Problem-Solving and Data Analysis Alıştırma Soruları — SAT — Sayfa 3

Soru 41Soru →

A technician at a chemistry lab prepares a solution by mixing $3$ milliliters of acid with every $7$ milliliters of distilled water. If the technician uses $35$ milliliters of distilled water to prepare the solution, how many milliliters of acid are needed?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The correct answer is 15.

To maintain the ratio of

3

milliliters of acid for every

7

milliliters of distilled water, the proportion

\frac{\text{acid}}{\text{water}} = \frac{3}{7} = \frac{x}{35}

is established. Multiplying both sides by

35

gives

x = 35 \times \frac{3}{7} = 15

milliliters of acid.

Adım Adım Çözüm

1

Set up the proportion relating acid to distilled water.

\frac{3}{7} = \frac{x}{35}

, where

x

is the required amount of acid in milliliters.

The ratio of acid to distilled water must remain constant.

2

Solve the proportion for

x

.

x = 15

Multiply both sides of the equation by 35 to isolate

x

.

Anahtar Kavram

Solving proportions to find an unknown quantity when the ratio is constant.

Alternatif Yöntem

Find the scale factor:

35

divided by

7

is

5

. Multiply the acid amount (

3

) by this scale factor of

5

to get

15

.

Tahmini Süre:45s

Soru 42Soru →

An agricultural drone sprays liquid fertilizer at a rate of $1.5$ fluid ounces per square yard. The drone travels at a constant speed of $4.5$ miles per hour, spraying a continuous path that is exactly $6$ feet wide. Which of the following is closest to the rate at which the drone sprays the fertilizer, in gallons per minute? (Given that $1\text{ mile} = 5,280\text{ feet}$ , $1\text{ yard} = 3\text{ feet}$ , and $1\text{ gallon} = 128\text{ fluid ounces}$ )

3.09

3.58

9.28

27.84

Cevabı ve açıklamayı göster

Cevap: 3.09

Cevap

The rate of

3.09

gallons per minute is closest to the drone's fertilizer spraying rate.

The correct answer is

3.09

because converting the speed of

4.5

miles per hour gives

396

feet per minute. Multiplying this by the width of

6

feet results in

2,376

square feet per minute. Dividing this by

9

converts the rate to

264

square yards per minute. Multiplying by the spray density of

1.5

fluid ounces per square yard gives

396

fluid ounces per minute. Finally, dividing by

128

fluid ounces per gallon yields approximately

3.09

gallons per minute.

Adım Adım Çözüm

1

Convert the speed of the drone from miles per hour to feet per minute.

4.5\text{ miles/hour} = 396\text{ feet/minute}

To find the rate of area covered per minute, the speed must first be expressed in feet per minute. Multiply

4.5

by

5,280

to convert to feet per hour, then divide by

60

to convert to feet per minute:

\frac{4.5 \times 5,280}{60} = 396

.

2

Calculate the area covered by the drone per minute in square feet.

396\text{ feet/minute} \times 6\text{ feet} = 2,376\text{ square feet/minute}

Multiplying the travel speed in feet per minute by the path width of

6

feet gives the area covered per minute in square feet.

3

Convert the area rate from square feet per minute to square yards per minute.

2,376\text{ square feet/minute} \div 9 = 264\text{ square yards/minute}

Since

1\text{ yard} = 3\text{ feet}

, a square yard is

3 \times 3 = 9

square feet. Therefore, divide the area in square feet by

9

to convert to square yards.

4

Determine the fertilizer spray rate in fluid ounces per minute.

264\text{ square yards/minute} \times 1.5\text{ fluid ounces/square yard} = 396\text{ fluid ounces/minute}

Multiplying the area covered per minute by the spray density yields the volume of fertilizer sprayed per minute.

5

Convert the spray rate from fluid ounces per minute to gallons per minute.

396\text{ fluid ounces/minute} \div 128 \approx 3.09\text{ gallons/minute}

Since

1\text{ gallon} = 128\text{ fluid ounces}

, divide the fluid ounces per minute by

128

to find the rate in gallons per minute.

Anahtar Kavram

Unit conversions involving compound rates and area scaling.

Tahmini Süre:3m 0s

Soru 43Soru →

At a certain car dealership, the number of hybrid vehicles sold in 2024 was $40\%$ of the number of gasoline vehicles sold in 2024. In 2025, the number of hybrid vehicles sold increased by $35\%$ compared to 2024, and the number of gasoline vehicles sold decreased by $10\%$ compared to 2024. If the dealership only sold hybrid and gasoline vehicles in both years, what percent of the total vehicles sold in 2025 were hybrid vehicles?

Cevabı ve açıklamayı göster

Cevap: 37.5

Cevap

37.5

To find the percentage of hybrid vehicles sold in 2025, we first express the number of vehicles sold in 2024 in terms of a single variable. Let the number of gasoline vehicles sold in 2024 be

g

. Since the hybrid sales were

40\%

of the gasoline sales, the number of hybrid vehicles sold in 2024 is

0.40g

. In 2025, hybrid sales increased by

35\%

, resulting in

0.40g \times 1.35 = 0.54g

hybrid vehicles sold. Gasoline sales decreased by

10\%

, resulting in

g \times 0.90 = 0.90g

gasoline vehicles sold. The total number of vehicles sold in 2025 is the sum of these two quantities:

0.54g + 0.90g = 1.44g

. The percentage of hybrid vehicles sold in 2025 is the ratio of hybrid sales to total sales:

\frac{0.54g}{1.44g} \times 100 = 37.5\%

.

Adım Adım Çözüm

1

Define variables for the number of vehicles sold in 2024.

Let the number of gasoline vehicles sold in 2024 be

g

. The number of hybrid vehicles sold in 2024 is

0.40g

.

Establishing initial values in terms of a single variable allows for comparison after percentage changes are applied.

2

Determine the number of hybrid and gasoline vehicles sold in 2025 after their respective percentage changes.

Hybrid vehicles sold in 2025:

0.40g \times (1 + 0.35) = 0.54g

. Gasoline vehicles sold in 2025:

g \times (1 - 0.10) = 0.90g

.

Applying the percent increase of

35\%

to the 2024 hybrid vehicles and the percent decrease of

10\%

to the 2024 gasoline vehicles.

3

Calculate the total number of vehicles sold in 2025.

Total vehicles sold in 2025:

0.54g + 0.90g = 1.44g

.

Summing the two types of vehicles sold in 2025 to find the new total.

4

Compute the percentage of hybrid vehicles out of the total vehicles sold in 2025.

\frac{0.54g}{1.44g} \times 100 = 37.5\%

.

Dividing the 2025 hybrid sales by the total 2025 sales and multiplying by 100 to convert to a percentage.

Anahtar Kavram

Calculating percent change and conditional percentages from initial relative ratios.

Soru 44Soru →

During a science experiment, a liquid is cooled at a constant rate of $0.5$ degrees Celsius per minute. What is the cooling rate of the liquid, in degrees Celsius per hour?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The cooling rate of the liquid is

30

degrees Celsius per hour.

To find the cooling rate in degrees Celsius per hour, we multiply the rate per minute by the conversion factor. There are

60

minutes in

1

hour. Multiplying

0.5

degrees Celsius per minute by

60

minutes per hour yields

30

degrees Celsius per hour.

Adım Adım Çözüm

1

Identify the initial cooling rate per minute.

0.5

degrees Celsius per minute

This is the rate given in the problem statement.

2

Convert minutes to hours by multiplying the rate by

60

.

30

degrees Celsius per hour

An hour contains

60

minutes, so the total temperature decrease in one hour will be

60

times the decrease in one minute.

Anahtar Kavram

Unit conversions involving rates

Soru 45Soru →

A local veterinarian clinic recorded the primary diets of 120 animals (cats and dogs). The results are summarized in the table below.

Animal	Dry food	Wet food	Total
Cats	35	25	60
Dogs	45	15	60
Total	80	40	120

If one of these animals is selected at random, what is the probability that the animal is a dog, given that the animal's primary diet is dry food?

\frac{9}{16}

\frac{3}{8}

\frac{3}{4}

\frac{5}{8}

Cevabı ve açıklamayı göster

Cevap: $\frac{9}{16}$

Cevap

The correct answer is the option representing

\frac{9}{16}

.

The correct answer is

\frac{9}{16}

. To find the probability that the selected animal is a dog given that its primary diet is dry food, we restrict our focus to the animals that eat dry food. According to the table, there are 80 animals in total whose primary diet is dry food. Among these 80 animals, 45 are dogs. Therefore, the probability is

\frac{45}{80}

, which simplifies to

\frac{9}{16}

.

Adım Adım Çözüm

1

Identify the given condition in the question.

The condition is that the selected animal's primary diet is dry food.

This restricts the sample space to only the animals in the 'Dry food' column.

2

Find the total number of animals that meet the given condition from the table.

The total number of animals whose primary diet is dry food is 80.

This value serves as the denominator for the conditional probability fraction.

3

Find the number of dogs within the conditional space identified in the previous step.

The number of dogs that eat dry food is 45.

This value serves as the numerator representing the favorable outcomes.

4

Calculate the conditional probability by dividing the favorable outcomes by the total outcomes in the restricted sample space.

\frac{45}{80} = \frac{9}{16}

Dividing the numerator by the denominator and simplifying by dividing both by 5 yields the final probability.

Anahtar Kavram

Conditional Probability from Two-Way Tables

Alternatif Yöntem

Instead of working directly with the counts in the table, you can write the conditional probability formula:

P(\text{Dog} \mid \text{Dry}) = \frac{P(\text{Dog} \cap \text{Dry})}{P(\text{Dry})} = \frac{45/120}{80/120} = \frac{45}{80} = \frac{9}{16}

.

Tahmini Süre:50s

Soru 46Soru →

A technology company has two regional offices, Office North and Office South. In 2024, the number of employees at Office North was $25\%$ greater than the number of employees at Office South. In 2025, the number of employees at Office North increased by $20\%$ , and the number of employees at Office South decreased by $10\%$ . If the total number of employees at both offices combined in 2025 was $432$ , what was the total number of employees at both offices combined in 2024?

360

405

225

450

Cevabı ve açıklamayı göster

Cevap: 405

Cevap

405

Let

S

represent the number of employees at Office South in 2024. Because Office North has

25\%

more employees than Office South in 2024, the count at Office North is

1.25S

. The combined total in 2024 is

S + 1.25S = 2.25S

. In 2025, Office North's staff increases by

20\%

, making it

1.20 \times 1.25S = 1.50S

. Office South's staff decreases by

10\%

, making it

0.90S

. The combined total in 2025 is

1.50S + 0.90S = 2.40S

. Given that the total in 2025 is

432

, we solve the equation

2.40S = 432

, which gives

S = 180

. Substituting

S = 180

into the expression for the 2024 total yields

2.25 \times 180 = 405

.

Adım Adım Çözüm

1

Define variables for the number of employees in 2024. Let

S

be the number of employees at Office South in 2024. Since Office North had

25\%

more employees, represent Office North as

1.25S

. Express the total number of employees in 2024 as

S + 1.25S = 2.25S

.

Total employees in 2024 is represented by

2.25S

.

To establish a baseline relationship between the employee counts at the two offices.

2

Express the number of employees in 2025 in terms of

S

by applying the percentage changes. Office North's employees increased by

20\%

, which is

1.20 \times 1.25S = 1.50S

. Office South's employees decreased by

10\%

, which is

0.90 \times S = 0.90S

. Find the total in 2025 by adding these values:

1.50S + 0.90S = 2.40S

.

Total employees in 2025 is represented by

2.40S

.

To determine the algebraic expression representing the combined count after the percent changes.

3

Set the algebraic expression for the 2025 total equal to the given value of

432

and solve for

S

.

2.40S = 432 \implies S = 180

To find the baseline number of employees at Office South in 2024.

4

Substitute

S = 180

back into the expression for the 2024 total (

2.25S

) to calculate the final answer.

2.25 \times 180 = 405

To calculate the combined total number of employees in 2024.

Anahtar Kavram

Percents and Percent Change

Tahmini Süre:1m 30s

Soru 47Soru →

To make a specific shade of green paint, a painter mixes yellow paint and blue paint in a ratio of $3$ to $5$ . If the painter wants to make a total of $40$ gallons of this green paint, how many gallons of yellow paint are needed?

12

15

24

25

Cevabı ve açıklamayı göster

Cevap: $15$

Cevap

The correct amount of yellow paint needed is

15

gallons.

The correct answer is

15

gallons. Since the ratio of yellow to blue paint is

3:5

, the total mixture consists of

3 + 5 = 8

equal parts. Yellow paint represents

3

of these

8

parts, or

\frac{3}{8}

of the total mixture. To find the amount of yellow paint needed for

40

gallons of green paint, multiply the fraction by the total volume:

\frac{3}{8} \times 40 = 15

gallons.

Adım Adım Çözüm

1

Determine the fraction of the total mixture that is yellow paint.

The ratio of yellow to blue paint is

3:5

, which gives a total of

3 + 5 = 8

parts. Therefore, yellow paint represents

\frac{3}{8}

of the total mixture.

To find the amount of one component in a given total volume, we must find the part-to-whole ratio of that component to the total mixture.

2

Multiply the fraction representing yellow paint by the total desired volume of green paint.

\frac{3}{8} \times 40 = 3 \times 5 = 15

gallons.

Multiplying the part-to-whole ratio by the total volume gives the specific volume of yellow paint needed.

Anahtar Kavram

Solving part-to-whole ratio problems by summing the ratio parts to find the total parts, then multiplying the corresponding fraction by the total quantity.

Soru 48Soru →

A deep space satellite transmits data to a ground station at a constant rate of $1.6$ megabits per second. The transmitted data consists of actual scientific data and protocol overhead. The protocol overhead accounts for $25\%$ of the total transmitted bits, meaning the actual scientific data constitutes only $75\%$ of the total transmitted bits. The ground station needs to receive $5$ scientific data files, each with a size of $1.08$ gigabytes. How many hours will it take to transmit all $5$ files? (Given that $1\\text{ byte} = 8\\text{ bits}$ , $1\\text{ megabit} = 10^6\\text{ bits}$ , and $1\\text{ gigabyte} = 10^9\\text{ bytes}$ .)

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

It will take 10 hours to transmit all 5 files.

The correct answer is 10. To find this, first calculate the total size of the scientific files: 5 files * 1.08 gigabytes per file = 5.4 gigabytes. Convert this to bytes: 5.4 * 10^9 bytes. Convert bytes to bits: 5.4 * 10^9 * 8 = 43.2 * 10^9 bits. Since protocol overhead is 25%, the scientific data is 75% of the total transmitted bits, so the total bits transmitted is 43.2 * 10^9 / 0.75 = 57.6 * 10^9 bits. The transmission rate is 1.6 megabits per second, or 1.6 * 10^6 bits per second. The transmission time in seconds is 57.6 * 10^9 / (1.6 * 10^6) = 36,000 seconds. Converting seconds to hours: 36,000 / 3,600 = 10 hours.

Adım Adım Çözüm

1

Calculate the total size of the scientific files in gigabytes.

5.4

gigabytes

To find the total amount of scientific data that must be received.

2

Convert the total scientific data size from gigabytes to bytes and then to bits.

43.2 \\times 10^9

bits

The transmission rate is given in megabits per second, so the data size must be converted to bits for unit consistency. Since

1\\text{ GB} = 10^9\\text{ bytes}

and

1\\text{ byte} = 8\\text{ bits}

, we have

5.4 \\times 10^9 \\times 8 = 43.2 \\times 10^9

bits.

3

Calculate the total number of bits transmitted, including protocol overhead.

57.6 \\times 10^9

bits

Since protocol overhead is

25\%

, the scientific data is only

75\%

of the total bits transmitted. Thus, we divide the scientific bits by

0.75

to find the total bits transmitted:

\\frac{43.2 \\times 10^9}{0.75} = 57.6 \\times 10^9

bits.

4

Calculate the transmission time in seconds.

36,000

seconds

Divide the total bits by the transmission rate in bits per second (

1.6\\text{ Mbps} = 1.6 \\times 10^6

bits per second):

\\frac{57.6 \\times 10^9}{1.6 \\times 10^6} = 36,000

seconds.

5

Convert the transmission time from seconds to hours.

10

hours

Since there are

3,600

seconds in one hour, divide the total seconds by

3,600

:

\\frac{36,000}{3,600} = 10

.

Anahtar Kavram

Multi-step dimensional analysis and compound rate conversions incorporating percentage overhead

Alternatif Yöntem

Instead of converting units step-by-step, dimensional analysis can be set up as a single product of conversion factors: 5 files * (1.08 GB / 1 file) * (10^9 bytes / 1 GB) * (8 bits / 1 byte) * (1 total bit / 0.75 scientific bits) * (1 second / 1.6 * 10^6 bits) * (1 hour / 3600 seconds) = 10 hours.

Tahmini Süre:3m 0s

Soru 49Soru →

A study by a wildlife biologist shows that the population of a certain species of songbird in a state park increases by $8\%$ of its previous year's population each year. The initial population of the songbirds in the park was $150$ . If $t$ represents the number of years since the study began, which of the following functions best models the population of songbirds, $P(t)$ , after $t$ years?

P(t) = 150 + 12t

P(t) = 162^t

P(t) = 150(1.08)^t

P(t) = 8 + 150t

Cevabı ve açıklamayı göster

Cevap: $P(t) = 150(1.08)^t$

Cevap

The correct function is

P(t) = 150(1.08)^t

.

The correct answer is the function showing the initial value of

150

multiplied by the growth factor

1.08

raised to the power of

t

. Since the population increases by a constant percentage (

8\%

) each year, the growth is exponential. The general form of an exponential growth model is

P(t) = P_0(1 + r)^t

, where

P_0 = 150

is the initial value and

r = 0.08

is the growth rate. Substituting these values gives

P(t) = 150(1.08)^t

.

Adım Adım Çözüm

1

Identify the initial value of the population.

The initial population of songbirds is

150

.

The initial value represents the population when

t = 0

, which corresponds to the coefficient

P_0

in the exponential growth model.

2

Determine the growth rate and calculate the growth factor.

The population increases by

8\%

each year, so the growth rate

r = 0.08

, and the growth factor is

1 + r = 1 + 0.08 = 1.08

.

An increase by a constant percentage each year represents exponential growth, where the growth factor is

1 + r

.

3

Formulate the exponential growth function.

P(t) = 150(1.08)^t

An exponential growth model is written in the form

P(t) = P_0(b)^t

, where

P_0

is the initial value and

b

is the growth factor.

Anahtar Kavram

Distinguishing between linear and exponential growth, and constructing exponential models from a percent growth rate.

Soru 50Soru →

The table below shows the daily high temperatures, in degrees Celsius, recorded in a city over a $9$ -day period.

Daily High Temperature (°C)	Number of Days
$18$	$3$
$19$	$1$
$20$	$2$
$21$	$3$

Based on the table, what was the median daily high temperature, in degrees Celsius, for the $9$ -day period?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The median daily high temperature for the 9-day period is

20

°C.

To find the median daily high temperature, we first sum the frequencies to find the total number of days:

3 + 1 + 2 + 3 = 9

. Because

9

is an odd number, the median is the middle value, which is the

5

th value when the data points are ordered from least to greatest. The first

3

values are

18

, the

4

th value is

19

, and the

5

th and

6

th values are

20

. Therefore, the

5

th value in the ordered list is

20

.

Adım Adım Çözüm

1

Determine the total number of observations (days) in the data set.

9

days

To find the position of the median value, we must first calculate the total number of data points by adding the frequencies:

3 + 1 + 2 + 3 = 9

.

2

Find the position of the median in the ordered data set.

5

th position

For an odd number of observations

n

, the median is located at the

\frac{n+1}{2}

position. With

n = 9

, this corresponds to the

\frac{9+1}{2} = 5

th value.

3

Identify the temperature at the

5

th position using the frequency table.

20

Listing the temperatures in ascending order gives:

18, 18, 18

(positions 1–3),

19

(position 4), and

20, 20

(positions 5–6). The value at the

5

th position is

20

.

Anahtar Kavram

Finding the median of a frequency distribution

Soru 51Soru →

A library administrator tracked a sample of $120$ books checked out last week, categorizing them by genre (fiction or nonfiction) and format (print or audiobook). The results are summarized in the table below.

Format	Fiction	Nonfiction
Print	$45$	$35$
Audiobook	$15$	$25$

Of the books in the sample that were checked out as audiobooks, what fraction were fiction?

Cevabı ve açıklamayı göster

Cevap: 0.375

Cevap

The fraction of audiobooks that were fiction is 3/8 (which can also be gridded as the decimal .375).

To find the fraction of audiobooks that were fiction, find the total number of audiobooks in the sample by adding the values in the audiobook row:

15 + 25 = 40

. Next, find the number of fiction audiobooks, which is

15

. The fraction of audiobooks that were fiction is the ratio of these two values:

\frac{15}{40}

, which simplifies to

\frac{3}{8}

(or

0.375

in decimal form).

Adım Adım Çözüm

1

Calculate the total number of audiobooks in the sample.

The total number of audiobooks is

15 + 25 = 40

.

The question asks for the fraction of books *that were checked out as audiobooks*, which means the denominator of our fraction must be the total number of audiobooks.

2

Identify the number of fiction audiobooks in the sample.

There are

15

fiction audiobooks.

The question asks what fraction of the audiobooks were *fiction*, so the numerator must be the number of books that are both audiobooks and fiction.

3

Write and simplify the fraction representing the conditional probability.

\frac{15}{40} = \frac{3}{8}

Divide the target number of books (15) by the total number of audiobooks (40). Simplify by dividing the numerator and the denominator by their greatest common divisor, which is 5.

Anahtar Kavram

Conditional Probability from a Two-Way Table

Tahmini Süre:1m 15s

Soru 52Soru →

A digital music playlist contains only rock songs and pop songs. At the beginning of the month, $60\%$ of the songs in the playlist were rock songs. During the month, the number of rock songs in the playlist increased by $10\%$ , and the number of pop songs in the playlist increased by $35\%$ . If there were $120$ rock songs in the playlist at the beginning of the month, what percent of the songs in the playlist at the end of the month were pop songs?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The percent of the songs in the playlist at the end of the month that were pop songs is 45.

To find the final percentage of pop songs, we first determine the initial number of songs. Since

120

rock songs represent

60\%

of the playlist, the initial total is

120 / 0.60 = 200

songs. This means there are

200 - 120 = 80

pop songs initially. During the month, the rock songs increase by

10\%

, yielding

120 \times 1.10 = 132

songs. The pop songs increase by

35\%

, yielding

80 \times 1.35 = 108

songs. The final total number of songs is

132 + 108 = 240

. The final percentage of pop songs is

\frac{108}{240} \times 100 = 45\%

. Thus, the correct value to grid in is 45.

Adım Adım Çözüm

1

Determine the initial total number of songs and initial pop songs.

The initial total number of songs is 200, and the initial number of pop songs is 80.

Since 120 rock songs represent 60% of the playlist, the initial total is 120 / 0.60 = 200. The remaining songs are pop songs: 200 - 120 = 80.

2

Calculate the updated number of rock songs and pop songs after the percentage increases.

The final rock song count is 132, and the final pop song count is 108.

The number of rock songs increased by 10%, which is 120 * 1.10 = 132. The number of pop songs increased by 35%, which is 80 * 1.35 = 108.

3

Calculate the final total number of songs and the final percentage of pop songs.

The final total is 240 songs, and the final pop song percentage is 45%.

The final total is 132 + 108 = 240. The final pop song percentage is (108 / 240) * 100 = 45%.

Anahtar Kavram

Calculating absolute quantities from initial percentages, applying percentage increases to separate groups, and finding a new conditional percentage based on the final total.

Soru 53Soru →

An industrial printing press uses ink at a constant rate of $0.05$ milliliters per square centimeter of printed paper. The press prints a continuous roll of paper that is $40$ centimeters wide at a constant speed of $2$ meters per second. At this rate, how many liters of ink does the press use during $1$ hour of continuous printing?

Cevabı ve açıklamayı göster

Cevap: 1440

Cevap

1440

The correct answer is 1440. First, convert the speed of the paper from meters per second to centimeters per second:

2 \text{ m/s} \times 100 \text{ cm/m} = 200 \text{ cm/s}

. Next, find the rate at which the paper surface area is printed:

40 \text{ cm} \times 200 \text{ cm/s} = 8,000 \text{ cm}^2/\text{s}

. Multiply this by the ink usage rate to find milliliters of ink per second:

8,000 \text{ cm}^2/\text{s} \times 0.05 \text{ mL/cm}^2 = 400 \text{ mL/s}

. Since there are

3,600

seconds in

1

hour, the press uses

400 \text{ mL/s} \times 3,600 \text{ s} = 1,440,000 \text{ mL}

of ink per hour. Finally, convert milliliters to liters:

1,440,000 \text{ mL} \div 1,000 \text{ mL/L} = 1,440

liters.

Adım Adım Çözüm

1

Convert the speed of the paper from meters per second to centimeters per second.

200 \text{ cm/s}

To align the speed unit with the paper width unit (

40 \text{ cm}

).

2

Calculate the surface area of paper printed per second.

8,000 \text{ cm}^2/\text{s}

Multiply the width of the paper (

40 \text{ cm}

) by the converted speed (

200 \text{ cm/s}

).

3

Calculate the volume of ink used per second in milliliters.

400 \text{ mL/s}

Multiply the area printed per second (

8,000 \text{ cm}^2/\text{s}

) by the ink consumption rate (

0.05 \text{ mL/cm}^2

).

4

Convert the ink volume rate from per second to per hour.

1,440,000 \text{ mL/h}

Multiply the rate per second by

3,600

, since there are

3,600

seconds in

1

hour.

5

Convert the final volume from milliliters to liters.

1440

Divide the total milliliters (

1,440,000

) by

1,000

, since

1

liter is equal to

1,000

milliliters.

Anahtar Kavram

Multi-step dimensional analysis and compound unit conversion (length, area, volume, and time).

Soru 54Soru →

A printer prints $24$ pages in $3$ minutes. At this constant rate, how many pages can the printer print in $10$ minutes?

1.25

72

80

240

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80

The correct answer is 80. The printer's rate of printing is determined by dividing 24 pages by 3 minutes, which is 8 pages per minute. Multiplying this rate of 8 pages per minute by 10 minutes gives a total of 80 pages printed.

Adım Adım Çözüm

1

Calculate the unit rate of pages printed per minute.

24 \div 3 = 8

pages per minute

Dividing the total pages by the total minutes gives the constant rate of printing.

2

Multiply the unit rate by the new amount of time.

8 \times 10 = 80

pages

Multiplying the pages per minute by 10 minutes gives the total number of pages printed in that time.

Anahtar Kavram

Calculating and applying a unit rate to solve a proportion.

Soru 55Soru →

A town's population was $12,000$ in the year $2010$ . Since $2010$ , the population of the town has increased by a constant amount of $350$ people each year. Based on this information, what was the population of the town in the year $2018$ ?

Cevabı ve açıklamayı göster

Cevap: 14800

Cevap

14,800

The population starts at

12,000

in

2010

and grows linearly by

350

people per year. The number of years from

2010

to

2018

is

2018 - 2010 = 8

years. The total increase in population is

350 \times 8 = 2,800

people. Adding this to the initial population of

12,000

gives a total population of

12,000 + 2,800 = 14,800

in the year

2018

.

Adım Adım Çözüm

1

Calculate the number of years that passed between

2010

and

2018

.

8

years

To determine the total population growth, we need the elapsed time in years.

2

Calculate the total population growth over this

8

-year period.

2,800

people

Since the growth is linear, the total growth is the constant annual rate of change multiplied by the number of years.

3

Add the total population growth to the initial population from

2010

.

14,800

The population in

2018

is the initial population plus the total growth over the

8

-year period.

Anahtar Kavram

Linear growth represents a quantity that increases by a constant amount per unit of time.

Soru 56Soru →

An agricultural researcher categorized a sample of $150$ tomato plants by their watering schedule (daily or weekly) and their fruit yield (high yield or average yield). The results are summarized in the table below.

Watering Schedule	High Yield	Average Yield	Total
Daily	$62$	$18$	$80$
Weekly	$28$	$42$	$70$
Total	$90$	$60$	$150$

Given that a tomato plant selected at random from the sample produced a high yield of fruit, what is the probability that the plant had a weekly watering schedule?

\frac{14}{75}

\frac{2}{5}

\frac{14}{45}

\frac{31}{45}

Cevabı ve açıklamayı göster

Cevap: $\frac{14}{45}$

Cevap

The correct answer is fourteen-forty-fifths (14/45). Given that the plant has a high yield, the sample space is restricted to the 90 plants in the High Yield column. Out of these 90 plants, 28 were on a weekly watering schedule, resulting in a probability of 28/90, which simplifies to 14/45.

To find the conditional probability that a tomato plant had a weekly watering schedule given that it produced a high yield of fruit, restrict the denominator to the total number of high-yield plants, which is 90. The numerator is the number of plants that meet both conditions (weekly watering and high yield), which is 28. The probability is 28/90, which simplifies to 14/45.

Adım Adım Çözüm

1

Identify the condition specified in the question stem to determine the subset of data to consider.

The condition is 'given that a tomato plant ... produced a high yield of fruit'. Thus, we restrict our focus to the 'High Yield' column.

This establishes the denominator for the conditional probability calculation.

2

Find the total number of plants in the 'High Yield' column (denominator) and the number of those plants that had a weekly watering schedule (numerator).

Total high-yield plants = 90. High-yield plants with a weekly watering schedule = 28.

These represent the total possible outcomes and the favorable outcomes within the restricted sample space.

3

Calculate and simplify the fraction representing the conditional probability.

Probability = \frac{28}{90} = \frac{14}{45}.

Expressing the final ratio in its simplest fractional form to match the standard exam options.

Anahtar Kavram

Two-Way Tables and Conditional Probability

Tahmini Süre:1m 30s

Soru 57Soru →

A study of $180$ college students categorized them by their major field of study (STEM or Humanities) and whether they participate in undergraduate research. The table below shows the partial results of the study, where $a$ , $b$ , $c$ , and $d$ represent the number of students in each category.

Major	Participates in Research	Does Not Participate in Research	Total
STEM	$a$	$b$	$110$
Humanities	$c$	$d$	$70$
Total	$80$	$100$	$180$

Given that a student selected at random is a Humanities major, the probability that the student participates in undergraduate research is $\frac{3}{7}$ . If a student who participates in undergraduate research is selected at random, what is the probability that the student is a STEM major?

\frac{5}{8}

\frac{5}{18}

\frac{5}{11}

\frac{1}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{8}$

Cevap

The correct probability is

\frac{5}{8}

.

To find the probability that a student is a STEM major given they participate in undergraduate research, we first determine the missing values in the table. The total number of Humanities majors is

70

. Since the probability that a Humanities major participates in research is

\frac{3}{7}

, the number of Humanities majors in research is

c = \frac{3}{7} \times 70 = 30

. Using the research column total, the number of STEM majors in research is

a = 80 - c = 80 - 30 = 50

. The probability that a student is a STEM major given that they participate in research is the ratio of STEM research participants to total research participants:

\frac{a}{a+c} = \frac{50}{80} = \frac{5}{8}

.

Adım Adım Çözüm

1

Find the number of Humanities majors who participate in research (

c

).

c = 30

The table shows there are

70

Humanities majors in total. We are given that the probability of a Humanities major participating in research is

\frac{3}{7}

. Therefore,

\frac{c}{70} = \frac{3}{7}

, which simplifies to

c = 30

.

2

Find the number of STEM majors who participate in research (

a

).

a = 50

The total number of students who participate in research is

80

. Since

a + c = 80

and

c = 30

, we have

a + 30 = 80

, which means

a = 50

.

3

Calculate the conditional probability that a student who participates in research is a STEM major.

\frac{5}{8}

We want to find the probability of selecting a STEM major given that the student participates in research. This is the ratio of STEM majors who participate in research (

a = 50

) to the total number of students who participate in research (

80

). The probability is

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

.

Anahtar Kavram

Conditional Probability in Two-Way Tables

Soru 58Soru →

The value of a financial asset, $V(t)$ , in dollars, is modeled as a function of time $t$ , in years, for $0 \le t \le 4$ . The growth of the asset's value is described by two different models over this period:

- For $0 \le t \le 2$ , the value of the asset increases by a constant percentage of $r\%$ per year.
- For $2 \le t \le 4$ , the value of the asset increases by a constant amount of $d$ dollars per year.

At $t = 0$ , the value of the asset is $100$ dollars, and at $t = 1$ , the value of the asset is $120$ dollars. If the average rate of change of the asset's value from $t = 0$ to $t = 4$ is $25$ dollars per year, what is the value of the asset, in dollars, at $t = 3$ ?

169

170

172

175

Cevabı ve açıklamayı göster

Cevap: 172

Cevap

172

To find the value of the asset at

t = 3

, we first determine its value at

t = 2

using the exponential growth model. Since the value increases by a constant percentage from

t=0

to

t=2

, the value is modeled by

V(t) = V(0) \times (1 + \frac{r}{100})^t

. Given

V(0) = 100

and

V(1) = 120

, the annual growth factor is

1.2

. Thus,

V(2) = 100 \times (1.2)^2 = 144

dollars.

Next, the linear growth from

t = 2

to

t = 4

is modeled by

V(t) = V(2) + d(t - 2) = 144 + d(t - 2)

, which gives

V(4) = 144 + 2d

dollars. The average rate of change of the asset's value from

t = 0

to

t = 4

is

25

dollars per year, so we set up the equation:

\frac{V(4) - V(0)}{4} = 25 \implies \frac{(144 + 2d) - 100}{4} = 25

. Solving for

d

gives

d = 28

dollars per year. Finally, we evaluate the value at

t = 3

:

V(3) = 144 + 28(3 - 2) = 172

dollars.

Adım Adım Çözüm

1

Identify the growth model for

0 \le t \le 2

and write the corresponding equation using the initial values.

Since the value increases by a constant percentage each year, the growth is exponential:

V(t) = V(0) \times (1 + \frac{r}{100})^t

. Given

V(0) = 100

and

V(1) = 120

, we have

120 = 100(1 + \frac{r}{100})^1 \implies 1 + \frac{r}{100} = 1.2

.

Establishing the exponential model allows us to find the growth factor and calculate the value of the asset at the boundary of the two models (

t = 2

).

2

Calculate the value of the asset at

t = 2

using the exponential growth model.

V(2) = 100 \times (1.2)^2 = 144

dollars.

The value at

t = 2

serves as the initial value for the linear growth model in the second interval.

3

Write the growth model for the interval

2 \le t \le 4

.

The growth is linear, so

V(t) = V(2) + d(t - 2) = 144 + d(t - 2)

for

t \ge 2

. At

t = 4

, the value is

V(4) = 144 + 2d

dollars.

Expressing the value at

t = 4

in terms of the constant rate

d

allows us to use the average rate of change information to solve for

d

.

4

Set up the equation for the average rate of change over the entire interval from

t = 0

to

t = 4

and solve for

d

.

The average rate of change is given by

\frac{V(4) - V(0)}{4 - 0} = 25

. Substituting the values, we get

\frac{(144 + 2d) - 100}{4} = 25 \implies \frac{44 + 2d}{4} = 25 \implies 44 + 2d = 100 \implies 2d = 56 \implies d = 28

dollars per year.

The average rate of change relates the final value at

t=4

to the initial value at

t=0

, enabling us to determine the linear rate of change

d

.

5

Calculate the value of the asset at

t = 3

using the linear model.

V(3) = V(2) + d(3 - 2) = 144 + 28(1) = 172

dollars.

Substituting

t = 3

into the linear growth model yields the final requested value.

Anahtar Kavram

Linear and Exponential Growth

Alternatif Yöntem

Instead of setting up the linear equation from

t = 2

to

t = 4

, we can write

V(4)

directly as

V(0) + 4 \times 25 = 200

dollars using the definition of average rate of change. Since

V(2) = 144

dollars, the total change over the linear interval

[2, 4]

is

200 - 144 = 56

dollars. Because the growth is linear during this interval, the change over one year (from

t = 2

to

t = 3

) must be exactly half of the change over two years (from

t = 2

to

t = 4

). Thus, the increase from

t = 2

to

t = 3

is

56 / 2 = 28

dollars, yielding

V(3) = 144 + 28 = 172

dollars.

Tahmini Süre:3m 0s

Soru 59Soru →

A biologist measures the heights, in centimeters, of five seedlings in a laboratory. The heights are $12$ , $15$ , $8$ , $21$ , and $14$ . What is the range, in centimeters, of these heights?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

To find the range of a dataset, subtract the minimum value from the maximum value. In the given dataset of seedling heights (

12

,

15

,

8

,

21

, and

14

), the maximum height is

21

centimeters and the minimum height is

8

centimeters. Calculating the difference gives

21 - 8 = 13

centimeters.

Adım Adım Çözüm

1

Identify the maximum and minimum values in the dataset.

The maximum value is

21

and the minimum value is

8

.

The range is defined as the difference between the largest (maximum) and smallest (minimum) values in a data distribution.

2

Subtract the minimum value from the maximum value to calculate the range.

21 - 8 = 13

Subtracting the minimum value of

8

from the maximum value of

21

yields the range of the dataset.

Anahtar Kavram

The range of a dataset measures the spread of the data and is calculated by subtracting the minimum value from the maximum value.

Soru 60Soru →

The list below shows the number of packages delivered to a local business on each of $8$ business days:

1, 2, 2, 3, 5, 6, 7, 10

What is the median number of packages delivered on these days?

2

3

4

4.5

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The correct answer is the median value of

4

. The list contains

8

values already sorted in ascending order:

1, 2, 2, 3, 5, 6, 7, 10

. Because the number of values is even, the median is the average of the two middle values (the

4

th and

5

th values, which are

3

and

5

). The average is calculated as

\frac{3 + 5}{2} = 4

.

Adım Adım Çözüm

1

Identify the two middle terms of the ordered dataset.

The two middle values in the ordered list of 8 numbers are the 4th term, which is

3

, and the 5th term, which is

5

.

Since the dataset has an even number of values (

8

), the median is the average of the two middle terms.

2

Calculate the average of the two middle terms.

\frac{3 + 5}{2} = \frac{8}{2} = 4

Finding the mean of these two middle values yields the median of the entire dataset.

Anahtar Kavram

Calculating the median of a dataset with an even number of values

Tahmini Süre:45s

Problem-Solving and Data Analysis

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram