All practice questions

Question 1561Question →

A positive integer $N$ has the prime factorization $N = 2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. The greatest common divisor of $N$ and $360$ has exactly $12$ positive factors, and the least common multiple of $N$ and $360$ has exactly $72$ positive factors. How many positive factors does $N$ have?

Show answer & explanation

Answer: 36

Answer

The positive integer

N

has exactly 36 positive factors.

By writing the prime factorization of

360

as

2^3 \times 3^2 \times 5^1

and expressing the GCD and LCM of

N

and

360

in terms of the minimum and maximum exponents of their prime factors, we establish a system of equations for the exponents. Solving this system yields two cases: either

a = 2

,

b = 1

, and

c = 5

, or

a = 1

with

(b + 1)(c + 1) = 18

. In both cases, the formula for the number of positive factors of

N

, which is

(a + 1)(b + 1)(c + 1)

, evaluates to exactly 36.

Step-by-Step Solution

1

Find the prime factorization of

360

.

360 = 2^3 \times 3^2 \times 5^1

.

This allows us to write the greatest common divisor (GCD) and least common multiple (LCM) of

N

and

360

in terms of their prime factors.

2

Express the number of factors of

\text{GCD}(N, 360)

using the minimum exponents.

(\min(a, 3) + 1)(\min(b, 2) + 1)(\min(c, 1) + 1) = 12

. Since

c \ge 1

, we have

\min(c, 1) = 1

, which simplifies the equation to

(\min(a, 3) + 1)(\min(b, 2) + 1) = 6

.

The GCD of two numbers is found by taking the minimum of their exponents for each prime factor.

3

Solve for the possible values of

a

and

b

from the GCD factor equation.

Since

a, b \ge 1

, the only integer pairs for the factors of

6

are: Case 1:

a = 2

and

b = 1

, or Case 2:

a = 1

and

b \ge 2

.

We analyze the possible values of

\min(a, 3)

and

\min(b, 2)

that multiply to

6

under the constraints

a, b \ge 1

.

4

Apply the LCM condition to find the number of factors of

N

for both cases.

For Case 1:

\text{LCM}(N, 360)

has factors count

(3+1)(2+1)(c+1) = 72 \implies c = 5

, giving

d(N) = (2+1)(1+1)(5+1) = 36

. For Case 2:

\text{LCM}(N, 360)

has factors count

(3+1)(b+1)(c+1) = 72 \implies (b+1)(c+1) = 18

, giving

d(N) = (1+1)(b+1)(c+1) = 2 \times 18 = 36

.

The LCM uses the maximum of the exponents, which allows us to relate the remaining unknown exponents to the total factor count.

Key Concept

Factors, Multiples, and Prime Factorization

Question 1562Question →

Early typewriters were prone to jamming because the metal typebars would collide if adjacent keys were pressed in rapid succession. To solve this problem, inventor Christopher Sholes rearranged the keyboard layout to separate the most frequently used letter pairs. Nevertheless, typists had to search longer for keys, which slowed their typing speed and prevented the mechanism from sticking.

Which choice best fits the underlined portion?

NO CHANGE

For instance,

Consequently,

Furthermore,

Show answer & explanation

Answer: Consequently,

Answer

Consequently,

The correct choice is 'Consequently,' because it logically connects the cause (rearranging the keyboard layout to separate common letter pairs) with its direct effect (typists having to search longer for keys and typing more slowly).

Step-by-Step Solution

1

Analyze the relationship between the sentence introducing Sholes's keyboard redesign and the sentence describing the effect on typists.

The first sentence explains that Sholes rearranged the layout to separate common letter pairs, while the second sentence describes how this forced typists to search longer and slowed their typing.

Understanding the logical connection between the two sentences is necessary to choose the correct transition.

2

Identify the type of logical connection between these two ideas.

Slowing down typing speed is a direct consequence of separating the keys. Therefore, a cause-and-effect transition is required.

Determining the correct transition category helps narrow down the choices.

3

Evaluate the given choices to find the one that establishes a cause-and-effect relationship.

The transition 'Consequently' establishes cause-and-effect. Transitions of contrast ('Nevertheless'), addition ('Furthermore'), and illustration ('For instance') do not fit the context.

Choosing the option that matches the identified logical connection yields the correct answer.

Key Concept

Sentence-Level Transitions

Estimated Time:1m 0s

Question 1563Question →

### Passage
The Sunless Abyss: How Deep-Sea Hydrothermal Vents Redefined the Boundaries of Life

For centuries, a fundamental tenet of biology was that the Sun served as the ultimate source of energy for all life on Earth. Through the process of photosynthesis, plants, algae, and cyanobacteria capture solar radiation and convert it into chemical energy, forming the indispensable foundation of global food webs. In ecosystems ranging from tropical rainforests to the open ocean’s photic zone, life was understood to be completely dependent on the penetration of sunlight. Even in the ocean’s twilight zones, organisms relied on the biological pump—a continuous drift of organic debris, colloquially known as "marine snow," falling from the sunlit waters above. It was widely accepted that without sunlight, complex ecological systems could not exist.

This biological dogma was shattered in 1977 during an expedition by the research submersible *Alvin* to the Galapagos Rift, a volcanic ridge in the eastern Pacific Ocean. Geologists searching for underwater hot springs discovered thriving communities of giant red-tipped tubeworms, large clams, and blind crabs clustered around hydrothermal vents nearly 2,500 meters below the surface. In this benthic realm of perpetual darkness, freezing temperatures, and crushing pressures, scientists did not find a sparse wasteland scavenging on scarce crumbs from above. Instead, they encountered a dense biomass that rivaled the productivity of coral reefs.

The existence of these vibrant deep-sea communities presented an immediate paradox: how could such an abundant array of complex life thrive in the complete absence of solar energy? The answer lay in a biological process previously considered minor and ecologically insignificant: chemosynthesis. Unlike photosynthesis, which uses light to synthesize organic compounds from carbon dioxide and water, chemosynthesis relies on the oxidation of inorganic molecules to produce energy. At hydrothermal vents, superheated water enriched with dissolved minerals spews from Earth’s crust. Among these minerals, hydrogen sulfide ( $H_2S$ )—a chemical highly toxic to most terrestrial and shallow-water organisms—serves as the primary energy currency.

Specialized, unicellular organisms known as chemolithoautotrophic bacteria are the engines of these ecosystems. These microbes oxidize hydrogen sulfide, utilizing the energy released from this chemical reaction to convert carbon dioxide into organic molecules. In doing so, these chemosynthetic bacteria serve as the primary producers of the vent communities, occupying the ecological niche filled by plants on land. Some bacteria form thick, mat-like colonies on rocky surfaces near the vents, which are grazed upon by small snails, limpets, and shrimp. Others have evolved highly specialized symbiotic relationships with larger organisms, most notably the giant tubeworm *Riftia pachyptila*.

The symbiotic relationship between *Riftia pachyptila* and chemosynthetic bacteria is one of the most remarkable evolutionary adaptations in the natural world. These tubeworms, which can grow up to eight feet in length, possess neither a mouth, a gut, nor an anus. Instead, their internal cavity is filled with a highly vascularized organ called a trophosome, which houses billions of symbiotic bacteria. The tubeworm's bright red plumes absorb dissolved oxygen, carbon dioxide, and hydrogen sulfide directly from the hydrothermal fluids and transport them via specialized hemoglobin to the trophosome. The bacteria oxidize the sulfide to produce organic carbon, which nourishes the host tubeworm, while the tubeworm provides the bacteria with a steady supply of reactants and a safe habitat.

Ultimately, the discovery of hydrothermal vent communities permanently transformed our understanding of life by demonstrating that complex, highly productive ecosystems can exist entirely independent of solar energy through the process of chemosynthesis. This paradigm shift has had profound implications extending far beyond marine biology. Astrobiologists, for instance, have revised their criteria for search zones of extraterrestrial life. Rather than limiting their search to planets with sunlit surfaces, they now look to icy moons within our solar system, such as Jupiter’s Europa or Saturn’s Enceladus, which are believed to possess sub-surface liquid oceans heated by tidal forces and volcanic activity, potentially hosting hydrothermal systems of their own. Furthermore, evolutionary biologists hypothesize that hydrothermal vents, with their rich chemical gradients and protected deep-sea environments, may have served as the cradle for the origin of life on Earth itself.

***

Based on the passage, is the following statement true or false?

"The discovery of deep-sea hydrothermal vent communities permanently transformed biological science by demonstrating that complex, highly productive ecosystems can exist entirely independent of solar energy."

True

False

Show answer & explanation

Answer: True

Answer

True

The statement is correct because the final paragraph explicitly states that the discovery of hydrothermal vent communities permanently transformed our understanding of life by showing that complex, highly productive ecosystems can exist entirely independent of solar energy. The question statement is a direct and accurate paraphrase of this central point.

Step-by-Step Solution

1

Locate the explicit discussion of the passage's main idea and significance in the text.

The final paragraph opens with the statement: 'Ultimately, the discovery of hydrothermal vent communities permanently transformed our understanding of life by demonstrating that complex, highly productive ecosystems can exist entirely independent of solar energy through the process of chemosynthesis.'

Identifying explicit main ideas requires finding the specific sentences where the author summarizes the core theme and overall takeaway of the text.

2

Compare the statement in the question with the located sentence from the passage.

The statement directly paraphrases the passage's explicit claim that the discovery of these vent communities permanently changed biology by showing ecosystems can exist completely independent of solar energy.

Comparing the question's statement to the text's explicit main idea confirms its factual accuracy according to the passage.

Key Concept

Identifying Explicit Passage-Level Main Ideas

Question 1564Question →

Passage Excerpt:
In 2009, chemist Mas Subramanian and his team at Oregon State University were researching materials for electronics applications. While heating manganese oxide along with other chemicals to high temperatures, they accidentally created a brilliant, durable new pigment: YInMn Blue. Chemists had long sought a blue pigment that was both stable and non-toxic, [X]

The writer wants to emphasize the safety benefits of YInMn Blue by contrasting it with the specific chemical hazards of traditional blue pigments. Given that all the choices are true, which one best accomplishes this goal?

since traditional options like cobalt blue and Prussian blue contained toxic elements such as cobalt and cyanide.

since traditional options like cobalt blue and Prussian blue were very expensive and highly sought after by painters.

since traditional options like cobalt blue and Prussian blue will contain toxic elements such as cobalt and cyanide.

traditional options like cobalt blue and Prussian blue contained toxic elements such as cobalt and cyanide, they were dangerous.

Show answer & explanation

Answer: since traditional options like cobalt blue and Prussian blue contained toxic elements such as cobalt and cyanide.

Answer

since traditional options like cobalt blue and Prussian blue contained toxic elements such as cobalt and cyanide.

The correct option, which states that traditional options contained toxic elements such as cobalt and cyanide, is grammatically correct and directly fulfills the writer's goal of highlighting safety benefits by contrasting YInMn Blue with the specific health hazards of older pigments.

Step-by-Step Solution

1

Identify the rhetorical goal specified in the prompt.

The goal is to highlight the safety benefits of the new pigment by referencing the specific chemical hazards of traditional blue pigments.

Recognizing the exact criteria prevents choosing options that are grammatically correct but rhetorically off-topic.

2

Check the options for grammatical correctness.

The option containing 'will contain' creates an illogical shift to future tense, and the option lacking a subordinating conjunction creates a comma splice.

Eliminating grammatically flawed options helps narrow down the choices.

3

Evaluate the remaining grammatically correct options against the rhetorical goal.

The option mentioning high costs does not address safety, whereas the option detailing toxic elements like cobalt and cyanide directly emphasizes the chemical hazards of traditional pigments.

The option that satisfies both grammatical and rhetorical requirements is the correct choice.

Key Concept

Meeting Specific Rhetorical Goals

Estimated Time:1m 0s

Question 1565Question →

For all real numbers $x$ and $y$ , two custom operations, $\oplus$ and $\otimes$ , are defined as follows:

x \oplus y = x + y - 1

x \otimes y = xy - x - y + 2

Which of the following statements about these operations must be true for all real numbers $a$ , $b$ , and $c$ ?

I. $a \oplus (b \oplus c) = (a \oplus b) \oplus c$
II. $a \otimes b = b \otimes a$
III. $a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c)$

I only

II only

I and II only

II and III only

I, II, and III

Show answer & explanation

Answer: I, II, and III

Answer

The statement containing all three Roman numerals, I, II, and III, is correct because the associative, commutative, and distributive properties all hold true under these custom definitions.

All three statements are true. First, the operation

\oplus

is associative because both

a \oplus (b \oplus c)

and

(a \oplus b) \oplus c

simplify to

a + b + c - 2

. Second, the operation

\otimes

is commutative because

a \otimes b = ab - a - b + 2

and

b \otimes a = ba - b - a + 2

are algebraically identical. Third, the operation

\otimes

distributes over

\oplus

because both

a \otimes (b \oplus c)

and

(a \otimes b) \oplus (a \otimes c)

simplify to

ab + ac - 2a - b - c + 3

. Therefore, the statement including all three Roman numerals is correct.

Step-by-Step Solution

1

Evaluate Statement I for associativity by expanding

a \oplus (b \oplus c)

and

(a \oplus b) \oplus c

.

Both expressions simplify to

a + b + c - 2

.

Since both groupings yield the same algebraic expression, the operation

\oplus

is associative.

2

Evaluate Statement II for commutativity by comparing

a \otimes b

and

b \otimes a

.

a \otimes b = ab - a - b + 2

and

b \otimes a = ba - b - a + 2

.

Since multiplication and addition of real numbers are commutative (

ab = ba

and

-a - b = -b - a

), the two expressions are equal, meaning the operation

\otimes

is commutative.

3

Evaluate Statement III for distributivity by expanding both sides of

a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c)

.

The left side expands to

ab + ac - 2a - b - c + 3

. The right side also expands to

ab + ac - 2a - b - c + 3

.

Because both sides simplify to the exact same expression, the operation

\otimes

distributes over

\oplus

.

Key Concept

Testing the definitions of the commutative, associative, and distributive properties using custom operations.

Question 1566Question →

Although nineteenth-century philologists could find no written records of the ancient tongue, they successfully reconstructed Proto-Indo-European, which was a prehistoric ancestral language from which later modern languages descended from in the past.

Which choice is the most effective and concise alternative to the underlined portion of the sentence?

which was a prehistoric ancestral language from which later modern languages descended from in the past

an ancestral tongue.

an ancestral tongue. Which served as the prehistoric parent language.

an ancestral tongue, later languages descended from it.

Show answer & explanation

Answer: an ancestral tongue.

Answer

The correct choice is the option that replaces the underlined text with 'an ancestral tongue.'

The correct choice is the option that reads 'an ancestral tongue.' This choice is grammatically correct and is the most concise way to express the relationship between Proto-Indo-European and later languages. It removes the unnecessary, repetitive descriptions of prehistoric time and descendant lineages present in the original sentence.

Step-by-Step Solution

1

Analyze the underlined portion for redundant phrasing and grammatical errors.

The original phrasing repeats the idea of 'past' (using both 'prehistoric' and 'in the past') and the concept of ancestry (using both 'ancestral' and 'from which later modern languages descended'). It also mistakenly duplicates the preposition 'from'.

Identifying redundancies and errors helps in eliminating incorrect distractors.

2

Evaluate the other choices for conciseness and grammatical correctness.

The option containing 'an ancestral tongue.' is concise and correct. The option containing 'Which served as...' creates a sentence fragment. The option containing 'later languages descended...' creates a comma splice.

The correct ACT English alternative must be the most concise option that is free of grammatical errors.

Key Concept

Eliminating Redundancy

Estimated Time:1m 0s

Question 1567Question →

By analyzing the calcified dental plaque of ancestral populations, biochemists can extract remarkably preserved proteins that reveal prehistoric dietary habits. This biochemical approach offers direct evidence of ancient nutrition, bypassing the need for speculative inferences. Scientists can now reconstruct prehistoric diets down to the nitty-gritty.

Which choice best maintains the established tone of the passage?

NO CHANGE

with unprecedented detail

with an ineffable level of specificity

on unprecedented detail

Show answer & explanation

Answer: with unprecedented detail

Answer

The correct choice is 'with unprecedented detail' because it maintains the formal, academic style of the passage and uses the correct preposition.

The passage describes a scientific process using formal, academic language. The phrase 'with unprecedented detail' fits this established tone perfectly while remaining grammatically correct and idiomatic.

Step-by-Step Solution

1

Analyze the tone of the surrounding sentences in the passage.

The passage uses formal, scientific language (e.g., 'calcified dental plaque', 'ancestral populations', 'speculative inferences') to describe archaeological biochemistry, establishing an academic and objective tone.

Identifying the established tone register of the passage is necessary to ensure the underlined portion matches it consistently.

2

Evaluate the options against the established academic tone.

The phrase 'down to the nitty-gritty' is too informal/colloquial. The phrase 'with an ineffable level of specificity' is too dramatic/pretentious. The phrase 'on unprecedented detail' uses an incorrect preposition. The phrase 'with unprecedented detail' is both grammatically correct and matches the academic tone.

Comparing the options allows us to eliminate choices that disrupt tone consistency or violate grammatical rules.

Key Concept

Tone Consistency

Question 1568Question →

A writer is editing an essay about early bridge engineering and has drafted the following paragraph:

James Finley, a judge in Pennsylvania, revolutionized transportation in $1801$ by building the first modern suspension bridge. Finley utilized iron chains to support a level wooden deck, allowing heavy carriages to cross Jacob's Creek safely. Prior to Finley's design, suspension bridges were constructed from rope or vines, which dipped significantly in the center and were highly unstable in strong winds. Finley patented his design in $1808$ , and he went on to write several influential pamphlets describing the structural engineering principles behind his bridge.

Should the writer keep or delete the underlined sentence?

Keep it, because it establishes the historical context that highlights the significance of Finley's structural innovation.

Keep it, because it details the geographical regions where rope and vine bridges were traditionally built.

Delete it, because it shifts the focus from Finley's engineering contributions to ancient bridge-building materials.

Delete it, because it repeats information about transport safety that is mentioned earlier in the paragraph.

Show answer & explanation

Answer: Keep it, because it establishes the historical context that highlights the significance of Finley's structural innovation.

Answer

Keep it, because it establishes the historical context that highlights the significance of Finley's structural innovation.

The correct option states that the sentence should be kept because it establishes the historical context that highlights the significance of Finley's structural innovation. The paragraph focuses on Finley's revolutionary bridge design, and contrasting it with the instability of prior designs helps the reader appreciate the importance of his invention.

Step-by-Step Solution

1

Identify the main focus of the paragraph.

The paragraph describes James Finley's development of the modern suspension bridge, focusing on its design innovations.

Understanding the paragraph's theme is necessary to evaluate the relevance of the sentence in question.

2

Analyze the function of the underlined sentence within the paragraph.

The sentence explains that prior bridges made of rope or vines were unstable and dipped, which contrasts with Finley's level and safe design.

This determines if the sentence contributes to the focus of the paragraph or introduces an irrelevant digression.

3

Evaluate the decision and choose the correct justification.

The sentence should be kept to provide historical context. The option containing 'Keep it' and mentioning 'historical context' is selected.

Only one option correctly identifies both the correct decision (keep) and the logical rhetorical purpose.

Key Concept

Evaluating the relevance of supporting details by assessing whether they contribute to the main topic of a paragraph or create an irrelevant shift in focus.

Question 1569Question →

A teacher has a bag of marbles to distribute to a class. If the marbles are divided equally among $8$ students, there are $6$ marbles left over. If the marbles are divided equally among $9$ students, there are $7$ marbles left over. What is the least possible number of marbles in the bag?

Show answer & explanation

Answer: 70

Answer

The least possible number of marbles in the bag is 70.

The correct answer is 70. The problem describes a situation where the number of marbles,

N

, leaves a remainder of

6

when divided by

8

, and a remainder of

7

when divided by

9

. This means

N + 2

is a multiple of both

8

and

9

. The smallest positive integer that is a multiple of both

8

and

9

is their least common multiple, which is

72

. Therefore,

N + 2 = 72

, which gives

N = 70

.

Step-by-Step Solution

1

Analyze the relationship between the divisors and their respective remainders.

For divisor

8

, the remainder is

6

, which is

8 - 6 = 2

less than the divisor. For divisor

9

, the remainder is

7

, which is

9 - 7 = 2

less than the divisor. Therefore, adding

2

to the total number of marbles creates a number that is exactly divisible by both

8

and

9

.

This establishes that the target number plus

2

must be a common multiple of the two divisors.

2

Calculate the least common multiple (LCM) of the divisors

8

and

9

.

The prime factorization of

8

is

2^3

and of

9

is

3^2

. The LCM is the product of the highest powers of all prime factors involved:

2^3 \times 3^2 = 8 \times 9 = 72

.

To find the smallest positive common multiple of

8

and

9

.

3

Subtract

2

from the LCM to find the least possible number of marbles.

72 - 2 = 70

.

Since the bag is

2

marbles short of having a multiple of both

8

and

9

, we subtract

2

from the least common multiple.

Key Concept

Least Common Multiple (LCM) application with remainders

Question 1570Question →

A woodworker has two wooden boards. One board is $42$ inches long, and the other is $70$ inches long. She wants to cut both boards into shorter pieces of equal length, with no wood left over. What is the greatest possible length, in inches, of each shorter piece?

2

7

14

28

210

Show answer & explanation

Answer: 14

Answer

14

To find the greatest possible length of the pieces, we need to find the greatest common factor (GCF) of the two board lengths,

42

and

70

. The prime factorization of

42

is

2 \times 3 \times 7

, and the prime factorization of

70

is

2 \times 5 \times 7

. The common prime factors are

2

and

7

. Multiplying these common prime factors gives the GCF:

2 \times 7 = 14

. Therefore, the greatest possible length of each piece is

14

inches.

Step-by-Step Solution

1

Identify the mathematical concept needed to solve the problem.

Since we need to cut two boards of lengths

42

and

70

into equal shorter pieces with no left over wood, we need to find a common factor of

42

and

70

. To find the greatest possible length, we must calculate their Greatest Common Factor (GCF).

This translates the word problem into a clear mathematical operation.

2

Find the prime factorizations of both numbers.

42 = 2 \times 3 \times 7

and

70 = 2 \times 5 \times 7

.

Prime factorization helps systematically identify all factors of the two numbers.

3

Determine the common prime factors and multiply them.

The common prime factors are

2

and

7

. Multiplying them gives

2 \times 7 = 14

.

The GCF is the product of all shared prime factors.

Key Concept

Greatest Common Factor (GCF)

Question 1571Question →

Let $f(n)$ represent the number of positive factors of a positive integer $n$ . If $n = 2^x \times 3^y$ , where $x$ and $y$ are positive integers such that $x + y = 7$ , what is the maximum possible value of $f(n)$ ?

Show answer & explanation

Answer: 20

Answer

The maximum possible value of

f(n)

is

20

.

The number of positive factors of

n = 2^x \times 3^y

is

(x + 1)(y + 1)

. Since

x

and

y

are positive integers that sum to 7, the possible pairs for

(x, y)

are

(1, 6)

,

(2, 5)

, and

(3, 4)

(and their reversals). The products

(x+1)(y+1)

for these pairs are

2 \times 7 = 14

,

3 \times 6 = 18

, and

4 \times 5 = 20

. The maximum value is 20.

Step-by-Step Solution

1

Determine the formula for the number of factors of

n

.

f(n) = (x + 1)(y + 1)

For any positive integer expressed in its prime factorization

p^a \times q^b

, the total number of positive factors is

(a + 1)(b + 1)

.

2

Identify the possible values for

x

and

y

given the constraints.

(x, y) \in \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\}

The exponents

x

and

y

must be positive integers (

x, y \geq 1

) that sum to 7.

3

Calculate

(x + 1)(y + 1)

for each pair to find the maximum product.

The maximum product is

4 \times 5 = 20

.

Evaluating the products:

2 \times 7 = 14

,

3 \times 6 = 18

, and

4 \times 5 = 20

. The product of two integers with a fixed sum is maximized when the integers are as close as possible.

Key Concept

Calculating the number of positive factors of an integer from its prime factorization and maximizing the count under exponent constraints.

Estimated Time:1m 30s

Question 1572Question →

The temperature at the top of a mountain is $-14^\circ\text{C}$ , and the temperature at the base of the mountain is $8^\circ\text{C}$ . What is the absolute difference, in degrees Celsius, between these two temperatures?

-22

-6

6

14

22

Show answer & explanation

Answer: 22

Answer

22

The correct answer is 22 because the absolute difference between

8

and

-14

is

|8 - (-14)| = |8 + 14| = |22| = 22

. This represents the total distance of 22 units between the two temperatures on a number line.

Step-by-Step Solution

1

Translate the problem into a mathematical expression using absolute value for the difference between the two temperatures.

The absolute difference is given by

|8 - (-14)|

or

|-14 - 8|

.

The absolute difference between two numbers

a

and

b

on a number line is defined as

|a - b|

.

2

Simplify the subtraction inside the absolute value.

8 - (-14) = 8 + 14 = 22

(or

-14 - 8 = -22

).

Subtracting a negative number is equivalent to adding its positive counterpart.

3

Evaluate the absolute value.

|22| = 22

(or

|-22| = 22

).

The absolute value of a number represents its non-negative distance from zero.

Key Concept

Integers, Absolute Value, and Number Lines

Estimated Time:45s

Question 1573Question →

Four mathematical expressions are shown below:

* Expression 1: $-3^2 - 12 \div 3 \times 2$
* Expression 2: $-| -4 - 2 \times 3 | + (-2)^3$
* Expression 3: $4 - 3(2 - 3^2)$
* Expression 4: $-2^4 \div 8 \times (-1 - 1)^2$

Order the expressions by their simplified numerical 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 value is Expression 2, Expression 1, Expression 4, and Expression 3.

Evaluating each expression using the correct order of operations yields: Expression 1 equals

-17

, Expression 2 equals

-18

, Expression 3 equals

25

, and Expression 4 equals

-8

. Arranging these values from least to greatest results in

-18 < -17 < -8 < 25

, corresponding to the sequence: Expression 2, Expression 1, Expression 4, Expression 3.

Step-by-Step Solution

1

Evaluate Expression 1.

-17

Apply the order of operations: first evaluate the exponent

-3^2 = -9

. Next, perform multiplication and division from left to right:

12 \div 3 = 4

, and

4 \times 2 = 8

. Finally, perform the subtraction:

-9 - 8 = -17

.

2

Evaluate Expression 2.

-18

Simplify inside the absolute value first by performing multiplication before subtraction:

2 \times 3 = 6

, and

-4 - 6 = -10

. The absolute value of

-10

is

10

, which becomes

-10

due to the negative sign outside. Next, evaluate the exponent:

(-2)^3 = -8

. Finally, add the terms:

-10 + (-8) = -18

.

3

Evaluate Expression 3.

25

Simplify inside the parentheses first, evaluating the exponent before subtracting:

3^2 = 9

, and

2 - 9 = -7

. Next, perform the multiplication:

3 \times (-7) = -21

. Finally, subtract:

4 - (-21) = 4 + 21 = 25

.

4

Evaluate Expression 4.

-8

Simplify the expression inside the parentheses:

-1 - 1 = -2

. Next, evaluate the exponents:

-2^4 = -16

and

(-2)^2 = 4

. Perform division and multiplication from left to right:

-16 \div 8 = -2

, and

-2 \times 4 = -8

.

5

Compare the simplified values to order them from least to greatest.

-18 < -17 < -8 < 25

Comparing the results shows that the values in ascending order are

-18

(Expression 2),

-17

(Expression 1),

-8

(Expression 4), and

25

(Expression 3).

Key Concept

Order of Operations and Number Properties

Estimated Time:2m 30s

Question 1574Question →

Passage Excerpt:
In 1885, Wilson Bentley, a farmer from Vermont, became the first person to photograph a single snowflake. Bentley accomplished this by attaching a microscope to his bellows camera. [X] Over his lifetime, he captured more than five thousand images of snowflakes, proving that no two are identical.

At this point in the passage, the writer wants to add a sentence that emphasizes the painstaking care and physical discomfort Bentley endured to capture his images. Which choice best accomplishes this goal?

He stood for hours in unheated woodsheds during Vermont's freezing winters, carefully transferring each fragile crystal with a feather so it would not melt.

His photographs were eventually published in scientific journals and purchased by several major universities.

He believed that capturing the beauty of nature was his true calling, despite the lack of initial financial reward.

Bentley's neighbors initially mocked his hobby as a waste of time, he persisted anyway because of his passion for nature.

Show answer & explanation

Answer: He stood for hours in unheated woodsheds during Vermont's freezing winters, carefully transferring each fragile crystal with a feather so it would not melt.

Answer

He stood for hours in unheated woodsheds during Vermont's freezing winters, carefully transferring each fragile crystal with a feather so it would not melt.

The correct option directly addresses both aspects of the writer's goal. It highlights physical discomfort by describing Bentley standing for hours in unheated woodsheds during freezing winters, and it illustrates painstaking care by noting how he carefully transferred each fragile crystal using a feather to prevent melting.

Step-by-Step Solution

1

Analyze the prompt to identify the specific rhetorical goals requested.

The goal is to find a sentence that emphasizes two specific elements: the painstaking care Bentley took and the physical discomfort he endured.

Understanding the precise criteria of the rhetorical goal is necessary to evaluate the options.

2

Evaluate each choice against the identified criteria to see which one contains details of both care and physical discomfort.

The statement describing standing in unheated woodsheds in freezing winters (discomfort) and transferring crystals with a feather (care) is the only option that fulfills both criteria.

Analyzing each option's semantic content ensures that the selected answer meets the specific prompt requirements.

3

Check the grammatically correct options that fail the goal and eliminate any choices containing grammatical errors.

The options about journal publications and natural calling are grammatically correct but miss the rhetorical goal. The choice about neighbors mocking him is a comma splice, making it both grammatically incorrect and rhetorically off-target.

Eliminating distractors based on both content mismatches and grammatical errors confirms the correct choice.

Key Concept

Meeting Specific Rhetorical Goals

Question 1575Question →

The following paragraph contains numbered sentences.

[1] In 1886, socialite Josephine Cochrane was frustrated by how often her servants chipped her fine heirlooms while washing them by hand. [2] Cochrane set to work in a shed behind her home, measuring the plates and constructing wire compartments customized to fit them. [3] She then placed these compartments inside a copper boiler that used pressurized hot water to spray the dishes clean. [4] This innovative design proved highly effective, eventually winning the highest award at the 1893 World's Columbian Exposition. [5] Dissatisfied with this manual chore, she first sought out mechanical alternatives, but she found that none existed.

For the sake of the logic and cohesion of the paragraph, where should Sentence 5 be placed?

where it is now

after Sentence 1

after Sentence 2

after Sentence 3

Show answer & explanation

Answer: after Sentence 1

Answer

after Sentence 1

Placing Sentence 5 after Sentence 1 is correct because the phrase 'this manual chore' refers directly back to the action of 'washing them by hand' mentioned in Sentence 1. Furthermore, explaining that she first sought out existing mechanical options but found none provides the necessary motivation for her decision to build her own device in Sentence 2, maintaining a logical narrative flow.

Step-by-Step Solution

1

Identify the logical and structural markers in the sentence to be placed (Sentence 5).

Sentence 5 begins with the phrase 'Dissatisfied with this manual chore,' where the demonstrative pronoun 'this' indicates a direct reference to a previously mentioned hand-based task.

Tracking pronoun-antecedent relationships is critical for identifying logical sentence order.

2

Locate the antecedent for 'this manual chore' in the paragraph.

The antecedent is 'washing them by hand' at the end of Sentence 1.

Placing Sentence 5 immediately after Sentence 1 keeps the pronoun close to its referent and maintains coherence.

3

Verify that the chronological order of the narrative remains intact.

Placing Sentence 5 after Sentence 1 describes her search for a machine, which logically explains why she decided to build one herself in Sentence 2. The rest of the paragraph then proceeds from design (Sentence 2) to assembly (Sentence 3) to public success (Sentence 4).

Chronological coherence ensures that the development of the invention flows in a logical sequence.

Key Concept

Sentence reordering requires tracking transition words, matching demonstrative pronouns to their antecedents, and maintaining chronological order.

Estimated Time:1m 0s

Question 1576Question →

Early cartographers faced immense challenges when mapping uncharted territories. Without modern satellite technology, they had to rely on celestial navigation and physical measurements. These early mapmakers had to be extremely picky about their calculations, as even a minor error could lead sailors miles off course.

Which choice is most precise and consistent with the tone of the passage?

NO CHANGE

precise

fussy

nice

Show answer & explanation

Answer: precise

Answer

precise

The word 'precise' is correct because it is the most accurate term to describe calculations requiring strict accuracy, and it matches the formal tone of the historical passage.

Step-by-Step Solution

1

Analyze the tone and style of the surrounding text.

The passage details the challenges of early cartographers, using academic language such as 'immense challenges' and 'celestial navigation'. The appropriate tone is formal.

Word choice must align with the established register and context of the passage.

2

Evaluate the choices based on precision and style.

'Picky' and 'fussy' are informal, and 'nice' is vague. 'Precise' is both formal and descriptive of mathematical exactness.

Choosing 'precise' ensures that the sentence conveys clear meaning while maintaining tone consistency.

Key Concept

Selecting words that are precise in meaning and consistent with the passage's register.

Question 1577Question →

Read the passage below about Roman engineering. Fill in the blank with the word that most precisely and clearly completes the sentence while maintaining the passage's formal tone.

Fill in the blanks below

To maintain a steady flow of water over miles of uneven terrain, Roman engineers constructed aqueducts with a remarkably consistent gradient. If the slope were too steep, the rushing water would the stone channels; if it were too flat, the water would stagnate.

Show answer & explanation

Answer

erode

The word 'erode' is the most precise and contextually appropriate choice because it specifically describes the physical process of water wearing away stone over time, aligning perfectly with the academic and technical tone of the passage.

Step-by-Step Solution

1

Analyze the surrounding context of the sentence.

The passage describes Roman aqueducts carrying rushing water through stone channels and discusses what happens if the slope is too steep.

Understanding the context helps identify the physical process being described.

2

Determine the necessary meaning for the blank.

The word must describe the gradual wearing away of the stone channels caused by the moving water.

The sentence sets up a contrast between water flowing too fast (wearing away the channels) and water flowing too slow (stagnating).

3

Select the word that is most precise and fits the formal, historical tone of the passage.

The verb 'erode' is the most precise geological and physical term for this process.

Choosing 'erode' maintains word precision and an academic register, unlike more general words like 'damage' or informal phrases like 'eat away'.

Key Concept

Word Precision and Clarity

Estimated Time:1m 0s

Question 1578Question →

A writer is editing a passage about the history of industrial chemistry and has drafted the following paragraph:

In 1856, eighteen-year-old chemist William Henry Perkin was attempting to synthesize quinine, a treatment for malaria, in his home laboratory. Instead of obtaining a clear drug, he produced a dark, sticky sludge. Cleaning his flask with alcohol, Perkin noticed that the mixture turned a vibrant, deep purple. [1] Perkin had accidentally created mauveine, the world’s first synthetic aniline dye. This new substance revolutionized the textile industry and birthed the modern chemical industry.

The writer is considering adding the following sentence at Point [1]:

"At the time, purple dye was traditionally extracted from sea snails, making it so expensive that only royalty could afford it."

Should the writer make this addition?

Yes, because it provides historical context that helps explain the revolutionary impact of Perkin's synthetic dye.

Yes, because it explains why Perkin was specifically trying to produce a purple dye rather than malaria medication.

No, because it interrupts the description of Perkin’s chemical experiment to introduce irrelevant details about historical fashion.

No, because it repeats information about the costs of textile manufacturing that is stated elsewhere in the paragraph.

Show answer & explanation

Answer: Yes, because it provides historical context that helps explain the revolutionary impact of Perkin's synthetic dye.

Answer

Yes, because it provides historical context that helps explain the revolutionary impact of Perkin's synthetic dye.

The correct option is correct because the proposed sentence explains why a synthetic purple dye was so groundbreaking. By explaining that natural purple dye was rare and extremely expensive, the addition provides the necessary context for the reader to understand why Perkin's discovery revolutionized the textile and chemical industries.

Step-by-Step Solution

1

Analyze the main focus and flow of the paragraph.

The paragraph describes William Henry Perkin's accidental discovery of the purple dye mauveine and notes that this discovery revolutionized the textile industry.

Understanding the paragraph's purpose helps determine if the new sentence supports or distracts from this purpose.

2

Evaluate the relevance of the proposed sentence.

The proposed sentence explains that natural purple dye was traditionally extremely expensive and reserved for royalty.

This establishes why a cheap, synthetic alternative (mauveine) would be highly significant and commercially revolutionary, directly connecting to the paragraph's conclusion.

3

Determine the correct decision and justification.

The sentence should be added (Yes) because it provides context that highlights the importance of the discovery, matching the option that states it provides historical context to explain the revolutionary impact.

Comparing the reasoning to the choices confirms the correct option.

Key Concept

Evaluating Relevance for Adding or Deleting Content

Estimated Time:1m 15s

Question 1579Question →

Four distinct integers, $p$ , $q$ , $r$ , and $s$ , are represented on a standard number line. The distance between $p$ and $q$ is 3, the distance between $q$ and $r$ is 4, and the distance between $r$ and $s$ is 5. What is the minimum possible distance between $p$ and $s$ ?

Show answer & explanation

Answer: 2

Answer

The minimum possible distance between

p

and

s

is 2.

By setting

p = 0

as a reference point, the coordinate for

q

must be either

3

or

-3

. Assuming

q = 3

by symmetry,

r

must be either

-1

or

7

because it is at a distance of 4 from

q

. From

r = 7

, a distance of 5 leads to

s

being at

2

or

12

. From

r = -1

, a distance of 5 leads to

s

being at

4

or

-6

. All of these positions result in four distinct integers. The distances between

p

and

s

are the absolute values of their coordinates, which are

12

,

2

,

6

, and

4

. The minimum of these distances is

2

.

Step-by-Step Solution

1

Establish a coordinate system on the number line.

Let

p

be at position 0. Since the distance between

p

and

q

is 3,

q

is at either

3

or

-3

. By symmetry, we assume

q = 3

.

Setting one point at the origin simplifies the relative distance calculations for all other points.

2

Find the possible coordinates of

r

.

Since the distance between

q

and

r

is 4,

r

is at

3 - 4 = -1

or

3 + 4 = 7

.

The absolute value equation

|q - r| = 4

has two solutions for

r

given

q = 3

.

3

Find the possible coordinates of

s

for each case of

r

.

If

r = 7

, then

s

can be at

7 - 5 = 2

or

7 + 5 = 12

. If

r = -1

, then

s

can be at

-1 - 5 = -6

or

-1 + 5 = 4

. All generated sets contain distinct values, satisfying the requirement.

The absolute value equation

|r - s| = 5

has two solutions for

s

for each candidate coordinate of

r

.

4

Determine the minimum distance between

p

and

s

.

The possible values for the distance

|p - s|

are

|0 - 2| = 2

,

|0 - 12| = 12

,

|0 - (-6)| = 6

, and

|0 - 4| = 4

. The minimum value is 2.

Comparing all possible valid configurations ensures we find the true minimum distance.

Key Concept

Representing distances between points on a number line using absolute values and resolving configurations for distinct integers.

Question 1580Question →

What is the value of the following expression?

-3^2 - \frac{16 - 4 \times 5}{2^3 - 10}

7

-10

-11

21

39

Show answer & explanation

Answer: $-11$

Answer

-11

The correct value is

-11

. By following the standard order of operations, we first evaluate the leading term

-3^2 = -9

. Next, we evaluate the numerator of the fraction (

16 - 4 \times 5 = 16 - 20 = -4

) and the denominator of the fraction (

2^3 - 10 = 8 - 10 = -2

). Simplifying the fraction gives

\frac{-4}{-2} = 2

. Finally, subtracting this value from the leading term yields

-9 - 2 = -11

.

Step-by-Step Solution

1

Evaluate the exponent term on the left side of the subtraction.

-3^2 = -(3 \times 3) = -9

According to the order of operations, the exponent applies only to the base

3

, not the negative sign, unless parentheses are present.

2

Evaluate the numerator of the fraction by performing multiplication before subtraction.

16 - 4 \times 5 = 16 - 20 = -4

Multiplication has a higher priority than subtraction in the standard order of operations.

3

Evaluate the denominator of the fraction by evaluating the exponent before subtraction.

2^3 - 10 = 8 - 10 = -2

Exponentiation must be performed before subtraction.

4

Divide the simplified numerator by the simplified denominator to find the value of the fraction.

\frac{-4}{-2} = 2

Dividing a negative number by a negative number yields a positive result.

5

Perform the final subtraction using the values from Step 1 and Step 4.

-9 - 2 = -11

Subtracting

2

from

-9

moves further left on the number line, resulting in

-11

.

Key Concept

Order of Operations (PEMDAS/BODMAS)

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept