Pre-Algebra Practice Questions — ACT — Page 15

Question 281Question →

A company surveyed a group of 25 employees about their daily commute times. The table below shows the frequency distribution of the commute times, where one frequency is represented by the variable $x$ .

Commute Time (minutes)	Number of Employees
10	5
20	8
30	$x$
45	4
60	3

If the mean commute time for all 25 employees is 28.8 minutes, what is the median commute time, in minutes, for these 25 employees?

5

20

25

28.8

30

Show answer & explanation

Answer: 20

Answer

The median commute time for the 25 employees is 20 minutes.

To find the median commute time, we first determine the missing frequency

x

. Since the total number of employees is 25, we set up the equation

5 + 8 + x + 4 + 3 = 25

, giving

x = 5

. With 25 total values ordered from smallest to largest, the median is the 13th value. Counting through the frequencies in order, the first 5 employees have a commute time of 10 minutes, and the next 8 employees (positions 6 through 13) have a commute time of 20 minutes. Therefore, the 13th employee's commute time is 20 minutes.

Step-by-Step Solution

1

Determine the value of the missing frequency

x

.

x = 5

The total number of surveyed employees is given as 25. Summing the frequencies:

5 + 8 + x + 4 + 3 = 25

, which simplifies to

20 + x = 25

, so

x = 5

.

2

Find the position of the median in the dataset.

13th position

For an odd number of data values (

N = 25

), the median is located at position

\frac{N + 1}{2} = \frac{25 + 1}{2} = 13

when the data is listed in ascending order.

3

Use cumulative frequencies to locate the 13th data value.

20 minutes

The cumulative frequencies are: 10 minutes (values 1–5) and 20 minutes (values 6–13). Since the 13th data value falls within the 20-minute category, the median commute time is 20 minutes.

Key Concept

Calculating the median from a frequency distribution table with a total count constraint.

Question 282Question →

A museum archivist digitizes a collection of historical glass-plate photographs at a constant rate of $p$ photographs per hour. If the archivist digitizes a total of $336$ photographs over a continuous $14$ -hour shift, which of the following equations can be solved to find $p$ ?

p + 14 = 336

\frac{p}{14} = 336

14p = 336

p - 14 = 336

14 + 336 = p

Show answer & explanation

Answer: $14p = 336$

Answer

The correct equation is

14p = 336

.

The total number of digitized photographs equals the rate per hour (

p

) multiplied by the number of hours (

14

). Setting this product equal to the total of

336

photographs yields

14p = 336

.

Step-by-Step Solution

1

Identify the relationship between rate, time, and total quantity.

\text{Total Quantity} = \text{Rate} \times \text{Time}

The total work done at a constant rate is found by multiplying the rate per unit of time by total time.

2

Substitute the given values into the relationship.

336 = p \times 14

, which simplifies to

14p = 336

.

The rate is

p

photographs per hour, time is

14

hours, and total quantity is

336

photographs.

Key Concept

Writing one-step multiplication equations from rate and time word problems

Estimated Time:1m 0s

Question 283Question →

A commercial bakery uses an industrial mixer that processes specialty flour at a constant rate. The mixer consumes $\frac{4}{7}$ of a full bag of flour in $12$ minutes. The equation $\frac{4}{7}b = 12$ represents this situation, where $b$ is the time, in minutes, it takes to process one full bag of flour. What is the value of $b$ ?

\frac{48}{7}

21

28

48

\frac{3}{7}

Show answer & explanation

Answer: $21$

Answer

The total time required to process one full bag of flour is 21 minutes.

To solve the one-step equation

\frac{4}{7}b = 12

, isolate

b

by multiplying both sides of the equation by the reciprocal of

\frac{4}{7}

, which is

\frac{7}{4}

. Computing

12 \times \frac{7}{4}

yields

\frac{84}{4} = 21

. Therefore, 21 minutes is the correct answer.

Step-by-Step Solution

1

Identify the given one-step equation

\frac{4}{7}b = 12

The equation models the relationship between the fraction of the bag used and the time taken.

2

Isolate the variable

b

by multiplying both sides by the reciprocal of

\frac{4}{7}

b = 12 \times \frac{7}{4}

Multiplying by the reciprocal undoes multiplication by a fraction.

3

Simplify the numerical calculation

b = \frac{12 \times 7}{4} = 3 \times 7 = 21

Dividing 12 by 4 yields 3, and multiplying 3 by 7 gives 21.

Key Concept

Solving One-Step Linear Equations with Fractional Coefficients

Estimated Time:1m 0s

Question 284Question →

At a local vehicle dealership, a buyer customizing a new car model can select $1$ exterior paint color from $5$ available colors, $1$ interior seat material from $3$ available materials, and $1$ transmission type from $2$ available options (manual or automatic). According to the Fundamental Counting Principle, how many different unique configurations of the car can a buyer create?

Show answer & explanation

Answer: 30

Answer

There are 30 different unique configurations possible.

According to the Fundamental Counting Principle, when making a series of independent choices, the total number of distinct outcomes is found by multiplying the number of options for each choice. Multiplying 5 exterior colors, 3 interior materials, and 2 transmission types gives

5 \times 3 \times 2 = 30

unique car configurations.

Step-by-Step Solution

1

Count the number of choices for each feature

Paint colors = 5, Interior materials = 3, Transmissions = 2

Each feature selection is an independent choice in the configuration process.

2

Multiply the number of options for all features

5 \times 3 \times 2 = 30

The Fundamental Counting Principle states that if there are

n_1

ways to make one choice,

n_2

ways to make a second, and

n_3

ways to make a third, the total number of combined outcomes is

n_1 \times n_2 \times n_3

.

Key Concept

Fundamental Counting Principle

Estimated Time:45s

Question 285Question →

A solar power microgrid generates electricity at a constant rate of $p$ kilowatts for each hour of active operation. On a day with $h$ total hours of daylight, cloud coverage reduces operation such that the microgrid operates at full capacity for $3$ hours less than half the total daylight hours. Which of the following equations represents the total electricity $E$ , in kilowatt-hours, generated by the microgrid on that day?

E = p\left(\frac{1}{2}h - 3\right)

E = p\left(3 - \frac{1}{2}h\right)

E = \frac{1}{2}ph - 3

E = p\left(\frac{h - 3}{2}\right)

E = p\left(\frac{1}{2}h + 3\right)

Show answer & explanation

Answer: $E = p\left(\frac{1}{2}h - 3\right)$

Answer

The correct equation is

E = p\left(\frac{1}{2}h - 3\right)

.

The expression for half the total daylight hours is

\frac{1}{2}h

. Taking '3 hours less than' this quantity requires subtracting 3, yielding an active time of

\left(\frac{1}{2}h - 3\right)

hours. Multiplying the hourly power rate

p

by this active time gives the total energy equation

E = p\left(\frac{1}{2}h - 3\right)

.

Step-by-Step Solution

1

Express half of the total daylight hours algebraically.

Half of

h

total hours is written as

\frac{1}{2}h

or

\frac{h}{2}

.

Dividing the total daylight duration

h

by 2 establishes the baseline time frame.

2

Translate '3 hours less than half the total daylight hours' into a time expression.

Operating time

= \frac{1}{2}h - 3

.

The phrase '3 less than X' means

X - 3

, so 3 is subtracted from

\frac{1}{2}h

.

3

Multiply the rate of generation by the active operating time to form the equation for total electricity

E

.

E = p\left(\frac{1}{2}h - 3\right)

.

Total energy generated is equal to the hourly rate

p

multiplied by the total active operating hours.

Key Concept

Translating verbal descriptions involving rates and modified time intervals into algebraic expressions.

Alternative Method

Substitute a concrete number for

h

to verify the translation. If

h = 10

, half the hours is 5, and 3 hours less is 2 hours. At rate

p

, energy generated is

2p

. Substituting

h = 10

into

E = p\left(\frac{1}{2}(10) - 3\right)

yields

p(5 - 3) = 2p

, confirming the correct formula.

Estimated Time:1m 30s

Question 286Question →

A bookshelf contains $4$ mystery novels, $6$ science fiction novels, and $5$ biography books. If one book is chosen at random from the bookshelf, what is the probability that the chosen book is NOT a science fiction novel?

\frac{3}{5}

\frac{2}{5}

\frac{2}{3}

\frac{3}{2}

\frac{11}{15}

Show answer & explanation

Answer: $\frac{3}{5}$

Answer

The probability that the chosen book is NOT a science fiction novel is

\frac{3}{5}

.

The correct answer is

\frac{3}{5}

. To find the probability of selecting a book that is not science fiction, divide the number of books that are not science fiction (

4

mystery +

5

biography =

9

) by the total number of books (

4 + 6 + 5 = 15

). This yields

\frac{9}{15}

, which simplifies to

\frac{3}{5}

.

Step-by-Step Solution

1

Find the total number of books on the shelf

Total books =

4 + 6 + 5 = 15

The sample space consists of all mystery, science fiction, and biography books combined.

2

Count the number of books that are NOT science fiction

Non-science fiction books =

4 + 5 = 9

The mystery novels (

4

) and biography books (

5

) are the favorable outcomes.

3

Calculate the probability and simplify the fraction

Probability =

\frac{9}{15} = \frac{3}{5}

Probability is the number of favorable outcomes divided by the total number of outcomes.

Key Concept

Basic Probability of Complementary Events

Estimated Time:45s

Question 287Question →

On a standard number line, point $m$ is located at coordinate $-9$ , and point $n$ is located to the right of point $m$ . If the distance between point $m$ and point $n$ is given by the expression $|3k - 6|$ where $k = -4$ , what is the coordinate of point $n$ ?

-27

-15

-3

9

27

Show answer & explanation

Answer: $9$

Answer

The coordinate of point

n

is

9

.

First, evaluate the distance expression

|3k - 6|

when

k = -4

:

3(-4) - 6 = -18

, and

|-18| = 18

. Since point

n

is located to the right of point

m

(which is at

-9

), add the distance of

18

to

-9

:

-9 + 18 = 9

. Thus, the coordinate of point

n

is

9

.

Step-by-Step Solution

1

Evaluate the expression inside the absolute value for

k = -4

3(-4) - 6 = -12 - 6 = -18

Substitute

k = -4

into

3k - 6

following the order of operations.

2

Calculate the absolute value to find the distance

|-18| = 18

Distance is non-negative, so taking the absolute value yields a distance of 18 units.

3

Find the position of point

n

relative to point

m

n = -9 + 18 = 9

Since point

n

lies to the right of point

m = -9

on the number line, add the distance of 18 units to

-9

.

Key Concept

Distance on a Number Line using Absolute Value

Estimated Time:1m 0s

Question 288Question →

A quality control engineer measured the lengths of precision bolts manufactured across five production lines. The table below shows the number of bolts tested from each line and the mean length, in millimeters, of the bolts from that line.

Production Line	Number of Bolts Tested	Mean Length (mm)
Line 1	$15$	$24.8$
Line 2	$25$	$25.2$
Line 3	$20$	$24.6$
Line 4	$10$	$25.4$
Line 5	$x$	$25.6$

If the overall mean length of all the bolts tested across all five lines is $25.1$ millimeters, what is the value of $x$ , the number of bolts tested from Line 5?

9

15

18

25

30

Show answer & explanation

Answer: $18$

Answer

18

To find the number of bolts

x

tested from Line 5, compute the sum of the bolt lengths for the first four lines by multiplying each count by its respective mean:

(15 \times 24.8) + (25 \times 25.2) + (20 \times 24.6) + (10 \times 25.4) = 372 + 630 + 492 + 254 = 1748

mm. The total number of bolts across all five lines is

15 + 25 + 20 + 10 + x = 70 + x

. Setting the weighted average equal to the target overall mean of

25.1

yields the equation

\frac{1748 + 25.6x}{70 + x} = 25.1

. Multiplying both sides by

(70 + x)

gives

1748 + 25.6x = 1757 + 25.1x

. Subtracting

25.1x

and

1748

from both sides gives

0.5x = 9

, which simplifies to

x = 18

.

Step-by-Step Solution

1

Calculate the total sum of lengths for Lines 1 through 4.

Line 1:

15 \times 24.8 = 372

; Line 2:

25 \times 25.2 = 630

; Line 3:

20 \times 24.6 = 492

; Line 4:

10 \times 25.4 = 254

. Total sum for Lines 1-4 =

372 + 630 + 492 + 254 = 1748

mm.

The total length of bolts in a line equals the line count multiplied by its mean length.

2

Determine the total count of bolts tested across all five lines.

Total count =

15 + 25 + 20 + 10 + x = 70 + x

.

Adding the number of bolts from each of the 5 production lines gives the overall sample size.

3

Set up the overall weighted mean equation and solve for

x

.

\frac{1748 + 25.6x}{70 + x} = 25.1 \implies 1748 + 25.6x = 25.1(70 + x) \implies 1748 + 25.6x = 1757 + 25.1x \implies 0.5x = 9 \implies x = 18

.

Overall mean is equal to the total sum of all bolt lengths divided by the overall number of bolts.

Key Concept

Weighted Mean and Combined Data Sets

Estimated Time:2m 0s

Question 289Question →

An environmental scientist measured the dissolved oxygen concentration, in milligrams per liter ( $\text{mg/L}$ ), at $6$ different monitoring locations along a river. The recorded concentrations for $5$ of the locations were $6.4$ , $7.2$ , $5.8$ , $8.0$ , and $7.6$ . If the mean dissolved oxygen concentration for all $6$ locations was exactly $7.1\text{ mg/L}$ , what was the dissolved oxygen concentration recorded at the $6\text{th}$ location?

0.5\text{ mg/L}

7.0\text{ mg/L}

7.2\text{ mg/L}

7.6\text{ mg/L}

8.5\text{ mg/L}

Show answer & explanation

Answer: $7.6\text{ mg/L}$

Answer

7.6\text{ mg/L}

To find the measurement at the sixth location, multiply the target mean by the total number of locations (

6 \times 7.1 = 42.6\text{ mg/L}

) to find the required sum of all measurements. Subtracting the sum of the five known measurements (

6.4 + 7.2 + 5.8 + 8.0 + 7.6 = 35.0\text{ mg/L}

) from

42.6\text{ mg/L}

gives

42.6 - 35.0 = 7.6\text{ mg/L}

.

Step-by-Step Solution

1

Calculate the total required sum for all 6 locations.

6 \times 7.1 = 42.6\text{ mg/L}

The mean of a dataset equals the sum of all values divided by the number of values (

N=6

).

2

Sum the 5 known location measurements.

6.4 + 7.2 + 5.8 + 8.0 + 7.6 = 35.0\text{ mg/L}

This determines the total concentration accounted for by the known locations.

3

Subtract the sum of the known measurements from the total required sum.

42.6 - 35.0 = 7.6\text{ mg/L}

The difference gives the exact measurement needed at the 6th location to achieve the target mean.

Key Concept

Finding a missing value given a target mean

Estimated Time:1m 0s

Question 290Question →

A municipal water treatment facility mixes three treatment chemicals—coagulant, disinfectant, and pH adjuster—in the ratio $3 : 5 : 4$ by volume. If the facility uses $120$ gallons of pH adjuster during a single shift, what is the total volume, in gallons, of all three chemicals combined used during that shift?

240

300

360

480

1,440

Show answer & explanation

Answer: 360

Answer

360 gallons

The ratio of coagulant to disinfectant to pH adjuster is

3 : 5 : 4

, giving a total of

3 + 5 + 4 = 12

parts. Since

4

parts of pH adjuster equal

120

gallons, each part corresponds to

\frac{120}{4} = 30

gallons. The total volume of all three chemicals combined is

12 \times 30 = 360

gallons.

Step-by-Step Solution

1

Determine the total number of ratio parts.

The total ratio parts are

3 + 5 + 4 = 12

parts.

Summing the individual terms of the ratio gives the total parts representing the entire chemical mixture.

2

Calculate the volume of one ratio part.

One part

= \frac{120\text{ gallons}}{4\text{ parts}} = 30\text{ gallons per part}

.

The pH adjuster accounts for 4 parts of the ratio and equals 120 gallons, so dividing 120 by 4 yields the value of 1 part.

3

Calculate the total volume of all three chemicals.

Total volume

= 12\text{ parts} \times 30\text{ gallons per part} = 360\text{ gallons}

.

Multiplying the volume of a single part by the total number of parts yields the total volume.

Key Concept

Part-to-Whole Ratios and Proportions

Question 291Question →

A community library received a bulk shipment of $b$ new books. The head librarian distributed the entire shipment equally among 7 reading rooms, resulting in each room receiving exactly 24 books. Which of the following equations can be solved to find $b$ , the total number of books in the shipment?

7b = 24

b - 7 = 24

\frac{b}{7} = 24

b + 7 = 24

\frac{24}{b} = 7

Show answer & explanation

Answer: $\frac{b}{7} = 24$

Answer

The equation that correctly models the situation is

\frac{b}{7} = 24

.

To represent distributing a total quantity

b

equally into 7 parts, the total must be divided by 7. Setting this quotient equal to the per-room amount of 24 gives the one-step equation

\frac{b}{7} = 24

.

Step-by-Step Solution

1

Identify the total quantity and the number of equal groups.

Total quantity =

b

books; Number of equal groups = 7 reading rooms.

Equal distribution of a total quantity requires dividing the total by the number of groups.

2

Set up the algebraic expression for the amount per group.

Amount per reading room =

\frac{b}{7}

.

Dividing the total number of books

b

by 7 yields the share each room receives.

3

Equate the expression to the given amount per room.

\frac{b}{7} = 24

.

Each room receives exactly 24 books, so the expression

\frac{b}{7}

must equal 24.

Key Concept

Translating verbal descriptions of equal sharing into one-step division equations

Estimated Time:1m 0s

Question 292Question →

On a temperature monitor scale represented as a horizontal number line, sensor $A$ is positioned at coordinate $-18$ and sensor $B$ is positioned at coordinate $14$ . A relay station $C$ is located on the line segment between sensor $A$ and sensor $B$ such that the distance from sensor $A$ to relay station $C$ is $\frac{3}{4}$ of the total distance between sensor $A$ and sensor $B$ . What is the absolute value of the coordinate of relay station $C$ ?

Show answer & explanation

Answer: 6

Answer

The absolute value of the coordinate of relay station

C

is

6

.

To find the distance between

A

(

-18

) and

B

(

14

), subtract

-18

from

14

to get

32

units. Three-fourths of

32

is

24

units. Adding

24

units to the starting coordinate

-18

yields

6

as the coordinate for

C

. The absolute value of

6

is

6

.

Step-by-Step Solution

1

Calculate the total distance between coordinates

-18

and

14

Distance =

|14 - (-18)| = |14 + 18| = 32

units

The distance between two points on a number line is given by the absolute value of the difference of their coordinates.

2

Calculate the distance from sensor

A

to relay station

C

Distance

AC = \frac{3}{4} \times 32 = 24

units

Relay station

C

is positioned

\frac{3}{4}

of the total distance away from

A

towards

B

.

3

Find the coordinate of relay station

C

Coordinate of

C = -18 + 24 = 6

Moving rightward from

-18

by

24

units gives the location of

C

.

4

Evaluate the absolute value of the coordinate of

C

|6| = 6

The problem asks for the absolute value of the coordinate of

C

.

Key Concept

Distance on a number line and absolute value evaluation

Estimated Time:1m 15s

Question 293Question →

A weather station recorded the daily high temperatures, in degrees Fahrenheit, for the first 10 days of a month. The mean high temperature for these 10 days was $72^\circ\text{F}$ . When the high temperatures for 2 additional days were included in the dataset, the mean high temperature for all 12 days increased to $74^\circ\text{F}$ . If the high temperature on the 12th day was $6^\circ\text{F}$ higher than the high temperature on the 11th day, what was the high temperature, in degrees Fahrenheit, on the 12th day?

81

84

87

90

93

Show answer & explanation

Answer: 87

Answer

87

To find the temperature on the 12th day, first calculate the sum of the first 10 days (

10 \times 72 = 720

) and the total sum of all 12 days (

12 \times 74 = 888

). The difference (

888 - 720 = 168

) is the sum of the high temperatures on the 11th and 12th days. Let

T_{12}

be the high temperature on the 12th day. Since the 12th day was

6^\circ\text{F}

warmer than the 11th day, the 11th day's temperature was

T_{12} - 6

. Substituting gives

(T_{12} - 6) + T_{12} = 168

, which simplifies to

2T_{12} = 174

, yielding

T_{12} = 87^\circ\text{F}

.

Step-by-Step Solution

1

Calculate the sum of high temperatures for the first 10 days.

10 \times 72 = 720^\circ\text{F}

The sum of values is equal to the number of data points multiplied by their mean.

2

Calculate the total sum of high temperatures for all 12 days.

12 \times 74 = 888^\circ\text{F}

The new mean for all 12 days is

74^\circ\text{F}

.

3

Determine the sum of the temperatures on the 11th and 12th days.

888 - 720 = 168^\circ\text{F}

Subtracting the sum of the first 10 days from the total sum gives the combined total of the 2 additional days.

4

Set up an equation to find the temperature of the 12th day (

T_{12}

).

T_{11} + T_{12} = 168

and

T_{11} = T_{12} - 6 \implies (T_{12} - 6) + T_{12} = 168 \implies 2T_{12} = 174 \implies T_{12} = 87^\circ\text{F}

The 12th day temperature is

6^\circ\text{F}

higher than the 11th day temperature, so substituting

T_{11}

allows us to solve directly for

T_{12}

.

Key Concept

Using the arithmetic mean to find missing data points in expanded datasets

Estimated Time:1m 30s

Question 294Question →

During a weekly training program, an athlete covers $\frac{2}{5}$ of her total distance by cycling and $35\%$ of her total distance by running. The remaining distance is completed by swimming. If she swims $15$ kilometers that week, what is her total training distance, in kilometers?

20

37.5

45

60

75

Show answer & explanation

Answer: $60$

Answer

The athlete's total training distance for the week is 60 kilometers.

Converting

\frac{2}{5}

to

40\%

shows that cycling (

40\%

) and running (

35\%

) together account for

75\%

of the total training distance. Swimming accounts for the remaining

25\%

(

100\% - 75\%

). Since

25\%

of the total distance is

15

kilometers, dividing

15

by

0.25

yields the total distance of

60

kilometers.

Step-by-Step Solution

1

Convert the cycling fraction to a percentage.

\frac{2}{5} = 0.40 = 40\%

Converting all portions to percentages allows for easy summation and subtraction.

2

Calculate the total percentage covered by cycling and running combined.

40\% + 35\% = 75\%

Adding the cycling and running portions yields the covered percentage before swimming.

3

Find the percentage of the total distance that swimming represents.

100\% - 75\% = 25\%

The remaining portion of the whole (

100\%

) must equal the swimming percentage.

4

Set up an equation relating the swimming percentage to the actual distance and solve for total distance

T

.

0.25 \times T = 15 \implies T = \frac{15}{0.25} = 60

Dividing the part by its corresponding percentage rate gives the whole total distance.

Key Concept

Solving multi-step part-to-whole word problems by combining fractions and percentages.

Question 295Question →

An electric vehicle charging hub has a total power output of $480$ kilowatts (kW). The total power output is equal to $4$ times the power output, $p$ , of a single ultra-fast charging stall. What is the value of $p$ , in kilowatts?

120

476

484

960

1,920

Show answer & explanation

Answer: 120

Answer

The power output of a single ultra-fast charging stall is 120 kilowatts.

Translating the verbal relationship into an equation gives

4p = 480

. Dividing both sides of the equation by 4 isolates

p

, giving

p = 120

kilowatts.

Step-by-Step Solution

1

Set up the algebraic equation based on the problem statement.

4p = 480

The problem states that 4 times the power output of a single stall,

p

, equals the total power output of 480 kW.

2

Isolate the variable

p

by applying the inverse operation.

p = \frac{480}{4}

To solve for

p

, divide both sides of the equation by 4.

3

Calculate the final numerical value.

p = 120

Dividing 480 by 4 yields 120.

Key Concept

Solving One-Step Multiplication Equations

Estimated Time:1m 0s

Question 296Question →

A theater group offers souvenir t-shirts in $4$ different sizes (Small, Medium, Large, and Extra-Large) and $5$ different colors. In addition, each t-shirt can be printed either with or without a custom emblem on the front. How many distinct t-shirt design options are available?

11

20

40

80

100

Show answer & explanation

Answer: $40$

Answer

There are 40 distinct t-shirt design options available.

According to the Fundamental Counting Principle, to find the total number of distinct outcomes from a series of independent choices, multiply the number of choices available for each feature. Since there are

4

sizes,

5

colors, and

2

emblem choices (with or without an emblem), the total number of distinct t-shirt options is

4 \times 5 \times 2 = 40

.

Step-by-Step Solution

1

Identify the number of independent choices for each attribute.

Size choices = 4; Color choices = 5; Emblem choices = 2 (with emblem or without emblem).

Each attribute represents an independent stage of selection.

2

Apply the Fundamental Counting Principle.

Total options =

4 \times 5 \times 2 = 40

.

When an event consists of multiple sequential independent choices, the total number of outcomes is the product of the number of options for each choice.

Key Concept

Fundamental Counting Principle

Estimated Time:45s

Question 297Question →

A customer service call center recorded the duration, rounded to the nearest minute, for a sample of $20$ technical support calls. The results are summarized in the frequency table below.

Call Duration (minutes)	Frequency (number of calls)
3	4
5	6
8	5
12	3
15	2

What is the positive difference between the mean call duration and the median call duration, in minutes?

0.6

0.9

1.9

2.1

2.4

Show answer & explanation

Answer: 0.9

Answer

The positive difference between the mean call duration and the median call duration is 0.9 minutes.

The mean is found by calculating the total sum of call durations (

(3 \times 4) + (5 \times 6) + (8 \times 5) + (12 \times 3) + (15 \times 2) = 148

) and dividing by the sample size (

20

), giving

7.4

minutes. Since there are

20

values, the median is the average of the 10th value (

5

) and 11th value (

8

), which is

6.5

minutes. The positive difference is

7.4 - 6.5 = 0.9

minutes.

Step-by-Step Solution

1

Calculate the total duration of all calls combined.

(3 \times 4) + (5 \times 6) + (8 \times 5) + (12 \times 3) + (15 \times 2) = 12 + 30 + 40 + 36 + 30 = 148\text{ minutes}

.

Each call duration must be multiplied by its corresponding frequency to find the total sum of all call durations.

2

Calculate the mean call duration.

\text{Mean} = \frac{148}{20} = 7.4\text{ minutes}

.

The mean is equal to the sum of all durations divided by the total number of calls (

20

).

3

Determine the median call duration.

\text{Median} = \frac{5 + 8}{2} = 6.5\text{ minutes}

.

For

20

sorted data points, the median is the average of the 10th and 11th values. The cumulative frequencies show that the 1st through 4th calls are

3

minutes, the 5th through 10th calls are

5

minutes, and the 11th through 15th calls are

8

minutes. Thus, the 10th value is

5

and the 11th value is

8

.

4

Compute the positive difference between the mean and the median.

7.4 - 6.5 = 0.9\text{ minutes}

.

Subtracting the median (

6.5

) from the mean (

7.4

) yields the required positive difference.

Key Concept

Calculating mean and median from a frequency table

Estimated Time:1m 15s

Question 298Question →

An environmental research vessel collects deep-sea water samples. A marine biologist determines that $\frac{4}{7}$ of the total capacity $V$ (in liters) of a specialized storage tank is occupied by seawater collected during a single dive. If the seawater collected during this dive measures exactly $28$ liters, which of the following equations correctly models this relationship, and what is the total capacity $V$ of the storage tank in liters?

\frac{4}{7}V = 28

;

V = 49

\frac{4}{7}V = 28

;

V = 16

V - \frac{4}{7} = 28

;

V = 28.57

\frac{7}{4}V = 28

;

V = 16

\frac{4}{7} + V = 28

;

V = 27.43

Show answer & explanation

Answer: $\frac{4}{7}V = 28$ ; $V = 49$

Answer

The correct equation is

\frac{4}{7}V = 28

, which yields a total tank capacity of

V = 49

liters.

The phrase '

\frac{4}{7}

of the total capacity

V

' signifies the product of the fraction and the variable

V

. Setting this product equal to the given volume of

28

liters yields the equation

\frac{4}{7}V = 28

. Isolating

V

requires dividing both sides by

\frac{4}{7}

, which is equivalent to multiplying

28

by the reciprocal

\frac{7}{4}

. Simplifying

28 \cdot \frac{7}{4}

gives

(28 \div 4) \cdot 7 = 7 \cdot 7 = 49

liters.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

\frac{4}{7}V = 28

The phrase '

\frac{4}{7}

of the total capacity

V

' means multiplying

\frac{4}{7}

by

V

, and this portion is equal to

28

liters.

2

Solve the one-step equation for

V

by multiplying both sides by the reciprocal of

\frac{4}{7}

.

V = 28 \times \frac{7}{4}

Multiplying by the reciprocal

\frac{7}{4}

isolates

V

on the left side of the equation.

3

Simplify the arithmetic calculation.

V = 7 \times 7 = 49

Dividing

28

by

4

gives

7

, and multiplying

7

by

7

results in

49

liters.

Key Concept

Formulating and solving one-step multiplication equations involving fractional coefficients.

Question 299Question →

A conference schedule allows attendees to select 1 morning workshop out of 6 options, 1 keynote session out of 3 options, and 1 afternoon panel out of 4 options. However, 2 specific morning workshops conflict with 1 specific afternoon panel and cannot be chosen together. How many different valid 3-session schedule combinations can an attendee create?

Show answer & explanation

Answer: 66

Answer

An attendee can create 66 different valid 3-session schedule combinations.

According to the Fundamental Counting Principle, the total unrestricted number of schedule combinations is

6 \times 3 \times 4 = 72

. The restriction specifies that 2 morning workshops cannot be paired with 1 specific afternoon panel. Since there are 3 keynote speaker options available for any schedule, the number of invalid combinations is

2 \times 3 \times 1 = 6

. Subtracting the invalid options from the total gives

72 - 6 = 66

valid schedule combinations.

Step-by-Step Solution

1

Calculate total unrestricted schedule combinations

72 combinations

Multiply the choices for each session:

6 \times 3 \times 4 = 72

.

2

Calculate the number of conflicting schedule combinations

6 invalid combinations

The 2 restricted morning workshops combined with 1 restricted afternoon panel can occur alongside any of the 3 keynote choices (

2 \times 3 \times 1 = 6

).

3

Subtract conflicting combinations from total combinations

66 valid combinations

Subtracting 6 invalid schedules from 72 total schedules leaves 66 allowable options.

Key Concept

Fundamental Counting Principle with Restrictions

Estimated Time:1m 30s

Question 300Question →

A youth athletic league has 140 total players registered for either soccer or basketball. Initially, the ratio of soccer players to basketball players is $4 : 3$ . If 15 additional basketball players join the league and no soccer players leave, what is the new ratio of soccer players to basketball players in simplest form?

8 : 15

16 : 15

4 : 5

16 : 19

3 : 5

Show answer & explanation

Answer: $16 : 15$

Answer

The new ratio of soccer players to basketball players in simplest form is

16 : 15

.

The initial ratio of

4 : 3

breaks 140 players into 7 equal parts of 20 players each, yielding 80 soccer players and 60 basketball players. Adding 15 basketball players gives 75 basketball players. The ratio of 80 soccer players to 75 basketball players simplifies to

16 : 15

.

Step-by-Step Solution

1

Determine the initial number of soccer and basketball players using the given ratio and total.

The total ratio parts equal

4 + 3 = 7

. Value per part

= 140 / 7 = 20

. Soccer players

= 4 \times 20 = 80

, Basketball players

= 3 \times 20 = 60

.

Converting ratio parts into actual counts is required before adjusting player totals.

2

Add 15 players to the basketball total.

New basketball player count

= 60 + 15 = 75

. Soccer player count remains

80

.

The problem states 15 new basketball players joined while soccer player count remained constant.

3

Form the new ratio of soccer players to basketball players and simplify.

80 : 75 = \frac{80}{5} : \frac{75}{5} = 16 : 15

.

Dividing both quantities by their greatest common factor, 5, expresses the ratio in simplest terms.

Key Concept

Solving multi-step ratio problems by determining actual quantities from a given total and updating quantities following a change.

Estimated Time:1m 30s

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