide

A^2

by

B

\frac{A^2}{B} = \frac{10.24 \times 10^{2n-10}}{8.0 \times 10^{n-3}} = \left(\frac{10.24}{8.0}\right) \times 10^{(2n-10) - (n-3)} = 1.28 \times 10^{n-7}

Divide coefficients

(10.24 / 8.0 = 1.28)

and subtract powers of ten exponents

((2n - 10) - (n - 3) = n - 7)

.

4

Equate the simplified expression to the given value and solve for

n

1.28 \times 10^{n-7} = 1.28 \times 10^{-4} \implies n - 7 = -4 \implies n = 3

Since the coefficients match (

1.28

), equate the exponents of

10

to solve for

n

.

Key Concept

Scientific Notation Operations and Exponent Laws

Estimated Time:2m 0s

Question 1848Question →

An investor divided a total capital of $\$20,000$ between two accounts. Account A pays a simple annual interest rate of $6\%$ , and Account B pays a simple annual interest rate of $9\%$ . If the total interest earned from both accounts at the end of one year was $\$1,470$ , how much money was invested in Account A?

\$8,000

\$9,000

\$9,800

\$11,000

\$12,000

Show answer & explanation

Answer: $\$11,000$

Answer

The amount invested in Account A was

\$11,000

.

Let

x

represent the amount invested in Account A. The remaining capital,

20,000 - x

, is invested in Account B. The total interest earned in one year is given by

0.06x + 0.09(20,000 - x) = 1,470

. Expanding gives

0.06x + 1,800 - 0.09x = 1,470

, which simplifies to

-0.03x = -330

. Dividing by

-0.03

yields

x = 11,000

. Thus,

\$11,000

was invested in Account A.

Step-by-Step Solution

1

Define the variable for the unknown quantity.

Let

x

be the amount in dollars invested in Account A. Then

(20,000 - x)

is the amount invested in Account B.

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

2

Formulate the total interest equation.

0.06x + 0.09(20,000 - x) = 1,470

Total interest is the sum of interest from Account A (

6\%

of

x

) and Account B (

9\%

of

20,000 - x

).

3

Expand and simplify the algebraic equation.

0.06x + 1,800 - 0.09x = 1,470 \implies -0.03x + 1,800 = 1,470

Distribute

0.09

across

(20,000 - x)

and combine like terms.

4

Isolate the variable

x

.

-0.03x = 1,470 - 1,800 \implies -0.03x = -330 \implies x = 11,000

Subtract

1,800

from both sides and divide by

-0.03

to find

x

.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 1849Question →

Complete the sentence by filling in the blank with the single word that best fits the passage based on its secondary academic meaning.

Fill in the blanks below

In her monograph on theoretical metaphysics, the philosopher sought to the deep structural origins of human cognition, moving past superficial explanations to examine its underlying mechanisms.

Show answer & explanation

Answer

The word 'plumb' correctly completes the passage, used in its secondary verb sense meaning to measure depth or investigate thoroughly.

The correct word 'plumb' functions as a verb meaning to measure depth or to sound out and examine thoroughly. The passage explicitly contrasts 'superficial explanations' with analyzing 'deep structural origins,' making 'plumb' the most contextually accurate fit.

Step-by-Step Solution

1

Analyze the surrounding context and contrast clues in the sentence.

The phrase 'moving past superficial explanations to examine its underlying mechanisms' indicates a need for a word that signifies deep, rigorous investigation.

Understanding the contextual contrast between 'superficial' and 'underlying mechanisms' determines the required semantic tone.

2

Identify the standard vocabulary word acting in its secondary usage.

While 'plumb' is most frequently encountered as a noun or adjective relating to verticality or pipework, its secondary verb definition means to measure the depth of water or to examine something deeply and thoroughly.

Using 'plumb' as a verb fits the philosophical context of delving beneath surface-level explanations.

Key Concept

Secondary and High-Frequency Alternative Meanings

Question 1850Question →

At the beginning of 2024, a university endowment fund was reallocated across various asset classes. Over the course of 2024, the total value of the endowment fund increased by $30\%$ . In 2025, market downturns caused the total value of the fund to decrease by $20\%$ relative to its value at the end of 2024. If the total value of the endowment fund at the end of 2025 was $\$520,000$ , what was the total value of the fund at the beginning of 2024?

\$455,000

\$468,000

\$480,000

\$499,200

\$500,000

Show answer & explanation

Answer: $\$500,000$

Answer

\$500,000

The option stating

\$500,000

is correct because a

30\%

increase followed by a

20\%

decrease corresponds to multiplying the initial value

V

by

1.30 \times 0.80 = 1.04

. Solving

1.04V = 520,000

yields

V = 500,000

.

Step-by-Step Solution

1

Define the initial value variable and express successive percent changes as growth/decay multipliers.

Let

V

be the initial value at the start of 2024. A

30\%

increase corresponds to a multiplier of

1 + 0.30 = 1.30

. A

20\%

decrease corresponds to a multiplier of

1 - 0.20 = 0.80

.

Successive percentage changes compound multiplicatively on the original base value.

2

Formulate the net equation for the final value.

V \times 1.30 \times 0.80 = 520,000 \implies V \times 1.04 = 520,000

Multiplying

1.30

by

0.80

yields a net factor of

1.04

, representing a net

4\%

increase over the two-year period.

3

Solve for the initial value

V

.

V = \frac{520,000}{1.04} = 500,000

Dividing the final amount by the combined multiplier isolates the original starting value.

Key Concept

Successive Percent Changes and Base Value Calculation

Estimated Time:1m 30s

Question 1851Question →

The computational linguist observed that incorporating syntactic tree constraints into the neural model did not merely improve translation accuracy; furthermore, by explicitly modeling grammatical structure, the architecture ________ a significant reduction in semantic errors.

Which two of the following options, when inserted into the blank, fit the meaning of the sentence as a whole and yield completed sentences that are logically equivalent?

Select all that apply

yielded

forestalled

generated

precluded

portended

presaged

Show answer & explanation

Answer: yielded; generated

Answer

The two correct choices are 'yielded' and 'generated', as both words indicate that the architectural enhancement directly produced or brought about the reduction in semantic errors.

The sentence uses the additive/continuation signal 'furthermore' following 'did not merely improve', indicating that the second clause reinforces and builds upon the positive outcome of the first. The words 'yielded' and 'generated' both mean produced or brought about, completing the causal chain logically and producing equivalent sentence meanings.

Step-by-Step Solution

1

Analyze sentence structure and structural transition signals.

The semi-colon followed by 'furthermore' and the clause 'did not merely... by explicitly modeling...' signals a positive continuation and causal reinforcement of the primary benefit.

Understanding structural signals determines whether the missing word must carry a positive, negative, or neutral causal meaning.

2

Determine the required contextual meaning and valence of the blank.

The blank requires a verb meaning 'produced', 'brought about', or 'resulted in' a positive reduction in errors.

The model's architectural improvement leads directly to the positive outcome of error reduction.

3

Evaluate option pairs for semantic equivalence and contextual fit.

'Yielded' and 'generated' both accurately express that the architecture produced the reduction. In contrast, 'forestalled'/'precluded' reverse the directional logic, while 'portended'/'presaged' incorrectly introduce speculative forecasting rather than direct causation.

GRE Sentence Equivalence requires choosing two words that create completed sentences with identical meaning.

Key Concept

Interpreting Continuation and Causal Structural Signals

Question 1852Question →

Match each formal GRE vocabulary word 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

'Profligate' pairs with 'Extravagant', 'Equanimous' pairs with 'Composed', 'Sycophantic' pairs with 'Obsequious', and 'Recondite' pairs with 'Esoteric'.

Each vocabulary term matches its exact semantic counterpart: 'profligate' denotes reckless wastefulness ('extravagant'); 'equanimous' denotes emotional stability under pressure ('composed'); 'sycophantic' denotes servile flattery ('obsequious'); and 'recondite' denotes specialized or obscure knowledge ('esoteric').

Step-by-Step Solution

1

Analyze the core definition of 'Profligate'

'Profligate' means recklessly extravagant or wasteful in the use of resources.

Find its semantic counterpart on the right, which is 'Extravagant'.

2

Analyze the core definition of 'Equanimous'

'Equanimous' means calm and composed, especially under stress.

Find its semantic counterpart on the right, which is 'Composed'.

3

Analyze the core definition of 'Sycophantic'

'Sycophantic' means behaving in an obsequious or fawning manner toward influential individuals.

Find its semantic counterpart on the right, which is 'Obsequious'.

4

Analyze the core definition of 'Recondite'

'Recondite' means dealing with profound, obscure, or little-known subject matter.

Find its semantic counterpart on the right, which is 'Esoteric'.

Key Concept

Identifying Synonym and Near-Synonym Pairs

Question 1853Question →

Complete the passage by filling in each blank with the word that best maintains the logical structure and contextual meaning of the text.

Fill in the blanks below

While early architectural historians dismissed the fortress's irregular outer walls as a structural design resulting from hastily improvised repairs, modern spatial analysis reveals that these asymmetric ramparts were systematically to the surrounding terrain to maximize defensive sightlines.

Show answer & explanation

Answer

Blank 1: haphazard; Blank 2: calibrated

The concession word 'While' establishes a opposition between early assumptions and modern findings. The descriptive clue 'hastily improvised repairs' dictates that Blank 1 must convey a lack of deliberate planning, which is precisely captured by 'haphazard'. Contrastingly, modern analysis demonstrates that the walls were not random, but 'systematically' fitted to the environment; hence 'calibrated' correctly completes Blank 2 to align with maximizing defensive sightlines.

Step-by-Step Solution

1

Identify structural transition signals and contrast clues.

The opening concession signal 'While' indicates a contrast between early historians' views and modern spatial analysis findings.

The sentence structure pits past misinterpretations against modern empirical insights.

2

Determine the contextual requirement for Blank 1 based on local clues.

The phrase 'hastily improvised repairs' suggests that early historians viewed the wall design as unplanned, random, or arbitrary; thus 'haphazard' fits Blank 1.

The word in Blank 1 must reflect the lack of order implied by hastily improvised work.

3

Determine the dependency between Blank 1 and Blank 2.

Modern analysis contradicts the initial belief by showing deliberate design. The adverb 'systematically' requires Blank 2 to express careful adjustment to the landscape, making 'calibrated' (or 'adapted') correct.

If the design was systematically executed to maximize sightlines, the ramparts were precisely fitted or calibrated to the topography.

Key Concept

Multi-Blank Dependency Tracking

Estimated Time:1m 15s

Question 1854Question →

A coffee merchant creates a $50$ -pound custom mixture combining Grade A coffee beans, which cost $\$8$ per pound, and Grade B coffee beans, which cost $\$14$ per pound. If the total cost of the mixture must be at least $\$460$ and at most $\$520$ , which of the following could be the weight, in pounds, of Grade A coffee beans used in the mixture? Select all such weights.

Select all that apply

25

32

35

40

45

Show answer & explanation

Answer: $32$ ; $35$ ; $40$

Answer

The weight of Grade A coffee beans can be 32 pounds, 35 pounds, or 40 pounds.

Let

x

represent the number of pounds of Grade A coffee beans. The remaining

50 - x

pounds consist of Grade B beans. The total cost of the mixture is given by

8x + 14(50 - x) = 700 - 6x

. Setting up the given inequality constraints, we have

460 \le 700 - 6x \le 520

. Subtracting 700 yields

-240 \le -6x \le -180

. Dividing by

-6

and reversing the inequality signs gives

30 \le x \le 40

. Therefore, any weight of Grade A beans between 30 and 40 pounds inclusive is valid. The values 32, 35, and 40 satisfy this condition.

Step-by-Step Solution

1

Define variables for the quantities of Grade A and Grade B beans.

Let

x

be the weight in pounds of Grade A beans. Then

50 - x

is the weight in pounds of Grade B beans.

The total weight of the mixture is fixed at 50 pounds.

2

Set up an expression for the total cost of the mixture in terms of

x

.

Total Cost =

8x + 14(50 - x) = 700 - 6x

.

Multiply the weight of each component by its price per pound.

3

Set up the compound inequality representing the given cost constraints.

460 \le 700 - 6x \le 520

.

The total cost must be at least

\$460

and at most

\$520

.

4

Solve the compound inequality for

x

.

Subtracting 700 from all parts gives

-240 \le -6x \le -180

. Dividing by

-6

and reversing the inequality signs yields

30 \le x \le 40

.

Dividing by a negative number reverses the direction of the inequality signs.

5

Identify which options fall within the valid range

[30, 40]

.

The values 32, 35, and 40 fall within the range

30 \le x \le 40

.

Any weight between 30 and 40 pounds inclusive satisfies the cost constraint.

Key Concept

Linear Modeling and Inequality Constraints in Mixture Problems

Estimated Time:1m 40s

Question 1855Question →

An investment fund allocated its total initial capital equally into two separate accounts, Account X and Account Y. Account X earns simple annual interest at a rate of $r\%$ , while Account Y earns compound annual interest at the same rate of $r\%$ , compounded annually. At the end of 2 years, the total interest earned by Account Y exceeded the total interest earned by Account X by an amount equal to $4\%$ of the initial amount placed in Account X. What was the net percentage increase in the fund's total combined capital over the 2-year period?

40%

42%

44%

60%

84%

Show answer & explanation

Answer: 42%

Answer

42%

The correct answer of 42% is found by recognizing that the 2-year difference between compound interest and simple interest at rate

r\%

equals

P(r/100)^2

. Setting this difference to

0.04P

yields

r = 20\%

. At

r = 20\%

, Account X produces

0.40P

in simple interest, while Account Y produces

0.44P

in compound interest. Together, they generate

0.84P

in interest on a total initial investment of

2P

, resulting in a net fund increase of

(0.84P / 2P) \times 100\% = 42\%

.

Step-by-Step Solution

1

Set up expressions for interest earned by each account after 2 years

Let

P

be the initial capital in Account X (so Account Y also starts with

P

, making total initial capital

2P

). Simple interest for Account X is

I_X = P \cdot \frac{2r}{100} = 0.02r P

. Compound interest for Account Y is

I_Y = P\left[\left(1 + \frac{r}{100}\right)^2 - 1\right] = P\left(\frac{2r}{100} + \frac{r^2}{10000}\right)

.

Establishes the interest expressions based on simple vs. compound interest formulas over a 2-year period.

2

Calculate the annual interest rate

r

Difference in interest:

I_Y - I_X = P\left(\frac{r^2}{10000}\right)

. Given this equals

4\%

of

P

(

0.04P

), we have

\frac{r^2}{10000} = 0.04 \implies r^2 = 400 \implies r = 20\%

.

Isolates the interest rate

r

using the difference between compound and simple interest.

3

Determine total interest earned across both accounts

For Account X:

I_X = 0.02(20)P = 0.40P

. For Account Y:

I_Y = 0.40P + 0.04P = 0.44P

. Total interest earned =

0.40P + 0.44P = 0.84P

.

Sums the returns of both portfolio components.

4

Compute the net percentage increase of the total fund

Total initial capital =

2P

. Net percentage increase =

\frac{\text{Total Interest}}{\text{Total Initial Capital}} \times 100\% = \frac{0.84P}{2P} \times 100\% = 42\%

.

Calculates the overall percent change using the correct part-to-whole base (total initial investment

2P

).

Key Concept

Simple vs. Compound Interest Base Shift and Net Percent Change

Question 1856Question →

Let $x$ , $y$ , and $z$ be non-zero integers that satisfy the following three conditions:

I. $(-1)^{x y + z} = -1$
II. $(-1)^{x z + y} = 1$
III. $x^3 y z^2 < 0$

Which of the following statements MUST be true?

x

and

y

are both even integers, and

x y < 0

x

is an odd integer, and

z

is an even integer

y

and

z

are both odd integers

x + z

is an even integer

x y z > 0

Show answer & explanation

Answer: $x$ and $y$ are both even integers, and $x y < 0$

Answer

x

and

y

are both even integers, and

x y < 0

From Condition I,

(-1)^{xy+z} = -1

implies

xy + z

is odd, so

xy

and

z

have opposite parity. From Condition II,

(-1)^{xz+y} = 1

implies

xz + y

is even, so

xz

and

y

have the same parity. If

x

were odd,

xy

would share the parity of

y

, and

xz

would share the parity of

z

. This would mean

y

and

z

must have opposite parity (from Condition I) and the same parity (from Condition II), which is impossible. Thus,

x

must be even. With

x

even,

xy

is even, so

z

must be odd for

xy + z

to be odd. Likewise,

xz

is even, so

y

must be even for

xz + y

to be even. Finally, from Condition III, since

z^2 > 0

for non-zero

z

,

x^3 y z^2 < 0

reduces to

x^3 y < 0

, which means

xy < 0

. Thus,

x

and

y

are both even integers, and

xy < 0

.

Step-by-Step Solution

1

Analyze Condition I for parity requirements

x y + z

must be an odd integer

Since

(-1)^k = -1

if and only if

k

is an odd integer,

x y + z

is odd. Thus,

x y

and

z

must have opposite parity.

2

Analyze Condition II for parity requirements

x z + y

must be an even integer

Since

(-1)^k = 1

if and only if

k

is an even integer,

x z + y

is even. Thus,

x z

and

y

must have the same parity.

3

Determine the parity of

x

,

y

, and

z

using proof by contradiction

x

is even,

y

is even, and

z

is odd

If

x

were odd, then

x y

would have the same parity as

y

, and

x z

would have the same parity as

z

. Condition I would require

y

and

z

to have opposite parity, while Condition II would require

y

and

z

to have the same parity, a contradiction. Therefore,

x

must be even. Since

x

is even,

x y

is even, making

z

odd (from Condition I). Similarly,

x z

is even, making

y

even (from Condition II).

4

Analyze Condition III for sign requirements

x y < 0

Since

z

is a non-zero integer,

z^2 > 0

. The inequality

x^3 y z^2 < 0

simplifies to

x^3 y < 0

. Since

x^3

has the same sign as

x

,

x^3 y < 0

implies

x y < 0

.

Key Concept

Parity rules for exponentiation and basic arithmetic operations, combined with sign rules for products

Question 1857Question →

A positive integer $n$ has the prime factorization $2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. Given that $\gcd(n, 7{}200) = 360$ and $\text{lcm}(n, 1{}080) = 3{}240$ , what is the value of $a + b + c$ ?

6

7

8

9

10

Show answer & explanation

Answer: 8

Answer

8

By prime factorizing each given term,

7{}200 = 2^5 \cdot 3^2 \cdot 5^2

,

360 = 2^3 \cdot 3^2 \cdot 5^1

,

1{}080 = 2^3 \cdot 3^3 \cdot 5^1

, and

3{}240 = 2^3 \cdot 3^4 \cdot 5^1

. Applying the definition of GCD as taking the minimum exponent gives

\min(a, 5) = 3 \implies a = 3

and

\min(c, 2) = 1 \implies c = 1

. Applying the definition of LCM as taking the maximum exponent gives

\max(b, 3) = 4 \implies b = 4

. Thus,

a + b + c = 3 + 4 + 1 = 8

.

Step-by-Step Solution

1

Find the prime factorizations of the given integers and the GCD/LCM values.

7{}200 = 2^5 \cdot 3^2 \cdot 5^2

,

360 = 2^3 \cdot 3^2 \cdot 5^1

,

1{}080 = 2^3 \cdot 3^3 \cdot 5^1

, and

3{}240 = 2^3 \cdot 3^4 \cdot 5^1

.

Decomposing into prime factors allows direct comparison of exponents using GCD (minimum exponent) and LCM (maximum exponent) rules.

2

Analyze the GCD condition

\gcd(n, 7{}200) = 360

.

\min(a, 5) = 3 \implies a = 3

,

\min(b, 2) = 2 \implies b \ge 2

, and

\min(c, 2) = 1 \implies c = 1

.

The exponent of each prime factor in

\gcd(x, y)

is the minimum of their respective exponents in

x

and

y

.

3

Analyze the LCM condition

\text{lcm}(n, 1{}080) = 3{}240

.

\max(a, 3) = 3 \implies a \le 3

,

\max(b, 3) = 4 \implies b = 4

, and

\max(c, 1) = 1 \implies c \le 1

.

The exponent of each prime factor in

\text{lcm}(x, y)

is the maximum of their respective exponents in

x

and

y

.

4

Combine the exponent constraints to determine

a

,

b

, and

c

, then calculate their sum.

a = 3

,

b = 4

, and

c = 1

, so

a + b + c = 3 + 4 + 1 = 8

.

Combining

a=3

,

b \ge 2

with

b=4

, and

c=1

with

c \le 1

uniquely specifies

(a,b,c) = (3,4,1)

.

Key Concept

Prime Exponent Analysis of Greatest Common Divisor and Least Common Multiple

Estimated Time:2m 0s

Question 1858Question →

A technology firm monitors its active user base and average monthly server cost per user across four consecutive quarters:

- In Quarter 1, the total server cost was $C$ .
- In Quarter 2, the active user base increased by $25\%$ relative to Quarter 1, while the average monthly server cost per user decreased by $20\%$ relative to Quarter 1.
- In Quarter 3, the active user base decreased by $x\%$ relative to Quarter 2, while the average monthly server cost per user increased by $x\%$ relative to Quarter 2, where $x > 0$ .
- In Quarter 4, the active user base increased by $20\%$ relative to Quarter 3, while the average monthly server cost per user decreased by $10\%$ relative to Quarter 3.

If the total server cost in Quarter 4 was $3.68\%$ greater than the total server cost in Quarter 1, what is the value of $x$ ?

4

15

20

25

40

Show answer & explanation

Answer: $20$

Answer

The value of

x

is

20

.

The correct response is 20. Total cost in Quarter 1 is

C_1 = N_1 P_1 = C

. In Quarter 2,

C_2 = (1.25 N_1)(0.80 P_1) = 1.00 C

. In Quarter 3, the user base and cost per user change by

-x\%

and

+x\%

, giving

C_3 = C \left(1 - \frac{x}{100}\right)\left(1 + \frac{x}{100}\right) = C \left(1 - \frac{x^2}{10,000}\right)

. In Quarter 4, the overall factor becomes

1.20 \times 0.90 = 1.08

, so

C_4 = 1.08 C \left(1 - \frac{x^2}{10,000}\right)

. Setting

1.08 \left(1 - \frac{x^2}{10,000}\right) = 1.0368

yields

1 - \frac{x^2}{10,000} = 0.96

, from which

\frac{x^2}{10,000} = 0.04

, giving

x^2 = 400

and

x = 20

.

Step-by-Step Solution

1

Define total cost variables for Quarter 1 and Quarter 2.

Quarter 1 cost

C_1 = N_1 P_1 = C

. Quarter 2 user base

N_2 = 1.25 N_1

and cost per user

P_2 = 0.80 P_1

, so

C_2 = (1.25 N_1)(0.80 P_1) = 1.00 N_1 P_1 = C

.

Total cost is the product of user count and cost per user; successive changes in Quarter 2 offset each other completely.

2

Calculate total cost in Quarter 3 in terms of

x

.

Quarter 3 user base

N_3 = N_2\left(1 - \frac{x}{100}\right)

and cost per user

P_3 = P_2\left(1 + \frac{x}{100}\right)

. Total cost

C_3 = N_3 P_3 = C_2 \left(1 - \frac{x}{100}\right)\left(1 + \frac{x}{100}\right) = C \left(1 - \frac{x^2}{10,000}\right)

.

Applying the difference of squares identity

(1 - u)(1 + u) = 1 - u^2

for compound percentage change.

3

Calculate total cost in Quarter 4 and set up the equation with Quarter 1 cost.

Quarter 4 user base

N_4 = 1.20 N_3

and cost per user

P_4 = 0.90 P_3

, giving

C_4 = (1.20)(0.90) C_3 = 1.08 C_3 = 1.08 C \left(1 - \frac{x^2}{10,000}\right)

. Since

C_4 = 1.0368 C

, we get

1.08 \left(1 - \frac{x^2}{10,000}\right) = 1.0368

.

Quarter 4 expenditure compounds the Quarter 3 cost by a factor of

1.20 \times 0.90 = 1.08

.

4

Solve for

x

.

1 - \frac{x^2}{10,000} = \frac{1.0368}{1.08} = 0.96 \implies \frac{x^2}{10,000} = 0.04 \implies x^2 = 400 \implies x = 20

.

Dividing

1.0368

by

1.08

yields

0.96

, giving

x^2 = 400

and

x = 20

since

x > 0

.

Key Concept

Compounding successive percent changes across multiple factors and using algebraic identities for net percent change.

Estimated Time:2m 30s

Question 1859Question →

Far from being a purely ______ shift in demographic distribution, the rapid growth of suburban micro-cities reflects profound underlying changes in workforce mobility and regional economics.

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

Select all that apply

superficial

aesthetic

cosmetic

artistic

ephemeral

spatial

Show answer & explanation

Answer: superficial; cosmetic

Answer

The words 'superficial' and 'cosmetic' complete the sentence to produce equivalent meanings.

The sentence relies on the contrast signal 'Far from being a purely ______ shift...' to oppose 'profound underlying changes'. The correct choices must both mean surface-level or minor. The words 'superficial' and 'cosmetic' both carry this meaning and yield completed sentences with equivalent sense.

Step-by-Step Solution

1

Analyze sentence syntax and structural transition clues

The opening concession 'Far from being a purely ______ shift' establishes a direct contrast with 'profound underlying changes' in the second clause.

The contrast pivot signals that the target word must mean shallow, surface-level, or minor.

2

Evaluate candidate vocabulary for contextual fit and synonymy

'Superficial' and 'cosmetic' both denote surface-level or non-essential characteristics, perfectly contrasting with 'profound'.

Both words complete the sentence with logically coherent and identical meanings.

3

Eliminate invalid distractors and false synonym pairs

'Aesthetic' and 'artistic' form a synonym pair but fail contextually. 'Ephemeral' fits tone but lacks a partner. 'Spatial' is topically relevant but lacks contrast and a pair.

GRE Sentence Equivalence requires both contextual fit and absolute semantic equivalence between the two chosen words.

Key Concept

Identifying Synonym and Near-Synonym Pairs in Context

Estimated Time:1m 0s

Question 1860Question →

A car travels a total distance of $120$ miles. For the first $60$ miles, the car travels at an average speed of $30$ miles per hour, and for the remaining $60$ miles, the car travels at an average speed of $60$ miles per hour. What is the average speed of the car, in miles per hour, for the entire $120$ -mile trip?

Show answer & explanation

Answer: 40

Answer

The average speed of the car for the entire trip is

40

miles per hour.

The overall average speed is determined by dividing total distance (

120

miles) by total time (

3

hours), yielding

40

miles per hour.

Step-by-Step Solution

1

Calculate time taken for the first segment

Time =

2

hours

Using

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

,

\frac{60}{30} = 2

hours.

2

Calculate time taken for the second segment

Time =

1

hour

Using

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

,

\frac{60}{60} = 1

hour.

3

Determine total distance and total time

Total distance =

120

miles, Total time =

3

hours

Add distances (

60 + 60 = 120

) and times (

2 + 1 = 3

).

4

Calculate overall average speed

Average speed =

40

miles per hour

Divide total distance by total time:

\frac{120}{3} = 40

.

Key Concept

Average Speed and Rates

Estimated Time:1m 0s

Answer

Step-by-Step Solution

Key Concept

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Common Mistakes

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