All practice questions

Question 4721Question →

A group of friends agrees to share the total cost, $d$ dollars, of renting a cabin equally. There are 6 friends in the group, and each friend's share of the cost is $145. Which of the following equations can be solved to find$ d$?

d + 6 = 145

d - 6 = 145

6d = 145

\frac{d}{6} = 145

\frac{6}{d} = 145

Show answer & explanation

Answer: $\frac{d}{6} = 145$

Answer

The equation

\frac{d}{6} = 145

can be solved to find the total cost.

Because the total cost of renting the cabin,

d

dollars, is split equally among 6 friends, each friend's individual share is represented by the total cost divided by 6, which is written as

\frac{d}{6}

. Since each friend's share is given as

145, this expression is set equal to 145, resulting in the equation

\frac{d}{6} = 145$.

Step-by-Step Solution

1

Determine the relationship between the total cost, the number of friends sharing the cost, and each friend's individual share.

Since the cost is shared equally, the individual share is the total cost,

d

, divided by the number of friends, 6.

This translates the sharing scenario into the algebraic expression

\frac{d}{6}

.

2

Set the algebraic expression equal to the known individual share value.

\frac{d}{6} = 145

The problem states that each friend's share is $145, so the expression representing each share must equal 145.

Key Concept

Translating real-world sharing scenarios into one-step equations involving division.

Question 4722Question →

In 1995, wildlife biologists reintroduced gray wolves to Yellowstone National Park, aiming to restore balance to an ecosystem that had suffered in their absence. Without an apex predator, the park's elk population had grown excessively, overgrazing the valley vegetation. Almost immediately after the wolves returned, they hunted the elk and begin to alter the physical geography of the park itself. As trees regenerated along the riverbanks, birds returned to the forests, and beavers built new dams in the restored waterways.

Which of the following choices best replaces the underlined word 'begin' to maintain a consistent and grammatically correct tense throughout the passage?

NO CHANGE

beginning

began

begins

Show answer & explanation

Answer: began

Answer

began

The past-tense verb 'began' is correct because the entire paragraph establishes a narrative set in the past, as shown by verbs like 'reintroduced,' 'returned,' and 'hunted.' Using the past tense maintains verb consistency throughout the passage.

Step-by-Step Solution

1

Analyze the context of the sentence and surrounding sentences to determine the overall narrative tense.

The verbs 'reintroduced', 'returned', 'hunted', 'regenerated', and 'built' establish that the passage is written in the past tense.

Verb tense consistency requires verbs describing actions in the same timeframe to match in tense unless there is a logical reason to shift.

2

Examine the grammatical structure of the sentence containing the underlined word.

The subject 'they' performs two actions in a compound predicate connected by 'and': they 'hunted' (past tense) and '[begin]' (underlined).

Verbs connected by coordinating conjunctions in a compound predicate should be parallel in tense and form.

3

Select the verb form that matches the past-tense requirement of the sentence.

The past-tense form 'began' matches 'hunted' and maintains the consistent past-tense narrative.

This establishes grammatical parallelism and maintains the temporal context of the passage.

Key Concept

Verb Tense and Consistency

Question 4723Question →

To calibrate a temperature sensor, a technician uses the formula $T_c = T_r - e$ , where $T_c$ is the calibrated temperature, $T_r$ is the raw sensor reading, and $e$ is the calibration error of the sensor. For a particular sensor, the calibration error is $-2.4^\circ\text{C}$ . If the calibrated temperature is measured to be $18.6^\circ\text{C}$ , which of the following equations can be solved to find the raw sensor reading, $T_r$ , in degrees Celsius?

18.6 = T_r - 2.4

18.6 = T_r + 2.4

18.6 = 2.4 - T_r

18.6 = -2.4 - T_r

18.6 = \\frac{T_r}{-2.4}

Show answer & explanation

Answer: $18.6 = T_r + 2.4$

Answer

The equation

18.6 = T_r + 2.4

To find the correct equation, substitute the given values into the formula

T_c = T_r - e

. Substituting

T_c = 18.6

and

e = -2.4

yields

18.6 = T_r - (-2.4)

. Subtracting a negative number is equivalent to addition, which simplifies the equation to

18.6 = T_r + 2.4

.

Step-by-Step Solution

1

Identify the given variables and their values from the problem statement.

The calibrated temperature

T_c = 18.6

and the calibration error

e = -2.4

.

Knowing the values allows for direct substitution into the formula.

2

Substitute the identified values into the calibration formula

T_c = T_r - e

.

The equation becomes

18.6 = T_r - (-2.4)

.

This establishes the relationship between the raw reading

T_r

and the known values.

3

Simplify the subtraction of the negative number on the right side of the equation.

The equation simplifies to

18.6 = T_r + 2.4

.

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

Key Concept

Substituting signed values into algebraic expressions and simplifying one-step linear equations.

Question 4724Question →

A company allocates its annual budget such that $\frac{1}{3}$ is spent on Research and Development, $\frac{1}{4}$ is spent on Marketing, and the remaining portion is spent on Operations. The Operations budget is then split, with $40\%$ of the Operations budget going to employee salaries and the rest to equipment. If the company spends $\$150,000$ on employee salaries in the Operations department, how much money does the company spend on Research and Development?

\$48,000

\$125,000

\$175,000

\$200,000

\$300,000

Show answer & explanation

Answer: $\$300,000$

Answer

\$300,000

The correct answer of

\$300,000

is found by first determining the Operations budget. Since

40\%

of the Operations budget is spent on salaries and this equals

\$150,000

, the Operations budget is

\frac{\$150,000}{0.40} = \$375,000

. The Research and Development (

\frac{1}{3}

) and Marketing (

\frac{1}{4}

) budgets combine for

\frac{7}{12}

of the total budget, leaving

\frac{5}{12}

for Operations. We set

\frac{5}{12}

of the total budget equal to

\$375,000

, which gives a total budget of

\$900,000

. Finally, the Research and Development budget is one-third of the total budget:

\frac{1}{3} \times \$900,000 = \$300,000

.

Step-by-Step Solution

1

Determine the fraction of the total budget allocated to Operations.

Operations receives

\frac{5}{12}

of the total budget.

The sum of the fractions for Research and Development and Marketing is

\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}

. The remaining fraction for Operations is

1 - \frac{7}{12} = \frac{5}{12}

.

2

Calculate the budget allocated to the Operations department.

Operations budget is

\$375,000

.

Employee salaries account for

40\%

(or

0.40

) of the Operations budget and equal

\$150,000

. Dividing

\$150,000

by

0.40

gives the total Operations budget:

\frac{\$150,000}{0.40} = \$375,000

.

3

Calculate the total company budget.

Total budget is

\$900,000

.

Since the Operations budget of

\$375,000

is

\frac{5}{12}

of the total budget

B

, we have

\frac{5}{12} B = \$375,000

. Multiplying both sides by

\frac{12}{5}

yields

B = \$375,000 \times \frac{12}{5} = \$900,000

.

4

Calculate the budget spent on Research and Development.

Research and Development budget is

\$300,000

.

Research and Development is allocated

\frac{1}{3}

of the total budget:

\frac{1}{3} \times \$900,000 = \$300,000

.

Key Concept

Solving multi-step word problems using fractions, decimals, and percentage allocations.

Question 4725Question →

A gymnast competed in 5 events during a state meet, earning scores of $14.2$ , $13.8$ , $14.5$ , $13.5$ , and $14.0$ . She has one event remaining. If her goal is to have an average (arithmetic mean) score of exactly $14.1$ for all 6 events, what score must she earn on her final event?

14.0

14.1

14.5

14.6

15.1

Show answer & explanation

Answer: 14.6

Answer

14.6

To find the score needed on the final event, we first calculate the total sum of the scores from the first 5 events:

14.2 + 13.8 + 14.5 + 13.5 + 14.0 = 70.0

. Next, we find the total sum required for all 6 events to average exactly

14.1

:

14.1 \times 6 = 84.6

. Finally, we subtract the sum of the first 5 events from this target total:

84.6 - 70.0 = 14.6

. Therefore, the required score on the final event is 14.6.

Step-by-Step Solution

1

Calculate the sum of the scores from the first 5 events.

14.2 + 13.8 + 14.5 + 13.5 + 14.0 = 70.0

To find the missing score, we need to compare the current total score to the target total score.

2

Calculate the target total score for all 6 events to achieve an average of

14.1

.

14.1 \times 6 = 84.6

Since the average is the sum divided by the number of events, the total sum is the average multiplied by 6.

3

Subtract the sum of the first 5 events from the target total sum to find the required final score.

84.6 - 70.0 = 14.6

The difference between the required total sum and the current sum represents the score needed on the final event.

Key Concept

Calculating a missing value in a data set to achieve a target arithmetic mean.

Question 4726Question →

The passage below describes how astronomers study the light from distant stars. Based on the context of the passage, which single, precise verb should be written in the blank to maintain a formal, academic tone and avoid repeating verbs already used in the paragraph?

Fill in the blanks below

Astronomers use spectrophotometry to analyze the light emitted by distant stars and galaxies. By splitting this light into a spectrum of colors, researchers can detect the presence of specific chemical elements. Each element absorbs and emits light at distinct wavelengths, leaving a unique signature similar to a fingerprint. This technique allows scientists to the chemical composition of celestial bodies billions of light-years away.

Show answer & explanation

Answer

determine (or identify, ascertain, deduce, establish)

The correct answer must be a precise, academic verb that fits the context of discovering the chemical makeup of distant stars from spectral data. Words such as 'determine', 'identify', 'ascertain', 'deduce', or 'establish' convey this action clearly, maintain a formal tone, and avoid redundancy with other verbs in the passage.

Step-by-Step Solution

1

Analyze the context of the sentence containing the blank to identify the core action being performed by the scientists.

The passage explains that spectrophotometry lets scientists discover or verify the chemical components of stars using spectral patterns.

Understanding the precise meaning intended by the sentence is necessary to select an appropriate word.

2

Evaluate word options that match the academic tone of the passage and check for redundant verbs.

Verbs like 'determine', 'identify', 'ascertain', 'deduce', or 'establish' fit the scientific context and avoid repeating words like 'analyze' or 'detect' which are already used in the paragraph.

ACT style requires precision, formal register, and the avoidance of repetitive phrasing.

Key Concept

Word Precision and Clarity

Estimated Time:1m 0s

Question 4727Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 - 8x - 6y = 0$ . A point on this circle has an $x$ -coordinate of $1$ and a positive $y$ -coordinate. What is the $y$ -coordinate of this point?

Show answer & explanation

Answer: 7

Answer

The positive y-coordinate of the point on the circle is 7.

Substituting

x = 1

into the circle's equation

x^2 + y^2 - 8x - 6y = 0

gives

1 + y^2 - 8 - 6y = 0

, which simplifies to the quadratic equation

y^2 - 6y - 7 = 0

. Factoring this equation yields

(y - 7)(y + 1) = 0

. The solutions are

y = 7

and

y = -1

. Because the y-coordinate must be positive, the correct value is 7.

Step-by-Step Solution

1

Substitute the x-coordinate

x = 1

into the circle's equation.

1^2 + y^2 - 8(1) - 6y = 0

This sets up the equation to solve for the corresponding y-coordinates.

2

Simplify the equation.

y^2 - 6y - 7 = 0

Combining the constant terms creates a standard quadratic equation in terms of y.

3

Solve the quadratic equation by factoring.

(y - 7)(y + 1) = 0

, yielding

y = 7

or

y = -1

Factoring allows us to find the two possible y-values that satisfy the equation.

4

Choose the positive y-coordinate.

y = 7

The question specifies that the y-coordinate must be positive.

Key Concept

Substituting a known coordinate into a circle's equation to find the other coordinate using quadratic equations.

Question 4728Question →

Carl Linnaeus, the Swedish botanist of the eighteenth century, revolutionized the biological sciences by creating a standardized system for naming organisms. Before Linnaeus, scientific names were long, descriptive Latin phrases that were difficult to remember and use. Linnaeus introduced binomial nomenclature, assigning each species a two-word Latin name consisting of its genus and specific epithet. While some contemporary scholars argued that this system was too simplistic, it quickly gained widespread acceptance because it was super easy for scientists to share their findings. Today, this elegant naming system remains the global standard for biological classification.

Which of the following choices best replaces the underlined portion to maintain the formal, academic tone of the passage?

NO CHANGE

it facilitated the expeditious dissemination of empirical observations among global researchers

it allowed scientists to collaborate together and share their findings with one another

it simplified the process of sharing scientific discoveries

Show answer & explanation

Answer: it simplified the process of sharing scientific discoveries

Answer

The correct option is the one that simplifies the process of sharing scientific discoveries, as it is both concise and stylistically appropriate for the formal tone of the passage.

The correct answer maintains the objective and academic tone of the biography by stating the result clearly and concisely without resorting to informal slang or overly pretentious jargon.

Step-by-Step Solution

1

Analyze the context and tone of the passage.

The passage is a formal, historical biography and scientific explanation, establishing an academic and objective tone.

Stylistic appropriateness requires matching the tone of the surrounding text.

2

Evaluate the underlined phrase and the options for tone, conciseness, and precision.

The original phrase contains the colloquialism 'super easy'. One option is overly wordy and redundant ('collaborate together'), while another is overly complex and pretentious ('facilitated the expeditious dissemination'). Only the option focusing on simplifying the process of sharing discoveries is precise, formal, and concise.

Eliminating inappropriate tone registers and wordiness leads to the best stylistic choice.

Key Concept

Stylistic Appropriateness

Question 4729Question →

A logistics company distributes $T$ tons of cargo among a fleet of trucks. A standard truck has a maximum capacity of $c$ tons, and a heavy-duty truck has a maximum capacity of twice that of a standard truck. The fleet consists of $x$ standard trucks and $y$ heavy-duty trucks, all loaded to their maximum capacities. If the ratio of standard trucks to heavy-duty trucks in the fleet is exactly $3:1$ , which of the following equations correctly expresses the capacity of a standard truck, $c$ , in terms of $T$ and $y$ ?

c = \frac{T}{4y}

c = \frac{3T}{7y}

c = \frac{T}{5y}

c = \frac{T}{2y + 3}

c = T - 5y

Show answer & explanation

Answer: $c = \frac{T}{5y}$

Answer

The equation

c = \frac{T}{5y}

correctly expresses the capacity of a standard truck.

The total cargo

T

is the sum of the capacities of all trucks. Since there are

x

standard trucks carrying

c

tons each and

y

heavy-duty trucks carrying

2c

tons each, the equation is

T = cx + 2cy

. The

3:1

ratio of standard trucks to heavy-duty trucks means

x = 3y

. Substituting this relationship into the cargo equation gives

T = c(3y) + 2cy = 5cy

. To isolate the capacity of a standard truck,

c

, divide both sides by

5y

to get

c = \frac{T}{5y}

.

Step-by-Step Solution

1

Write the equation for the total cargo weight by summing the capacities of all trucks in the fleet.

T = cx + 2cy

The total cargo

T

is distributed among

x

standard trucks, each carrying

c

tons, and

y

heavy-duty trucks, each carrying

2c

tons.

2

Translate the ratio of standard trucks to heavy-duty trucks into an algebraic relationship.

x = 3y

The ratio of standard trucks (

x

) to heavy-duty trucks (

y

) is

3:1

, meaning there are 3 times as many standard trucks as heavy-duty trucks.

3

Substitute the ratio relationship

x = 3y

into the total cargo equation.

T = c(3y) + 2cy

Replacing

x

with

3y

eliminates the variable

x

and expresses

T

in terms of

c

and

y

.

4

Combine like terms to simplify the expression.

T = 5cy

Adding

3cy

and

2cy

yields

5cy

.

5

Isolate the variable

c

by dividing both sides of the equation by

5y

.

c = \frac{T}{5y}

Division is the inverse operation of multiplication, allowing us to solve the one-step equation

T = (5y)c

for

c

.

Key Concept

Translating verbal descriptions into algebraic equations and solving one-step literal equations

Estimated Time:2m 0s

Question 4730Question →

A certain region has a total land area of $2.5 \times 10^4$ square miles and a population of $8.0 \times 10^5$ people. What is the average population density of this region, in people per square mile?

3.2 \times 10^1

3.125 \times 10^{-2}

2.0 \times 10^{10}

3.2 \times 10^9

3.2 \times 10^{20}

Show answer & explanation

Answer: $3.2 \times 10^1$

Answer

The population density is

3.2 \times 10^1

people per square mile.

To find the average population density, we divide the population by the land area:

\frac{8.0 \times 10^5}{2.5 \times 10^4}

. Dividing the coefficients gives

\frac{8.0}{2.5} = 3.2

. Subtracting the exponents of the powers of 10 gives

10^{5-4} = 10^1

. Combining these results yields the correct density of

3.2 \times 10^1

people per square mile.

Step-by-Step Solution

1

Set up the ratio for population density.

\text{Density} = \frac{8.0 \times 10^5 \text{ people}}{2.5 \times 10^4 \text{ square miles}}

Population density is defined as population divided by land area.

2

Divide the numerical coefficients.

\frac{8.0}{2.5} = 3.2

To divide numbers in scientific notation, first divide their lead coefficients.

3

Divide the powers of 10 by subtracting the exponent of the denominator from the exponent of the numerator.

\frac{10^5}{10^4} = 10^{5-4} = 10^1

According to the quotient rule of exponents,

\frac{x^a}{x^b} = x^{a-b}

.

4

Combine the simplified coefficient and power of 10.

3.2 \times 10^1

This is the standard scientific notation form of the final answer.

Key Concept

Operations with numbers in scientific notation

Estimated Time:1m 0s

Question 4731Question →

A student volunteer recorded the hours they spent working at a community garden and the number of plants watered each day, as shown in the table below:

Day	Hours Volunteered	Plants Watered
Monday	4	32
Tuesday	5	45
Wednesday	6	48
Thursday	4	36
Friday	8	80

What was the average (mean) number of hours volunteered per day by the student during this five-day period?

5.0

5.4

20.6

27.0

48.2

Show answer & explanation

Answer: 5.4

Answer

5.4 hours per day

The correct answer is 5.4 hours. To find the mean, sum the daily hours volunteered (

4 + 5 + 6 + 4 + 8 = 27

) and divide by the number of days (

5

), resulting in

5.4

hours per day.

Step-by-Step Solution

1

Calculate the total hours volunteered by summing the values in the Hours Volunteered column.

4 + 5 + 6 + 4 + 8 = 27

hours

To find the average, we first find the sum of all daily hours.

2

Divide the total hours by the number of days, which is 5.

27 \div 5 = 5.4

hours per day

The mean is defined as the sum of the data values divided by the total number of data points.

Key Concept

Calculating the arithmetic mean from a frequency table or data set.

Question 4732Question →

An architectural firm is building a scale model of a new office building. The ratio of the height of the model to the height of the actual building is $1:50$ . The actual building will have a rectangular base with a perimeter of $270$ meters. If the ratio of the length of the base to the width of the base is $5:4$ , what is the length, in centimeters, of the base of the scale model?

1.5

120

150

300

1,500

Show answer & explanation

Answer: 150

Answer

The length of the base of the scale model is

150

centimeters.

The correct answer represents the scaled-down length of the base converted to centimeters. First, the sum of the actual length and width is half the perimeter:

270 \div 2 = 135

meters. Since the ratio of length to width is

5:4

, we divide

135

meters into

9

equal parts of

15

meters each. The actual length is

5

parts, which equals

75

meters. Applying the scale of

1:50

gives a model length of

75 \div 50 = 1.5

meters. Finally, converting this to centimeters yields

1.5 \times 100 = 150

centimeters.

Step-by-Step Solution

1

Calculate the sum of the actual length and width from the given perimeter.

L + W = 135

meters.

The perimeter of a rectangle is

2(L + W)

, so the sum of one length and one width is half of the perimeter:

270 \div 2 = 135

meters.

2

Use the length-to-width ratio to find the actual length of the building's base.

L = 75

meters.

The ratio

L:W = 5:4

means

L = 5x

and

W = 4x

. Substituting these into the sum gives

5x + 4x = 135 \Rightarrow 9x = 135 \Rightarrow x = 15

. Therefore, the actual length is

5 \times 15 = 75

meters.

3

Scale the actual length down to the model's scale.

Model length

= 1.5

meters.

The scale of the model is

1:50

, so the dimensions of the model are found by dividing the actual dimensions by

50

:

75 \text{ meters} \div 50 = 1.5

meters.

4

Convert the model length from meters to centimeters.

Model length

= 150

centimeters.

Since

1

meter is equal to

100

centimeters, multiplying the length in meters by

100

gives the length in centimeters:

1.5 \times 100 = 150

centimeters.

Key Concept

Solving multi-step ratio and proportion problems involving scale drawings, geometric perimeter, and unit conversions.

Alternative Method

Instead of finding the actual dimensions first, you can scale the total perimeter first: the perimeter of the model is

270 \text{ meters} \div 50 = 5.4 \text{ meters} = 540 \text{ centimeters}

. Since the ratio of length to width is

5:4

, the perimeter consists of

2(5 + 4) = 18

equal parts. The length represents

5

of these parts, so the model length is

\frac{5}{18} \times 540 = 150

centimeters.

Estimated Time:1m 30s

Question 4733Question →

The passage below is from an article about biotechnology in prosthetics.

Although synthetic materials are commonly used in modern prosthetic design, researchers are experimenting with mycelium, the root structure of fungi. This organic substance is incredibly durable and lightweight; moreover, because it is fully biodegradable, it offers a sustainable alternative to plastics, it can be easily grown into custom shapes. This development could significantly reduce medical waste in clinics worldwide.

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

NO CHANGE

moreover, because it is fully biodegradable. It offers a sustainable alternative to plastics, and it can be easily grown into custom shapes.

moreover, because it is fully biodegradable, it offers a sustainable alternative to plastics; it can be easily grown into custom shapes.

moreover, because it is fully biodegradable; it offers a sustainable alternative to plastics, it can be easily grown into custom shapes.

Show answer & explanation

Answer: moreover, because it is fully biodegradable, it offers a sustainable alternative to plastics; it can be easily grown into custom shapes.

Answer

The grammatically correct revision is the option that uses a semicolon to separate the two independent clauses ('it offers a sustainable alternative to plastics' and 'it can be easily grown into custom shapes') while keeping the introductory dependent clause ('because it is fully biodegradable') properly attached with a comma.

The correct option resolves the comma splice by using a semicolon to separate the two independent clauses ('it offers a sustainable alternative to plastics' and 'it can be easily grown into custom shapes'). Semicolons are grammatically appropriate for connecting closely related independent clauses without coordinating conjunctions.

Step-by-Step Solution

1

Identify the clause boundaries in the underlined portion of the sentence.

The underlined text contains a dependent clause ('because it is fully biodegradable') and two independent clauses ('it offers a sustainable alternative to plastics' and 'it can be easily grown into custom shapes').

Understanding the structure of the sentence is necessary to identify the punctuation errors present.

2

Diagnose the error in the original sentence.

The two independent clauses are joined only by a comma (', it can be easily grown...'), which is a comma splice error.

Identifying the error helps eliminate options that do not correct it or that introduce new grammatical errors.

3

Evaluate the choices to find the correct punctuation that resolves the comma splice without creating new errors.

Using a semicolon after 'plastics' correctly separates the two independent clauses. Separating the dependent clause with a semicolon is incorrect (semicolon fragment), as is separating the dependent clause with a period (sentence fragment).

A semicolon is the standard way to connect two independent clauses when no coordinating conjunction is used, and a comma must separate the introductory dependent clause from its main clause.

Key Concept

Run-on Sentences and Comma Splices

Estimated Time:1m 0s

Question 4734Question →

The following passage describes the origin of Mary Shelley's famous novel, Frankenstein. Read the passage and determine the correct form of the verb in parentheses to maintain a consistent tense throughout the text. Which form of the verb 'begin' correctly completes the sentence to maintain verb tense consistency?

Fill in the blanks below

In the gloomy summer of 1816, Mary Shelley and her companions gathered at Lake Geneva, where unusually cold weather confined them indoors. To pass the time, Lord Byron proposed a contest to see who could write the best ghost story. Shelley struggled for days to find inspiration, but a vivid waking dream eventually sparked her imagination. She immediately (begin) drafting her masterpiece, establishing a tale that would endure for centuries.

Show answer & explanation

Answer

began

The verb 'began' is the correct simple past tense form of 'begin', maintaining consistency with the other past tense verbs in the passage ('gathered', 'confined', 'proposed', 'struggled', 'sparked').

Step-by-Step Solution

1

Analyze the context of the surrounding sentences to determine the overall timeframe of the passage.

The verbs throughout the passage ('gathered', 'confined', 'proposed', 'struggled', 'sparked') are all in the simple past tense, indicating that the entire narrative takes place in the past.

Maintaining tense consistency requires that verbs describing actions in the same timeframe use the same tense unless there is a logical reason to shift.

2

Identify the subject and base verb that need to be conjugated for the sentence.

The subject is 'She' (Mary Shelley) and the base verb is 'begin'.

The sentence describes an action that she took immediately after her dream sparked her imagination.

3

Conjugate the base verb 'begin' into the simple past tense.

The simple past tense of 'begin' is 'began'.

This maintains the past tense timeline established by the surrounding sentences.

Key Concept

Verb Tense and Consistency

Estimated Time:1m 0s

Question 4735Question →

A highway is constructed up a mountain pass at a constant incline. At a distance of $3.6$ miles from the starting toll gate, the elevation of the highway is $1,420$ feet above sea level. At a distance of $8.4$ miles from the toll gate, the elevation is $3,100$ feet above sea level. What is the slope of the highway's elevation profile, in feet per mile?

Show answer & explanation

Answer: 350

Answer

The slope of the highway's elevation profile is 350 feet per mile.

The slope represents the constant rate of change of elevation per mile. By defining the coordinates as

(x_1, y_1) = (3.6, 1420)

and

(x_2, y_2) = (8.4, 3100)

, we calculate the slope using the slope formula. Substituting the values gives

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8} = 350

.

Step-by-Step Solution

1

Represent the given data as coordinate points where the x-coordinate represents the distance in miles and the y-coordinate represents the elevation in feet.

The coordinate points are

(3.6, 1420)

and

(8.4, 3100)

.

Slope is the change in the dependent variable (elevation) per unit change in the independent variable (distance).

2

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to find the rate of change.

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8}

The slope represents the constant rate of change of elevation relative to the distance traveled.

3

Perform the division to find the numerical slope.

350

Dividing 1680 by 4.8 yields the exact value of 350.

Key Concept

The slope of a line represents a constant rate of change between two variables, calculated as the change in the vertical coordinate divided by the change in the horizontal coordinate.

Question 4736Question →

During the restoration of the historic library, the preservationists uncovered a valuable, hand-painted mural, it had been hidden behind plaster for over a century.

Which of the following options represents the correct punctuation for the underlined portion?

NO CHANGE

valuable, hand-painted mural; it

valuable, hand-painted mural; which

valuable, hand-painted mural, that

Show answer & explanation

Answer: valuable, hand-painted mural; it

Answer

The correct option is 'valuable, hand-painted mural; it' because it correctly punctuates the coordinate adjectives with a comma and uses a semicolon to connect the two independent clauses.

The correct option correctly places a comma between the coordinate adjectives 'valuable' and 'hand-painted' because they are parallel descriptors of equal weight that independently modify 'mural'. Additionally, the independent clauses 'the preservationists uncovered a valuable, hand-painted mural' and 'it had been hidden behind plaster for over a century' are joined by a semicolon, which is the correct grammatical way to link two closely related independent clauses without a coordinating conjunction.

Step-by-Step Solution

1

Determine if the adjectives modifying the noun are coordinate.

The adjectives 'valuable' and 'hand-painted' are coordinate because they can be reversed ('hand-painted, valuable mural') and joined by 'and' ('valuable and hand-painted mural') while retaining their meaning. Therefore, a comma is required between them.

To ensure correct punctuation between adjectives modifying a single noun.

2

Identify the clause boundaries in the sentence.

The sentence contains two independent clauses: 'During the restoration of the historic library, the preservationists uncovered a valuable, hand-painted mural' and 'it had been hidden behind plaster for over a century.'

To determine what punctuation is required to connect the two clauses.

3

Evaluate the punctuation options to connect the independent clauses.

A comma alone creates a comma splice. A semicolon correctly separates two independent clauses. A relative pronoun like 'which' makes the second clause dependent, but a semicolon cannot precede it. A restrictive relative clause starting with 'that' cannot be preceded by a comma.

To choose the option that correctly separates the two clauses without introducing grammatical errors.

Key Concept

Punctuation conventions for coordinate adjectives and independent clause boundaries

Question 4737Question →

An agricultural irrigation system draws water from a main tank at a constant rate to supply three fields: Field X, Field Y, and Field Z. The ratio of the water volume distributed daily to Field X and Field Y is $3:4$ , and the ratio of the water volume distributed daily to Field Y and Field Z is $3:5$ . If Field Z receives $2,200$ gallons more water daily than Field X, what is the total number of gallons of water distributed daily to all three fields combined?

4,000

5,800

8,200

13,200

11,275

Show answer & explanation

Answer: $8,200$

Answer

8,200

gallons

The correct answer is

8,200

gallons. By finding a common multiple for the Field Y term in the two ratios, we align the proportions to get a joint ratio of

9:12:20

for Field X, Field Y, and Field Z. Let the volumes be

9k

,

12k

, and

20k

. The difference between Field Z and Field X is

20k - 9k = 11k = 2,200

gallons, giving

k = 200

. The combined volume is

9k + 12k + 20k = 41k = 41 \times 200 = 8,200

gallons.

Step-by-Step Solution

1

Find a common unit of comparison for the ratio of water distributed to all three fields.

The joint ratio is

9:12:20

.

We are given Field X to Field Y is

3:4

and Field Y to Field Z is

3:5

. To combine these ratios, we find the least common multiple of the parts representing Field Y, which is

\text{LCM}(4, 3) = 12

. Multiplying the first ratio by 3 gives

9:12

, and multiplying the second ratio by 4 gives

12:20

.

2

Set up an equation using the given difference between the water volume of Field Z and Field X.

k = 200

Let the daily volumes of Field X, Field Y, and Field Z be represented by

9k

,

12k

, and

20k

respectively. The difference between Field Z and Field X is

20k - 9k = 11k

. We are given that this difference is

2,200

gallons, so

11k = 2,200

, which simplifies to

k = 200

.

3

Calculate the combined daily volume for all three fields.

8,200

gallons

The total volume distributed to all three fields is

9k + 12k + 20k = 41k

. Substituting

k = 200

gives

41 \times 200 = 8,200

gallons.

Key Concept

Solving multi-part ratio word problems by establishing a common ratio and setting up algebraic equations.

Question 4738Question →

A hiker starts at a base camp at an elevation of $120$ feet. She hikes up to an overlook and then down to a valley at an elevation of $-45$ feet. The elevation of the overlook is higher than both the base camp and the valley. On a vertical number line representing elevations, the distance between the overlook's coordinate and the valley's coordinate is exactly twice the distance between the overlook's coordinate and the base camp's coordinate. What is the elevation, in feet, of the overlook?

65

70

95

195

285

Show answer & explanation

Answer: 285

Answer

285 feet

The correct answer is

285

feet. Let

x

represent the elevation of the overlook. Because the overlook is higher than both the base camp at

120

feet and the valley at

-45

feet, we know that

x > 120

. The distance between the overlook and the valley is

x - (-45) = x + 45

, and the distance between the overlook and the base camp is

x - 120

. Setting up the relationship where the distance to the valley is twice the distance to the base camp gives the equation

x + 45 = 2(x - 120)

. Solving this equation yields

x + 45 = 2x - 240

, which simplifies to

x = 285

. This value is indeed higher than both elevations, and we can verify that the distance to the valley (

285 - (-45) = 330

) is twice the distance to the base camp (

285 - 120 = 165

).

Step-by-Step Solution

1

Define the variable and write the expressions for the distances on a number line.

Let

x

represent the elevation of the overlook in feet. The distance between the overlook and the base camp (

120

feet) is

|x - 120|

. The distance between the overlook and the valley (

-45

feet) is

|x - (-45)| = |x + 45|

.

Distance between two points

a

and

b

on a number line is defined by the absolute value of their difference,

|a - b|

.

2

Set up the equation based on the given ratio.

|x + 45| = 2|x - 120|

The problem states that the distance to the valley is exactly twice the distance to the base camp.

3

Apply the condition that the overlook is higher than both locations to simplify the absolute value expressions.

Since

x > 120

and

x > -45

, both terms inside the absolute values are positive. Therefore,

|x + 45| = x + 45

and

|x - 120| = x - 120

. The equation simplifies to:

x + 45 = 2(x - 120)

.

For any expression

y > 0

,

|y| = y

. Establishing the location of

x

relative to

120

and

-45

allows us to remove the absolute value bars.

4

Solve the simplified linear equation for

x

.

x + 45 = 2x - 240 \Rightarrow x = 285

Distribute the

2

on the right side and isolate

x

by subtracting

x

and adding

240

to both sides.

Key Concept

Representing distances between points on a number line using absolute value equations.

Alternative Method

Instead of setting up equations, you can test the options starting from the constraint that the elevation must be greater than

120

feet. This eliminates

65

,

70

, and

95

. Test the remaining options: for

195

, the distance to

120

is

75

and the distance to

-45

is

240

(not twice

75

). For

285

, the distance to

120

is

165

and the distance to

-45

is

330

, which is exactly twice

165

.

Estimated Time:1m 30s

Question 4739Question →

An online retailer determines the sale price of a promotional item by reducing the original price, $P$ dollars, by a discount of $D$ dollars. The discount is defined as $D = \frac{1}{6} P$ . If the sale price of the item is $125$ dollars, which of the following equations can be solved to find the original price of the item, in dollars?

\frac{1}{6} P = 125

P - \frac{1}{6} = 125

\frac{5}{6} P = 125

P + \frac{1}{6} P = 125

\frac{6}{5} P = 125

Show answer & explanation

Answer: $\frac{5}{6} P = 125$

Answer

The equation stating that five-sixths of the original price equals one hundred twenty-five is correct.

The correct equation is found by expressing the sale price as the original price minus the discount. The discount is one-sixth of the original price, which is represented by the expression containing five-sixths of the original price. Setting this expression equal to the given sale price of one hundred twenty-five dollars results in the correct equation.

Step-by-Step Solution

1

Write the relationship for the sale price based on the scenario.

S = P - D

The sale price (

S

) is the original price (

P

) minus the discount amount (

D

).

2

Substitute the given expression for the discount (

D = \frac{1}{6} P

) into the sale price equation.

S = P - \frac{1}{6} P

This expresses the sale price in terms of the single variable

P

representing the original price.

3

Simplify the algebraic expression by combining the like terms.

S = \frac{5}{6} P

Combining

1P

and

-\frac{1}{6} P

gives

\frac{5}{6} P

because

1 - \frac{1}{6} = \frac{5}{6}

.

4

Substitute the known sale price of

125

dollars into the simplified equation.

\frac{5}{6} P = 125

This sets up the final one-step equation needed to find the original price.

Key Concept

Translating verbal descriptions of discounts into simplified one-step linear equations.

Estimated Time:1m 30s

Question 4740Question →

A manufacturer of solar panels inspects their production in three stages. In the first stage, $\frac{1}{12}$ of the total panels produced are identified as defective and recycled. In the second stage, $20\%$ of the remaining non-defective panels are found to have cosmetic flaws and are sold at a discount. In the third stage, $\frac{1}{5}$ of the panels that passed the first two stages without any defects or flaws are selected for quality testing. If $1,760$ panels passed the first two stages but were NOT selected for quality testing, what was the total number of panels originally produced?

Show answer & explanation

Answer: 3000

Answer

The total number of panels originally produced was

3,000

.

The correct answer of

3,000

is determined by calculating the cumulative fraction of panels that are not defective, do not have cosmetic flaws, and are not selected for testing. In Stage 1, subtracting

\frac{1}{12}

leaves

\frac{11}{12}

of the original panels. In Stage 2, subtracting

20\%

of those leaves

80\%

(or

\frac{4}{5}

), resulting in

\frac{11}{15}

of the original panels. In Stage 3, subtracting

\frac{1}{5}

of these leaves

\frac{4}{5}

, resulting in

\frac{44}{75}

of the original panels. Setting

\frac{44}{75}

of the original panels equal to

1,760

and solving for the total original panels yields

3,000

.

Step-by-Step Solution

1

Calculate the fraction of panels remaining after the first stage of inspection.

\frac{11}{12}

of the original panels

Since

\frac{1}{12}

of the total panels are defective and recycled, the remaining portion is

1 - \frac{1}{12} = \frac{11}{12}

.

2

Calculate the fraction of the original panels that pass the second stage without defects or flaws.

\frac{11}{15}

of the original panels

Of the remaining

\frac{11}{12}

of the panels,

20\%

have cosmetic flaws, meaning

100\% - 20\% = 80\%

(or

\frac{4}{5}

) do not have flaws. Multiplying these fractions gives

\frac{4}{5} \times \frac{11}{12} = \frac{11}{15}

.

3

Calculate the fraction of original panels that pass the first two stages but are not selected for testing.

\frac{44}{75}

of the original panels

Since

\frac{1}{5}

of the flawless panels are selected for testing, the remaining

1 - \frac{1}{5} = \frac{4}{5}

are not selected. Multiplying this by the fraction from Step 2 gives

\frac{4}{5} \times \frac{11}{15} = \frac{44}{75}

.

4

Solve for the original number of panels,

P

, using the given value of

1,760

.

P = 3,000

panels

Set up the equation

\frac{44}{75} P = 1,760

. Solving for

P

gives

P = 1,760 \times \frac{75}{44} = 40 \times 75 = 3,000

.

Key Concept

Applying successive fractions and percentages to resolve a multi-stage word problem by working backward to the initial value.

Answer

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