Problem-Solving and Data Analysis Practice Questions — SAT — Page 6

Question 101Question →

A dataset consists of $15$ positive integers. The median of the dataset is $24$ , and the mean is $20$ . The dataset has a unique mode of $32$ , which occurs exactly $5$ times. What is the maximum possible range of the dataset?

Show answer & explanation

Answer: 81

Answer

The maximum possible range of the dataset is 81.

The total sum of the 15 elements is 300. With the median (8th term) equal to 24, we minimize the first 7 terms to 10 by using four 1s and three 2s (since no element other than 32 can repeat 5 or more times). Setting the 9th term to 24 and the 10th through 14th terms to the mode of 32 minimizes the sum of the first 14 terms to 218. This leaves a maximum possible value of 82 for the 15th term. The range is the difference between the maximum and minimum values: 82 - 1 = 81.

Step-by-Step Solution

1

Calculate the total sum of the dataset.

The sum of all 15 elements is

300

.

Since the mean of 15 elements is 20, the sum must be

15 \times 20 = 300

.

2

Identify the median and set up the sorted terms.

The 8th term

x_8 = 24

.

For 15 sorted elements, the median is the 8th term.

3

Minimize the sum of the first 7 terms.

The minimum sum of

x_1

through

x_7

is

10

, with

x_1 = 1

.

To minimize the sum, we use the smallest positive integers. However, no value other than 32 can appear 5 or more times. Thus, we can have at most four 1s and three 2s:

4(1) + 3(2) = 10

.

4

Minimize the 9th term.

x_9 = 24

.

Since the dataset is sorted,

x_9 \geq x_8 = 24

. To minimize the sum of the other terms and maximize

x_{15}

, we set

x_9 = 24

.

5

Account for the mode of 32.

x_{10} = x_{11} = x_{12} = x_{13} = x_{14} = 32

.

The mode 32 appears exactly 5 times. Since it is greater than the median 24, these 5 occurrences must be in the upper half of the sorted list.

6

Calculate the maximum possible value of the 15th term and the range.

x_{15} = 82

, and the range is

81

.

Subtracting the minimum sum of the first 14 terms from the total sum:

300 - (10 + 24 + 24 + 160) = 82

. The range is

x_{15} - x_1 = 82 - 1 = 81

.

Key Concept

Calculating measures of center (mean, median, mode) and variability (range) under constrained datasets.

Estimated Time:3m 0s

Question 102Question →

An analyst recorded the number of years of experience for $25$ employees at a software company. The distribution of the employees' experience is shown in the table below:

Years of Experience	Number of Employees
$1$	$5$
$2$	$8$
$3$	$6$
$4$	$4$
$15$	$2$

If the $2$ employees with $15$ years of experience are removed from the dataset, which of the following statements is true?

The mean of the dataset will decrease, and the median of the dataset will remain the same.

Both the mean and the median of the dataset will decrease.

The median of the dataset will decrease, and the mean of the dataset will remain the same.

The mean of the dataset will decrease, and the median of the dataset will increase.

Show answer & explanation

Answer: The mean of the dataset will decrease, and the median of the dataset will remain the same.

Answer

The mean of the dataset will decrease, and the median of the dataset will remain the same.

Removing the two extreme values of

15

years of experience significantly reduces the sum of the data, causing the mean to decrease from

3.4

to approximately

2.39

. However, the median remains

2

because both the

13^{\text{th}}

value of the original

25

-element dataset and the

12^{\text{th}}

value of the new

23

-element dataset fall within the category of

2

years of experience.

Step-by-Step Solution

1

Calculate the initial mean and identify the initial median of the dataset.

Initial mean is

3.4

years, and initial median is

2

years.

To establish the baseline values. The total number of employees is

5 + 8 + 6 + 4 + 2 = 25

. The sum of the years of experience is

(1 \times 5) + (2 \times 8) + (3 \times 6) + (4 \times 4) + (15 \times 2) = 5 + 16 + 18 + 16 + 30 = 85

. The initial mean is

\frac{85}{25} = 3.4

. Since there are

25

data points, the median is the

13^{\text{th}}

data point when ordered. Cumulatively counting the frequencies: the first

5

values are

1

, and the next

8

values (positions

6

to

13

) are

2

. Therefore, the

13^{\text{th}}

value is

2

.

2

Calculate the new mean and identify the new median after removing the two employees with

15

years of experience.

New mean is approximately

2.39

years, and new median is

2

years.

To find the new measures of center after the outliers are removed. The new total number of employees is

25 - 2 = 23

. The new sum of experience is

85 - 30 = 55

. The new mean is

\frac{55}{23} \approx 2.39

. With

23

data points, the median is the

12^{\text{th}}

data point when ordered. Cumulatively, the first

5

values are

1

, and the next

8

values (positions

6

to

13

) are

2

. Therefore, the

12^{\text{th}}

value is still

2

.

3

Compare the initial and new values of the mean and median.

The mean decreases from

3.4

to approximately

2.39

, while the median remains

2

.

To determine which option correctly describes the relationship between the changes.

Key Concept

The mean is sensitive to extreme values (outliers) and changes when they are removed. The median is a resistant measure of center and remains unchanged if the middle position of the ordered data stays within the same value class.

Question 103Question →

A hydroponics facility uses two nutrient solutions, Solution X and Solution Y. In a standard growth tank, the ratio of the volume of Solution X to the volume of Solution Y is $3$ to $5$ . In a specialized high-yield tank, the ratio of the volume of Solution X to the volume of Solution Y is $4$ to $7$ . A technician mixes $24$ gallons of the mixture from the standard growth tank with $44$ gallons of the mixture from the high-yield tank. What is the ratio of the volume of Solution X to the volume of Solution Y in the final mixture?

7 to 12

25 to 43

43 to 25

5 to 9

Show answer & explanation

Answer: 25 to 43

Answer

25 to 43

To find the combined ratio, we must first calculate the actual volume of each solution in both mixtures. For the standard mixture, the ratio of Solution X to Solution Y is

3:5

, meaning Solution X represents

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

of the total volume, and Solution Y represents

\frac{5}{8}

. In

24

gallons of this mixture, there are

\frac{3}{8} \times 24 = 9

gallons of Solution X and

\frac{5}{8} \times 24 = 15

gallons of Solution Y. For the high-yield mixture, the ratio is

4:7

, so Solution X represents

\frac{4}{4+7} = \frac{4}{11}

of the total volume, and Solution Y represents

\frac{7}{11}

. In

44

gallons of this mixture, there are

\frac{4}{11} \times 44 = 16

gallons of Solution X and

\frac{7}{11} \times 44 = 28

gallons of Solution Y. Combining the mixtures yields a total of

9 + 16 = 25

gallons of Solution X and

15 + 28 = 43

gallons of Solution Y. Therefore, the ratio of the volume of Solution X to the volume of Solution Y in the final mixture is

25

to

43

.

Step-by-Step Solution

1

Determine the volume of Solution X and Solution Y in the

24

gallons of standard growth tank mixture.

Solution X volume is

9

gallons; Solution Y volume is

15

gallons.

The ratio of Solution X to Solution Y is

3:5

, which means there are

3 + 5 = 8

total parts. Solution X makes up

\frac{3}{8}

of the mixture, so its volume is

\frac{3}{8} \times 24 = 9

gallons. Solution Y makes up

\frac{5}{8}

of the mixture, so its volume is

\frac{5}{8} \times 24 = 15

gallons.

2

Determine the volume of Solution X and Solution Y in the

44

gallons of specialized high-yield tank mixture.

Solution X volume is

16

gallons; Solution Y volume is

28

gallons.

The ratio of Solution X to Solution Y is

4:7

, which means there are

4 + 7 = 11

total parts. Solution X makes up

\frac{4}{11}

of the mixture, so its volume is

\frac{4}{11} \times 44 = 16

gallons. Solution Y makes up

\frac{7}{11}

of the mixture, so its volume is

\frac{7}{11} \times 44 = 28

gallons.

3

Calculate the total volumes of Solution X and Solution Y in the final combined mixture.

Total Solution X =

25

gallons; Total Solution Y =

43

gallons.

Adding the respective volumes from both mixtures gives: Total Solution X =

9 + 16 = 25

gallons. Total Solution Y =

15 + 28 = 43

gallons.

4

Find the ratio of the total volume of Solution X to the total volume of Solution Y.

The final ratio is

25

to

43

.

Comparing the total volume of Solution X (

25

gallons) to Solution Y (

43

gallons) yields the ratio

25:43

(or

25

to

43

).

Key Concept

Combining mixtures with different component ratios by calculating the absolute amounts of each component using part-to-whole fractions.

Question 104Question →

A team of $10$ workers, all working at the same constant rate, can complete a project in $18$ days. After working together for $6$ days, additional workers are hired, all of whom work at the same rate as the original workers. If the remaining portion of the project is completed in $8$ days, how many additional workers were hired?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5. Five additional workers were hired to complete the remaining work in 8 days.

The correct answer is

5

. To find the number of additional workers hired, we first calculate the total work required for the project in worker-days. Since

10

workers can complete the project in

18

days, the total work is

10 \times 18 = 180

worker-days. In the first

6

days, the

10

workers complete

10 \times 6 = 60

worker-days of work, leaving

180 - 60 = 120

worker-days of work. To complete this remaining work in

8

days, the total number of workers required is

\frac{120}{8} = 15

workers. Subtracting the original

10

workers gives

15 - 10 = 5

additional workers.

Step-by-Step Solution

1

Calculate the total amount of work required for the project.

180

worker-days

Since

10

workers can complete the project in

18

days, the total work is the product of the number of workers and the number of days:

10 \times 18 = 180

worker-days.

2

Calculate the work completed in the first

6

days.

60

worker-days

The original

10

workers worked for

6

days, completing

10 \times 6 = 60

worker-days of work.

3

Find the remaining work to be done.

120

worker-days

Subtracting the completed work from the total work gives the remaining work:

180 - 60 = 120

worker-days.

4

Find the total number of workers required to finish the remaining work in

8

days.

15

workers

Dividing the remaining work of

120

worker-days by the target time of

8

days gives the total number of workers needed:

\frac{120}{8} = 15

workers.

5

Calculate the number of additional workers hired.

5

workers

Subtract the original

10

workers from the total

15

workers required for the second phase:

15 - 10 = 5

.

Key Concept

Inverse variation and rate-time-work relationships, specifically using the concept of worker-days to solve multi-stage rate problems.

Estimated Time:2m 30s

Question 105Question →

At an agricultural research station, the nitrogen level of two soil plots, Plot A and Plot B, is monitored over several weeks. The nitrogen level of Plot A increases linearly at a constant rate, and the nitrogen level of Plot B increases exponentially at a constant percentage rate. At week $t = 0$ , Plot A contains $150\text{ grams}$ of nitrogen and Plot B contains $80\text{ grams}$ of nitrogen. At week $t = 2$ , both plots contain the same amount of nitrogen, which is $180\text{ grams}$ . At week $t = 4$ , Plot B contains $N\text{ grams}$ of nitrogen and Plot A contains $A\text{ grams}$ of nitrogen. What is the value of $N - A$ ?

Show answer & explanation

Answer: 195

Answer

195

The correct answer is

195

. At

t = 0

, Plot A contains

150\text{ grams}

and Plot B contains

80\text{ grams}

. Since Plot A increases linearly and reaches

180\text{ grams}

at

t = 2

, its weekly rate of change is

\frac{180 - 150}{2} = 15\text{ grams per week}

. Thus, at week

t = 4

, Plot A contains

150 + 15(4) = 210\text{ grams}

. Since Plot B increases exponentially and reaches

180\text{ grams}

at

t = 2

, its weekly growth factor

b

satisfies

80b^2 = 180

, which gives

b^2 = 2.25

and

b = 1.5

. Thus, at week

t = 4

, Plot B contains

80(1.5)^4 = 405\text{ grams}

. The difference

N - A

is

405 - 210 = 195

.

Step-by-Step Solution

1

Find the nitrogen level of Plot A at week

t = 4

using a linear model.

A = 210

Plot A grows linearly from an initial

150\text{ grams}

at

t = 0

to

180\text{ grams}

at

t = 2

. The rate of increase is

\frac{180 - 150}{2} = 15\text{ grams per week}

. Thus, at

t = 4

, the nitrogen level is

150 + 15(4) = 210\text{ grams}

.

2

Find the nitrogen level of Plot B at week

t = 4

using an exponential model.

N = 405

Plot B grows exponentially from an initial

80\text{ grams}

at

t = 0

to

180\text{ grams}

at

t = 2

. The weekly growth factor

b

satisfies

80b^2 = 180

, so

b^2 = 2.25

and

b = 1.5

. Thus, at

t = 4

, the nitrogen level is

80(1.5)^4 = 80(5.0625) = 405\text{ grams}

.

3

Calculate the difference between the two nitrogen levels at week

t = 4

.

195

Subtracting

A

from

N

yields

N - A = 405 - 210 = 195

.

Key Concept

Modeling linear growth (constant rate of change) and exponential growth (constant percentage rate of change or growth factor) over time.

Question 106Question →

The table below shows the distribution of a group of $180$ commuters surveyed about their primary mode of transportation and their daily commuting distance.

Commuting Distance	Public Transit	Personal Vehicle
Under 10 miles	45	35
10 miles or more	60	40

Based on the table, if a commuter who primarily uses a personal vehicle is selected at random, what is the probability that this commuter has a daily commuting distance of under 10 miles?

\frac{7}{36}

\frac{7}{16}

\frac{7}{15}

\frac{5}{12}

Show answer & explanation

Answer: $\frac{7}{15}$

Answer

The probability that the selected commuter has a daily commuting distance of under 10 miles, given that they primarily use a personal vehicle, is

\frac{7}{15}

.

To find the probability that a commuter has a daily commuting distance of under 10 miles given that they primarily use a personal vehicle, we limit our sample space to only the commuters who use a personal vehicle. The total number of commuters who primarily use a personal vehicle is

35 + 40 = 75

. Within this group, the number of commuters with a daily commuting distance of under 10 miles is 35. Therefore, the probability is the fraction

\frac{35}{75}

, which simplifies to

\frac{7}{15}

.

Step-by-Step Solution

1

Identify the total number of commuters who satisfy the condition of primarily using a personal vehicle.

The total number of commuters who primarily use a personal vehicle is

35 + 40 = 75

.

This sum represents the total number of possible outcomes (the denominator) because the selection is restricted to this group.

2

Identify the number of commuters within that subset who have a daily commuting distance of under 10 miles.

The number of personal vehicle commuters with a daily commuting distance of under 10 miles is 35.

This represents the number of favorable outcomes (the numerator).

3

Divide the number of favorable outcomes by the total number of possible outcomes and simplify the resulting fraction.

The probability is

\frac{35}{75} = \frac{7}{15}

.

Dividing the numerator and denominator by their greatest common divisor, 5, gives the simplified probability.

Key Concept

Calculating conditional probability from a two-way table by identifying the correct subset of the population to use as the denominator.

Estimated Time:1m 30s

Question 107Question →

A dataset consists of $11$ positive integers. The mean of the dataset is $14$ , and the median is $12$ . The range of the dataset is $20$ . If the maximum possible value of the largest integer in the dataset is $L$ , what is the value of $L$ ?

Show answer & explanation

Answer: 32

Answer

The maximum possible value of the largest integer in the dataset is

32

.

The correct answer is

32

. Let the 11 integers in nondecreasing order be

x_1 \le x_2 \le \dots \le x_{11}

. Since the mean of the 11 integers is

14

, their sum is

11 \times 14 = 154

. Since the median is

12

, the 6th integer is

x_6 = 12

. The range is

20

, so

x_{11} - x_1 = 20

, which gives

x_1 = x_{11} - 20

. To maximize the largest integer

x_{11}

, we need to observe that

x_1 \le x_6 = 12

. Substituting

x_1 = x_{11} - 20

into this inequality yields

x_{11} - 20 \le 12

, which simplifies to

x_{11} \le 32

. We can verify that

x_{11} = 32

is achievable with the dataset:

12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 32

. Therefore, the maximum possible value of the largest integer is

32

.

Step-by-Step Solution

1

Calculate the sum of the 11 integers in the dataset.

The sum of the 11 integers is

154

.

Since the mean of 11 values is 14, their sum must be

11 \times 14 = 154

.

2

Identify the median and establish the order constraint.

The 6th value in the ordered list of 11 integers is

12

.

The median of an odd number of sorted values is the middle value, which is the 6th value (

x_6 = 12

).

3

Relate the minimum and maximum values using the range.

The minimum value

x_1

is related to the maximum value

x_{11}

by

x_1 = x_{11} - 20

.

The range of the dataset is 20, which is the difference between the maximum and minimum values.

4

Find the maximum possible value of the largest integer.

The maximum value is

32

.

Since the dataset is sorted, the minimum value

x_1

cannot exceed the median

x_6 = 12

. Therefore,

x_1 \le 12 \implies x_{11} - 20 \le 12 \implies x_{11} \le 32

.

Key Concept

Using measures of center (mean, median) and spread (range) to determine bounds on data values.

Question 108Question →

A municipal research group conducted a survey of a random sample of $n_1$ registered voters in a city to estimate the proportion of voters who support a proposed school bond. In this survey, the proportion of voters who supported the bond was $0.52$ , with an associated margin of error of $0.06$ at a $95\%$ confidence level. The group plans to conduct a follow-up survey of a random sample of $n_2$ registered voters from the same population. Assuming the sample proportion remains approximately the same, and the group wants the margin of error for the follow-up survey to be $0.02$ at the same $95\%$ confidence level, what is the value of the ratio $\frac{n_2}{n_1}$ ?

Show answer & explanation

Answer: 9

Answer

9

The margin of error for a sample proportion is inversely proportional to the square root of the sample size. Since the target margin of error is one-third of the initial margin of error (from

0.06

to

0.02

), the sample size must be increased by a factor of

3^2 = 9

. Thus, the ratio of the new sample size to the original sample size is

9

.

Step-by-Step Solution

1

State the formula for the margin of error of a sample proportion.

E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

This formula defines the mathematical relationship between the margin of error, the confidence level (via the critical value

z^*

), the sample proportion

\hat{p}

, and the sample size

n

.

2

Determine the relationship between margin of error and sample size when other variables are constant.

E \propto \frac{1}{\sqrt{n}}

Since the confidence level remains at

95\%

(constant

z^*

) and the sample proportion remains approximately the same (constant

\hat{p}

), the margin of error is inversely proportional to the square root of the sample size.

3

Set up the ratio equation for the two surveys.

\frac{E_1}{E_2} = \sqrt{\frac{n_2}{n_1}}

Dividing the expression for the first margin of error by the expression for the second margin of error simplifies to the square root of the ratio of the second sample size to the first sample size.

4

Substitute the given margins of error and solve for the ratio.

\frac{n_2}{n_1} = 9

Substitute

E_1 = 0.06

and

E_2 = 0.02

to get

\frac{0.06}{0.02} = 3 = \sqrt{\frac{n_2}{n_1}}

. Squaring both sides yields the ratio of the sample sizes, which is

9

.

Key Concept

The relationship between sample size and margin of error in statistical studies.

Question 109Question →

A financial technology platform offers two merchant rewards programs, Program $P$ and Program $E$ , based on a merchant's monthly transaction volume. The table below shows the monthly reward for both programs at a transaction volume of $\$50,000$ .

Program	Growth Model	Reward at $\$50,000$ Volume
Program $P$	Linear	$\$800$
Program $E$	Exponential	$\$800$

For every increase of $\$10,000$ in monthly transaction volume:
- Under Program $P$ , the monthly reward increases by $\$150$ .
- Under Program $E$ , the monthly reward increases by $5\%$ .

If a merchant's monthly transaction volume increases from $\$50,000$ to $\$90,000$ , what is the closest dollar amount to the difference between the monthly rewards the merchant would receive under the two programs?

$159

$440

$428

$1,178

Show answer & explanation

Answer: $428

Answer

$428

The correct answer of

\$428

represents the difference between the rewards under the two models at a monthly transaction volume of

\$90,000

. The number of

\$10,000

increments from

\$50,000

to

\$90,000

is

4

. The linear reward under Program P increases by

\$150

per increment, resulting in a reward of

800 + 4(150) = 1400

. The exponential reward under Program E increases by

5\%

per increment, resulting in a reward of

800 \times (1.05)^4 \approx 972.41

. The difference is

1400 - 972.41 = 427.59

, which is closest to

\$428

.

Step-by-Step Solution

1

Determine the number of

\$10,000

increments from the initial transaction volume to the target volume.

4

increments

The transaction volume increases from

\$50,000

to

\$90,000

, a change of

\$40,000

. Since the growth rates are defined per

\$10,000

increase, the number of increments is

\frac{40,000}{10,000} = 4

.

2

Calculate the monthly reward under the linear model (Program P) after

4

increments.

1400

dollars

Program P adds a constant

\$150

per increment. The reward after

4

increments is

800 + 4(150) = 800 + 600 = 1400

.

3

Calculate the monthly reward under the exponential model (Program E) after

4

increments.

972.41

dollars

Program E increases by a constant percent (

5\%

) per increment, which corresponds to a growth factor of

1.05

. The reward after

4

increments is

800 \times (1.05)^4 \approx 800 \times 1.2155 = 972.41

.

4

Find the difference between the two monthly rewards.

427.59

dollars, which rounds to

428

dollars

Subtract the reward under Program E from the reward under Program P:

1400 - 972.41 = 427.59

.

Key Concept

Distinguishing between linear and exponential growth models and evaluating them at specific values in a real-world context.

Estimated Time:2m 30s

Question 110Question →

An aquarium filter recirculates water at a constant rate of $250$ milliliters per second. The filter cartridge must be replaced after it has processed a total of $540$ cubic meters of water. For how many hours of continuous operation can the filter run before the cartridge must be replaced? ( $1\text{ cubic meter} = 1,000\text{ liters}$ )

Show answer & explanation

Answer: 600

Answer

The filter can run for

600

hours of continuous operation before the cartridge must be replaced.

To find the number of hours the filter can run, we first convert the flow rate to cubic meters per hour. The flow rate of

250

milliliters per second is equal to

0.25

liters per second. Multiplying by

3,600

seconds per hour gives a flow rate of

900

liters per hour. Since

1\text{ cubic meter} = 1,000\text{ liters}

,

900

liters per hour is equivalent to

0.9

cubic meters per hour. Finally, dividing the total capacity of

540

cubic meters by the flow rate of

0.9

cubic meters per hour yields a duration of

600

hours.

Step-by-Step Solution

1

Convert the water recirculating rate from milliliters per second to liters per second.

0.25

liters per second

To express the flow rate in terms of liters before converting to cubic meters.

2

Convert the rate from liters per second to liters per hour.

900

liters per hour

To align the time unit of the rate with the requested time unit (hours).

3

Convert the rate from liters per hour to cubic meters per hour.

0.9

cubic meters per hour

To match the volume unit of the filter cartridge capacity (cubic meters).

4

Divide the total cartridge capacity by the hourly flow rate of the filter.

600

hours

To find the total duration of continuous operation before replacement.

Key Concept

Unit Conversions

Question 111Question →

A group of $25$ students took a history quiz, and their scores are summarized in the frequency table below.

Score	Frequency
6	2
7	4
8	10
9	6
10	3

The teacher decides to remove the scores of the $2$ students who scored $6$ because they were absent on the day of the quiz and took it later under different conditions. Which of the following statements describes the effect of removing these two scores on the mean, median, and standard deviation of the quiz scores?

The mean increases, the median remains unchanged, and the standard deviation decreases.

The mean increases, the median increases, and the standard deviation decreases.

The mean remains unchanged, the median remains unchanged, and the standard deviation increases.

The mean increases, the median remains unchanged, and the standard deviation increases.

Show answer & explanation

Answer: The mean increases, the median remains unchanged, and the standard deviation decreases.

Answer

The mean increases, the median remains unchanged, and the standard deviation decreases.

The correct answer is the statement indicating that the mean increases, the median remains unchanged, and the standard deviation decreases. Removing the two lowest scores (

6

) eliminates the values that pull the average down, resulting in an increased mean. The median remains at

8

because the middle position of the sorted scores still lies within the class of

8

s. The standard deviation decreases because the overall spread of the data is reduced when the lowest and most distant values are removed.

Step-by-Step Solution

1

Calculate the original mean and locate the original median.

Original Mean =

8.16

; Original Median =

8

.

The original sum of scores is

(2 \times 6) + (4 \times 7) + (10 \times 8) + (6 \times 9) + (3 \times 10) = 204

. With

25

students, the mean is

204 / 25 = 8.16

. The median is the

13\text{th}

value in the sorted list of

25

scores. Looking at the cumulative frequencies, the first

2

scores are

6

, the next

4

are

7

(cumulative

6

), and the next

10

are

8

(cumulative

16

). Thus, the

13\text{th}

value is

8

.

2

Calculate the new mean and locate the new median after removing the two scores of

6

.

New Mean

\approx 8.35

; New Median =

8

.

Removing the two scores of

6

reduces the sum by

12

(new sum =

192

) and the count of students to

23

. The new mean is

192 / 23 \approx 8.35

, which is an increase. The new median is the

12\text{th}

value of the remaining

23

sorted scores. The first

4

scores are

7

, and the next

10

are

8

(cumulative

14

). The

12\text{th}

value is still

8

, so the median remains unchanged.

3

Analyze the change in the standard deviation.

The standard deviation decreases.

Standard deviation measures the spread of the data. The scores of

6

are the minimum scores and are the furthest from the mean of the distribution. Removing these extreme values reduces the variability and concentrates the remaining scores closer to the mean, which decreases the standard deviation.

Key Concept

Analyzing the effects of removing extreme data values on the mean, median, and standard deviation of a dataset.

Question 112Question →

A laboratory uses two types of equipment, Instrument A and Instrument B, which both have an initial value of $V$ dollars when purchased at the same time ( $t = 0$ years). The value of Instrument A decreases linearly at a constant rate over time. At $t = 3$ years, the value of Instrument A is $\$12,000$ , and at $t = 6$ years, its value is $\$8,000$ . The value of Instrument B decreases exponentially at a constant annual rate. At $t = 2$ years, the value of Instrument B is $\$4,000$ . What is the value, in dollars, of Instrument B at $t = 4$ years?

Show answer & explanation

Answer: 1000

Answer

1000

The correct answer is 1000. By defining the value of Instrument A linearly, we obtain the system

V - 3d = 12000

and

V - 6d = 8000

. Solving this system yields an initial value of

V = 16000

. Instrument B decreases exponentially with the model

V_B(t) = 16000 \cdot b^t

. Given

V_B(2) = 4000

, we have

16000 \cdot b^2 = 4000

, which gives

b^2 = 0.25

. The value at

t = 4

is then calculated as

V_B(4) = V_B(2) \cdot b^2 = 4000 \cdot 0.25 = 1000

.

Step-by-Step Solution

1

Set up a system of linear equations for Instrument A's value.

V - 3d = 12000

and

V - 6d = 8000

Instrument A decreases linearly, meaning its value is modeled by

V_A(t) = V - dt

.

2

Solve for the constant annual rate of decrease,

d

, and the initial purchase value,

V

.

d = \frac{4000}{3}

and

V = 16000

Subtracting the two linear equations eliminates

V

, allowing us to solve for

d

and subsequently substitute it back to find

V

.

3

Apply the initial value

V = 16000

to Instrument B's exponential decay model at

t = 2

.

16000 \cdot b^2 = 4000

Instrument B decreases exponentially starting from the same initial value

V

, so its model is

V_B(t) = V \cdot b^t

.

4

Solve for the 2-year decay factor,

b^2

.

b^2 = 0.25

Dividing both sides of the equation by

16000

isolates the term representing the growth or decay factor over a two-year interval.

5

Calculate the value of Instrument B at

t = 4

years.

1000

Multiply the value at

t = 2

by the 2-year decay factor

b^2

, which yields

4000 \cdot 0.25 = 1000

.

Key Concept

Distinguishing between linear decay (constant absolute rate of change) and exponential decay (constant percentage rate of change) to set up and solve coupled system models.

Question 113Question →

A group of $15$ students recorded the number of books they read over the summer. The table below shows the distribution of the number of books read.

Number of books	Number of students
$1$	$1$
$2$	$3$
$3$	$5$
$4$	$5$
$18$	$1$

If the student who read $18$ books is removed from the dataset, which of the following statements best describes the effect on the mean and the median of the dataset?

The mean decreases by

1

, and the median remains the same.

The mean remains the same, and the median decreases by

1

.

Both the mean and the median decrease by

1

.

Both the mean and the median remain the same.

Show answer & explanation

Answer: The mean decreases by $1$ , and the median remains the same.

Answer

The mean decreases by

1

, and the median remains the same.

The correct answer states that the mean decreases by

1

and the median remains the same. Removing the outlier of

18

reduces the sum from

60

to

42

and the number of terms from

15

to

14

, resulting in a new mean of

3

(down from

4

). The median is determined by the middle values of the dataset; both the

8\text{th}

value of the

15

-element set and the average of the

7\text{th}

and

8\text{th}

values of the

14

-element set are

3

, meaning the median does not change.

Step-by-Step Solution

1

Calculate the mean of the original dataset containing

15

students.

The sum of the books read is

(1 \times 1) + (2 \times 3) + (3 \times 5) + (4 \times 5) + (18 \times 1) = 1 + 6 + 15 + 20 + 18 = 60

. The mean is

\frac{60}{15} = 4

.

To find the initial baseline mean before the outlier is removed.

2

Find the median of the original dataset.

With

15

data points, the median is the

8\text{th}

value when ordered. Counting from the smallest value, the first value is

1

, the next

3

values are

2

, and the next

5

values are

3

(covering the

5\text{th}

through

9\text{th}

positions). Thus, the

8\text{th}

value is

3

, so the median is

3

.

To find the initial baseline median before the outlier is removed.

3

Calculate the new mean after removing the student who read

18

books.

The new number of students is

14

. The new sum of books read is

60 - 18 = 42

. The new mean is

\frac{42}{14} = 3

. The mean decreased by

4 - 3 = 1

.

To determine the change in the mean.

4

Find the new median after removing the student who read

18

books.

With

14

data points, the median is the average of the

7\text{th}

and

8\text{th}

values. Both the

7\text{th}

and

8\text{th}

values are in the group of

3

s (covering the

5\text{th}

through

9\text{th}

positions). Therefore, the new median is

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

, meaning the median remains the same.

To determine the change in the median.

Key Concept

Understanding how outliers affect measures of center, specifically the mean (which is sensitive to outliers) and the median (which is resistant to outliers).

Question 114Question →

A wind turbine generates electricity at a constant rate of $1.5\text{ megawatt-hours (MWh)}$ per day. If $1\text{ megawatt-hour}$ is equivalent to $3.6\text{ gigajoules (GJ)}$ , how many gigajoules of energy does the wind turbine generate in $8\text{ hours}$ ?

16.2

1.8

43.2

0.14

Show answer & explanation

Answer: 1.8

Answer

1.8

The correct answer is 1.8. To find the total energy generated in gigajoules over 8 hours, the daily rate of

1.5\text{ MWh}

is first converted to an hourly rate by dividing by 24 hours, yielding

0.0625\text{ MWh}

per hour. Multiplying this rate by 8 hours results in

0.5\text{ MWh}

of energy generated. Finally, multiplying the energy in MWh by the conversion factor of

3.6\text{ GJ}

per MWh yields

1.8\text{ GJ}

.

Step-by-Step Solution

1

Convert the daily generation rate of

1.5\text{ MWh}

to an hourly generation rate by dividing by 24 hours.

0.0625\text{ MWh per hour}

Since there are 24 hours in a day, dividing the daily rate by 24 yields the rate of energy generated per hour.

2

Multiply the hourly generation rate by 8 hours to find the total energy generated in megawatt-hours.

0.5\text{ MWh}

Multiplying the rate per hour by the duration of 8 hours gives the cumulative energy generated over that period.

3

Convert the energy generated from megawatt-hours to gigajoules by multiplying by the conversion factor of

3.6\text{ GJ}

per MWh.

1.8\text{ GJ}

Multiplying the energy in MWh by

3.6\text{ GJ/MWh}

converts the unit to gigajoules.

Key Concept

Unit Conversions

Question 115Question →

An administrator at a large university with 12,000 enrolled students wants to evaluate student satisfaction with campus services. The administrator designs two separate surveys:

* Survey A: A survey is sent to a random sample of 1,000 students selected from the registrar's list of all 12,000 enrolled students. In this sample, 60% of the students report being satisfied with campus services.
* Survey B: A survey is sent to a random sample of 250 students selected from the 3,000 students who live in on-campus dormitories. In this sample, 70% of the students report being satisfied with campus services.

Which of the following statements must be true?

I. The results of Survey B can be generalized to estimate the satisfaction of all 12,000 enrolled students at the university.
II. If the administrator wants to reduce the margin of error of Survey A to approximately half of its current value, the sample size of Survey A should be increased to 4,000 students.
III. Based on the results of Survey B, it is estimated that 2,100 students who live in on-campus dormitories are satisfied with campus services.

II only

II and III only

I and II only

I, II, and III

Show answer & explanation

Answer: II and III only

Answer

II and III only

The correct option is 'II and III only'. Statement II is true because the margin of error is inversely proportional to the square root of the sample size (

n

). To reduce the margin of error by half (a factor of

\frac{1}{2}

), the sample size must be multiplied by

2^2 = 4

. Thus, increasing the sample size of Survey A from 1,000 to 4,000 students will halve its margin of error. Statement III is true because Survey B selected a random sample of 250 students from the population of 3,000 students living in on-campus dormitories. Since the sample is representative of this population, the sample proportion of 70% can be generalized to the 3,000 dormitory residents, yielding an estimate of

0.70 \times 3,000 = 2,100

satisfied students. Statement I is false because the sample for Survey B was drawn exclusively from students living in on-campus dormitories, who may have different satisfaction levels than off-campus students; therefore, the results cannot be generalized to all 12,000 enrolled students.

Step-by-Step Solution

1

Evaluate Statement I by checking the target population of Survey B's sampling frame.

Statement I is false.

Survey B's sample was drawn specifically from the 3,000 students living in on-campus dormitories, not from the entire student body of 12,000. Therefore, the findings of Survey B can only be generalized to on-campus dormitory residents, not to the entire university.

2

Evaluate Statement II by analyzing the relationship between sample size and margin of error.

Statement II is true.

The margin of error for a sample proportion is inversely proportional to the square root of the sample size (

n

). To reduce the margin of error to

\frac{1}{2}

of its current value, the sample size must be increased by a factor of

2^2 = 4

. Since the original sample size of Survey A is 1,000, increasing it to

1,000 \times 4 = 4,000

will halve the margin of error.

3

Evaluate Statement III by calculating the estimated population value from the sample proportion.

Statement III is true.

Since Survey B used a random sample of the 3,000 dormitory residents, the sample proportion of 70% (

0.70

) can be used to estimate the number of satisfied dormitory residents in the population:

0.70 \times 3,000 = 2,100

students.

4

Combine the evaluations to select the correct option.

Only Statement II and Statement III must be true.

Since Statement I is false and both Statement II and Statement III are true, the correct choice is the option containing 'II and III only'.

Key Concept

Generalization limits of survey results based on sampling frames, and the mathematical relationship between sample size and margin of error.

Estimated Time:2m 30s

Question 116Question →

A water desalination facility uses two types of filtration systems, System X and System Y. System X processes saltwater at a constant rate of $15$ liters per minute, producing purified water and brine in a ratio of $4:1$ . System Y processes saltwater at a constant rate of $21$ liters per minute, producing purified water and brine in a ratio of $5:2$ . If both systems operate simultaneously, how many liters of brine will be produced in total during the time it takes the two systems combined to produce $324$ liters of purified water?

81

108

126

432

Show answer & explanation

Answer: 108

Answer

108

The correct value of 108 is found by calculating the rate of brine production for each system. System X produces brine at a rate of

15 \times \frac{1}{5} = 3

liters per minute, and System Y produces brine at

21 \times \frac{2}{7} = 6

liters per minute, giving a combined brine rate of

9

liters per minute. Meanwhile, the combined purified water rate is

(15 \times \frac{4}{5}) + (21 \times \frac{5}{7}) = 12 + 15 = 27

liters per minute. It takes the systems

\frac{324}{27} = 12

minutes to produce the purified water. During these 12 minutes, the total brine produced is

12 \times 9 = 108

liters.

Step-by-Step Solution

1

Find the purified water and brine production rates for System X.

Purified water rate =

12

liters per minute; Brine rate =

3

liters per minute.

Since the ratio of purified water to brine is

4:1

, there are

4 + 1 = 5

parts in total. System X processes

15

liters of saltwater per minute, so the purified water rate is

15 \times \frac{4}{5} = 12

liters per minute and the brine rate is

15 \times \frac{1}{5} = 3

liters per minute.

2

Find the purified water and brine production rates for System Y.

Purified water rate =

15

liters per minute; Brine rate =

6

liters per minute.

Since the ratio of purified water to brine is

5:2

, there are

5 + 2 = 7

parts in total. System Y processes

21

liters of saltwater per minute, so the purified water rate is

21 \times \frac{5}{7} = 15

liters per minute and the brine rate is

21 \times \frac{2}{7} = 6

liters per minute.

3

Calculate the combined purified water and brine rates.

Combined purified rate =

27

liters per minute; Combined brine rate =

9

liters per minute.

Adding the individual rates of System X and System Y gives the combined rates of

12 + 15 = 27

liters per minute for purified water and

3 + 6 = 9

liters per minute for brine.

4

Determine the time required to produce the target amount of purified water.

Time =

12

minutes.

Divide the target purified water volume of

324

liters by the combined purified water rate of

27

liters per minute to find the operating time:

\frac{324}{27} = 12

minutes.

5

Calculate the total amount of brine produced during this time.

Total brine =

108

liters.

Multiply the operating time of

12

minutes by the combined brine production rate of

9

liters per minute:

12 \times 9 = 108

liters.

Key Concept

Combining rates derived from part-to-whole ratios to solve multi-step rate problems.

Question 117Question →

A marine biologist records the dive depths, in meters, of a seal on 6 consecutive dives. The recorded depths are 52, 68, 57, 65, 48, and $x$ . The median of these 6 dive depths is 58 meters. What is the value of $x$ ?

Show answer & explanation

Answer: 59

Answer

59

To find the median of a set of 6 numbers, the numbers must be listed in ascending order, and the median will be the average of the 3rd and 4th numbers. The five known numbers in ascending order are 48, 52, 57, 65, and 68. If the unknown value

x

is less than or equal to 57, the 3rd and 4th numbers in the sorted list would be at most 57, which would make the median at most 57. If

x

is greater than or equal to 65, the 3rd and 4th numbers in the sorted list would be 57 and 65, which would make the median

\frac{57+65}{2} = 61

. Since the median is given as 58,

x

must be between 57 and 65. Thus, when the 6 numbers are sorted, the 3rd number is 57 and the 4th number is

x

. Setting their average equal to 58 gives

\frac{57+x}{2} = 58

. Multiplying both sides by 2 yields

57 + x = 116

. Subtracting 57 from both sides gives

x = 59

.

Step-by-Step Solution

1

Order the five known depths from least to greatest.

48, 52, 57, 65, 68

To find or use the median of a dataset, the data points must first be arranged in ascending order.

2

Analyze the position of the median in a dataset of 6 values.

The median is the average of the 3rd and 4th values in the ordered list.

For an even number of data points, the median is the arithmetic mean of the two middle values.

3

Determine the constraints on

x

to achieve a median of 58.

The value of

x

must be between 57 and 65, making 57 the 3rd value and

x

the 4th value.

If

x \le 57

, the 3rd and 4th values are both 57 or less, yielding a median of at most 57. If

x \ge 65

, the 3rd and 4th values are 57 and 65, yielding a median of 61. Thus,

x

must be the 4th value.

4

Set up and solve the equation for the median.

\frac{57 + x}{2} = 58 \implies 57 + x = 116 \implies x = 59

Since the 3rd value is 57 and the 4th value is

x

, their average must equal the given median of 58.

Key Concept

Calculating and interpreting the median of a dataset with an even number of values, including handling unknown variables.

Question 118Question →

A scientist surveyed $15$ different forest regions and recorded the number of endangered plant species in each region. The numbers of endangered species in $14$ of the regions are:

12, 15, 15, 16, 18, 20, 20, 22, 22, 22, 25, 28, 30, 34

The number of endangered species in the $15\text{th}$ region is $x$ , where $x$ is an integer greater than $25$ . If the median of the numbers of endangered species for all $15$ regions is equal to the mean, what is the value of $x$ ?

Show answer & explanation

Answer: 31

Answer

The correct answer is 31.

The correct answer is 31. The sum of the 14 known values is 299. Since the 15th value

x

is greater than 25, it must be placed at or after the 12th position when all 15 values are sorted. Thus, the 8th value (the median of the 15 values) remains the same as the 8th value of the known 14 sorted values, which is 22. Setting the mean equal to the median gives

\frac{299 + x}{15} = 22

, which simplifies to

299 + x = 330

, resulting in

x = 31

. Since

31 > 25

, this is the correct value.

Step-by-Step Solution

1

Find the sum of the 14 known values.

The sum of the 14 known values is 299.

This is needed to construct the expression for the mean of the 15 values.

2

Determine the median of the 15 values given the constraint

x > 25

.

The median is 22.

Since

x > 25

, it will be sorted after the first 11 values (the 11th value is 25). The median of 15 values is the 8th value, which remains 22.

3

Set the mean equal to the median and solve for

x

.

\frac{299 + x}{15} = 22 \implies 299 + x = 330 \implies x = 31

.

This yields the value of

x

that satisfies the condition that the mean equals the median.

Key Concept

Calculating and comparing the mean and median of a dataset, and analyzing how a new data point affects these measures.

Question 119Question →

An agricultural drone sprays a liquid fertilizer at a constant rate of $120$ milliliters per second. The drone flies at a constant speed, covering $1.5$ hectares per hour. If the drone operates continuously at this rate, how many liters of fertilizer are sprayed per hectare?

(Note: $1$ liter = $1,000$ milliliters)

4.8

288

432

648

Show answer & explanation

Answer: 288

Answer

The drone sprays 288 liters of fertilizer per hectare.

To find the number of liters sprayed per hectare, the rate of spraying must be converted from milliliters per second to liters per hour, and then divided by the coverage rate of hectares per hour. First, multiply the spray rate of

120

milliliters per second by the number of seconds in an hour (

3,600

seconds) to find that the drone sprays

432,000

milliliters per hour. Next, convert this volume to liters by dividing by

1,000

milliliters per liter, which gives

432

liters per hour. Finally, divide this hourly spray volume by the coverage rate of

1.5

hectares per hour to obtain

288

liters per hectare.

Step-by-Step Solution

1

Calculate the volume of fertilizer sprayed in one hour in milliliters.

120\text{ mL/s} \times 3,600\text{ s/h} = 432,000\text{ mL/h}

There are

3,600

seconds in one hour, so multiplying the per-second rate by

3,600

gives the volume sprayed per hour.

2

Convert the hourly volume from milliliters to liters.

\frac{432,000\text{ mL/h}}{1,000\text{ mL/L}} = 432\text{ L/h}

Since

1

liter is equivalent to

1,000

milliliters, divide the volume in milliliters by

1,000

to get the volume in liters.

3

Calculate the volume of fertilizer sprayed per hectare.

\frac{432\text{ L/h}}{1.5\text{ hectares/h}} = 288\text{ L/hectare}

Divide the total liters sprayed in one hour by the total hectares covered in that same hour to find the rate per hectare.

Key Concept

Unit conversion involving compound rates of volume, time, and area.

Question 120Question →

A paper mill machine produces paper at a constant rate of $24$ square feet per second. If $1\text{ square yard} = 9\text{ square feet}$ , how many square yards of paper does the machine produce in $15$ minutes?

40

2,400

21,600

194,400

Show answer & explanation

Answer: 2,400

Answer

2,400

The correct answer of 2,400 is found by first converting the time of 15 minutes to seconds:

15 \times 60 = 900

seconds. Next, calculate the total area produced in square feet by multiplying the production rate by the time:

24 \times 900 = 21,600

square feet. Finally, convert this area to square yards by dividing by 9:

21,600 / 9 = 2,400

square yards.

Step-by-Step Solution

1

Convert the time from minutes to seconds.

15\text{ minutes} \times 60\text{ seconds/minute} = 900\text{ seconds}

The production rate is given in square feet per second, so the time duration must be in seconds.

2

Calculate the total area of paper produced in square feet.

24\text{ square feet/second} \times 900\text{ seconds} = 21,600\text{ square feet}

Multiplying the rate by the total time gives the total amount produced in square feet.

3

Convert the total area from square feet to square yards.

21,600\text{ square feet} \div 9\text{ square feet/square yard} = 2,400\text{ square yards}

Since

1\text{ square yard} = 9\text{ square feet}

, dividing the total square feet by 9 converts the value to square yards.

Key Concept

Multi-step unit conversions involving rates of change and area scaling.

Problem-Solving and Data Analysis

Answer

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