All practice questions

Question 1641Question →

To preserve medieval parchment manuscripts, conservators must maintain a stable environment with precise humidity levels. When the air becomes too dry, the animal skin shrinks and warps, potentially flaking off the delicate gold leaf and pigments. Therefore, museums utilize state-of-the-art climate control systems to keep the ancient documents in tip-top shape.

Which choice best maintains the established tone of the passage?

NO CHANGE

in optimal condition

in a state of absolute, unblemished pristine splendor

in compliance to standard guidelines

Show answer & explanation

Answer: in optimal condition

Answer

in optimal condition

The option 'in optimal condition' is correct because it maintains the formal, academic, and objective tone established throughout the passage. It fits naturally into the sentence and is grammatically correct.

Step-by-Step Solution

1

Analyze the established tone of the passage.

The passage uses formal, objective, and academic language (e.g., 'preserve,' 'conservators,' 'stable environment,' 'utilize').

Understanding the surrounding context is necessary to determine the correct register.

2

Evaluate the underlined phrase and the choices for tone and grammatical correctness.

The original phrase 'in tip-top shape' is too colloquial. The choice 'in a state of absolute, unblemished pristine splendor' is overly dramatic and pretentious. The choice 'in compliance to standard guidelines' is unidiomatic because it uses 'compliance to' instead of 'compliance with'. The choice 'in optimal condition' fits both the formal tone and the sentence structure perfectly.

Selecting the option that maintains the correct style and is grammatically correct satisfies the conventions of the exam.

Key Concept

Tone Consistency

Estimated Time:45s

Question 1642Question →

The passage below is adapted from an article about the history of synthetic chemistry.

In 1856, eighteen-year-old chemist William Henry Perkin accidentally discovered mauveine, the first synthetic organic chemical dye, while attempting to synthesize quinine. This vibrant purple substance, which resisted fading from washing and light, quickly became a fashion sensation. Perkin's breakthrough not only revolutionized the textile market but also instigated the growth of the modern chemical industry.

Which option is the most precise and clear choice for the underlined portion in the passage?

NO CHANGE

catalyzed

kicked off

provoked

Show answer & explanation

Answer: catalyzed

Answer

catalyzed

The word 'catalyzed' is the most precise and appropriate choice. Although it is a chemical term meaning to speed up a reaction, it is commonly used in a broader sense to mean bringing about or accelerating a change. In this context, it perfectly aligns with the theme of chemical discovery and precisely describes the acceleration of the modern chemical industry while maintaining a formal, academic tone.

Step-by-Step Solution

1

Analyze the underlined word 'instigated' in the context of the sentence.

The sentence describes how a chemical breakthrough led to the growth of a new industry. 'Instigated' means to initiate or start, but it is almost always associated with negative actions (e.g., instigating a fight or riot).

To determine if the original word is appropriate or needs to be changed.

2

Evaluate the other options for tone, register, and precision.

'Kicked off' is too colloquial for a formal science passage. 'Provoked' is imprecise because it has a negative connotation (meaning to irritate or incite). 'Catalyzed' means to accelerate or cause a reaction, which fits both the scientific theme and the context of rapid industrial growth.

To eliminate incorrect options based on register and connotation.

3

Select the option that is most precise, grammatically correct, and matches the formal tone.

'Catalyzed' is the best choice.

It maintains the academic tone and offers the most precise and contextually rich meaning.

Key Concept

Word Precision and Clarity

Estimated Time:1m 0s

Question 1643Question →

Arrange the following mathematical expressions in order of their simplified values from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

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

-3 - (4 - 2)

, then

-3^2 + 4 \times 2

, then

(-3)^2 - 4 \times 2

, and finally

-(3 - 4 \times 2)

.

Evaluating each expression using the correct order of operations yields

-5

for

-3 - (4 - 2)

,

-1

for

-3^2 + 4 \times 2

,

1

for

(-3)^2 - 4 \times 2

, and

5

for

-(3 - 4 \times 2)

. Ordering these results from least to greatest yields the sequence:

-3 - (4 - 2)

, followed by

-3^2 + 4 \times 2

, then

(-3)^2 - 4 \times 2

, and finally

-(3 - 4 \times 2)

.

Step-by-Step Solution

1

Simplify the expression

-3 - (4 - 2)

.

-3 - (4 - 2) = -3 - 2 = -5

Perform the operation inside the parentheses first, then subtract the result from the leading term.

2

Simplify the expression

-3^2 + 4 \times 2

.

-3^2 + 4 \times 2 = -9 + 8 = -1

Evaluate the exponent first, noting that the negative sign is outside the base (

-3^2 = -9

), then multiply, and finally add the terms.

3

Simplify the expression

(-3)^2 - 4 \times 2

.

(-3)^2 - 4 \times 2 = 9 - 8 = 1

Evaluate the term inside parentheses raised to the power first (

(-3)^2 = 9

), then perform multiplication, and subtract.

4

Simplify the expression

-(3 - 4 \times 2)

.

-(3 - 4 \times 2) = -(3 - 8) = -(-5) = 5

Inside the parentheses, perform multiplication before subtraction, then apply the negative sign outside the parentheses.

5

Compare the simplified values of the four expressions.

-5 < -1 < 1 < 5

Arrange the resulting values from least to greatest to determine the correct sequence.

Key Concept

Applying the correct order of operations (PEMDAS/GEMS) to evaluate numerical expressions, specifically distinguishing between

-a^2

and

(-a)^2

.

Question 1644Question →

Arrange the following mathematical values in order from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is

4 \times 10^{-2}

,

2^{-3}

,

\sqrt{0.09}

, and

\frac{3^2}{2 \times 10^1}

.

By converting each value to a decimal, we find

4 \times 10^{-2} = 0.04

,

2^{-3} = 0.125

,

\sqrt{0.09} = 0.3

, and

\frac{3^2}{2 \times 10^1} = 0.45

. Ordering these decimals from smallest to largest yields the correct sequence.

Step-by-Step Solution

1

Convert the scientific notation expression

4 \times 10^{-2}

to its decimal form.

4 \times 10^{-2} = 0.04

Multiplying by

10^{-2}

is equivalent to moving the decimal point of the coefficient

4

two places to the left.

2

Convert the negative exponent expression

2^{-3}

to a fraction and then to a decimal.

2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125

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

3

Evaluate the square root of the decimal

\sqrt{0.09}

.

\sqrt{0.09} = 0.3

The square root of a decimal less than

1

results in a larger decimal value because

0.3 \times 0.3 = 0.09

.

4

Evaluate the exponential fraction expression

\frac{3^2}{2 \times 10^1}

.

\frac{3^2}{2 \times 10^1} = \frac{9}{20} = 0.45

Simplify the numerator to

9

and the denominator to

20

, then divide to find the decimal equivalent.

5

Compare the evaluated decimal values to determine the correct order from least to greatest.

0.04 < 0.125 < 0.3 < 0.45

, which correspond to

4 \times 10^{-2} < 2^{-3} < \sqrt{0.09} < \frac{3^2}{2 \times 10^1}

.

Comparing decimal place values shows that

0.04

is the smallest and

0.45

is the largest.

Key Concept

Converting and comparing exponential, radical, and scientific notation expressions to a common decimal format.

Question 1645Question →

The following passage contains an underlined portion. Choose the alternative that is most concise, grammatically correct, and preserves the original meaning of the sentence.

In 1921, Frederick Banting and Charles Best successfully isolated insulin at the University of Toronto. Because of the fact that insulin is a hormone that regulates glucose levels in the bloodstream of the human body, their discovery transformed diabetes from a fatal disease into a manageable condition.

Which of the following alternatives to the underlined portion is the most concise and grammatically correct?

NO CHANGE

Because insulin regulates blood glucose levels

Since insulin, which regulates blood glucose levels

Insulin regulates blood glucose levels

Show answer & explanation

Answer: Because insulin regulates blood glucose levels

Answer

Because insulin regulates blood glucose levels

The correct option is the most concise way to express the cause-and-effect relationship. It eliminates redundant and wordy phrases like 'because of the fact that' and 'in the bloodstream of the human body' while maintaining perfect grammatical structure.

Step-by-Step Solution

1

Identify the relationship and meaning in the original sentence.

The sentence establishes a cause-and-effect relationship: because insulin regulates glucose, the discovery of insulin was major.

To ensure the correct option preserves the original meaning.

2

Evaluate the choices for conciseness and wordiness.

The phrase 'Because insulin regulates blood glucose levels' is much more concise than the original 'Because of the fact that insulin is a hormone that regulates glucose levels in the bloodstream of the human body'.

ACT English prioritizes the most concise correct phrasing that avoids redundancy.

3

Verify the grammatical correctness of the concise choice.

Plugging 'Because insulin regulates blood glucose levels' into the sentence creates a grammatically correct structure with a dependent introductory clause followed by an independent clause.

Eliminating wordy options must not introduce a new grammatical error.

Key Concept

Eliminating Wordiness

Question 1646Question →

First discovered in 1977 near the Galápagos Rift, hydrothermal vents are fissures on the seafloor that release geothermally heated water. These vents support unique ecosystems that thrive in complete darkness, and they survive and exist independently of sunlight for energy.

Which of the following choices is the most effective and concise replacement for the underlined portion?

NO CHANGE

existing independently of

they exist independently, they do not rely on

which exist independently because they do not rely on

Show answer & explanation

Answer: existing independently of

Answer

The most concise and grammatically correct option is 'existing independently of', which eliminates the redundant verbs and smoothly integrates into the sentence.

The correct answer is the option that reads 'existing independently of'. This option is the most concise and grammatically correct choice because it avoids repeating the redundant ideas of surviving and existing, which are already implied by the word 'thrive' earlier in the sentence. It also uses a participle phrase to cleanly modify the preceding clause without introducing comma splices or fragments.

Step-by-Step Solution

1

Identify the redundancy in the underlined portion.

The verbs 'survive' and 'exist' in 'and they survive and exist independently of' repeat the same concept of survival or existence, which is already established by 'thrive' in the sentence.

Redundancy must be eliminated to make the sentence concise and clear.

2

Evaluate the grammar and structure of each option.

The option containing 'they exist independently, they do not rely on' creates a comma splice. The option containing 'which exist independently because they do not rely on' is unnecessarily wordy and creates a clunky dependent clause fragment. The option 'existing independently of' is concise and uses a participle phrase that functions correctly as an adverbial modifier.

The correct option must be both grammatically correct and concise.

Key Concept

Eliminating Redundancy

Question 1647Question →

An artist mixes blue, yellow, and white paint in a ratio of $5:6:3$ , respectively. To create a new shade, she adds $8\text{ ounces}$ of blue paint and $8\text{ ounces}$ of white paint to the mixture, but adds no yellow paint. If the ratio of yellow paint to white paint in the new mixture is $6:5$ , how many ounces of blue paint are in the new mixture?

20

24

28

52

72

Show answer & explanation

Answer: 28

Answer

28

The correct answer is

28

ounces. Let the initial quantities of blue, yellow, and white paint be

5x

,

6x

, and

3x

, respectively. Adding

8

ounces of blue and

8

ounces of white paint gives new amounts of

5x + 8

for blue and

3x + 8

for white. The amount of yellow paint remains

6x

. The new ratio of yellow to white paint is

\frac{6x}{3x + 8} = \frac{6}{5}

. Cross-multiplying yields

30x = 18x + 48

, which simplifies to

12x = 48

, giving

x = 4

. Substituting

x = 4

into the expression for the new amount of blue paint gives

5(4) + 8 = 28

ounces.

Step-by-Step Solution

1

Define variables for the initial quantities of paint using a common multiplier

x

.

Blue paint =

5x

, yellow paint =

6x

, and white paint =

3x

.

Since the ratio of blue to yellow to white paint is

5:6:3

, the actual amounts are multiples of these ratio parts.

2

Express the new amounts of paint after adding the additional ounces.

New blue paint =

5x + 8

, new yellow paint =

6x

, and new white paint =

3x + 8

.

The artist adds

8\text{ ounces}

of blue and

8\text{ ounces}

of white paint, with no change to the yellow paint.

3

Set up a proportion using the ratio of yellow paint to white paint in the new mixture.

\frac{6x}{3x + 8} = \frac{6}{5}

The problem states that the ratio of yellow paint to white paint in the new mixture is

6:5

.

4

Solve the equation for the multiplier

x

.

30x = 6(3x + 8) \implies 30x = 18x + 48 \implies 12x = 48 \implies x = 4

.

Cross-multiplying allows us to solve the linear equation for the scale factor.

5

Find the final amount of blue paint in the new mixture by substituting

x = 4

.

5(4) + 8 = 28\text{ ounces}

.

The question asks for the ounces of blue paint in the new mixture, which is represented by

5x + 8

.

Key Concept

Setting up and solving algebraic proportions from multi-part ratios when quantities are modified.

Question 1648Question →

An astronomical unit (AU) is a unit of length equal to approximately $1.50 \times 10^8$ kilometers. A research probe travels through space at a constant speed of $2.50 \times 10^4$ meters per second. Which of the following is closest to the number of hours it will take the probe to travel a distance of $5.00$ AU?

8.33 \times 10^0

8.33 \times 10^3

8.33 \times 10^4

3.00 \times 10^7

1.08 \times 10^{11}

Show answer & explanation

Answer: $8.33 \times 10^3$

Answer

The correct answer is

8.33 \times 10^3

hours.

To find the travel time in hours, the distance of

5.00

AU is first converted to meters:

5.00 \times 1.50 \times 10^8\text{ km} \times 10^3\text{ m/km} = 7.50 \times 10^{11}\text{ m}

. Dividing this distance by the speed of

2.50 \times 10^4\text{ m/s}

yields a time of

3.00 \times 10^7\text{ seconds}

. Dividing this time by the

3,600\text{ seconds}

in an hour results in

\frac{3.00 \times 10^7}{3,600} \approx 8,333.33\text{ hours}

, which is written in standard scientific notation as

8.33 \times 10^3\text{ hours}

.

Step-by-Step Solution

1

Convert the total distance from astronomical units (AU) to meters.

Distance =

7.50 \times 10^{11}

meters

Since

1\text{ AU} \approx 1.50 \times 10^8\text{ km}

and

1\text{ km} = 10^3\text{ m}

, then

1\text{ AU} \approx 1.50 \times 10^{11}\text{ m}

. Therefore, a distance of

5.00\text{ AU} = 5.00 \times (1.50 \times 10^{11}\text{ m}) = 7.50 \times 10^{11}\text{ m}

.

2

Calculate the travel time in seconds.

Time =

3.00 \times 10^7

seconds

Dividing the total distance in meters by the speed in meters per second gives:

\frac{7.50 \times 10^{11}\text{ m}}{2.50 \times 10^4\text{ m/s}} = 3.00 \times 10^7\text{ seconds}

.

3

Convert the travel time from seconds to hours.

Time

\approx 8.33 \times 10^3

hours

Since

1\text{ hour} = 3,600\text{ seconds}

(which is

3.60 \times 10^3\text{ seconds}

), we divide the time in seconds by

3,600

:

\frac{3.00 \times 10^7}{3.60 \times 10^3} \approx 0.833 \times 10^4 = 8.33 \times 10^3\text{ hours}

.

Key Concept

Applying division and unit conversion with numbers written in scientific notation.

Estimated Time:2m 0s

Question 1649Question →

Four students are comparing the portion of their homework assignments they have completed:

* Jamie has completed $35\%$ of his assignment.
* Kayla has completed $\frac{3}{8}$ of her assignment.
* Leo has completed $0.4$ of his assignment.
* Mia has completed $\frac{1}{2}$ of her assignment.

Arrange the students by the portion of their homework completed, from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), and Mia (

\frac{1}{2}

)

Converting all values to decimals yields

35\% = 0.35

,

\frac{3}{8} = 0.375

,

0.4 = 0.40

, and

\frac{1}{2} = 0.50

. Comparing these decimals gives

0.35 < 0.375 < 0.40 < 0.50

, which corresponds to Jamie, Kayla, Leo, and Mia.

Step-by-Step Solution

1

Convert each portion to a decimal value to make them easy to compare.

Jamie:

35\% = 0.35

Kayla:

\frac{3}{8} = 0.375

Leo:

0.4 = 0.40

Mia:

\frac{1}{2} = 0.50

Converting all values to a common format (decimals) allows for direct numerical comparison.

2

Compare the decimal values from least to greatest.

0.35 < 0.375 < 0.40 < 0.50

This establishes the relative sizes of the portions.

3

Match each decimal back to the student and order them.

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), Mia (

\frac{1}{2}

)

This provides the final ordered sequence.

Key Concept

Converting fractions, decimals, and percents to a single format to compare their values.

Question 1650Question →

Carved directly into the vibrant red sandstone cliffs of southern Jordan, the ancient city of Petra served as the capital of the Nabataean Empire. For centuries, the Nabataeans controlled the regional trade routes, accumulating immense wealth and creating a sophisticated water management system in the arid desert. However, over time, changes in trade routes and a series of devastating earthquakes led to the city's gradual decline. By the late Byzantine era, Petra was largely abandoned and forgotten by the Western world, hanging out in obscurity until Swiss explorer Johann Ludwig Burckhardt rediscovered it in 1812.

Which choice most effectively maintains the tone of the passage?

NO CHANGE

remaining in obscurity

remaining hidden from view and not seen by many people

languishing in a state of unremittingly desolate oblivion

Show answer & explanation

Answer: remaining in obscurity

Answer

remaining in obscurity

The correct answer provides a neutral, formal, and concise phrasing that matches the academic tone of the passage. It avoids slang, wordiness, and overly dramatic language.

Step-by-Step Solution

1

Analyze the established tone and style of the passage.

The passage uses a formal, academic, and historical tone to describe the history of the ancient city of Petra.

Identifying the established tone helps determine which register is appropriate for the underlined portion.

2

Evaluate the underlined phrase for stylistic appropriateness.

The phrase 'hanging out in obscurity' is colloquial and informal, creating a tone clash with the academic nature of the text.

The target phrase must be replaced because it deviates from the formal register.

3

Assess the remaining options to find a choice that is grammatically correct, concise, and stylistically consistent.

The choice 'remaining in obscurity' matches the neutral, academic tone. The choice 'remaining hidden from view and not seen by many people' is wordy and redundant, and 'languishing in a state of unremittingly desolate oblivion' is overly dramatic and pretentious.

The correct choice must maintain the passage's formal style without redundancy or excessive drama.

Key Concept

Stylistic Appropriateness

Question 1651Question →

What is the value of the expression $2^3 \cdot 2^2 - \sqrt{81}$ ?

-40

1

23

55

-49

Show answer & explanation

Answer: 23

Answer

23

To evaluate the expression, follow the order of operations. First, simplify the exponential multiplication:

2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32

. Next, evaluate the square root:

\sqrt{81} = 9

. Finally, perform the subtraction:

32 - 9 = 23

. Thus, the correct value is 23.

Step-by-Step Solution

1

Simplify the exponential term

2^3 \cdot 2^2

2^5 = 32

When multiplying exponential terms with the same base, add their exponents:

3 + 2 = 5

.

2

Evaluate the square root term

\sqrt{81}

9

The square root of 81 is the positive number that, when multiplied by itself, equals 81, which is 9.

3

Perform the subtraction:

32 - 9

23

Subtract 9 from 32 to find the final value of the expression.

Key Concept

Simplifying numerical expressions using laws of exponents, square roots, and order of operations

Question 1652Question →

If $a = 3$ and $b = 2$ , what is the value of the expression $\sqrt{a^4 - b^5}$ ?

Show answer & explanation

Answer: 7

Answer

The value of the expression is 7.

Evaluating the exponents yields

3^4 = 81

and

2^5 = 32

. Substituting these values under the radical gives

\sqrt{81 - 32} = \sqrt{49}

. Evaluating the square root of 49 gives the correct value of 7.

Step-by-Step Solution

1

Evaluate the terms with exponents using the given values

a = 3

and

b = 2

.

a^4 = 3^4 = 81

and

b^5 = 2^5 = 32

Exponent operations must be evaluated before performing subtraction or taking the root.

2

Substitute the evaluated terms back into the radical expression and subtract.

\sqrt{81 - 32} = \sqrt{49}

Simplifying the expression under the radical is necessary before taking the square root.

3

Find the square root of the simplified value.

\sqrt{49} = 7

Evaluating the square root of 49 completes the simplification of the expression.

Key Concept

Evaluating expressions involving exponents, subtraction, and square roots

Question 1653Question →

The following passage is adapted from an article about historical audio preservation. Read the passage and fill in the blank with a word that precisely and clearly completes the sentence while maintaining the academic tone of the passage.

Fill in the blanks below

To recover the voices of the past, audio preservationists must carefully digitize delicate wax cylinders from the early twentieth century. Because these cylinders degrade with each playback, engineers use specialized optical scanners to map the grooves without physical contact. This non-invasive method successfully the acoustic data while preventing further wear to the fragile media.

Show answer & explanation

Answer

The blank should be filled with a precise, academic verb such as 'extracts', 'retrieves', 'captures', or 'recovers' to describe obtaining data from the wax cylinder grooves.

The correct verb must precisely describe how the optical scanning method obtains data from the physical grooves of a wax cylinder without physical contact, while maintaining a formal, academic tone. Verbs such as 'extracts', 'retrieves', 'captures', or 'recovers' are precise, fit the context of data acquisition, and align with the passage's professional register.

Step-by-Step Solution

1

Analyze the context of the sentence containing the blank to determine the required part of speech and meaning.

The blank requires a third-person singular verb that describes the action the 'non-invasive method' performs on the 'acoustic data'.

This establishes the grammatical framework for the word choice.

2

Evaluate the surrounding sentences to understand the technical process and the appropriate register.

The passage describes engineers using optical scanners to map grooves and obtain sound without touching the cylinder. The word must be precise and match the formal, academic tone.

This guides the selection of a word that is neither too informal nor semantically contradictory.

3

Select a verb that precisely denotes the acquisition of data through a technical process.

Words like 'extracts', 'retrieves', 'captures', or 'recovers' accurately describe getting data from a physical medium using technology.

These choices maintain the formal tone and provide the necessary clarity and precision.

Key Concept

Word Precision and Clarity

Question 1654Question →

At Oakridge High School, $\frac{3}{5}$ of the students participate in extracurricular sports. Of the students who participate in extracurricular sports, $25\%$ are members of the track and field team. What percent of the total students at Oakridge High School are on the track and field team?

Show answer & explanation

Answer: 15

Answer

The correct answer is 15.

To find the portion of the total students on the track team, convert the fraction of students in sports to a decimal, which is 0.60. Then, multiply this by the track team percentage expressed as a decimal, which is 0.25. The product is 0.15, which represents 15% of the total student body.

Step-by-Step Solution

1

Convert the fraction of students playing sports to a decimal

0.60

To easily multiply the two portions, convert the fraction to a decimal:

\frac{3}{5} = 0.60

.

2

Multiply the sports-participating portion by the track team percentage

0.15

To find a percentage of a decimal, convert the percentage to a decimal (

25\% = 0.25

) and multiply:

0.60 \times 0.25 = 0.15

.

3

Convert the final decimal back to a percentage

15\%

Multiply the decimal by 100 to express the final portion as a percentage:

0.15 \times 100 = 15\%

.

Key Concept

Fractions, Decimals, and Percentages

Estimated Time:45s

Question 1655Question →

The prime factorization of a positive integer $N$ is of the form $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are non-negative integers. The number $N$ has exactly $12$ positive factors. If $N$ is a multiple of $4$ but is not divisible by $3$ , what is the smallest possible value of $N$ ?

Show answer & explanation

Answer: 160

Answer

The smallest possible value of

N

is 160.

The correct value is 160 because it satisfies all the conditions: its prime factorization has only 2 and 5 (no 3, so not divisible by 3), the exponent of 2 is 5 (which is greater than or equal to 2, so it is a multiple of 4), the number of factors is

(5+1)(1+1) = 12

, and it is the smallest such integer.

Step-by-Step Solution

1

Set up the factor counting formula using the prime factorization.

The number of positive factors of

N = 2^a \times 3^b \times 5^c

is

(a+1)(b+1)(c+1) = 12

.

The number of positive factors of any integer is found by adding one to each exponent in its prime factorization and multiplying the results.

2

Apply the given divisibility conditions to simplify the exponents.

Since

N

is not divisible by

3

,

b = 0

. Since

N

is a multiple of

4

,

a \ge 2

. The equation simplifies to

(a+1)(c+1) = 12

with

a+1 \ge 3

.

A number is not divisible by a prime if its exponent in the prime factorization is 0. A number is a multiple of 4 if the exponent of 2 in its prime factorization is at least 2.

3

List all possible pairs of

(a+1, c+1)

that multiply to 12 where

a+1 \ge 3

, and compute the corresponding values of

N

.

Case 1:

a+1 = 3 \implies a = 2, c = 3 \implies N = 2^2 \times 5^3 = 500

. Case 2:

a+1 = 4 \implies a = 3, c = 2 \implies N = 2^3 \times 5^2 = 200

. Case 3:

a+1 = 6 \implies a = 5, c = 1 \implies N = 2^5 \times 5^1 = 160

. Case 4:

a+1 = 12 \implies a = 11, c = 0 \implies N = 2^{11} \times 5^0 = 2048

.

Testing all possible divisor pairs of 12 that satisfy the constraint on

a

allows us to find all possible candidate values for

N

.

4

Determine the smallest value of

N

from the candidate values.

The smallest candidate is 160.

Comparing the values 500, 200, 160, and 2048 shows that 160 is the minimum.

Key Concept

Factors, Multiples, and Prime Factorization

Question 1656Question →

A student simplifies the expression $7 + 3(x + 4) - 7$ in three steps as shown below:

Step 1: $7 - 7 + 3(x + 4)$
Step 2: $0 + 3(x + 4)$
Step 3: $3x + 12$

Which of the following lists the mathematical properties of real numbers that justify Step 1 and Step 3, respectively?

Commutative property of addition; distributive property

Associative property of addition; distributive property

Commutative property of addition; associative property of multiplication

Associative property of addition; commutative property of multiplication

Distributive property; commutative property of addition

Show answer & explanation

Answer: Commutative property of addition; distributive property

Answer

Commutative property of addition; distributive property

In Step 1, the student reorders the terms from

7 + 3(x+4) - 7

to

7 - 7 + 3(x+4)

. Since changing the order of elements being added is governed by the commutative property of addition, this property justifies Step 1. In Step 3, the expression

3(x + 4)

is expanded to

3x + 12

by multiplying

3

by both

x

and

4

. This operation is justified by the distributive property. Combining these, the correct pair of properties is the commutative property of addition and the distributive property.

Step-by-Step Solution

1

Analyze the transition from the initial expression to Step 1:

7 + 3(x + 4) - 7 \rightarrow 7 - 7 + 3(x + 4)

The terms are rearranged such that the constant term

-7

is moved next to the initial term

7

.

The commutative property of addition states that the order in which real numbers are added does not affect the sum (

a + b = b + a

).

2

Analyze the transition from Step 2 to Step 3:

3(x + 4) \rightarrow 3x + 12

The coefficient

3

is multiplied by both terms inside the parentheses (

x

and

4

).

The distributive property states that multiplying a sum by a number is equivalent to multiplying each term in the sum individually (

a(b + c) = ab + ac

).

Key Concept

Identifying commutative and distributive properties of real numbers.

Estimated Time:1m 0s

Question 1657Question →

Three distinct integers, $a$ , $b$ , and $c$ , lie on a standard number line such that $a < b < c$ . The distance between $a$ and $b$ is $3$ times the distance between $b$ and $c$ . If $|a| = 15$ , $|c| = 7$ , and $b < 0$ , what is the value of $b$ ?

Show answer & explanation

Answer: -9

Answer

The value of

b

is

-9

.

The correct answer is

-9

. By interpreting

|a| = 15

and

|c| = 7

with the constraint

a < b < c

, we find

a = -15

. Testing the possible values for

c

, when

c = -7

, we set up the distance equation

b - (-15) = 3(-7 - b)

, which simplifies to

b + 15 = -21 - 3b

, giving

4b = -36

and

b = -9

. This satisfies all constraints, including

b

being a negative integer.

Step-by-Step Solution

1

Find the possible coordinates for

a

and

c

based on their absolute values.

a \in \{-15, 15\}

and

c \in \{-7, 7\}

.

The absolute value of a number represents its distance from zero, so

|x| = d \implies x = \pm d

.

2

Use the ordering condition

a < b < c

to eliminate invalid combinations.

a = -15

and

c \in \{-7, 7\}

.

Since

a

must be less than

c

,

a

cannot be

15

because both possible values of

c

(

-7

and

7

) are less than

15

.

3

Set up an equation representing the distance relationship on the number line.

b + 15 = 3(c - b) \implies 4b = 3c - 15

.

For points on a number line ordered

a < b < c

, the distance between

a

and

b

is

b - a

, and the distance between

b

and

c

is

c - b

.

4

Substitute each possible value of

c

and solve for

b

to find the one that yields a negative integer.

For

c = -7

,

b = -9

.

If

c = 7

,

b = 1.5

, which is not an integer. If

c = -7

,

b = -9

, which is a negative integer, satisfying all given conditions.

Key Concept

Using absolute values and relative order to determine integer positions and distances on a number line.

Question 1658Question →

Evaluate the mathematical expressions below and arrange them in order from least to greatest value.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is

\sqrt{25}

,

2^3

, and

1.2 \times 10^1

.

Evaluating each expression gives

\sqrt{25} = 5

,

2^3 = 8

, and

1.2 \times 10^1 = 12

. Comparing these values gives

5 < 8 < 12

, which yields the sequence:

\sqrt{25}

,

2^3

, and

1.2 \times 10^1

.

Step-by-Step Solution

1

Evaluate the square root expression.

\sqrt{25} = 5

Identify the non-negative number that, when multiplied by itself, equals

25

.

2

Evaluate the exponential expression.

2^3 = 8

Multiply the base,

2

, by itself three times (

2 \times 2 \times 2

).

3

Evaluate the expression in scientific notation.

1.2 \times 10^1 = 12

Multiply

1.2

by

10

to shift the decimal point one spot to the right.

4

Order the resulting values from least to greatest.

5 < 8 < 12

, which corresponds to the expressions:

\sqrt{25}

,

2^3

, and

1.2 \times 10^1

.

Compare the numbers

5

,

8

, and

12

to establish their relative sizes.

Key Concept

Evaluating basic exponents, square roots, and scientific notation, then ordering their values.

Question 1659Question →

Bioluminescence, the production and emission of light by a living organism, is a widespread phenomenon in the deep ocean, where sunlight cannot penetrate. In these pitch-black depths, organisms utilize light for communication, camouflage, and attracting prey. While some fish possess specialized organs containing symbiotic bacteria that generate light, others produce the necessary chemicals themselves. Scientists studying these deep-sea creatures have documented how they [underlined]whip up some pretty wild light shows[/underlined] to startle predators.

Which choice best maintains the established tone and style of the passage?

NO CHANGE

emit sudden flashes of light

generate repetitive and redundant displays of bright visual light

effectuate manifestations of radiant luminescence

Show answer & explanation

Answer: emit sudden flashes of light

Answer

emit sudden flashes of light

The correct option, which proposes 'emit sudden flashes of light', is appropriate because it uses a neutral, precise, and objective tone that matches the academic and scientific register of the passage. It conveys the scientific phenomenon clearly and concisely without unnecessary embellishment.

Step-by-Step Solution

1

Analyze the context and establish the tone of the surrounding passage.

The passage is a scientific description of deep-sea bioluminescence, utilizing an objective, formal, and academic tone.

Stylistic appropriateness questions require alignment with the established register of the text.

2

Evaluate the underlined portion for appropriateness.

The phrase 'whip up some pretty wild light shows' is slang-like and informal, creating a clear register clash.

Identifying the error in the original phrasing confirms that NO CHANGE is incorrect.

3

Evaluate the remaining options for tone, precision, and conciseness.

The option proposing 'emit sudden flashes of light' maintains the academic tone and is concise. The option with 'repetitive and redundant' is excessively wordy and repetitive. The option with 'effectuate manifestations' is overly bombastic and pretentious.

The correct choice must fit the scientific context without being wordy or overly dramatic.

Key Concept

Stylistic Appropriateness

Question 1660Question →

To protect their nests from hungry predators, leafcutter ants build complex underground chambers. These industrious insects, which are known to be extremely hard-working, carry fragments of foliage deep into their colonies to cultivate a specialized fungus.

Which of the following choices is the most concise and grammatically correct way to write the underlined portion of the sentence?

NO CHANGE

These industrious insects

Since these industrious insects are known to be hard-working,

These industrious insects are hard-working, they

Show answer & explanation

Answer: These industrious insects

Answer

The correct choice is the most concise version that removes the redundant clause and maintains grammatical correctness.

The correct choice is the most concise option. Because the adjective 'industrious' already means hard-working, the phrase 'which are known to be extremely hard-working' is redundant. Removing this phrase leaves a clean, grammatically correct subject that flows directly into the verb.

Step-by-Step Solution

1

Analyze the underlined portion for redundancy.

The word 'industrious' and the phrase 'known to be extremely hard-working' are synonyms that express the same idea.

ACT English questions require eliminating unnecessary words or repetitive ideas to achieve the most concise phrasing.

2

Evaluate the grammar of the remaining choices.

The most concise option without redundancy must also function grammatically. The option beginning with 'Since' creates a fragment, and the option ending with 'they' creates a comma splice.

Eliminating redundancy must not introduce new grammatical errors.

Key Concept

Eliminating Redundancy

Estimated Time:1m 0s

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