All practice questions

Question 361Question →

A technology startup needs to form a 4-member project team selected from a pool of 5 front-end developers and 5 back-end developers. However, the selection is subject to one condition: if a specific front-end developer, Elena, is selected for the team, then a specific back-end developer, Marcus, must also be selected for the team. How many different 4-member teams can be formed under this condition?

Show answer & explanation

Answer: 154

Answer

154 different 4-member teams can be formed under the given condition.

The total number of ways to select a 4-member team from 10 developers without restrictions is

\binom{10}{4} = 210

. The conditional restriction requires that whenever Elena is on the team, Marcus must also be on the team. The only selection that violates this rule is one where Elena is included but Marcus is excluded. For this invalid scenario, Elena takes 1 spot and Marcus is removed from consideration, leaving

\binom{8}{3} = 56

ways to pick the remaining 3 members from the remaining 8 developers. Subtracting these 56 invalid teams from the 210 total teams gives 154 valid teams.

Step-by-Step Solution

1

Calculate the total possible 4-member teams from the 10 developers without any restrictions.

Total combinations =

\binom{10}{4} = 210

.

Choosing any 4 members out of a total pool of 10 developers (5 front-end and 5 back-end).

2

Identify and calculate the number of invalid team combinations.

Invalid combinations =

\binom{8}{3} = 56

.

The conditional rule is violated only if Elena is selected while Marcus is excluded. Selecting Elena fills 1 spot and excluding Marcus leaves 8 developers from which to choose the remaining 3 team members.

3

Subtract the invalid combinations from the total unconstrained combinations.

210 - 56 = 154

.

Using complementary counting yields the number of valid team selections directly and efficiently.

Key Concept

Combinations with Conditional Restrictions (Complementary Counting)

Estimated Time:1m 30s

Question 362Question →

A regional healthcare provider operates two outpatient facilities, Facility X and Facility Y. Facility X manages $60$ patient appointments per day with an average consultation time of $15$ minutes per patient. Facility Y manages $40$ patient appointments per day with an average consultation time of $25$ minutes per patient. What is the overall average consultation time, in minutes per patient, for all $100$ appointments combined across both facilities?

Show answer & explanation

Answer: 19

Answer

The overall average consultation time across both facilities is

19

minutes per patient.

The total consultation time across both facilities is

60 \times 15 + 40 \times 25 = 900 + 1000 = 1900

minutes. Dividing this by the total number of appointments (

60 + 40 = 100

) gives a weighted average of

\frac{1900}{100} = 19

minutes.

Step-by-Step Solution

1

Calculate the total consultation minutes contributed by Facility X.

60 \times 15 = 900

minutes

Total time for a set is the number of items multiplied by the mean of that set.

2

Calculate the total consultation minutes contributed by Facility Y.

40 \times 25 = 1000

minutes

Total time for the second set is the number of items multiplied by its mean.

3

Find the grand total consultation time and total number of patients.

Total minutes =

900 + 1000 = 1900

; Total patients =

60 + 40 = 100

Combined sets require aggregating both total values and total counts.

4

Compute the combined weighted average.

\frac{1900}{100} = 19

minutes

The weighted average is total accumulated value divided by total count.

Key Concept

Weighted Average of Combined Sets

Estimated Time:1m 30s

Question 363Question →

A cybersecurity firm must select a 4-tool diagnostic suite from a pool of 6 network scanners and 4 code analyzers. If the suite must contain at least 1 network scanner and at least 1 code analyzer, how many different 4-tool suites can be selected?

Show answer & explanation

Answer: 194

Answer

The total number of different 4-tool suites that can be selected is 194.

The total number of ways to select any 4 tools from the pool of 10 is

\binom{10}{4} = 210

. The only selections that violate the requirement of having at least one tool of each type are choosing 4 scanners and 0 analyzers (

\binom{6}{4} = 15

ways) or 0 scanners and 4 analyzers (

\binom{4}{4} = 1

way). Subtracting these 16 invalid outcomes from the total gives

210 - 16 = 194

valid 4-tool suites.

Step-by-Step Solution

1

Calculate the total unconstrained selections

210

There are

6 + 4 = 10

total tools, and 4 must be chosen.

\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210

.

2

Determine the number of invalid combinations violating the constraint

16 invalid combinations

The constraint requires at least 1 scanner and 1 analyzer. Invalid suites consist entirely of scanners (

\binom{6}{4} = 15

) or entirely of analyzers (

\binom{4}{4} = 1

).

3

Apply complementary counting to find the valid combinations

194 valid combinations

Subtract the invalid combinations from total unconstrained combinations:

210 - (15 + 1) = 194

.

Key Concept

Combinations with Group Selections and Complementary Counting

Question 364Question →

A citrus orchard harvested two varieties of oranges: Valencia and Navel. The harvest consisted of $800$ crates of Valencia oranges with an average weight of $45$ pounds per crate, and $1,200$ crates of Navel oranges with an average weight of $50$ pounds per crate. What is the average weight, in pounds per crate, for the combined harvest of all crates?

Show answer & explanation

Answer: 48

Answer

The average weight of the combined harvest is 48 pounds per crate.

The weighted average is found by calculating the total overall weight divided by the total number of crates. Since Valencia oranges account for 36,000 pounds (800 crates at 45 lbs/crate) and Navel oranges account for 60,000 pounds (1,200 crates at 50 lbs/crate), the total weight is 96,000 pounds across 2,000 crates. Dividing 96,000 by 2,000 gives exactly 48 pounds per crate.

Step-by-Step Solution

1

Calculate the total weight contributed by each variety of oranges.

Valencia weight =

800 \times 45 = 36,000

lbs; Navel weight =

1,200 \times 50 = 60,000

lbs.

Total weight equals the product of the number of units and the average weight per unit for each group.

2

Determine the total combined weight and the total number of crates.

Total weight =

36,000 + 60,000 = 96,000

lbs; Total crates =

800 + 1,200 = 2,000

crates.

Combined weighted averages require summing total values across all subsets.

3

Divide total weight by total number of crates to find the overall weighted average.

Weighted average =

\frac{96,000}{2,000} = 48

lbs per crate.

Weighted mean formula:

\bar{x} = \frac{\sum (n_i \cdot x_i)}{\sum n_i}

.

Key Concept

Weighted Average of Combined Sets

Estimated Time:1m 30s

Question 365Question →

A boutique dairy farm produced $4,000$ liters of organic milk in 2021. The annual milk production increased by $15\%$ from 2021 to 2022, decreased by $10\%$ from 2022 to 2023, and then increased by $20\%$ from 2023 to 2024. How many liters of organic milk did the farm produce in 2024?

Show answer & explanation

Answer: 4968

Answer

The farm produced 4,968 liters of organic milk in 2024.

To calculate the final quantity after sequential percent changes, multiply the starting amount by the multiplier for each period. The net expression is

4,000 \times 1.15 \times 0.90 \times 1.20

. Stepwise,

4,000 \times 1.15 = 4,600

, then

4,600 \times 0.90 = 4,140

, and finally

4,140 \times 1.20 = 4,968

liters.

Step-by-Step Solution

1

Calculate milk production in 2022 after a 15% increase.

4,000 \times 1.15 = 4,600

liters

An increase of 15% corresponds to multiplying the initial base value by

1 + 0.15 = 1.15

.

2

Calculate milk production in 2023 after a 10% decrease.

4,600 \times 0.90 = 4,140

liters

A decrease of 10% corresponds to multiplying the new base value of 4,600 by

1 - 0.10 = 0.90

.

3

Calculate milk production in 2024 after a 20% increase.

4,140 \times 1.20 = 4,968

liters

An increase of 20% corresponds to multiplying the current base value of 4,140 by

1 + 0.20 = 1.20

.

Key Concept

Successive Percent Change

Question 366Question →

A beverage manufacturer creates a functional tea blend by combining Extract $A$ , which contains $12\%$ polyphenols by volume, and Extract $B$ , which contains $28\%$ polyphenols by volume. If the manufacturer needs to produce a $200$ -liter batch of tea with a total polyphenol concentration of $18\%$ by volume, how many liters of Extract $A$ must be included in the mixture?

Show answer & explanation

Answer: 125

Answer

The manufacturer must include

125

liters of Extract

A

in the mixture.

By setting up the linear system

x + y = 200

and

0.12x + 0.28y = 36

, substituting

y = 200 - x

leads directly to

-0.16x = -20

, giving

x = 125

liters.

Step-by-Step Solution

1

Formulate a system of two linear equations representing the total volume and polyphenol volume.

x + y = 200

and

0.12x + 0.28y = 36

The sum of the individual extract volumes equals the total batch volume, and the sum of the pure polyphenols from each extract equals the total pure polyphenols in the final mixture.

2

Substitute

y = 200 - x

into the polyphenol equation to solve for

x

.

0.12x + 0.28(200 - x) = 36 \implies 0.12x + 56 - 0.28x = 36

Substituting one variable reduces the system to a single linear equation in terms of

x

.

3

Combine like terms and solve for

x

.

-0.16x = -20 \implies x = \frac{-20}{-0.16} = 125

Isolating

x

yields the required volume of Extract

A

.

Key Concept

Systems of Linear Equations in Applied Mixture Problems

Question 367Question →

The table below displays the vehicle fleet composition across four regional divisions of a logistics corporation.

Division	Electric Vehicles	Gasoline Vehicles	Total Vehicles
North	120	280	400
South	160	240	400
East	90	210	300
West	230	270	500

What percentage of the total fleet across all four divisions consists of electric vehicles?

Show answer & explanation

Answer: 37.5

Answer

The percentage of the total fleet consisting of electric vehicles across all four divisions is

37.5\%

.

To find the overall percentage of electric vehicles, sum the number of electric vehicles across all divisions (

120 + 160 + 90 + 230 = 600

) and divide by the overall total number of vehicles (

400 + 400 + 300 + 500 = 1600

). Dividing

600

by

1600

gives

0.375

, which equals

37.5\%

.

Step-by-Step Solution

1

Calculate the total number of electric vehicles across all divisions.

120 + 160 + 90 + 230 = 600

electric vehicles.

Summing the electric vehicle column values gives the total count of electric vehicles.

2

Calculate the total vehicle count across all divisions.

400 + 400 + 300 + 500 = 1600

total vehicles.

Summing the total vehicle column values provides the aggregate denominator.

3

Compute the required percentage.

\frac{600}{1600} \times 100\% = 37.5\%

.

Dividing total electric vehicles by total vehicles yields the aggregate proportion.

Key Concept

Ratio, Proportion, and Percent Calculations

Estimated Time:1m 0s

Question 368Question →

The table below provides operational data for eight solar power facilities across two regions:

Facility Name	Region	Capacity (MW)	Operational Efficiency (%)
Solaria Alpha	North	120	88
Solaria Beta	South	150	91
Solaria Gamma	North	200	97
Solaria Delta	North	110	82
Solaria Epsilon	South	180	89
Solaria Zeta	North	135	92
Solaria Eta	North	160	94
Solaria Theta	North	140	85

What is the median operational efficiency, in percent, of the solar power facilities located in the North region?

Show answer & explanation

Answer: 90

Answer

The median operational efficiency of the solar power facilities in the North region is 90%.

Filtering the table for facilities in the North region yields six efficiency values: 82%, 85%, 88%, 92%, 94%, and 97%. Because the dataset contains an even count of six values, the median is the average of the two middle values (88% and 92%). Calculating

\frac{88 + 92}{2}

gives 90%.

Step-by-Step Solution

1

Filter the table to isolate facilities in the North region.

Identified 6 facilities with operational efficiency values: 88%, 97%, 82%, 92%, 94%, and 85%.

The stem specifies evaluating descriptive statistics for the North region subset only.

2

Sort the efficiency values of the filtered subset in ascending order.

Ordered set: {82, 85, 88, 92, 94, 97}.

Calculating the median requires ordering data points from lowest to highest.

3

Compute the arithmetic mean of the two central terms in the 6-element dataset.

\frac{88 + 92}{2} = 90

.

For an even-count set of

N = 6

items, the median is defined as the average of the

\frac{N}{2}

-th (3rd) and

(\frac{N}{2} + 1)

-th (4th) terms.

Key Concept

Descriptive Statistics Interpretation (Median of Even-Count Subsets)

Question 369Question →

The table below presents operational metrics and financial allocations for six municipal water treatment facilities in 2025.

Facility	Total Water Processed (Million Gallons)	Recycled Water (Million Gallons)	Distribution Loss Rate (%)	Annual Operating Budget ($ Millions)
Alton	450	90	8%	18.0
Belton	600	150	12%	21.0
Colton	350	70	6%	14.0
Dalton	700	210	15%	22.4
Easton	500	130	10%	17.5
Fulton	400	80	5%	16.0

For the facility with the lowest operating budget per million gallons of total water processed, what percentage of its total processed water was recycled?

Show answer & explanation

Answer: 30

Answer

30%

Evaluating the unit budget per million gallons processed across all facilities reveals that Dalton has the lowest operating rate at

22.4\text{M} / 700\text{ MG} = \$0.0320\text{M / MG}

. Computing the percentage of total processed water that was recycled for Dalton gives

(210 / 700) \times 100\% = 30\%

.

Step-by-Step Solution

1

Calculate operating budget per million gallons processed for each facility to determine unit cost.

Alton = 0.0400; Belton = 0.0350; Colton = 0.0400; Dalton = 0.0320; Easton = 0.0350; Fulton = 0.0400 ($M per MG).

To identify which facility operates with the lowest financial cost per unit of water processed.

2

Identify the minimum unit cost value among all six facilities.

Dalton has the lowest operating budget per million gallons ($0.0320M per MG).

Dalton satisfies the condition of having the lowest operating budget per million gallons processed.

3

Divide Dalton's recycled water volume by its total processed water volume and express as a percentage.

(210 / 700) * 100% = 30%.

To find the specific proportion of recycled water relative to total processed water at the identified facility.

Key Concept

Unit Rate Comparison and Percentage Calculation from Tabular Data

Question 370Question →

A chemist mixes $x$ liters of a $20\%$ acid solution with $y$ liters of a $50\%$ acid solution to produce $30$ liters of a $40\%$ acid solution. How many liters of the $20\%$ acid solution were used in the mixture?

Show answer & explanation

Answer: 10

Answer

10 liters of the 20% acid solution were used.

Solving the system of simultaneous linear equations

x + y = 30

and

0.20x + 0.50y = 12

yields

x = 10

liters.

Step-by-Step Solution

1

Formulate the linear system of equations representing total volume and total pure acid content.

x + y = 30

and

0.20x + 0.50y = 12

.

The sum of component volumes equals total volume, and the sum of pure acid contents equals total pure acid.

2

Substitute

y = 30 - x

into the pure acid equation.

0.20x + 0.50(30 - x) = 12

Reduces the two-variable system to a single linear equation in terms of

x

.

3

Expand and solve for

x

.

15 - 0.30x = 12 \implies -0.30x = -3 \implies x = 10

Isolates the target variable

x

representing the volume of the 20% acid solution.

Key Concept

Simultaneous Linear Equations in Mixture Problems

Estimated Time:1m 0s

Question 371Question →

The table lists performance and cost metrics for 10 municipal water treatment facilities during the 2025 fiscal year:

Facility ID	Region	Daily Capacity (MGD)	Average Turbidity (NTU)	Operating Cost ($/thousand gallons)
W-01	North	24.0	0.12	1.42
W-02	South	18.0	0.25	1.85
W-03	North	32.0	0.08	1.28
W-04	Central	12.0	0.19	1.65
W-05	South	28.0	0.15	1.56
W-06	Central	40.0	0.10	1.18
W-07	North	15.0	0.22	1.70
W-08	South	22.0	0.14	1.50
W-09	Central	35.0	0.11	1.34
W-10	North	10.0	0.28	1.92

Based on the data provided, what is the median operating cost, in dollars per thousand gallons, for the facilities that have a daily capacity greater than 20.0 MGD?

Show answer & explanation

Answer: 1.38

Answer

The median operating cost for facilities with a daily capacity greater than 20.0 MGD is $1.38 per thousand gallons.

Filtering the table for facilities with Daily Capacity > 20.0 MGD yields 6 facilities (W-01, W-03, W-05, W-06, W-08, W-09). Sorting their operating costs in ascending order produces:

1.18,

1.28,

1.34,

1.42,

1.50,

1.56. Because the count of elements is even (6), the median is the average of the 3rd element (

1.34) and the 4th element (

1.42), which equals $1.38.

Step-by-Step Solution

1

Filter the dataset by the given capacity constraint.

Identified 6 facilities with Daily Capacity > 20.0 MGD: W-01 (24.0 MGD), W-03 (32.0 MGD), W-05 (28.0 MGD), W-06 (40.0 MGD), W-08 (22.0 MGD), and W-09 (35.0 MGD).

Only facilities meeting the criterion 'Daily Capacity > 20.0 MGD' must be evaluated.

2

Extract and order the operating costs for the filtered subset.

Operating costs in ascending order:

1.18,

1.28,

1.34,

1.42,

1.50,

1.56.

Determining the median of a dataset requires arranging values sequentially.

3

Calculate the median for an even-numbered dataset.

Average of the 3rd and 4th values: $\frac{1.34 + 1.42}{2} = 1.38$.

When a dataset contains an even number of elements $N = 6$, the median is the arithmetic mean of the two central numbers at positions $N/2 = 3$ and $N/2 + 1 = 4$.

Key Concept

Calculating the median of a filtered even-count subset from tabular data.

Alternative Method

As a verification step, sum all operating costs in the filtered set:

1.18 + 1.28 + 1.34 + 1.42 + 1.50 + 1.56 = 8.28

. The mean is

8.28 / 6 = 1.38

. In this symmetric subset, the mean and median coincide at

1.38

.

Estimated Time:2m 0s

Question 372Question →

Machine A produces $120$ widgets in $4$ hours at a constant rate, and Machine B produces $150$ widgets in $3$ hours at a constant rate. If both machines work simultaneously at their respective constant rates, how many total widgets will they produce in $2$ hours?

Show answer & explanation

Answer: 160

Answer

The total number of widgets produced by both machines in 2 hours is 160.

Machine A produces

30

widgets per hour (

120 \div 4

), and Machine B produces

50

widgets per hour (

150 \div 3

). Together, their combined rate is

80

widgets per hour (

30 + 50

). Working together for

2

hours, they produce a total of

160

widgets (

80 \times 2

).

Step-by-Step Solution

1

Find the hourly rate of Machine A

Machine A produces

\frac{120}{4} = 30

widgets per hour.

Dividing total output by time gives the unit work rate per hour.

2

Find the hourly rate of Machine B

Machine B produces

\frac{150}{3} = 50

widgets per hour.

Dividing total output by time gives the unit work rate per hour.

3

Calculate the combined production rate per hour

The combined rate is

30 + 50 = 80

widgets per hour.

When working simultaneously, individual rates add together.

4

Multiply combined rate by the target time

In

2

hours, total production is

80 \times 2 = 160

widgets.

Total work equals rate multiplied by time.

Key Concept

Combined Work Rates

Estimated Time:45s

Question 373Question →

The table below presents subscriber numbers and annual operating revenue data for five regional divisions of a telecommunications provider in fiscal year 2025.

Region	Total Broadband Subscribers (thousands)	Fiber-Optic Subscribers (thousands)	Total Operating Revenue ($ millions)
North	250	150	45.0
South	400	200	64.0
East	180	90	27.0
West	320	224	57.6
Central	500	350	80.0

Across the two regional divisions with the highest total operating revenue combined, fiber-optic subscribers represent what percentage of total broadband subscribers? Express your answer to the nearest tenth of a percent.

Show answer & explanation

Answer: 61.1

Answer

61.1%

To find the proportion of fiber-optic subscribers across the top two revenue-generating divisions, first rank the divisions by Total Operating Revenue: Central (

80.0M) and South (

64.0M) are the highest. Combine their subscriber counts to find the appropriate base: Central has 500 thousand total broadband subscribers and South has 400 thousand, giving a total base of

500 + 400 = 900

thousand. Next, combine their fiber-optic subscribers: Central has 350 thousand and South has 200 thousand, giving

350 + 200 = 550

thousand fiber-optic subscribers. The combined percentage is

\frac{550}{900} \times 100\% \approx 61.111\%

, which rounds to

61.1\%

.

Step-by-Step Solution

1

Identify the top two divisions by operating revenue

Central (

80.0M) and South (

64.0M)

The question specifically restricts the calculation to the two divisions with the highest operating revenue.

2

Sum total broadband subscribers for Central and South

500 + 400 = 900 thousand subscribers

This establishes the correct aggregate base value for the percentage calculation.

3

Sum fiber-optic subscribers for Central and South

350 + 200 = 550 thousand subscribers

This gives the total combined numerator value for the fiber-optic subgroup.

4

Calculate percentage and round to the nearest tenth

(550 / 900) * 100% = 61.111...% -> 61.1%

Dividing the target subgroup total by the base total yields the requested proportion.

Key Concept

Ratio, Proportion, and Percent Calculations across Filtered Aggregate Groups

Estimated Time:1m 30s

Question 374Question →

Two server clusters, Alpha and Beta, process data jobs of Type 1 and Type 2. Cluster Alpha processes $15$ Type 1 jobs and $25$ Type 2 jobs in a total time of $210$ minutes. Cluster Beta processes $25$ Type 1 jobs and $15$ Type 2 jobs in a total time of $190$ minutes. Assuming constant processing rates per job type across both clusters, how many total minutes will it take to process a workload consisting of $30$ Type 1 jobs and $10$ Type 2 jobs?

Show answer & explanation

Answer: 180

Answer

The total processing time required is 180 minutes.

Let

x

be the processing time in minutes for a Type 1 job, and

y

be the processing time in minutes for a Type 2 job. Translating the given data gives the simultaneous equations

15x + 25y = 210

and

25x + 15y = 190

. Dividing both equations by

5

yields

3x + 5y = 42

and

5x + 3y = 38

. Adding these simplified equations yields

8x + 8y = 80

, so

x + y = 10

. Subtracting the first from the second gives

2x - 2y = -4

, so

x - y = -2

. Solving

x + y = 10

and

x - y = -2

gives

x = 4

and

y = 6

. The required workload duration is

30(4) + 10(6) = 120 + 60 = 180

minutes.

Step-by-Step Solution

1

Set up a linear system of simultaneous equations based on the rates given.

15x + 25y = 210

and

25x + 15y = 190

, where

x

and

y

are the processing times per Type 1 and Type 2 job.

Establishing accurate algebraic expressions translates the real-world scenario into a solvable model.

2

Simplify the system by dividing through by common factors.

3x + 5y = 42

and

5x + 3y = 38

.

Simplification reduces arithmetic complexity and minimizes computational errors.

3

Solve for variables x and y using symmetric combinations (adding/subtracting equations).

x = 4

and

y = 6

.

Symmetric linear systems can be solved efficiently by taking the sum and difference of the equations.

4

Evaluate the target expression

30x + 10y

.

30(4) + 10(6) = 180

.

Multiplying individual job times by requested quantities yields the total workload time.

Key Concept

Simultaneous Linear Systems and Symmetric Reduction

Question 375Question →

The table below details operational metrics for 10 wind turbines managed by a renewable energy utility during the previous calendar year:

Turbine ID	Region	Capacity (MW)	Availability (%)	Unplanned Maintenance (hours)
T-01	Offshore	6.0	94.2	48
T-02	Onshore	4.0	97.5	18
T-03	Offshore	8.0	91.0	84
T-04	Onshore	3.5	98.1	12
T-05	Offshore	6.0	95.8	36
T-06	Offshore	8.0	93.5	62
T-07	Onshore	4.0	96.0	24
T-08	Offshore	6.0	92.4	70
T-09	Onshore	3.5	98.8	8
T-10	Offshore	8.0	96.5	28

Based on the table, what is the median number of unplanned maintenance hours for turbines located in the Offshore region?

Show answer & explanation

Answer: 55

Answer

The median number of unplanned maintenance hours for turbines in the Offshore region is 55 hours.

Filtering the table for Offshore turbines yields 6 entries with unplanned maintenance hours of 48, 84, 36, 62, 70, and 28. Arranging these values in ascending order gives {28, 36, 48, 62, 70, 84}. Because the count is even (N = 6), the median is the average of the 3rd and 4th elements: (48 + 62) / 2 = 55 hours.

Step-by-Step Solution

1

Filter the dataset by Region

Identified 6 turbines in the Offshore region: T-01 (48 hrs), T-03 (84 hrs), T-05 (36 hrs), T-06 (62 hrs), T-08 (70 hrs), and T-10 (28 hrs).

The question specifically restricts the calculation to turbines operating in the Offshore region.

2

Sort the filtered unplanned maintenance hours in ascending order

Ordered set: 28, 36, 48, 62, 70, 84.

Finding the median requires data elements to be arranged sequentially.

3

Calculate the median of the even-count dataset (N = 6)

The two middle values are the 3rd element (48) and the 4th element (62). Mean = (48 + 62) / 2 = 55.

When a dataset contains an even number of observations, the median is the arithmetic average of the two central terms.

Key Concept

Descriptive Statistics Interpretation (Median of Even-Count Filtered Subsets)

Estimated Time:1m 30s

Question 376Question →

The table below presents the annual municipal water supply metrics and resident population for four administrative districts in 2025. Note that $1\text{ cubic meter } (\text{m}^3) = 1,000\text{ liters}$ .

District	Annual Water Supplied (in millions of $\text{m}^3$ )	Total Revenue (in tens of thousands of USD)	Resident Population (in thousands)
Metro North	21.9	876	200
Riverdale	8.76	438	120
Coastal Bay	14.6	730	160
Highland Peak	13.14	525.6	150

Based on the data provided, what was the average daily water consumption per resident in Coastal Bay in 2025, expressed in liters per person per day? (Assume a 365-day year.)

Show answer & explanation

Answer: 250

Answer

The average daily water consumption per resident in Coastal Bay in 2025 was 250 liters per person per day.

To find the average daily water consumption per resident in Coastal Bay, first read the values for Coastal Bay from the table: Annual Water Supplied is

14.6\text{ million m}^3

and Resident Population is

160\text{ thousand}

. Convert water volume to liters:

14.6 \times 10^6 \times 1,000 = 14,600,000,000\text{ liters}

. Convert population to individual count:

160 \times 1,000 = 160,000\text{ residents}

. Calculate total annual resident-days:

160,000 \times 365 = 58,400,000\text{ resident-days}

. Finally, divide total liters by total resident-days:

\frac{14,600,000,000}{58,400,000} = 250\text{ liters per person per day}

.

Step-by-Step Solution

1

Table Lookup and Scale Expansion for Water Supplied

Water supplied =

14.6 \times 10^6 \text{ m}^3 = 14,600,000 \text{ m}^3

The table header specifies that values are given in millions of cubic meters.

2

Unit Conversion from Cubic Meters to Liters

Total liters =

14,600,000 \text{ m}^3 \times 1,000 \text{ L/m}^3 = 14,600,000,000 \text{ L}

Each cubic meter contains

1,000

liters as stated in the problem note.

3

Population Scale Expansion and Resident-Days Calculation

Total resident-days =

(160 \times 1,000 \text{ people}) \times 365 \text{ days} = 58,400,000 \text{ person-days}

The population header specifies values in thousands, and a full year comprises 365 days.

4

Calculate Per Capita Daily Rate

Daily per capita rate =

\frac{14,600,000,000 \text{ L}}{58,400,000 \text{ person-days}} = 250 \text{ L/person/day}

Dividing total annual volume by total person-days yields the average daily consumption per person.

Key Concept

Unit Conversion and Scale Interpretation in Table Analysis

Question 377Question →

A biotechnology laboratory operates two automated synthesis workflows, System X and System Y, to manufacture two compounds, Alpha ( $A$ ) and Beta ( $B$ ). System X operates for $x$ hours, producing $A$ at a rate of $3\text{ mg/hr}$ and $B$ at a rate of $5\text{ mg/hr}$ . System Y operates for $y$ hours, producing $A$ at a rate of $15\text{ mg/hr}$ and $B$ at a rate of $8\text{ mg/hr}$ . The combined mass of compounds $A$ and $B$ produced across both workflows is exactly $340\text{ mg}$ . Furthermore, the system's operating efficiency requires that the ratio of operating hours $\frac{x}{y}$ equals the ratio of total mass produced $\frac{B}{A}$ . What is the total operating time, $x + y$ , in hours?

Show answer & explanation

Answer: 20

Answer

The total operating time of both workflows is 20 hours.

Representing the total production of compounds A and B as functions of hours x and y yields the linear total mass equation 8x + 23y = 340 and the ratio relation x/y = B/A, which simplifies to the homogeneous quadratic 3x^2 + 10xy - 8y^2 = 0. Factoring gives (3x - 2y)(x + 4y) = 0. Since operating hours are strictly positive, x = (2/3)y. Substituting this into the linear equation yields (85/3)y = 340, giving y = 12 hours and x = 8 hours. The total operating time is 8 + 12 = 20 hours.

Step-by-Step Solution

1

Set up algebraic expressions for total mass of each compound produced.

Compound A mass =

3x + 15y

mg; Compound B mass =

5x + 8y

mg.

Mass equals production rate multiplied by total operating hours.

2

Formulate the total combined production equation.

8x + 23y = 340

.

The sum of all Compound A and Compound B produced across both systems is given as 340 mg.

3

Set up and simplify the ratio equality

\frac{x}{y} = \frac{B}{A}

.

3x^2 + 10xy - 8y^2 = 0

.

Cross-multiplying

\frac{x}{y} = \frac{5x + 8y}{3x + 15y}

gives

x(3x + 15y) = y(5x + 8y)

.

4

Factor the homogeneous quadratic equation.

(3x - 2y)(x + 4y) = 0 \implies x = \frac{2}{3}y

.

Because operating hours

x

and

y

must be positive,

x + 4y > 0

, forcing

3x - 2y = 0

.

5

Substitute

x = \frac{2}{3}y

into

8x + 23y = 340

.

y = 12

hours and

x = 8

hours.

Solving

\frac{85}{3}y = 340

gives

y = 12

, and substituting back into

x = \frac{2}{3}y

gives

x = 8

.

6

Compute total operating time

x + y

.

8 + 12 = 20

hours.

The question asks for the combined total operating time of both systems.

Key Concept

Solving non-linear systems of simultaneous equations involving ratio constraints and quadratic factorization

Estimated Time:2m 30s

Question 378Question →

A telecommunications satellite utilizes two distinct signal amplifier modules, Module A and Module B, to process high-frequency and low-frequency data streams. Each Module A processes $18\text{ Gbps}$ of high-frequency data and $12\text{ Gbps}$ of low-frequency data, consuming $150\text{ watts}$ of operational power. Each Module B processes $30\text{ Gbps}$ of high-frequency data and $6\text{ Gbps}$ of low-frequency data, consuming $180\text{ watts}$ of operational power. During a peak transmission test, the active modules processed a combined total of $354\text{ Gbps}$ of high-frequency data and $138\text{ Gbps}$ of low-frequency data. What is the total operational power, in watts, consumed by all the active modules during this test?

Show answer & explanation

Answer: 2460

Answer

The total operational power consumed by all active modules during the test is 2,460 watts.

By setting up a system of simultaneous equations representing the total high-frequency (

18a + 30b = 354

) and low-frequency (

12a + 6b = 138

) bandwidths, solving yields

a = 8

active Module A units and

b = 7

active Module B units. Substituting these module counts into the total power equation

150(8) + 180(7)

gives exactly 2,460 watts.

Step-by-Step Solution

1

Define variables for the unknown counts of modules.

Let

a

represent the number of Module A units and

b

represent the number of Module B units.

Establishing explicit variables allows representing the data throughput and power constraints as algebraic equations.

2

Formulate equations for high-frequency and low-frequency data streams based on total throughput.

High-frequency equation:

18a + 30b = 354

. Low-frequency equation:

12a + 6b = 138

.

Summing the data contributions from each module type gives the total processed bandwidth for each frequency band.

3

Solve the system of linear equations.

Simplifying the low-frequency equation yields

2a + b = 23 \implies b = 23 - 2a

. Substituting into the simplified high-frequency equation (

3a + 5b = 59

) gives

3a + 5(23 - 2a) = 59 \implies -7a = -56 \implies a = 8

. Substituting

a = 8

back into

b = 23 - 2(8)

gives

b = 7

.

Determining the exact number of active modules of each type is required before calculating overall energy usage.

4

Calculate the total power consumption.

Total Power =

150a + 180b = 150(8) + 180(7) = 1,200 + 1,260 = 2,460\text{ watts}

.

Multiplying the module counts by their respective wattage ratings yields the total operational power.

Key Concept

Solving systems of simultaneous linear equations with two unknowns and applying the solution to calculate a weighted linear total.

Estimated Time:2m 30s

Question 379Question →

An urban courier service dispatches two types of electric cargo bikes, Cargo-X and Cargo-Y, for package deliveries. On Monday, a fleet of 3 Cargo-X bikes and 4 Cargo-Y bikes completed a total of 38 package deliveries. On Tuesday, a fleet of 5 Cargo-X bikes and 2 Cargo-Y bikes completed a total of 40 package deliveries. Assuming each bike of a given type completes a constant number of deliveries per day, how many total package deliveries will a combined fleet of 4 Cargo-X bikes and 3 Cargo-Y bikes complete in one day?

Show answer & explanation

Answer: 39

Answer

A combined fleet of 4 Cargo-X bikes and 3 Cargo-Y bikes will complete 39 package deliveries in one day.

Solving the linear system

3x + 4y = 38

and

5x + 2y = 40

gives

x = 6

deliveries per Cargo-X bike and

y = 5

deliveries per Cargo-Y bike. Substituting these rates into

4x + 3y

yields

4(6) + 3(5) = 39

total deliveries.

Step-by-Step Solution

1

Formulate a system of linear equations representing the delivery totals.

System established:

3x + 4y = 38

and

5x + 2y = 40

, where

x

and

y

represent the daily delivery capacities of Cargo-X and Cargo-Y bikes.

Translates contextual problem statements into quantitative algebraic equations.

2

Solve the system using elimination or substitution.

x = 6

deliveries per Cargo-X bike and

y = 5

deliveries per Cargo-Y bike.

Determines the exact rate per vehicle type.

3

Evaluate the target linear combination

4x + 3y

.

4(6) + 3(5) = 39

.

Computes the total capacity for the requested fleet combination.

Key Concept

Solving systems of simultaneous linear equations in two variables

Question 380Question →

A manufacturing facility uses two processing lines, Line A and Line B, to produce liquid chemical compounds. Line A operates at a constant rate of $r_A$ liters per hour and yields a mixture containing $15\%$ active solute by volume. Line B operates at a constant rate of $r_B$ liters per hour and yields a mixture containing $40\%$ active solute by volume.

When Line A operates for 4 hours and Line B operates for 6 hours, the combined output is $1,600$ liters of a solution containing $30\%$ active solute by volume.

If Line A and Line B are run simultaneously for 5 hours, how many liters of active solute will be produced in total?

Show answer & explanation

Answer: 440

Answer

440 liters of active solute will be produced in total.

By setting up a system of two linear equations relating the individual line flow rates (

r_A

and

r_B

) to total volume and total solute, we find

r_A = 160

L/hr and

r_B = 160

L/hr. Running Line A and Line B together yields

0.15(160) + 0.40(160) = 88

liters of active solute per hour. Over 5 hours, this produces

88 \times 5 = 440

liters of active solute.

Step-by-Step Solution

1

Formulate total volume equation

4r_A + 6r_B = 1600 \implies 2r_A + 3r_B = 800

Line A runs for 4 hours and Line B runs for 6 hours to produce a total volume of 1,600 liters.

2

Formulate active solute equation

0.15(4r_A) + 0.40(6r_B) = 0.30(1600) \implies 0.6r_A + 2.4r_B = 480 \implies r_A + 4r_B = 800

The total solute produced is

30\%

of the 1,600-liter mixture, which equals 480 liters.

3

Solve the system of linear equations

r_A = 160

liters/hour and

r_B = 160

liters/hour

Substituting

r_A = 800 - 4r_B

into

2r_A + 3r_B = 800

yields

5r_B = 800 \implies r_B = 160

and

r_A = 160

.

4

Calculate solute rate per hour when both run simultaneously

0.15(160) + 0.40(160) = 24 + 64 = 88

liters of solute per hour

Each line produces solute at its respective concentration percentage of its hourly fluid rate.

5

Compute total solute produced in 5 hours

88 \times 5 = 440

liters

Multiplying the combined solute production rate of 88 liters per hour by 5 hours gives the final quantity.

Key Concept

Simultaneous equations combining volumetric rates and mixture percentage concentrations.

Estimated Time:2m 30s

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

Alternative Method

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