Two-Part Analysis Practice Questions — GMAT — Page 2

Question 21Question →

An investment firm allocates a total of $\$60,000$ between two funds, Fund X and Fund Y, in the ratio of $3:2$ , respectively. Fund X yields an annual return of $10\%$ , while Fund Y yields an annual return of $15\%$ .

Which of the following correctly pairs the amount invested in Fund X with the total dollar return generated from both funds combined after one year?

\ halt{\$36,000}

invested in Fund X, and

\$7,200

total return

\$24,000

invested in Fund X, and

\$7,200

total return

\$36,000

invested in Fund X, and

\$6,000

total return

\$40,000

invested in Fund X, and

\$8,000

total return

\$36,000

invested in Fund X, and

\$9,000

total return

Show answer & explanation

Answer: $\ halt{\$36,000}$ invested in Fund X, and $\$7,200$ total return

Answer

The investment in Fund X is

\$36,000

, and the total dollar return from both funds combined is

\$7,200

.

The option stating '

\$36,000

invested in Fund X, and

\$7,200

total return' is correct because Fund X receives

3/5

of the

\$60,000

total (

\$36,000

), generating

\$3,600

at

10\%

. Fund Y receives

2/5

of the total (

\$24,000

), generating

\$3,600

at

15\%

. The combined return is

\$3,600 + \$3,600 = \$7,200

.

Step-by-Step Solution

1

Determine the dollar amount allocated to each fund using the given ratio.

Fund X allocation:

\frac{3}{3+2} \times \$60,000 = \frac{3}{5} \times \$60,000 = \$36,000

. Fund Y allocation:

\frac{2}{5} \times \$60,000 = \$24,000

.

The total ratio parts equal

3 + 2 = 5

parts.

2

Calculate the annual return for Fund X.

Fund X return:

10\% \text{ of } \$36,000 = 0.10 \times \$36,000 = \$3,600

.

Fund X yields a

10\%

return on its invested capital.

3

Calculate the annual return for Fund Y.

Fund Y return:

15\% \text{ of } \$24,000 = 0.15 \times \$24,000 = \$3,600

.

Fund Y yields a

15\%

return on its invested capital.

4

Sum the returns from both funds to find the total return.

Total combined return: \$3,600 + \$3,600 = \$7,200.

The total return is the sum of returns from Fund X and Fund Y.

Key Concept

Ratio Allocation and Weighted Percentage Rates

Question 22Question →

A corporate health study observed a strong positive correlation between weekly consumption of energy drinks and reported workplace stress levels among software engineers. The researchers hypothesized that the high caffeine content in energy drinks directly causes increased workplace stress.

Which of the following statements, if true, provide either valid evidence supporting the researchers' causal hypothesis or a valid alternative causal explanation for the observed correlation? Select all that apply.

Select all that apply

Engineers who participated in a four-week trial eliminating energy drinks reported a statistically significant decrease in their workplace stress levels.

Engineers experiencing severe workplace stress often consume energy drinks as a coping mechanism to sustain focus during long hours.

Higher workplace stress levels cause engineers to develop metabolic changes that reduce the physiological impact of caffeine.

Energy drinks are generally more expensive than other caffeinated beverages such as coffee or tea.

Software engineers who report low stress levels tend to prefer working during morning hours rather than late nights.

Show answer & explanation

Answer: Engineers who participated in a four-week trial eliminating energy drinks reported a statistically significant decrease in their workplace stress levels.; Engineers experiencing severe workplace stress often consume energy drinks as a coping mechanism to sustain focus during long hours.

Answer

The correct selections are the statement demonstrating that eliminating energy drinks reduces stress levels, and the statement explaining that workplace stress leads engineers to consume energy drinks.

The correct options are the statement showing that removing energy drinks lowers stress levels (which directly supports the causal hypothesis) and the statement revealing that high stress leads to energy drink consumption (which presents a valid reverse-causality alternative explanation for the observed correlation).

Step-by-Step Solution

1

Identify the core causal hypothesis in the stem.

Hypothesis: Energy drink consumption (Cause) leads to increased workplace stress (Effect).

Establishing the directional claim made by researchers is necessary to evaluate supporting or alternative evidence.

2

Evaluate statements for direct empirical support.

The trial showing reduced stress after eliminating energy drinks supports the causal claim by showing that removing the cause reduces the effect.

Demonstrating cause-and-effect control in a trial strengthens a causal hypothesis.

3

Evaluate statements for alternative causal explanations.

The statement that stressed engineers consume energy drinks as a coping mechanism provides a reverse causality explanation.

Reverse causality explains the correlation while challenging the original directional claim.

Key Concept

Evaluating Causal Arguments and Alternative Explanations

Question 23Question →

A metropolitan transportation authority implemented a peak-hour congestion pricing toll for private vehicles entering District X. Over the following 18 months, private vehicle traffic entering District X decreased by $22\%$ , while retail foot traffic in District X increased by $14\%$ . The transit board concluded that the congestion toll directly caused the increase in retail foot traffic by eliminating traffic congestion and encouraging suburban shoppers to visit downtown via public transit.

Match each logical argument role below with the statement that best exemplifies that role when evaluating the transit board's causal claim.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The statement describing retail merchant lobbying matches the reverse causal relationship role; the statement concerning the launch of a zero-fare light rail line matches the confounding variable role; and the statement regarding transit lines operating at 99% capacity matches the invalidation of a necessary assumption role.

Each argument role correctly aligns with its corresponding logical effect: the merchant lobbying statement reverses the temporal and causal sequence; the zero-fare light rail statement introduces a concurrent external factor driving both observed trends; and the 99% transit capacity statement demonstrates a physical bottleneck that breaks a required link in the board's proposed mechanism.

Step-by-Step Solution

1

Analyze the board's core causal hypothesis.

Cause = Congestion pricing toll → Effect = Reduced car traffic & increased retail foot traffic via public transit conversion.

Establishing the explicit mechanism (toll → less car congestion → suburban shoppers take transit → higher foot traffic) is necessary to evaluate logical flaws.

2

Evaluate the statement concerning merchant lobbying.

This statement indicates that higher pedestrian traffic came first and caused the political demand for the toll.

This directly reverses the direction of causality (

Y \rightarrow X

instead of

X \rightarrow Y

), matching the reverse causal relationship role.

3

Evaluate the statement concerning the zero-fare light rail extension.

The light rail extension introduces an external event occurring at the same time that independently drives both observed trends.

A third factor (

Z

) that causes both decreased vehicle traffic and increased foot traffic constitutes a classic confounding variable alternative explanation.

4

Evaluate the statement concerning transit capacity operating at 99%.

Because transit was already saturated and received no upgrades, it physically could not carry new shoppers into District X.

The board's conclusion hinges on the unstated premise that transit could accommodate modal shift; proving capacity was zero invalidates this necessary assumption.

Key Concept

Evaluating Causal Arguments in Two-Part Analysis

Question 24Question →

A corporate study evaluated the relationship between flexible remote work policies and employee turnover in technology firms. Researchers observed that firms introducing remote work experienced a 20% drop in voluntary departures, leading them to conclude that remote work directly reduces employee burnout and thereby improves retention.

Match each statement below with the logical role it plays in evaluating this causal argument.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The statement regarding employee self-selection reverses the direction of causality. The statement regarding concurrent salary increases introduces a confounding variable. The statement regarding commute stress elaborates on the causal mechanism.

Each statement directly matches its precise logical role in causal analysis: reversing cause and effect exposes selection bias; introducing concurrent pay raises presents an alternative explanation; and detailing commute reduction establishes the causal mechanism.

Step-by-Step Solution

1

Analyze the core causal argument

Cause: Remote work policy -> Intermediary Effect: Reduced burnout -> Final Effect: Reduced voluntary turnover.

Establishing the precise structure of the argument is essential to determine how new statements alter its validity.

2

Evaluate the statement on employee self-selection

Shows that low burnout precedes placement in remote work positions.

When the presumed effect (low burnout) is the condition that leads to the presumed cause (remote work), the causal direction is reversed.

3

Evaluate the statement on salary increases

Identifies an unaddressed parallel factor capable of reducing turnover.

An independent, concurrent change offers an alternative causal explanation for the outcome, serving as a confounding variable.

4

Evaluate the statement on commute strain

Provides the step-by-step physical explanation connecting remote work to burnout reduction.

Detailing the intermediate pathway strengthens a causal claim by demonstrating how the cause produces the effect.

Key Concept

Causal Arguments: Reverse Causality, Confounding Variables, and Underlying Mechanisms

Question 25Question →

A municipal environmental agency conducted a study across 50 comparable mid-sized cities over a four-year period. The study observed that cities that introduced a mandatory pay-as-you-throw (PAYT) unit-pricing system—charging households per bag of municipal solid waste disposed—experienced a 35 percent decrease in landfilled waste and a 20 percent increase in curbside recycling rates. The environmental director concluded that the financial burden of the PAYT fees directly caused households to adopt waste-minimization habits and actively divert recyclable materials from the waste stream.

Which two of the following statements, if true, most strongly support the environmental director's causal conclusion?

Select all that apply

In the cities implementing PAYT, the reduction in landfilled waste began immediately following the introduction of per-bag fees rather than during prior years when public awareness campaigns were active.

Similar mid-sized cities in the same geographic region that did not implement a PAYT system experienced no significant change in landfilled waste volumes or recycling rates over the four-year study period.

Cities with residents who already exhibited above-average baseline recycling rates and high environmental consciousness were significantly more likely to adopt the PAYT pricing system.

Concurrently with the introduction of PAYT, the participating cities enacted strict municipal bans on single-use commercial plastic packaging.

Municipalities that increased municipal trash collection frequency saw a temporary 10 percent rise in total waste volume handled by private waste management contractors.

Show answer & explanation

Answer: In the cities implementing PAYT, the reduction in landfilled waste began immediately following the introduction of per-bag fees rather than during prior years when public awareness campaigns were active.; Similar mid-sized cities in the same geographic region that did not implement a PAYT system experienced no significant change in landfilled waste volumes or recycling rates over the four-year study period.

Answer

The two statements that most strongly support the causal conclusion are: (1) the statement indicating that waste reduction began immediately after per-bag fees were introduced rather than during prior awareness campaigns, and (2) the statement showing that control cities without PAYT experienced no significant changes in waste or recycling levels.

To strengthen a causal claim of the form 'X caused Y', an argument must demonstrate that Y occurred specifically following X (temporal precedence) and that Y does not occur in the absence of X (control comparison), while eliminating alternative explanations. The statement noting that waste reduction began immediately after per-bag fees were introduced establishes clear temporal precedence, ruling out earlier public awareness campaigns as the cause. The statement confirming that control cities without PAYT saw no changes in waste or recycling rules out broader regional macro-trends.

Step-by-Step Solution

1

Identify the director's core causal claim

The core claim is that mandatory per-bag PAYT fees directly caused households to decrease landfilled waste and increase recycling.

Evaluating causal arguments requires isolating the proposed cause (unit-pricing fees) and effect (waste reduction/recycling increase).

2

Evaluate potential strengtheners for temporal precedence and control group validity

The statement showing that waste reduction immediately followed fee implementation demonstrates temporal precedence over alternative factors like public awareness campaigns. The statement establishing that non-PAYT cities saw no changes provides a control group ruling out regional trends.

Showing that the effect occurs specifically when and where the cause is present (and not when/where it is absent) reinforces the causal link.

3

Identify and eliminate weakeners and irrelevant statements

The option describing pre-existing high recycling rates suggests selection bias (reverse/alternative causation). The option mentioning single-use plastic bans introduces a confounding variable. The option regarding collection frequency is irrelevant.

Alternative explanations and confounding variables undermine the claim that PAYT fees were the primary driver of the observed outcome.

Key Concept

Causal Argument Evaluation: Establishing Control Groups and Temporal Precedence

Question 26Question →

A marine biology study observed that coral reefs with higher populations of herbivorous parrotfish experienced significantly lower levels of algae overgrowth over a five-year monitoring period. Based on these results, the lead researchers concluded that the presence of parrotfish prevents algae overgrowth on coral reefs.

Which of the following correctly pairs the proposed independent cause and the dependent effect in the researchers' conclusion?

Cause: Presence of herbivorous parrotfish | Effect: Lower levels of algae overgrowth

Cause: Lower levels of algae overgrowth | Effect: Presence of herbivorous parrotfish

Cause: Five-year monitoring period | Effect: Presence of herbivorous parrotfish

Cause: Overall coral reef growth | Effect: Increased algae overgrowth

Cause: Lower levels of algae overgrowth | Effect: Five-year monitoring period

Show answer & explanation

Answer: Cause: Presence of herbivorous parrotfish | Effect: Lower levels of algae overgrowth

Answer

Cause: Presence of herbivorous parrotfish | Effect: Lower levels of algae overgrowth

The conclusion asserts that the presence of parrotfish prevents algae overgrowth. Therefore, the presence of parrotfish is the independent cause, and the lower level of algae overgrowth is the dependent effect.

Step-by-Step Solution

1

Analyze the causal claim in the conclusion.

The claim states that 'the presence of parrotfish prevents algae overgrowth on coral reefs.'

Identifying the explicit conclusion isolates the variables being evaluated.

2

Determine which variable acts as the independent cause.

The presence of herbivorous parrotfish is the action/agent producing the change.

In causal reasoning, the cause is the independent variable introduced or observed first that influences the outcome.

3

Determine which variable acts as the dependent effect.

Lower levels of algae overgrowth is the outcome resulting from the action.

The effect is the measured response or dependent variable produced by the cause.

Key Concept

Distinguishing Cause and Effect in Causal Arguments

Estimated Time:45s

Question 27Question →

A logistics company operates three delivery routes with specific distances and constant average speeds. Match each delivery route to its corresponding total travel time.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Route X matches 3 hours, Route Y matches 4 hours, and Route Z matches 5 hours.

Each route's travel time is accurately determined by dividing the given distance by the constant average speed: Route X takes

150 / 50 = 3

hours, Route Y takes

240 / 60 = 4

hours, and Route Z takes

225 / 45 = 5

hours.

Step-by-Step Solution

1

Calculate travel time for Route X

\text{Time}_X = \frac{150}{50} = 3\text{ hours}

Using the rate formula

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

for Route X.

2

Calculate travel time for Route Y

\text{Time}_Y = \frac{240}{60} = 4\text{ hours}

Using the rate formula

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

for Route Y.

3

Calculate travel time for Route Z

\text{Time}_Z = \frac{225}{45} = 5\text{ hours}

Using the rate formula

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

for Route Z.

Key Concept

Distance, Rate, and Time Relationship

Estimated Time:45s

Question 28Question →

Two server clusters, Alpha and Beta, process data jobs of Type 1 and Type 2. Cluster Alpha processes $15$ Type 1 jobs and $25$ Type 2 jobs in a total time of $210$ minutes. Cluster Beta processes $25$ Type 1 jobs and $15$ Type 2 jobs in a total time of $190$ minutes. Assuming constant processing rates per job type across both clusters, how many total minutes will it take to process a workload consisting of $30$ Type 1 jobs and $10$ Type 2 jobs?

Show answer & explanation

Answer: 180

Answer

The total processing time required is 180 minutes.

Let

x

be the processing time in minutes for a Type 1 job, and

y

be the processing time in minutes for a Type 2 job. Translating the given data gives the simultaneous equations

15x + 25y = 210

and

25x + 15y = 190

. Dividing both equations by

5

yields

3x + 5y = 42

and

5x + 3y = 38

. Adding these simplified equations yields

8x + 8y = 80

, so

x + y = 10

. Subtracting the first from the second gives

2x - 2y = -4

, so

x - y = -2

. Solving

x + y = 10

and

x - y = -2

gives

x = 4

and

y = 6

. The required workload duration is

30(4) + 10(6) = 120 + 60 = 180

minutes.

Step-by-Step Solution

1

Set up a linear system of simultaneous equations based on the rates given.

15x + 25y = 210

and

25x + 15y = 190

, where

x

and

y

are the processing times per Type 1 and Type 2 job.

Establishing accurate algebraic expressions translates the real-world scenario into a solvable model.

2

Simplify the system by dividing through by common factors.

3x + 5y = 42

and

5x + 3y = 38

.

Simplification reduces arithmetic complexity and minimizes computational errors.

3

Solve for variables x and y using symmetric combinations (adding/subtracting equations).

x = 4

and

y = 6

.

Symmetric linear systems can be solved efficiently by taking the sum and difference of the equations.

4

Evaluate the target expression

30x + 10y

.

30(4) + 10(6) = 180

.

Multiplying individual job times by requested quantities yields the total workload time.

Key Concept

Simultaneous Linear Systems and Symmetric Reduction

Question 29Question →

An industrial research group analyzed 80 semiconductor manufacturing facilities over a three-year period to evaluate the impact of an Automated Predictive Maintenance (APM) system on equipment failure rates. The study recorded that facilities operating APM software experienced 40% less unplanned downtime than facilities operating without APM.

Critics argue that this observed correlation does not prove causation, hypothesizing instead that facilities electing to adopt APM already possessed superior quality-control protocols and newer machinery, which independently caused the lower downtime.

Based on the information above, which of the following joint selections correctly identifies both a finding that most strongly supports the claim that APM adoption directly caused the reduction in downtime (Column 1) AND a finding that most strongly supports the critics' counter-explanation (Column 2)?

Column 1: Facilities adopting APM mid-study experienced an immediate 40% drop in downtime while keeping machinery age and maintenance protocols constant; Column 2: Facilities with older machinery that adopted APM experienced no reduction in downtime relative to non-adopting facilities of identical machinery age.

Column 1: Facilities with older machinery that adopted APM experienced no reduction in downtime relative to non-adopting facilities of identical machinery age; Column 2: Facilities adopting APM mid-study experienced an immediate 40% drop in downtime while keeping machinery age and maintenance protocols constant.

Column 1: High baseline downtime directly compelled facility managers to purchase APM software; Column 2: Facilities adopting APM mid-study experienced an immediate 40% drop in downtime while keeping machinery age constant.

Column 1: Facilities with newer machinery recorded significantly lower downtime before APM was commercially released; Column 2: Facilities across all machinery age brackets experienced an identical 40% drop in downtime following APM installation.

Column 1: Facilities adopting APM mid-study experienced an immediate 40% drop in downtime while keeping machinery age constant; Column 2: Sharp increases in equipment downtime caused plant managers to deactivate APM modules.

Show answer & explanation

Answer: Column 1: Facilities adopting APM mid-study experienced an immediate 40% drop in downtime while keeping machinery age and maintenance protocols constant; Column 2: Facilities with older machinery that adopted APM experienced no reduction in downtime relative to non-adopting facilities of identical machinery age.

Answer

Column 1: Facilities adopting APM mid-study experienced an immediate 40% drop in downtime while keeping machinery age and maintenance protocols constant; Column 2: Facilities with older machinery that adopted APM experienced no reduction in downtime relative to non-adopting facilities of identical machinery age.

The correct selection properly assigns the logical roles. To strengthen the claim that APM adoption directly causes lower downtime (Column 1), one must rule out alternative explanations by demonstrating an immediate drop in downtime following APM installation while holding pre-existing variables constant. To support the critics' counter-explanation (Column 2), one must show that APM adoption provides no benefit in the absence of superior machinery/protocols, confirming that the pre-existing factors were driving the observed performance.

Step-by-Step Solution

1

Analyze the core argument and the critics' counter-hypothesis.

Primary claim: APM adoption directly reduces unplanned downtime. Critics' counter-explanation: Pre-existing factors (newer machinery, stricter protocols) caused lower downtime, making the APM correlation spurious.

Establishing the exact logical roles required for Column 1 (Strengthens APM causation) and Column 2 (Supports critics' counter-explanation).

2

Identify the finding that strengthens the APM causal claim for Column 1.

A finding showing that when APM is introduced while controlling for confounding variables (machinery age and protocols remain constant), downtime drops immediately.

Controlling for confounding variables isolates APM as the sole cause of the downtime reduction.

3

Identify the finding that supports the critics' counter-explanation for Column 2.

A finding showing that when APM is introduced into facilities without newer machinery or strict protocols, it fails to produce a downtime reduction.

This demonstrates that APM alone is ineffective without the underlying pre-existing facility characteristics, validating the critics' argument.

4

Match findings to Column 1 and Column 2 in correct order.

Column 1 must contain the controlled mid-study drop evidence, and Column 2 must contain the ineffective APM on older equipment evidence.

Ensures column assignments are not reversed.

Key Concept

Causal Arguments and Confounding Variable Isolation in Two-Part Analysis

Question 30Question →

An epidemiological task force conducted a two-year study across 50 regional hospital wards to evaluate the efficacy of newly installed continuous-discharge ultraviolet-C (UV-C) air disinfection units. During the study period, the incidence rate of hospital-acquired airborne respiratory infections in these 50 wards decreased by $40\%$ . Based on this observation, the hospital administration concluded that the installation of the UV-C units directly caused the reduction in airborne infection rates.

Match each logical role on the left with the statement on the right that best fulfills that role in evaluating the hospital administration's causal argument.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match: (1) the confounding variable role with the statement regarding the simultaneous implementation of a mandatory surface decontamination protocol; (2) the causal direction reversal role with the statement regarding system deployment being triggered by prior infection spikes and staff vigilance; (3) the necessary assumption role with the statement that ventilation rates and physical layouts remained constant; and (4) the strengthening statement role with the observation that infection rates in non-participating control wards did not decrease.

The correct pairings accurately assign each logical role to its corresponding statement based on standard GMAT causal analysis principles. The co-occurring decontamination protocol serves as a confounding variable because it introduces an unmeasured alternative cause for lower infection rates. The deployment of UV-C units in response to prior infection spikes illustrates causal direction reversal by demonstrating that infection conditions triggered the intervention. The constancy of mechanical ventilation rates represents a required assumption because significant changes in ventilation would invalidate the claim that UV-C units were responsible. Finally, the lack of infection reduction in control wards strengthens the causal claim by demonstrating that the outcome does not occur in the absence of the intervention.

Step-by-Step Solution

1

Analyze the core causal argument in the stem.

Premise: UV-C units installed

\rightarrow

40\%

drop in airborne infections. Conclusion: UV-C units directly caused the reduction.

Deconstructing the argument into cause, effect, and underlying assumptions is necessary to evaluate statements against specific logical roles.

2

Identify the confounding variable (left_1).

The statement describing the simultaneous implementation of a surface decontamination protocol introduces an alternative explanatory variable introduced at the same time as the UV-C units.

A confounding variable is an external factor that correlates with both the presumed cause and effect, offering an alternative explanation for the outcome.

3

Identify the causal direction reversal (left_2).

The statement noting that UV-C deployment was approved in response to severe infection spikes reveals that high infection rates drove the administrative decision and staff vigilance.

Reversing causal direction involves showing that the outcome or pre-existing state influenced the implementation of the condition rather than the condition causing the outcome.

4

Identify the required assumption (left_3).

The statement asserting that mechanical ventilation rates and ward physical layout remained constant is a necessary assumption.

If ventilation rates had significantly increased, that environmental change could explain the infection drop; assuming stability in these factors is required for the argument to hold.

5

Identify the strengthening statement (left_4).

The statement showing that non-participating control wards experienced no infection drop reinforces the causal claim.

Demonstrating that the effect is absent when the cause is absent (control group comparison) rules out general background trends and strengthens the causal link.

Key Concept

Causal Reasoning in Two-Part Analysis: Confounding Factors, Reverse Causality, Necessary Assumptions, and Control Comparison

Question 31Question →

An automated algorithmic system updates a portfolio's Index ( $V$ ) and Stabilization Pool ( $P$ ) across sequential stages $k = 1, 2, 3, 4$ .

Initial State (Stage 0): $V_0 = 100$ and $P_0 = 20$ .

For each Stage $k$ (where $k = 1, 2, 3, 4$ ):
1. Sub-stage A (Temporary Index Calculation): Calculate $V_k^{\text{temp}} = V_{k-1} + 0.5 \times P_{k-1}$ .
2. Sub-stage B (State Adjustment):
- If $V_k^{\text{temp}} \le 110$ : set $P_k = P_{k-1} + 15$ and final stage index $V_k = V_k^{\text{temp}} + 10$ .
- If $V_k^{\text{temp}} > 110$ : set $P_k = P_{k-1} - 10$ and final stage index $V_k = V_k^{\text{temp}} - 5$ .

Arrange the following four state transition events in chronological order from earliest to latest occurrence.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct chronological order of events is: (1) The stabilization pool

P

reaches its peak value of

35

, (2) The temporary index

V^{\text{temp}}

first exceeds

130

, (3) The stabilization pool

P

drops to

25

, and (4) The final stage index

V

reaches

140

.

Tracing the state transitions step-by-step reveals:
- At Sub-stage 1B,

P_1 = 35

, which is the peak value of

P

.
- At Sub-stage 2A,

V_2^{\text{temp}} = 137.5

, marking the first time

V^{\text{temp}}

exceeds

130

.
- At Sub-stage 2B,

P_2 = 25

.
- At Sub-stage 3B,

V_3 = 140.0

.

Thus, the event where the pool reaches

35

happens first (Sub-stage 1B), followed by

V^{\text{temp}}

exceeding

130

(Sub-stage 2A), then the pool dropping to

25

(Sub-stage 2B), and finally

V

reaching

140

(Sub-stage 3B).

Step-by-Step Solution

1

Evaluate Stage 1 state transitions

Sub-stage 1A:

V_1^{\text{temp}} = 100 + 0.5(20) = 110

. Since

V_1^{\text{temp}} \le 110

, Sub-stage 1B sets

P_1 = 20 + 15 = 35

and

V_1 = 110 + 10 = 120

.

This establishes that the event 'The stabilization pool

P

reaches its peak value of

35

' occurs at Sub-stage 1B.

2

Evaluate Stage 2 state transitions

Sub-stage 2A:

V_2^{\text{temp}} = 120 + 0.5(35) = 137.5

. Here,

V^{\text{temp}}

first exceeds

130

. Sub-stage 2B: Since

V_2^{\text{temp}} = 137.5 > 110

, set

P_2 = 35 - 10 = 25

and

V_2 = 137.5 - 5 = 132.5

.

This establishes that 'The temporary index

V^{\text{temp}}

first exceeds

130

' occurs at Sub-stage 2A, followed immediately by 'The stabilization pool

P

drops to

25

' at Sub-stage 2B.

3

Evaluate Stage 3 state transitions

Sub-stage 3A:

V_3^{\text{temp}} = 132.5 + 0.5(25) = 145.0

. Sub-stage 3B: Since

V_3^{\text{temp}} = 145.0 > 110

, set

P_3 = 25 - 10 = 15

and

V_3 = 145.0 - 5 = 140.0

.

This confirms that 'The final stage index

V

reaches

140

' occurs at Sub-stage 3B.

4

Synthesize chronological sequence of all four events

Sequence: Sub-stage 1B (

P = 35

)

\rightarrow

Sub-stage 2A (

V^{\text{temp}} = 137.5

)

\rightarrow

Sub-stage 2B (

P = 25

)

\rightarrow

Sub-stage 3B (

V = 140

).

Matching the chronological execution order yields the correct item ordering.

Key Concept

Sequential Processes and State Transitions

Question 32Question →

A distribution network manages inventory between two regional facilities, Facility A and Facility B. At the beginning of Stage 1 ( $k=1$ ), Facility A holds $216$ units of product and Facility B holds $160$ units of product.

In each stage $k$ (for $k = 1, 2, 3$ ), the inventory levels undergo four sequential state transitions in the exact order listed below:

1. Step 1: Facility A dispatches $\frac{1}{2}$ of its current inventory.
2. Step 2: Facility B dispatches $\frac{1}{2}$ of its current inventory.
3. Step 3: Facility A transfers $\frac{1}{3}$ of its remaining inventory to Facility B.
4. Step 4: Facility B receives $28$ new units from an external manufacturer.

Which of the following statements regarding the inventory levels at the end of Stage 2 and Stage 3 are correct? Select all that apply.

Select all that apply

At the end of Stage 2, Facility B holds exactly 112 units.

At the end of Stage 3, the total combined inventory across both facilities is 96 units.

At the end of Stage 2, Facility A holds 36 units.

At the end of Stage 3, Facility B holds 60 units.

The total combined inventory decreases by 40 units from the end of Stage 2 to the end of Stage 3.

Show answer & explanation

Answer: At the end of Stage 2, Facility B holds exactly 112 units.; At the end of Stage 3, the total combined inventory across both facilities is 96 units.; The total combined inventory decreases by 40 units from the end of Stage 2 to the end of Stage 3.

Answer

The correct statements are: (1) At the end of Stage 2, Facility B holds exactly 112 units; (2) At the end of Stage 3, the total combined inventory across both facilities is 96 units; and (3) The total combined inventory decreases by 40 units from the end of Stage 2 to the end of Stage 3.

Tracking all 4 transition steps systematically across 3 stages yields the exact state vectors: Stage 1

(72, 144)

, Stage 2

(24, 112)

, and Stage 3

(8, 88)

. Comparing these state vectors confirms that Facility B has 112 units at Stage 2 end, total inventory at Stage 3 end is

8 + 88 = 96

units, and the change from Stage 2 end to Stage 3 end is

136 - 96 = 40

units.

Step-by-Step Solution

1

Calculate state transitions for Stage 1 (

k=1

)

A_1 = 72

,

B_1 = 144

Start at

A_0 = 216, B_0 = 160

. Step 1:

A \to 216 - 108 = 108

. Step 2:

B \to 160 - 80 = 80

. Step 3:

A

transfers

\frac{1}{3}(108) = 36

to

B

, so

A \to 108 - 36 = 72

and

B \to 80 + 36 = 116

. Step 4:

B

receives 28, so

B \to 116 + 28 = 144

.

2

Calculate state transitions for Stage 2 (

k=2

)

A_2 = 24

,

B_2 = 112

,

\text{Total}_2 = 136

Start at

A_1 = 72, B_1 = 144

. Step 1:

A \to 72 - 36 = 36

. Step 2:

B \to 144 - 72 = 72

. Step 3:

A

transfers

\frac{1}{3}(36) = 12

to

B

, so

A \to 36 - 12 = 24

and

B \to 72 + 12 = 84

. Step 4:

B

receives 28, so

B \to 84 + 28 = 112

.

3

Calculate state transitions for Stage 3 (

k=3

)

A_3 = 8

,

B_3 = 88

,

\text{Total}_3 = 96

Start at

A_2 = 24, B_2 = 112

. Step 1:

A \to 24 - 12 = 12

. Step 2:

B \to 112 - 56 = 56

. Step 3:

A

transfers

\frac{1}{3}(12) = 4

to

B

, so

A \to 12 - 4 = 8

and

B \to 56 + 4 = 60

. Step 4:

B

receives 28, so

B \to 60 + 28 = 88

.

4

Evaluate the given statements against calculated stage states

Statements A, B, and E are true; C and D are false.

Facility B holds 112 units at Stage 2 end. Total inventory at Stage 3 end is

8 + 88 = 96

units. Total inventory change between Stage 2 and Stage 3 is

136 - 96 = 40

units.

Key Concept

Sequential Multi-Stage State Transitions

Estimated Time:2m 30s

Question 33Question →

A financial compliance department classifies portfolio companies into three regulatory monitoring tiers—Tier Alpha, Tier Beta, and Tier Gamma—based on three financial criteria:

1. Debt-to-Equity ( $D/E$ ) ratio $< 1.5$
2. Net Profit Margin $> 12\%$
3. Audit Rating of 'Unqualified'

The classification rules are defined as follows:
- Tier Alpha: The company satisfies at least two of the three criteria.
- Tier Gamma: The company satisfies fewer than two of the three criteria AND meets at least one of the following risk triggers: an Audit Rating of 'Adverse' OR a $D/E$ ratio $> 2.5$ .
- Tier Beta: Any company that does not meet the criteria for either Tier Alpha or Tier Gamma.

The relevant metrics for four portfolio companies are summarized in the table below:

Company	Debt-to-Equity ( $D/E$ )	Net Profit Margin	Audit Rating
Company W	$1.2$	$15\%$	Adverse
Company X	$2.8$	$10\%$	Unqualified
Company Y	$1.8$	$14\%$	Qualified
Company Z	$1.4$	$11\%$	Adverse

Based on the classification rules, which of the following correctly pairs the regulatory monitoring tiers for Company W and Company Y, respectively?

Company W: Tier Alpha; Company Y: Tier Beta

Company W: Tier Alpha; Company Y: Tier Gamma

Company W: Tier Gamma; Company Y: Tier Beta

Company W: Tier Beta; Company Y: Tier Gamma

Company W: Tier Beta; Company Y: Tier Beta

Show answer & explanation

Answer: Company W: Tier Alpha; Company Y: Tier Beta

Answer

Company W is classified under Tier Alpha and Company Y is classified under Tier Beta.

To classify Company W, we evaluate the three criteria:

D/E = 1.2 < 1.5

(satisfied), Margin

= 15\% > 12\%

(satisfied), and Audit Rating = 'Adverse' (not satisfied). Because Company W satisfies exactly 2 criteria, it meets the Tier Alpha definition of satisfying at least two criteria.

To classify Company Y, we evaluate its criteria:

D/E = 1.8 < 1.5

(not satisfied), Margin

= 14\% > 12\%

(satisfied), and Audit Rating = 'Qualified' (not satisfied). Company Y satisfies 1 criterion (fewer than 2), so it is not Tier Alpha. Next, we test the Tier Gamma conditions: it has fewer than 2 criteria satisfied, but its Audit Rating is not 'Adverse' and its

D/E = 1.8 \le 2.5

. Since neither risk trigger is present, it does not meet the Tier Gamma criteria. By elimination, Company Y must be in Tier Beta.

Step-by-Step Solution

1

Evaluate the 3 core criteria for Company W

Criterion 1 (

D/E < 1.5

):

1.2 < 1.5

(True). Criterion 2 (Margin

> 12\%

):

15\% > 12\%

(True). Criterion 3 (Audit 'Unqualified'): 'Adverse' (False). Total satisfied = 2.

Determining how many criteria are satisfied dictates tier eligibility.

2

Classify Company W based on rules

Company W satisfies 2 of 3 criteria

\ge 2

, so Company W is Tier Alpha.

The Tier Alpha rule requires satisfying at least two of the three criteria.

3

Evaluate the 3 core criteria for Company Y

Criterion 1 (

D/E < 1.5

):

1.8 < 1.5

(False). Criterion 2 (Margin

> 12\%

):

14\% > 12\%

(True). Criterion 3 (Audit 'Unqualified'): 'Qualified' (False). Total satisfied = 1.

Fewer than 2 conditions satisfied rules out Tier Alpha.

4

Check Tier Gamma risk triggers for Company Y

Risk Trigger 1 (Audit 'Adverse'): 'Qualified' (False). Risk Trigger 2 (

D/E > 2.5

):

1.8 > 2.5

(False). Neither risk trigger is met.

Tier Gamma requires satisfying fewer than two conditions AND meeting at least one risk trigger.

5

Classify Company Y based on rules

Company Y is neither Tier Alpha nor Tier Gamma, so it falls into Tier Beta.

Tier Beta contains all remaining companies.

Key Concept

Multi-attribute categorical classification using nested logical conjunctions (AND/OR) and threshold rules.

Estimated Time:2m 30s

Question 34Question →

An international logistics terminal classifies shipments into four distinct processing protocols—Priority Express, Standard Clearance, Enhanced Inspection, and Quarantine Hold—based on three categorical attributes: Destination Type (Inland Hub or Local Metro), Content Hazard Tier (Tier 1: Hazardous, Tier 2: Perishable, or Tier 3: General), and Customs Certification Status (Certified or Non-Certified).

The classification rules are defined as follows:
- Priority Express: Assigned to any shipment that is Certified AND has either an Inland Hub destination OR Tier 3 (General) content.
- Quarantine Hold: Assigned to any shipment that has Tier 1 (Hazardous) content AND is Non-Certified.
- Enhanced Inspection: Assigned to any shipment that is Non-Certified and is NOT assigned to Quarantine Hold.
- Standard Clearance: Assigned to all remaining shipments that do not meet the criteria for any of the above protocols.

Match each shipment profile on the left with its correct processing protocol on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Shipment Alpha matches Priority Express; Shipment Beta matches Quarantine Hold; Shipment Gamma matches Enhanced Inspection; Shipment Delta matches Standard Clearance.

Each shipment profile is mapped by rigorously applying the conditional priority rules step-by-step: Certified shipments meeting specific destination/tier thresholds take Priority Express, Non-Certified hazardous shipments go to Quarantine Hold, other Non-Certified shipments undergo Enhanced Inspection, and remaining Certified shipments default to Standard Clearance.

Step-by-Step Solution

1

Evaluate Shipment Alpha against protocol criteria

Certified = TRUE; (Inland Hub OR Tier 3) = TRUE (Inland Hub). Qualifies for Priority Express.

Priority Express requires Certified status along with either Inland Hub destination or Tier 3 content.

2

Evaluate Shipment Beta against protocol criteria

Non-Certified = TRUE; Tier 1 (Hazardous) = TRUE. Qualifies for Quarantine Hold.

Quarantine Hold requires both Tier 1 content and Non-Certified status.

3

Evaluate Shipment Gamma against protocol criteria

Non-Certified = TRUE; Tier 1 = FALSE (Tier 2). Not Quarantine Hold, so it goes to Enhanced Inspection.

Enhanced Inspection applies to all Non-Certified shipments that do not meet the Quarantine Hold threshold.

4

Evaluate Shipment Delta against protocol criteria

Certified = TRUE; (Inland Hub OR Tier 3) = FALSE (Local Metro & Tier 2). Does not meet Priority Express or Non-Certified protocols; defaults to Standard Clearance.

Shipments not meeting specific priority or inspection rules default to Standard Clearance.

Key Concept

Categorical Classification & Multi-Attribute Boolean Logic Hierarchy

Question 35Question →

A biotechnology research facility classifies five experimental compounds—Candidate V, Candidate W, Candidate X, Candidate Y, and Candidate Z—for clinical trial priority according to a strict multi-attribute precedence rule.

Compounds are evaluated on three metrics:
- Toxicity Index (measured from $1.0$ to $5.0$ ; lower indicates safer compound)
- Receptor Affinity (measured from $1$ to $100$ ; higher indicates stronger binding)
- Bioavailability (measured as a percentage from $0\%$ to $100\%$ ; higher indicates better absorption)

Precedence Sorting Hierarchy:
1. Primary Criterion: Any compound with a Toxicity Index strictly less than $3.0$ takes absolute precedence over any compound with a Toxicity Index of $3.0$ or greater.
2. Secondary Criterion: Within the same toxicity category, compounds are sorted in descending order of Receptor Affinity.
3. Tertiary Criterion: If two compounds within the same toxicity category have identical Receptor Affinity scores, the compound with the higher Bioavailability is ranked higher.

Compound Metrics:

Compound	Toxicity Index	Receptor Affinity	Bioavailability
Candidate V	$2.4$	$85$	$75\%$
Candidate W	$2.8$	$90$	$60\%$
Candidate X	$3.2$	$95$	$80\%$
Candidate Y	$2.4$	$85$	$82\%$
Candidate Z	$3.5$	$90$	$95\%$

Based on the classification and sorting criteria above, arrange the candidates in order from highest priority (Rank 1) to lowest priority (Rank 5).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from highest priority (Rank 1) to lowest priority (Rank 5) is Candidate W, Candidate Y, Candidate V, Candidate X, Candidate Z.

The sequence Candidate W → Candidate Y → Candidate V → Candidate X → Candidate Z correctly applies the three-tiered sorting logic. Primary tiering separates Candidates V, W, and Y (Toxicity < 3.0) from Candidates X and Z (Toxicity ≥ 3.0). Candidate W leads Tier 1 with the highest affinity (90). Candidate Y beats Candidate V for second place due to higher bioavailability (82% vs. 75%). In Tier 2, Candidate X leads Candidate Z due to higher affinity (95 vs. 90).

Step-by-Step Solution

1

Partition the compounds into primary classification tiers based on the Toxicity Index threshold (

3.0

).

Tier 1 (Toxicity < 3.0): Candidate V (

2.4

), Candidate W (

2.8

), Candidate Y (

2.4

). Tier 2 (Toxicity ≥ 3.0): Candidate X (

3.2

), Candidate Z (

3.5

). Tier 1 compounds take absolute precedence over Tier 2 compounds.

The primary criterion establishes strict category partitioning before evaluating secondary attributes.

2

Sort Tier 1 candidates (Candidate V, Candidate W, Candidate Y) using secondary and tertiary criteria.

Candidate W has Receptor Affinity

90

(highest in Tier 1), securing Rank 1. Candidate V and Candidate Y both have Receptor Affinity

85

. Comparing Bioavailability: Candidate Y (

82\%

) > Candidate V (

75\%

). Thus, Rank 2 is Candidate Y and Rank 3 is Candidate V.

Secondary sorting uses descending Receptor Affinity. Equal affinity scores trigger tertiary sorting by descending Bioavailability.

3

Sort Tier 2 candidates (Candidate X, Candidate Z) using secondary criteria.

Candidate X has Receptor Affinity

95

, while Candidate Z has Receptor Affinity

90

. Therefore, Candidate X gets Rank 4 and Candidate Z gets Rank 5.

Within Tier 2, higher Receptor Affinity determines higher rank position.

Key Concept

Categorical hierarchical sorting requires applying criteria in strict sequential order (primary tiering before secondary/tertiary attribute ranking), avoiding premature overall sorting across non-equivalent tiers.

Question 36Question →

A hybrid microgrid facility generates power using Solar Arrays and Wind Turbines to supply a regional grid.

- Solar Arrays generate electricity at a constant rate of $S$ megawatts per hour (MW/h), and $15\%$ of the generated solar energy is lost during battery storage transfer.
- Wind Turbines generate electricity at a constant rate of $W$ megawatts per hour (MW/h), and $10\%$ of the generated wind energy is lost during battery storage transfer.

During a continuous 10-hour operating period, both sources operate simultaneously. The total usable energy delivered to the battery storage after accounting for transfer losses is $435$ megawatt-hours (MWh). Furthermore, the ratio of raw solar energy generated to raw wind energy generated during this 10-hour period is $3 : 2$ .

Based on this information, which of the following correctly identifies the generation rates $S$ and $W$ , in MW/h, for the Solar Arrays and Wind Turbines, respectively?

Solar Rate (

S

) = 30 MW/h; Wind Rate (

W

) = 20 MW/h

Solar Rate (

S

) = 20 MW/h; Wind Rate (

W

) = 30 MW/h

Solar Rate (

S

) = 27 MW/h; Wind Rate (

W

) = 18 MW/h

Solar Rate (

S

) = 300 MW/h; Wind Rate (

W

) = 200 MW/h

Solar Rate (

S

) = 36 MW/h; Wind Rate (

W

) = 24 MW/h

Show answer & explanation

Answer: Solar Rate ( $S$ ) = 30 MW/h; Wind Rate ( $W$ ) = 20 MW/h

Answer

Solar Rate (S) = 30 MW/h; Wind Rate (W) = 20 MW/h

The option specifying Solar Rate (

S

) = 30 MW/h and Wind Rate (

W

) = 20 MW/h is correct because over 10 hours, raw generation totals are

300

MWh for solar and

200

MWh for wind, satisfying the

3 : 2

ratio. Applying the respective

85\%

and

90\%

storage efficiencies gives

300 \times 0.85 = 255

MWh and

200 \times 0.90 = 180

MWh. The combined usable energy is

255 + 180 = 435

MWh, perfectly matching all given constraints.

Step-by-Step Solution

1

Formulate the ratio equation for raw generation rates.

Since raw solar generation is to raw wind generation as

3 : 2

over the same 10-hour duration,

\frac{10S}{10W} = \frac{3}{2} \implies S = 1.5W

.

Establishes the proportional relationship between the solar and wind rates.

2

Express usable energy accounting for efficiency loss.

Solar efficiency is

100\% - 15\% = 85\%

(

0.85

). Wind efficiency is

100\% - 10\% = 90\%

(

0.90

). Total usable energy stored in 10 hours is

10(0.85S) + 10(0.90W) = 8.5S + 9W = 435

.

Translates loss percentages into usable energy multipliers over the 10-hour period.

3

Substitute

S = 1.5W

into the usable energy equation and solve for

W

and

S

.

8.5(1.5W) + 9W = 435 \implies 12.75W + 9W = 435 \implies 21.75W = 435 \implies W = 20

MW/h. Consequently,

S = 1.5(20) = 30

MW/h.

Calculates the exact hourly rates for both energy sources.

Key Concept

Simultaneous Rate and Efficiency Equations in Two-Part Analysis

Estimated Time:2m 0s

Question 37Question →

An urban ecology study across 40 metropolitan districts evaluated the impact of municipal sanitation infrastructure on local avian populations over a five-year period. In districts that replaced traditional open trash receptacles with sealed, animal-proof containment bins, researchers recorded a 40% reduction in local crow populations within 18 months. The researchers hypothesized that restricting access to human food waste directly caused the decline in the local crow population.

Consider the following two logical roles regarding the researchers' causal hypothesis:
- Column 1: A statement that, if true, most strongly SUPPORTS the hypothesis.
- Column 2: A statement that, if true, most strongly WEAKENS the hypothesis by demonstrating a causal direction reversal or an unaddressed alternative cause.

Which of the following joint selections correctly identifies a supporting statement for Column 1 and a weakening statement for Column 2?

Column 1: Adjacent control districts maintaining open receptacles showed stable crow populations, while tagged crows in sealed-bin districts exhibited severe caloric deficits; Column 2: Municipalities installed sealed bins specifically in response to a pre-existing drop in crow numbers caused by an avian pox outbreak.

Column 1: Municipalities installed sealed bins specifically in response to a pre-existing drop in crow numbers caused by an avian pox outbreak; Column 2: Adjacent control districts maintaining open receptacles showed stable crow populations, while tagged crows in sealed-bin districts exhibited severe caloric deficits.

Column 1: Crows in sealed-bin districts shifted their primary foraging grounds to neighboring open-receptacle districts, causing local counts in sealed-bin districts to drop; Column 2: Open trash receptacles attract a wide variety of urban scavengers in addition to crows.

Column 1: Adjacent control districts maintaining open receptacles showed stable crow populations, while tagged crows in sealed-bin districts exhibited severe caloric deficits; Column 2: Sealed bins also reduced the local population of urban rodents that rely on spilled organic waste.

Column 1: The overall mass of organic waste collected in sealed-bin districts increased during the study period; Column 2: Crows are highly intelligent birds capable of learning new foraging behaviors.

Show answer & explanation

Answer: Column 1: Adjacent control districts maintaining open receptacles showed stable crow populations, while tagged crows in sealed-bin districts exhibited severe caloric deficits; Column 2: Municipalities installed sealed bins specifically in response to a pre-existing drop in crow numbers caused by an avian pox outbreak.

Answer

Column 1: Adjacent control districts maintaining open receptacles showed stable crow populations, while tagged crows in sealed-bin districts exhibited severe caloric deficits; Column 2: Municipalities installed sealed bins specifically in response to a pre-existing drop in crow numbers caused by an avian pox outbreak.

The correct selection properly aligns both logical roles. For Column 1, evidence that control districts with open bins maintained stable populations while crows in sealed-bin districts suffered caloric deficits directly supports the claim that food waste restriction caused the population decline. For Column 2, evidence that bin installation occurred in response to a pre-existing decline caused by avian pox undermines the hypothesis by reversing the temporal sequence (showing the population drop came first) and introducing an alternative cause.

Step-by-Step Solution

1

Analyze the causal claim in the stem.

Cause: Restricting access to human food waste via sealed bins. Effect: 40% reduction in local crow population.

To evaluate support and weaken statements, the precise independent cause and dependent effect must be clearly isolated.

2

Identify the criteria for Column 1 (Support).

Column 1 requires evidence showing that the absence of the cause (control districts with open bins) prevents the effect, and that the proposed mechanism (caloric restriction) is active in the treatment group.

Demonstrating control stability alongside evidence of the hypothesized mechanism strongly reinforces a causal link.

3

Identify the criteria for Column 2 (Weaken via Causal Reversal / Alternative Cause).

Column 2 requires a statement showing either that the decline in crow population occurred prior to bin installation (causal direction reversal) or was driven by an unmeasured confounder (avian pox outbreak).

If the decline in crows caused the installation of bins rather than vice versa, the researchers' causal hypothesis is completely invalidated.

4

Evaluate the joint options to find the correct pairing.

The option placing the control group / caloric deficit evidence in Column 1 and the pre-existing disease / reversed timeline evidence in Column 2 is the unique correct pairing.

Only this option places both logically complementary statements in their correct designated columns without reversal or irrelevance.

Key Concept

Two-Part Analysis Causal Argument Evaluation & Reversal Identification

Question 38Question →

A small workshop produces two items, Item X and Item Y. Each unit of Item X requires $2$ hours of labor and yields a profit of $\$30$ . Each unit of Item Y requires $4$ hours of labor and yields a profit of $\$50$ . The workshop has a maximum of $16$ labor hours available daily and must produce a combined total of at least $5$ units per day. Assuming only whole units can be produced, which combination of (Item X, Item Y) maximizes total daily profit while satisfying all given constraints?

6 units of Item X and 1 unit of Item Y

4 units of Item X and 2 units of Item Y

2 units of Item X and 3 units of Item Y

0 units of Item X and 4 units of Item Y

1 unit of Item X and 4 units of Item Y

Show answer & explanation

Answer: 6 units of Item X and 1 unit of Item Y

Answer

6 units of Item X and 1 unit of Item Y

The pair consisting of 6 units of Item X and 1 unit of Item Y uses

2(6) + 4(1) = 16

labor hours (satisfying the maximum

16

-hour limit) and produces

6 + 1 = 7

units (satisfying the minimum

5

-unit limit). It yields a profit of

30(6) + 50(1) = \$230

, which is higher than all other feasible options.

Step-by-Step Solution

1

Formulate the constraint inequalities and profit objective function.

Labor constraint:

2X + 4Y \le 16

. Minimum unit constraint:

X + Y \ge 5

, where

X, Y \ge 0

are integers. Profit function:

P = 30X + 50Y

.

Setting up the mathematical relations allows systematic evaluation of candidate pairs.

2

Evaluate feasible integer combinations

(X, Y)

against both constraints.

Combination

(6, 1)

: Labor

= 16 \le 16

, Units

= 7 \ge 5

. Profit

= 30(6) + 50(1) = \$230

.
Combination

(4, 2)

: Labor

= 16 \le 16

, Units

= 6 \ge 5

. Profit

= 30(4) + 50(2) = \$220

.
Combination

(2, 3)

: Labor

= 16 \le 16

, Units

= 5 \ge 5

. Profit

= 30(2) + 50(3) = \$210

.

Evaluating all valid pairs determines which produces the peak profit value.

3

Compare profit outcomes across feasible pairs.

The combination of

6

units of Item X and

1

unit of Item Y achieves the highest valid profit of

\$230

.

The option

(6, 1)

optimizes the objective function under all bounded constraints.

Key Concept

Optimization under linear integer and inequality bounds

Question 39Question →

A renewable energy facility operates two models of solar panels: Type P and Type Q. A total of $100$ Type P panels and $200$ Type Q panels generate a combined total of $1,400$ kilowatt-hours (kWh) of electricity per day. If the ratio of the daily output of a single Type P panel to that of a single Type Q panel is $3 : 2$ , which of the following represents the daily energy output per panel, in kWh, for Type P and Type Q, respectively?

6\text{ kWh/day}

for Type P and

4\text{ kWh/day}

for Type Q

4\text{ kWh/day}

for Type P and

6\text{ kWh/day}

for Type Q

3\text{ kWh/day}

for Type P and

2\text{ kWh/day}

for Type Q

9\text{ kWh/day}

for Type P and

6\text{ kWh/day}

for Type Q

7\text{ kWh/day}

for Type P and

3.5\text{ kWh/day}

for Type Q

Show answer & explanation

Answer: $6\text{ kWh/day}$ for Type P and $4\text{ kWh/day}$ for Type Q

Answer

6\text{ kWh/day}

for Type P and

4\text{ kWh/day}

for Type Q

Expressing the rates as

r_P = 3k

and

r_Q = 2k

allows us to model total production as

100(3k) + 200(2k) = 700k = 1,400

. Solving gives

k = 2

, yielding

r_P = 6\text{ kWh/day}

and

r_Q = 4\text{ kWh/day}

, which satisfies both the

3:2

ratio and the total output constraint.

Step-by-Step Solution

1

Set up equations based on the ratio constraint.

Let

r_P = 3k

and

r_Q = 2k

for some positive constant

k

.

The given ratio of outputs

r_P : r_Q

is

3 : 2

.

2

Set up the total daily output equation.

100(3k) + 200(2k) = 1,400

There are

100

Type P panels and

200

Type Q panels generating

1,400\text{ kWh}

in total.

3

Solve for the multiplier

k

.

300k + 400k = 1,400 \implies 700k = 1,400 \implies k = 2

Combine like terms and divide by

700

.

4

Calculate the individual panel outputs.

Type P rate

r_P = 3(2) = 6\text{ kWh/day}

; Type Q rate

r_Q = 2(2) = 4\text{ kWh/day}

.

Substitute

k = 2

back into the expressions for

r_P

and

r_Q

.

Key Concept

Work Rates and Ratios

Estimated Time:1m 0s

Question 40Question →

A specialty coffee roaster produces two custom coffee blends, Blend Alpha and Blend Beta, using Arabica and Robusta beans. Each bag of Blend Alpha requires $300\text{ grams}$ of Arabica beans and $100\text{ grams}$ of Robusta beans. Each bag of Blend Beta requires $200\text{ grams}$ of Arabica beans and $200\text{ grams}$ of Robusta beans. On a given day, the roaster used a total of $14,000\text{ grams}$ of Arabica beans and $8,000\text{ grams}$ of Robusta beans to produce bags of these two blends, using all available inventory.

In the table below, select the number of bags of Blend Alpha produced and the number of bags of Blend Beta produced that are consistent with this information.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Bags of Blend Alpha produced = 30; Bags of Blend Beta produced = 25

By representing the number of bags of Blend Alpha as

x

and Blend Beta as

y

, the constraints form the system

3x + 2y = 140

and

x + 2y = 80

. Subtracting the equations yields

2x = 60

, so

x = 30

. Substituting

x = 30

gives

y = 25

. Thus, Blend Alpha produced is 30 and Blend Beta produced is 25.

Step-by-Step Solution

1

Define variables for the unknowns.

Let

x

be the number of bags of Blend Alpha produced, and

y

be the number of bags of Blend Beta produced.

Setting up explicit variables allows formulation of linear algebraic equations.

2

Set up equations based on total bean usage.

For Arabica beans:

300x + 200y = 14,000

, which simplifies to

3x + 2y = 140

.
For Robusta beans:

100x + 200y = 8,000

, which simplifies to

x + 2y = 80

.

Translating the physical constraints into two linear equations in two variables.

3

Solve the system by subtracting the second equation from the first.

(3x + 2y) - (x + 2y) = 140 - 80 \implies 2x = 60 \implies x = 30

.

Subtracting eliminates

y

directly because both equations have identical

2y

terms.

4

Substitute

x = 30

into the simplified Robusta equation to find

y

.

30 + 2y = 80 \implies 2y = 50 \implies y = 25

.

Determines the remaining unknown variable.

Key Concept

Simultaneous Linear Equations in Two Variables

Two-Part Analysis

Answer

Step-by-Step Solution

Key Concept

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