Pre-Algebra Practice Questions — ACT — Page 17

Question 321Question →

A high school student council must select a 3-member executive committee consisting of a President, a Vice President, and a Treasurer from a group of 6 juniors and 4 seniors. If the positions are filled sequentially at random and no student can hold more than one position, what is the probability that the President is a senior, the Vice President is a junior, and the Treasurer is a senior?

\frac{1}{10}

\frac{3}{10}

\frac{2}{15}

\frac{12}{125}

\frac{17}{120}

Show answer & explanation

Answer: $\frac{1}{10}$

Answer

The probability that the President is a senior, the Vice President is a junior, and the Treasurer is a senior is

\frac{1}{10}

.

To find the probability of dependent sequential events, multiply the probability of each event given that the preceding events have occurred. First, selecting a senior President has a probability of

\frac{4}{10}

. Next, selecting a junior Vice President from the remaining 9 students has a probability of

\frac{6}{9}

. Finally, selecting another senior as Treasurer from the remaining 8 students (which now contains 3 seniors) has a probability of

\frac{3}{8}

. Multiplying these probabilities yields

\frac{4}{10} \times \frac{6}{9} \times \frac{3}{8} = \frac{72}{720} = \frac{1}{10}

.

Step-by-Step Solution

1

Determine the probability of choosing a senior as President.

There are 4 seniors out of 10 total students, so

P(\text{President is Senior}) = \frac{4}{10}

.

The first selection is made from the full group of 10 students.

2

Determine the probability of choosing a junior as Vice President given the first event.

There are 6 juniors remaining out of 9 total remaining students, so

P(\text{VP is Junior} \mid \text{President is Senior}) = \frac{6}{9}

.

One student (a senior) has already been selected, reducing the total count to 9.

3

Determine the probability of choosing a senior as Treasurer given the first two events.

There are 3 seniors remaining out of 8 total remaining students, so

P(\text{Treasurer is Senior} \mid \text{first two selections}) = \frac{3}{8}

.

Two students have now been selected, leaving 8 total students and 3 remaining seniors.

4

Multiply the sequential conditional probabilities together.

\frac{4}{10} \times \frac{6}{9} \times \frac{3}{8} = \frac{72}{720} = \frac{1}{10}

.

According to the Multiplication Rule for Dependent Events, the joint probability is the product of the sequential probabilities.

Key Concept

Probability of Dependent Compound Events

Estimated Time:1m 30s

Question 322Question →

A municipal transit system tracks its monthly revenue from three fare categories: Single-Ride passes, Weekly passes, and Monthly subscriptions. Single-Ride passes account for $\frac{9}{20}$ of the total monthly revenue, and Weekly passes account for $30\%$ of the total monthly revenue. If the remaining revenue from Monthly subscriptions is $\$21,000$ , what was the total monthly revenue, in dollars, for the transit system?

$28,000

$35,000

$63,000

$84,000

$105,000

Show answer & explanation

Answer: $84,000

Answer

The total monthly revenue for the transit system is $84,000.

Single-Ride passes account for

\frac{9}{20} = 45\%

of revenue, and Weekly passes account for

30\%

. Together, they account for

45\% + 30\% = 75\%

of total revenue. The remaining

25\%

(

100\% - 75\%

) corresponds to Monthly subscriptions, which equals

\$21,000

. Dividing

\$21,000

by

0.25

gives the total monthly revenue of

\$84,000

.

Step-by-Step Solution

1

Convert the fraction portion into a percentage.

\frac{9}{20} = \frac{45}{100} = 45\%

Converting all proportions to percentages allows for direct comparison and addition.

2

Calculate the combined percentage accounted for by Single-Ride and Weekly passes.

45\% + 30\% = 75\%

Summing the two given parts determines the total portion of revenue accounted for so far.

3

Find the percentage corresponding to Monthly subscriptions.

100\% - 75\% = 25\%

The three fare categories make up

100\%

of the total monthly revenue.

4

Set up an equation to solve for the total revenue

T

.

0.25 T = 21,000 \implies T = \frac{21,000}{0.25} = 84,000

Dividing the dollar amount of the remaining category by its percentage equivalent yields the total revenue.

Key Concept

Combining fractions and percentages to solve for an unknown total value.

Estimated Time:1m 30s

Question 323Question →

A community orchard initially planted apple, peach, and plum trees in the ratio $4 : 3 : 5$ , respectively. After a severe frost, $12$ plum trees died and were removed from the orchard, while all apple and peach trees survived. If the new ratio of apple trees to remaining plum trees became $2 : 1$ , how many total trees were originally in the orchard?

20

36

40

48

60

Show answer & explanation

Answer: $48$

Answer

The total number of trees originally in the orchard was 48.

By representing the initial tree counts with the multiplier

x

as

4x

(apple),

3x

(peach), and

5x

(plum), the total original tree count is

12x

. Setting up the proportion

\frac{4x}{5x - 12} = \frac{2}{1}

gives

4x = 10x - 24

, which simplifies to

6x = 24

, or

x = 4

. Multiplying

x = 4

by the total ratio units (

12

) yields

48

original trees.

Step-by-Step Solution

1

Define variables using the initial ratio.

Let the original number of apple, peach, and plum trees be

4x

,

3x

, and

5x

, respectively. The original total number of trees is

4x + 3x + 5x = 12x

.

Ratios allow quantities to be represented in terms of a common multiplier

x

.

2

Set up an equation based on the updated number of plum trees and the new ratio.

After removing

12

plum trees, the number of plum trees is

5x - 12

. The ratio of apple trees to remaining plum trees is

\frac{4x}{5x - 12} = \frac{2}{1}

.

The number of apple trees remains

4x

, and the problem states the new apple-to-plum ratio is

2 : 1

.

3

Solve the proportion for

x

.

4x = 2(5x - 12) \implies 4x = 10x - 24 \implies 6x = 24 \implies x = 4

.

Cross-multiplying eliminates fractions and isolates the variable

x

.

4

Calculate the original total number of trees.

Original total trees

= 12x = 12(4) = 48

.

Substituting

x = 4

back into the original total expression yields the requested answer.

Key Concept

Solving multi-part ratio problems with altered quantities by establishing algebraic proportions.

Estimated Time:1m 15s

Question 324Question →

A software development team consists of $4$ front-end developers, $5$ back-end developers, and $3$ quality assurance (QA) engineers. The project manager needs to select a $3$ -member subcommittee to present a new product feature. If the subcommittee must contain at least $1$ front-end developer and at least $1$ back-end developer, how many different $3$ -member subcommittees can be formed?

Show answer & explanation

Answer: 130

Answer

The total number of valid 3-member subcommittees that can be formed is 130.

To form a 3-member committee with at least 1 front-end developer and at least 1 back-end developer from 4 front-end, 5 back-end, and 3 QA engineers, we break the problem into three mutually exclusive valid scenarios: selecting 1 member from each role gives

4 \times 5 \times 3 = 60

ways; selecting 2 front-end and 1 back-end developer gives

C(4,2) \times C(5,1) = 6 \times 5 = 30

ways; and selecting 1 front-end and 2 back-end developers gives

C(4,1) \times C(5,2) = 4 \times 10 = 40

ways. Summing these possibilities gives

60 + 30 + 40 = 130

. Alternatively, subtracting invalid committees (those with no front-end developers:

C(8,3) = 56

, no back-end developers:

C(7,3) = 35

, minus double-counted 3 QA engineers:

1

) from total 3-member committees (

C(12,3) = 220

) yields

220 - (56 + 35 - 1) = 130

.

Step-by-Step Solution

1

Determine the required subcommittee size and role counts

The team has 4 front-end developers, 5 back-end developers, and 3 QA engineers (12 members total). A subcommittee of size 3 is required.

Establishing the total population and category counts is necessary before computing combination constraints.

2

Enumerate the mutually exclusive cases that satisfy all constraints

Case 1: (1 front-end, 1 back-end, 1 QA)
Case 2: (2 front-end, 1 back-end, 0 QA)
Case 3: (1 front-end, 2 back-end, 0 QA)

The committee requires at least 1 front-end and at least 1 back-end member in a 3-person group.

3

Compute combinations for each valid case using the combination formula

C(n, k) = \frac{n!}{k!(n-k)!}

Case 1:

C(4,1) \times C(5,1) \times C(3,1) = 4 \times 5 \times 3 = 60

Case 2:

C(4,2) \times C(5,1) \times C(3,0) = 6 \times 5 \times 1 = 30

Case 3:

C(4,1) \times C(5,2) \times C(3,0) = 4 \times 10 \times 1 = 40

Order of selection does not matter when forming a committee, so combinations are used.

4

Add the counts from all mutually exclusive cases

Total valid subcommittees =

60 + 30 + 40 = 130

According to the addition rule of counting, the total number of outcomes across disjoint cases is the sum of their individual counts.

Key Concept

Counting combinations across multiple groups with specific distribution constraints

Question 325Question →

A municipal recycling center processes $p$ tons of plastic waste per hour. During a standard 14-hour operation cycle, the center processes a total of 119 tons of plastic waste. Which of the following equations represents this scenario, and what is the value of $p$ ?

14p = 119

;

p = 8.5

\frac{p}{14} = 119

;

p = 1,666

p + 14 = 119

;

p = 105

\frac{14}{p} = 119

;

p = 0.118

14p = 119

;

p = 8.25

Show answer & explanation

Answer: $14p = 119$ ; $p = 8.5$

Answer

The equation representing the situation is

14p = 119

, which solves to

p = 8.5

tons per hour.

The total amount of plastic waste processed is determined by multiplying the rate per hour (

p

) by the total number of hours (14), yielding the equation

14p = 119

. Dividing both sides of this equation by 14 isolates

p

, giving

p = 8.5

tons of plastic processed per hour.

Step-by-Step Solution

1

Set up the algebraic equation for total processing amount

14p = 119

Total plastic processed equals the hourly processing rate (

p

) multiplied by the total hours of operation (14).

2

Solve the one-step linear equation for

p

p = \frac{119}{14} = 8.5

Divide both sides of the equation by 14 to isolate the variable

p

.

Key Concept

Formulating and solving one-step linear equations from real-world rate scenarios

Estimated Time:1m 0s

Question 326Question →

A local fitness center recorded the number of weekly workout sessions attended by a sample of $20$ members over the past month. The results are summarized in the frequency table below:

Sessions per Week	Number of Members
$1$	$4$
$2$	$6$
$3$	$5$
$4$	$3$
$5$	$2$

What is the positive difference between the mean and the median number of weekly workout sessions for these $20$ members?

0.15

0.35

0.50

0.65

8.10

Show answer & explanation

Answer: $0.15$

Answer

The positive difference between the mean and median number of weekly workout sessions is

0.15

.

To find the mean, multiply each number of sessions by its frequency, sum the products, and divide by the total number of members (

20

):

(4 + 12 + 15 + 12 + 10) / 20 = 53 / 20 = 2.65

. To find the median for

20

members, take the average of the 10th and 11th values in order. Looking at cumulative frequencies, the 5th through 10th entries are

2

and the 11th through 15th entries are

3

. The median is

(2 + 3)/2 = 2.5

. The positive difference is

2.65 - 2.5 = 0.15

.

Step-by-Step Solution

1

Calculate the total number of sessions.

(1 \times 4) + (2 \times 6) + (3 \times 5) + (4 \times 3) + (5 \times 2) = 4 + 12 + 15 + 12 + 10 = 53

Multiply each session value by its corresponding frequency to find the total sum of sessions.

2

Calculate the mean number of sessions.

\text{Mean} = \frac{53}{20} = 2.65

Divide the total number of sessions by the total number of members (

20

).

3

Find the median number of sessions.

\text{Median} = \frac{2 + 3}{2} = 2.5

With

20

total data points, the median is the average of the 10th and 11th ordered values. The cumulative frequencies show that the first

4

values are

1

, the next

6

values (5th through 10th) are

2

, and the next

5

values (11th through 15th) are

3

. The 10th value is

2

and the 11th value is

3

.

4

Calculate the positive difference between the mean and median.

|2.65 - 2.5| = 0.15

Subtract the median from the mean to find the difference.

Key Concept

Calculating mean and median from a frequency table

Estimated Time:1m 30s

Question 327Question →

A wildlife conservation team tags and monitors sea turtle nests along a coastal beach. The team divides a total of $n$ nesting data loggers equally among $6$ field researchers. If each researcher receives $14$ data loggers, which of the following equations correctly models this situation to find $n$ ?

6n = 14

n - 6 = 14

\frac{n}{6} = 14

n + 6 = 14

\frac{14}{n} = 6

Show answer & explanation

Answer: $\frac{n}{6} = 14$

Answer

The correct equation is

\frac{n}{6} = 14

.

Dividing the total quantity

n

equally among

6

researchers is algebraically expressed as

\frac{n}{6}

. Setting this equal to the

14

loggers each researcher receives yields

\frac{n}{6} = 14

.

Step-by-Step Solution

1

Identify the total quantity and how it is divided.

The total number of loggers is

n

, and it is split equally among

6

researchers.

Dividing a total quantity into equal groups corresponds to the division operation

\frac{n}{6}

.

2

Equate the algebraic expression to the given result per group.

\frac{n}{6} = 14

Each researcher receives

14

loggers, so the share per researcher,

\frac{n}{6}

, must equal

14

.

Key Concept

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

Estimated Time:1m 0s

Question 328Question →

An art gallery features a collection of $60$ paintings, each categorized by its medium (oil or watercolor) and its historical period ( $19\text{th}$ -century or $20\text{th}$ -century). Among the collection, $35$ paintings are $19\text{th}$ -century works, $40$ paintings are oil paintings, and $10$ paintings are $20\text{th}$ -century watercolor works. If a single painting is selected at random from the collection, what is the probability that the selected painting is a $19\text{th}$ -century oil painting?

\frac{1}{6}

\frac{1}{4}

\frac{7}{18}

\frac{5}{12}

\frac{1}{2}

Show answer & explanation

Answer: $\frac{5}{12}$

Answer

\frac{5}{12}

To find the probability of choosing a

19\text{th}

-century oil painting, determine the number of paintings that fit both criteria. Out of

60

paintings,

40

are oil, meaning

20

are watercolors. Given that

10

watercolors are

20\text{th}

-century, the remaining

10

watercolors must be

19\text{th}

-century. Since there are

35

total

19\text{th}

-century paintings, subtracting the

10

19\text{th}

-century watercolors yields

25

19\text{th}

-century oil paintings. The probability is therefore

\frac{25}{60}

, which simplifies to

\frac{5}{12}

.

Step-by-Step Solution

1

Find the total number of watercolor paintings in the collection.

Since there are

60

total paintings and

40

oil paintings, the number of watercolor paintings is

60 - 40 = 20

.

Paintings are categorized into oil or watercolor, so the remaining paintings must be watercolors.

2

Find the number of

19\text{th}

-century watercolor paintings.

There are

20

total watercolor paintings and

10

are from the

20\text{th}

century, so the number of

19\text{th}

-century watercolors is

20 - 10 = 10

.

Subtracting the

20\text{th}

-century watercolors from the total watercolors gives the

19\text{th}

-century watercolors.

3

Calculate the number of

19\text{th}

-century oil paintings.

There are

35

total

19\text{th}

-century paintings. Subtracting the

10

19\text{th}

-century watercolors leaves

35 - 10 = 25

19\text{th}

-century oil paintings.

All

19\text{th}

-century paintings are either oil or watercolor.

4

Compute the probability of picking a

19\text{th}

-century oil painting.

The probability is

\frac{25}{60} = \frac{5}{12}

.

Divide the number of favorable outcomes (

25

) by the total number of paintings in the sample space (

60

).

Key Concept

Two-way categorization and conditional counting for compound probabilities

Estimated Time:2m 0s

Question 329Question →

A software development company allocated its annual operating budget across three departments: Engineering, Marketing, and Customer Support. Exactly $\frac{2}{5}$ of the total budget was allocated to Engineering, and $35\%$ of the total budget was allocated to Marketing. If the remaining budget of $\$45,000$ was allocated to Customer Support, what was the company's total annual operating budget?

\$60,000

\$112,500

\$125,000

\$180,000

\$225,000

Show answer & explanation

Answer: $\$180,000$

Answer

The total annual operating budget of the company was

\$180,000

.

To find the total budget, first convert

\frac{2}{5}

to a percentage, which is

40\%

. Adding Engineering's

40\%

and Marketing's

35\%

gives

75\%

of the total budget. This leaves

100\% - 75\% = 25\%

for Customer Support. Since Customer Support received

\$45,000

, setting

0.25 T = 45,000

yields

T = \frac{45,000}{0.25} = \$180,000

.

Step-by-Step Solution

1

Convert the fraction allocated to Engineering into a percentage.

\frac{2}{5} = \frac{40}{100} = 40\%

Converting all departmental allocations to percentages allows for direct comparison and summation.

2

Calculate the combined percentage allocated to Engineering and Marketing.

40\% + 35\% = 75\%

Adding the two known departmental shares determines the total fraction of the budget accounted for.

3

Find the percentage allocated to Customer Support.

100\% - 75\% = 25\%

The remaining portion of the

100\%

total budget belongs to Customer Support.

4

Set up an equation relating Customer Support's dollar allocation to the total budget

T

and solve for

T

.

0.25 T = 45,000 \implies T = \frac{45,000}{0.25} = 180,000

Dividing the dollar amount by its corresponding decimal percentage yields the original total budget.

Key Concept

Solving word problems involving mixed representations of fractions, decimals, and percentages to find an unknown initial total.

Estimated Time:1m 30s

Question 330Question →

In a wildlife sanctuary habitat, the ratio of finches to sparrows is $3 : 5$ , and the ratio of sparrows to thrushes is $2 : 3$ . If there are $45$ thrushes in the habitat, what is the total number of finches, sparrows, and thrushes?

48

93

102

125

165

Show answer & explanation

Answer: 93

Answer

The total number of birds in the habitat is 93.

To find the total number of birds, first combine the finch-to-sparrow ratio (

3 : 5

) and the sparrow-to-thrush ratio (

2 : 3

) into a single three-part ratio. Since sparrows are common to both, scale the ratios so that sparrows have the same number of parts:

3 : 5 = 6 : 10

and

2 : 3 = 10 : 15

. The combined ratio of finches to sparrows to thrushes is

6 : 10 : 15

. Since there are

45

thrushes corresponding to

15

parts,

1

part represents

45 \div 15 = 3

birds. The total number of parts is

6 + 10 + 15 = 31

, so the total number of birds is

31 \times 3 = 93

.

Step-by-Step Solution

1

Unify the two given ratios using a common term for sparrows.

The ratio of finches to sparrows is

3 : 5 = 6 : 10

. The ratio of sparrows to thrushes is

2 : 3 = 10 : 15

. Combined ratio of finches to sparrows to thrushes is

6 : 10 : 15

.

Since sparrows appear in both ratios with values

5

and

2

, finding the least common multiple (

10

) allows all three quantities to be expressed in a single triple ratio.

2

Determine the value of one ratio unit using the known quantity of thrushes.

Thrushes correspond to

15

ratio parts. Value per unit

= \frac{45}{15} = 3

birds.

Given that there are

45

thrushes and they represent

15

units in the combined ratio, dividing the total thrushes by

15

gives the multiplier for each unit.

3

Calculate the total number of birds across all three species.

Total ratio units

= 6 + 10 + 15 = 31

units. Total birds

= 31 \times 3 = 93

.

Multiplying the total number of ratio units by the value of a single unit yields the overall population of birds in the habitat.

Key Concept

Combining Compound Ratios and Scaling Proportions

Estimated Time:1m 30s

Question 331Question →

A customer satisfaction survey asked $25$ patrons at a local restaurant to rate their dining experience on a scale from $1$ (poor) to $5$ (excellent). The frequency table below summarizes the ratings collected:

Rating	Number of Patrons
1	3
2	5
3	8
4	6
5	3

What is the positive difference between the mean rating and the median rating of these $25$ patrons?

0.04

0.00

1.96

12.20

58.60

Show answer & explanation

Answer: $0.04$

Answer

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

0.04

.

To find the positive difference, first compute the mean by calculating the sum of products of each rating and its frequency:

(1 \times 3) + (2 \times 5) + (3 \times 8) + (4 \times 6) + (5 \times 3) = 76

. Dividing

76

by the total number of patrons (

25

) yields a mean of

3.04

. Next, find the median by identifying the

13\text{th}

ordered rating out of

25

. The cumulative frequency reaches

8

after rating 2 and

16

after rating 3, so the

13\text{th}

entry is

3

. The positive difference between the mean (

3.04

) and median (

3.00

) is

0.04

.

Step-by-Step Solution

1

Calculate the total sum of all ratings.

Sum

= (1 \times 3) + (2 \times 5) + (3 \times 8) + (4 \times 6) + (5 \times 3) = 3 + 10 + 24 + 24 + 15 = 76

.

Each rating value must be multiplied by its frequency to find the total value of all responses.

2

Calculate the mean rating.

Mean

= \frac{76}{25} = 3.04

.

Divide the total sum of all ratings by the total number of patrons (

25

).

3

Determine the median rating.

Median

= 3

.

With

25

data points arranged in ascending order, the median is the

\frac{25 + 1}{2} = 13\text{th}

score. Accumulating frequencies: Ratings 1 and 2 cover the first

3 + 5 = 8

scores, while Rating 3 covers the

9\text{th}

through

16\text{th}

scores. Thus, the

13\text{th}

score is

3

.

4

Find the positive difference between the mean and median.

Difference

= 3.04 - 3.00 = 0.04

.

Subtract the median rating (

3

) from the mean rating (

3.04

).

Key Concept

Calculating Mean and Median from a Frequency Table

Estimated Time:1m 30s

Question 332Question →

A solar-powered charging station operates using two types of solar panels, Type A and Type B. The ratio of Type A panels to Type B panels installed at the station is $5 : 3$ . Each Type A panel generates $24$ watt-hours of energy per hour, and each Type B panel generates $40$ watt-hours of energy per hour. If there are a total of $48$ solar panels installed at the station, how many total watt-hours of energy do all the panels generate combined in one hour?

240

1,536

1,440

1,632

1,920

Show answer & explanation

Answer: $1,440$

Answer

The total energy generated by all panels in one hour is

1,440

watt-hours.

The correct answer is

1,440

watt-hours. The ratio

5 : 3

indicates that out of every 8 panels (

5 + 3

), 5 are Type A and 3 are Type B. Converting these to fractions of the total

48

panels gives

48 \times \frac{5}{8} = 30

Type A panels and

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

Type B panels. Multiplying each quantity by its respective hourly output yields

30 \times 24 = 720

watt-hours for Type A panels and

18 \times 40 = 720

watt-hours for Type B panels. Combining these gives

720 + 720 = 1,440

watt-hours.

Step-by-Step Solution

1

Calculate the total number of ratio parts

The total ratio parts equal

5 + 3 = 8

.

To divide the total number of panels into proportional parts, add the terms of the ratio together.

2

Find the quantity of each type of panel installed

Type A panels =

48 \times \frac{5}{8} = 30

; Type B panels =

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

.

Multiply the total number of panels (

48

) by each panel type's fraction of the total ratio.

3

Calculate the energy generated by each panel type in one hour

Type A energy =

30 \times 24 = 720

watt-hours; Type B energy =

18 \times 40 = 720

watt-hours.

Multiply the count of each panel type by its specific hourly generation rate.

4

Sum the energy outputs to find the combined total

Total combined energy =

720 + 720 = 1,440

watt-hours.

Add the hourly energy outputs of Type A and Type B panels together.

Key Concept

Solving multi-step word problems involving part-to-whole ratios and rate calculations

Estimated Time:1m 30s

Question 333Question →

A container holds $40$ marbles, each of which is red, blue, or green. The probability of randomly drawing a red marble from the container is $\frac{2}{5}$ . Among the remaining marbles, the ratio of blue marbles to green marbles is $1:3$ . If one marble is selected at random from the container, what is the probability that it is green?

\frac{3}{20}

\frac{1}{5}

\frac{9}{20}

\frac{3}{5}

\frac{3}{4}

Show answer & explanation

Answer: $\frac{9}{20}$

Answer

The probability that the randomly selected marble is green is

\frac{9}{20}

.

The correct answer is

\frac{9}{20}

. The probability of drawing a non-red marble is

1 - \frac{2}{5} = \frac{3}{5}

. Given that the non-red marbles are split in a

1:3

ratio between blue and green, green marbles represent

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

of the non-red group. Multiplying the probability of drawing a non-red marble by the proportion of green marbles within that group gives

\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}

.

Step-by-Step Solution

1

Find the total number of red marbles and the number of remaining marbles.

Since the probability of drawing a red marble is

\frac{2}{5}

, the number of red marbles is

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

. The remaining number of marbles (blue and green) is

40 - 16 = 24

.

Determining the count of non-red marbles isolates the sample space for blue and green marbles.

2

Use the ratio of blue to green marbles to find the number of green marbles.

The ratio of blue to green is

1:3

, meaning there are

1 + 3 = 4

equal parts. Each part contains

\frac{24}{4} = 6

marbles. Therefore, there are

3 \times 6 = 18

green marbles.

Converting a part-to-part ratio into actual counts allows calculation of the favorable outcome.

3

Calculate the probability of randomly drawing a green marble.

The probability is the number of green marbles divided by the total number of marbles:

\frac{18}{40} = \frac{9}{20}

.

Basic probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.

Key Concept

Combining complement probability rules with part-to-whole ratio conversions to compute event probabilities.

Estimated Time:1m 30s

Question 334Question →

A solar power plant generated a total of $480\text{ megawatt-hours (MWh)}$ of energy in one month. Of this total, $\frac{2}{5}$ was stored in battery banks, $0.35$ was supplied directly to an industrial facility, and the remaining portion was delivered to residential homes. How many megawatt-hours of energy were delivered to residential homes?

168 MWh

192 MWh

120 MWh

216 MWh

360 MWh

Show answer & explanation

Answer: 120 MWh

Answer

120 MWh

Converting

\frac{2}{5}

to

0.40

allows us to sum the battery and industrial portions (

0.40 + 0.35 = 0.75

). The remaining residential portion is

1 - 0.75 = 0.25

, or

25\%

. Taking

25\%

of

480\text{ MWh}

gives

120\text{ MWh}

.

Step-by-Step Solution

1

Convert the fraction to a decimal

\frac{2}{5} = 0.40

Converting all given values to a common format (decimals) simplifies addition.

2

Sum the non-residential portions of energy

0.40 + 0.35 = 0.75

Combining the battery storage fraction and industrial supply fraction determines the total non-residential portion.

3

Find the remaining fraction for residential homes

1.00 - 0.75 = 0.25

Subtracting the combined non-residential portion from the whole (1.00) gives the residential fraction.

4

Calculate the megawatt-hours delivered to residential homes

0.25 \times 480 = 120\text{ MWh}

Multiplying the residential fraction by the total energy generated gives the target quantity.

Key Concept

Combining fractions and decimals to determine a remaining percentage of a whole quantity

Estimated Time:1m 0s

Question 335Question →

A community garden supervisor distributes a total of $d$ pounds of organic compost equally among $8$ raised garden beds. If each bed receives exactly $14.5$ pounds of compost, which of the following equations can be solved to find $d$ ?

\frac{d}{8} = 14.5

8d = 14.5

d - 8 = 14.5

d + 8 = 14.5

\frac{14.5}{d} = 8

Show answer & explanation

Answer: $\frac{d}{8} = 14.5$

Answer

The equation representing the relationship is

\frac{d}{8} = 14.5

.

The correct equation represents taking the total weight of compost,

d

, and dividing it equally into

8

parts to get

14.5

pounds per bed, which yields

\frac{d}{8} = 14.5

.

Step-by-Step Solution

1

Identify the total quantity and how it is divided.

Total compost =

d

pounds, divided equally among

8

beds.

Equal sharing corresponds to division of the total quantity by the number of shares.

2

Set up the algebraic expression for the amount per bed.

Amount per bed =

\frac{d}{8}

.

Dividing total pounds

d

by

8

gives the weight allocated to each individual bed.

3

Equate the expression to the given quantity per bed.

\frac{d}{8} = 14.5

.

The problem states that each bed receives

14.5

pounds of compost.

Key Concept

Translating equal division word problems into one-step algebraic equations

Estimated Time:1m 0s

Question 336Question →

A commercial diver descends at a constant rate of $2.5$ meters per second. If the diver's total change in depth after $t$ seconds is $-85$ meters, which of the following equations can be solved to find $t$ ?

-2.5t = -85

\frac{t}{-2.5} = -85

t - 2.5 = -85

-2.5 + t = -85

-85t = -2.5

Show answer & explanation

Answer: $-2.5t = -85$

Answer

-2.5t = -85

The rate of depth change is

-2.5

meters per second. Over

t

seconds, the total distance descended is given by the product of the unit rate and time,

-2.5t

. Equating this to the given total change in depth of

-85

meters gives

-2.5t = -85

.

Step-by-Step Solution

1

Identify the given rate, total quantity, and variable.

Rate of change =

-2.5

m/s, time =

t

seconds, total depth change =

-85

m.

Establishing knowns and unknown variables is the first step in setting up a one-step algebraic equation.

2

Formulate the relationship using the rate formula

(\text{Rate}) \times (\text{Time}) = \text{Total Change}

.

-2.5 \times t = -85

, which simplifies to

-2.5t = -85

.

Since the descent occurs at a constant rate per second, total change is the product of the rate per second and total seconds.

Key Concept

Formulating one-step linear equations from rate word problems

Question 337Question →

A solar power system generated a total of $450$ kilowatt-hours (kWh) of electricity during a 3-day period. On Day 1, the system generated $\frac{2}{9}$ of the total 3-day electricity. On Day 2, it generated $40\%$ of the remaining electricity after Day 1. How many kilowatt-hours of electricity were generated on Day 3?

Show answer & explanation

Answer: 210

Answer

210 kWh

To find the electricity generated on Day 3, first determine the Day 1 portion by multiplying

\frac{2}{9}

by

450

, which gives

100

kWh. The amount remaining for Days 2 and 3 is

450 - 100 = 350

kWh. Day 2 accounts for

40\%

of this remaining amount, which is

0.40 \times 350 = 140

kWh. Finally, subtract Day 2's portion from the remaining amount:

350 - 140 = 210

kWh.

Step-by-Step Solution

1

Calculate the amount of electricity generated on Day 1.

100

kWh

Multiply the fraction

\frac{2}{9}

by the total electricity (

450

kWh).

2

Determine the remaining electricity after Day 1.

350

kWh

Subtract Day 1's generation (

100

kWh) from the total amount (

450

kWh).

3

Calculate the amount of electricity generated on Day 2.

140

kWh

Convert

40\%

to a decimal (

0.40

) and multiply by the remaining

350

kWh.

4

Calculate the amount of electricity generated on Day 3.

210

kWh

Subtract Day 2's generation (

140

kWh) from the

350

kWh remaining after Day 1.

Key Concept

Multi-step word problems involving fractions, percentage calculations, and remaining quantities.

Estimated Time:1m 15s

Question 338Question →

A community library's summer program requires participants to create a 3-book reading list consisting of exactly $1$ biography, $1$ science fiction novel, and $1$ history book selected from a featured list. The featured list contains $5$ biographies (2 of which have over $400$ pages), $6$ science fiction novels (3 of which have over $400$ pages), and $4$ history books (1 of which has over $400$ pages). If a participant selects $1$ book of each genre at random from the featured list, what is the probability that at least $1$ of the selected books has over $400$ pages? Express your answer as a decimal.

Show answer & explanation

Answer: 0.775

Answer

0.775

To find the probability of selecting at least one book over

400

pages, it is most efficient to use the complement rule:

P(\text{at least one}) = 1 - P(\text{none})

. There are

5 \times 6 \times 4 = 120

total 3-book combinations. The number of books with

400

pages or fewer in each category are

3

biographies,

3

sci-fi novels, and

3

history books. Thus, there are

3 \times 3 \times 3 = 27

combinations with no books over

400

pages. The probability of choosing no books over

400

pages is

\frac{27}{120} = 0.225

. Subtracting this from

1

gives

1 - 0.225 = 0.775

.

Step-by-Step Solution

1

Calculate the total number of possible combinations of selecting 1 book from each genre

Total combinations =

5 \times 6 \times 4 = 120

By the Fundamental Counting Principle, multiplying the number of choices in each independent category yields the total outcomes.

2

Determine the complement event: selecting a reading list where NO book has over 400 pages

Available books of 400 pages or fewer: 3 biographies, 3 sci-fi novels, and 3 history books

Subtracting the number of books over 400 pages from the total in each category gives the count of books with 400 pages or fewer.

3

Calculate the number of combinations consisting entirely of books with 400 pages or fewer

Complement combinations =

3 \times 3 \times 3 = 27

Applying the Fundamental Counting Principle to the non-qualifying choices gives the total outcomes for the complement event.

4

Find the probability of the complement event and subtract from 1 to find the target probability

P(\text{at least one over } 400) = 1 - \frac{27}{120} = 1 - 0.225 = 0.775

The complement rule states that

P(A) = 1 - P(A')

, which is much more efficient than calculating probabilities for 1, 2, or 3 long books separately.

Key Concept

Complementary Probability and Fundamental Counting Principle

Estimated Time:1m 30s

Question 339Question →

A high school science club organized a daily recycling drive over a 10-day period. The frequency table below summarizes the number of aluminum cans collected per day:

Cans Collected Per Day	Number of Days
10	2
20	3
30	3
40	1
50	1

What is the positive difference between the mean number of cans collected per day and the median number of cans collected per day?

0

1

4

5

6

Show answer & explanation

Answer: 1

Answer

The positive difference between the mean and median is 1.

To find the mean, calculate the weighted sum of the products of each daily quantity and its frequency: (10 × 2 + 20 × 3 + 30 × 3 + 40 × 1 + 50 × 1) = 260. Dividing 260 by the total 10 days gives a mean of 26. To find the median, list the 10 data points in order: 10, 10, 20, 20, 20, 30, 30, 30, 40, 50. The middle two numbers are the 5th and 6th terms (20 and 30), so the median is (20 + 30) / 2 = 25. Subtracting 25 from 26 yields a positive difference of 1.

Step-by-Step Solution

1

Calculate the total number of cans collected and the total number of days.

Total days = 2 + 3 + 3 + 1 + 1 = 10. Total cans = (10 × 2) + (20 × 3) + (30 × 3) + (40 × 1) + (50 × 1) = 20 + 60 + 90 + 40 + 50 = 260.

To find the mean from a frequency table, multiply each data value by its frequency and sum the results, then divide by the total frequency.

2

Calculate the mean number of cans collected per day.

Mean = 260 / 10 = 26.

Divide the total sum of collected cans by the total number of days.

3

Find the median of the data set.

The 10 ordered data points are: 10, 10, 20, 20, 20, 30, 30, 30, 40, 50. The 5th value is 20 and the 6th value is 30. Median = (20 + 30) / 2 = 25.

For an even number of data points (

N = 10

), the median is the arithmetic mean of the two middle values (

N/2

th and

(N/2 + 1)

th terms).

4

Compute the positive difference between the mean and median.

Difference = 26 - 25 = 1.

Subtract the median from the mean to find the requested difference.

Key Concept

Weighted Mean and Median from Frequency Tables

Estimated Time:1m 15s

Question 340Question →

A community theater group allocated a total budget of $\$1,500$ to create costumes for an upcoming production. Exactly $\frac{1}{4}$ of the budget was spent on fabric, $0.30$ of the budget was spent on specialized footwear, and $\frac{1}{5}$ of the budget was spent on sewing accessories. The remaining amount of the budget was spent on theatrical wigs. What amount, in dollars, did the theater group spend on theatrical wigs?

\$300

\$375

\$450

\$975

\$1,125

Show answer & explanation

Answer: $\$375$

Answer

The theater group spent

\$375

on theatrical wigs.

The correct answer of

\$375

is found by determining each individual expense and subtracting their sum from the total budget. Fabric accounts for

\frac{1}{4}

of

\$1,500

, which is

\$375

. Footwear accounts for

0.30

of

\$1,500

, which is

\$450

. Sewing accessories account for

\frac{1}{5}

of

\$1,500

, which is

\$300

. Combined, these three categories equal

\$375 + \$450 + \$300 = \$1,125

. Subtracting

\$1,125

from the total budget of

\$1,500

leaves

\$375

for theatrical wigs.

Step-by-Step Solution

1

Calculate the dollar amount spent on fabric

\frac{1}{4} \times \$1,500 = \$375

Multiply the fraction representing fabric by the total budget.

2

Calculate the dollar amount spent on footwear

0.30 \times \$1,500 = \$450

Multiply the decimal representing footwear by the total budget.

3

Calculate the dollar amount spent on sewing accessories

\frac{1}{5} \times \$1,500 = \$300

Multiply the fraction representing accessories by the total budget.

4

Sum the three known category expenses

\$375 + \$450 + \$300 = \$1,125

Add the expenses together to determine the total spent before wigs.

5

Subtract the sum of known expenses from the total budget

\$1,500 - \$1,125 = \$375

The remaining portion of the budget is allocated to theatrical wigs.

Key Concept

Converting fractions and decimals to quantities of a total amount and calculating remaining portions.

Pre-Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept