All practice questions

Question 5401Question →

A container holds $6$ red tiles, $4$ blue tiles, and $5$ green tiles. If $2$ tiles are selected at random from the container one after another without replacement, what is the probability that both selected tiles are blue?

\frac{7}{29}

\frac{2}{35}

\frac{6}{55}

\frac{16}{225}

\frac{101}{210}

Show answer & explanation

Answer: $\frac{2}{35}$

Answer

The probability that both selected tiles are blue is

\frac{2}{35}

.

The correct answer is determined by calculating the probability of each draw sequentially. The probability of drawing a blue tile first is

\frac{4}{15}

. Because the tile is not replaced,

3

blue tiles and

14

total tiles remain for the second draw, giving a probability of

\frac{3}{14}

. Multiplying these probabilities yields

\frac{4}{15} \times \frac{3}{14} = \frac{12}{210}

, which simplifies to

\frac{2}{35}

.

Step-by-Step Solution

1

Find the total number of tiles in the container.

Total tiles =

6 + 4 + 5 = 15

.

The sample space for the first selection consists of all available tiles.

2

Calculate the probability that the first tile selected is blue.

Probability of first blue tile =

\frac{4}{15}

.

There are

4

blue tiles out of a total of

15

tiles.

3

Determine the remaining number of blue tiles and total tiles for the second selection.

Remaining blue tiles =

3

, remaining total tiles =

14

.

Since the first tile was selected without replacement, both the count of blue tiles and total tiles decrease by

1

.

4

Calculate the probability that the second tile selected is blue given the first was blue.

Probability of second blue tile =

\frac{3}{14}

.

There are now

3

blue tiles remaining out of

14

total remaining tiles.

5

Multiply the probabilities of the sequential dependent events.

\frac{4}{15} \times \frac{3}{14} = \frac{12}{210} = \frac{2}{35}

.

By the multiplication rule for probability,

P(\text{both blue}) = P(\text{1st blue}) \times P(\text{2nd blue | 1st blue})

.

Key Concept

Basic Probability of Dependent Sequential Events (Without Replacement)

Practice More

Try solving a similar question where 3 tiles are selected sequentially without replacement.

Alternative Method

Calculate using combinations: The total ways to choose any 2 tiles out of 15 is

\binom{15}{2} = \frac{15 \times 14}{2} = 105

. The total ways to choose 2 blue tiles out of 4 is

\binom{4}{2} = \frac{4 \times 3}{2} = 6

. The probability is

\frac{6}{105} = \frac{2}{35}

.

Estimated Time:1m 0s

Question 5402Question →

When analyzing low-frequency gravitational waves from merging supermassive black holes, astrophysicists processed complex, massive data sets, identified subtle timing anomalies, and published their findings in a peer-reviewed journal.

Which choice provides the correct punctuation for the underlined portion?

NO CHANGE

complex massive data sets, identified subtle timing anomalies and published

complex, massive data sets, identified subtle timing anomalies; and published

complex, massive data sets, they identified subtle timing anomalies, and published

Show answer & explanation

Answer: NO CHANGE

Answer

The original sentence correctly separates the coordinate adjectives with a comma and uses serial commas between items in a three-part list.

The original text accurately uses a comma between the coordinate adjectives 'complex' and 'massive' because both words independently modify the noun 'data sets'. Furthermore, commas correctly separate the three parallel verb phrases in the series ('processed complex, massive data sets', 'identified subtle timing anomalies', and 'published their findings').

Step-by-Step Solution

1

Evaluate the adjectives preceding 'data sets'.

'Complex' and 'massive' are coordinate adjectives because they can be joined by 'and' ('complex and massive data sets') or reordered ('massive, complex data sets') without altering the sentence's meaning.

Coordinate adjectives require a comma separating them.

2

Identify the list structure in the sentence.

The sentence contains three parallel action predicates: 'processed...', 'identified...', and 'published...'

Items in a series of three or more must be separated by commas, including a serial comma before the final coordinating conjunction.

3

Verify clause boundaries.

All three items share the single subject 'astrophysicists', so no semicolon or subject insertion is needed.

Semicolons are reserved for separating independent clauses or items containing internal punctuation.

Key Concept

Commas with Coordinate Adjectives and Parallel Lists

Question 5403Question →

A regional sports club assigns identification codes to all of its members. Each code consists of $1$ letter chosen from the set $\{K, L, M, N\}$ , followed by $2$ digits chosen from $\{1, 2, 3, 4, 5\}$ such that no digit is repeated within a code, followed by $1$ symbol chosen from $\{*, \#\}$ . How many unique identification codes can be created using this system?

Show answer & explanation

Answer: 160

Answer

The total number of unique identification codes that can be created is

160

.

According to the Fundamental Counting Principle, to find the total number of multi-stage outcomes, multiply the number of choices at each stage. For the letter slot, there are

4

choices. For the two digit slots without repetition, there are

5 \times 4 = 20

choices. For the symbol slot, there are

2

choices. Multiplying these gives

4 \times 20 \times 2 = 160

unique identification codes.

Step-by-Step Solution

1

Determine the number of available letter choices for the first slot.

4 options

The set of allowed letters

\{K, L, M, N\}

contains 4 distinct elements.

2

Calculate the number of permutations for the two-digit section without repetition.

20 options

Choosing 2 distinct digits from 5 options gives

5 \times 4 = 20

possible outcomes.

3

Determine the number of available symbol choices for the last slot.

2 options

The set of allowed symbols

\{*, \#\}

contains 2 elements.

4

Multiply the number of choices for each slot using the Fundamental Counting Principle.

160 unique codes

Total codes =

4 \times 20 \times 2 = 160

.

Key Concept

Fundamental Counting Principle and Permutations without Repetition

Estimated Time:1m 0s

Question 5404Question →

Experiment 1
Students assembled a continuous-flow photocatalytic reactor system to evaluate the degradation kinetics of an aqueous organic pollutant under ultraviolet ( $\text{UV}$ ) light. The system consists of four primary components: a peristaltic metering pump, a quartz reaction column coated internally with titanium dioxide ( $\text{TiO}_2$ ), a degassing bubbler module supplying compressed air, and an inline UV-Vis flow-through spectrophotometer.

Match each apparatus component used in Experiment 1 to its primary function within the experimental setup.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Quartz Reaction Column matches with transmitting incident UV radiation efficiently; Degassing Bubbler Module matches with supplying dissolved oxygen as an electron acceptor; Inline UV-Vis Flow-Through Spectrophotometer matches with providing real-time measurement of pollutant absorbance; Peristaltic Metering Pump matches with regulating volumetric flow rate to control residence time.

Each apparatus component is correctly paired with its operational function in the flow setup: quartz material ensures high UV light transmission to the catalyst, the bubbler provides oxygen required for electron capture, the inline spectrophotometer measures effluent absorbance continuously in real time, and the metering pump sets the volumetric flow rate to regulate light exposure duration.

Step-by-Step Solution

1

Analyze the optical requirements of the reaction vessel material.

Standard glass absorbs UV radiation, whereas quartz transmits UV wavelengths efficiently to excite the titanium dioxide catalyst.

Photocatalytic oxidation requires unattenuated UV photon flux reaching the active catalyst surface.

2

Determine the chemical role of the gas bubbling module.

Sparging air/gas dissolves oxygen into the liquid substrate prior to entering the reactor.

Dissolved oxygen traps conduction-band electrons generated on titanium dioxide, preventing charge recombination.

3

Identify the analytical measurement tool function.

The flow-through UV-Vis cell measures effluent absorbance continuously.

This non-invasive monitoring provides concentration data over time without withdrawing discrete samples or stopping flow.

4

Evaluate the hydrodynamic control component.

The peristaltic pump dictates the volumetric displacement rate through the column.

Flow rate determines the residence time of the pollutant inside the reaction zone under light exposure.

Key Concept

Analyzing apparatus component functions and experimental control in continuous-flow photocatalytic systems.

Estimated Time:1m 30s

Question 5405Question →

A chemistry lab technician measures the concentration of sugar by weight in four distinct liquid samples:

• Sample P: $\frac{7}{16}$
• Sample Q: $0.425$
• Sample R: $44\%$
• Sample S: $\frac{4}{9}$

Which of the following lists the four samples in order from least sugar concentration to greatest sugar concentration?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest concentration is Sample Q, Sample P, Sample R, and Sample S.

To arrange the values from least to greatest, convert all numbers to decimals: Sample Q is

0.425

; Sample P is

\frac{7}{16} = 0.4375

; Sample R is

44\% = 0.44

; and Sample S is

\frac{4}{9} = 0.4444\dots

. Comparing these decimals gives

0.4250 < 0.4375 < 0.4400 < 0.4444\dots

, which corresponds to the sequence Sample Q, Sample P, Sample R, Sample S.

Step-by-Step Solution

1

Convert each fraction, decimal, and percentage to a common decimal representation.

Sample P = 7/16 = 0.4375; Sample Q = 0.425; Sample R = 44% = 0.44; Sample S = 4/9 ≈ 0.4444...

Converting all numbers to decimals allows for direct place-value comparison.

2

Compare the converted decimal values from left to right.

0.4250 < 0.4375 < 0.4400 < 0.4444...

Comparing digits at the hundredths and thousandths places establishes the relative sizes.

3

Map the ordered decimal values back to their original sample names.

Sample Q < Sample P < Sample R < Sample S

Ordering matches the decimal values calculated: 0.425 < 0.4375 < 0.44 < 0.4444...

Key Concept

Converting mixed numerical forms (fractions, decimals, percentages) to a single decimal format for ordering.

Question 5406Question →

A community garden allocates space among three categories: vegetable beds, herb gardens, and flower borders. Vegetable beds take up $\frac{3}{8}$ of the total area, herb gardens take up $0.25$ of the total area, and flower borders occupy the remaining 1,800 square feet. What is the total area, in square feet, of the community garden?

2,700

2,880

4,800

675

7,200

Show answer & explanation

Answer: 4,800

Answer

4,800 square feet

To find the total area, first convert

0.25

to

\frac{2}{8}

. Summing the vegetable portion (

\frac{3}{8}

) and the herb portion (

\frac{2}{8}

) gives

\frac{5}{8}

of the garden. The flower section takes up the remaining

1 - \frac{5}{8} = \frac{3}{8}

of the area. Setting

\frac{3}{8}

of the total area equal to 1,800 square feet and dividing 1,800 by

\frac{3}{8}

gives

1,800 \times \frac{8}{3} = 4,800

square feet.

Step-by-Step Solution

1

Convert the decimal portion to a fraction with a common denominator.

0.25 = \frac{1}{4} = \frac{2}{8}

Converting all given portions into fractions with a common denominator allows direct addition.

2

Calculate the combined fraction for vegetable beds and herb gardens.

\frac{3}{8} + \frac{2}{8} = \frac{5}{8}

Adding the two known fractions identifies the combined portion of the garden accounted for by vegetables and herbs.

3

Find the fraction representing the flower borders.

1 - \frac{5}{8} = \frac{3}{8}

Subtracting the combined fraction from the whole (1) yields the fractional part representing the remaining 1,800 square feet.

4

Solve for the total garden area.

\text{Total Area} = 1,800 \div \frac{3}{8} = 1,800 \times \frac{8}{3} = 4,800\text{ sq ft}

Dividing the part by its corresponding fractional portion calculates the whole quantity.

Key Concept

Combining fractions and decimals to solve part-to-whole problems

Estimated Time:1m 30s

Question 5407Question →

Study 1
Students constructed a horizontal choice-chamber apparatus to investigate the light wavelength preferences of *Daphnia magna* (water fleas). The acrylic tube apparatus was divided into two equal volume sections—Chamber A (fitted with a 450 nm blue optical filter) and Chamber B (fitted with a 650 nm red optical filter)—separated by a central removable barrier. An overhead white light source provided uniform light intensity across both chambers.

For each trial, 40 *Daphnia* were placed into the central barrier compartment and kept in complete darkness for 5 minutes. The central barrier was then raised simultaneously with switching on the overhead light source. After 10 minutes, the barrier was lowered to trap the organisms in their respective chambers, and the number of *Daphnia* in each chamber was recorded.

What was the primary procedural purpose of keeping the *Daphnia magna* in complete darkness for 5 minutes prior to raising the central barrier and activating the light source?

To standardize the baseline physiological state of the organisms and eliminate bias from previous ambient light exposure

To calibrate the optical transmission wavelengths of the red and blue filters prior to the start of the trial

To act as the negative control setup that measures the mortality rate of Daphnia when permanently deprived of light

To modify the independent variable by systematically decreasing the luminous intensity of the overhead bulb

Show answer & explanation

Answer: To standardize the baseline physiological state of the organisms and eliminate bias from previous ambient light exposure

Answer

The primary purpose of the 5-minute dark incubation step is to standardize the baseline physiological state of the organisms and prevent prior light exposure from biasing their choice between the two test filters.

In behavioral choice assays, placing test organisms in a uniform, neutral environment (such as darkness) prior to introducing choice stimuli standardizes their baseline sensory state. This prevents lingering adaptation or stress from previous room lighting from skewing their movement toward one filter over the other.

Step-by-Step Solution

1

Analyze the experimental sequence described in Study 1

Identified that the 5-minute dark period occurs immediately after loading the organisms into the apparatus and before exposing them to the colored filters.

Understanding when a step occurs in a protocol clarifies whether it serves as a preparation/acclimation phase or an experimental measurement phase.

2

Evaluate the biological rationale for pre-test acclimation in behavioral choice experiments

Organisms previously exposed to ambient room light may have adapted photoreceptors or directional bias. Storing them in darkness ensures all subjects begin the test in an identical baseline physiological state.

Controlling pre-test environmental factors eliminates confounding variables that could influence the choice between Chamber A and Chamber B.

3

Distinguish the acclimation step from variable manipulations and apparatus calibration

The dark step does not alter light intensity during the test phase nor calibrate physical equipment.

Ensures that procedural steps designed for biological standardization are not confused with equipment setup or independent variable adjustments.

Key Concept

Acclimation and Baseline Standardization in Behavioral Experiments

Estimated Time:1m 30s

Question 5408Question →

When analyzing ocean sediment samples, marine geologists identified resilient, adaptive microfossils, cataloged mineral compositions, and recorded thermal fluctuations.

Which choice best maintains standard written English in the underlined portion?

NO CHANGE

resilient adaptive microfossils; cataloged mineral compositions, and recorded

resilient, adaptive microfossils cataloged mineral compositions and recorded

resilient, adaptive microfossils, they cataloged mineral compositions, and recorded

Show answer & explanation

Answer: NO CHANGE

Answer

The original phrasing 'resilient, adaptive microfossils, cataloged mineral compositions, and recorded' is correct.

The original text correctly places a comma between the coordinate adjectives 'resilient' and 'adaptive' and correctly uses serial commas to separate the three parallel verb phrases that complete the compound predicate.

Step-by-Step Solution

1

Evaluate coordinate adjectives

The adjectives 'resilient' and 'adaptive' both modify 'microfossils' independently (e.g., 'resilient and adaptive microfossils'), so a comma is required between them.

Coordinate adjectives modifying the same noun require a comma separator.

2

Identify items in a series

The sentence lists three parallel past-tense verb phrases: (1) identified resilient, adaptive microfossils, (2) cataloged mineral compositions, and (3) recorded thermal fluctuations.

A list of three or more items requires commas separating each item, including before the coordinating conjunction.

3

Select the option that applies both comma rules accurately

The unchanged phrasing contains the necessary comma between coordinate adjectives and all required serial commas.

It maintains proper grammatical structure without creating fragments or comma splices.

Key Concept

Commas with Coordinate Adjectives and Items in a Series

Question 5409Question →

A coffee roasting company creates a custom espresso blend using three types of coffee beans: Arabica, Robusta, and Liberica. In the blend, $\frac{3}{8}$ of the total weight is Arabica beans and $40\%$ of the total weight is Robusta beans. The remaining weight of the blend consists of Liberica beans. If a batch of this custom blend contains $18$ ounces of Liberica beans, what is the total weight, in ounces, of the coffee bean batch?

30

48

62

72

80

Show answer & explanation

Answer: 80

Answer

The total weight of the coffee bean batch is 80 ounces.

To find the total weight of the batch, first express each given portion as a decimal or percentage of the total weight. The Arabica beans make up

\frac{3}{8} = 0.375

or

37.5\%

of the total weight, and Robusta beans make up

40\%

of the total weight. Together, Arabica and Robusta represent

37.5\% + 40\% = 77.5\%

of the total weight. Consequently, Liberica beans represent the remaining

100\% - 77.5\% = 22.5\%

(or

0.225

) of the batch. Since

18

ounces equals

22.5\%

of the total weight

W

, we set up the equation

0.225 W = 18

and solve for

W

:

W = \frac{18}{0.225} = 80

ounces.

Step-by-Step Solution

1

Convert the fractional share of Arabica beans into a decimal or percentage.

\frac{3}{8} = 0.375 = 37.5\%

Converting all components to a common format (percentages or decimals) allows easy addition of the component shares.

2

Find the combined percentage share of Arabica and Robusta beans.

37.5\% + 40\% = 77.5\%

Adding the individual percentages determines the total fraction of the batch occupied by Arabica and Robusta.

3

Calculate the percentage share representing Liberica beans.

100\% - 77.5\% = 22.5\%

(or

0.225

in decimal form)

The remaining percentage of the batch must equal the Liberica portion.

4

Set up and solve the equation to find the total batch weight

W

.

0.225 \times W = 18 \implies W = \frac{18}{0.225} = 80

ounces

Dividing the given weight of Liberica beans by its fractional representation yields the total weight of the entire coffee blend.

Key Concept

Combining fractions and percentages to solve part-to-whole word problems

Estimated Time:1m 30s

Question 5410Question →

A catering service allows customers to build a custom buffet meal by selecting $1$ appetizer from $4$ available options, $1$ main course from $5$ available options, and $2$ distinct side dishes from $6$ available options. However, due to preparation constraints, $2$ specific side dishes cannot be selected together. How many different valid buffet meal combinations can a customer choose?

23

120

280

300

600

Show answer & explanation

Answer: $280$

Answer

The total number of valid buffet meal combinations is

280

.

There are

4

appetizer choices and

5

main course choices. Choosing

2

side dishes out of

6

options gives

\binom{6}{2} = 15

total pairs. Subtracting the

1

disallowed pair leaves

14

valid side dish pairs. Multiplying these independent choices gives

4 \times 5 \times 14 = 280

valid buffet menu combinations.

Step-by-Step Solution

1

Calculate the total number of ways to select 2 distinct side dishes from 6 options.

\binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15

combinations.

Since the order in which the 2 side dishes are picked does not matter, combinations are used.

2

Subtract the restricted pair of side dishes from the total side dish combinations.

15 - 1 = 14

valid side dish combinations.

The problem specifies that 2 specific side dishes cannot be selected together, eliminating 1 specific pair.

3

Apply the Fundamental Counting Principle to find the overall number of valid buffet menu choices.

4 \text{ (appetizers)} \times 5 \text{ (main courses)} \times 14 \text{ (valid side pairs)} = 280

.

The choices for appetizer, main course, and side dish pair are independent, so their individual number of possibilities are multiplied.

Key Concept

Fundamental Counting Principle and Combinations with Restrictions

Estimated Time:1m 15s

Question 5411Question →

In July, a company's IT department logged a set of service tickets. Of these tickets, $\frac{3}{10}$ were categorized as hardware issues, $45\%$ were categorized as software issues, and the remaining 60 tickets were categorized as network issues. What was the total number of service tickets logged by the IT department in July?

Show answer & explanation

Answer: 240

Answer

The total number of service tickets logged in July was 240.

Converting

\frac{3}{10}

gives

30\%

. Combining hardware (

30\%

) and software (

45\%

) issues yields

75\%

of the total tickets. The network tickets make up the remaining

25\%

(

100\% - 75\%

). Setting

25\%

of the total equal to 60 tickets gives a total of

\frac{60}{0.25} = 240

service tickets.

Step-by-Step Solution

1

Convert the fraction of hardware tickets to a percentage

\frac{3}{10} = 0.30 = 30\%

Expressing all portions as percentages enables direct combination.

2

Sum the percentages for hardware and software tickets

30\% + 45\% = 75\%

This determines the combined percentage of the non-network tickets.

3

Determine the remaining percentage corresponding to network tickets

100\% - 75\% = 25\%

The total of all categories must sum to 100%.

4

Calculate the total number of service tickets

\frac{60}{0.25} = 240

Dividing the quantity of network tickets by their decimal equivalent of 0.25 gives the total count.

Key Concept

Combining fractions and percentages to solve for an unknown whole amount

Estimated Time:1m 30s

Question 5412Question →

A fair spinner is divided into $8$ congruent sectors numbered $1$ through $8$ . A player spins the spinner twice in succession. How many of the $64$ possible outcomes result in a sum of the two spins that is strictly greater than $12$ ?

Show answer & explanation

Answer: 10

Answer

There are 10 outcomes that yield a sum strictly greater than 12.

Systematically listing all ordered pairs

(x, y)

from

\{1, 2, \dots, 8\} \times \{1, 2, \dots, 8\}

such that

x + y > 12

yields 4 outcomes for sum 13, 3 outcomes for sum 14, 2 outcomes for sum 15, and 1 outcome for sum 16, totaling 10 valid outcomes.

Step-by-Step Solution

1

Determine the acceptable sums for the two spins

The possible sums strictly greater than 12 are 13, 14, 15, and 16.

Since each spin has a maximum value of 8, the maximum possible sum is 8 + 8 = 16.

2

Count the ordered pairs (spin 1, spin 2) for each valid sum

4 outcomes for sum 13, 3 outcomes for sum 14, 2 outcomes for sum 15, and 1 outcome for sum 16.

First and second spins are ordered, so (5,8) and (8,5) represent distinct outcomes.

3

Sum the outcome counts across all valid cases

4 + 3 + 2 + 1 = 10 outcomes.

The sets of outcomes for distinct sums are mutually exclusive.

Key Concept

Basic Probability and Counting Sample Space Outcomes

Estimated Time:1m 15s

Question 5413Question →

The following sentence contains an underlined portion. Choose the option that best maintains standard written English conventions:

Although ambient noise in urban environments often disrupts wildlife communication, migratory songbirds produce louder, higher-pitched calls, and they alter their singing schedules during morning traffic hours.

Which choice produces the most grammatically correct sentence?

NO CHANGE

louder higher-pitched calls, and they alter

louder, higher-pitched calls and they alter

louder, higher-pitched calls; and they alter

Show answer & explanation

Answer: NO CHANGE

Answer

The original sentence ('NO CHANGE') is correct because it properly uses a comma between coordinate adjectives and places a comma before the coordinating conjunction joining two independent clauses.

The original phrasing correctly applies punctuation rules for both coordinate adjectives and clause boundary coordination. The adjectives 'louder' and 'higher-pitched' modify 'calls' independently, requiring a separating comma. Additionally, a comma is correctly placed before 'and' because it joins two independent clauses ('migratory songbirds produce...' and 'they alter...').

Step-by-Step Solution

1

Evaluate the adjectives 'louder' and 'higher-pitched'.

They are coordinate adjectives because their order can be reversed ('higher-pitched, louder calls') and 'and' can be inserted between them ('louder and higher-pitched calls'). Therefore, a comma must separate them.

Coordinate adjectives modifying the same noun require a comma.

2

Analyze the clause structure on either side of the conjunction 'and'.

'migratory songbirds produce louder, higher-pitched calls' is an independent clause, and 'they alter their singing schedules during morning traffic hours' is also an independent clause.

Two independent clauses joined by a coordinating conjunction (FANBOYS) require a comma before the conjunction.

Key Concept

Commas with Coordinate Adjectives and Independent Clauses

Estimated Time:45s

Question 5414Question →

A fitness center surveyed its members regarding their primary exercise routine. Of the members surveyed, $\frac{1}{4}$ chose strength training, $0.35$ chose cardio workouts, and $15\%$ chose group fitness classes. The remaining $75$ members chose swimming. What is the total number of members surveyed by the fitness center?

100

200

250

300

500

Show answer & explanation

Answer: 300

Answer

300

Converting

\frac{1}{4}

to

25\%

and

0.35

to

35\%

, the three specified groups represent

25\% + 35\% + 15\% = 75\%

of the total members. Consequently, the remaining

25\%

of members choose swimming. Since

25\%

(or

\frac{1}{4}

) of the total surveyed group equals 75 members, multiplying 75 by 4 gives a total of 300 members.

Step-by-Step Solution

1

Convert all given proportions to a common format (percentages or decimals)

Strength training =

\frac{1}{4} = 25\%

; Cardio workouts =

0.35 = 35\%

; Group fitness =

15\%

Converting all amounts to percentages allows for direct addition.

2

Calculate the total percentage accounted for by the first three categories

25\% + 35\% + 15\% = 75\%

Combining these parts determines what fraction of the total surveyed group is already accounted for.

3

Determine the percentage corresponding to the remaining members who chose swimming

100\% - 75\% = 25\%

The remaining members represent the remaining portion of the whole.

4

Set up an equation to find the total number of members (

T

)

0.25 T = 75 \implies T = \frac{75}{0.25} = 300

Dividing the part by its corresponding decimal percentage yields the total population.

Key Concept

Converting between fractions, decimals, and percentages to calculate an unknown total quantity

Estimated Time:1m 15s

Question 5415Question →

Biophysicists conducted an experiment measuring the action potential conduction velocity ( $v$ , in $\text{m/s}$ ) in unmyelinated giant nerve fibers as a function of fiber diameter ( $d$ , in $\mu\text{m}$ ) at a constant temperature of $20^\circ\text{C}$ . The recorded data is presented in Table 1.

Fiber Diameter ( $d$ , $\mu\text{m}$ )	Conduction Velocity ( $v$ , $\text{m/s}$ )
100	5.0
200	8.0
400	14.0
600	18.0
900	21.0

Based on Table 1, what is the average rate of change in conduction velocity, in $\text{m/s}$ per $100\,\mu\text{m}$ increase in fiber diameter, over the interval from $d = 200\,\mu\text{m}$ to $d = 600\,\mu\text{m}$ ?

Show answer & explanation

Answer: 2.5

Answer

The average rate of change in conduction velocity over the specified interval is

2.5\,\text{m/s per }100\,\mu\text{m}

increase in fiber diameter.

To find the average rate of change in conduction velocity per

100\,\mu\text{m}

increase in fiber diameter between

d = 200\,\mu\text{m}

and

d = 600\,\mu\text{m}

, subtract the initial velocity (

8.0\,\text{m/s}

) from the final velocity (

18.0\,\text{m/s}

) to get

\Delta v = 10.0\,\text{m/s}

. Divide by the change in diameter

\Delta d = 600 - 200 = 400\,\mu\text{m}

to obtain

0.025\,\text{m/s per }\mu\text{m}

. Multiplying by

100

yields

2.5\,\text{m/s per }100\,\mu\text{m}

.

Step-by-Step Solution

1

Locate data points for

d = 200\,\mu\text{m}

and

d = 600\,\mu\text{m}

in Table 1.

At

d = 200\,\mu\text{m}

,

v = 8.0\,\text{m/s}

. At

d = 600\,\mu\text{m}

,

v = 18.0\,\text{m/s}

.

These data points define the boundaries of the interval specified in the question.

2

Calculate the overall changes in velocity (

\Delta v

) and diameter (

\Delta d

).

\Delta v = 18.0 - 8.0 = 10.0\,\text{m/s}

and

\Delta d = 600 - 200 = 400\,\mu\text{m}

.

Calculating the rate of change requires dividing the change in the dependent variable by the change in the independent variable.

3

Scale the rate of change to a

100\,\mu\text{m}

diameter interval.

\frac{10.0\,\text{m/s}}{400\,\mu\text{m}} \times 100\,\mu\text{m} = 2.5\,\text{m/s per }100\,\mu\text{m}

.

The question asks specifically for the rate per

100\,\mu\text{m}

increase in diameter.

Key Concept

Calculating average rate of change and trend slopes from quantitative scientific data tables.

Question 5416Question →

An algebraic function

F(a, b)

is defined as:

F(a, b) = \frac{a^2 - 3b}{a b^{-1} + 2}

The values for variables $a$ and $b$ are given in the table below:

Variable	Value
$a$	$-4$
$b$	$-\frac{1}{2}$

What is the value of $F(a, b)$ for these given values?

-\frac{29}{20}

\frac{29}{20}

\frac{7}{4}

\frac{35}{8}

-\frac{35}{12}

Show answer & explanation

Answer: $\frac{7}{4}$

Answer

The value of

F(a, b)

is

\frac{7}{4}

.

Substituting

a = -4

and

b = -1/2

into the expression gives a numerator of

(-4)^2 - 3(-1/2) = 16 + 3/2 = 35/2

and a denominator of

(-4)(-2) + 2 = 8 + 2 = 10

. Dividing

35/2

by

10

simplifies to

35/20

, which equals

7/4

.

Step-by-Step Solution

1

Evaluate the reciprocal term

b^{-1}

b^{-1} = \left(-\frac{1}{2}\right)^{-1} = -2

A negative exponent indicates taking the reciprocal of the base.

2

Evaluate the numerator

a^2 - 3b

(-4)^2 - 3\left(-\frac{1}{2}\right) = 16 + \frac{3}{2} = \frac{35}{2}

Squaring a negative number yields a positive value (

(-4)^2 = 16

), and subtracting a negative value is equivalent to addition.

3

Evaluate the denominator

a b^{-1} + 2

(-4)(-2) + 2 = 8 + 2 = 10

Multiplying two negative numbers yields a positive product.

4

Divide the numerator by the denominator and simplify

\frac{\frac{35}{2}}{10} = \frac{35}{20} = \frac{7}{4}

Dividing

\frac{35}{2}

by

10

gives

\frac{35}{20}

, which simplifies to

\frac{7}{4}

when dividing both numerator and denominator by

5

.

Key Concept

Evaluating algebraic expressions with negative numbers, fractions, and negative exponents.

Estimated Time:1m 30s

Question 5417Question →

A school library receives a shipment of $12$ new books, consisting of $5$ science fiction novels, $4$ historical fiction novels, and $3$ biography books. A librarian wants to choose $3$ books to feature on a display shelf. How many different combinations of $3$ books can be formed that contain exactly $2$ science fiction novels and $1$ biography book?

Show answer & explanation

Answer: 30

Answer

The total number of combinations containing exactly 2 science fiction novels and 1 biography book is 30.

To find the total number of combinations, first determine the number of ways to choose

2

science fiction novels from

5

available:

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

. Next, determine the number of ways to choose

1

biography book from

3

available:

\binom{3}{1} = 3

. Finally, apply the Fundamental Counting Principle by multiplying the counts together:

10 \times 3 = 30

.

Step-by-Step Solution

1

Calculate the combinations for selecting the science fiction novels.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

Select

2

novels out of the

5

available science fiction novels without regard to order.

2

Calculate the combinations for selecting the biography book.

\binom{3}{1} = 3

Select

1

book out of the

3

available biography books.

3

Apply the Fundamental Counting Principle to find total combinations.

10 \times 3 = 30

Multiply the combinations calculated for each independent selection.

Key Concept

Combinations and the Fundamental Counting Principle

Question 5418Question →

A project team of $3$ software developers is to be randomly selected from a pool of $4$ frontend developers and $6$ backend developers. What is the probability that the selected team will consist of exactly $1$ frontend developer and $2$ backend developers?

\frac{1}{6}

\frac{3}{10}

\frac{19}{120}

\frac{1}{2}

\frac{3}{25}

Show answer & explanation

Answer: $\frac{1}{2}$

Answer

The probability that the team consists of exactly 1 frontend developer and 2 backend developers is

\frac{1}{2}

.

To find the probability, divide the number of ways to select 1 frontend developer and 2 backend developers by the total number of ways to choose any 3 developers from 10. There are

\binom{4}{1} = 4

ways to select 1 frontend developer and

\binom{6}{2} = 15

ways to select 2 backend developers. Multiplying these yields

4 \times 15 = 60

favorable team combinations. The total number of ways to choose 3 developers from 10 is

\binom{10}{3} = 120

. Thus, the probability is

\frac{60}{120} = \frac{1}{2}

.

Step-by-Step Solution

1

Calculate the total number of possible 3-developer teams from the pool of 10 developers.

\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120

total outcomes.

The total number of ways to choose 3 people out of 10 without regard to order is given by combinations.

2

Calculate the number of ways to select 1 frontend developer from 4 and 2 backend developers from 6.

\binom{4}{1} \times \binom{6}{2} = 4 \times \frac{6 \times 5}{2 \times 1} = 4 \times 15 = 60

favorable outcomes.

By the Fundamental Counting Principle, multiply the number of ways to choose the frontend developer by the number of ways to choose the backend developers.

3

Divide the number of favorable outcomes by the total possible outcomes to find the probability.

\text{Probability} = \frac{60}{120} = \frac{1}{2}

.

Probability is defined as the ratio of favorable outcomes to total outcomes in a uniform sample space.

Key Concept

Probability of Compound Events using Combinations

Question 5419Question →

A bakery allows customers to customize a fruit tart by selecting $1$ crust type from $3$ available choices, $1$ filling type from $4$ available choices, and $2$ different fruit toppings from $5$ available choices. How many different fruit tart combinations are possible?

17

60

120

240

300

Show answer & explanation

Answer: $120$

Answer

The total number of different fruit tart combinations possible is 120.

To find the total number of unique fruit tart combinations, apply the Fundamental Counting Principle across all decision stages. Selecting 1 crust out of 3 yields 3 options. Selecting 1 filling out of 4 yields 4 options. Selecting 2 different fruit toppings out of 5 requires choosing combinations without regard to order, which is calculated as 5 choose 2 = 10. Multiplying the options from each stage gives 3 * 4 * 10 = 120 total combinations.

Step-by-Step Solution

1

Determine the number of ways to choose the crust and filling.

There are

3

crust choices and

4

filling choices.

Each selection stage is independent.

2

Calculate the number of ways to choose

2

different fruit toppings from

5

available options.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

choices.

The order in which the two toppings are selected does not matter, so combinations must be used.

3

Apply the Fundamental Counting Principle to find the total combinations.

3 \times 4 \times 10 = 120

.

Multiply the number of choices for each independent step.

Key Concept

Fundamental Counting Principle and Combinations

Estimated Time:1m 0s

Question 5420Question →

An art gallery manager is arranging 6 unique paintings side by side in a single line on a display wall. Two specific paintings created by the same artist must be placed directly next to each other. How many different line arrangements of the 6 paintings are possible?

Show answer & explanation

Answer: 240

Answer

The total number of different line arrangements is 240.

To satisfy the constraint that two specific paintings must be adjacent, treat them as a single combined block. This reduces the problem to arranging 5 items (the single block plus the other 4 paintings), which can be ordered in 5! = 120 ways. Within the block, the two paintings can be ordered in 2! = 2 ways. Applying the Fundamental Counting Principle gives 120 * 2 = 240 distinct arrangements.

Step-by-Step Solution

1

Group the two specific paintings together as a single block.

There are 5 units to arrange (1 block of 2 paintings plus 4 individual paintings).

Since the two paintings must remain adjacent, treating them as a single item guarantees they will stay together.

2

Calculate the permutations of the 5 units.

5! = 5 * 4 * 3 * 2 * 1 = 120 ways.

There are 5 distinct units being arranged in a straight line.

3

Determine the internal arrangements of the grouped pair.

2! = 2 * 1 = 2 ways.

The two specific paintings within the block can switch positions (Painting A then B, or Painting B then A).

4

Multiply the arrangements together.

120 * 2 = 240 total arrangements.

By the Fundamental Counting Principle, total arrangements equal the number of ways to place the units multiplied by the internal arrangements of the restricted pair.

Key Concept

Permutations with Restrictions (Grouped Elements)

Estimated Time:1m 15s

Answer

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Key Concept

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