All practice questions

Question 1861Question →

A sequence of numbers begins with $t_1 = 4$ . For all integers $n \geq 2$ , each term is defined by $t_n = 3t_{n-1} - 2$ . What is the value of $t_5$ ?

Show answer & explanation

Answer: 244

Answer

The 5th term of the sequence,

t_5

, is equal to 244.

Applying the recursive relation

t_n = 3t_{n-1} - 2

step-by-step starting from

t_1 = 4

gives

t_2 = 10

,

t_3 = 28

,

t_4 = 82

, and finally

t_5 = 244

.

Step-by-Step Solution

1

Find the second term,

t_2

, using

t_1 = 4

.

t_2 = 3(4) - 2 = 12 - 2 = 10

Substitute

n = 2

into the recursive rule

t_n = 3t_{n-1} - 2

.

2

Find the third term,

t_3

, using

t_2 = 10

.

t_3 = 3(10) - 2 = 30 - 2 = 28

Substitute

n = 3

into the recursive rule.

3

Find the fourth term,

t_4

, using

t_3 = 28

.

t_4 = 3(28) - 2 = 84 - 2 = 82

Substitute

n = 4

into the recursive rule.

4

Find the fifth term,

t_5

, using

t_4 = 82

.

t_5 = 3(82) - 2 = 246 - 2 = 244

Substitute

n = 5

into the recursive rule.

Key Concept

Evaluating terms in a recursively defined sequence

Question 1862Question →

Which word best completes the blank in the passage below based on the structural contrast clue provided?

Fill in the blanks below

While early models of planetary formation posited that giant planets formed in stable, immutable orbits, recent astrophysical simulations suggest that their trajectories are actually quite , shifting significantly over cosmic time.

Show answer & explanation

Answer

dynamic (or fluid / variable / unstable / volatile / changeable)

The initial contrast signal 'While' sets up an opposition between the early view of planetary orbits as 'stable, immutable' and the view supported by recent simulations. Therefore, the blank requires a word denoting changeability or instability, such as 'dynamic' or 'variable', which is reinforced by the elaboration that trajectories are 'shifting significantly over cosmic time'.

Step-by-Step Solution

1

Identify the structural contrast signal in the sentence.

The sentence begins with the contrast pivot 'While'.

Subordinating conjunctions like 'While' signal a contrast or semantic reversal between the two clauses of the sentence.

2

Determine the meaning of the reference concept in the first clause.

The first clause describes early planetary orbits as 'stable, immutable'.

To complete the reversal, the blank must express a property opposite to being stable and unchanging.

3

Deduce the word required for the blank.

A word such as 'dynamic' or 'variable' fits the required contrast and aligns with the contextual clue 'shifting significantly over cosmic time'.

This completes the semantic reversal dictated by the contrast pivot while maintaining contextual coherence.

Key Concept

Using contrast and reversal signals to infer contextually opposite vocabulary in Text Completion

Question 1863Question →

Which vocabulary choices best complete the passage below so that the blanks maintain logical coherence and inter-blank dependency?

Fill in the blanks below

Far from demonstrating that the exoplanet's atmosphere was entirely , the atmospheric spectroscopy data actually revealed a complex equilibrium maintained by photochemical cycles, which in turn suggested that previous models overemphasizing chemical stagnation were fundamentally .

Show answer & explanation

Answer

The first blank requires a word meaning inactive or motionless (such as 'inert', 'quiescent', or 'static'), which sets up the contrast introduced by 'Far from' against 'a complex equilibrium maintained by photochemical cycles'. The second blank requires a word meaning incorrect or unsound (such as 'flawed', 'erroneous', or 'misguided'), which logically follows from discovering that previous models overemphasized chemical stagnation.

The sentence relies on a contrast pivot ('Far from') and a causal result signal ('which in turn suggested'). The first blank must contrast with the discovery of active 'photochemical cycles', making words denoting inactivity such as 'inert' or 'static' correct. The second blank evaluates previous models that assumed 'chemical stagnation'; because those models are disproved by the new data, terms denoting invalidity such as 'flawed' or 'erroneous' complete the logical relationship.

Step-by-Step Solution

1

Analyze structural transition signals in the sentence

Identified the pivot phrase 'Far from demonstrating that...', which establishes a direct contrast between the concept in the first blank and the subsequent finding of 'a complex equilibrium maintained by photochemical cycles'.

Understanding the contrast signal is necessary to determine the required polarity for the first blank.

2

Determine the required meaning for the first blank based on inter-blank relationship

Since the atmosphere reveals active photochemical cycles, the contrasting notion in the first blank must denote a lack of activity or chemical motionlessness, pointing to terms such as 'inert' or 'static'.

The phrase 'Far from' reverses the meaning of the clause relative to the observed active equilibrium.

3

Determine the required meaning for the second blank

The clause states that previous models overemphasized 'chemical stagnation'—a view contradicted by the new spectroscopic findings. Thus, these previous models must be characterized as incorrect, leading to terms like 'flawed' or 'erroneous'.

The phrase 'which in turn suggested' establishes a causal relationship between the new evidence of dynamic equilibrium and the invalidation of earlier static models.

Key Concept

Multi-Blank Dependency Tracking

Estimated Time:1m 30s

Question 1864Question →

At the beginning of a quarter, the cost of raw material for a manufacturing plant was $x$ dollars per ton. During the first month, the cost per ton increased by $15\%$ . During the second month, the cost per ton decreased by $20\%$ relative to the first month's price. If the cost per ton at the end of the second month was $\$552$ , what was the initial cost $x$ , in dollars?

Show answer & explanation

Answer: 600

Answer

600

A

15\%

increase followed by a

20\%

decrease yields a combined multiplier of

(1 + 0.15)(1 - 0.20) = 1.15 \times 0.80 = 0.92

. Setting

0.92x = 552

gives

x = \frac{552}{0.92} = 600

.

Step-by-Step Solution

1

Calculate the price multiplier for the first month

Price after month 1 is

1.15x

A

15\%

increase scales the initial value by

(1 + 0.15) = 1.15

.

2

Calculate the price multiplier for the second month

Price after month 2 is

1.15x \times 0.80 = 0.92x

A

20\%

decrease applies to the new base price, scaling it by

(1 - 0.20) = 0.80

.

3

Solve for the initial price

x

x = 600

Divide the given final price of

\$552

by the overall multiplier

0.92

.

Key Concept

Successive Percent Changes and Base Shifts

Estimated Time:1m 30s

Question 1865Question →

Read the passage below:

For decades, historical demographers treated nineteenth-century municipal censuses as remarkably (i)________ records, assuming that civic authorities recorded births and deaths with meticulous objectivity. Recent archival analysis, however, reveals that local magistrates frequently manipulated mortality figures to avoid public panic during epidemic outbreaks. Consequently, far from providing an unvarnished mirror of demographic reality, these registries offer a deeply (ii)________ account, requiring modern scholars to approach official statistics with unprecedented (iii)________.

Which of the following combinations of terms best completes the passage to preserve its logical and cross-sentence contextual coherence?

(i) infallible, (ii) tendentious, (iii) circumspection

(i) suspect, (ii) tendentious, (iii) enthusiasm

(i) infallible, (ii) comprehensive, (iii) credulity

(i) haphazard, (ii) objective, (iii) circumspection

(i) infallible, (ii) tendentious, (iii) indifference

Show answer & explanation

Answer: (i) infallible, (ii) tendentious, (iii) circumspection

Answer

The correct completion is (i) infallible, (ii) tendentious, (iii) circumspection.

The combination consisting of 'infallible', 'tendentious', and 'circumspection' maintains precise thematic and logical coherence throughout the passage. 'Infallible' reflects the initial belief in meticulous objectivity; 'tendentious' captures the biased nature of records resulting from deliberate manipulation; and 'circumspection' accurately describes the cautious posture modern researchers must adopt when evaluating flawed archival data.

Step-by-Step Solution

1

Analyze Blank (i) using the immediate context clue.

The initial assumption is described as believing authorities recorded data with 'meticulous objectivity'. Therefore, Blank (i) must mean faultless or trustworthy, making 'infallible' the fitting choice.

Establish the baseline historical perspective before the contrast signal.

2

Evaluate Blank (ii) across sentence boundaries using structural pivot words.

The contrast transition 'however' and the evidence that magistrates 'manipulated mortality figures' indicate that the accounts were not objective but deliberately biased, making 'tendentious' the appropriate fit.

Ensure cross-sentence coherence between the cause (manipulated figures) and its characterization.

3

Determine Blank (iii) based on logical inference from the passage's overall argument.

Because the records are unreliable and biased, scholars must view them with cautious scrutiny, which is expressed by 'circumspection'.

Complete the logical outcome dictated by the revelation of corrupted historical data.

Key Concept

Cross-Sentence Contextual Coherence in Multi-Blank Text Completion

Estimated Time:1m 30s

Question 1866Question →

A plumbing service charges a flat diagnostic fee of $\$45$ plus $\$65$ for each hour of repair work. If the total bill for a repair job was $\$305$ , how many hours of repair work were performed?

Show answer & explanation

Answer: 4

Answer

The total number of hours of repair work performed was 4.

To determine the number of repair hours, represent the scenario with the linear equation

45 + 65h = 305

, where

h

is the number of hours. Subtracting 45 from both sides yields

65h = 260

. Dividing 260 by 65 gives

h = 4

.

Step-by-Step Solution

1

Formulate the linear equation from the word problem context.

45 + 65h = 305

The total charge consists of a one-time fixed fee of

45 plus a variable fee of

65 per hour

h

.

2

Isolate the variable term by subtracting 45 from both sides of the equation.

65h = 260

Subtracting the constant fee isolates the total cost incurred from hourly work.

3

Divide both sides by the coefficient of the variable to solve for

h

.

h = 4

Dividing the total labor charge (

260) by the hourly rate (

65) yields the number of hours worked.

Key Concept

Linear Equations in One Variable

Question 1867Question →

A contractor prepares a concrete mix using gravel and sand in a ratio of $4 : 7$ by weight. If the contractor uses $28$ tons of sand for a project, what is the total weight, in tons, of the concrete mix?

16

21

44

49

77

Show answer & explanation

Answer: $44$

Answer

The total weight of the concrete mix is

44

tons.

The ratio of gravel to sand is

4 : 7

. Sand corresponds to

7

parts of the total mixture and weighs

28

tons, meaning

1

part represents

4

tons. The total mixture comprises

4 + 7 = 11

parts. Multiplying

11

parts by

4

tons per part gives a total weight of

44

tons.

Step-by-Step Solution

1

Determine the weight per ratio part using the known quantity of sand.

Since

7

parts equal

28

tons of sand,

1

part

= \frac{28}{7} = 4

tons.

The ratio specifies that sand accounts for

7

equal parts of the total mixture.

2

Calculate the weight of the gravel component.

Gravel weight

= 4 \text{ parts} \times 4 \text{ tons/part} = 16

tons.

The ratio specifies that gravel accounts for

4

parts of the mixture.

3

Compute the total weight of the concrete mixture.

Total weight

= 16 \text{ tons (gravel)} + 28 \text{ tons (sand)} = 44

tons.

The total weight of the mixture is the sum of the weights of all individual ingredients.

Key Concept

Solving Proportions from Given Component Values

Question 1868Question →

Read the following passage from an essay on architectural conservation:

'In mid-twentieth-century architectural conservation, practitioners frequently lauded concrete for its remarkable structural durability. Contemporary conservators, however, evaluate the material through a more dynamic lens, emphasizing instead its remarkably plastic nature—its capacity to deform under persistent loads and accommodate subtle seismic shifts over decades without catastrophic fracturing.'

In the context of the passage, which of the following words best captures the meaning of 'plastic' as it is used by contemporary conservators?

synthetic

malleable

rigid

fragile

ornamental

Show answer & explanation

Answer: malleable

Answer

malleable

The correct answer accurately reflects the secondary definition of 'plastic'—meaning pliant, adaptable, or capable of being deformed under pressure—which is directly established by the em-dash structural clue explaining the material's capacity to deform and accommodate seismic shifts without fracturing.

Step-by-Step Solution

1

Identify structural transition signals and contrast markers in the passage.

The pivot word 'however' signals a shift from viewing concrete as merely static and durable to evaluating it through a 'more dynamic lens.'

Contrast signals indicate that the target word will express a property distinct from unyielding durability.

2

Analyze the punctuation-based structural clue following the target word.

The em-dash ('—') introduces a direct restatement/elaboration: 'its capacity to deform under persistent loads and accommodate subtle seismic shifts... without catastrophic fracturing.'

Em-dashes frequently introduce definitive contextual explanations of preceding terms.

3

Select the option that aligns with the structural definition rather than the primary colloquial meaning.

'Malleable' (pliant, adaptable, capable of deformation) precisely matches the structural definition provided by the author.

Relying on contextual structural clues resolves the secondary polysemous meaning of the vocabulary word.

Key Concept

Deciphering Vocabulary Meaning via Em-Dash Elaborations and Contrast Signals

Estimated Time:1m 30s

Question 1869Question →

Read the passage below and select the word that logically completes the sentence.

Paradoxically, while early musicologists dismissed the traditional singers' performances as ________, recent acoustic re-examinations reveal that the musicians adhered to an astonishing degree of structural discipline and rhythmic precision.

Which of the following words best completes the blank?

fastidious

desultory

exemplary

pellucid

immutable

Show answer & explanation

Answer: desultory

Answer

desultory

The sentence relies on the contrast signal 'Paradoxically, while' to set up an opposition between the early dismissal of the performances and the recent finding that they possessed 'structural discipline and rhythmic precision.' The word completing the blank must describe a lack of order, planning, or discipline. 'Desultory' precisely fits this definition, meaning lacking a definite plan, purpose, or regularity.

Step-by-Step Solution

1

Analyze the sentence structure and identify key transition structural clues.

The pivot phrase 'Paradoxically, while' signals a contrast between how early musicologists dismissed the performances and what recent re-examinations actually demonstrated.

Reversal clues dictate that the target word must express a quality opposite in valence to the description in the main clause.

2

Determine the required semantic tone and meaning for the blank.

The main clause states that the musicians maintained 'structural discipline and rhythmic precision' (highly organized and deliberate). Therefore, the blank describing the early dismissal must mean disorganized, planless, or erratic.

The word must satisfy the negative framing of early dismissal while contrasting directly with organized discipline.

3

Evaluate the choices to find the word matching 'lacking plan or discipline'.

'Desultory' specifically means lacking a definite plan, enthusiasm, or purpose (haphazard). This perfectly contrasts with 'structural discipline'.

Selecting 'desultory' satisfies both the negative sentiment of the dismissal and the direct semantic reversal dictated by the structural pivot.

Key Concept

Identifying Contrast and Reversal Structural Signals

Estimated Time:1m 15s

Question 1870Question →

Complete the passage by determining which terms best fill each corresponding blank to maintain cross-sentence contextual coherence.

Fill in the blanks below

For decades, historians of economic thought portrayed early modern mercantile guilds as thoroughly entities, bound to archaic trade protections and hostile to commercial novelties. Yet re-examinations of ledger archives demonstrate that their regulatory frameworks were surprisingly , adjusting seamlessly to turbulent international monetary shifts. Far from inducing market , these institutional adaptations enabled sustained commercial growth across diverse urban jurisdictions.

Show answer & explanation

Answer

The passage is completed correctly by selecting 'sclerotic' for the first blank, 'malleable' for the second blank, and 'stagnation' for the third blank.

The passage relies on clear cross-sentence structural signals to establish logical coherence. The first sentence establishes traditional views of guilds as unyielding and hostile to innovation ('sclerotic'). The contrast signal 'Yet' in the second sentence introduces recent archival evidence showing their rules were adaptable and responsive to monetary shifts ('malleable'). The third sentence uses 'Far from inducing' to negate a negative outcome, showing that rather than causing market paralysis ('stagnation'), these institutional adaptations actually facilitated commercial growth.

Step-by-Step Solution

1

Analyze the context of the first sentence to determine the meaning needed for the first blank.

The first sentence explains that guilds were 'bound to archaic trade protections and hostile to commercial novelties.' This indicates a characterization defined by rigidity, unresponsiveness, and resistance to change, pointing to 'sclerotic'.

The surrounding descriptive clauses directly elaborate on the negative, rigid portrait painted by traditional historians.

2

Examine the shift in perspective signaled at the beginning of the second sentence for the second blank.

The structural signal 'Yet' marks a reversal of the previous view, and the sentence describes frameworks 'adjusting seamlessly to turbulent international monetary shifts.' This requires a word meaning adaptable or flexible, pointing to 'malleable'.

The contrast transition 'Yet' combined with evidence of seamless adjustment demands a term opposite in tone to 'sclerotic'.

3

Evaluate the rhetorical structure of the third sentence for the third blank.

The phrase 'Far from inducing market...' introduces a negative condition that is contrasted with 'enabled sustained commercial growth.' The correct word must mean economic immobility or decline, pointing to 'stagnation'.

The preconditional construction 'Far from inducing' sets up an inverse relationship with the positive outcome of commercial growth.

Key Concept

Multi-Blank Text Completion: Cross-Sentence Contextual Coherence

Estimated Time:1m 30s

Question 1871Question →

On a real number line, point $A$ has coordinate $-5$ and point $B$ has coordinate $11$ . Point $C$ has coordinate $x$ such that the distance between $C$ and the midpoint of line segment $AB$ is equal to $\frac{1}{3}$ of the distance between $C$ and point $B$ . What is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 5

Answer

The maximum possible value of

x

is

5

.

The midpoint of

A(-5)

and

B(11)

is

M = 3

. Setting up the distance equation

|x - 3| = \frac{1}{3}|x - 11|

leads to

3|x - 3| = |x - 11|

. Evaluating the two cases

3(x - 3) = x - 11

and

3(x - 3) = -(x - 11)

yields solutions

x = -1

and

x = 5

. The maximum possible value among these is 5.

Step-by-Step Solution

1

Calculate the coordinate of the midpoint of line segment

AB

.

Midpoint coordinate

M = 3

.

The midpoint of two coordinates

a

and

b

on a number line is given by

\frac{a + b}{2} = \frac{-5 + 11}{2} = 3

.

2

Formulate the distance equation using absolute values.

3|x - 3| = |x - 11|

.

The distance between

x

and

3

is

|x - 3|

and the distance between

x

and

11

is

|x - 11|

. The given condition is

|x - 3| = \frac{1}{3}|x - 11|

.

3

Solve the absolute value equation for all possible values of

x

.

x = -1

and

x = 5

.

Splitting

3(x - 3) = x - 11

yields

x = -1

, and splitting

3(x - 3) = -(x - 11)

yields

x = 5

.

4

Select the maximum value among all valid solutions.

5

Comparing

x = -1

and

x = 5

, the maximum value is

5

.

Key Concept

Distance on a number line using absolute value and midpoint formula

Estimated Time:2m 0s

Question 1872Question →

For how many integer values of $x$ does the inequality $||2x - 7| - 5| \le 4$ hold true?

Show answer & explanation

Answer: 10

Answer

10

To solve

||2x - 7| - 5| \le 4

, break the outer absolute value into

-4 \le |2x - 7| - 5 \le 4

. Adding 5 to all parts yields

1 \le |2x - 7| \le 9

. This produces two simultaneous conditions:

|2x - 7| \le 9

, which gives

-1 \le x \le 8

, and

|2x - 7| \ge 1

, which gives

x \le 3

or

x \ge 4

. Intersecting these solution sets yields the real intervals

[-1, 3]

and

[4, 8]

. The integers contained within these intervals are

-1, 0, 1, 2, 3

and

4, 5, 6, 7, 8

, giving a total of 10 valid integer values.

Step-by-Step Solution

1

Unpack the outer absolute value inequality

-4 \le |2x - 7| - 5 \le 4

An inequality of the form

|u| \le k

for

k \ge 0

is equivalent to

-k \le u \le k

.

2

Isolate the inner absolute value term

1 \le |2x - 7| \le 9

Adding 5 across the compound inequality isolates the term

|2x - 7|

.

3

Solve the upper bound inequality

|2x - 7| \le 9

-1 \le x \le 8

-9 \le 2x - 7 \le 9 \implies -2 \le 2x \le 16 \implies -1 \le x \le 8

.

4

Solve the lower bound inequality

|2x - 7| \ge 1

x \le 3

or

x \ge 4

The inequality

|u| \ge 1

splits into

u \ge 1

or

u \le -1

, yielding

2x - 7 \ge 1 \implies x \ge 4

or

2x - 7 \le -1 \implies x \le 3

.

5

Determine the overlapping interval and count integer solutions

10 integer solutions

The intersection of

[-1, 8]

with

((-\infty, 3] \cup [4, \infty))

is

[-1, 3] \cup [4, 8]

. The integers in this domain are

-1, 0, 1, 2, 3, 4, 5, 6, 7, 8

, which equals 10 integer values.

Key Concept

Solving nested absolute value inequalities on the real number line

Question 1873Question →

At a public university, the ratio of the number of undergraduate students to the number of graduate students at the beginning of an academic year was $4$ to $1$ . By the end of the academic year, the number of undergraduate students had increased by $15\%$ , and the total student population (undergraduate and graduate combined) had increased by $18\%$ . By what percent did the number of graduate students increase?

Show answer & explanation

Answer: 30

Answer

30%

Representing the initial populations by their ratio components (

4

parts undergraduate,

1

part graduate,

5

parts total), an overall growth of

18\%

across

5

total parts yields

5 \times 0.18 = 0.90

total parts of growth. The undergraduate population grew by

15\%

, contributing

4 \times 0.15 = 0.60

parts of growth. The remaining

0.90 - 0.60 = 0.30

parts of growth must come from the graduate student component. Since the initial graduate population was

1

part, a growth of

0.30

parts corresponds to a

30\%

increase.

Step-by-Step Solution

1

Set up initial component values using the ratio

4:1

Undergraduates =

400

, Graduates =

100

, Total =

500

Choosing convenient numbers compatible with the

4:1

ratio simplifies percentage calculations.

2

Calculate the updated undergraduate count after a

15\%

increase

New Undergraduates =

460

An increase of

15\%

means the new value is

115\%

of the original

400

.

3

Calculate the updated total count after an

18\%

increase

New Total =

590

An overall increase of

18\%

means the new total is

118\%

of the original

500

.

4

Subtract the new undergraduate count from the new total count to find the new graduate count

New Graduates =

130

The total population is the sum of undergraduate and graduate populations.

5

Calculate the percent change for the graduate population

30\%

The increase of

30

graduate students over the initial

100

graduate students represents a

30\%

increase.

Key Concept

Weighted percentage change across subgroup components

Estimated Time:2m 0s

Question 1874Question →

Far from viewing the early naturalist's preliminary logs as (i)________ speculations lacking empirical rigor, contemporary historians of science maintain that these tentative observations were (ii)________ to the ultimate formulation of her unified theory; without these initial records, the overarching paradigm could never have been consolidated.

Which of the following combinations of words best completes the passage in a logically coherent manner?

(i) specious, (ii) integral

(i) specious, (ii) peripheral

(i) meticulous, (ii) peripheral

(i) meticulous, (ii) integral

(i) essential, (ii) detrimental

Show answer & explanation

Answer: (i) specious, (ii) integral

Answer

(i) specious, (ii) integral

The combination of '(i) specious' and '(ii) integral' creates logical coherence. The phrase 'Far from viewing... as (i) specious' indicates that historians do not consider the notes misleading or lacking foundation; instead, they view them as '(ii) integral' (essential), a stance supported by the concluding assertion that the theory could not have been consolidated without them.

Step-by-Step Solution

1

Analyze structural transition signals and inter-blank dependencies

The opening transition 'Far from viewing [X] as (i)____' sets up a contrast between a common misconception and the actual perspective of contemporary historians. The semicolon and concluding phrase 'without these initial records, the overarching paradigm could never have been consolidated' provide a firm clue for Blank (ii).

Tracking dependencies between clauses reveals that Blank (ii) must denote essentiality, which in turn clarifies the false view negated in Blank (i).

2

Evaluate Blank (ii) using the explicit cause-effect clue

Since the paradigm could not be consolidated without the logs, the logs were indispensable. 'Integral' accurately captures this required meaning.

The final clause directly defines the relationship between the preliminary logs and the final theory.

3

Evaluate Blank (i) based on the contrast with Blank (ii)

Because historians view the logs as vital ('integral'), the view being rejected ('Far from viewing... as') must have framed them as unsound or misleading ('specious').

The sentence structure requires Blank (i) to express a negative assessment of validity that stands in opposition to the positive value identified in Blank (ii).

Key Concept

Multi-Blank Dependency Tracking

Question 1875Question →

In the $xy$ -plane, line $L$ is given by the equation $4x + 3y = 24$ . Line $M$ passes through the origin $(0,0)$ and is perpendicular to line $L$ . Which of the following statements must be true? Select all that apply.

Select all that apply

Line

M

passes through the point

(8, 6)

.

The point of intersection of line

L

and line

M

lies in Quadrant I.

The slope of line

M

is

-\frac{3}{4}

.

The area of the triangular region bounded by line

L

and the coordinate axes is

48

square units.

The

x

-intercept of line

L

is

(0, 8)

.

Show answer & explanation

Answer: Line $M$ passes through the point $(8, 6)$ .; The point of intersection of line $L$ and line $M$ lies in Quadrant I.

Answer

The statements confirming that line

M

passes through

(8, 6)

and that the intersection point of lines

L

and

M

lies in Quadrant I are correct.

The correct statements accurately calculate the negative reciprocal slope of line

M

as

\frac{3}{4}

, verify that

(8,6)

satisfies

y = \frac{3}{4}x

, and correctly find that the intersection of lines

L

and

M

has positive

x

and

y

coordinates in Quadrant I.

Step-by-Step Solution

1

Find the slope and equation of line

L

.

Line

L

in slope-intercept form is

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

, with slope

m_L = -\frac{4}{3}

,

x

-intercept

(6,0)

, and

y

-intercept

(0,8)

.

Converting to slope-intercept form

y = mx + b

exposes the slope and intercepts of line

L

.

2

Determine the slope and equation of perpendicular line

M

.

Slope

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

. Line

M

passes through the origin, so its equation is

y = \frac{3}{4}x

.

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

3

Evaluate the statement regarding point

(8,6)

on line

M

.

y = \frac{3}{4}(8) = 6

. The point

(8,6)

satisfies the equation of line

M

.

Substituting the coordinates into the line equation verifies point membership.

4

Find the intersection point of line

L

and line

M

.

Substituting

y = \frac{3}{4}x

into

4x + 3y = 24

yields

4x + 3\left(\frac{3}{4}x\right) = 24 \implies \frac{25}{4}x = 24 \implies x = \frac{96}{25}

. Then

y = \frac{72}{25}

.

Both

x

and

y

coordinates are positive, which confirms the intersection lies in Quadrant I.

Key Concept

Perpendicular Line Slopes and Intersections in Coordinate Geometry

Question 1876Question →

At a logistics company, fuel expenditure accounted for $40\%$ of total operational costs in Year 1, while maintenance expenditure accounted for the remaining $60\%$ . In Year 2, fuel expenditure increased by $25\%$ , and maintenance expenditure decreased by $15\%$ . If there were no other operational costs in either year, by what percentage did the total operational cost change from Year 1 to Year 2?

An increase of

1\%

An increase of

5\%

An increase of

9\%

An increase of

10\%

A decrease of

1\%

Show answer & explanation

Answer: An increase of $1\%$

Answer

An increase of

1\%

The total operational cost increases by

1\%

because fuel cost increases by

0.40 \times 25\% = +10\%

of the total cost, while maintenance cost decreases by

0.60 \times 15\% = -9\%

of the total cost, yielding a net change of

+10\% - 9\% = +1\%

.

Step-by-Step Solution

1

Assign an arbitrary base value or variable for Year 1 total cost

Let the total operational cost in Year 1 be

C = 100

. Thus, fuel expenditure is

40

and maintenance expenditure is

60

.

Choosing a concrete baseline value makes multi-step percentage change calculations clear and straightforward.

2

Calculate Year 2 cost for each component

Year 2 fuel cost

= 40 \times (1 + 0.25) = 50

. Year 2 maintenance cost

= 60 \times (1 - 0.15) = 51

.

Apply the individual percentage increase (

+25\%

) and decrease (

-15\%

) to the respective Year 1 component values.

3

Sum the Year 2 component costs and compute total percentage change

Total Year 2 cost

= 50 + 51 = 101

. Percent change

= \frac{101 - 100}{100} \times 100\% = +1\%

.

Compare the new total operational cost to the original baseline cost to determine net percentage change.

Key Concept

Weighted Percentage Change

Estimated Time:1m 30s

Question 1877Question →

An astronomical digital sensor captures $4.5 \times 10^7$ raw data packets during an observation run, where each packet has a size of $1.6 \times 10^{-4}$ megabytes. A compression algorithm reduces the total volume of these combined packets by $75\%$ . What is the total volume, in megabytes, of the compressed data expressed in scientific notation?

1.8 \times 10^3

5.4 \times 10^3

7.2 \times 10^3

1.8 \times 10^4

7.2 \times 10^{-3}

Show answer & explanation

Answer: $1.8 \times 10^3$

Answer

1.8 \times 10^3

megabytes

Multiplying

4.5 \times 10^7

by

1.6 \times 10^{-4}

gives

7.2 \times 10^3

megabytes. Decreasing this total by

75\%

retains

25\%

of the volume, which equals

0.25 \times 7.2 \times 10^3 = 1.8 \times 10^3

megabytes.

Step-by-Step Solution

1

Calculate the total uncompressed size by multiplying the packet count by individual packet size.

(4.5 \times 10^7) \times (1.6 \times 10^{-4}) = (4.5 \times 1.6) \times 10^{7 + (-4)} = 7.2 \times 10^3

megabytes.

Total volume is the product of quantity and unit size.

2

Determine the remaining fraction of data after a

75\%

size reduction.

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

A

75\%

reduction leaves

25\%

of the original data.

3

Multiply the uncompressed volume by

0.25

and write the result in standard scientific notation.

0.25 \times (7.2 \times 10^3) = 1.8 \times 10^3

megabytes.

The coefficient

1.8

meets the condition

1 \leq a < 10

for scientific notation

a \times 10^n

.

Key Concept

Decimals and Scientific Notation

Question 1878Question →

At the beginning of Year 1, a commercial real estate portfolio consisted of two properties, Property A and Property B, where the value of Property A was $50\%$ greater than the value of Property B. During Year 1, the value of Property A increased by $20\%$ while the value of Property B decreased by $20\%$ . During Year 2, the value of Property A decreased by $x\%$ while the value of Property B increased by $x\%$ . If the combined value of the two properties at the end of Year 2 was equal to the combined value of the two properties at the beginning of Year 1, what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

10

Tracking base value changes across both properties yields a Year 1 ending total of

1.8B + 0.8B = 2.6B

. Applying Year 2 percentage adjustments gives a total value of

1.8B(1 - 0.01x) + 0.8B(1 + 0.01x) = 2.6B - 0.01Bx

. Setting this equal to the initial portfolio value of

2.5B

gives

0.1B = 0.01Bx

, which solves to

x = 10

.

Step-by-Step Solution

1

Define initial values for both properties relative to a single variable at the beginning of Year 1

Property B value =

B

, Property A value =

1.5B

, and total portfolio value =

2.5B

Setting Property B as

B

allows all subsequent values and combined sums to be expressed cleanly in terms of

B

.

2

Calculate individual property values at the end of Year 1 after applying respective

+20\%

and

-20\%

changes

Property A value =

1.5B \times 1.20 = 1.8B

; Property B value =

B \times 0.80 = 0.8B

Year 1 percentage changes must be calculated using each property's initial starting base.

3

Formulate algebraic expressions for the property values at the end of Year 2

Total Year 2 end value =

1.8B \left(1 - \frac{x}{100}\right) + 0.8B \left(1 + \frac{x}{100}\right) = 2.6B - 0.01Bx

Year 2 percentage shifts apply to the new updated base values (

1.8B

and

0.8B

) obtained at the end of Year 1.

4

Set the Year 2 end total value equal to the Year 1 start total value and solve for

x

2.6B - 0.01Bx = 2.5B \implies 0.1B = 0.01Bx \implies x = 10

Equating the two expressions eliminates

B

from both sides, leaving a linear equation in

x

.

Key Concept

Successive Percentage Changes and Base Shifts across Multiple Portfolio Assets

Question 1879Question →

Three automated production lines—Line A, Line B, and Line C—run continuous maintenance cycles every $45$ minutes, $60$ minutes, and $75$ minutes, respectively. All three lines completed a maintenance cycle simultaneously at 8:00 a.m. on Monday. A supervisory inspection is triggered whenever at least two of the three lines complete a maintenance cycle at the exact same time. Between 8:01 a.m. on Monday and 8:00 a.m. on Tuesday, inclusive, how many supervisory inspections will be triggered?

Show answer & explanation

Answer: 16

Answer

16

To find the number of times at least two lines finish maintenance simultaneously, we determine the LCM for each pair of intervals (180, 225, and 300 minutes) and calculate how many multiples of these values occur within 1,440 minutes. Within this duration, there are 8 multiples of 180, 6 multiples of 225, and 4 multiples of 300. The single time all three coincide (at 900 minutes) is included in all three pairwise counts. Subtracting 2 instances of this overlap yields 8 + 6 + 4 - 2 = 16 total supervisory inspections.

Step-by-Step Solution

1

Prime factorize each interval length

45 = 3^2 × 5; 60 = 2^2 × 3 × 5; 75 = 3 × 5^2

Prime factorization allows systematic calculation of pairwise and triple LCMs.

2

Compute pairwise and triple LCMs

lcm(45, 60) = 180; lcm(45, 75) = 225; lcm(60, 75) = 300; lcm(45, 60, 75) = 900

The LCM determines the interval between simultaneous completions.

3

Count occurrences within the 1,440-minute time window

A & B: 8 times; A & C: 6 times; B & C: 4 times; All three: 1 time

Dividing the total time window by each LCM yields the frequency of simultaneous maintenance events.

4

Apply Principle of Inclusion-Exclusion for sets

8 + 6 + 4 - 2(1) = 16 distinct inspection events

Events where all three lines coincide (at t = 900) are counted in all three pair counts and must be adjusted so that the time point is counted exactly once.

Key Concept

Least Common Multiple (LCM) and Inclusion-Exclusion Principle

Question 1880Question →

On the real number line, $x$ is a real number such that its distance from $1$ is at most $6$ , and its distance from $-2$ is at least $4$ . Which of the following inequalities represents all possible values of $x$ ?

2 \le x \le 7

-5 \le x \le 7

-6 \le x \le 7

-5 \le x \le 2

-5 \le x \le -2 \text{ or } 2 \le x \le 7

Show answer & explanation

Answer: $2 \le x \le 7$

Answer

The inequality

2 \le x \le 7

represents all possible values of

x

.

The distance condition 'at most 6 units from 1' translates to

|x - 1| \le 6

, yielding the interval

[-5, 7]

. The condition 'at least 4 units from -2' translates to

|x + 2| \ge 4

, yielding

x \le -6

or

x \ge 2

. Finding the values that belong to both conditions requires taking the intersection of

[-5, 7]

and

(-\infty, -6] \cup [2, \infty)

. Since

[-5, 7]

does not overlap with

(-\infty, -6]

, the only overlapping region is

[2, 7]

, which corresponds to

2 \le x \le 7

.

Step-by-Step Solution

1

Express the geometric distance conditions as absolute value inequalities

Condition 1:

|x - 1| \le 6

. Condition 2:

|x - (-2)| = |x + 2| \ge 4

.

The distance between two numbers

a

and

b

on the number line is given by

|a - b|

.

2

Solve the first inequality

|x - 1| \le 6

-6 \le x - 1 \le 6 \implies -5 \le x \le 7

.

An absolute value inequality of the form

|u| \le k

(for

k \ge 0

) unwraps to

-k \le u \le k

.

3

Solve the second inequality

|x + 2| \ge 4

x + 2 \ge 4

or

x + 2 \le -4 \implies x \ge 2

or

x \le -6

.

An absolute value inequality of the form

|u| \ge k

(for

k \ge 0

) unwraps to

u \ge k

or

u \le -k

.

4

Find the intersection of the two solution sets

Combine

[-5, 7]

with

(-\infty, -6] \cup [2, \infty)

. The intersection of

[-5, 7]

and

(-\infty, -6]

is empty since

-5 > -6

. The intersection of

[-5, 7]

and

[2, \infty)

is

[2, 7]

, which means

2 \le x \le 7

.

The variable

x

must satisfy both conditions simultaneously.

Key Concept

Distance on a number line expressed as absolute value inequalities and finding overlapping intervals

Estimated Time:2m 0s

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