All practice questions

Question 1801Question →

In the $xy$ -plane, line $k$ is defined by the equation $2x - y = 8$ . Line $m$ is perpendicular to line $k$ and passes through the points $(-1, 9)$ and $(t, 5)$ . What is the value of $t$ ?

-9

-3

1

7

9

Show answer & explanation

Answer: $7$

Answer

The value of

t

is

7

.

The line

k

has equation

y = 2x - 8

, giving a slope of

2

. A line perpendicular to it must have a slope of

-\frac{1}{2}

. Using the slope formula for line

m

passing through

(-1, 9)

and

(t, 5)

yields

\frac{5 - 9}{t - (-1)} = -\frac{1}{2}

, which simplifies to

\frac{-4}{t + 1} = -\frac{1}{2}

, leading directly to

t = 7

.

Step-by-Step Solution

1

Find the slope of line

k

.

Converting

2x - y = 8

to slope-intercept form gives

y = 2x - 8

, so the slope of line

k

is

m_k = 2

.

The slope-intercept form

y = mx + b

directly identifies the slope

m

of a line.

2

Determine the slope of line

m

.

Since line

m

is perpendicular to line

k

, its slope is

m_m = -\frac{1}{m_k} = -\frac{1}{2}

.

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

3

Set up the slope formula for line

m

using the given points and solve for

t

.

\frac{5 - 9}{t - (-1)} = -\frac{1}{2} \implies \frac{-4}{t + 1} = -\frac{1}{2} \implies \frac{4}{t + 1} = \frac{1}{2} \implies t + 1 = 8 \implies t = 7

.

The slope between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

Key Concept

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

Estimated Time:1m 30s

Question 1802Question →

Fill in the blank with the single word that best completes the passage based on its secondary academic meaning.

Fill in the blanks below

Recognizing that the initial proposal had drawn widespread criticism from committee members, the envoy chose to a thoroughly revised draft, hoping that formally submitting the new terms would restore confidence in the negotiation process.

Show answer & explanation

Answer

The word 'tender' best completes the blank, functioning as a verb that means to offer or submit formally.

In formal academic and legal contexts, 'tender' functions as a verb meaning to present, offer, or submit officially (such as tendering a document or tendering one's resignation). The sentence specifically refers to 'formally submitting the new terms', making 'tender' the precise word required.

Step-by-Step Solution

1

Analyze the context clues in the sentence.

The phrase 'formally submitting the new terms' directly elaborates on the action performed by the envoy regarding the revised draft.

Contextual clues in GRE fill-in-the-blank questions often provide an explicit restatement or definition of the target word.

2

Identify the required part of speech and tone.

The blank requires an infinitive verb following 'chose to' in a formal diplomatic context.

Grammatical fit narrows down candidate words to verbs signifying formal presentation.

3

Select the high-frequency secondary meaning word.

While 'tender' is most commonly known as an adjective meaning gentle or soft, its secondary usage as a verb means to offer, proffer, or submit formally (e.g., to tender a resignation or tender a proposal).

This matches the target subtopic requirement of testing secondary definitions of everyday words.

Key Concept

Secondary and High-Frequency Alternative Meanings

Estimated Time:1m 0s

Question 1803Question →

In the geometric configuration formed by adjacent triangles $ABC$ and $ACD$ sharing segment $AC$ , $\angle ABC = 90^\circ$ and $\angle ACD = 90^\circ$ . The lengths of the sides of triangle $ABC$ are $AB = 9$ and $BC = 12$ . If $\angle ADC = 30^\circ$ , what is the length of segment $AD$ ?

Show answer & explanation

Answer: 30

Answer

30

Applying the Pythagorean theorem to right triangle ABC gives AC = sqrt(9^2 + 12^2) = 15. In right triangle ACD with angle ADC = 30 degrees, side AC is opposite the 30-degree angle. By the 30-60-90 triangle side ratio (1 : sqrt(3) : 2), the hypotenuse AD is equal to twice the short leg AC, giving AD = 2 * 15 = 30.

Step-by-Step Solution

1

Calculate hypotenuse AC of right triangle ABC using the Pythagorean theorem.

AC = 15

In right triangle ABC with legs 9 and 12, AC^2 = 9^2 + 12^2 = 225, so AC = 15.

2

Identify the side relationships in special right triangle ACD.

AC is the shorter leg opposite the 30-degree angle ADC, and AD is the hypotenuse.

Since angle ACD is 90 degrees and angle ADC is 30 degrees, triangle ACD is a 30-60-90 right triangle.

3

Compute the hypotenuse AD from short leg AC.

AD = 30

In any 30-60-90 right triangle, the hypotenuse is twice the length of the leg opposite the 30-degree angle.

Key Concept

Pythagorean Theorem and Special Right Triangles

Estimated Time:1m 30s

Question 1804Question →

An investor allocated a total principal of $\$10,000$ between two accounts, Account X and Account Y. Account X pays simple annual interest at a rate of $6\%$ , while Account Y pays simple annual interest at a rate of $8\%$ . Let $x$ represent the amount of money, in dollars, invested in Account X, and let $d$ represent the total annual interest earned, in dollars, from both accounts after one year. Which of the following statements must be true? Select all that apply.

Select all that apply

The total annual interest

d

satisfies

600 \le d \le 800

.

The amount invested in Account X can be represented as

x = \frac{800 - d}{0.02}

.

If

d = 750

, the amount invested in Account X is greater than the amount invested in Account Y.

If equal amounts were invested in both accounts, the total annual interest earned is

\$700

.

If

d = 680

, the ratio of the amount invested in Account X to the amount invested in Account Y is

2

to

3

.

Show answer & explanation

Answer: The total annual interest $d$ satisfies $600 \le d \le 800$ .; The amount invested in Account X can be represented as $x = \frac{800 - d}{0.02}$ .; If equal amounts were invested in both accounts, the total annual interest earned is $\$700$ .

Answer

The true statements are that the total annual interest

d

satisfies

600 \le d \le 800

, the amount in Account X is given by

x = \frac{800 - d}{0.02}

, and equal investment in both accounts yields

\$700

in total interest.

The total interest equation

d = 800 - 0.02x

dictates all valid relationships. Because

0 \le x \le 10,000

, the bounds for

d

are strictly between

600

and

800

. Rearranging the equation yields

x = \frac{800 - d}{0.02}

, which correctly models the Account X investment. Substituting

x = 5,000

yields

d = 700

, confirming the equal-allocation scenario.

Step-by-Step Solution

1

Formulate the total interest model as a linear equation in terms of

x

.

d = 0.06x + 0.08(10,000 - x) = 800 - 0.02x

The interest earned from Account X is

0.06x

and the interest from Account Y is

0.08(10,000 - x)

.

2

Determine the range of possible interest values for

0 \le x \le 10,000

.

When

x = 10,000

,

d = 600

. When

x = 0

,

d = 800

. Thus

600 \le d \le 800

.

The minimum and maximum interest values occur at the extreme allocation bounds.

3

Rearrange the interest equation to express

x

in terms of

d

.

0.02x = 800 - d \implies x = \frac{800 - d}{0.02}

Isolating

x

provides a formula for calculating the Account X principal directly from the total interest.

4

Evaluate the specific numerical cases given in the options.

Equal investment (

x = 5,000

) gives

d = 800 - 100 = 700

. For

d = 750

,

x = 2,500

(Account Y =

7,500

). For

d = 680

,

x = 6,000

(Account Y =

4,000

, ratio

3:2

).

Substituting specific values tests the validity of each conditional statement.

Key Concept

Linear Modeling and Algebraic Rate Allocation

Question 1805Question →

A librarian is arranging $7$ distinct books on a display shelf: $4$ history books and $3$ science books. If all $3$ science books must be placed adjacent to one another, in how many different linear orders can the $7$ books be arranged?

720

144

5,040

120

35

Show answer & explanation

Answer: 720

Answer

720 linear arrangements

To arrange items with an adjacency restriction, bundle the restricted items (the 3 science books) into 1 block. This leaves 4 history books and 1 science block, totaling 5 units to arrange linearly in 5! = 120 ways. Within the science block, the 3 distinct books can be arranged in 3! = 6 ways. By the Fundamental Counting Principle, the total number of linear arrangements is 5! × 3! = 120 × 6 = 720.

Step-by-Step Solution

1

Group the constrained items into a single block.

Treat the

3

distinct science books as a single unit or block, denoted as

[S]

.

Since the science books must always stand next to each other, they move as a single combined item.

2

Count the total number of items to arrange linearly.

We have

4

individual history books plus

1

science block, giving

4 + 1 = 5

items.

The

5

units can be ordered among themselves in

5!

ways.

3

Calculate the outer arrangements of the 5 units.

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

.

This accounts for all position permutations of the history books and the block.

4

Calculate internal permutations within the science block and multiply.

3! = 6

ways within the block. Total arrangements =

120 \times 6 = 720

.

By the Fundamental Counting Principle, total ways equals outer permutations multiplied by inner permutations.

Key Concept

Permutations with Adjacency Restrictions (Block Method)

Question 1806Question →

A quality assurance auditor reviewed the processing times, in seconds, for a batch of 7 completed tasks: $22, 10, 24, 18, 30, 22,$ and $14$ . An 8th task with a processing time of $x$ seconds, where $x$ is a positive integer, is added to the batch. Which of the following statements must be true? Select all such statements.

Select all that apply

If

x = 22

, the median of the 8 processing times is equal to 22.

If the arithmetic mean of the 8 processing times is equal to 21, then

x = 28

.

If

x = 18

, the median of the 8 processing times is equal to 21.

If

x = 30

, the arithmetic mean of the dataset increases by 5.

If

x = 10

, the mode of the 8 processing times becomes 10.

Show answer & explanation

Answer: If $x = 22$ , the median of the 8 processing times is equal to 22.; If the arithmetic mean of the 8 processing times is equal to 21, then $x = 28$ .

Answer

The correct statements are those asserting that if

x=22

, the median of the 8 processing times is 22, and if the mean of the 8 processing times is 21, then

x=28

.

The statement regarding

x=22

is correct because inserting 22 places 22 at both the 4th and 5th positions of the 8-element ordered list, yielding a median of 22. The statement regarding a mean of 21 is correct because the required total sum for 8 items is

8 \times 21 = 168

, which requires

x = 168 - 140 = 28

.

Step-by-Step Solution

1

Order the original dataset and calculate initial metrics.

Sorted original list:

10, 14, 18, 22, 22, 24, 30

. Original sum = 140, original mean =

\frac{140}{7} = 20

, original median = 22 (4th term).

Establishing the initial baseline values is necessary to evaluate statements about changes in mean and median.

2

Evaluate the statement regarding median when

x=22

.

Inserting

x=22

yields sorted list:

10, 14, 18, 22, 22, 22, 24, 30

. The 4th and 5th items are both 22, giving median

\frac{22+22}{2} = 22

.

Confirms the first statement is true.

3

Evaluate the statement regarding mean equal to 21.

Target sum for 8 items with mean 21 is

8 \times 21 = 168

. Setting

140 + x = 168

yields

x = 28

.

Confirms the second statement is true.

4

Evaluate remaining incorrect statements.

For

x=18

, ordered list is

10, 14, 18, 18, 22, 22, 24, 30

, giving median

\frac{18+22}{2} = 20 \neq 21

. For

x=30

, new mean is

21.25

, an increase of

1.25 \neq 5

. For

x=10

, 10 and 22 both appear twice, making it bimodal rather than having a single mode of 10.

Verifies that all other statements are false.

Key Concept

Measures of Central Tendency (Mean, Median, Mode) for modified datasets

Question 1807Question →

If $3x + y = 14$ and $x - 2y = -7$ , what is the value of $x + y$ ?

3

5

8

11

2

Show answer & explanation

Answer: 8

Answer

8

Solving the linear system yields

x = 3

and

y = 5

. Adding these values together gives

3 + 5 = 8

, which is the correct answer.

Step-by-Step Solution

1

Express

y

in terms of

x

using the first equation

y = 14 - 3x

Isolating one variable allows for direct substitution into the second equation.

2

Substitute

y = 14 - 3x

into the second equation

x - 2y = -7

x - 2(14 - 3x) = -7 \implies x - 28 + 6x = -7 \implies 7x = 21 \implies x = 3

Solving the single-variable equation gives the exact value for

x

.

3

Calculate

y

using

x = 3

y = 14 - 3(3) = 5

Substituting

x = 3

back into

y = 14 - 3x

gives the value for

y

.

4

Compute the sum

x + y

x + y = 3 + 5 = 8

The question specifically asks for the sum of both variables.

Key Concept

Solving a 2x2 System of Linear Equations by Substitution or Elimination

Estimated Time:45s

Question 1808Question →

In the $xy$ -plane, line $L_1$ is defined by the equation $y = 3x - 4$ . Line $L_2$ is perpendicular to line $L_1$ and passes through the point $(6, 2)$ . What is the $x$ -intercept of line $L_2$ ?

Show answer & explanation

Answer: 12

Answer

12

Line

L_1

has a slope of

3

. A line perpendicular to

L_1

must have a slope equal to the negative reciprocal of

3

, which is

-\frac{1}{3}

. Substituting the point

(6, 2)

into the point-slope form gives

y - 2 = -\frac{1}{3}(x - 6)

, which simplifies to

y = -\frac{1}{3}x + 4

. Setting

y = 0

yields

0 = -\frac{1}{3}x + 4

, giving

x = 12

as the

x

-intercept.

Step-by-Step Solution

1

Find the slope of line

L_1

Slope

m_1 = 3

The equation

y = 3x - 4

is in slope-intercept form

y = mx + b

, where

m = 3

.

2

Determine the slope of perpendicular line

L_2

Slope

m_2 = -\frac{1}{3}

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

m_1 \cdot m_2 = -1

).

3

Find the equation of line

L_2

y = -\frac{1}{3}x + 4

Substitute point

(6, 2)

into point-slope formula

y - 2 = -\frac{1}{3}(x - 6)

.

4

Calculate the

x

-intercept of line

L_2

x = 12

Set

y = 0

in the line equation and solve for

x

:

0 = -\frac{1}{3}x + 4 \implies x = 12

.

Key Concept

Perpendicular lines have negative reciprocal slopes (

m_1 \cdot m_2 = -1

). The

x

-intercept is the point where

y = 0

.

Question 1809Question →

On the real number line, $p$ and $q$ are real numbers such that $|3 - 2p| \le 5$ and $|q + 4| = 2$ . What is the maximum possible value of $|p^2 - q|$ ?

22

18

14

10

7

Show answer & explanation

Answer: 22

Answer

22

Solving

|3 - 2p| \le 5

gives

-5 \le 3 - 2p \le 5 \implies -1 \le p \le 4

, so

p^2

can range from

0

up to

16

. Solving

|q + 4| = 2

yields

q = -2

or

q = -6

. To maximize

|p^2 - q|

, we combine the maximum possible value of

p^2

(

16

) with

q = -6

, obtaining

|16 - (-6)| = 22

.

Step-by-Step Solution

1

Solve the absolute value inequality

|3 - 2p| \le 5

for

p

.

-1 \le p \le 4

-5 \le 3 - 2p \le 5 \implies -8 \le -2p \le 2

. Dividing by

-2

and reversing the inequality direction gives

-1 \le p \le 4

.

2

Determine the range of possible values for

p^2

.

0 \le p^2 \le 16

Since

p

spans from

-1

to

4

(which includes

0

), the minimum square is

0^2 = 0

and the maximum square is

4^2 = 16

.

3

Solve the absolute value equation

|q + 4| = 2

for

q

.

q = -2

or

q = -6

q + 4 = 2 \implies q = -2

, and

q + 4 = -2 \implies q = -6

.

4

Find the combination of

p^2

and

q

that maximizes

|p^2 - q|

.

Maximum value is 22

Pairing

p^2 = 16

with

q = -6

yields

|16 - (-6)| = 22

, which is greater than

|16 - (-2)| = 18

.

Key Concept

Absolute value inequalities, real number bounds, and distance optimization.

Question 1810Question →

In convex quadrilateral $ABCD$ , $\angle ABC = 90^\circ$ and $\angle ADC = 90^\circ$ . If $AB = BC$ , $AD = 6$ , and $CD = 8$ , what is the area of quadrilateral $ABCD$ ?

48

49

55

73

74

Show answer & explanation

Answer: 49

Answer

49

The correct answer is 49. Dividing quadrilateral

ABCD

along diagonal

AC

creates two right triangles:

\triangle ADC

with legs 6 and 8, and

\triangle ABC

with hypotenuse

AC

and equal legs. Using the Pythagorean theorem on

\triangle ADC

gives hypotenuse

AC = \sqrt{6^2 + 8^2} = 10

, and its area is

\frac{1}{2}(6)(8) = 24

. For isosceles right triangle

\triangle ABC

,

AB^2 + BC^2 = 10^2 \implies 2(AB^2) = 100 \implies AB^2 = 50

, so its area is

\frac{1}{2}(50) = 25

. Adding both triangle areas yields

24 + 25 = 49

.

Step-by-Step Solution

1

Divide the quadrilateral into two right triangles using diagonal

AC

.

Quadrilateral

ABCD

is split into

\triangle ADC

and

\triangle ABC

, both of which are right-angled triangles sharing hypotenuse

AC

.

Diagonal

AC

connects the vertices opposite the

90^\circ

angles.

2

Calculate the length of diagonal

AC

using

\triangle ADC

.

AC^2 = AD^2 + CD^2 = 6^2 + 8^2 = 36 + 64 = 100 \implies AC = 10

.

\triangle ADC

is a right triangle with legs of length 6 and 8.

3

Find the area of

\triangle ADC

.

\text{Area}(\triangle ADC) = \frac{1}{2} \times AD \times CD = \frac{1}{2} \times 6 \times 8 = 24

.

The area of a right triangle is half the product of its legs.

4

Determine the area of isosceles right triangle

\triangle ABC

.

Let

AB = BC = s

. Since

s^2 + s^2 = AC^2

, we have

2s^2 = 100 \implies s^2 = 50

. Thus,

\text{Area}(\triangle ABC) = \frac{1}{2} s^2 = \frac{1}{2} \times 50 = 25

.

\triangle ABC

is a right triangle with equal legs

s

and hypotenuse

AC = 10

.

5

Sum the areas of the two triangles to get the total area.

\text{Total Area} = 24 + 25 = 49

.

The total area of the quadrilateral is the sum of the areas of its non-overlapping constituent triangles.

Key Concept

Polygon area decomposition using diagonal partitioning and the Pythagorean theorem

Estimated Time:1m 30s

Question 1811Question →

A medical laboratory uses an automated analyzer to screen blood samples for two distinct markers, Marker A and Marker B. The probability that a randomly selected sample contains Marker A is $0.40$ , and the probability that it contains Marker B is $0.25$ . If the presence of Marker A and the presence of Marker B are independent events, what is the probability that a randomly selected sample contains at least one of these two markers?

0.10

0.45

0.55

0.65

0.70

Show answer & explanation

Answer: 0.55

Answer

The probability that a randomly selected sample contains at least one of the two markers is

0.55

.

To find the probability that a sample contains at least one marker, apply the general addition rule

P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})

. Because the events are independent,

P(\text{A and B}) = P(\text{A}) \times P(\text{B}) = 0.40 \times 0.25 = 0.10

. Substituting the values gives

0.40 + 0.25 - 0.10 = 0.55

. Alternatively, using the complementary probability rule yields

1 - P(\text{neither}) = 1 - (1 - 0.40)(1 - 0.25) = 1 - (0.60 \times 0.75) = 1 - 0.45 = 0.55

.

Step-by-Step Solution

1

Identify the given probabilities and event relationship.

P(\text{A}) = 0.40

,

P(\text{B}) = 0.25

, and events A and B are independent.

Establishing the parameters is necessary to apply the appropriate probability formulas.

2

Calculate the joint probability

P(\text{A and B})

.

P(\text{A and B}) = P(\text{A}) \times P(\text{B}) = 0.40 \times 0.25 = 0.10

.

For independent events, the probability of both events occurring simultaneously is the product of their individual probabilities.

3

Apply the general addition rule for probability to find

P(\text{A or B})

.

P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B}) = 0.40 + 0.25 - 0.10 = 0.55

.

The probability of at least one event occurring requires subtracting the overlapping joint probability to avoid double-counting.

Key Concept

Probability of Independent Events and the General Addition Rule

Question 1812Question →

In the $xy$ -plane, line $k$ passes through the points $(-2, 5)$ and $(4, -7)$ . Which of the following statements about line $k$ must be true? Select all that apply.

Select all that apply

Line

k

has a slope of

-2

.

Line

k

passes through the point

(3, -5)

.

Line

k

does not pass through Quadrant III.

Line

k

has an

x

-intercept at

(-2, 0)

.

A line perpendicular to line

k

has a slope of

2

.

Show answer & explanation

Answer: Line $k$ has a slope of $-2$ .; Line $k$ passes through the point $(3, -5)$ .; Line $k$ does not pass through Quadrant III.

Answer

The correct statements are: line

k

has a slope of

-2

, line

k

passes through the point

(3, -5)

, and line

k

does not pass through Quadrant III.

The slope of line

k

is

m = \frac{-7 - 5}{4 - (-2)} = -2

. The line equation is

y = -2x + 1

. Substituting

x = 3

yields

y = -5

, confirming

(3, -5)

is on the line. For all negative

x

-values (

x < 0

),

y = -2x + 1

remains strictly positive (

y > 1

), so no point on the line has both negative coordinates; hence line

k

never passes through Quadrant III.

Step-by-Step Solution

1

Calculate the slope of line

k

slope

m = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2

The slope formula is

m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Determine the equation of line

k

y = -2x + 1

Using point-slope form

y - y_1 = m(x - x_1)

with point

(-2, 5)

gives

y - 5 = -2(x + 2) \implies y = -2x + 1

.

3

Test point

(3, -5)

and calculate the

x

-intercept

Point

(3, -5)

lies on the line since

-2(3) + 1 = -5

. Setting

y = 0

gives

0 = -2x + 1 \implies x = \frac{1}{2}

.

A point lies on a line if its coordinates satisfy the line equation. The

x

-intercept occurs where

y = 0

.

4

Analyze quadrant passage and perpendicular slope

For

x < 0

,

y = -2x + 1 > 0

, so points in Quadrant III (

x < 0, y < 0

) are never reached. Perpendicular slope is

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

.

Quadrant III requires both coordinates to be negative. Perpendicular lines have negative reciprocal slopes.

Key Concept

Coordinate Geometry of Lines: Slope, Line Equations, Intercepts, Quadrant Passage, and Perpendicular Slopes

Question 1813Question →

Despite the conservators' initial belief that the medieval manuscript was exceptionally _______, subsequent microscopic analysis revealed that centuries of exposure to fluctuating humidity had rendered its vellum fibers surprisingly brittle and prone to disintegration.

Which of the following words best completes the sentence?

friable

delicate

robust

immutable

esoteric

Show answer & explanation

Answer: robust

Answer

The correct completion is 'robust'.

The sentence relies on the contrast signal 'Despite', which creates an opposition between the conservators' initial assumption and the true physical state of the manuscript. Since the manuscript was found to be 'brittle and prone to disintegration', the conservators must have initially thought it was durable and strong. The word 'robust' means sturdy and healthy in condition, directly fulfilling this semantic contrast.

Step-by-Step Solution

1

Identify the structural contrast signal in the sentence.

The word 'Despite' at the beginning of the sentence indicates that the initial belief about the manuscript contrasts directly with the empirical findings that follow.

Contrast pivots signal a semantic reversal between the two clauses.

2

Analyze the context clues in the non-blank clause.

The second clause states that microscopic analysis revealed the vellum fibers were 'surprisingly brittle and prone to disintegration'.

The target word must describe a physical characteristic opposite to being brittle and easily disintegrated.

3

Determine the required tone and definition for the blank.

The missing word must mean strong, durable, or resilient to contrast with 'brittle'.

'Robust' precisely fits the meaning of sturdy or durable, fulfilling the contrast requirement.

Key Concept

Contrast and Reversal Clues

Question 1814Question →

A chemical laboratory starts with a liquid solution having a total volume of $3.2 \times 10^{-4}$ cubic meters. The solution is partitioned equally into $400$ identical micro-vials. Subsequently, heat treatment causes the volume of liquid in each micro-vial to evaporate, reducing its volume by $75\%$ . What is the final volume of liquid remaining in a single micro-vial, expressed in scientific notation?

2.0 \times 10^{-7}

cubic meters

8.0 \times 10^{-7}

cubic meters

6.0 \times 10^{-7}

cubic meters

2.0 \times 10^{-5}

cubic meters

2.0 \times 10^{-6}

cubic meters

Show answer & explanation

Answer: $2.0 \times 10^{-7}$ cubic meters

Answer

2.0 \times 10^{-7}

cubic meters

Dividing the initial total volume of

3.2 \times 10^{-4}

cubic meters by

400

(

4 \times 10^2

) gives an initial volume of

8.0 \times 10^{-7}

cubic meters per micro-vial. Reducing this volume by

75\%

means

25\%

of the liquid remains. Multiplying

8.0 \times 10^{-7}

by

0.25

yields

2.0 \times 10^{-7}

cubic meters.

Step-by-Step Solution

1

Express the number of micro-vials in scientific notation.

400 = 4 \times 10^2

Converting into scientific notation simplifies division involving powers of 10.

2

Calculate the initial volume per micro-vial before evaporation.

\frac{3.2 \times 10^{-4}}{4 \times 10^2} = \left(\frac{3.2}{4}\right) \times 10^{-4 - 2} = 0.8 \times 10^{-6} = 8.0 \times 10^{-7}

cubic meters

Equal partitioning requires dividing the total volume by the total number of containers.

3

Determine the remaining fraction of liquid after a

75\%

volume reduction.

Remaining fraction

= 100\% - 75\% = 25\% = 0.25

A reduction by

75\%

leaves

25\%

of the liquid volume in the vial.

4

Multiply the volume per micro-vial by the remaining fraction.

8.0 \times 10^{-7} \times 0.25 = 2.0 \times 10^{-7}

cubic meters

Applying the remaining fraction yields the final volume in standard scientific notation format

a \times 10^n

where

1 \le a < 10

.

Key Concept

Operations with Decimals, Place Value, and Scientific Notation

Estimated Time:2m 0s

Question 1815Question →

A technician charges a one-time fixed diagnostic fee of $\$35$ plus $\$25$ for each hour of repair work. If the total bill for a repair job was $\$160$ , how many hours of repair work were performed?

3

4

5

6

7

Show answer & explanation

Answer: $5$

Answer

5 hours

Let

h

represent the number of hours of repair work. The total charge is given by the sum of the fixed fee (

\$35

) and the hourly rate times hours worked (

25h

). Setting up the linear equation

35 + 25h = 160

and isolating

h

gives

25h = 125

, which simplifies to

h = 5

. Thus, 5 hours of work were performed.

Step-by-Step Solution

1

Define the variable and set up the linear equation

35 + 25h = 160

, where

h

is the number of repair hours.

The total cost is the sum of the fixed diagnostic fee and the hourly charge multiplied by hours worked.

2

Subtract the fixed fee from both sides of the equation

25h = 160 - 35 \implies 25h = 125

Isolate the variable term containing

h

.

3

Divide both sides by the hourly rate coefficient

h = \frac{125}{25} = 5

Solve for

h

.

Key Concept

Formulating and solving linear equations in one variable from real-world scenarios

Question 1816Question →

What is the value of $\frac{2^7 + 2^7}{2^4}$ ?

1

8

16

128

1024

Show answer & explanation

Answer: 16

Answer

16

Combining the numerator yields

2^7 + 2^7 = 2(2^7) = 2^8

. Dividing

2^8

by

2^4

using the quotient rule gives

2^{8-4} = 2^4 = 16

.

Step-by-Step Solution

1

Simplify the numerator by factoring out common terms

2^7 + 2^7 = 2 \times 2^7 = 2^{1+7} = 2^8

Adding two identical exponential terms is equivalent to multiplying one term by 2.

2

Apply the quotient rule of exponents

\frac{2^8}{2^4} = 2^{8-4} = 2^4

When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

3

Evaluate the power

2^4 = 16

Compute the numerical value of 2 raised to the 4th power.

Key Concept

Combining like exponential terms and applying the quotient rule of exponents.

Question 1817Question →

If

x

satisfies the linear equation

\frac{3(x - 4)}{4} - \frac{2x + 1}{6} = \frac{x + 2}{3} - 5

what is the value of

2x + 5

?

-31

-23

-14

41

87

Show answer & explanation

Answer: $-23$

Answer

The value of

2x + 5

is

-23

.

Clearing denominators by multiplying the entire equation by

12

gives

9(x - 4) - 2(2x + 1) = 4(x + 2) - 60

. Expanding both sides yields

9x - 36 - 4x - 2 = 4x + 8 - 60

, which simplifies to

5x - 38 = 4x - 52

. Isolating

x

gives

x = -14

. Substituting

x = -14

into the target expression

2x + 5

yields

2(-14) + 5 = -23

.

Step-by-Step Solution

1

Clear the denominators by multiplying every term on both sides of the equation by the least common multiple of

4, 6,

and

3

, which is

12

.

12 \cdot \frac{3(x - 4)}{4} - 12 \cdot \frac{2x + 1}{6} = 12 \cdot \frac{x + 2}{3} - 12 \cdot 5

, leading to

9(x - 4) - 2(2x + 1) = 4(x + 2) - 60

.

Clearing denominators simplifies the multi-step fractional equation into an integer linear equation.

2

Expand all grouping symbols and combine like terms on both sides.

9x - 36 - 4x - 2 = 4x + 8 - 60

, which simplifies to

5x - 38 = 4x - 52

.

Distributing terms carefully ensures proper sign distribution, especially for negative signs across parentheses.

3

Isolate the variable

x

on one side of the equation.

5x - 4x = -52 + 38

, giving

x = -14

.

Subtracting

4x

and adding

38

isolates

x

.

4

Evaluate the target expression

2x + 5

using the solved value

x = -14

.

2(-14) + 5 = -28 + 5 = -23

.

The question asks for the value of

2x + 5

, not

x

.

Key Concept

Solving linear equations in one variable with fractional coefficients and evaluating algebraic expressions

Estimated Time:1m 30s

Question 1818Question →

A municipal library system allocated a total annual acquisition budget of $B$ dollars between fiction and non-fiction titles. In the first half of the year, $60\%$ of the total budget was spent, with $70\%$ of that spent amount allocated to fiction books and the remainder to non-fiction books. In the second half of the year, the remaining budget was fully spent such that equal dollar amounts were allocated to fiction and non-fiction books. What percentage of the total annual budget $B$ was spent on fiction books?

42\%

50\%

62\%

72\%

82\%

Show answer & explanation

Answer: $62\%$

Answer

The total percentage of budget

B

spent on fiction books is

62\%

.

To find the total percentage of budget

B

spent on fiction books, we analyze the two periods separately relative to the total budget

B

. In the first half,

60\%

of

B

is spent, and

70\%

of this spent amount goes to fiction, giving

0.70 \times 0.60 B = 0.42 B

(

42\%

of

B

). In the second half, the remaining

40\%

of

B

is spent equally between fiction and non-fiction, so fiction receives

50\%

of

40\%

, which is

0.50 \times 0.40 B = 0.20 B

(

20\%

of

B

). Adding both periods yields

42\% + 20\% = 62\%

of the total annual budget

B

.

Step-by-Step Solution

1

Calculate the percentage of total budget

B

spent on fiction in the first half of the year.

0.70 \times 0.60 B = 0.42 B

, which is

42\%

of

B

.

The library spent

60\%

of budget

B

, and

70\%

of that spent portion went to fiction books.

2

Determine the remaining budget percentage for the second half of the year.

100\% - 60\% = 40\%

of

B

.

The remaining budget is the total budget minus the portion spent during the first half.

3

Calculate the percentage of total budget

B

spent on fiction in the second half of the year.

0.50 \times 0.40 B = 0.20 B

, which is

20\%

of

B

.

The remaining

40\%

budget was split equally between fiction and non-fiction.

4

Sum the fiction expenditures from both halves of the year.

42\% + 20\% = 62\%

of

B

.

Adding the contributions from both periods gives the total fraction of budget

B

allocated to fiction books.

Key Concept

Multi-stage percentage change and part-of-a-part base shift calculations.

Estimated Time:1m 30s

Question 1819Question →

In her analysis of late nineteenth-century labor movements, the historian observed that the rapid proliferation of local trade unions did not merely unite disparate worker factions; furthermore, by establishing a consolidated platform for collective bargaining, these coalitions systematically ___________ the negotiating power of the working class.

Which two of the following options, when inserted into the sentence, produce completed sentences that are logically consistent and produce equivalent meanings?

Select all that apply

augmented

enhanced

curtailed

diminished

documented

ratified

Show answer & explanation

Answer: augmented; enhanced

Answer

The correct options are the words meaning to increase or bolster ('augmented' and 'enhanced'). Both words logically complete the sentence to show that establishing a consolidated bargaining platform increased the working class's power, keeping the sentence coherent and equivalent in meaning.

The sentence uses the transition 'did not merely...; furthermore' alongside a causal phrase ('by establishing a consolidated platform') to indicate that the outcome was an expansion of power. The words meaning to increase or strengthen ('augmented' and 'enhanced') fit the blank precisely and produce sentences with identical meaning.

Step-by-Step Solution

1

Analyze sentence structure and transition signals.

The clause contains 'did not merely...; furthermore', which is an additive continuation signal establishing that the second clause reinforces and builds upon the positive outcome mentioned in the first clause.

Structural signals dictate whether the blank requires a word with positive, negative, or contrasting valence.

2

Determine the required contextual meaning and polarity for the blank.

The phrase 'by establishing a consolidated platform for collective bargaining' provides a positive causal reason. Therefore, the blank must express an increase, expansion, or strengthening of negotiating power.

The causal relationship specifies how the platform affected negotiating power.

3

Evaluate option pairs for semantic equivalence and contextual fit.

'Augmented' and 'enhanced' both mean to increase or intensify. When placed into the blank, both yield identical meaning and logical sentence completion.

GRE Sentence Equivalence requires two words that independently complete the sentence with equivalent overall meaning.

Key Concept

Identifying continuation and causal signals (e.g., 'furthermore', 'by [action]') to determine sentence polarity and select contextual synonym pairs.

Question 1820Question →

If $n$ is a positive integer that is divisible by $6$ but not divisible by $4$ , and $n^2$ has exactly $15$ positive divisors, what is the value of $n$ ?

6

12

18

24

36

Show answer & explanation

Answer: $18$

Answer

The value of

n

is

18

.

Because

n

is divisible by

6

, it must have prime factors

2

and

3

. Since

n

is not divisible by

4

, the exponent of

2

in

n

is exactly

1

, meaning

n^2

has

2^2

as a factor. The divisor count formula for

n^2

requires

(2+1)(2b+1) = 15

, which gives

2b+1 = 5

, so

b=2

. Thus,

n = 2^1 \cdot 3^2 = 18

.

Step-by-Step Solution

1

Analyze the prime factorization structure of

n

based on given divisibility conditions.

Since

n

is divisible by

6 = 2 \cdot 3

, its prime factorization must contain at least one factor of

2

and one factor of

3

. Since

n

is not divisible by

4 = 2^2

, the exponent of

2

in the prime factorization of

n

must be exactly

1

. Thus,

n = 2^1 \cdot 3^b \cdot k

, where

k

contains no factors of

2

or

3

.

Establishing the exponent of

2

narrows down the search space for prime factor exponents.

2

Express

n^2

in terms of its prime factors and write the formula for its number of positive divisors.

n^2 = 2^2 \cdot 3^{2b} \cdot k^2

. The number of positive divisors of

n^2

is given by

d(n^2) = (2 + 1)(2b + 1) \cdot d(k^2) = 3(2b + 1) \cdot d(k^2) = 15

.

The total number of positive divisors of a number

p_1^{a_1} p_2^{a_2} \dots

is

(a_1 + 1)(a_2 + 1) \dots

.

3

Solve for

b

and determine if

k

has any additional prime factors.

Dividing

15

by

3

gives

(2b + 1) \cdot d(k^2) = 5

. Since

5

is prime, we must have

d(k^2) = 1

(meaning

k = 1

) and

2b + 1 = 5

, which yields

2b = 4 \implies b = 2

.

Determining the exponent of

3

fixes the exact prime factorization of

n

.

4

Calculate

n

.

n = 2^1 \cdot 3^2 = 18

.

Multiplying the prime factors together yields the target integer.

Key Concept

Prime Factorization and Divisor Count Formula

Estimated Time:1m 30s

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