All practice questions

Question 221Question →

Arrange the following segments to construct a clear, grammatically correct sentence that avoids misplaced and dangling modifiers.

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Answer

Having analyzed the tree rings of ancient bristlecone pines, climatologists reconstructed with remarkable precision regional temperature fluctuations that occurred over the past two millennia.

The correct order builds a logical and grammatically sound sentence where each modifier is positioned adjacent to its target. The introductory phrase is followed by the subject 'climatologists' (avoiding a dangling modifier), the adverbial modifier 'with remarkable precision' is kept next to the verb 'reconstructed', and the relative clause is kept next to the noun 'fluctuations'.

Step-by-Step Solution

1

Identify the introductory modifying phrase.

The phrase 'Having analyzed the tree rings of ancient bristlecone pines,' is identified as an introductory modifier.

This phrase describes an action and requires a logical subject to follow it directly.

2

Determine the logical subject that performed the action.

The subject 'climatologists' is the actor, so the segment 'climatologists reconstructed with remarkable precision' must follow the introductory phrase.

Placing the subject immediately after the introductory modifier avoids a dangling modifier.

3

Identify the direct object and its modifier.

The direct object is 'regional temperature fluctuations,' which is modified by the clause 'that occurred over the past two millennia.'

The direct object must follow the verb, and the relative clause must immediately follow the noun it modifies to avoid a misplaced modifier.

Key Concept

Modifiers must be placed next to the words they modify. Introductory participial phrases must be followed by the subject performing the action to avoid dangling modifiers. Adjectival clauses and prepositional phrases must be adjacent to their target nouns or verbs to avoid misplaced modifiers.

Estimated Time:1m 30s

Question 222Question →

The following sentences are from a paragraph about computer pioneer Grace Hopper. Arrange the sentences in the most logical order to create a cohesive paragraph, paying close attention to which sentences function as the most effective introduction and conclusion.

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Answer

The paragraph begins with the statement introducing modern high-level programming and Grace Hopper, moves to the historical context of tedious binary programming, transitions to the creation of the A-0 compiler, describes the immediate benefits of faster coding, and concludes with a summary of the compiler's lasting legacy.

The correct order establishes a logical progression: it begins with a general modern context, introduces a historical problem, explains the solution, details the immediate benefit of that solution, and concludes with the long-term impact, neatly returning to the modern context mentioned in the introduction.

Step-by-Step Solution

1

Identify the introductory sentence.

The sentence introducing the contrast between modern high-level programming and the work of Grace Hopper is selected.

An effective introduction must establish the paragraph's main subject and context without referencing prior sentences.

2

Select the sentence establishing the historical problem.

The sentence about early programmers working in tedious binary language is placed next.

This provides the necessary historical background to understand why Hopper's later work was necessary.

3

Connect the problem to the solution.

The sentence introducing the A-0 compiler as a response to 'this limitation' is placed third.

The phrase 'this limitation' forms a logical cohesive link back to the tedious binary language described in the previous sentence.

4

Trace the immediate results of the solution.

The sentence stating that 'this breakthrough' allowed faster coding is placed fourth.

The phrase 'this breakthrough' directly references the newly introduced A-0 compiler.

5

Identify the concluding sentence.

The sentence summarizing how the invention laid the groundwork for all modern programming languages is placed last.

An effective concluding sentence wraps up the narrative arc, uses transition words indicating finality, and echoes the introductory focus on modern high-level languages.

Key Concept

Determining logical paragraph flow by identifying sentences that establish initial context (introductions) and summarize the overall significance of the topic (conclusions).

Estimated Time:1m 30s

Question 223Question →

A conservation center recently announced a major milestone in its endangered species program. In what sequence must the following segments be arranged to construct a grammatically correct second sentence that avoids misplaced and dangling modifiers?

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Answer

Although nearly extinct in the wild, the Siberian tiger, which has since been released back into its natural habitat, was successfully bred by the zoologists in a specialized conservation facility.

The correct sequence places the introductory modifier 'Although nearly extinct in the wild' next to its logical target, 'the Siberian tiger'. It then correctly positions the relative clause ', which has since been released back into its natural habitat,' immediately after 'the Siberian tiger' to avoid a misplaced modifier. The sentence is completed by the main predicate 'was successfully bred by the zoologists in a specialized conservation facility.'

Step-by-Step Solution

1

Identify the introductory modifying phrase and its logical target.

The phrase 'Although nearly extinct in the wild' must modify the subject 'the Siberian tiger'.

Placing any other noun (like 'the zoologists' or 'conservation facility') immediately after this phrase would create a dangling modifier, as people or buildings cannot be extinct species.

2

Determine the correct placement of the relative clause.

The relative clause ', which has since been released back into its natural habitat,' must be placed directly after 'the Siberian tiger'.

If placed elsewhere, such as after 'conservation facility' or 'zoologists', it would create a misplaced modifier, implying the facility or the researchers were released into the wild.

3

Arrange the remaining elements of the main independent clause.

Combine the subject 'the Siberian tiger' with the verb phrase 'was successfully bred by the zoologists' and the location 'in a specialized conservation facility.'

This logical sequence ensures the main action of the sentence is clear and grammatically correct.

Key Concept

Modifier Placement

Question 224Question →

The following sentences are from a paragraph about restoring historical sound recordings. Arrange the sentences in the most logical order to create a cohesive paragraph that begins with the most effective introduction and ends with the most effective conclusion.

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Answer

The logical order of the paragraph begins with the introduction of the archivist's dilemma with fragile media, followed by the introduction of the optical scanning solution, then the explanation of how the computer reconstructs the sound, and finally the concluding statement about the overall impact of this non-invasive technology.

The correct order begins by establishing the central historical challenge (the fragility of wax cylinders and other recordings), introduces the digital camera solution (the IRENE system), details how the software converts the digital images back into sound waves, and concludes with a summary of the technology's overall impact on historical preservation.

Step-by-Step Solution

1

Identify the most effective opening sentence.

The sentence about archivists struggling to play damaged historical recordings because of physical needles is the only one that introduces a general topic and sets up a problem without relying on prior context.

An effective introduction must establish the subject matter and provide the context necessary for the rest of the paragraph to make sense.

2

Determine the logical flow of the body sentences.

The sentence about the IRENE system must follow because it begins with 'To solve this problem,' referring directly to the needles destroying the soft media. The sentence about the computer program analyzing the images must come next, as it explains the subsequent step of how the scans are converted back into sound waves.

Cohesive paragraphs maintain a clear, step-by-step logical progression where each sentence builds directly on the information presented in the preceding one.

3

Identify the most effective concluding sentence.

The sentence beginning with 'Consequently' summarizes the ultimate success and significance of this technology, closing the loop on the initial problem of preserving fragile recordings without damaging them.

An effective conclusion uses transitional elements to signal the end of the discussion and summarizes the main point or impact of the paragraph.

Key Concept

Identifying the logical flow of a paragraph by determining which sentence serves as the best introduction to establish the topic and which sentence serves as the best conclusion to summarize the main point.

Estimated Time:1m 30s

Question 225Question →

The following sentences are from a paragraph about the engineering feat of reversing the Chicago River. Arrange the sentences in the most logical order to create a cohesive paragraph that begins with a clear introduction of the problem and ends with an effective concluding sentence that summarizes the project's impact.

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Answer

The logical order begins with the introduction of the public health crisis, followed by the decision to reverse the river, then the details of how the canal redirected the water, and ends with the concluding sentence summarizing the project's impact.

The correct sequence begins with the sentence introducing Chicago's public health crisis, which provides the background context. Next comes the sentence explaining the city's decision to reverse the river to address this threat. This is followed by the sentence detailing how the canal was excavated to redirect the flow. Finally, the paragraph concludes with a sentence summarizing the dual positive outcomes of the project.

Step-by-Step Solution

1

Identify the introductory sentence.

The sentence introducing Chicago's late nineteenth-century public health crisis is the logical starting point because it sets the context without relying on preceding information.

An effective introduction must establish the core topic and problem before any actions or solutions are discussed.

2

Find the logical transition and development.

The sentence beginning with 'To address this threat' must follow the introduction, followed by the sentence describing how engineers achieved the feat using the canal.

The demonstrative pronoun 'this threat' requires the antecedent 'public health crisis,' and the details of the canal explain how the reversal project ('this feat') was physically accomplished.

3

Identify the concluding sentence.

The sentence beginning 'By redirecting the river's flow' wraps up the paragraph by summarizing the overall results and significance of the project.

A strong conclusion should tie the narrative together and show the broader implications of the topic discussed.

Key Concept

Establishing logical paragraph flow using clear opening sentences, transition words, and summarizing concluding sentences.

Question 226Question →

The following sentences are from a paragraph about William Henry Perkin's accidental invention. Arrange the sentences in the most logical order to create a cohesive paragraph that begins with a clear introduction and ends with a strong concluding sentence.

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Answer

The most logical order of the sentences is: first, the sentence introducing William Henry Perkin's attempt to synthesize quinine; second, the sentence describing the resulting dark, sticky residue; third, the sentence about the residue dissolving into a purple liquid; fourth, the sentence detailing his realization that the solution could be used as a dye; and finally, the sentence concluding with the historical impact of the discovery.

The correct order follows a clear chronological and logical progression: introducing Perkin's initial experiment (quinine synthesis), describing the failed result (sticky residue), explaining how cleaning the residue produced a new substance (purple liquid), identifying the commercial potential of that substance (textile dye), and concluding with the global impact of the discovery (revolutionizing the fashion and chemical industries).

Step-by-Step Solution

1

Identify the introductory sentence.

The sentence introducing William Henry Perkin and his goal of synthesizing quinine is selected as the starting point.

An effective introduction must establish context (who, when, what) without relying on antecedent pronouns or references to previous information.

2

Find the immediate transition describing the experiment's outcome.

The sentence about the 'dark, sticky residue' is placed second.

The reference to 'the desired clear medicine' connects directly to the attempt to synthesize 'quinine' in the first sentence.

3

Identify the chemical reaction description.

The sentence describing the substance dissolving into a purple liquid is placed third.

The noun phrase 'the substance' needs an antecedent, which is the 'dark, sticky residue' from the second sentence.

4

Locate the transition from discovery to application.

The sentence explaining how the liquid could be a textile dye is placed fourth.

The phrase 'this accidental solution' refers to the 'purple liquid' described in the third sentence.

5

Determine the concluding sentence.

The sentence reflecting on the long-term impact on the fashion and chemical industries is placed last.

A strong conclusion should summarize the significance of the event and provide a sense of closure to the passage.

Key Concept

Logical paragraph flow relies on clear transitions, referencing pronouns (like 'this accidental solution' or 'the substance') to their proper antecedents, and placing general introductory and concluding thoughts at the boundaries of the paragraph.

Question 227Question →

Evaluate each of the following numerical expressions using the standard order of operations. Arrange the expressions in order of their simplified values from least to greatest.

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Answer

The correct order of the expressions from least to greatest simplified value is:

12 - 3 \times | -4 + 1 |

(value of 3),

18 \div 3 \times 2 - 2^3

(value of 4),

(6 - 8)^2 \div 2 + 3

(value of 5), and

-2^2 + 5 \times (3 - 1)

(value of 6).

Evaluating each expression according to the correct order of operations results in values of 3, 4, 5, and 6, respectively.

Step-by-Step Solution

1

Evaluate the expression

12 - 3 \times | -4 + 1 |

.

3

Simplify inside the absolute value first:

-4 + 1 = -3

. Taking the absolute value gives

|-3| = 3

. Perform multiplication before subtraction:

3 \times 3 = 9

. Subtract:

12 - 9 = 3

.

2

Evaluate the expression

18 \div 3 \times 2 - 2^3

.

4

Evaluate the exponent first:

2^3 = 8

. Perform multiplication and division from left to right:

18 \div 3 = 6

and

6 \times 2 = 12

. Subtract:

12 - 8 = 4

.

3

Evaluate the expression

(6 - 8)^2 \div 2 + 3

.

5

Simplify inside parentheses:

6 - 8 = -2

. Apply the exponent:

(-2)^2 = 4

. Divide:

4 \div 2 = 2

. Add:

2 + 3 = 5

.

4

Evaluate the expression

-2^2 + 5 \times (3 - 1)

.

6

Apply the exponent to the base 2:

-2^2 = -4

. Evaluate inside parentheses:

3 - 1 = 2

. Multiply:

5 \times 2 = 10

. Add:

-4 + 10 = 6

.

Key Concept

Using the standard order of operations (PEMDAS/GEMS) to evaluate numerical expressions containing exponents, grouping symbols, absolute values, multiplication, division, addition, and subtraction.

Question 228Question →

Arrange the following numbers in order from least to greatest: $\frac{2}{5}$ , $0.38$ , $45\%$ , and $0.42$ .

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Answer

The correct order of the values from least to greatest is

0.38

,

\frac{2}{5}

,

0.42

, and

45\%

.

Converting all numbers to decimals allows for direct comparison:

0.38

remains

0.38

; the fraction

\frac{2}{5}

equals

0.40

;

0.42

remains

0.42

; and

45\%

equals

0.45

. Comparing these decimal values, we get

0.38 < 0.40 < 0.42 < 0.45

. Therefore, the correct order from least to greatest is

0.38

,

\frac{2}{5}

,

0.42

, and

45\%

.

Step-by-Step Solution

1

Convert the fraction to a decimal.

\frac{2}{5} = 0.40

Converting all values to decimals makes them easier to compare.

2

Convert the percentage to a decimal.

45\% = 0.45

Percentages represent values out of 100, so

45\% = \frac{45}{100} = 0.45

.

3

Compare the four decimal numbers.

0.38 < 0.40 < 0.42 < 0.45

By comparing the tenths and hundredths place, we find the correct order from smallest to largest.

Key Concept

Comparing and ordering fractions, decimals, and percents by converting them to a common decimal format.

Question 229Question →

The passage below describes a famous historical site. The second sentence of the passage has been split into four segments.

'Overlooking the Urubamba River Valley, the ruins of a fifteenth-century Inca citadel continue to fascinate historians. [Segment 1] [Segment 2] [Segment 3] [Segment 4]'

Arrange the four segments below in the correct order to form a grammatically correct second sentence that avoids misplaced or dangling modifiers.

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Answer

The correct sequence of segments to form a grammatically correct sentence is: 'the ancient city of Machu Picchu', followed by 'which is located in the Andes mountains', followed by 'attracts thousands of tourists', and ending with 'who hope to see the sunrise over the ruins'.

The correct order constructs the sentence: 'The ancient city of Machu Picchu, which is located in the Andes mountains, attracts thousands of tourists who hope to see the sunrise over the ruins.' This placement keeps the non-restrictive relative clause next to 'the ancient city of Machu Picchu' and the restrictive relative clause next to 'tourists', ensuring proper modifier alignment and subject-verb agreement throughout the sentence.

Step-by-Step Solution

1

Identify the main subject of the sentence.

'the ancient city of Machu Picchu' is identified as the subject because it is the only noun phrase that can logically start the sentence and serve as the subject for the main verb.

Establishing the main subject provides the grammatical foundation for the rest of the sentence.

2

Determine the placement of the relative clause starting with 'which'.

'which is located in the Andes mountains' must be placed immediately after 'the ancient city of Machu Picchu'.

To avoid a misplaced modifier, the relative clause must be adjacent to the noun it modifies. Placing it elsewhere creates a subject-verb agreement error with the singular verb 'is' and makes the meaning illogical.

3

Locate the main verb phrase.

'attracts thousands of tourists' follows the modified subject.

The main verb must follow the complete subject noun phrase to complete the independent clause.

4

Place the final relative clause modifying the object.

'who hope to see the sunrise over the ruins' must follow 'tourists'.

The relative pronoun 'who' refers to people ('tourists') and must be placed immediately after its antecedent to prevent ambiguity or misplacement.

Key Concept

Modifier Placement

Question 230Question →

Four investment options, each with a different starting value (expressed as a fraction, decimal, or percentage of a standard initial budget $B$ ) and a specified first-year growth rate, are described below. Arrange the options in order from the lowest final value to the highest final value after the first year.

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Answer

Option Y, Option Z, Option X, Option W

The correct order is Option Y, Option Z, Option X, Option W. Converting each initial value to a decimal fraction of

B

and applying the respective growth multipliers yields: Option Y is

0.90 \times 1.15 = 1.035B

, Option Z is

0.875 \times 1.184 = 1.036B

, Option X is

0.85 \times 1.22 = 1.037B

, and Option W is

0.80 \times 1.30 = 1.040B

. Comparing these decimal values gives

1.035 < 1.036 < 1.037 < 1.040

.

Step-by-Step Solution

1

Calculate the final value of Option W relative to

B

.

0.80 \times 1.30 \times B = 1.040B

Convert the starting fraction

\frac{4}{5}

to the decimal

0.80

. A

30\%

increase corresponds to a multiplier of

1 + 0.30 = 1.30

. Multiply these values together:

0.80 \times 1.30 = 1.040

.

2

Calculate the final value of Option X relative to

B

.

0.85 \times 1.22 \times B = 1.037B

The starting decimal is

0.85

. A

22\%

increase corresponds to a multiplier of

1 + 0.22 = 1.22

. Multiply these values together:

0.85 \times 1.22 = 1.037

.

3

Calculate the final value of Option Y relative to

B

.

0.90 \times 1.15 \times B = 1.035B

Convert the starting percentage

90\%

to the decimal

0.90

. A

15\%

increase corresponds to a multiplier of

1 + 0.15 = 1.15

. Multiply these values together:

0.90 \times 1.15 = 1.035

.

4

Calculate the final value of Option Z relative to

B

.

0.875 \times 1.184 \times B = 1.036B

Convert the starting fraction

\frac{7}{8}

to the decimal

0.875

. An

18.4\%

increase corresponds to a multiplier of

1 + 0.184 = 1.184

. Multiply these values together:

0.875 \times 1.184 = 1.036

.

5

Compare the resulting multipliers to order the final values from lowest to highest.

1.035 < 1.036 < 1.037 < 1.040

, which translates to Option Y < Option Z < Option X < Option W.

By arranging the computed multipliers in ascending order, we find the correct sequence.

Key Concept

Converting fractions, decimals, and percentages to common decimal representations to compute and compare multi-step percentage increases.

Question 231Question →

Arrange the following four mathematical expressions in order from least to greatest value:

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Answer

The correct order from least to greatest is

S

,

P

,

R

, and

Q

, which correspond to the decimal values

0.16 < 0.25 < 0.28 < 0.3

.

Evaluating each term yields decimal values:

S = 0.16

,

P = 0.25

,

R = 0.28

, and

Q = 0.3

. Ordering these from least to greatest gives

0.16 < 0.25 < 0.28 < 0.3

, which corresponds to the sequence

S, P, R, Q

.

Step-by-Step Solution

1

Convert the fractional squared expression

S

to a decimal.

S = \left(\frac{2}{5}\right)^2 = \frac{4}{25} = 0.16

Squaring a fraction involves squaring both the numerator and denominator, then converting the resulting fraction to decimal form.

2

Convert the negative exponent expression

P

to a decimal.

P = 4^{-1} = \frac{1}{4} = 0.25

A negative exponent represents the reciprocal of the base raised to the positive power.

3

Convert the scientific notation expression

R

to a decimal.

R = 2.8 \times 10^{-1} = 0.28

Multiplying by

10^{-1}

shifts the decimal point one position to the left.

4

Convert the square root expression

Q

to a decimal.

Q = \sqrt{0.09} = 0.3

Finding the square root of a decimal less than

1

results in a value larger than the original decimal, since

0.3 \times 0.3 = 0.09

.

5

Compare the resulting decimal values to order them from least to greatest.

0.16 < 0.25 < 0.28 < 0.3

, so the order is

S < P < R < Q

.

Ordering the converted decimal values yields the correct final arrangement.

Key Concept

Converting exponential, radical, and scientific notation expressions to decimals to compare and order their values.

Question 232Question →

The following sentence segments, when combined, should form a single, grammatically correct sentence. Arrange the segments in the correct order to construct a sentence that contains no misplaced or dangling modifiers.

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Answer

The correct order of the sentence segments is: 'Dressed in a sleek tuxedo, the master of ceremonies warmly greeted the patrons who were entering the theater.'

The correct order places the introductory participial phrase 'Dressed in a sleek tuxedo,' directly before the noun it modifies, 'the master of ceremonies.' It then follows with the main verb phrase 'warmly greeted the patrons' and concludes with the relative clause 'who were entering the theater,' which correctly modifies 'the patrons.' This arrangement ensures that all modifiers are adjacent to their targets and the sentence structure is clear and grammatical.

Step-by-Step Solution

1

Identify the introductory modifier.

The phrase 'Dressed in a sleek tuxedo,' is identified as an introductory participial phrase.

Participial phrases at the beginning of a sentence must immediately precede the noun they modify to avoid a dangling modifier.

2

Determine the subject of the modifier.

The noun phrase 'the master of ceremonies' is the logical subject who is dressed in the tuxedo.

Placing this noun phrase directly after the introductory phrase ensures the modifier is placed correctly.

3

Connect the verb and object to complete the main clause.

The verb phrase 'warmly greeted the patrons' follows the subject.

This establishes the core action of the sentence in a natural subject-verb-object order.

4

Place the relative clause next to its antecedent.

The relative clause 'who were entering the theater.' modifies 'the patrons'.

Placing this clause immediately after 'the patrons' prevents it from mismodifying other nouns like the master of ceremonies or the tuxedo.

Key Concept

Modifiers must be placed as close as possible to the words they are intended to modify to ensure clarity and grammatical correctness.

Estimated Time:1m 0s

Question 233Question →

Arrange the following mathematical expressions in order from least to greatest value:

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Answer

The correct order from least to greatest value is

(2.5 \times 10^1)^2

, followed by

\sqrt[3]{3.43 \times 10^8}

, then

\sqrt{6.4 \times 10^5}

, and finally

\frac{5.4 \times 10^6}{6 \times 10^3}

.

Each expression is simplified to a standard integer representation:

(2.5 \times 10^1)^2 = 625

,

\sqrt[3]{3.43 \times 10^8} = 700

,

\sqrt{6.4 \times 10^5} = 800

, and

\frac{5.4 \times 10^6}{6 \times 10^3} = 900

. Comparing these integers from smallest to largest yields the correct sequence.

Step-by-Step Solution

1

Evaluate the expression

\sqrt{6.4 \times 10^5}

by converting the radicand to a standard square value.

\sqrt{64 \times 10^4} = 8 \times 10^2 = 800

Converting the base to 64 and reducing the exponent of 10 to an even number allows easy extraction of the square root.

2

Evaluate the expression

(2.5 \times 10^1)^2

by simplifying the terms inside the parentheses first.

(25)^2 = 625

Multiplying 2.5 by 10 simplifies the base to 25 before applying the exponent.

3

Evaluate the expression

\frac{5.4 \times 10^6}{6 \times 10^3}

using rules of scientific division.

0.9 \times 10^3 = 900

Dividing the coefficients (

5.4 \div 6 = 0.9

) and subtracting the exponent in the denominator from the numerator (

10^{6-3} = 10^3

) yields the result.

4

Evaluate the expression

\sqrt[3]{3.43 \times 10^8}

by converting the radicand to a standard cube value.

\sqrt[3]{343 \times 10^6} = 7 \times 10^2 = 700

Converting the base to 343 and adjusting the exponent to 6 (a multiple of 3) allows easy extraction of the cube root.

5

Compare the evaluated integers to determine the correct order from least to greatest.

625 < 700 < 800 < 900

Comparing the simplified values reveals that the correct sequence is

(2.5 \times 10^1)^2

,

\sqrt[3]{3.43 \times 10^8}

,

\sqrt{6.4 \times 10^5}

, and

\frac{5.4 \times 10^6}{6 \times 10^3}

.

Key Concept

Simplification of exponential, radical, and scientific notation expressions to comparable decimal values.

Alternative Method

Instead of converting all expressions to standard numbers, convert them all to scientific notation with a base power of

10^2

(i.e.,

6.25 \times 10^2

,

7.0 \times 10^2

,

8.0 \times 10^2

, and

9.0 \times 10^2

) to compare their coefficients directly.

Estimated Time:2m 0s

Question 234Question →

Four different brands of a sports drink contain different proportions of electrolytes by weight. The proportions for each brand are listed below:

* Brand P: $\frac{5}{12}$ of its weight is electrolytes
* Brand Q: $41.8\%$ of its weight is electrolytes
* Brand R: $0.416$ of its weight is electrolytes
* Brand S: $\frac{13}{30}$ of its weight is electrolytes

Arrange the brands in order from the least proportion of electrolytes to the greatest proportion of electrolytes.

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Answer

The correct order from least to greatest proportion of electrolytes is Brand R, Brand P, Brand Q, and Brand S.

To compare the values, converting each to a decimal is the most reliable strategy. Brand R is already a decimal (

0.416

). Converting Brand P yields

\frac{5}{12} \approx 0.4167

. Converting Brand Q yields

41.8\% = 0.418

. Converting Brand S yields

\frac{13}{30} \approx 0.4333

. Comparing these decimals reveals that

0.416 < 0.4167 < 0.418 < 0.4333

, corresponding to the sequence: Brand R, Brand P, Brand Q, Brand S.

Step-by-Step Solution

1

Identify and list all values given for each brand.

Brand P:

\frac{5}{12}

, Brand Q:

41.8\%

, Brand R:

0.416

, Brand S:

\frac{13}{30}

.

This establishes the raw numbers that need comparison.

2

Convert each value into a decimal format to allow direct comparison, rounding to four decimal places where necessary.

Brand R:

0.4160

; Brand P:

5 \div 12 \approx 0.4167

; Brand Q:

41.8\% = 0.4180

; Brand S:

13 \div 30 \approx 0.4333

.

Converting to decimals standardizes the formats and makes close comparisons straightforward.

3

Compare the standardized decimal values from smallest to largest.

0.4160 < 0.4167 < 0.4180 < 0.4333

, which corresponds to the order: Brand R, Brand P, Brand Q, Brand S.

Ordering the values correctly identifies the brands from the least proportion of electrolytes to the greatest.

Key Concept

Comparing and ordering rational numbers by converting fractions, decimals, and percents into a single consistent format.

Alternative Method

Instead of converting to decimals, all values can be converted to percentages: Brand R is

41.6\%

, Brand P is

\frac{5}{12} \times 100\% \approx 41.67\%

, Brand Q is

41.8\%

, and Brand S is

\frac{13}{30} \times 100\% \approx 43.33\%

. Comparing these values gives

41.6\% < 41.67\% < 41.8\% < 43.33\%

.

Estimated Time:1m 30s

Question 235Question →

Simplify the four mathematical expressions below. How should these expressions be ordered from least to greatest based on their simplified values?

* Expression A: $\sqrt{2.25 \times 10^4} \times \left(2.0 \times 10^{-1}\right)^{-2}$
* Expression B: $\frac{\left(3.0 \times 10^2\right)^3}{6.0 \times 10^3}$
* Expression C: $\sqrt[3]{2.7 \times 10^7} \times \left(4.0 \times 10^{-2}\right)^{-1/2}$
* Expression D: $\left(6.4 \times 10^5\right)^{1/2} \times 5.0 \times 10^0$

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Answer

Expression C, Expression A, Expression D, Expression B

The correct order is determined by fully simplifying each expression: Expression C simplifies to

1,500

, Expression A simplifies to

3,750

, Expression D simplifies to

4,000

, and Expression B simplifies to

4,500

. Arranging these values in ascending order gives the correct sequence starting with Expression C, followed by Expression A, then Expression D, and ending with Expression B.

Step-by-Step Solution

1

Simplify Expression C:

\sqrt[3]{2.7 \times 10^7} \times \left(4.0 \times 10^{-2}\right)^{-1/2}

C = 1,500

First, rewrite

2.7 \times 10^7

as

27 \times 10^6

to compute the cube root easily:

\sqrt[3]{27 \times 10^6} = 3 \times 10^2 = 300

. Second, evaluate the term with the rational exponent:

\left(4.0 \times 10^{-2}\right)^{-1/2} = \frac{1}{\sqrt{4.0 \times 10^{-2}}} = \frac{1}{2.0 \times 10^{-1}} = \frac{1}{0.2} = 5

. Finally, multiply the two parts:

300 \times 5 = 1,500

.

2

Simplify Expression A:

\sqrt{2.25 \times 10^4} \times \left(2.0 \times 10^{-1}\right)^{-2}

A = 3,750

First, calculate the square root:

\sqrt{2.25 \times 10^4} = \sqrt{2.25} \times \sqrt{10^4} = 1.5 \times 10^2 = 150

. Second, evaluate the term with the negative exponent:

\left(2.0 \times 10^{-1}\right)^{-2} = (0.2)^{-2} = \frac{1}{0.04} = 25

. Finally, multiply the two simplified values:

150 \times 25 = 3,750

.

3

Simplify Expression D:

\left(6.4 \times 10^5\right)^{1/2} \times 5.0 \times 10^0

D = 4,000

First, simplify the square root by rewriting the radicand:

\left(6.4 \times 10^5\right)^{1/2} = \sqrt{64 \times 10^4} = 8 \times 10^2 = 800

. Second, evaluate

5.0 \times 10^0 = 5.0 \times 1 = 5

. Finally, multiply the results:

800 \times 5 = 4,000

.

4

Simplify Expression B:

\frac{\left(3.0 \times 10^2\right)^3}{6.0 \times 10^3}

B = 4,500

First, raise the numerator to the power of three:

\left(3.0 \times 10^2\right)^3 = 3^3 \times \left(10^2\right)^3 = 27 \times 10^6

. Next, divide by the denominator:

\frac{27 \times 10^6}{6.0 \times 10^3} = \frac{27}{6.0} \times 10^3 = 4.5 \times 10^3 = 4,500

.

5

Compare the simplified values of the four expressions.

1,500 < 3,750 < 4,000 < 4,500

, which corresponds to Expression C, Expression A, Expression D, then Expression B.

Sorting the values in ascending numerical order reveals the correct order from least to greatest.

Key Concept

Simplifying complex numeric expressions containing combinations of square roots, cube roots, negative exponents, fractional exponents, and scientific notation.

Estimated Time:3m 0s

Question 236Question →

Four prototype solar cells are tested for their efficiency in converting solar energy into electricity. The efficiency of each solar cell is recorded as follows:

* Cell W: $\frac{4}{25}$ of the incoming energy is converted.
* Cell X: $16.5\%$ of the incoming energy is converted.
* Cell Y: $0.1605$ of the incoming energy is converted.
* Cell Z: $\frac{13}{80}$ of the incoming energy is converted.

Based on this data, how should these solar cells be ordered from least efficient to most efficient?

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Answer

The correct ordering from least efficient to most efficient is Cell W, Cell Y, Cell Z, and Cell X.

Converting all efficiency values to decimals facilitates comparison. Cell W is

\frac{4}{25} = 0.1600

. Cell Y is

0.1605

. Cell Z is

\frac{13}{80} = 0.1625

. Cell X is

16.5\% = 0.1650

. Comparing these decimals shows that

0.1600 < 0.1605 < 0.1625 < 0.1650

, which matches the order: Cell W, Cell Y, Cell Z, Cell X.

Step-by-Step Solution

1

Convert the efficiency of Cell W to a decimal.

\frac{4}{25} = 0.16 = 0.1600

To easily compare the values, convert the fraction to a decimal by dividing the numerator by the denominator.

2

Convert the efficiency of Cell X to a decimal.

16.5\% = 0.165 = 0.1650

Convert the percentage to a decimal by dividing by 100.

3

Convert the efficiency of Cell Z to a decimal.

\frac{13}{80} = 0.1625

Divide 13 by 80 to obtain its decimal representation.

4

Compare all four decimal values:

0.1600

(Cell W),

0.1650

(Cell X),

0.1605

(Cell Y), and

0.1625

(Cell Z).

0.1600 < 0.1605 < 0.1625 < 0.1650

Align the decimals by place value to determine their correct ascending order.

Key Concept

Comparing and ordering fractions, decimals, and percentages by converting them to a common format (decimals).

Estimated Time:1m 30s

Question 237Question →

Arrange the following numbers in order from least to greatest: $\left(\frac{1}{4}\right)^3$ , $\sqrt{0.0004}$ , $2.4 \times 10^{-2}$ , and $5^{-2}$ .

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Answer

The correct order of the values from least to greatest is

\left(\frac{1}{4}\right)^3

, followed by

\sqrt{0.0004}

, then

2.4 \times 10^{-2}

, and finally

5^{-2}

.

Converting all expressions to decimals allows direct comparison:

\left(\frac{1}{4}\right)^3 = 0.015625

,

\sqrt{0.0004} = 0.02

,

2.4 \times 10^{-2} = 0.024

, and

5^{-2} = 0.04

. Arranging these decimals from smallest to largest results in the sequence

0.015625 < 0.02 < 0.024 < 0.04

.

Step-by-Step Solution

1

Convert each of the exponential, root, and scientific notation expressions into decimal values to make comparison straightforward.

The expressions convert as follows:
-

\left(\frac{1}{4}\right)^3 = \frac{1}{64} = 0.015625

-

\sqrt{0.0004} = \sqrt{4 \times 10^{-4}} = 2 \times 10^{-2} = 0.02

-

2.4 \times 10^{-2} = 0.024

-

5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04

Expressing all values in standard decimal notation eliminates different bases and representations, allowing direct comparison.

2

Compare and arrange the resulting decimals from the smallest value to the largest value.

0.015625 < 0.02 < 0.024 < 0.04

By aligning the place values, we can see that

0.015625

(one hundredth plus fractional parts) is less than

0.02

(two hundredths), which is less than

0.024

(twenty-four thousandths), which is less than

0.04

(four hundredths).

Key Concept

Comparing numerical expressions involving negative exponents, fractional powers, roots, and scientific notation by converting them to a common decimal representation.

Question 238Question →

The following segments can be arranged to form a single, grammatically correct sentence. Arrange the segments in the correct order to construct a sentence that avoids any misplaced or dangling modifiers.

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Answer

The correct order of the sentence segments is: 'The museum curator displayed', 'the fragile medieval tapestry,', 'which had been restored over several months,', and 'in a temperature-controlled glass case.'

The correct order ensures that the non-restrictive relative clause starting with 'which' is adjacent to 'tapestry', the noun it logically modifies, preventing any ambiguity or misplaced modification.

Step-by-Step Solution

1

Identify the subject and main verb of the sentence.

The segment 'The museum curator displayed' is identified as the opening clause.

It contains the capitalized subject and the main active verb of the sentence.

2

Locate the direct object that is acted upon by the verb.

The segment 'the fragile medieval tapestry,' is placed immediately after the verb.

The transitive verb 'displayed' requires a direct object to follow it without awkward separation.

3

Position the relative clause modifying the object.

The clause 'which had been restored over several months,' is placed right after 'tapestry,'.

Placing it here ensures that the modifier correctly describes the tapestry, preventing a misplaced modifier.

4

Add the final prepositional phrase describing location.

The segment 'in a temperature-controlled glass case.' is placed at the end.

It provides contextual detail and contains the terminal period for the sentence.

Key Concept

Modifier Placement

Question 239Question →

The following passage is adapted from an article on desert ecology and bio-inspired engineering:

In the hyper-arid Namib Desert along the southwestern coast of Africa, annual rainfall is exceedingly rare. Yet, the Namib Desert beetle (*Stenocara gracilipes*) thrives by extracting moisture directly from morning ocean fogs. The key to this remarkable survival strategy lies in the micro-structure of the beetle's hardened forewings, or elytra.

The surface of the elytra features a microscopic array of smooth, hydrophilic (water-attracting) bumps interspersed throughout a wax-coated, hydrophobic (water-repelling) trough region. When fog rolls across the dunes, the beetle tilts its body upward into the wind—a behavior known as fog-basking. Droplets of airborne water accumulate on the hydrophilic peaks of the bumps. Because the surrounding troughs repel water, the droplets remain anchored to the peaks, growing progressively larger as more fog condenses. Once a droplet reaches a critical diameter of approximately five millimeters, its weight overcomes the electrostatic forces holding it to the peak. The droplet detaches and rolls down the hydrophobic trough directly into the beetle’s mouthparts.

Based on the passage, arrange the steps of the Namib Desert beetle's water-collection process in the exact chronological order in which they explicitly occur, from first to last.

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Answer

The correct chronological order of events is: 1) The beetle tilts its body upward into the wind in a posture known as fog-basking; 2) Airborne water droplets condense and collect on the hydrophilic peaks of the elytra; 3) Growing droplets reach a size where their weight overcomes electrostatic attraction to the peak; 4) Detached water droplets travel down the wax-coated hydrophobic troughs to the beetle's mouthparts.

The passage outlines the process in explicit chronological order: first, the beetle tilts upward into the wind ('fog-basking'); second, water droplets collect on the hydrophilic peaks; third, the droplets grow until their weight overcomes electrostatic forces; fourth, the released droplets roll down the hydrophobic troughs to the mouthparts.

Step-by-Step Solution

1

Locate the beginning of the water-harvesting process described in the text.

The text states that when fog arrives, 'the beetle tilts its body upward into the wind—a behavior known as fog-basking.'

This behavioral positioning must happen first before water can condense on the shell.

2

Identify the immediate next explicit detail following fog-basking.

The text states: 'Droplets of airborne water accumulate on the hydrophilic peaks of the bumps.'

Water gathering on the peaks is the direct result of fog making contact with the tilted beetle.

3

Track what happens as the droplets continue to gather water on the peaks.

The text explains that 'droplets remain anchored... growing progressively larger' until 'its weight overcomes the electrostatic forces holding it to the peak.'

The growth and detachment of the droplet due to weight occurs after initial accumulation.

4

Identify the final explicit outcome of droplet detachment.

The text states: 'The droplet detaches and rolls down the hydrophobic trough directly into the beetle’s mouthparts.'

Rolling down to the mouthparts completes the collection process.

Key Concept

Identifying Stated Sequence of Events

Question 240Question →

Based on the passage below, place the recorded milestones of Basil Brown's archaeological work at Sutton Hoo in the exact chronological order in which they occurred.

In May 1938, self-taught archaeologist Basil Brown arrived at the Sutton Hoo estate in Suffolk, England, commissioned by landowner Edith Pretty to investigate several mysterious earth mounds on her property. Brown initially focused his efforts on Mound 3, where he excavated damaged grave goods including an axe head and textile fragments. Satisfied with his preliminary survey there, he moved to Mound 2 in July 1938, where his trench work revealed scattered iron ship rivets and fragmented human remains, confirming the presence of disturbed early Anglo-Saxon burials.

Recognizing the potential significance of the site, formal support was arranged for a second digging season. On May 11, 1939, Brown returned to begin excavating the untried Mound 1. Within hours of opening the trench, he uncovered a row of corroded iron ship rivets still set in their original positions within the soil matrix. Realizing that the mound concealed an intact ship burial of exceptional size, Brown proceeded with meticulous care. By June 14, 1939, his careful removal of sand and earth fully exposed the eighty-nine-foot impression of a seventh-century wooden vessel, establishing Sutton Hoo as one of the most remarkable medieval archaeological discoveries in Europe.

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Answer

The correct chronological sequence of events is: excavating grave goods from Mound 3 (May 1938), uncovering iron ship rivets and human remains in Mound 2 (July 1938), unearthing the first corroded rivets in Mound 1 (May 11, 1939), and exposing the full 89-foot vessel impression in Mound 1 (June 14, 1939).

The passage outlines the archaeological discoveries in clear chronological order: Brown began with Mound 3 in May 1938, moved to Mound 2 in July 1938, returned to start Mound 1 on May 11, 1939, and fully exposed the ship impression by June 14, 1939.

Step-by-Step Solution

1

Identify the date associated with Mound 3 excavation in the passage.

The text explicitly states Brown began excavating Mound 3 and found an axe head in May 1938.

Establishes the starting event of Brown's fieldwork.

2

Locate the details describing Mound 2 excavation.

The text notes Brown shifted to Mound 2 in July 1938, discovering rivets and human remains.

Places this finding directly after the Mound 3 work in the summer of 1938.

3

Track the sequence of discoveries made in Mound 1.

Brown began on Mound 1 on May 11, 1939, unearthing initial rivets immediately, followed by fully uncovering the 89-foot ship impression by June 14, 1939.

Determines the third and fourth steps in the timeline based on explicit month and day references.

Key Concept

Identifying explicitly stated chronological sequence of events from text details.

Answer

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