All practice questions

Question 361Question →

A remote Arctic weather station relies on an automated hybrid power system combining a wind turbine and a diesel backup generator. During a severe blizzard, heavy icing freezes the turbine blades, immediately halting all wind-generated current to the primary battery bank. Telemetry indicates that the battery bank is discharging rapidly to sustain critical heating circuits. System protocol dictates that once battery reserves fall below 30%, the automatic transfer switch engages the diesel pre-heater for five minutes before cranking the engine starter. However, if the ambient intake temperature remains below -40°F, fuel line viscosity sensors will automatically trigger an auxiliary line-heating cycle, delaying the engine crank until fuel flows freely. Station sensors currently report an ambient temperature of -48°F as battery reserves decline from 31% toward 28%.

Based on the conditions and procedural rules described in the passage, arrange the projected future events in the correct chronological and logical order in which they will occur.

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Answer

The correct sequence begins with the battery level dropping below 30% to trigger the transfer switch, followed by the activation of the five-minute pre-heater, then the engagement of the auxiliary fuel line heater due to the sub -40°F temperature, and finally the cranking of the diesel engine starter.

The passage outlines a clear causal chain governed by numerical thresholds. First, the declining battery reserve must fall below 30% to trip the automatic transfer switch. Second, this switch initiates the standard five-minute diesel pre-heater. Third, because the ambient temperature of -48°F is colder than the -40°F threshold, the auxiliary fuel line heating cycle must be triggered. Finally, once proper fuel viscosity is ensured, the starter motor cranks the engine.

Step-by-Step Solution

1

Identify the immediate triggering threshold from the current system state.

The battery level is decreasing from 31% toward 28%, meaning it must first cross below the 30% mark to trigger automated response protocols.

System logic requires battery reserves under 30% before the automatic transfer switch can engage.

2

Determine the direct consequence of the transfer switch activation.

The transfer switch initiates the five-minute diesel pre-heating cycle.

The passage establishes that transfer switch engagement activates the pre-heater prior to engine cranking.

3

Evaluate conditional environmental overrides on the starting sequence.

Because ambient temperature is -48°F (colder than the -40°F limit), viscosity sensors trigger an auxiliary line-heating cycle that delays engine cranking.

The text specifies that sub -40°F conditions mandate auxiliary heating before allowing the starter motor to turn.

4

Establish the concluding milestone once all procedural conditions are resolved.

The starter motor cranks the diesel engine to restore facility power.

Engine cranking is the culminating event that only occurs after fuel viscosity and pre-heating criteria are fulfilled.

Key Concept

Predicting multi-step future outcomes by synthesizing explicit procedural thresholds and conditional environmental overrides.

Estimated Time:1m 30s

Question 362Question →

Read the following technical passage about engine maintenance:

In turbocharged engines, improper shutdown procedures can cause severe damage through a process known as oil coking. When an engine is turned off immediately following high-load operation, the active circulation of oil and coolant ceases abruptly. However, immense residual heat from the exhaust manifold continues to soak into the turbocharger's center housing. Because the stationary oil inside the bearings is exposed to this extreme temperature, it undergoes thermal breakdown and bakes into abrasive carbon deposits. Consequently, these hardened deposits restrict internal oil passages, leading to bearing starvation and mechanical seizure.

Based on the causal relationships described in the passage, arrange the following events in the order of cause to final effect.

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Answer

The correct sequence from primary cause to ultimate effect is: cessation of oil and coolant circulation, followed by residual heat soaking into the center housing, leading to thermal breakdown of stationary oil into carbon deposits, and culminating in restricted oil passages that cause bearing starvation and seizure.

The correct sequence mirrors the explicit cause-and-effect progression described in the text: stopping fluid circulation initiates heat absorption into the housing, which causes stagnant oil to bake into carbon deposits, which in turn obstructs passages and causes bearing starvation.

Step-by-Step Solution

1

Identify the primary initiating action stated in the passage.

The text states that turning off the engine immediately causes active fluid circulation to cease abruptly.

This establishes the initial condition that permits subsequent thermal damage.

2

Trace the immediate physical consequence of stopping fluid circulation while the turbocharger is hot.

Residual heat from the exhaust manifold soaks into the center housing.

Without active coolant flow to carry heat away, the housing absorbs excess thermal energy.

3

Extract the direct effect of heat exposure on the trapped lubricant.

Stationary oil breaks down thermally into hard, abrasive carbon deposits.

The passage explicitly links the thermal breakdown of stagnant oil to extreme temperature exposure.

4

Determine the final consequence resulting from the accumulated deposits.

Hardened deposits block oil passages, causing bearing starvation and seizure.

The passage signals the end result with 'Consequently,' linking carbon build-up to final component failure.

Key Concept

Extracting explicitly stated multi-step causal chains and sequence of mechanisms from technical text.

Estimated Time:1m 0s

Question 363Question →

A municipal water treatment facility utilizes a multi-stage emergency filtration protocol during heavy storm runoff. When incoming turbidity sensors detect sediment levels exceeding safe operating limits, the intake valve automatically restricts raw water volume to prevent filter clogging. Once the intake is restricted, chemical coagulants are injected at an elevated rate to clump the remaining suspended particles. After coagulation occurs, the water passes into the settling basins where the heavy sediment drops to the bottom before the water is directed to the secondary ultraviolet disinfection tanks.

Based on the passage, in what chronological order will the system and water treatment processes respond to a sudden surge in sediment levels?

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Answer

The correct sequence begins with the intake valve restricting raw water inflow, followed by the injection of chemical coagulants, then sediment settling in the basins, and finally routing the water to the secondary ultraviolet disinfection tanks.

The passage outlines a direct causal and chronological progression: detection leads immediately to intake restriction, which is followed by coagulant injection, which then allows sediment to settle in the basins, and finally leads to routing the water into the secondary ultraviolet disinfection tanks.

Step-by-Step Solution

1

Identify the initial trigger and immediate physical response.

Excessive turbidity causes the intake valve to restrict incoming water volume first.

The passage establishes that intake restriction is the immediate automatic reaction to prevent filter clogging.

2

Trace the subsequent chemical treatment action.

Chemical coagulants are injected at an elevated rate.

The text states that coagulant injection occurs 'Once the intake is restricted'.

3

Determine the physical separation stage that follows chemical clumping.

Sediment settles to the bottom of the settling basins.

The passage indicates that water moves to the settling basins 'After coagulation occurs'.

4

Identify the final destination of the clarified water.

Water is directed to the secondary ultraviolet disinfection tanks.

The passage explicitly notes that settling occurs before the water reaches the secondary disinfection tanks.

Key Concept

Predicting the chronological sequence of future events and operational outcomes based on stated causal relationships and sequential transition markers in a text.

Estimated Time:45s

Question 364Question →

Read the following technical passage:

In industrial fluid distribution networks, a sudden closure of a downstream valve causes a rapid deceleration of the moving liquid column. Because liquid is nearly incompressible, this abrupt momentum stoppage generates a localized high-pressure shock wave upstream of the valve. As this pressure surge travels backward through the piping system, it exerts severe mechanical stress against internal pipe walls and fittings. Consequently, the excessive radial strain leads to joint rupture and catastrophic structural failure of the conduit.

Based on the passage, arrange the events in the exact causal sequence leading to conduit failure, from initial cause to final effect.

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Answer

The correct causal sequence begins with the sudden closure of a downstream valve, followed by the generation of an upstream high-pressure shock wave, leading to severe mechanical stress on pipe walls, and culminating in joint rupture and conduit failure.

The correct arrangement directly mirrors the chronological and causal progression stated in the passage: downstream valve closure initiates fluid deceleration, the stoppage creates an upstream high-pressure shock wave, the backward-traveling wave exerts mechanical stress on the pipe walls, and the resulting strain causes pipe joints to rupture.

Step-by-Step Solution

1

Identify the primary initiating cause from the text.

The passage identifies 'a sudden closure of a downstream valve' as the starting action causing deceleration.

Every subsequent effect in the chain stems from this initial physical disruption.

2

Trace the immediate causal link indicated by causal signal words.

Because the liquid cannot compress, momentum stoppage creates a high-pressure shock wave upstream.

The text explicitly links the stoppage directly to shock wave formation using 'Because'.

3

Determine the intermediate mechanical effect of the shock wave.

As the surge travels backward, it places severe mechanical stress and radial strain on internal walls.

The passage details that backward propagation directly stresses internal conduit surfaces.

4

Locate the ultimate outcome signaled by concluding conjunctions.

Excessive radial strain leads directly to joint rupture and catastrophic structural failure.

The word 'Consequently' marks joint rupture and structural failure as the end result of the radial strain.

Key Concept

Extracting explicitly stated multi-step cause-and-effect sequences from technical text using transitional markers.

Question 365Question →

Read the following passage:

Ultrasonic phased array testing has become the primary non-destructive evaluation technique for detecting subsurface flaws in modern carbon-fiber aircraft fuselages. By directing high-frequency acoustic waves into composite laminates, the equipment captures internal reflection patterns that reveal structural anomalies. Specifically, internal delaminations between laminate plies create distinct acoustic impedance boundaries, which attenuate wave penetration and produce early echo returns. Technicians analyze these anomalous echo signals on calibrated A-scan and B-scan displays to map the exact depth and lateral boundary of the ply separation.

Arrange the following statements in hierarchical order, from the overarching main idea of the passage (broadest claim) to the most specific subordinate detail.

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Answer

The correct hierarchical sequence from broadest main idea to most specific subordinate detail is: 1) Ultrasonic testing identifies composite aircraft flaws, 2) Acoustic waves generate reflection patterns revealing anomalies, 3) Inter-ply delaminations cause acoustic impedance boundaries and early echoes, and 4) Technicians use A-scan and B-scan displays to measure flaw depth and boundaries.

The correct order follows a clear deductive hierarchy: the statement introducing ultrasonic phased array testing as the primary evaluation method establishes the central thesis; the general acoustic reflection mechanism provides broad supporting evidence; the formation of impedance boundaries at delaminations narrows the focus to a specific flaw type; and the use of A-scan and B-scan displays represents the most localized procedural detail.

Step-by-Step Solution

1

Identify the overarching main idea that covers the entire passage scope.

The statement declaring ultrasonic phased array testing as the primary non-destructive evaluation technique for aircraft composite inspection is the central thesis.

The main idea must unify all subsequent sentences under a single governing claim.

2

Determine the major conceptual mechanism that supports the main idea.

The general explanation of how transmitting acoustic waves into laminates produces diagnostic reflection patterns acts as the primary supporting principle.

This concept explains the general working mechanism before examining specific flaw types or readout methods.

3

Isolate the intermediate detail describing a specific defect mechanism.

The description of how delamination between plies creates an acoustic impedance boundary causing early echoes is identified.

This narrows the focus from general acoustic reflection down to the behavior of a specific flaw condition.

4

Locate the narrowest operational detail.

The procedural interpretation of A-scan and B-scan display data to record exact dimensions and depths is ranked as the most specific subordinate detail.

This point focuses on granular instrumentation readouts and precise data mapping.

Key Concept

Distinguishing the central thesis from major supporting concepts and granular operational details based on scope and structural hierarchy.

Estimated Time:1m 15s

Question 366Question →

Arrange the following calculation steps in the correct order to evaluate the mathematical expression according to the order of operations:

15 + (6 - 2) \times 3^2

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Answer

The correct sequence is to first subtract 2 from 6 (resolving the parentheses), then square the number 3 (evaluating the exponent), multiply those two results together, and finally add 15.

The correct order strictly follows the PEMDAS sequence: Parentheses (

6 - 2

) must be resolved first, followed by Exponents (

3^2

). After those are evaluated, Multiplication (

4 \times 9

) is performed. Finally, Addition (

15 + 36

) is completed.

Step-by-Step Solution

1

Evaluate the operation inside the parentheses.

15 + (4) \times 3^2

Parentheses (P) have the highest priority in the order of operations.

2

Evaluate the exponent.

15 + 4 \times 9

Exponents (E) are evaluated after parentheses but before multiplication.

3

Perform the multiplication.

15 + 36

Multiplication (M) takes precedence over addition.

4

Perform the final addition.

51

Addition (A) is the final operation left in this expression.

Key Concept

Order of Operations

Question 367Question →

An auto detailing shop keeps different formulations of car wash soap in various containers. To properly inventory their stock, the manager asks a technician to arrange $4$ containers based on their capacity.

Arrange the following containers in order from the smallest volume to the largest volume.

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Answer

The correct sequence from smallest to largest is:

5

quarts,

11

pints,

1.5

gallons, and

200

fluid ounces.

By converting every measurement to fluid ounces, we find the exact comparative sizes:

5

quarts equals

160

fluid ounces,

11

pints equals

176

fluid ounces,

1.5

gallons equals

192

fluid ounces, and the final container is exactly

200

fluid ounces. Arranging these calculated amounts from smallest to largest gives the correct sequence.

Step-by-Step Solution

1

Identify the conversion factors needed to change all units to the smallest unit given, which is fluid ounces.

1 \text{ gallon} = 128 \text{ fluid ounces}

,

1 \text{ quart} = 32 \text{ fluid ounces}

, and

1 \text{ pint} = 16 \text{ fluid ounces}

.

To accurately compare different volumes, they must all be converted to a single common unit.

2

Convert the volume of the quarts container into fluid ounces.

5 \text{ quarts} \times 32 \text{ fluid ounces/quart} = 160 \text{ fluid ounces}

.

Standardizing the unit for accurate comparison.

3

Convert the volume of the pints container into fluid ounces.

11 \text{ pints} \times 16 \text{ fluid ounces/pint} = 176 \text{ fluid ounces}

.

Standardizing the unit for accurate comparison.

4

Convert the volume of the gallons container into fluid ounces.

1.5 \text{ gallons} \times 128 \text{ fluid ounces/gallon} = 192 \text{ fluid ounces}

.

Standardizing the unit for accurate comparison.

5

Compare all four quantities in fluid ounces and order them from smallest to largest.

160 \text{ fl oz} < 176 \text{ fl oz} < 192 \text{ fl oz} < 200 \text{ fl oz}

.

Sorting the standard numerical values provides the final, correct arrangement.

Key Concept

Converting various US customary volume measurements into a common unit to compare and order capacities.

Estimated Time:1m 30s

Question 368Question →

Arrange the following mathematical operations in the exact sequence they must be performed to correctly evaluate the expression below:

40 - [2 \times (3 + 4)^2] \div 7

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Answer

The correct sequence is: Add the numbers inside the innermost parentheses, apply the exponent of 2, multiply the squared result by 2, divide the value of the brackets by 7, and finally subtract the quotient from 40.

Evaluating the expression following the Order of Operations (PEMDAS) requires these steps in sequence:
1) Add (3 + 4) inside the parentheses.
2) Square that sum.
3) Multiply the squared value by 2 to clear the brackets.
4) Divide that result by 7.
5) Subtract the quotient from 40.

Step-by-Step Solution

1

Identify the innermost grouping symbols (parentheses) and perform the addition.

40 - [2 \times (7)^2] \div 7

Parentheses (P) are evaluated first in the order of operations.

2

Evaluate the exponent applied to the innermost group.

40 - [2 \times 49] \div 7

Exponents (E) are evaluated immediately after parentheses.

3

Complete the operations inside the brackets by performing multiplication.

40 - [98] \div 7

The brackets act as secondary grouping symbols and must be fully simplified to a single value.

4

Perform the division located outside the brackets.

40 - 14

Between division and subtraction, division (D) always takes precedence.

5

Perform the final subtraction.

26

Subtraction (S) is evaluated last.

Key Concept

Order of Operations (PEMDAS)

Question 369Question →

Read the following passage:

Modern gas turbine engines rely on ceramic thermal barrier coatings to protect high-pressure turbine blades from extreme combustion temperatures. By depositing a micro-porous ceramic layer over a metallic bond coat, these coatings significantly reduce heat transfer to the underlying superalloy. For instance, a 200-micrometer ceramic layer can lower the metal substrate temperature by over 100°C. This thermal protection allows engines to operate at combustion temperatures exceeding the melting point of the blade metal, directly improving thermodynamic efficiency and reducing overall fuel consumption.

Based on the passage, arrange the following statements in hierarchical order from the primary main idea (broadest claim) to the most specific supporting detail.

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Answer

The correct hierarchy places the overarching claim regarding turbine efficiency and thermal protection first, followed by the general insulating mechanism, then the thermal gradient effect, and concludes with the specific 200-micrometer numerical measurement.

Hierarchical comprehension requires organizing text elements from macro-level synthesis to micro-level evidence. The statement summarizing turbine efficiency and thermal limits represents the central thesis because all other sentences exist to explain or prove it. The description of applying ceramic layers over a bond coat provides the primary supporting method. The creation of a thermal gradient is an intermediate functional outcome, and the exact 200-micrometer temperature reduction serves as the narrowest quantitative supporting detail.

Step-by-Step Solution

1

Identify the primary thesis of the passage

The statement expressing how thermal barrier coatings allow engines to operate at higher efficiency and withstand extreme combustion temperatures encompasses the entire passage.

The main idea must cover the overall purpose and outcome described across the whole text.

2

Identify the primary supporting method or mechanism

The statement describing the deposition of a micro-porous ceramic layer over a bond coat directly supports the main idea by explaining how the barrier is constructed.

Major supporting ideas explain the general method used to achieve the central thesis.

3

Identify the intermediate physical detail

The statement explaining the thermal gradient between the combustion gas and the metal substrate is a narrower elaboration of the insulating mechanism.

Subordinate points clarify the physical process resulting from the primary mechanism.

4

Identify the narrowest empirical evidence

The statement citing the 200-micrometer layer reducing temperature by over 100°C is a single quantitative example.

Specific numerical measurements serve as narrow evidentiary support rather than broad conceptual ideas.

Key Concept

Distinguishing hierarchical levels of text structure by separating overarching claims, functional mechanisms, and narrow evidentiary details.

Estimated Time:1m 15s

Question 370Question →

Determine the correct chronological sequence of steps required to evaluate the following mathematical expression according to the standard order of operations:

\frac{ -3 \times \left[ 4 - \left( 5 - 2^3 \right) \right] }{ -7 + 4 }

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Answer

The correct sequence begins with the innermost exponent, resolves the nested grouping symbols outward, simplifies the numerator and denominator independently, and finishes with division.

The correct evaluation sequence strictly follows the order of operations (PEMDAS). It begins by isolating the innermost grouping symbols and addressing the exponent. Once the innermost parentheses are reduced to a single value, the surrounding brackets are resolved, carefully tracking negative signs. The large fraction bar acts as an invisible grouping symbol, requiring the entire numerator and denominator to be independently simplified before performing the final division.

Step-by-Step Solution

1

Identify innermost grouping symbols and evaluate exponents.

The expression

(5 - 2^3)

becomes

(5 - 8)

.

Parentheses and Exponents have the highest priority in the order of operations.

2

Complete operations inside the innermost parentheses.

The expression

(5 - 8)

evaluates to

-3

.

The innermost group must be resolved to a single number before moving outward to the next level of grouping.

3

Evaluate operations inside the outer square brackets.

The expression

[4 - (-3)]

becomes

[4 + 3]

, which simplifies to

7

.

Square brackets function as the next level of grouping and must be completely simplified before multiplying.

4

Simplify the remainder of the numerator and the denominator.

The numerator becomes

-3 \times 7 = -21

. The denominator becomes

-7 + 4 = -3

.

A fraction bar acts as a grouping symbol, meaning the top and bottom expressions must be fully evaluated independently before division occurs.

5

Perform the final division.

-21 \div -3 = 7

.

Division is the final operation combining the fully simplified numerator and denominator.

Key Concept

Order of Operations (PEMDAS) with nested grouping symbols, negative numbers, and fraction bars.

Question 371Question →

Arrange the following calculation steps in the exact chronological sequence they must be performed relative to each other when evaluating the mathematical expression below using the standard order of operations:

\frac{-2[(-3)^2 - 4(5 - 7)^3]}{|-9| - (8 - 1)} - 9

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Answer

The correct sequence starts with the innermost exponents and grouping symbols, resolves the bracket, evaluates the full numerator and denominator, divides, and finally subtracts 9.

The correct order strictly follows the standard order of operations (PEMDAS). Operations within grouping symbols (innermost parentheses first, then brackets) must be evaluated before external operations. Exponents are applied immediately after their base is simplified. The fraction bar acts as a major grouping symbol, requiring the numerator and denominator to be fully evaluated before dividing. The final subtraction occurs last.

Step-by-Step Solution

1

Simplify the innermost parentheses and absolute value

$5 - 7 = -2$ and $8 - 1 = 7$ and $|-9| = 9$

Parentheses and absolute value brackets (P in PEMDAS) are evaluated first.

2

Evaluate the exponents

$(-3)^2 = 9$ and $(-2)^3 = -8$

Exponents (E in PEMDAS) follow parentheses.

3

Perform multiplication inside the square brackets

4 \times (-8) = -32

Multiplication (M in PEMDAS) occurs before subtraction inside the grouping symbols.

4

Perform subtraction inside the square brackets

$9 - (-32) = 41$

Subtraction is the final operation needed to simplify the inside of the square bracket.

5

Multiply the bracket by the external coefficient

-2 \times 41 = -82

This completes the evaluation of the numerator.

6

Simplify the denominator

$9 - 7 = 2$

The denominator must be evaluated to a single number before division can occur.

7

Divide the numerator by the denominator

-82 \div 2 = -41

The fraction bar represents division of the evaluated top and bottom parts.

8

Perform the final subtraction

$-41 - 9 = -50$

Subtraction outside the fraction is the last operational step.

Key Concept

Order of Operations with Nested Grouping Symbols and Fractions

Question 372Question →

Passage: A remote mountain telecommunications relay station relies on a primary solar array and a backup diesel generator equipped with an automated power management unit. When solar output drops below critical operating thresholds for more than thirty consecutive minutes, the unit is programmed to initiate a tiered conservation and failover sequence. First, auxiliary climate control and non-essential telemetry transmitters are powered down to preserve stored battery reserves. Next, if solar irradiance does not recover within ten minutes of initial load shedding, the system triggers the generator's pre-heat cycle and starter motor. Once the diesel generator stabilizes at its rated rotational speed, the automated transfer switch disconnects the depleted battery bank and routes emergency electrical power directly to the primary communications transceivers.

Based on the passage, arrange the projected automated sequence of events in chronological order from first to last following a prolonged drop in solar output.

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Answer

The correct chronological order of projected events is: (1) Auxiliary climate control and non-essential telemetry systems are powered down, (2) The power management unit engages the diesel generator's pre-heat cycle and starter motor, (3) The backup diesel engine reaches and stabilizes at its rated rotational operating speed, and (4) The automated transfer switch routes generator power directly to the primary transceivers.

The sequence follows the explicit tiered timeline outlined in the passage: non-essential loads are shed first to conserve remaining battery power; after ten minutes without solar recovery, the engine begins cranking; once the engine stabilizes at rated speed, the transfer switch completes the process by supplying generator power to the primary transceivers.

Step-by-Step Solution

1

Identify the initial conservation trigger and response in the text.

The text states that auxiliary climate controls and non-essential telemetry are deactivated first upon reaching the thirty-minute threshold.

This establishes the first stage of the automated failover sequence.

2

Determine the conditional threshold for generator activation.

If solar irradiance remains low ten minutes after shedding loads, the unit triggers the generator's pre-heat cycle and starter motor.

Starting the generator represents the second sequential action.

3

Analyze the mechanical requirement before electrical power transfer.

The engine must reach and stabilize at its rated rotational speed.

Physical stabilization must occur prior to engaging the electrical transfer switch.

4

Determine the final failover state.

The automated transfer switch disconnects the battery bank and routes power to the primary transceivers.

This completes the failover sequence based on the stated protocol.

Key Concept

Predicting Sequential Future Outcomes from Procedural Text

Estimated Time:1m 0s

Question 373Question →

Evaluate each of the following arithmetic expressions. Then, arrange the expressions in order from the smallest resulting value to the largest resulting value.

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Answer

The correct sequence from smallest to largest is:

10 - 2 \times (4 - 1)^2

,

10 - (2 \times 4) - 1^2

,

10 - 2 \times 4 - (-1)^3

, and

(10 - 2) \times 4 - 1^2

.

Evaluating each expression using the correct order of operations yields -8, 31, 1, and 3 respectively. Arranging these integer values from smallest to largest dictates the correct order.

Step-by-Step Solution

1

Evaluate the first expression:

10 - 2 \times (4 - 1)^2

-8

Applying PEMDAS: solve inside parentheses

(3)

, apply the exponent

(3^2 = 9)

, multiply

(2 \times 9 = 18)

, and subtract

(10 - 18 = -8)

.

2

Evaluate the second expression:

(10 - 2) \times 4 - 1^2

31

Applying PEMDAS: solve inside parentheses

(8)

, apply the exponent

(1^2 = 1)

, multiply

(8 \times 4 = 32)

, and subtract

(32 - 1 = 31)

.

3

Evaluate the third expression:

10 - (2 \times 4) - 1^2

1

Applying PEMDAS: multiply inside parentheses

(8)

, apply the exponent

(1^2 = 1)

, and subtract from left to right

(10 - 8 = 2; 2 - 1 = 1)

.

4

Evaluate the fourth expression:

10 - 2 \times 4 - (-1)^3

3

Applying PEMDAS: apply the exponent

((-1)^3 = -1)

, multiply

(2 \times 4 = 8)

, and subtract from left to right

(10 - 8 = 2; 2 - (-1) = 3)

.

5

Arrange the evaluated numbers from smallest to largest

-8 < 1 < 3 < 31

Comparing the integer values yields the final logical order of the respective expressions.

Key Concept

Evaluating arithmetic expressions sequentially by applying the correct order of operations (PEMDAS) and then sorting the numerical values.

Question 374Question →

Arrange the mathematical steps in the correct sequence to evaluate the following expression according to the standard order of operations (PEMDAS):

12 - 4 \times \left| -5 + 2^3 \right|

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Answer

The correct sequence starts with evaluating the exponent inside the absolute value, followed by addition, resolving the absolute value, multiplication, and finally subtraction.

Following the standard order of operations (PEMDAS), operations inside grouping symbols are prioritized. The absolute value bars act as a grouping symbol. Inside, the exponent is evaluated first, then the addition. The absolute value is then applied to the result. Finally, outside the grouping symbol, multiplication takes precedence over subtraction.

Step-by-Step Solution

1

Evaluate the exponent inside the absolute value.

12 - 4 \times \left| -5 + 8 \right|

Exponents within grouping symbols (absolute value) must be evaluated first.

2

Perform the addition inside the absolute value.

12 - 4 \times \left| 3 \right|

Complete the operations inside the grouping symbols.

3

Resolve the absolute value.

12 - 4 \times 3

The absolute value of

3

is

3

.

4

Perform the multiplication.

12 - 12

Multiplication precedes subtraction.

5

Perform the subtraction.

0

Subtraction is the final remaining operation.

Key Concept

Order of Operations (PEMDAS)

Question 375Question →

A military supply officer is inventorying emergency drinking water supplies stored in four different transport containers. To properly balance the cargo load for an upcoming transport flight, the officer needs to evaluate their relative capacities. Arrange the following containers from least to greatest volume of water.

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Answer

The correct sequence from least to greatest volume is Container C, Container D, Container A, and Container B.

To accurately compare the volumes, each measurement must be standardized to a common unit, such as liters. Container C converts to

450

liters (

450,000 \div 1,000

). Container D converts to

465

liters (

46,500 \div 100

). Container A converts to

480

liters (

0.48 \times 1,000

). Container B is already given as

490

liters. Arranging these converted values from least to greatest yields Container C, Container D, Container A, and Container B.

Step-by-Step Solution

1

Convert the volume of Container A to liters.

0.48 \text{ kL} \times 1,000 = 480 \text{ L}

The prefix kilo- means 1,000. Multiplying kiloliters by 1,000 converts the measurement to the base unit of liters.

2

Convert the volume of Container C to liters.

450,000 \text{ mL} \div 1,000 = 450 \text{ L}

The prefix milli- means one-thousandth. Dividing milliliters by 1,000 converts the measurement to liters.

3

Convert the volume of Container D to liters.

46,500 \text{ cL} \div 100 = 465 \text{ L}

The prefix centi- means one-hundredth. Dividing centiliters by 100 converts the measurement to liters.

4

Compare the standardized values to establish the final order.

450 \text{ L} < 465 \text{ L} < 480 \text{ L} < 490 \text{ L}

Ordering the converted numerical values directly dictates the arrangement from least to greatest capacity.

Key Concept

Converting metric capacity units (kiloliters, liters, centiliters, milliliters) to a common base unit to compare magnitudes.

Question 376Question →

Arrange the following mathematical values related to factors, multiples, and primes in ascending order (from least to greatest).

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Answer

The correct ascending sequence is: the total number of positive factors of 120 (16), followed by the sum of the distinct prime factors of 330 (21), followed by the GCF of 84 and 140 (28), and finally the LCM of 2, 3, 4, and 5 (60).

Evaluating each statement yields the numerical values

16

,

21

,

28

, and

60

. Arranging these from least to greatest gives the correct order: Factor count of

120

(16), Prime sum of

330

(21), GCF of

84

and

140

(28), and LCM of

2, 3, 4,

and

5

(60).

Step-by-Step Solution

1

Calculate the total number of positive factors of

120

.

16

By finding the prime factorization (

120 = 2^3 \times 3^1 \times 5^1

) and adding

1

to each exponent before multiplying them, we get

(3+1) \times (1+1) \times (1+1) = 4 \times 2 \times 2 = 16

.

2

Calculate the sum of the distinct prime factors of

330

.

21

The prime factorization is

330 = 33 \times 10 = 3 \times 11 \times 2 \times 5

. The distinct primes are

2, 3, 5,

and

11

. Their sum is

2 + 3 + 5 + 11 = 21

.

3

Determine the greatest common factor (GCF) of

84

and

140

.

28

Using prime factorizations (

84 = 2^2 \times 3 \times 7

and

140 = 2^2 \times 5 \times 7

), the common prime factors are

2^2

and

7

. Multiplying these yields

4 \times 7 = 28

.

4

Determine the least common multiple (LCM) of

2, 3, 4,

and

5

.

60

The LCM must include the highest power of each prime factor present in the numbers:

2^2

(from

4

),

3^1

, and

5^1

.

4 \times 3 \times 5 = 60

.

5

Order the evaluated results from least to greatest.

16 < 21 < 28 < 60

Sorting the calculated values provides the final correct arrangement of the mathematical statements.

Key Concept

Applying Prime Factorization to Calculate Factor Counts, GCF, and LCM

Alternative Method

Instead of full prime factorization for the GCF, the Euclidean algorithm can be used:

140 - 84 = 56

. Next,

84 - 56 = 28

. Next,

56 - 28 = 28

. The GCF is

28

.

Estimated Time:2m 30s

Question 377Question →

Determine the total number of positive factors for each of the following integers. Then, place the integers in order from the one with the fewest factors to the one with the most factors.

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Show answer & explanation

Answer

The correct sequence is: The integer

7

(2 factors), The integer

9

(3 factors), The integer

8

(4 factors), and The integer

12

(6 factors).

By counting the positive divisors of each number, we find that

7

has two factors,

9

has three factors,

8

has four factors, and

12

has six factors. Placing them in ascending order of their factor counts yields

7, 9, 8

, and

12

.

Step-by-Step Solution

1

List all positive factors for the integer

12

.

The factors of

12

are

1, 2, 3, 4, 6

, and

12

. There are

6

factors in total.

Finding the total count of factors is required to compare and order the integers.

2

List all positive factors for the integer

7

.

The factors of

7

are

1

and

7

. There are

2

factors in total.

Determining the factor count for each number allows for proper sorting.

3

List all positive factors for the integer

8

.

The factors of

8

are

1, 2, 4

, and

8

. There are

4

factors in total.

We must evaluate every integer provided in the list.

4

List all positive factors for the integer

9

.

The factors of

9

are

1, 3

, and

9

. There are

3

factors in total.

This completes the factor counting for all items.

5

Sort the integers based on their factor counts from fewest to most.

The sequence is

2

factors (The integer

7

), then

3

factors (The integer

9

), then

4

factors (The integer

8

), and finally

6

factors (The integer

12

).

The question asks to place the integers in order from fewest factors to most factors.

Key Concept

Factors and Prime Numbers

Estimated Time:45s

Question 378Question →

Consider the process of evaluating the numerical expression below using the standard order of operations:

14 + 3 \times (7 - 2^2) - 8 \div 4

How should the following intermediate simplified forms be arranged to represent the exact chronological sequence of the step-by-step evaluation?

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Show answer & explanation

Answer

The expressions should be ordered starting with the exponent evaluation, followed by the parentheses, then multiplication and division, and finally addition.

The proper application of the order of operations (PEMDAS/BODMAS) ensures that we first simplify inside the parentheses (handling the exponent first, then subtraction), next process multiplication and division from left to right, and finally address addition and subtraction from left to right. This produces the exact sequence of intermediate expressions shown in the correct arrangement.

Step-by-Step Solution

1

Evaluate the exponent inside the grouping symbols.

2^2

simplifies to

4

, changing the expression to

14 + 3 \times (7 - 4) - 8 \div 4

.

Order of operations dictates that exponents inside parentheses take precedence over other operations.

2

Complete the operation inside the parentheses.

7 - 4

simplifies to

3

, yielding

14 + 3 \times 3 - 8 \div 4

.

All operations within grouping symbols must be fully resolved before proceeding to external operations.

3

Evaluate multiplication and division.

3 \times 3

becomes

9

and

8 \div 4

becomes

2

, resulting in

14 + 9 - 2

.

Multiplication and division share the same priority and are executed from left to right before addition or subtraction.

4

Perform the addition.

14 + 9

becomes

23

, leaving

23 - 2

.

Addition and subtraction share the lowest priority and are evaluated from left to right.

5

Perform the final subtraction.

23 - 2

simplifies to the final answer of

21

.

This is the final remaining mathematical operation.

Key Concept

Order of Operations (PEMDAS)

Question 379Question →

Arrange the following fractions and mixed numbers in ascending order (from least to greatest).

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Show answer & explanation

Answer

The correct ascending order is

2 \frac{7}{12}

,

2 \frac{5}{8}

,

\frac{8}{3}

, and

\frac{11}{4}

.

By converting all values to mixed numbers with a common denominator of 24, we find their fractional parts are

\frac{14}{24}

,

\frac{15}{24}

,

\frac{16}{24}

, and

\frac{18}{24}

. Because all whole numbers are 2, the order of the original fractions matches the ascending order of these fractional parts.

Step-by-Step Solution

1

Convert the improper fractions to mixed numbers to simplify comparison.

\frac{11}{4} = 2 \frac{3}{4}

and

\frac{8}{3} = 2 \frac{2}{3}

. The other numbers are already mixed numbers:

2 \frac{5}{8}

and

2 \frac{7}{12}

.

Because all the whole numbers are 2, we only need to compare their fractional parts to determine the order.

2

Identify the least common denominator (LCD) for the fractional parts:

\frac{3}{4}

,

\frac{5}{8}

,

\frac{7}{12}

, and

\frac{2}{3}

.

The denominators are 4, 8, 12, and 3. The lowest number they all divide into evenly is 24.

A common denominator is required to accurately compare the sizes of fractions with unlike denominators.

3

Convert each fractional part to an equivalent fraction with a denominator of 24.

\frac{3}{4} = \frac{18}{24}

,

\frac{5}{8} = \frac{15}{24}

,

\frac{7}{12} = \frac{14}{24}

, and

\frac{2}{3} = \frac{16}{24}

.

Multiplying both the numerator and denominator by the same value preserves the fraction's overall mathematical value.

4

Order the converted fractions from least to greatest based on their numerators.

\frac{14}{24} < \frac{15}{24} < \frac{16}{24} < \frac{18}{24}

.

When denominators are equal, a larger numerator represents a larger overall value.

5

Match the ordered fractions back to their original given forms.

2 \frac{7}{12}

(from

\frac{14}{24}

),

2 \frac{5}{8}

(from

\frac{15}{24}

),

\frac{8}{3}

(from

\frac{16}{24}

), and

\frac{11}{4}

(from

\frac{18}{24}

).

The final answer must present the exact numbers provided in the original stem.

Key Concept

Comparing and Ordering Fractions with Unlike Denominators

Question 380Question →

Arrange the following values from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from least to greatest is:

620

,

8,500

,

1.2 \times 10^4

,

3.0 \times 10^5

.

By converting all values to standard form, it is clear that

620

is the smallest, followed by

8,500

, then

12,000

(which is

1.2 \times 10^4

), and finally

300,000

(which is

3.0 \times 10^5

).

Step-by-Step Solution

1

Convert all values to standard numerical form to easily compare their sizes.

The values become:

12,000

(from item 1),

620

(from item 2),

300,000

(from item 3), and

8,500

(from item 4).

Translating scientific notation into standard form allows for direct visual comparison of digit lengths and magnitudes.

2

Group and compare the numbers by their total number of digits.

620

has three digits,

8,500

has four digits,

12,000

has five digits, and

300,000

has six digits.

For whole positive numbers, a value with fewer digits is always smaller than a value with more digits.

3

Order the values sequentially starting from the smallest.

The final order is

620

<

8,500

<

12,000

<

300,000

.

This establishes the ascending sequence requested by the question.

Key Concept

Comparing Magnitudes in Standard and Scientific Notation

Alternative Method

Instead of using standard form, you can convert all numbers to scientific notation:

6.2 \times 10^2

,

8.5 \times 10^3

,

1.2 \times 10^4

,

3.0 \times 10^5

. Because the exponents (

2, 3, 4, 5

) are already in ascending order, you can immediately determine the correct sequence without looking at the coefficients.

Estimated Time:45s

Answer

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