Problem-Solving and Data Analysis Practice Questions — SAT — Page 5

Question 81Question →

An organic farm harvested a certain amount of wheat in 2024. In 2025, due to favorable weather, the amount of wheat harvested was $25\%$ greater than in 2024. In 2026, due to a dry season, the harvest was $12\%$ less than in 2025. If the farm harvested $880$ bags of wheat in 2026, how many bags of wheat did it harvest in 2024?

Show answer & explanation

Answer: 800

Answer

800

The correct answer is 800. Let the 2024 harvest be

w

bags. A

25\%

increase in 2025 means the harvest became

1.25w

. A subsequent

12\%

decrease in 2026 means the harvest became

0.88

of the 2025 value, which is

1.25w \times 0.88 = 1.1w

. Given that the 2026 harvest was

880

bags, we solve

1.1w = 880

to find

w = 800

.

Step-by-Step Solution

1

Define the variable for the 2024 harvest and write an expression for the 2025 harvest.

The harvest in 2025 is

1.25w

, where

w

is the harvest in 2024.

An increase of 25% is equivalent to multiplying the original value by

1 + 0.25 = 1.25

.

2

Write an expression for the 2026 harvest in terms of the 2024 harvest variable.

The harvest in 2026 is

1.1w

.

A decrease of 12% is equivalent to multiplying the 2025 value by

1 - 0.12 = 0.88

. Multiplying the factors gives

1.25 \times 0.88 = 1.1

.

3

Set up an equation using the 2026 harvest total of 880 and solve for the 2024 harvest variable.

w = 800

Dividing the 2026 harvest value of 880 by the cumulative rate of 1.1 yields the initial 2024 value.

Key Concept

Solving for initial values in multi-step percent change scenarios.

Alternative Method

Work backward from the final year. The 2026 harvest of

880

bags is

12\%

less than the 2025 harvest, meaning it represents

88\%

of the 2025 harvest. Thus, the 2025 harvest was

880 / 0.88 = 1000

bags. The 2025 harvest of

1000

bags was

25\%

greater than the 2024 harvest, meaning it represents

125\%

of the 2024 harvest. Thus, the 2024 harvest was

1000 / 1.25 = 800

bags.

Estimated Time:1m 30s

Question 82Question →

A quality control manager at a manufacturing plant selected a random sample of $150$ lightbulbs from a batch of $3,000$ lightbulbs produced on a certain day. The manager found that $6$ of the selected lightbulbs were defective. Based on this sample, what is the estimated number of defective lightbulbs in the entire batch of $3,000$ ?

Show answer & explanation

Answer: 120

Answer

120

The correct answer is

120

. Since the sample of lightbulbs was selected at random, the proportion of defective lightbulbs in the sample can be used to estimate the proportion of defective lightbulbs in the entire batch. The proportion of defective lightbulbs in the sample is

\frac{6}{150} = 0.04

. Multiplying this proportion by the total number of lightbulbs in the batch gives the estimated number of defective lightbulbs:

0.04 \times 3,000 = 120

.

Step-by-Step Solution

1

Calculate the proportion of defective lightbulbs in the sample.

The sample proportion is

\frac{6}{150} = 0.04

.

This establishes the rate of defects in the sample to be generalized to the entire batch.

2

Estimate the total number of defective lightbulbs in the batch.

The estimate is

0.04 \times 3,000 = 120

.

Scaling the sample proportion to the population size yields the expected number of defective lightbulbs in the entire batch.

Key Concept

Generalizing results from a random sample to estimate a population parameter

Question 83Question →

A certain antique watch increases in value by a constant percent each year. The watch was purchased for $150$ , and its value after $2$ years is $216$ . What is the annual percent increase in the value of the watch?

Show answer & explanation

Answer: 20

Answer

The annual percent increase in the value of the watch is 20.

Because the watch increases in value by a constant percent each year, its value can be modeled by the exponential growth equation

V(t) = P(1+r)^t

, where

P

is the initial purchase price,

r

is the annual growth rate as a decimal, and

t

is the number of years. Substituting the given values

P = 150

,

t = 2

, and

V(2) = 216

yields

150(1+r)^2 = 216

. Dividing both sides by

150

gives

(1+r)^2 = 1.44

. Taking the positive square root of both sides results in

1+r = 1.2

, which simplifies to

r = 0.2

. Converting

0.2

to a percentage gives an annual percent increase of

20\%

. Therefore, the correct numeric answer is

20

.

Step-by-Step Solution

1

Set up the exponential growth equation using the given values.

150(1+r)^2 = 216

Since the value increases by a constant percent each year, the growth is exponential and modeled by

V(t) = P(1+r)^t

.

2

Divide both sides of the equation by

150

to isolate the exponential base term.

(1+r)^2 = 1.44

Dividing

216

by

150

simplifies the equation to solve for the growth factor.

3

Take the positive square root of both sides of the equation.

1 + r = 1.2

Taking the square root of

1.44

isolates the expression

1+r

.

4

Solve for

r

and convert the decimal to a percentage.

r = 0.2

, which is

20\%

Subtracting

1

from both sides gives the annual rate

r = 0.2

, and multiplying by

100

gives the percent increase.

Key Concept

Solving for the growth rate in an exponential growth model

Alternative Method

Alternatively, since the question asks for a simple percent increase, you can test a few straightforward percentages. For example, if the annual increase were

10\%

, the value after 1 year would be

150 \times 1.10 = 165

, and after 2 years would be

165 \times 1.10 = 181.5

, which is too low. Testing

20\%

, the value after 1 year is

150 \times 1.20 = 180

, and after 2 years is

180 \times 1.20 = 216

. This matches the given value exactly.

Estimated Time:45s

Question 84Question →

An industrial water filtration system filters water at a constant rate of $0.5\text{ liters per second}$ . The system consumes a chemical powder at a rate of $2.5\text{ milligrams}$ per gram of impurities removed from the water. If the water contains an average of $0.12\text{ grams}$ of impurities per liter, how many grams of the chemical powder does the system consume during $4\text{ hours}$ of continuous operation?

Show answer & explanation

Answer: 2.16

Answer

The system consumes

2.16\text{ grams}

of chemical powder during the

4

hours of operation.

The correct answer is

2.16

. First, convert the

4

hours of continuous operation into seconds:

4 \times 3,600 = 14,400

seconds. Next, multiply this duration by the filtration rate to find the total water filtered:

14,400 \times 0.5 = 7,200

liters. Multiply the volume by the impurity concentration to find the total mass of impurities:

7,200 \times 0.12 = 864

grams. Then, find the mass of chemical powder consumed:

864 \times 2.5 = 2,160

milligrams. Finally, convert milligrams to grams by dividing by

1,000

, resulting in

2.16

grams.

Step-by-Step Solution

1

Convert the operating time from hours to seconds.

14,400\text{ seconds}

Since the filtration rate is given in liters per second, the total operating time must be converted from hours to seconds to align units. There are

3,600

seconds in one hour.

2

Calculate the total volume of water filtered during the operation.

7,200\text{ liters}

Multiply the operating time in seconds by the filtration rate of

0.5

liters per second.

3

Calculate the total mass of impurities removed from the filtered water.

864\text{ grams}

Multiply the total volume of filtered water by the concentration rate of

0.12

grams of impurities per liter.

4

Determine the mass of chemical powder consumed in milligrams.

2,160\text{ milligrams}

Multiply the total mass of impurities in grams by the consumption rate of

2.5

milligrams of chemical powder per gram of impurities.

5

Convert the mass of chemical powder from milligrams to grams.

2.16\text{ grams}

Divide the mass in milligrams by

1,000

because there are

1,000

milligrams in one gram.

Key Concept

Dimensional analysis and multi-step unit conversion involving rates

Question 85Question →

The table below summarizes the results of a survey about extracurricular participation among 11th-grade and 12th-grade students at a high school.

Grade	Participates	Does not participate
11th Grade	$45$	$30$
12th Grade	$55$	$20$

Based on the table, if a student who participates in extracurricular activities is selected at random, what is the probability that the student is in the 11th grade? (Express your answer as a decimal or a fraction.)

Show answer & explanation

Answer: 0.45

Answer

0.45 (or 9/20)

To find the probability that a student is in the 11th grade given that they participate in extracurricular activities, the sample space is restricted to only those students who participate. According to the table, the total number of students who participate is

45 + 55 = 100

. Out of these

100

students,

45

are in the 11th grade. The probability is therefore

\frac{45}{100}

, which can be written as

\frac{9}{20}

or

0.45

.

Step-by-Step Solution

1

Determine the size of the restricted sample space by finding the total number of students who participate in extracurricular activities.

100

The probability is conditional on selecting a student who participates in extracurricular activities, so only the 'Participates' column is considered.

2

Identify the number of students within this restricted group who are in the 11th grade.

45

We count the number of students who are both in the 11th grade and participate in extracurricular activities.

3

Calculate the probability by dividing the number of favorable outcomes by the size of the restricted sample space.

45/100 = 9/20 = 0.45

The probability is the ratio of 11th-grade participants to the total participants.

Key Concept

Conditional Probability from a Two-Way Table

Question 86Question →

A municipal parking garage tracked the charging times of $200$ electric vehicles over a one-week period. The vehicles were classified by type—Battery Electric Vehicles (BEVs) or Plug-in Hybrid Electric Vehicles (PHEVs)—and by the time of day they primarily charged: Daytime (6:00 AM to 6:00 PM) or Nighttime (6:00 PM to 6:00 AM). The results are summarized in the table below.

Vehicle Type	Daytime Charging	Nighttime Charging	Total
BEV	$45$	$75$	$120$
PHEV	$55$	$25$	$80$
Total	$100$	$100$	$200$

If one of these electric vehicles is selected at random, and it is found to have charged during the nighttime, what is the probability that the vehicle is a PHEV?

\frac{1}{8}

\frac{1}{4}

\frac{5}{16}

\frac{3}{8}

Show answer & explanation

Answer: $\frac{1}{4}$

Answer

\frac{1}{4}

The correct answer is the simplified fraction representing the number of nighttime-charging PHEVs (

25

) divided by the total number of nighttime-charging vehicles (

100

), which is

\frac{25}{100} = \frac{1}{4}

.

Step-by-Step Solution

1

Identify the total number of vehicles in the restricted sample space.

The total number of vehicles that charged during the nighttime is

100

.

The question specifies that the vehicle 'is found to have charged during the nighttime', which limits the denominator to the 'Nighttime Charging' column total.

2

Determine the number of favorable outcomes within this restricted sample space.

The number of PHEVs that charged during the nighttime is

25

.

We need to find the count in the intersection of the 'PHEV' row and the 'Nighttime Charging' column.

3

Calculate the conditional probability as a fraction.

The probability is

\frac{25}{100} = \frac{1}{4}

.

Dividing the number of favorable vehicles (

25

) by the conditional total (

100

) yields the final probability, which simplifies to

\frac{1}{4}

.

Key Concept

Conditional probability calculations from a two-way table involve restricting the sample space to a specific row or column total rather than the grand total.

Estimated Time:1m 30s

Question 87Question →

In a laboratory experiment, a wet soil sample has a mass of $800$ grams. During a drying process, the sample loses $20\%$ of its mass. After the drying process, a nutrient solution is added, which increases the sample's mass by $15\%$ of its dried mass. What is the final mass, in grams, of the soil sample after the solution is added?

544

655

736

760

Show answer & explanation

Answer: 736

Answer

736

To find the final mass of the soil sample, first calculate the mass after it loses

20\%

of its initial

800

grams:

800 \times (1 - 0.20) = 640

grams. Next, calculate the final mass after a

15\%

increase is applied to this intermediate mass:

640 \times (1 + 0.15) = 736

grams. This shows that the final mass of the sample is

736

grams.

Step-by-Step Solution

1

Calculate the mass of the soil sample after losing

20\%

of its initial mass of

800

grams.

Dried mass =

800 \times (1 - 0.20) = 640

grams.

This establishes the intermediate baseline mass before the nutrient solution is added.

2

Calculate the final mass of the soil sample after a

15\%

increase is applied to the dried mass of

640

grams.

Final mass =

640 \times (1 + 0.15) = 736

grams.

The

15\%

increase must be calculated using the dried mass of

640

grams as the new base.

Key Concept

Multi-step percent change and identifying the correct base value for sequential percentage increases or decreases.

Alternative Method

You can express the sequential percent changes as a single product:

800 \times 0.80 \times 1.15

. Multiplying these values directly yields

800 \times 0.92 = 736

grams.

Estimated Time:1m 15s

Question 88Question →

A botanist conducted an experiment to study the germination rates of seeds in two different soil mixtures, Mixture A and Mixture B. The results of the experiment are partially shown in the table below, where $x$ is a positive constant.

Soil Mixture	Germinated	Did Not Germinate
Mixture A	$x$	$12$
Mixture B	$30$	$x + 6$

Of the seeds in the experiment that did not germinate, the probability that the seed was planted in Mixture A is $\frac{1}{3}$ . Given that a seed selected at random from the experiment germinated, what is the probability that it was planted in Mixture A?

Show answer & explanation

Answer: 0.375

Answer

3/8 (or 0.375)

To find the probability that a germinated seed was planted in Mixture A, we must first solve for the variable

x

. The number of seeds in Mixture A that did not germinate is

12

, and the total number of seeds that did not germinate is

12 + (x + 6) = x + 18

. We are given that the probability a non-germinated seed was planted in Mixture A is

\frac{1}{3}

. Setting up the equation:

\frac{12}{x + 18} = \frac{1}{3}

yields

x + 18 = 36

, so

x = 18

. Substituting this value back into the table, the number of germinated seeds in Mixture A is

18

. The total number of germinated seeds is the sum of germinated seeds in Mixture A and Mixture B:

18 + 30 = 48

. Thus, the probability that a germinated seed was planted in Mixture A is

\frac{18}{48}

, which simplifies to

\frac{3}{8}

(or

0.375

).

Step-by-Step Solution

1

Set up the conditional probability equation for seeds that did not germinate to find

x

.

x = 18

From the table, the number of seeds in Mixture A that did not germinate is

12

, and the number of seeds in Mixture B that did not germinate is

x + 6

. The total number of seeds that did not germinate is

12 + (x + 6) = x + 18

. The probability that a seed that did not germinate was from Mixture A is

\frac{12}{x + 18} = \frac{1}{3}

. Solving this equation gives

36 = x + 18

, which simplifies to

x = 18

.

2

Find the number of germinated seeds in Mixture A and the total number of germinated seeds.

Mixture A germinated =

18

, Total germinated =

48

Substituting

x = 18

, the number of germinated seeds in Mixture A is

18

. The number of germinated seeds in Mixture B is given as

30

. The total number of germinated seeds is

18 + 30 = 48

.

3

Calculate the conditional probability that a germinated seed was planted in Mixture A.

\frac{3}{8}

or

0.375

The probability is the number of germinated seeds in Mixture A divided by the total number of germinated seeds, which is

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

, or

0.375

as a decimal.

Key Concept

Conditional Probability in Two-Way Tables

Question 89Question →

A researcher at a university wants to estimate the average number of hours that students at the university spend studying per week. The researcher randomly selects $200$ students from a database of all students enrolled at the university to participate in a survey. Which of the following is the largest population to which the results of the survey can be most reasonably generalized?

All students enrolled at the university

All college students in the nation

Only the 200 students who were selected for the survey

All students who study more than the average number of hours

Show answer & explanation

Answer: All students enrolled at the university

Answer

All students enrolled at the university

The correct option is the one stating that the results can be generalized to all students enrolled at the university. In an observational study or survey, results can be generalized to the population from which the sample was randomly selected. Here, the database used for selection included all students enrolled at the university, making them the target population.

Step-by-Step Solution

1

Identify the population from which the sample was drawn.

The sample of

200

students was randomly selected from a database containing all students enrolled at the university.

Generalization of survey results requires that the sample be randomly selected from the target population.

2

Determine the scope of the generalization based on the random sampling design.

The results can be generalized to the entire population of students enrolled at the university, but not to broader populations (like all college students in the nation) or restricted biased subgroups.

Random selection ensures that the sample is representative of the population from which it was drawn.

Key Concept

Generalizing results from a random sample to the parent population

Question 90Question →

A dataset consists of 10 positive integers: $4, 6, 8, 8, 10, 12, 12, 14, 16$ , and $x$ . If the median of the dataset is equal to the mean of the dataset, and $x > 16$ , what is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

20

The correct answer is 20. When the 10 positive integers are sorted in ascending order, the condition

x > 16

ensures that

x

is the largest value and occupies the final position:

4, 6, 8, 8, 10, 12, 12, 14, 16, x

. The median of this 10-value dataset is the average of the 5th and 6th values, which are 10 and 12. Thus, the median is

\frac{10 + 12}{2} = 11

. The mean of the dataset is the sum of the 10 values divided by 10. The sum of the 9 known values is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

, so the mean is

\frac{90 + x}{10}

. Setting the mean equal to the median gives the equation

\frac{90 + x}{10} = 11

. Multiplying both sides by 10 gives

90 + x = 110

, and subtracting 90 yields

x = 20

. This value is consistent with the condition

x > 16

.

Step-by-Step Solution

1

Sort the dataset including the variable

x

.

The sorted dataset of 10 integers is

4, 6, 8, 8, 10, 12, 12, 14, 16, x

.

Since

x > 16

, it must be the largest value in the dataset and will be positioned at the end of the sorted list.

2

Find the median of the sorted dataset.

The median is 11.

For an even number of values (10), the median is the average of the two middle values (the 5th and 6th values), which are 10 and 12:

(10 + 12) / 2 = 11

.

3

Find the sum of the known values and write an expression for the mean.

The mean is

\frac{90 + x}{10}

.

The sum of the 9 known integers is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

. Adding

x

gives a total sum of

90 + x

, and dividing by the total count of 10 gives the mean.

4

Equate the mean to the median and solve for

x

.

x = 20

Setting the mean expression equal to the median yields

\frac{90 + x}{10} = 11

. Multiplying by 10 gives

90 + x = 110

, which solves to

x = 20

.

Key Concept

Calculating and equating the mean and median of a dataset containing a variable.

Question 91Question →

A fitness center surveyed a random sample of its members about their primary workout time and their membership type. The results of the survey are partially shown in the table below.

Primary Workout Time	Basic Membership	Premium Membership	Total
Morning	$60$	$a$
Afternoon	$b$	$20$	$60$
Evening	$60$	$140$
Total	$180$	$300$

Given that a randomly selected member from this survey has a Basic membership, what is the probability that this member's primary workout time is in the evening?

\frac{4}{15}

\frac{4}{9}

\frac{4}{7}

\frac{2}{9}

Show answer & explanation

Answer: $\frac{4}{9}$

Answer

\frac{4}{9}

To find the probability that a randomly selected member's primary workout time is in the evening given that they have a Basic membership, we restrict the denominator to the total number of Basic members, which is

180

. The number of Basic members who work out in the afternoon is

60 - 20 = 40

. Subtracting the morning (

60

) and afternoon (

40

) Basic members from the total Basic members (

180

) yields

180 - 60 - 40 = 80

Basic members who work out in the evening. The probability is

\frac{80}{180}

, which simplifies to

\frac{4}{9}

.

Step-by-Step Solution

1

Find the total number of members with a Premium membership.

The total number of Premium members is

120

.

The grand total of surveyed members is

300

, and the total number of Basic members is

180

. Subtracting the Basic members from the grand total gives the Premium members:

300 - 180 = 120

.

2

Find the value of

b

, which is the number of Basic members whose primary workout time is in the afternoon.

b = 40

The total number of members who work out in the afternoon is

60

, and

20

of them have a Premium membership. Subtracting the Premium members from the afternoon total gives the Basic members:

60 - 20 = 40

.

3

Find the number of Basic members whose primary workout time is in the evening.

There are

80

Basic members who work out in the evening.

The total number of Basic members is

180

. Subtracting the Basic members who work out in the morning (

60

) and afternoon (

40

) from the total number of Basic members gives the remaining Basic members who work out in the evening:

180 - 60 - 40 = 80

.

4

Calculate the conditional probability that a randomly selected member works out in the evening, given they have a Basic membership.

The probability is

\frac{4}{9}

.

The condition restricts the sample space to the

180

members with a Basic membership. Of these

180

members,

80

work out in the evening. The probability is the ratio of these two values:

\frac{80}{180} = \frac{4}{9}

.

Key Concept

Conditional Probability in Two-Way Tables

Alternative Method

Instead of subtracting from the column total to find the missing Basic Evening value, one could first find all Premium values: the Premium Total is

300 - 180 = 120

. Then, the Premium Morning value is

120 - 20 - 60 = 40

. This allows calculation of the Morning Total (

60 + 40 = 100

). The Evening Total is given as

140

, and the Premium Evening value is

60

, which directly gives the Basic Evening value as

140 - 60 = 80

. Finally, the probability is

\frac{80}{180} = \frac{4}{9}

.

Estimated Time:2m 0s

Question 92Question →

At the beginning of 2015, City A and City B each had a daily water consumption of $1,200,000$ gallons. The daily water consumption of City A decreased linearly by $24,000$ gallons each year, and the daily water consumption of City B decreased exponentially by a constant percentage each year. At the beginning of 2025, the daily water consumption of City A was equal to the daily water consumption of City B. If these trends continue, what will be the daily water consumption of City B, in gallons, at the beginning of 2035?

Show answer & explanation

Answer: 768000

Answer

768,000

To find the daily water consumption of City B at the beginning of 2035, we first determine the consumption of both cities at the beginning of 2025 (

t = 10

years after the beginning of 2015). For City A, which decreases linearly, the consumption is

1,200,000 - 10 \times 24,000 = 960,000

gallons. Since City B's consumption is equal to City A's at this time and decreases exponentially, we set up the equation

1,200,000 \cdot b^{10} = 960,000

, where

b

is the annual decay factor. This gives

b^{10} = 0.8

. The consumption of City B at the beginning of 2035 (

t = 20

) is given by

1,200,000 \cdot b^{20}

. Using exponent rules, we can rewrite

b^{20}

as

(b^{10})^2

. Substituting

b^{10} = 0.8

gives

1,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000

.

Step-by-Step Solution

1

Calculate the consumption of City A at the beginning of 2025.

960,000

gallons

City A decreases linearly by a constant

24,000

gallons per year for

10

years starting from

1,200,000

gallons.

2

Set up the exponential model for City B at the beginning of 2025.

b^{10} = 0.8

City B's consumption decreases exponentially, so it is modeled by

W_B(t) = 1,200,000 \cdot b^t

. Since it equals City A's consumption at

t = 10

,

1,200,000 \cdot b^{10} = 960,000

.

3

Determine the consumption of City B at the beginning of 2035.

768,000

gallons

At the beginning of 2035 (

t = 20

), City B's consumption is

1,200,000 \cdot b^{20}

. Since

b^{20} = (b^{10})^2

, we substitute

0.8

for

b^{10}

to get

1,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000

.

Key Concept

Distinguishing between linear and exponential models and applying exponent properties to solve exponential growth/decay problems.

Alternative Method

Alternatively, you can solve for the annual decay factor

b

directly using a calculator. Since

b^{10} = 0.8

, we take the 10th root of both sides to get

b = 0.8^{0.1} \approx 0.97793

. The water consumption at

t = 20

is then calculated as

1,200,000 \cdot (0.97793)^{20} \approx 768,000

gallons. Recognizing the algebraic shortcut

(b^{10})^2 = b^{20}

avoids decimal approximations and is much faster.

Estimated Time:2m 30s

Question 93Question →

A technology company surveyed a random sample of $128$ of its employees to determine their primary operating system and their job role. The table below summarizes the results of the survey.

Job Role	macOS	Windows	Total
Developer	$30$	$50$	$80$
Designer	$30$	$18$	$48$
Total	$60$	$68$	$128$

If an employee from the survey whose job role is developer is selected at random, what is the probability that the employee's primary operating system is Windows?

Show answer & explanation

Answer: 0.625

Answer

5/8 (or 0.625)

The question asks for the probability that a randomly selected employee uses Windows, given that their job role is developer. This restricts the sample space to only the developers. From the table, there are

80

total developers, which serves as the denominator. Within this group of developers,

50

use Windows, which serves as the numerator. The probability is

\frac{50}{80}

, which simplifies to

\frac{5}{8}

or

0.625

. Both

5/8

and

.625

are correct formats that fit the standard SAT grid-in requirements.

Step-by-Step Solution

1

Identify the subgroup defined by the condition.

The total number of developers is

80

.

The probability is conditional on the employee being a developer, which restricts the denominator of the probability to the total number of developers.

2

Identify the favorable outcomes within the subgroup.

There are

50

developers whose primary operating system is Windows.

This value represents the numerator for our conditional probability calculation.

3

Calculate the conditional probability.

The probability is

\frac{50}{80} = \frac{5}{8}

(or

0.625

).

Dividing the favorable outcomes by the total outcomes in the restricted sample space gives the correct conditional probability.

Key Concept

Conditional Probability in Two-Way Tables

Question 94Question →

A technology company monitored the annual carbon emissions of its data centers. In 2025, the data centers' carbon emissions decreased by $12\%$ compared to the emissions in 2024. In 2026, the emissions were $387.2$ metric tons, which represented a $10\%$ increase compared to the emissions in 2025. What were the data centers' carbon emissions, in metric tons, in 2024?

380

394

396

400

Show answer & explanation

Answer: 400

Answer

400

The correct answer is 400. To find the carbon emissions in 2024, we can set up the relationship using the variable

E

for the 2024 emissions. A

12\%

decrease in 2025 means the emissions were

0.88E

. A

10\%

increase in 2026 means the emissions were

1.10 \times 0.88E = 0.968E

. Since the 2026 emissions were

387.2

metric tons, we solve

0.968E = 387.2

to get

E = 400

.

Step-by-Step Solution

1

Set up an equation for the 2025 emissions in terms of the 2024 emissions (

E

).

2025\text{ emissions} = E \times (1 - 0.12) = 0.88E

The emissions in 2025 decreased by

12\%

compared to the emissions in 2024.

2

Set up an equation for the 2026 emissions in terms of the 2025 emissions.

2026\text{ emissions} = 2025\text{ emissions} \times (1 + 0.10) = 1.10 \times 0.88E = 0.968E

The emissions in 2026 represented a

10\%

increase compared to the emissions in 2025.

3

Equate the expression for 2026 emissions to the given value and solve for

E

.

0.968E = 387.2 \implies E = \frac{387.2}{0.968} = 400

The 2026 emissions are given as

387.2

metric tons.

Key Concept

Solving multi-step percent change problems by working backward using the correct base values.

Alternative Method

Instead of setting up an algebraic equation, we can work backward step-by-step. First, find the 2025 emissions by reversing the

10\%

increase from 2025 to 2026:

\frac{387.2}{1.10} = 352

metric tons. Next, find the 2024 emissions by reversing the

12\%

decrease from 2024 to 2025:

\frac{352}{0.88} = 400

metric tons.

Estimated Time:1m 15s

Question 95Question →

Two startup companies, Company X and Company Y, begin operations at the same time. At the start of operations ( $m = 0$ ), both companies have a monthly revenue of $R$ dollars.

* The monthly revenue of Company X increases by a constant amount of $0.12R$ dollars every $3$ months.
* The monthly revenue of Company Y increases exponentially, growing by a constant percent of $6\%$ every $2$ months.

Let $X(m)$ and $Y(m)$ represent the monthly revenue of Company X and Company Y, respectively, after $m$ months of operation. Which of the following functions represents the ratio $\frac{Y(m)}{X(m)}$ for any positive integer $m$ that is a multiple of $6$ ?

\frac{1 + 0.03m}{1 + 0.04m}

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.12m}

\frac{(1.06)^m}{1 + 0.04m}

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

Show answer & explanation

Answer: $\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}$

Answer

The expression

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

The correct answer is the expression

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

. This is found by representing the linear growth of Company X as

X(m) = R(1 + 0.04m)

because its revenue increases by a constant amount of

0.04R

per month (since

0.12R

every 3 months is

\frac{0.12R}{3} = 0.04R

). The exponential growth of Company Y is represented as

Y(m) = R(1.06)^{\frac{m}{2}}

because its revenue increases by

6\%

every 2 months, which yields a growth factor of

1.06

applied

\frac{m}{2}

times over

m

months. The ratio

\frac{Y(m)}{X(m)}

is then

\frac{R(1.06)^{\frac{m}{2}}}{R(1 + 0.04m)}

, which simplifies to the correct expression after canceling the common factor

R

.

Step-by-Step Solution

1

Determine the expression for Company X's monthly revenue,

X(m)

.

X(m) = R(1 + 0.04m)

Since Company X's revenue increases by a constant amount of

0.12R

every 3 months, it grows linearly. The rate of increase per month is

\frac{0.12R}{3} = 0.04R

dollars. Thus, after

m

months, the revenue is

X(m) = R + 0.04R \cdot m = R(1 + 0.04m)

.

2

Determine the expression for Company Y's monthly revenue,

Y(m)

.

Y(m) = R(1.06)^{\frac{m}{2}}

Since Company Y's revenue grows by a constant percent of

6\%

every 2 months, it grows exponentially. The growth factor for each 2-month period is

1 + 0.06 = 1.06

. In

m

months, the number of 2-month compounding periods is

\frac{m}{2}

. Therefore, the revenue is

Y(m) = R(1.06)^{\frac{m}{2}}

.

3

Find the ratio of

Y(m)

to

X(m)

.

\frac{Y(m)}{X(m)} = \frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

Divide the expression for

Y(m)

by the expression for

X(m)

:

\frac{Y(m)}{X(m)} = \frac{R(1.06)^{\frac{m}{2}}}{R(1 + 0.04m)}

. The constant initial revenue factor

R

cancels out, simplifying to the final ratio.

Key Concept

Linear and Exponential Growth

Estimated Time:2m 30s

Question 96Question →

A quality control analyst records the battery life, in hours, of 11 distinct prototype laptop batteries. The shortest battery life recorded is 10 hours, the median battery life is 50 hours, and the range of the battery lives is 74 hours. If each battery life is a whole number of hours, what is the maximum possible mean battery life, in hours, of the 11 prototypes?

50

60

61

65

Show answer & explanation

Answer: 60

Answer

The maximum possible mean battery life of the 11 prototypes is 60 hours.

The maximum possible mean is achieved by maximizing the sum of the 11 distinct integers under the given constraints. By setting the elements below the median to the largest possible distinct integers less than 50 (46, 47, 48, 49) and the elements above the median to the largest possible distinct integers less than the maximum value of 84 (80, 81, 82, 83), we find the maximum sum to be 660. Dividing this sum by the 11 elements yields a maximum mean of 60.

Step-by-Step Solution

1

Define the variable terms representing the sorted battery lives.

Let the 11 battery lives in sorted order be

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10} < x_{11}

.

Establishing the order of elements helps apply the median and distinct integer constraints.

2

Identify the values of the minimum, maximum, and median elements.

The minimum value is

x_1 = 10

. The median of 11 elements is the 6th element, so

x_6 = 50

. The range is 74, which means the maximum value is

x_{11} = 10 + 74 = 84

.

These fixed values act as boundaries for the remaining elements.

3

Maximize the remaining elements under the distinct integer constraint.

To maximize the mean, we must maximize the sum

S = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10} + x_{11}

. The elements below the median (

x_2, x_3, x_4, x_5

) must be distinct integers strictly less than 50, so their maximum values are 46, 47, 48, and 49. The elements above the median (

x_7, x_8, x_9, x_{10}

) must be distinct integers strictly less than 84, so their maximum values are 80, 81, 82, and 83.

Maximizing the individual elements maximizes the total sum, which in turn maximizes the mean.

4

Calculate the maximum sum and the resulting maximum mean.

The maximum sum is

10 + 46 + 47 + 48 + 49 + 50 + 80 + 81 + 82 + 83 + 84 = 660

. The maximum possible mean is

\frac{660}{11} = 60

.

Dividing the maximum possible sum by the number of elements gives the maximum possible mean.

Key Concept

Maximizing the mean of a dataset with constraints on distinct values, median, and range.

Question 97Question →

A 3D printer uses a plastic filament at a constant rate of $2.5\text{ grams}$ per minute of printing time. The filament has a density of $1.25\text{ grams per cubic centimeter}$ . If the printer runs continuously for $4\text{ hours}$ , what is the total volume, in cubic centimeters, of the filament used?

8

480

600

750

Show answer & explanation

Answer: 480

Answer

480

The correct answer is 480. First, convert the printing time from hours to minutes: 4 hours multiplied by 60 minutes per hour equals 240 minutes. Next, calculate the total mass of the filament used by multiplying the printing time by the usage rate: 240 minutes multiplied by 2.5 grams per minute equals 600 grams. Finally, convert the mass to volume by dividing the mass by the density: 600 grams divided by 1.25 grams per cubic centimeter equals 480 cubic centimeters.

Step-by-Step Solution

1

Convert the printing time from hours to minutes.

240\text{ minutes}

Since the filament consumption rate is given in grams per minute, the total time must be in minutes. Multiplying

4\text{ hours}

by

60\text{ minutes per hour}

yields

240\text{ minutes}

.

2

Calculate the total mass of the filament used.

600\text{ grams}

Multiplying the total printing time of

240\text{ minutes}

by the consumption rate of

2.5\text{ grams per minute}

gives a total mass of

600\text{ grams}

.

3

Calculate the total volume of the filament used.

480\text{ cubic centimeters}

Divide the total mass of

600\text{ grams}

by the density of

1.25\text{ grams per cubic centimeter}

to find the volume:

\frac{600}{1.25} = 480\text{ cubic centimeters}

.

Key Concept

Multi-unit dimensional analysis involving rates, time, and density.

Question 98Question →

A forestry service worker randomly selected $80$ oak trees in a state park containing $2,000$ oak trees. The worker found that $12$ of the selected oak trees showed signs of a leaf disease. Based on this sample, what is the estimated number of oak trees in the entire state park that show signs of the leaf disease?

Show answer & explanation

Answer: 300

Answer

300

Because the sample of

80

oak trees was selected at random, it can be assumed to be representative of the population of

2,000

oak trees in the state park. The proportion of diseased trees in the sample is

\frac{12}{80} = 0.15

. Multiplying this proportion by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

.

Step-by-Step Solution

1

Calculate the sample proportion of diseased trees.

0.15

Dividing the

12

diseased trees by the sample size of

80

trees gives the proportion of diseased trees in the sample:

\frac{12}{80} = 0.15

.

2

Generalize the sample proportion to the entire population of oak trees in the park.

300

Multiplying the sample proportion of

0.15

by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

.

Key Concept

Generalizing results from a representative random sample to estimate a population parameter.

Question 99Question →

A food safety laboratory tested $200$ samples of organic and conventional produce to detect the presence of a specific agricultural residue. The table below summarizes the results.

Produce Type	Residue Detected	Residue Not Detected	Total
Organic	$15$	$65$	$80$
Conventional	$45$	$75$	$120$
Total	$60$	$140$	$200$

If one of the tested samples is selected at random, given that the sample had no residue detected, what is the probability that the selected sample is conventional produce?

\frac{3}{8}

\frac{5}{8}

\frac{15}{28}

\frac{13}{28}

Show answer & explanation

Answer: $\frac{15}{28}$

Answer

The correct probability is

\frac{15}{28}

.

The correct answer is

\frac{15}{28}

. To find the conditional probability that a randomly selected sample is conventional produce given that it has no residue detected, we restrict our focus to the column labeled 'Residue Not Detected'. The total number of samples in this column is

140

(

65

organic +

75

conventional). Out of these

140

samples,

75

are conventional produce. The probability is therefore

\frac{75}{140}

, which simplifies to

\frac{15}{28}

by dividing both the numerator and the denominator by

5

.

Step-by-Step Solution

1

Identify the total number of outcomes that satisfy the given condition.

The total number of samples with no residue detected is

65 + 75 = 140

.

Since the question asks for the probability 'given that the sample had no residue detected', the sample space is restricted to only the column 'Residue Not Detected'.

2

Identify the number of favorable outcomes within the restricted sample space.

Within the group of samples with no residue detected, the number of conventional produce samples is

75

.

We need to find the count where the sample is conventional produce under the condition that no residue was detected.

3

Calculate the conditional probability and simplify the fraction.

\text{Probability} = \frac{75}{140} = \frac{15}{28}

.

The probability is the ratio of favorable outcomes to the total outcomes in the restricted sample space. Dividing the numerator and denominator by their greatest common divisor,

5

, yields

\frac{15}{28}

.

Key Concept

Conditional Probability and Two-Way Tables

Estimated Time:1m 15s

Question 100Question →

An electric vehicle consumes $0.35$ kilowatt-hours ( $\text{kWh}$ ) of electricity per mile traveled. The vehicle travels at a constant speed of $48$ miles per hour. What is the vehicle's rate of electricity consumption, in $\text{kWh}$ per minute?

Show answer & explanation

Answer: 0.28

Answer

0.28

To find the rate of electricity consumption in kilowatt-hours (

\text{kWh}

) per minute, we must convert the given rate of

0.35\text{ kWh}

per mile into

\text{kWh}

per minute. First, convert the speed of

48\text{ miles per hour}

to miles per minute:

\frac{48\text{ miles}}{60\text{ minutes}} = 0.8\text{ miles per minute}

. Next, multiply the energy consumed per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

. Therefore, the vehicle's rate of electricity consumption is

0.28\text{ kWh}

per minute.

Step-by-Step Solution

1

Convert the vehicle's speed from miles per hour to miles per minute.

0.8

miles per minute

Since there are 60 minutes in 1 hour, dividing the speed in miles per hour by 60 gives the speed in miles per minute:

\frac{48\text{ miles}}{1\text{ hour}} \times \frac{1\text{ hour}}{60\text{ minutes}} = 0.8\text{ miles per minute}

.

2

Calculate the rate of electricity consumption in kWh per minute.

0.28

kWh per minute

Multiply the consumption rate per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

.

Key Concept

Unit conversion involving rates and compound units.

Alternative Method

First, find the total electricity consumed in one hour of driving by multiplying the consumption rate per mile by the miles traveled in one hour:

0.35\text{ kWh per mile} \times 48\text{ miles} = 16.8\text{ kWh}

. Then, convert this hourly rate to a minute rate by dividing by 60 minutes:

\frac{16.8\text{ kWh}}{60\text{ minutes}} = 0.28\text{ kWh per minute}

.

Estimated Time:1m 15s

Problem-Solving and Data Analysis

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