Pre-Algebra Practice Questions — ACT — Page 14

Question 261Question →

A commercial printing facility uses three high-speed printers—Printer X, Printer Y, and Printer Z—to produce advertising brochures. The ratio of the number of brochures printed by Printer X to Printer Y during a morning shift is $3:5$ . During the same shift, the ratio of the number of brochures printed by Printer Y to Printer Z is $4:3$ . If Printer Z printed $1,800$ brochures during the shift, what is the total number of brochures printed by all three printers combined?

3,840

3,960

5,640

6,600

7,050

Show answer & explanation

Answer: $5,640$

Answer

The total number of brochures printed by all three printers combined is

5,640

.

To find the total printed brochures, determine the output of each printer step by step. Using the ratio of Printer Y to Printer Z (

4:3

), since Printer Z printed

1,800

brochures, Printer Y printed

\frac{4}{3} \times 1,800 = 2,400

brochures. Using the ratio of Printer X to Printer Y (

3:5

), Printer X printed

\frac{3}{5} \times 2,400 = 1,440

brochures. Adding the amounts for all three printers gives

1,440 + 2,400 + 1,800 = 5,640

brochures.

Step-by-Step Solution

1

Calculate the output of Printer Y using the ratio of Printer Y to Printer Z.

Printer Y printed

\frac{4}{3} \times 1,800 = 2,400

brochures.

The ratio of Printer Y to Printer Z is

4:3

, so Printer Y produced

\frac{4}{3}

times as many brochures as Printer Z.

2

Calculate the output of Printer X using the ratio of Printer X to Printer Y.

Printer X printed

\frac{3}{5} \times 2,400 = 1,440

brochures.

The ratio of Printer X to Printer Y is

3:5

, so Printer X produced

\frac{3}{5}

times as many brochures as Printer Y.

3

Sum the outputs of all three printers.

1,440 + 2,400 + 1,800 = 5,640

brochures.

The question asks for the total combined production of Printer X, Printer Y, and Printer Z.

Key Concept

Combining rates or proportions involving shared quantities by setting up equivalent proportional relationships.

Estimated Time:1m 30s

Question 262Question →

What is the sum of all integer solutions to the equation $|2x - 8| = 14$ ?

-3

8

11

14

22

Show answer & explanation

Answer: 8

Answer

The sum of all integer solutions is 8.

To solve

|2x - 8| = 14

, consider both possible cases:

2x - 8 = 14

and

2x - 8 = -14

. The first equation gives

2x = 22

, so

x = 11

. The second equation gives

2x = -6

, so

x = -3

. Adding these two solutions yields

11 + (-3) = 8

.

Step-by-Step Solution

1

Set up the two linear equations represented by the absolute value equation.

2x - 8 = 14

or

2x - 8 = -14

The absolute value expression

|x| = k

for

k > 0

splits into

x = k

and

x = -k

.

2

Solve the first equation for

x

.

2x = 22 \implies x = 11

Add 8 to both sides and divide by 2.

3

Solve the second equation for

x

.

2x = -6 \implies x = -3

Add 8 to both sides and divide by 2.

4

Calculate the sum of both integer solutions.

11 + (-3) = 8

The problem asks for the sum of all integer solutions.

Key Concept

Solving Absolute Value Equations with Integers

Estimated Time:1m 0s

Question 263Question →

A university library's archival team is digitizing rare manuscripts using three scanner models: Model X, Model Y, and Model Z. The ratio of the scanning speed of Model X to Model Y is $3:5$ , and the ratio of the scanning speed of Model Y to Model Z is $2:3$ . If Model X and Model Z together scan a total of $420$ pages in $1$ hour, how many total pages will all three scanner models combined scan in $3$ hours?

620

1,240

1,860

2,310

2,520

Show answer & explanation

Answer: 1,860

Answer

All three scanner models combined will scan 1,860 pages in 3 hours.

To find the total number of pages scanned, first unify the given ratios using Model Y as the common bridge: Model X to Model Y is

3:5

(

6:10

) and Model Y to Model Z is

2:3

(

10:15

), yielding an extended ratio of

6:10:15

. Since Model X and Model Z together scan

420

pages in

1

hour,

6k + 15k = 420

, which yields

k = 20

. The total hourly rate for all three models is

(6 + 10 + 15) \times 20 = 620

pages per hour. Over

3

hours, the total volume scanned is

620 \times 3 = 1,860

pages.

Step-by-Step Solution

1

Unify the two ratios into a single three-part ratio for Model X : Model Y : Model Z.

Model X : Model Y =

3 : 5 = 6 : 10

and Model Y : Model Z =

2 : 3 = 10 : 15

. Thus, Model X : Model Y : Model Z =

6 : 10 : 15

.

Model Y is the common element between both given ratios, so we scale both ratios so that Model Y has the same value (

10

) in both.

2

Set up an equation based on the combined hourly output of Model X and Model Z.

Let

k

be the multiplier. Rate of X

= 6k

, Rate of Z

= 15k

.

6k + 15k = 420 \implies 21k = 420 \implies k = 20

.

Model X and Model Z together produce

420

pages per hour, which allows us to solve for the constant ratio multiplier

k

.

3

Calculate the combined hourly rate for all three scanner models.

Combined Rate

= (6 + 10 + 15)k = 31k = 31 \times 20 = 620

pages per hour.

Summing the ratio parts of all three models gives the total rate per hour.

4

Multiply the combined hourly rate by 3 hours to find the total production.

620 \text{ pages/hour} \times 3 \text{ hours} = 1,860 \text{ pages}

.

The question asks for the total output over a 3-hour period.

Key Concept

Unifying compound ratios into a single extended ratio and scaling proportional rates over time.

Estimated Time:2m 0s

Question 264Question →

A renewable energy facility operates three solar arrays: Alpha, Beta, and Gamma. The ratio of energy generated by Array Alpha to Array Beta is $4:5$ , and the ratio of energy generated by Array Beta to Array Gamma is $3:2$ . On a given day, Array Gamma generated $120$ kilowatt-hours (kWh) less energy than Array Alpha. What is the total energy, in kWh, generated by all three arrays combined on that day?

660

900

1,320

2,220

4,440

Show answer & explanation

Answer: $2,220$

Answer

The total energy generated by all three arrays combined on that day is

2,220

kWh.

To solve this problem, first express the energy outputs as a unified ratio Alpha : Beta : Gamma. Since Alpha : Beta =

4:5

and Beta : Gamma =

3:2

, convert both so that Beta has a common ratio value of

15

. This gives Alpha : Beta : Gamma =

12 : 15 : 10

. Define the outputs in terms of a multiplier

x

: Alpha =

12x

, Beta =

15x

, and Gamma =

10x

. The problem states that Gamma generates

120

kWh less than Alpha, so

12x - 10x = 120

, which simplifies to

2x = 120

or

x = 60

. The total output for all three arrays combined is

12x + 15x + 10x = 37x

. Multiplying

37 \times 60

yields

2,220

kWh.

Step-by-Step Solution

1

Unify the two separate ratios into a single continuous ratio for Alpha : Beta : Gamma.

The ratio Alpha : Beta is

4:5

and Beta : Gamma is

3:2

. The common term is Beta. Multiply

4:5

by

3

to get

12:15

, and multiply

3:2

by

5

to get

15:10

. The combined ratio is Alpha : Beta : Gamma =

12 : 15 : 10

.

Array Beta must have the same relative value in both ratios to enable direct comparison across all three arrays.

2

Set up an equation for the given difference between Array Alpha and Array Gamma.

Let

x

be the common multiplier. Array Alpha produces

12x

kWh and Array Gamma produces

10x

kWh. The difference is

12x - 10x = 2x

. We are given

2x = 120

, so

x = 60

.

Using a single variable

x

allows conversion from ratio units to actual kilowatt-hours.

3

Calculate the total energy generated by summing the parts of all three arrays.

Total energy =

12x + 15x + 10x = 37x

. Substituting

x = 60

:

37 \times 60 = 2,220

kWh.

The question asks for the total energy produced by all three arrays combined.

Key Concept

Combining Three-Part Ratios and Proportions

Question 265Question →

Arrange the following four mathematical expressions in order from least to greatest value.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest value is

\frac{9 \times 10^{-4}}{0.5}

,

\left(5 \times 10^{-2}\right)^2

,

3.2 \times 10^{-3}

, and

\sqrt{1.6 \times 10^{-5}}

.

Converting each expression to standard scientific notation with base power

10^{-3}

allows for direct comparison of coefficients:

\frac{9 \times 10^{-4}}{0.5} = 1.8 \times 10^{-3}

,

\left(5 \times 10^{-2}\right)^2 = 2.5 \times 10^{-3}

,

3.2 \times 10^{-3}

is already in standard form, and

\sqrt{1.6 \times 10^{-5}} = \sqrt{16 \times 10^{-6}} = 4.0 \times 10^{-3}

. Comparing coefficients

1.8 < 2.5 < 3.2 < 4.0

confirms the order from least to greatest.

Step-by-Step Solution

1

Evaluate the expression

\frac{9 \times 10^{-4}}{0.5}

1.8 \times 10^{-3}

(or

0.0018

)

Dividing the coefficient

9

by

0.5

gives

18

. Then convert

18 \times 10^{-4}

to standard scientific notation

1.8 \times 10^{-3}

.

2

Evaluate the expression

\left(5 \times 10^{-2}\right)^2

2.5 \times 10^{-3}

(or

0.0025

)

Apply the power of a product rule:

5^2 \times (10^{-2})^2 = 25 \times 10^{-4} = 2.5 \times 10^{-3}

.

3

Evaluate the expression

\sqrt{1.6 \times 10^{-5}}

4.0 \times 10^{-3}

(or

0.0040

)

Rewrite

1.6 \times 10^{-5}

as

16 \times 10^{-6}

so the exponent of

10

is even. Taking the square root gives

\sqrt{16} \times \sqrt{10^{-6}} = 4 \times 10^{-3}

.

4

Compare all expressions in standard scientific notation with power

10^{-3}

1.8 \times 10^{-3} < 2.5 \times 10^{-3} < 3.2 \times 10^{-3} < 4.0 \times 10^{-3}

Since all numbers share the same power of ten (

10^{-3}

), compare their coefficients:

1.8 < 2.5 < 3.2 < 4.0

.

Key Concept

Comparing quantities in scientific notation by simplifying exponent and radical operations to standard forms.

Question 266Question →

A track athlete recorded the following times, in seconds, for five 100-meter sprint trials:

12.4,\ 11.8,\ 13.1,\ 12.0,\ 12.2

What is the median time, in seconds, for these five trials?

1.3

12.0

12.2

12.3

13.1

Show answer & explanation

Answer: 12.2

Answer

The median time for the five trials is 12.2 seconds.

To find the median, first arrange the five times in ascending order: 11.8, 12.0, 12.2, 12.4, and 13.1. Since there are 5 values, the median is the middle (3rd) value, which is 12.2.

Step-by-Step Solution

1

Order the data set in ascending numerical order.

11.8, 12.0, 12.2, 12.4, 13.1

The median is defined as the middle value of an ordered data set.

2

Identify the middle (third) position in the 5-item ordered list.

The third value is

12.2

.

For an odd number of data points (

n = 5

), the median is the value at position

\frac{n+1}{2} = 3

.

Key Concept

Calculating the median of a finite data set

Question 267Question →

An agricultural research facility divides a field with a total area of $A$ acres into $n$ individual test plots of equal size after reserving an area of 12 acres for administrative buildings. Which of the following equations correctly expresses the area, $p$ , in acres, of each individual test plot in terms of $A$ and $n$ ?

p = \frac{A - 12}{n}

p = \frac{A}{n} - 12

p = \frac{A + 12}{n}

p = n(A - 12)

p = \frac{n}{A - 12}

Show answer & explanation

Answer: $p = \frac{A - 12}{n}$

Answer

The equation representing the area of each test plot is

p = \frac{A - 12}{n}

.

Reserving 12 acres leaves

A - 12

acres available for testing. Dividing this remaining quantity equally into

n

test plots results in an area per plot of

p = \frac{A - 12}{n}

.

Step-by-Step Solution

1

Determine the remaining area allocated for test plots.

Available area =

A - 12

acres.

The 12 acres used for administrative buildings must be subtracted from the total field area

A

.

2

Divide the remaining area by the number of equal test plots

n

.

p = \frac{A - 12}{n}

.

Dividing the remaining acreage equally among

n

plots gives the area

p

of a single plot.

Key Concept

Translating contextual verbal scenarios into basic algebraic expressions using grouping and division.

Question 268Question →

A microchip manufacturer produces $2.4 \times 10^8$ components per day across 4 identical production lines. Each line operates for 20 hours per day. On average, how many components are produced per hour on a single production line?

3.0 \times 10^5

3.0 \times 10^6

1.2 \times 10^7

3.0 \times 10^7

4.8 \times 10^9

Show answer & explanation

Answer: $3.0 \times 10^6$

Answer

3.0 \times 10^6

components per hour

To find the hourly output of a single line, first divide the total components (

2.4 \times 10^8

) by the 4 lines to get

6.0 \times 10^7

components per line per day. Then, divide by the 20 operating hours (

2.0 \times 10^1

) to obtain

3.0 \times 10^6

components per hour per line.

Step-by-Step Solution

1

Calculate daily production per single line

\frac{2.4 \times 10^8}{4} = 0.6 \times 10^8 = 6.0 \times 10^7

components per line per day

The total daily output is split equally among 4 identical lines.

2

Calculate hourly production per single line

\frac{6.0 \times 10^7}{20} = \frac{6.0 \times 10^7}{2.0 \times 10^1} = 3.0 \times 10^6

components per hour

Dividing daily output per line by the 20 operating hours per day gives the hourly rate per line.

Key Concept

Division of numbers expressed in scientific notation

Question 269Question →

On a vertical number line representing elevation relative to sea level ( $0$ meters), Submersible $X$ is located at a coordinate of $-45$ meters. Submersible $Y$ is located at a depth such that the distance between Submersible $X$ and Submersible $Y$ is $30$ meters. If Submersible $Y$ is closer to sea level than Submersible $X$ , what is the value of $|2Y - X|$ ?

15

60

75

105

135

Show answer & explanation

Answer: $15$

Answer

The value of

|2Y - X|

is

15

.

Submersible

X

is at

-45

. Being

30

meters away means Submersible

Y

is at either

-15

or

-75

. Since Submersible

Y

is closer to sea level (

0

), its coordinate is

-15

. Substituting

X = -45

and

Y = -15

into

|2Y - X|

gives

|2(-15) - (-45)| = |-30 + 45| = |15| = 15

.

Step-by-Step Solution

1

Determine the possible coordinates for Submersible

Y

.

Since

X = -45

and the distance between

X

and

Y

is

30

,

Y = -45 + 30 = -15

or

Y = -45 - 30 = -75

.

Distance on a number line from coordinate

x

is given by

x \pm d

.

2

Select the correct coordinate for

Y

based on the given constraint.

Y = -15

, because

|-15| = 15 < 45 = |-45|

, meaning

Y

is closer to sea level (

0

).

The problem states that Submersible

Y

is closer to sea level than Submersible

X

.

3

Substitute

X = -45

and

Y = -15

into the expression

|2Y - X|

and evaluate.

|2(-15) - (-45)| = |-30 + 45| = |15| = 15

.

Multiplying

2

by

-15

yields

-30

, and subtracting

-45

is equivalent to adding

45

.

Key Concept

Distance on a Number Line and Absolute Value Evaluation

Estimated Time:1m 15s

Question 270Question →

A cafe offers a lunch special where a customer selects $1$ sandwich from $4$ available options, $1$ side dish from $3$ available options, and $1$ drink from $5$ available options. How many different total lunch combinations consisting of one sandwich, one side dish, and one drink can a customer choose?

Show answer & explanation

Answer: 60

Answer

60 combinations

According to the Fundamental Counting Principle, if one event can occur in

m

ways, a second in

n

ways, and a third in

p

ways, the total number of combinations for all three events occurring together is

m \times n \times p

. Multiplying

4

sandwiches by

3

sides by

5

drinks yields

4 \times 3 \times 5 = 60

distinct lunch combinations.

Step-by-Step Solution

1

Identify the number of options available for each independent selection.

4 sandwiches, 3 side dishes, 5 drinks

The total outcomes depend on making one selection from each distinct category.

2

Apply the Fundamental Counting Principle.

4 × 3 × 5 = 60

The total number of outcomes for independent sequential choices is the product of the number of options for each choice.

Key Concept

Fundamental Counting Principle

Estimated Time:45s

Question 271Question →

A commercial bakery mixes three types of flour—whole wheat, rye, and oat—to create a specialty bread blend. Initially, the ratio of whole wheat flour to rye flour to oat flour by weight is $5:3:2$ . The baker then adds 12 pounds of oat flour and 6 pounds of rye flour to the batch while leaving the amount of whole wheat flour unchanged. As a result, the ratio of whole wheat flour to oat flour in the new mixture becomes $1:1$ . What was the total weight, in pounds, of the initial flour mixture before any additional flour was added?

20

32

40

58

60

Show answer & explanation

Answer: 40

Answer

The total weight of the initial flour mixture was 40 pounds.

Representing the initial amounts of whole wheat, rye, and oat flour as

5x

,

3x

, and

2x

establishes that the initial total weight is

10x

. Adding 12 pounds to the oat flour makes the new oat weight

2x + 12

. Because the new ratio of whole wheat to oat is

1:1

, we set

5x = 2x + 12

, which yields

x = 4

. Multiplying the multiplier

x = 4

by the total number of parts (

10

) gives the original total weight of 40 pounds.

Step-by-Step Solution

1

Express the initial weights using a common multiplier

x

.

Whole Wheat =

5x

, Rye =

3x

, Oat =

2x

. The total initial weight is

5x + 3x + 2x = 10x

.

An extended ratio of

5:3:2

means each component is a multiple of a shared constant

x

.

2

Set up an equation based on the new ratio of whole wheat to oat flour.

Whole Wheat =

5x

, New Oat =

2x + 12

. Since the ratio is

1:1

,

5x = 2x + 12

.

Adding 12 pounds of oat flour changes the oat amount to

2x + 12

, and a

1:1

ratio means the two amounts are equal.

3

Solve for the multiplier

x

.

3x = 12 \Rightarrow x = 4

.

Subtract

2x

from both sides and divide by 3.

4

Calculate the total initial weight.

Total initial weight =

10x = 10(4) = 40

pounds.

Substitute

x = 4

back into the total initial expression

10x

.

Key Concept

Solving multi-part ratio change problems by expressing quantities in terms of a common variable.

Estimated Time:1m 30s

Question 272Question →

A local library recorded the number of books read by students participating in a summer reading program. The results are summarized in the frequency table below.

Number of Books Read	Number of Students
$2$	$4$
$3$	$6$
$5$	$5$
$8$	$3$
$10$	$2$

What is the mean number of books read per student for this group?

4.75

5.00

5.60

19.00

4.00

Show answer & explanation

Answer: $4.75$

Answer

The mean number of books read per student is

4.75

.

The value

4.75

is correct because finding the mean from a frequency table requires calculating the weighted sum of all data values divided by the total frequency. Multiplying each book count by its student count gives

2 \times 4 + 3 \times 6 + 5 \times 5 + 8 \times 3 + 10 \times 2 = 95

total books. Dividing

95

by the total student count of

20

yields

4.75

.

Step-by-Step Solution

1

Calculate the total number of students.

Total students =

4 + 6 + 5 + 3 + 2 = 20

.

To find the mean per student, the total number of participants is needed for the denominator.

2

Calculate the total number of books read by multiplying each value by its frequency and summing.

Total books =

(2 \times 4) + (3 \times 6) + (5 \times 5) + (8 \times 3) + (10 \times 2) = 8 + 18 + 25 + 24 + 20 = 95

.

The mean calculation requires the aggregate total of all books read across all students.

3

Divide the total number of books by the total number of students.

Mean =

\frac{95}{20} = 4.75

.

The mean of a frequency distribution is given by the total sum divided by the total frequency.

Key Concept

Weighted Mean from a Frequency Table

Estimated Time:1m 30s

Question 273Question →

A box contains $5$ red tickets, $7$ blue tickets, and $8$ green tickets. If one ticket is selected at random from the box, what is the probability that the selected ticket is NOT blue?

\frac{7}{20}

\frac{13}{20}

\frac{7}{13}

\frac{3}{5}

\frac{13}{7}

Show answer & explanation

Answer: $\frac{13}{20}$

Answer

The probability that the selected ticket is not blue is

\frac{13}{20}

.

To find the probability of selecting a ticket that is not blue, divide the number of non-blue tickets by the total number of tickets. There are

5 + 7 + 8 = 20

total tickets, of which

5 + 8 = 13

are not blue. Thus, the probability is

\frac{13}{20}

.

Step-by-Step Solution

1

Find the total number of outcomes (total tickets in the box).

Total tickets =

5 + 7 + 8 = 20

.

Basic probability requires dividing the number of favorable outcomes by the total sample space.

2

Calculate the number of favorable outcomes (tickets that are NOT blue).

Favorable tickets =

5\text{ (red)} + 8\text{ (green)} = 13

(or

20 - 7 = 13

).

The complement of choosing a blue ticket is choosing a red or green ticket.

3

Form the probability ratio of favorable outcomes to total outcomes.

Probability =

\frac{13}{20}

.

Probability is defined as

\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}

.

Key Concept

Basic Probability of Complementary Events

Estimated Time:45s

Question 274Question →

A municipal water authority pumps water into a storage reservoir. During a drought conservation phase, the authority reduces its standard daily pumping volume by $35\%$ . If the resulting reduced daily pumping volume is $W$ acre-feet, which of the following equations correctly expresses the standard daily pumping volume, $S$ , in acre-feet, in terms of $W$ ?

S = \frac{W}{0.65}

S = 0.65W

S = \frac{W}{0.35}

S = 0.35W

S = W + 0.35

Show answer & explanation

Answer: $S = \frac{W}{0.65}$

Answer

The standard daily pumping volume is given by

S = \frac{W}{0.65}

.

Reducing the standard volume

S

by

35\%

leaves

65\%

of

S

, which can be written algebraically as

0.65S

. Since this reduced amount equals

W

, the relationship is

0.65S = W

. Dividing both sides by

0.65

isolates

S

to give the one-step equation solution

S = \frac{W}{0.65}

.

Step-by-Step Solution

1

Express the reduced volume in terms of the standard volume

S

Reduced volume

= S - 0.35S = 0.65S

A

35\%

reduction leaves

100\% - 35\% = 65\%

of the original standard volume

S

.

2

Set up the one-step algebraic equation matching the given variable

W

0.65S = W

The reduced volume is defined as

W

acre-feet.

3

Solve the one-step equation for

S

S = \frac{W}{0.65}

Divide both sides of the equation by

0.65

to isolate

S

.

Key Concept

Formulating and solving a one-step algebraic equation involving a percentage decrease.

Alternative Method

Convert

65\%

to the fraction

\frac{13}{20}

. Then

\frac{13}{20}S = W

, which yields

S = \frac{20}{13}W = \frac{W}{0.65}

.

Estimated Time:1m 30s

Question 275Question →

A community theater company uses a combination of red, yellow, and blue light filters to create a specific lighting effect on stage. The ratio of the number of red filters to yellow filters is $4:3$ , and the ratio of the number of yellow filters to blue filters is $2:5$ . If the lighting setup uses a total of $58$ filters across these three colors, how many blue filters are used in the setup?

10

12

16

28

30

Show answer & explanation

Answer: 30

Answer

30

To find the number of blue filters, we must write a single combined ratio for Red : Yellow : Blue. Since Red : Yellow =

4 : 3

and Yellow : Blue =

2 : 5

, we convert both ratios so that Yellow has the same term. Multiples of

3

and

2

give a common value of

6

. Scaling Red : Yellow by

2

gives

8 : 6

, and scaling Yellow : Blue by

3

gives

6 : 15

. Thus, Red : Yellow : Blue =

8 : 6 : 15

. The total number of ratio units is

8 + 6 + 15 = 29

. Each ratio unit represents

58 \div 29 = 2

filters. Therefore, the number of blue filters is

15 \times 2 = 30

.

Step-by-Step Solution

1

Find a common value for the yellow ratio component to combine the two ratios into a single three-part ratio.

The ratio of Red to Yellow is

4:3 = 8:6

. The ratio of Yellow to Blue is

2:5 = 6:15

. Therefore, Red : Yellow : Blue =

8 : 6 : 15

.

Yellow is the shared quantity between both given ratios, so its ratio value must be equal in both expressions to combine them.

2

Calculate the total number of ratio parts and determine the value of one part.

Total parts =

8 + 6 + 15 = 29

parts. Value per part =

\frac{58}{29} = 2

filters per part.

Dividing the total filter quantity by the total number of parts yields the scale factor for the ratio.

3

Multiply the blue filter ratio parts by the value per part.

15 \text{ parts} \times 2 = 30

blue filters.

The question specifically asks for the total quantity of blue filters.

Key Concept

Three-Quantity Ratio Combination and Scaling

Question 276Question →

A biology class recorded the height, in centimeters, of $15$ plant seedlings after two weeks of growth. The frequency table below summarizes the recorded measurements:

Height (cm)	Number of Seedlings
$3$	$2$
$4$	$4$
$5$	$5$
$6$	$3$
$7$	$1$

What is the positive difference between the mean height and the median height, in centimeters, of these $15$ seedlings?

0.0

0.2

0.5

0.8

1.2

Show answer & explanation

Answer: $0.2$

Answer

The positive difference between the mean height and the median height is

0.2

centimeters.

To calculate the mean height, multiply each height by its frequency, sum the products (

72

), and divide by the total number of seedlings (

15

), giving

4.8

cm. To find the median, locate the middle value (

8^{\text{th}}

value) in the ordered dataset of

15

measurements, which is

5.0

cm. The positive difference between

5.0

cm and

4.8

cm is

0.2

cm.

Step-by-Step Solution

1

Calculate the total sum of all seedling heights using the frequency table

(3 \times 2) + (4 \times 4) + (5 \times 5) + (6 \times 3) + (7 \times 1) = 6 + 16 + 25 + 18 + 7 = 72

cm

Each height value must be multiplied by its corresponding frequency count to find the total sum.

2

Calculate the weighted mean height

Mean =

\frac{72}{15} = 4.8

cm

The sum of all heights (

72

) is divided by the total number of seedlings (

15

).

3

Determine the median height of the

15

seedlings

Median =

5.0

cm

For

15

ordered values, the median is the

8^{\text{th}}

data point. Accumulating counts from smallest height:

2

(at height 3)

+ 4

(at height 4)

= 6

values. The next

5

values (positions 7 through 11) all have a height of 5. Therefore, the

8^{\text{th}}

value is

5.0

cm.

4

Find the positive difference between the mean and the median

|4.8 - 5.0| = 0.2

cm

Subtracting the mean (

4.8

) from the median (

5.0

) gives a positive difference of

0.2

cm.

Key Concept

Descriptive Statistics and Data Representations

Estimated Time:1m 30s

Question 277Question →

A commercial bakery uses an automated mixing system to prepare large batches of dough. The system processes flour at a constant rate such that $F$ kilograms of flour are mixed over $m$ minutes. If the mixer consumes flour at a rate of $\frac{3}{8}$ kilogram per minute, which of the following equations correctly expresses $m$ , the total mixing time in minutes, in terms of $F$ ?

m = \frac{3F}{8}

m = \frac{8F}{3}

m = F - \frac{3}{8}

m = \frac{3}{8F}

m = \frac{8}{3F}

Show answer & explanation

Answer: $m = \frac{8F}{3}$

Answer

The correct equation expressing

m

in terms of

F

is

m = \frac{8F}{3}

.

The total amount of flour used is determined by multiplying the rate per minute by the number of minutes, giving the equation

F = \frac{3}{8}m

. To express

m

in terms of

F

, solve this one-step equation for

m

by dividing both sides by

\frac{3}{8}

. Dividing by

\frac{3}{8}

is mathematically equivalent to multiplying by its reciprocal,

\frac{8}{3}

. Therefore,

m = \frac{8F}{3}

.

Step-by-Step Solution

1

Set up the one-step relationship between rate, total flour, and time.

F = \frac{3}{8}m

Total flour

F

equals the rate per minute multiplied by total minutes

m

.

2

Isolate the variable

m

by dividing both sides of the equation by

\frac{3}{8}

.

m = \frac{F}{\frac{3}{8}}

To solve a one-step multiplication equation, apply the inverse operation (division).

3

Simplify the fraction division by multiplying by the reciprocal of

\frac{3}{8}

.

m = F \cdot \frac{8}{3} = \frac{8F}{3}

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Key Concept

Solving One-Step Equations with Fractional Coefficients

Estimated Time:1m 0s

Question 278Question →

A landscaping company prepares a custom soil mixture by combining topsoil, compost, and coarse sand in the ratio $2 : 3 : 7$ by weight. A project requires a total of $360\text{ pounds}$ of this soil mixture. The cost per pound for each component is as follows:

- Topsoil: $\$0.40$ per pound
- Compost: $\$0.60$ per pound
- Coarse sand: $\$0.25$ per pound

What is the total cost of the components required to make the $360\text{ pounds}$ of soil mixture?

$106.50

$124.50

$130.50

$150.00

$450.00

Show answer & explanation

Answer: $130.50

Answer

The total cost of the components required is $130.50.

To find the total cost, first determine the total number of parts in the ratio:

2 + 3 + 7 = 12

parts. Divide the total weight of

360\text{ pounds}

by

12

to find that each ratio part corresponds to

30\text{ pounds}

. The weight of topsoil is

2 \times 30 = 60\text{ pounds}

, compost is

3 \times 30 = 90\text{ pounds}

, and coarse sand is

7 \times 30 = 210\text{ pounds}

. Multiplying each weight by its unit price yields: topsoil at

60 \times \$0.40 = \$24.00

, compost at

90 \times \$0.60 = \$54.00

, and sand at

210 \times \$0.25 = \$52.50

. Adding these amounts together gives a total cost of

\$130.50

.

Step-by-Step Solution

1

Find the total number of ratio parts.

2 + 3 + 7 = 12 parts

The total weight of the mixture is distributed across the sum of all parts in the ratio.

2

Determine the weight per single ratio part.

360 pounds / 12 parts = 30 pounds per part

Dividing the total weight by the total number of parts yields the scaling factor.

3

Calculate the weight of each component.

Topsoil: 2 x 30 = 60 pounds; Compost: 3 x 30 = 90 pounds; Coarse sand: 7 x 30 = 210 pounds

Multiply each component's ratio value by the weight per part.

4

Calculate the cost of each component and sum them up.

Topsoil: 60 x

0.40 =

24.00; Compost: 90 x

0.60 =

54.00; Sand: 210 x

0.25 =

52.50. Total cost =

24.00 +

54.00 +

52.50 =

130.50

Multiply the weight of each component by its price per pound and add the individual costs.

Key Concept

Solving multi-part ratio problems by converting part-to-part ratios into part-to-whole fractions to scale quantities and compute weighted totals.

Question 279Question →

A jar contains $3$ green marbles, $5$ yellow marbles, and $12$ purple marbles. If one marble is drawn at random from the jar, what is the probability that the drawn marble is NOT purple? Express your answer as a decimal.

Show answer & explanation

Answer: 0.4

Answer

The probability that the drawn marble is NOT purple is 0.4.

To find the probability of drawing a non-purple marble, identify the number of non-purple marbles (3 green + 5 yellow = 8) and divide by the total number of marbles in the jar (3 + 5 + 12 = 20). The resulting fraction 8/20 simplifies to 0.4.

Step-by-Step Solution

1

Calculate the total number of outcomes in the sample space.

Total marbles = 3 + 5 + 12 = 20

The total sample space includes all marbles contained in the jar.

2

Determine the number of favorable outcomes.

Non-purple marbles = 3 + 5 = 8

A marble that is not purple must be either green or yellow.

3

Compute the probability.

Probability = 8 / 20 = 0.4

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

Key Concept

Basic Probability of Complementary Events

Question 280Question →

On a standard number line, point $P$ is located at $-17$ and point $Q$ is located at $31$ . Point $R$ lies between $P$ and $Q$ such that the ratio of the distance between $P$ and $R$ to the distance between $R$ and $Q$ is $3:5$ . What is the value of $|R - (-5)|$ ?

Show answer & explanation

Answer: 6

Answer

The value of

|R - (-5)|

is

6

.

The total distance between point

P (-17)

and point

Q (31)

is

31 - (-17) = 48

. Since point

R

divides segment

PQ

in a

3:5

ratio, the distance from

P

to

R

is

\frac{3}{8} \times 48 = 18

. Adding this distance to

-17

gives

R = 1

. Substituting

R = 1

into

|R - (-5)|

yields

|1 - (-5)| = |6| = 6

.

Step-by-Step Solution

1

Calculate the total distance between points

P

and

Q

Distance

PQ = 31 - (-17) = 48

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

2

Determine the coordinate of point

R

Coordinate of

R = 1

The ratio of

PR

to

RQ

is

3:5

, meaning

PR

is

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

of the total distance

PQ

. Adding

\frac{3}{8} \times 48 = 18

to the starting coordinate

-17

gives

R = 1

.

3

Evaluate the absolute value expression

|R - (-5)|

|1 - (-5)| = |6| = 6

Substitute

R = 1

into the target expression and simplify the double negative before taking the absolute value.

Key Concept

Calculating distance on a number line, segment ratio partitioning, and absolute value evaluation.

Estimated Time:1m 30s

Pre-Algebra

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