All practice questions

Question 4701Question →

What is the value of the expression $\frac{\sqrt{3^8 \times 2^6}}{6^2}$ ?

11

18

36

108

144

Show answer & explanation

Answer: 18

Answer

18

To evaluate the expression, first simplify the square root in the numerator. Since the square root of a product is the product of the square roots of its factors, we have

\sqrt{3^8 \times 2^6} = 3^{8/2} \times 2^{6/2} = 3^4 \times 2^3

. Next, rewrite the base in the denominator in terms of its prime factors:

6^2 = (3 \times 2)^2 = 3^2 \times 2^2

. Now, divide the simplified terms using the quotient rule for exponents:

\frac{3^4 \times 2^3}{3^2 \times 2^2} = 3^{4-2} \times 2^{3-2} = 3^2 \times 2^1

. Evaluating this gives

9 \times 2 = 18

.

Step-by-Step Solution

1

Simplify the square root in the numerator.

\sqrt{3^8 \times 2^6} = \sqrt{3^8} \times \sqrt{2^6} = 3^{8/2} \times 2^{6/2} = 3^4 \times 2^3

.

Applying the square root to factors with even exponents reduces their exponents by half.

2

Rewrite the denominator

6^2

in terms of prime bases

2

and

3

.

6^2 = (2 \times 3)^2 = 2^2 \times 3^2

.

Expressing the bases in prime factors allows the use of exponent rules to simplify the fraction.

3

Divide the simplified numerator by the simplified denominator using the quotient rule.

\frac{3^4 \times 2^3}{3^2 \times 2^2} = 3^{4-2} \times 2^{3-2} = 3^2 \times 2^1 = 9 \times 2 = 18

.

Subtracting exponents of like bases simplifies the rational expression to a final numerical value.

Key Concept

Simplifying radical expressions and applying laws of exponents for multiplication and division of powers.

Estimated Time:1m 0s

Question 4702Question →

Deep within the ocean, hydrothermal vents support unique ecosystems that thrive without sunlight. Currently, these specialized communities are at this present time dependent on chemical synthesis rather than photosynthesis.

Which of the following options is the best version of the underlined portion of the sentence?

Currently, these specialized communities are at this present time dependent

These specialized communities depend

Because these specialized communities depend

These specialized communities depend, they rely

Show answer & explanation

Answer: These specialized communities depend

Answer

These specialized communities depend

The correct option is the most concise and direct way to express the sentence's meaning. By replacing 'Currently, these specialized communities are at this present time dependent' with 'These specialized communities depend', it successfully eliminates the redundant temporal references ('currently' and 'at this present time') and uses a strong, active verb ('depend') to connect the subject to the prepositional phrase.

Step-by-Step Solution

1

Identify the redundancy in the underlined portion of the original sentence.

The words 'Currently' and the phrase 'at this present time' both convey the same meaning, making them redundant when used together.

ACT English questions require eliminating unnecessary words that repeat the same information.

2

Evaluate the remaining options for conciseness and grammatical correctness.

The option starting with 'Because' creates a dependent clause fragment. The option with 'they rely' creates a comma splice. The option 'These specialized communities depend' is both grammatically correct and concise.

The correct answer must be free of redundancy, punctuation errors, and sentence structure issues.

Key Concept

Eliminating Redundancy

Question 4703Question →

In the standard $(x, y)$ coordinate plane, a line with a slope of $\frac{3}{2}$ passes through the points $(k, -2)$ and $(2k, 4)$ . Is the statement ' $k = 4$ ' true or false?

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because solving the slope equation with the given coordinates and slope value yields

k = 4

.

The correct option is true because substituting

k = 4

into the coordinates yields the points

(4, -2)

and

(8, 4)

. Using the slope formula, the slope is calculated as

\frac{4 - (-2)}{8 - 4} = \frac{6}{4} = \frac{3}{2}

, which matches the given slope value.

Step-by-Step Solution

1

State the formula for the slope

m

of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

.

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

This formula defines the constant rate of change (slope) between any two points on a non-vertical line.

2

Substitute the coordinates

(k, -2)

and

(2k, 4)

and the slope value

\frac{3}{2}

into the formula.

\\frac{3}{2} = \frac{4 - (-2)}{2k - k}

This creates an algebraic equation in terms of

k

that represents the given geometric conditions.

3

Simplify the numerator and denominator of the fraction and solve the resulting equation for

k

.

\frac{3}{2} = \frac{6}{k} \Rightarrow 3k = 12 \Rightarrow k = 4

Simplification reduces the algebraic expression, allowing the variable

k

to be isolated and solved.

Key Concept

Calculating the slope of a line from two coordinate points containing an algebraic parameter.

Estimated Time:1m 30s

Question 4704Question →

A tablet battery charges at a constant rate of $r$ percentage points per hour. If the battery charge increases by $48$ percentage points in $3$ hours, which of the following equations can be used to find $r$ ?

r + 3 = 48

3r = 48

\frac{r}{3} = 48

r - 3 = 48

\frac{3}{r} = 48

Show answer & explanation

Answer: $3r = 48$

Answer

The equation

3r = 48

can be used to find the charging rate.

The battery charges at a constant rate of

r

percentage points per hour. Over

3

hours, the total increase is the rate multiplied by the time, which is

3r

. Since the total increase is given as

48

percentage points, the correct relationship is expressed by the equation

3r = 48

.

Step-by-Step Solution

1

Identify the relationship between the charging rate, time, and total charge increase.

The total charge increase is equal to the charging rate multiplied by the number of hours.

Since the battery charges at a constant rate of

r

percentage points per hour, in

3

hours it will charge a total of

3 \times r

or

3r

percentage points.

2

Set the expression for the total charge increase equal to the given total increase value.

3r = 48

The total charge increase is given as

48

percentage points, so we set the expression

3r

equal to

48

.

Key Concept

Translating a real-world scenario involving a constant rate into a one-step linear equation.

Alternative Method

You can also solve for the rate first by dividing the total increase of

48

by

3

hours to get

r = 16

percentage points per hour. Substituting

r = 16

into the equations shows that only

3(16) = 48

is a true statement.

Estimated Time:45s

Question 4705Question →

If $a$ and $b$ are integers such that $-5 < a < -1$ and $2 < b < 6$ , what is the difference between the maximum possible value of $|a - b|$ and the minimum possible value of $|a + b|$ ?

2

4

8

9

11

Show answer & explanation

Answer: 9

Answer

The difference between the maximum possible value of

|a - b|

and the minimum possible value of

|a + b|

is 9.

The maximum possible value of

|a - b|

is 9, which occurs when

a = -4

and

b = 5

, giving

|-4 - 5| = 9

. The minimum possible value of

|a + b|

is 0, which occurs when

a = -3

and

b = 3

(or

a = -4

and

b = 4

), giving

|-3 + 3| = 0

. The difference between these two values is

9 - 0 = 9

.

Step-by-Step Solution

1

Identify the possible integer values for

a

and

b

from the strict inequalities.

The integers satisfying

-5 < a < -1

are

a \in \{-4, -3, -2\}

. The integers satisfying

2 < b < 6

are

b \in \{3, 4, 5\}

.

Since the inequalities are strict (

<

), the endpoints

-5, -1, 2,

and

6

are excluded.

2

Determine the maximum possible value of

|a - b|

by selecting the values of

a

and

b

that maximize their distance.

Using

a = -4

and

b = 5

yields

|-4 - 5| = |-9| = 9

.

The absolute value of the difference is maximized when

a

is as small (most negative) as possible and

b

is as large (most positive) as possible.

3

Determine the minimum possible value of

|a + b|

by finding values of

a

and

b

that are closest to being additive opposites.

Using

a = -3

and

b = 3

(or

a = -4

and

b = 4

) yields

|-3 + 3| = |0| = 0

.

The absolute value of any real number is at least 0. Since we can choose integers that sum to exactly 0, the minimum possible value is 0.

4

Calculate the difference between the two extreme values found in the previous steps.

The difference is

9 - 0 = 9

.

We subtract the minimum value of the second expression from the maximum value of the first expression.

Key Concept

Absolute value represents distance from zero, and finding extreme values of absolute value expressions involving restricted integer sets requires testing boundary combinations and understanding additive inverses.

Alternative Method

Instead of checking every pair, we can analyze the extreme values of the intervals. Since

a \in [-4, -2]

and

b \in [3, 5]

, the difference

a - b

ranges from

-4 - 5 = -9

to

-2 - 3 = -5

. The absolute value

|a - b|

therefore ranges from 5 to 9, making the maximum value 9. For the sum, since the interval of

-a

is

[2, 4]

and overlaps with the interval of

b

which is

[3, 5]

, they can be equal (specifically at 3 and 4). When

-a = b

, we have

a + b = 0

, so the minimum value of

|a + b|

must be 0. Subtracting the two values gives

9 - 0 = 9

.

Estimated Time:1m 30s

Question 4706Question →

During a science experiment, a student measured the temperature changes of a liquid in degrees Celsius at seven different intervals. The recorded changes were $15^\circ\text{C}$ , $6^\circ\text{C}$ , $20^\circ\text{C}$ , $24^\circ\text{C}$ , $7^\circ\text{C}$ , $4^\circ\text{C}$ , and $8^\circ\text{C}$ . What is the positive difference between the median and the mean of these seven recorded temperatures?

4

6

8

12

16

Show answer & explanation

Answer: 4

Answer

4

To find the correct positive difference, we first find the mean by dividing the sum of the temperatures by 7:

(15 + 6 + 20 + 24 + 7 + 4 + 8) / 7 = 84 / 7 = 12

. We then sort the temperatures in ascending order:

4, 6, 7, 8, 15, 20, 24

. The median is the middle value of this sorted list, which is 8. The positive difference between the median and the mean is

|8 - 12| = 4

.

Step-by-Step Solution

1

Calculate the mean of the temperatures.

Mean = 12

Sum the seven values (

15 + 6 + 20 + 24 + 7 + 4 + 8 = 84

) and divide by the total number of data points (7) to get

84 / 7 = 12

.

2

Find the median of the temperatures.

Median = 8

Sort the values in ascending order:

4, 6, 7, 8, 15, 20, 24

. Since there are 7 data points, the median is the fourth value, which is 8.

3

Calculate the positive difference between the median and the mean.

Positive difference = 4

Subtract the median from the mean (or vice versa) and take the absolute value:

|8 - 12| = 4

.

Key Concept

Calculating and comparing the mean and median of a dataset, including the necessity of sorting data before finding the median.

Question 4707Question →

In 132 CE, the Chinese polymath Zhang Heng introduced a device that could detect earthquakes from hundreds of miles away. This invention, a large bronze vessel adorned with dragons and frogs, was initially met with skepticism by the imperial court. However, its utility became clear when one of the dragons dropped a bronze ball into a frog's mouth, indicating an earthquake had occurred to the west. A messenger arrived days later to verify the disaster, proving the device's ability to spot seismic activity that could not be felt locally.

Which choice provides the most precise and clear wording for the underlined portion?

NO CHANGE

perceive

register

catch

Show answer & explanation

Answer: register

Answer

The word 'register' provides the most precise and clear wording for the passage.

The word 'register' is the most precise and appropriate choice because it specifically describes an instrument detecting or recording physical data. This fits the formal and scientific register of the passage.

Step-by-Step Solution

1

Analyze the context of the sentence and the established tone of the passage.

The passage discusses a historical scientific invention (Zhang Heng's seismograph) in a formal, academic tone.

Identifying the register and context helps narrow down the appropriate vocabulary choices.

2

Evaluate the underlined word and each alternative choice for precision, clarity, and register.

The original word 'spot' and the alternative 'catch' are too informal. The word 'perceive' implies conscious sensory experience, which is incorrect for a mechanical instrument. The word 'register' is precise and fits the scientific context of a device recording data.

This determines which word provides the most accurate and stylistically appropriate meaning.

Key Concept

Word Precision and Clarity

Estimated Time:1m 0s

Question 4708Question →

A web designer is creating a layout with three sections: a header, a main content area, and a sidebar. The ratio of the area of the header to the area of the main content area is $1:3$ . The ratio of the area of the main content area to the area of the sidebar is $4:3$ . If the total area of the layout is $2,000$ square pixels, what is the area, in square pixels, of the sidebar?

320

720

960

1,000

1,500

Show answer & explanation

Answer: 720

Answer

720

The correct answer is 720 square pixels. To find this, combine the ratios

H:M = 1:3

(which scales to

4:12

) and

M:S = 4:3

(which scales to

12:9

) using their common term

M

. This yields a joint ratio of

H:M:S = 4:12:9

. Let the areas be

4x

,

12x

, and

9x

. The sum is

25x = 2,000

, which gives

x = 80

. The sidebar area is

9x = 9(80) = 720

square pixels.

Step-by-Step Solution

1

Express the ratios with a common term.

The ratio of the header to the main content is

H:M = 1:3

, which is equivalent to

4:12

. The ratio of the main content to the sidebar is

M:S = 4:3

, which is equivalent to

12:9

. Combining these gives a joint ratio of

H:M:S = 4:12:9

.

Finding a common term for the main content area (

M

) allows us to express all three sections in a single ratio.

2

Set up an equation for the total area in terms of a multiplier

x

.

4x + 12x + 9x = 2,000 \Rightarrow 25x = 2,000

The sum of the individual areas must equal the total area of

2,000

square pixels.

3

Solve for

x

.

x = 80

This determines the value of one part of the ratio.

4

Calculate the area of the sidebar.

9 \times 80 = 720

The sidebar's area is represented by

9x

, so we multiply

9

by

80

.

Key Concept

Combining multiple ratios through a common term to solve multi-part ratio problems.

Question 4709Question →

A brief passage from an essay on forestry management is shown below.

In the field of restoration ecology, forest managers use controlled burns to clear underbrush and promote new growth. Although this practice mimics natural fire cycles that have sustained ecosystems for millennia, many residents express concern about smoke inhalation. Consequently, local agencies must carefully coordinate burn schedules, they monitor wind patterns to ensure smoke disperses away from populated areas.

For the underlined portion, which option represents the most grammatically correct choice?

NO CHANGE

schedules; monitoring

schedules. Because they monitor

schedules; they monitor

Show answer & explanation

Answer: schedules; they monitor

Answer

The correct option replaces the comma with a semicolon to properly separate the two independent clauses: 'schedules; they monitor'.

The correct option correctly uses a semicolon to separate two complete independent clauses. Since both 'Consequently, local agencies must carefully coordinate burn schedules' and 'they monitor wind patterns to ensure smoke disperses away from populated areas' can stand alone as sentences, a semicolon is a grammatically correct way to link them.

Step-by-Step Solution

1

Analyze the clauses around the underlined portion to determine their grammatical roles.

The first clause ('Consequently, local agencies must carefully coordinate burn schedules') and the second clause ('they monitor wind patterns to ensure smoke disperses away from populated areas') are both independent clauses.

Identifying clause types is necessary because independent clauses must be joined using appropriate punctuation and conjunctions.

2

Evaluate the original phrasing (Option A) for grammatical errors.

The original phrasing attempts to connect two independent clauses with only a comma, resulting in a comma splice.

A comma alone cannot join two independent clauses without a coordinating conjunction.

3

Evaluate the remaining options to find the correct correction.

Replacing the comma with a semicolon correctly links the two independent clauses. The options introducing a subordinating conjunction or a participle phrase create fragments.

A semicolon is a standard and correct way to join two independent clauses without a coordinating conjunction.

Key Concept

Run-on Sentences and Comma Splices

Estimated Time:1m 0s

Question 4710Question →

What is the area, in square units, of the circle defined by the equation $x^2 + y^2 + 8x - 10y + 5 = 0$ in the standard $(x, y)$ coordinate plane?

36\pi

12\pi

6\pi

46\pi

5\pi

Show answer & explanation

Answer: $36\pi$

Answer

The area of the circle is

36\pi

square units.

To find the area of the circle, we first rewrite the given equation

x^2 + y^2 + 8x - 10y + 5 = 0

in standard form

(x-h)^2 + (y-k)^2 = r^2

. Rearranging the terms yields

(x^2 + 8x) + (y^2 - 10y) = -5

. Completing the square for

x

requires adding

(\frac{8}{2})^2 = 16

to both sides, and completing the square for

y

requires adding

(\frac{-10}{2})^2 = 25

to both sides. This gives

(x^2 + 8x + 16) + (y^2 - 10y + 25) = -5 + 16 + 25

, which simplifies to

(x + 4)^2 + (y - 5)^2 = 36

. Thus, the squared radius is

r^2 = 36

. The area of the circle is

\pi r^2 = 36\pi

square units.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms and move the constant to the right side of the equation.

(x^2 + 8x) + (y^2 - 10y) = -5

This prepares the equation for completing the square.

2

Complete the square for both the

x

and

y

terms by adding

(\frac{8}{2})^2 = 16

and

(\frac{-10}{2})^2 = 25

to both sides.

(x^2 + 8x + 16) + (y^2 - 10y + 25) = -5 + 16 + 25

This allows the quadratic expressions to be written as perfect squares.

3

Simplify the equation into the standard form of a circle

(x-h)^2 + (y-k)^2 = r^2

.

(x + 4)^2 + (y - 5)^2 = 36

This standard form directly reveals that the radius squared (

r^2

) is

36

.

4

Calculate the area of the circle using the formula

A = \pi r^2

.

A = \pi (36) = 36\pi

The area of a circle is calculated by multiplying

\pi

by the square of the radius.

Key Concept

Converting the general equation of a circle to standard form by completing the square to determine its radius and calculate its area.

Estimated Time:1m 30s

Question 4711Question →

A commuter train travels at a constant speed to complete a trip of $D$ miles in $t$ hours. Due to track maintenance, the train's speed is decreased by 15 miles per hour for a 20-minute portion of the trip. Which of the following expressions represents the distance, in miles, the train traveled during this 20-minute portion of the trip?

\frac{D}{t} - 5

\frac{D}{3t} - 15

\frac{D}{3t} - 5

\frac{D}{3t} - 45

\frac{3D}{t} - 5

Show answer & explanation

Answer: $\frac{D}{3t} - 5$

Answer

The expression

\frac{D}{3t} - 5

represents the distance traveled during the 20-minute portion of the trip.

The train's original speed is

\frac{D}{t}

miles per hour. During the maintenance segment, its speed is reduced by 15 miles per hour, giving a speed of

\frac{D}{t} - 15

miles per hour. To find the distance traveled over a 20-minute interval (which represents

\frac{20}{60} = \frac{1}{3}

of an hour), the reduced speed is multiplied by the elapsed time. Applying the distributive property to

\left(\frac{D}{t} - 15\right) \times \frac{1}{3}

results in

\frac{D}{3t} - 5

.

Step-by-Step Solution

1

Determine the train's original speed.

Original speed is

\frac{D}{t}

miles per hour.

Speed is defined as distance divided by time.

2

Express the reduced speed during the track maintenance portion.

Reduced speed is

\frac{D}{t} - 15

miles per hour.

The speed was decreased by 15 miles per hour from the original speed.

3

Convert the duration of the maintenance portion from minutes to hours.

20 minutes is equal to

\frac{20}{60} = \frac{1}{3}

hour.

Since the speed is in miles per hour, the time must be converted to hours to ensure unit consistency.

4

Calculate the distance traveled during the 20-minute portion.

Distance is

\left(\frac{D}{t} - 15\right) \times \frac{1}{3} = \frac{D}{3t} - 5

miles.

Distance is equal to the product of speed and time. The distributive property must be applied to both terms in the speed expression.

Key Concept

Translating a verbal rate and time relationship into a simplified algebraic expression using the distributive property and unit conversion.

Alternative Method

Alternatively, you can assign plug-in values to make the problem concrete. Let the total distance

D = 120

miles and the total time

t = 2

hours. This gives an original speed of

60

miles per hour. The reduced speed is

60 - 15 = 45

miles per hour. Traveling at

45

miles per hour for

20

minutes (

\frac{1}{3}

hour) results in a distance of

45 \times \frac{1}{3} = 15

miles. Substituting

D = 120

and

t = 2

into the correct expression gives

\frac{120}{3(2)} - 5 = \frac{120}{6} - 5 = 20 - 5 = 15

miles, which matches our result.

Estimated Time:1m 30s

Question 4712Question →

While adjusting the telescope during the meteor shower, the astronomer captured a series of sharp, breathtaking, photographs of the celestial event, and she immediately sent them to her colleagues for analysis.

Which of the following choices best punctuates the underlined portion of the sentence?

sharp, breathtaking, photographs of the celestial event, and she

sharp, breathtaking photographs of the celestial event, and she

sharp, breathtaking photographs of the celestial event, she

sharp, breathtaking photographs; of the celestial event, she

Show answer & explanation

Answer: sharp, breathtaking photographs of the celestial event, and she

Answer

sharp, breathtaking photographs of the celestial event, and she

The correct option correctly places a comma between the coordinate adjectives 'sharp' and 'breathtaking' without separating the final adjective from the noun 'photographs'. It also correctly uses a comma followed by the coordinating conjunction 'and' to join two independent clauses.

Step-by-Step Solution

1

Determine if the adjectives 'sharp' and 'breathtaking' are coordinate adjectives modifying 'photographs'.

They are coordinate adjectives because they can be reversed ('breathtaking, sharp photographs') and joined by 'and' ('sharp and breathtaking photographs'). Therefore, they require a comma between them, but no comma should separate the final adjective 'breathtaking' from the noun 'photographs'.

To apply the correct punctuation rules for coordinate adjectives.

2

Analyze the clause structure of the sentence around the underline.

The sentence consists of an introductory dependent clause followed by two independent clauses: 'the astronomer captured a series of photographs' and 'she immediately sent them to her colleagues'.

To identify the clause boundaries and determine appropriate punctuation and conjunctions.

3

Select the correct punctuation to join the two independent clauses.

Two independent clauses must be joined by a comma and a coordinating conjunction (such as 'and'), or by a semicolon. The option using 'and' preceded by a comma is grammatically correct.

To avoid run-on sentences and comma splices.

Key Concept

Punctuation with coordinate adjectives and joining independent clauses.

Question 4713Question →

In 1947, computer pioneer Grace Hopper was working on the Harvard Mark II computer when her team encountered a mysterious system failure. Upon investigating the hardware, they discovered a moth trapped in a relay, impeding the electrical signals. Hopper taped the insect into the logbook, labeling it the 'first actual case of bug being found.' Throughout her career, Hopper remained a trailblazer, continuously seeking new ways to make technology accessible. She believed that programming languages should be written in English rather than machine code. Hopper's work revolutionized the programming landscape, enabling developers to write code using English-like commands instead of binary numbers. This innovation meant that program design was no longer confined to mathematical specialists, allowing regular folks to construct software on their own. Ultimately, this breakthrough laid the foundation for modern programming languages.

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

NO CHANGE

permitting individuals who were not mathematical specialists to create programs independently by themselves.

allowing a wider range of people to write software.

empowering a broader demographic of individuals to execute the fabrication of software applications.

Show answer & explanation

Answer: allowing a wider range of people to write software.

Answer

The option that reads 'allowing a wider range of people to write software.' is correct because it is concise, direct, and maintains the professional yet accessible style established throughout the passage.

The correct option preserves the professional and clear tone of the passage while expressing the idea concisely and precisely, without repeating information or introducing overly complex or informal phrasing.

Step-by-Step Solution

1

Analyze the tone and style of the surrounding passage.

The passage uses an informative, objective, and professional historical tone.

Establishing the tone of the passage helps identify whether a candidate choice matches the register or creates a stylistic clash.

2

Evaluate the underlined original phrase for stylistic fit.

The original phrase containing 'regular folks' and 'on their own' is too informal and conversational for a biographical text.

Identifying that the original text has a tone clash indicates that 'NO CHANGE' is not the correct choice.

3

Evaluate the remaining options to select the one that is concise, precise, and matches the established register.

The option beginning with 'permitting individuals' is redundant and wordy. The option beginning with 'empowering a broader' is overly inflated and pretentious. The option beginning with 'allowing a wider' is concise, direct, and matches the tone.

Eliminating options that introduce wordiness, redundancy, or inappropriate tone registers leaves the correct, stylistically appropriate answer.

Key Concept

Stylistic Appropriateness

Estimated Time:1m 0s

Question 4714Question →

A courier uses three modes of transportation to deliver packages: a bicycle, an electric scooter, and a delivery van. The ratio of the average speed of the electric scooter to the average speed of the bicycle is $3:2$ , and the ratio of the average speed of the delivery van to the average speed of the electric scooter is $5:2$ . If it takes the courier $60$ minutes to complete a delivery route using the bicycle, how many minutes would it take the courier to complete the exact same delivery route using the delivery van, assuming all average speeds remain constant?

16

24

36

100

225

Show answer & explanation

Answer: 16

Answer

16 minutes

Because travel time is inversely proportional to speed when distance is constant, the time required decreases as speed increases. With a scooter-to-bicycle speed ratio of

3:2

, the scooter takes

\frac{2}{3}

of the bicycle's time:

60 \times \frac{2}{3} = 40

minutes. With a van-to-scooter speed ratio of

5:2

, the van takes

\frac{2}{5}

of the scooter's time:

40 \times \frac{2}{5} = 16

minutes.

Step-by-Step Solution

1

Determine the relationship between the speed of the bicycle and the speed of the electric scooter.

v_{\text{scooter}} = \frac{3}{2}v_{\text{bicycle}}

The given ratio of the average speed of the scooter to the bicycle is

3:2

.

2

Calculate the time it would take to complete the route using the electric scooter.

40

minutes

Since speed and travel time are inversely proportional for a constant distance, the scooter's time is

60 \times \frac{2}{3} = 40

minutes.

3

Determine the relationship between the speed of the electric scooter and the delivery van.

v_{\text{van}} = \frac{5}{2}v_{\text{scooter}}

The given ratio of the average speed of the van to the scooter is

5:2

.

4

Calculate the time it would take to complete the route using the delivery van.

16

minutes

Since speed and travel time are inversely proportional, the van's time is

40 \times \frac{2}{5} = 16

minutes.

Key Concept

Inverse proportionality in rate-time-distance relationships and compounding multiple ratios

Question 4715Question →

During his expedition to the Arctic in 1909, Robert Peary claimed to have reached the North Pole, a monumental achievement that was met with both skepticism and acclaim. At that period of time, many scholars disputed his measurements, leading to a decades-long debate.

Which option is the best replacement for the underlined portion of the sentence?

At that period of time, many scholars disputed

Many scholars disputed

Because many scholars disputed

Many scholars disputed, they questioned

Show answer & explanation

Answer: Many scholars disputed

Answer

Many scholars disputed

The correct option is the most concise choice that avoids redundant phrasing. Since the preceding sentence already specifies the year 1909, repeating the timeframe with 'At that period of time' is unnecessary. Furthermore, 'period' and 'time' are repetitive synonyms. Simply starting the clause with the subject and verb provides a clear, grammatically sound sentence.

Step-by-Step Solution

1

Analyze the underlined portion for redundant phrasing.

The phrase 'At that period of time' contains 'period' and 'time', which are redundant synonyms. Moreover, the context has already specified the year as 1909.

ACT English requires the most concise phrasing that preserves grammatical correctness and meaning.

2

Evaluate the grammar of the remaining choices to ensure they do not introduce new errors.

Removing the redundant phrase leaves 'Many scholars disputed', which is grammatically correct and forms a complete independent clause. Introducing 'Because' creates a fragment, and adding 'they questioned' creates a comma splice.

Eliminating redundancy must not come at the cost of sentence structure and punctuation rules.

Key Concept

Eliminating Redundancy

Question 4716Question →

The diameter of a human red blood cell is approximately $7.0 \times 10^{-6}$ meters, and the diameter of a typical influenza virus is approximately $1.4 \times 10^{-7}$ meters. How many times larger is the diameter of the red blood cell than the diameter of the influenza virus?

Show answer & explanation

Answer: 50

Answer

50

To determine how many times larger the diameter of the red blood cell is than the diameter of the influenza virus, divide the larger diameter by the smaller diameter:

\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}

. Dividing the coefficients gives

\frac{7.0}{1.4} = 5

. Dividing the powers of ten using the quotient rule for exponents gives

\frac{10^{-6}}{10^{-7}} = 10^{-6 - (-7)} = 10^1

. Multiplying these results gives

5 \times 10^1 = 50

.

Step-by-Step Solution

1

Set up the ratio of the larger diameter to the smaller diameter

\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}

To find how many times larger one quantity is than another, divide the larger quantity by the smaller quantity.

2

Divide the decimal coefficients

5.0

Dividing

7.0

by

1.4

simplifies the numerical coefficient.

3

Divide the exponential terms using exponent properties

10^1

Using the quotient rule for exponents,

\frac{10^a}{10^b} = 10^{a - b}

, so

10^{-6 - (-7)} = 10^1

.

4

Combine and simplify the final value

50

Multiplying the coefficient by the simplified power of ten yields

5.0 \times 10 = 50

.

Key Concept

Division of numbers in scientific notation using properties of exponents

Estimated Time:1m 30s

Question 4717Question →

A geology student collected a group of rocks and recorded their weights, in grams, in the table below. One frequency, $f$ , is missing.

Weight (grams)	Frequency
25	5
10	4
20	$f$
30	2
15	6

If the mean weight of the rocks in the group is exactly $18.75$ grams, what is the median weight, in grams, of the rocks?

3

10

15

17.5

20

Show answer & explanation

Answer: 17.5

Answer

17.5

The correct answer is 17.5. By setting up the weighted mean equation, we find that the missing frequency

f

is 3. This means there are 20 rocks in total. When sorted in ascending order, the weights are four 10s, six 15s, three 20s, five 25s, and two 30s. The 10th rock weighs 15 grams and the 11th rock weighs 20 grams. The median is the average of these two middle values: (15 + 20) / 2 = 17.5 grams.

Step-by-Step Solution

1

Set up an equation for the mean weight using the frequencies and weights from the table.

The sum of the weights is

25(5) + 10(4) + 20(f) + 30(2) + 15(6) = 315 + 20f

. The total number of rocks is

5 + 4 + f + 2 + 6 = 17 + f

. The mean is

\frac{315 + 20f}{17 + f} = 18.75

.

To find the missing frequency

f

using the given mean.

2

Solve the equation for

f

.

315 + 20f = 18.75(17 + f) \implies 315 + 20f = 318.75 + 18.75f \implies 1.25f = 3.75 \implies f = 3

.

To determine the number of rocks that weigh 20 grams.

3

Find the total number of rocks and identify the median position.

The total number of rocks is

17 + 3 = 20

. Since the total number is even, the median is the average of the 10th and 11th values when the weights are sorted in ascending order.

The median of an even number of data points is the average of the two middle values.

4

List the sorted weights and find the 10th and 11th values.

The sorted weights are: four 10s, six 15s (making 10 rocks total), followed by three 20s. Thus, the 10th rock weighs 15 grams and the 11th rock weighs 20 grams.

To find the values of the two middle rocks.

5

Calculate the average of the 10th and 11th values.

Median =

\frac{15 + 20}{2} = 17.5

grams.

To find the median weight.

Key Concept

Calculating the median from a frequency distribution table after finding a missing frequency using the mean.

Estimated Time:2m 30s

Question 4718Question →

A scientist records the temperature of a gas compound inside a chamber as it is heated at a constant rate. At a time of $t = 3$ minutes, the temperature of the compound is $-8^\circ\text{C}$ . At $t = 15$ minutes, the temperature is $28^\circ\text{C}$ . What is the rate of change of the temperature with respect to time, in degrees Celsius per minute, of the compound?

\frac{1}{3}

-3

3

\frac{5}{3}

\frac{3}{5}

Show answer & explanation

Answer: $3$

Answer

The rate of change of the temperature of the compound is

3

degrees Celsius per minute.

The rate of change is equivalent to the slope of the line passing through the points

(3, -8)

and

(15, 28)

. Using the slope formula, we compute the ratio of the change in temperature to the change in time:

\frac{28 - (-8)}{15 - 3} = \frac{36}{12} = 3

degrees Celsius per minute.

Step-by-Step Solution

1

Identify the coordinate points from the given information.

The two points are

(3, -8)

and

(15, 28)

, where the first coordinate represents time

t

in minutes and the second coordinate represents temperature

T

in degrees Celsius.

To find the constant rate of change, we need to determine the change in temperature per unit of time, which corresponds to the slope between these two points.

2

Apply the slope formula

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

to calculate the rate of change.

Substituting the coordinates into the formula gives

\frac{28 - (-8)}{15 - 3}

.

The slope formula measures the vertical change (rise) over the horizontal change (run).

3

Simplify the expression to find the final value.

\frac{28 + 8}{12} = \frac{36}{12} = 3

.

Simplifying the numerator and denominator gives the constant rate of change.

Key Concept

The slope of a line represents the constant rate of change of the dependent variable with respect to the independent variable.

Question 4719Question →

A scientist monitors the temperature of two research chambers. Chamber A is kept at $-5^\circ\text{C}$ and Chamber B is kept at $7^\circ\text{C}$ . The scientist sets a third chamber, Chamber C, to a temperature of $T^\circ\text{C}$ such that the distance between $T$ and the temperature of Chamber A on the Celsius scale is exactly $3$ times the distance between $T$ and the temperature of Chamber B. If the temperature of Chamber C is warmer than Chamber A but colder than Chamber B, what is the value of $T$ ?

Show answer & explanation

Answer: 4

Answer

The correct temperature value of Chamber C is 4.

The temperature of Chamber C must be

4^\circ\text{C}

because the distance from

4

to

-5

is

|4 - (-5)| = 9

and the distance from

4

to

7

is

|4 - 7| = 3

. The distance of

9

is exactly

3

times the distance of

3

. Furthermore,

4

lies between

-5

and

7

, satisfying the condition that Chamber C is warmer than Chamber A but colder than Chamber B.

Step-by-Step Solution

1

Represent the distances on the number line using absolute value expressions.

The distance to Chamber A is

|T - (-5)| = |T + 5|

and the distance to Chamber B is

|T - 7|

.

Distance between two points

x

and

y

on a number line is represented by

|x - y|

.

2

Set up the algebraic equation reflecting the relationship between the distances.

|T + 5| = 3|T - 7|

The problem states the distance to Chamber A is exactly 3 times the distance to Chamber B.

3

Solve the absolute value equation by considering both positive and negative cases.

Case 1:

T + 5 = 3(T - 7) \Rightarrow T = 13

. Case 2:

T + 5 = -3(T - 7) \Rightarrow T = 4

.

The equation

|x| = |y|

implies

x = y

or

x = -y

.

4

Verify which solution satisfies the temperature boundary condition.

Since Chamber C must be warmer than

-5^\circ\text{C}

but colder than

7^\circ\text{C}

, the only valid value is

T = 4

.

The value

T = 13

is warmer than both chambers and does not lie between them.

Key Concept

Using absolute value to represent distance on a number line and solving absolute value equations with boundary conditions.

Alternative Method

Alternatively, visualize this on a number line. The total distance between Chamber A (

-5

) and Chamber B (

7

) is

12

units. Since Chamber C lies between them and the distance from C to A is

3

times the distance from C to B, we can divide the

12

-unit interval into

3 + 1 = 4

equal parts. Each part is

12 \div 4 = 3

units. Chamber C is located

1

part away from Chamber B (towards Chamber A), which places it at

7 - 3 = 4

.

Estimated Time:1m 30s

Question 4720Question →

The following passage is about the history of a unique musical instrument.

Invented in 1920 by Russian physicist Lev Sergeyevich Termen, the theremin is one of the earliest electronic musical instruments. Controlled without physical contact by the performer, the instrument's sound is produced by two metal antennas that sense the relative position of the player's hands. Because of its eerie, otherworldly tone, it is often featured in science fiction film soundtracks.

Which choice most effectively corrects the underlined portion of the sentence?

NO CHANGE

the instrument, which relies on two metal antennas, make sound by sensing the player's hands.

the instrument produces sound through two metal antennas that sense the relative position of the player's hands.

the instrument produces sound, two metal antennas sense the relative position of the player's hands.

Show answer & explanation

Answer: the instrument produces sound through two metal antennas that sense the relative position of the player's hands.

Answer

the instrument produces sound through two metal antennas that sense the relative position of the player's hands.

The correct option resolves the dangling modifier by placing the logical subject 'the instrument' immediately after the introductory modifier. It is grammatically correct and correctly describes the relationship between the performer and the instrument.

Step-by-Step Solution

1

Identify the introductory modifier and determine its logical target.

The phrase 'Controlled without physical contact by the performer' must modify a noun representing what is actually being controlled. The instrument itself is controlled, not 'the instrument's sound'.

Introductory modifiers must be immediately followed by the noun they describe to avoid dangling modifiers.

2

Evaluate the choices to find which one begins with the logical subject.

The choices starting with 'the instrument' correctly place the logical subject after the modifier, which eliminates the dangling modifier present in the original sentence.

This step narrows the choices down to options that resolve the modifier placement issue.

3

Verify the remaining choices for other grammatical errors.

The option containing the subject-verb agreement error ('the instrument... make') and the option containing the comma splice are incorrect. Only the option stating 'the instrument produces sound through two metal antennas that sense the relative position of the player's hands' has no grammatical errors.

The correct answer must be grammatically correct in all aspects, not just in modifier placement.

Key Concept

Modifier Placement

Estimated Time:1m 0s

Answer

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