Problem-Solving and Data Analysis Alıştırma Soruları — SAT — Sayfa 9

Soru 161Soru →

A local utility company is evaluating two different methods, Method A and Method B, to reduce electricity distribution losses. The table below shows the modeled annual electricity losses, in megawatt-hours (MWh), for each method over the first two years ( $t = 0$ to $t = 2$ ), where $t$ is the number of years since the evaluation began.

Year ( $t$ )	Method A losses (MWh)	Method B losses (MWh)
0	8,000	8,000
1	7,400	7,200
2	6,800	6,480

One method is modeled by a linear function, and the other method is modeled by an exponential function. What is the difference, in MWh, between the modeled annual electricity losses for the two methods at $t = 3$ years?

368

600

732

1,800

Cevabı ve açıklamayı göster

Cevap: 368

Cevap

368

To find the difference between the modeled annual electricity losses at

t = 3

, we first identify the type of model for each method. For Method A, the losses decrease by a constant amount of

600\text{ MWh}

each year (

8,000 - 7,400 = 600

and

7,400 - 6,800 = 600

), which indicates a linear relationship. Thus, the model for Method A is

L_A(t) = 8,000 - 600t

. At

t = 3

, the losses for Method A are

L_A(3) = 8,000 - 600(3) = 6,200\text{ MWh}

. For Method B, the losses decrease by a constant percentage of

10\%

each year, meaning each year's value is

90\%

of the previous year's value (

\frac{7,200}{8,000} = 0.9

and (

\frac{6,480}{7,200} = 0.9

). This indicates an exponential relationship with a decay factor of

0.9

. Thus, the model for Method B is

L_B(t) = 8,000(0.9)^t

. At

t = 3

, the losses for Method B are

L_B(3) = 8,000(0.9)^3 = 8,000(0.729) = 5,832\text{ MWh}

. The difference between the modeled losses is

6,200 - 5,832 = 368\text{ MWh}

.

Adım Adım Çözüm

1

Identify the type of model for Method A by examining its rate of change.

Method A decreases by a constant value of

600\text{ MWh}

each year (

8,000 - 7,400 = 600

and

7,400 - 6,800 = 600

), which means it is a linear function.

Linear functions have a constant first difference (constant rate of change).

2

Identify the type of model for Method B by examining its ratios.

Method B decreases by a constant ratio of

0.9

each year (

\frac{7,200}{8,000} = 0.9

and (

\frac{6,480}{7,200} = 0.9

), which means it is an exponential function.

Exponential functions have a constant ratio between consecutive terms.

3

Formulate the equations for both models.

L_A(t) = 8,000 - 600t

and

L_B(t) = 8,000(0.9)^t

.

These equations represent the linear decrease and exponential decay starting from

8,000

.

4

Calculate the modeled losses for both methods at

t = 3

.

L_A(3) = 8,000 - 600(3) = 6,200\text{ MWh}

and

L_B(3) = 8,000(0.9)^3 = 8,000(0.729) = 5,832\text{ MWh}

.

To find the values at

t = 3

, substitute

3

for

t

in both equations.

5

Calculate the difference between the two values.

6,200 - 5,832 = 368\text{ MWh}

.

Subtract the smaller loss value from the larger loss value to find the difference.

Anahtar Kavram

Identifying and modeling linear growth/decay (constant change per unit time) versus exponential growth/decay (constant percent change per unit time) from data tables.

Soru 162Soru →

A food packaging company produces a trail mix containing almonds, cashews, and raisins. The ratio of the weight of almonds to the weight of cashews is $3:2$ , and the ratio of the weight of cashews to the weight of raisins is $5:4$ . If a batch of trail mix contains $360$ ounces of raisins, what is the total weight, in ounces, of the trail mix?

840

1,125

1,385

1,485

Cevabı ve açıklamayı göster

Cevap: $1,485$

Cevap

1,485

To find the total weight of the trail mix, we determine the weights of cashews and almonds using the given ratios and the weight of raisins. The weight of cashews is calculated as

\frac{5}{4} \times 360 = 450

ounces. Using the weight of cashews, the weight of almonds is calculated as

\frac{3}{2} \times 450 = 675

ounces. Summing the weights of all three components yields

675 + 450 + 360 = 1,485

ounces.

Adım Adım Çözüm

1

Use the given ratio of cashews to raisins to find the weight of the cashews.

The weight of the cashews is

450

ounces.

Since the ratio of the weight of cashews to raisins is

5:4

, we set up the proportion

\frac{\text{cashews}}{360} = \frac{5}{4}

. Solving for cashews gives

\text{cashews} = 360 \times \frac{5}{4} = 450

ounces.

2

Use the ratio of almonds to cashews to find the weight of the almonds.

The weight of the almonds is

675

ounces.

Since the ratio of the weight of almonds to cashews is

3:2

, we set up the proportion

\frac{\text{almonds}}{450} = \frac{3}{2}

. Solving for almonds gives

\text{almonds} = 450 \times \frac{3}{2} = 675

ounces.

3

Sum the weights of all three ingredients to find the total weight of the trail mix.

The total weight is

1,485

ounces.

The total weight is the sum of the weights of almonds, cashews, and raisins:

675 + 450 + 360 = 1,485

ounces.

Anahtar Kavram

Solving multi-step ratio problems by linking parts using a common element to find the total quantity.

Soru 163Soru →

At the beginning of 2015, Forest A had 4,000 trees and Forest B had 3,000 trees. The number of trees in Forest A increases by 150 trees each year. The number of trees in Forest B increases by 4% each year. If $A(t)$ and $B(t)$ represent the number of trees in Forest A and Forest B, respectively, $t$ years after the beginning of 2015, what is the value of $A(5) - B(5)$ , rounded to the nearest whole number?

Cevabı ve açıklamayı göster

Cevap: 1100

Cevap

The correct value of

A(5) - B(5)

is 1100.

The correct value is 1100. By defining Forest A's population with the linear function

A(t) = 4000 + 150t

and Forest B's population with the exponential function

B(t) = 3000(1.04)^t

, evaluating both functions at

t = 5

yields

A(5) = 4750

and

B(5) \approx 3650

. The difference is

4750 - 3650 = 1100

.

Adım Adım Çözüm

1

Formulate the growth model for Forest A.

A(t) = 4000 + 150t

Forest A grows linearly with a constant increase of 150 trees per year from an initial value of 4,000 trees.

2

Formulate the growth model for Forest B.

B(t) = 3000(1.04)^t

Forest B grows exponentially with a constant percent increase of 4% per year (growth factor of

1 + 0.04 = 1.04

) from an initial value of 3,000 trees.

3

Calculate the population of each forest after 5 years (

t = 5

).

A(5) = 4750

and

B(5) \approx 3650

Substitute

t = 5

into both models:

A(5) = 4000 + 150(5) = 4750

and

B(5) = 3000(1.04)^5 \approx 3649.96

.

4

Calculate the difference

A(5) - B(5)

and round to the nearest whole number.

1100

Subtract:

4750 - 3649.96 = 1100.04

. Rounding 1100.04 to the nearest whole number yields 1100.

Anahtar Kavram

Linear vs. Exponential Growth Models

Alternatif Yöntem

Instead of evaluating each function separately, you can construct the difference expression directly as

D(t) = (4000 + 150t) - 3000(1.04)^t

and substitute

t = 5

into

D(t)

to compute

D(5) = 4750 - 3000(1.04)^5 \approx 1100.04

, which rounds to 1100.

Tahmini Süre:1m 30s

Soru 164Soru →

An agricultural irrigation system uses two types of sprinklers, Type A and Type B. Water flows through $3$ Type A sprinklers at a combined rate of $15$ gallons per minute. Water flows through $5$ Type B sprinklers at a combined rate of $18$ gallons per minute. If a farmer runs $8$ Type A sprinklers and $10$ Type B sprinklers simultaneously, what is the total rate of water flow, in gallons per minute?

Cevabı ve açıklamayı göster

Cevap: 76

Cevap

The total rate of water flow is 76 gallons per minute.

To find the total water flow rate, determine the rate of a single sprinkler of each type first. For Type A, the rate is

15 \text{ gallons per minute} \div 3 = 5 \text{ gallons per minute}

. For Type B, the rate is

18 \text{ gallons per minute} \div 5 = 3.6 \text{ gallons per minute}

. Running

8

Type A sprinklers and

10

Type B sprinklers simultaneously yields a total rate of

(8 \times 5) + (10 \times 3.6) = 40 + 36 = 76

gallons per minute.

Adım Adım Çözüm

1

Find the rate of water flow for a single Type A sprinkler.

Each Type A sprinkler has a water flow rate of

5

gallons per minute.

Dividing the combined rate of

15

gallons per minute by the

3

sprinklers gives the individual rate.

2

Find the rate of water flow for a single Type B sprinkler.

Each Type B sprinkler has a water flow rate of

3.6

gallons per minute.

Dividing the combined rate of

18

gallons per minute by the

5

sprinklers gives the individual rate.

3

Calculate the total water flow rate for the combined group of active sprinklers.

The total combined flow rate is

76

gallons per minute.

Multiplying the individual rate of each type by the respective count of active sprinklers and summing the results:

(8 \times 5) + (10 \times 3.6) = 40 + 36 = 76

.

Anahtar Kavram

Combining rates of individual components to find a total rate.

Tahmini Süre:1m 30s

Soru 165Soru →

A marine biologist wants to estimate the number of blue crabs in a bay that are infected with a specific parasite. The biologist captures a random sample of $250$ blue crabs from the bay, finds that $15$ of the crabs are infected, and then releases them. Based on this sample, if there is a total population of $12,000$ blue crabs in the bay, what is the best estimate of the total number of blue crabs infected with the parasite?

Cevabı ve açıklamayı göster

Cevap: 720

Cevap

The best estimate of the total number of blue crabs infected with the parasite is 720.

The correct answer is found by establishing the sample proportion of infected crabs, which is 15 out of 250, or 6%. Multiplying this rate by the total estimated population of 12,000 crabs yields 720.

Adım Adım Çözüm

1

Calculate the proportion of infected crabs in the sample.

\frac{15}{250} = 0.06

To determine the rate of infection in the representative sample.

2

Multiply the sample proportion by the total population of blue crabs in the bay.

0.06 \times 12,000 = 720

To generalize the sample rate to the entire population of crabs in the bay.

Anahtar Kavram

Generalizing a sample proportion to estimate a population parameter.

Soru 166Soru →

A dermatologist at a medical clinic wants to estimate the proportion of residents in a city who experience seasonal dry skin. The dermatologist surveys a random sample of 150 patients who visited the clinic in December and finds that 90 of them (60%) reported experiencing seasonal dry skin. Which of the following is the most appropriate conclusion based on this study?

Approximately 60% of all residents in the city experience seasonal dry skin.

Approximately 60% of all residents in the city who visit any medical clinic in December experience seasonal dry skin.

No reliable conclusion can be drawn about the proportion of all residents in the city who experience seasonal dry skin because the sample is not representative of the city's population.

Exactly 60% of the residents who visit this specific clinic throughout the entire year experience seasonal dry skin.

Cevabı ve açıklamayı göster

Cevap: No reliable conclusion can be drawn about the proportion of all residents in the city who experience seasonal dry skin because the sample is not representative of the city's population.

Cevap

No reliable conclusion can be drawn about the proportion of all residents in the city who experience seasonal dry skin because the sample is not representative of the city's population.

The correct conclusion is that no reliable conclusion can be drawn about all residents of the city. For a sample's findings to be generalized to a larger population, the sample must be randomly selected from that population. In this study, the sample is composed of patients at a dermatology clinic, who are more likely to have skin concerns, and the study was done in December, when dry skin is seasonally more common. Therefore, the sample is not representative of the city's general population, and we cannot generalize the 60% statistic to the entire city.

Adım Adım Çözüm

1

Identify the population of interest and the sample group.

The population of interest is all residents of the city. The sample group is 150 patients visiting a dermatology clinic in December.

Understanding the relationship between the sample and the population is necessary to determine if generalization is valid.

2

Evaluate whether the sample is representative of the population of interest.

The sample is not representative because dermatology patients are more likely to have skin issues than the average resident, and December is a winter month when dry skin is more common.

A sample must be randomly selected and representative of the target population to allow for valid generalization.

3

Select the conclusion that reflects the limitation of the sample.

The only appropriate conclusion is that no reliable conclusion can be drawn about the general city population from this biased sample.

Since the sample is biased and non-representative, generalizing the results would lead to an incorrect conclusion about the population of all city residents.

Anahtar Kavram

Generalizability of study results based on sample selection and representativeness.

Soru 167Soru →

An educational organization wants to estimate the number of high school seniors in a school district who plan to major in a STEM field. A random sample of $320$ high school seniors in the district was surveyed, and $35\%$ of the surveyed students reported that they plan to major in a STEM field. The survey has an associated margin of error of $4\%$ . There are $4,500$ high school seniors in the school district. Based on these results, what is the maximum estimated number of high school seniors in the district who plan to major in a STEM field?

Cevabı ve açıklamayı göster

Cevap: 1755

Cevap

1755

The survey estimate for the proportion of students who plan to major in a STEM field is

35\%

with a margin of error of

4\%

. This means the true proportion is estimated to be between

35\% - 4\% = 31\%

and

35\% + 4\% = 39\%

. To find the maximum estimated number of students in the population of

4,500

, we use the maximum estimated proportion of

39\%

(or

0.39

). Multiplying

0.39

by

4,500

yields

1,755

.

Adım Adım Çözüm

1

Calculate the maximum estimated percentage of students by adding the margin of error to the sample percentage.

39\%

(or

0.39

)

The margin of error gives the range of values above and below the sample estimate that represents the true population proportion. The maximum value of this range is found by adding the margin of error.

2

Multiply the maximum estimated proportion by the total number of high school seniors in the district.

1,755

To generalize the sample results to the entire population of

4,500

seniors, multiply the maximum proportion by the total population size.

Anahtar Kavram

Estimating population parameters using sample proportions and margin of error

Soru 168Soru →

A juice mixture is made by combining orange concentrate and water in a ratio of $2$ to $5$ by volume. To make a large batch of this mixture, a worker uses a total of $35$ liters of water. If the worker wants to dilute the mixture so that the new ratio of orange concentrate to water is $1$ to $4$ by volume, how many additional liters of water must be added?

5

7

21

56

Cevabı ve açıklamayı göster

Cevap: 21

Cevap

21

The correct answer is 21. First, we find the volume of orange concentrate in the initial mixture. The ratio of orange concentrate to water is 2 to 5, and the volume of water is 35 liters. Setting up the proportion

\frac{2}{5} = \frac{\text{concentrate}}{35}

gives a concentrate volume of 14 liters. When the mixture is diluted, the amount of orange concentrate remains 14 liters, but the ratio of concentrate to water becomes 1 to 4. Setting up the new proportion

\frac{1}{4} = \frac{14}{\text{new water}}

gives a new water volume of 56 liters. The additional water required is the difference between the final and initial water volumes, which is

56 - 35 = 21

liters.

Adım Adım Çözüm

1

Determine the volume of orange concentrate in the initial mixture.

The initial ratio of orange concentrate to water is

2:5

. Since the mixture contains

35

liters of water, the volume of orange concentrate,

x

, can be found using the proportion

\frac{2}{5} = \frac{x}{35}

. Solving for

x

gives

x = \frac{2 \times 35}{5} = 14

liters.

The amount of orange concentrate must be determined because it remains constant during dilution.

2

Calculate the total volume of water needed for the diluted mixture.

In the diluted mixture, the ratio of orange concentrate to water is

1:4

. Since the amount of orange concentrate is still

14

liters, the new volume of water,

y

, can be found using the proportion

\frac{1}{4} = \frac{14}{y}

. Solving for

y

gives

y = 14 \times 4 = 56

liters.

The final volume of water is required to determine the additional amount of water that needs to be added.

3

Find the additional volume of water that must be added.

Subtract the initial volume of water from the final volume of water:

56 - 35 = 21

liters.

Subtracting the starting amount of water gives the volume of new water needed to achieve the target ratio.

Anahtar Kavram

Using proportions to solve multi-step mixture and dilution problems.

Soru 169Soru →

A consumer electronics reviewer is tracking the resale value of two different smartphone models. At the time of purchase ( $t = 0$ years), Smartphone A and Smartphone B each have a retail value of $1,000$ dollars. The value of Smartphone A decreases by $10\%$ of its value from the previous year each year. The value of Smartphone B decreases by a constant amount of $100$ dollars each year. What is the difference, in dollars, between the values of the two smartphones $3$ years after they were purchased?

0

20

429

29

Cevabı ve açıklamayı göster

Cevap: 29

Cevap

The difference between the values of the two smartphones is 29 dollars.

The correct answer is the option showing a difference of

29

dollars. To solve this, we model each smartphone's resale value over time. Smartphone A experiences exponential decay with a decay factor of

0.90

. Its value after

3

years is

1,000(0.90)^3 = 729

dollars. Smartphone B experiences linear decay with a constant rate of decrease of

100

dollars per year. Its value after

3

years is

1,000 - 100(3) = 700

dollars. The difference between these two values is

729 - 700 = 29

dollars.

Adım Adım Çözüm

1

Find the value of Smartphone A at

t = 3

years.

The resale value of Smartphone A is

1,000(1 - 0.10)^3 = 1,000(0.90)^3 = 1,000(0.729) = 729

dollars.

Smartphone A decreases by a constant percentage of

10\%

each year, which represents exponential decay.

2

Find the value of Smartphone B at

t = 3

years.

The resale value of Smartphone B is

1,000 - 100(3) = 1,000 - 300 = 700

dollars.

Smartphone B decreases by a constant absolute amount of

100

dollars each year, which represents linear decay.

3

Calculate the difference between the two values at

t = 3

years.

729 - 700 = 29

dollars.

Subtract the smaller value from the larger value to find the positive difference.

Anahtar Kavram

Linear and Exponential Growth and Decay

Soru 170Soru →

A network router transmits data from two sources, Server $A$ and Server $B$ , at a constant ratio of $5$ megabytes ( $\text{MB}$ ) from Server $A$ for every $3\text{ MB}$ from Server $B$ . In a single session, the router transmitted a combined total of $160\text{ MB}$ of data from both servers. During the next session, the router transmits data at the same ratio, but the amount of data transmitted from Server $A$ is $20\%$ greater than the amount of data transmitted from Server $A$ in the first session. How many megabytes of data are transmitted from Server $B$ in the second session?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

The amount of data transmitted from Server B in the second session is 72 megabytes.

To find the data transmitted from Server B in the second session, we first determine the individual data amounts from the first session. Given the ratio of data from Server A to Server B is

5:3

and the total is

160\text{ MB}

, we can express their data amounts as

5x

and

3x

. Solving

5x + 3x = 160

yields

x = 20

. Thus, Server A transmitted

5 \times 20 = 100\text{ MB}

and Server B transmitted

3 \times 20 = 60\text{ MB}

in the first session. In the second session, the data from Server A increased by

20\%

, yielding

100 \times 1.20 = 120\text{ MB}

. Since the transmission ratio between Server A and Server B remains constant at

5:3

, we can set up the proportion

\frac{120}{B} = \frac{5}{3}

. Cross-multiplying gives

5B = 360

, which solves to

B = 72

megabytes.

Adım Adım Çözüm

1

Set up an equation for the total data in the first session.

5x + 3x = 160 \implies 8x = 160 \implies x = 20

The ratio of data from Server A to Server B is

5:3

, meaning the amounts can be represented as

5x

and

3x

for some multiplier

x

.

2

Calculate the data transmitted from Server A in the first session.

A_1 = 5(20) = 100\text{ MB}

Multiply the ratio term for Server A by the scale factor

x

to find its initial data contribution.

3

Calculate the increased data from Server A in the second session.

A_2 = 100 \times (1 + 0.20) = 120\text{ MB}

Increase the first session's data from Server A by

20\%

.

4

Use the constant ratio to find the data from Server B in the second session.

B_2 = 72\text{ MB}

Set up the proportion

\frac{120}{B_2} = \frac{5}{3}

and solve for

B_2

by cross-multiplying.

Anahtar Kavram

Solving part-to-part and part-to-whole ratio problems under proportional changes

Soru 171Soru →

An agricultural researcher wants to evaluate the effectiveness of a new organic fertilizer. The researcher selects a random sample of $80$ tomato plants of a specific variety from a greenhouse containing $1,200$ tomato plants of this variety. The researcher randomly assigns $40$ of the selected plants to receive the organic fertilizer and the remaining $40$ plants to receive a standard chemical fertilizer. At the end of the study, the plants that received the organic fertilizer produced a significantly higher average yield of tomatoes than the plants that received the standard chemical fertilizer. Based on this information, which of the following is the most appropriate conclusion?

The organic fertilizer causes a higher average yield of tomatoes than the standard chemical fertilizer for all tomato plants of this variety.

The organic fertilizer causes a higher average yield of tomatoes than the standard chemical fertilizer for the

1,200

tomato plants of this variety in the greenhouse.

The organic fertilizer causes a higher average yield of tomatoes than the standard chemical fertilizer for all plants in the greenhouse.

There is an association between the organic fertilizer and a higher average yield of tomatoes, but it cannot be concluded that the fertilizer caused the higher yield for the plants in the greenhouse.

Cevabı ve açıklamayı göster

Cevap: The organic fertilizer causes a higher average yield of tomatoes than the standard chemical fertilizer for the $1,200$ tomato plants of this variety in the greenhouse.

Cevap

The organic fertilizer causes a higher average yield of tomatoes than the standard chemical fertilizer for the

1,200

tomato plants of this variety in the greenhouse.

The study design includes both random sampling from a defined population (the

1,200

tomato plants of this variety in the greenhouse) and random assignment of treatments (the fertilizers). Random assignment allows the researcher to conclude that the organic fertilizer caused the difference in yield, while random sampling allows this causal conclusion to be generalized to the entire population of

1,200

tomato plants.

Adım Adım Çözüm

1

Determine if a cause-and-effect relationship can be established.

A cause-and-effect relationship can be established because the treatments (organic fertilizer vs. standard chemical fertilizer) were randomly assigned to the selected plants.

Random assignment of treatments minimizes the influence of confounding variables, allowing the researcher to conclude that differences in the outcome are caused by the treatments.

2

Identify the population to which the causal conclusion can be generalized.

The causal conclusion can be generalized only to the

1,200

tomato plants of this specific variety in this greenhouse.

The sample of

80

plants was randomly selected from this specific population of

1,200

plants in the greenhouse. Generalizing to all tomato plants of this variety everywhere is not valid because the sample may not represent plants in other environments. Generalizing to other species of plants in the greenhouse is not valid because only one specific variety of tomato plant was studied.

Anahtar Kavram

Generalizability and causal inference in statistical studies.

Soru 172Soru →

An older printing press can produce $420$ brochures every $15$ minutes. A newer printing press can produce brochures at a rate that is $1.5$ times the rate of the older press. If both presses run simultaneously at their respective constant rates, how many minutes will it take them to produce a total of $2,100$ brochures?

35

50

30

75

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The correct answer is

30

minutes.

To find the time needed, first calculate the individual rates in brochures per minute. The older press prints

420

brochures in

15

minutes, which is

\frac{420}{15} = 28

brochures per minute. The newer press prints at

1.5

times this rate, which is

1.5 \times 28 = 42

brochures per minute. Working together, their combined rate is

28 + 42 = 70

brochures per minute. Dividing the total target of

2,100

brochures by the combined rate of

70

brochures per minute yields

\frac{2,100}{70} = 30

minutes. Thus, the correct option is

30

.

Adım Adım Çözüm

1

Calculate the rate of the older printing press in brochures per minute.

The older press prints at a rate of

\frac{420}{15} = 28

brochures per minute.

Finding the unit rate allows us to easily combine and compare the performance of both presses.

2

Calculate the rate of the newer printing press.

The newer press prints at a rate of

28 \times 1.5 = 42

brochures per minute.

The problem states the newer press operates at

1.5

times the rate of the older press.

3

Determine the combined rate of both presses working together.

The combined rate is

28 + 42 = 70

brochures per minute.

When both presses run simultaneously, their rates add up to form the total production rate.

4

Calculate the time required to produce

2,100

brochures.

The required time is

\frac{2,100}{70} = 30

minutes.

Dividing the total work by the combined rate gives the total time needed to complete the task.

Anahtar Kavram

Solving rate problems by finding individual unit rates and combining them to find the total time required for a joint task.

Soru 173Soru →

At a currency exchange booth, $4$ US dollars (USD) can be exchanged for $5$ local credits, and $6$ local credits can be exchanged for $15$ reward coupons. Based on these rates, what is the total number of reward coupons that can be obtained in exchange for $24$ USD?

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

75

To find the number of reward coupons obtained from

24

USD, we perform a two-step rate conversion. First, convert USD to local credits:

24\text{ USD} \times \frac{5\text{ local credits}}{4\text{ USD}} = 30\text{ local credits}

. Second, convert local credits to reward coupons:

30\text{ local credits} \times \frac{15\text{ coupons}}{6\text{ local credits}} = 75\text{ coupons}

. Thus, the final value is

75

.

Adım Adım Çözüm

1

Set up a proportion to find the number of local credits equivalent to

24

USD.

30

local credits

Using the exchange rate of

4

USD to

5

local credits, multiply the starting value of

24

USD by the conversion factor

\frac{5}{4}

.

2

Set up a proportion to find the number of coupons equivalent to

30

local credits.

75

coupons

Using the exchange rate of

6

local credits to

15

coupons, multiply the

30

local credits by the conversion factor

\frac{15}{6}

.

Anahtar Kavram

Multi-step rate conversion and unit analysis

Soru 174Soru →

In a science department, the ratio of the number of students enrolled in Chemistry to the number of students enrolled in Physics is $5$ to $3$ . The ratio of the number of students enrolled in Physics to the number of students enrolled in Biology is $2$ to $7$ . If no student is enrolled in more than one of these courses and there are $105$ students enrolled in Biology, what is the total number of students enrolled in these three courses?

135

153

185

225

Cevabı ve açıklamayı göster

Cevap: 185

Cevap

185

To find the total number of students, we must determine the enrollment in each course. The number of Biology students is given as

105

. The ratio of Physics to Biology students is

2

to

7

, which means the number of Physics students is

105 \times \frac{2}{7} = 30

. The ratio of Chemistry to Physics students is

5

to

3

, so the number of Chemistry students is

30 \times \frac{5}{3} = 50

. Summing these values gives the total enrollment:

50 + 30 + 105 = 185

.

Adım Adım Çözüm

1

Calculate the number of students enrolled in Physics

30 students

Since the ratio of Physics to Biology is

2

to

7

and there are

105

students in Biology, we set up the proportion

\frac{\text{Physics}}{105} = \frac{2}{7}

, which yields

\text{Physics} = 105 \times \frac{2}{7} = 30

.

2

Calculate the number of students enrolled in Chemistry

50 students

Since the ratio of Chemistry to Physics is

5

to

3

and there are

30

students in Physics, we set up the proportion

\frac{\text{Chemistry}}{30} = \frac{5}{3}

, which yields

\text{Chemistry} = 30 \times \frac{5}{3} = 50

.

3

Calculate the total number of students enrolled in the three courses

185 students

Add the students enrolled in all three courses:

50 \text{ (Chemistry)} + 30 \text{ (Physics)} + 105 \text{ (Biology)} = 185

.

Anahtar Kavram

Solving multi-step ratio problems by finding a common variable to link multiple ratios.

Soru 175Soru →

A solar energy storage system collects electricity from two solar panel arrays, Array $X$ and Array $Y$ . Array $X$ produces $3$ kilowatt-hours (kWh) of electricity for every $4$ hours of direct sunlight. Array $Y$ produces $5$ kWh of electricity for every $6$ hours of direct sunlight. If both arrays receive direct sunlight simultaneously, how many hours of direct sunlight are required for the two arrays to produce a combined total of $38$ kWh of electricity?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

The correct answer is

24

. First, find the rate of each array in kWh per hour: Array

X

has a rate of

\frac{3}{4}

kWh/hour and Array

Y

has a rate of

\frac{5}{6}

kWh/hour. Since the arrays operate simultaneously, add their individual rates to find the combined rate of

\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}

kWh/hour. To find the total hours

t

needed to produce

38

kWh, set up the equation

\frac{19}{12}t = 38

. Solving for

t

gives

t = 38 \times \frac{12}{19} = 24

hours.

Adım Adım Çözüm

1

Calculate the individual hourly rates of electricity production for Array

X

and Array

Y

.

Array

X

produces at a rate of

\frac{3}{4}

kWh per hour, and Array

Y

produces at a rate of

\frac{5}{6}

kWh per hour.

Converting the given production rates to unit rates (kWh per hour) allows them to be directly compared and combined.

2

Add the individual rates to find the combined rate of both arrays operating simultaneously.

Combined rate is

\frac{19}{12}

kWh per hour.

Since both arrays operate simultaneously, their rates of production add together. The common denominator for

4

and

6

is

12

, so

\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}

.

3

Determine the time required to produce a combined total of

38

kWh.

24

hours.

Divide the total target production of

38

kWh by the combined rate of

\frac{19}{12}

kWh per hour:

38 \div \frac{19}{12} = 38 \times \frac{12}{19} = 2 \times 12 = 24

.

Anahtar Kavram

Ratios, Rates, and Proportions

Tahmini Süre:1m 30s

Soru 176Soru →

A quality control inspector at an electronics factory found that in a large shipment of smartphone screens, the ratio of the number of damaged screens to the number of undamaged screens was $3$ to $22$ . If the shipment contained a total of $6,600$ smartphone screens, how many of the screens in the shipment were damaged?

762

792

900

5,808

Cevabı ve açıklamayı göster

Cevap: 792

Cevap

792

The correct value of

792

is obtained by first determining the total number of parts in the ratio, which is

3 + 22 = 25

. The fraction of damaged screens in the shipment is therefore

\frac{3}{25}

. Multiplying this fraction by the total number of screens in the shipment (

6,600

) yields

\frac{3}{25} \times 6,600 = 792

.

Adım Adım Çözüm

1

Determine the total number of parts in the ratio.

3 + 22 = 25

total parts

To find the fraction of the total shipment that is damaged, we must add the parts of the damaged and undamaged screens.

2

Set up the fraction of the shipment that is damaged.

\frac{3}{25}

Since there are

3

damaged screens for every

25

total screens, the proportion of damaged screens is

\frac{3}{25}

.

3

Multiply the fraction by the total number of screens in the shipment.

\frac{3}{25} \times 6,600 = 792

Multiplying the proportion of damaged screens by the total size of the shipment gives the total number of damaged screens.

Anahtar Kavram

Applying part-to-whole ratios to solve for a specific quantity in a given total.

Tahmini Süre:1m 15s

Soru 177Soru →

To create a specific type of mortar, a builder mixes cement and sand in a ratio of $2$ to $5$ by weight. To create a concrete mixture, the builder mixes this mortar with gravel in a ratio of $7$ to $3$ by weight. If the builder prepares $300$ kilograms of the concrete mixture, how many kilograms of sand are needed?

90

120

150

210

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

150

To find the weight of the sand needed, first determine the weight of the mortar in the concrete mixture. The ratio of mortar to gravel is

7

to

3

, meaning mortar represents

\frac{7}{10}

of the total

300

-kilogram concrete mixture. The weight of the mortar is

\frac{7}{10} \times 300 = 210

kilograms. Next, since mortar consists of cement and sand in a ratio of

2

to

5

, the sand represents

\frac{5}{7}

of the mortar's weight. Calculating

\frac{5}{7}

of

210

kilograms yields

150

kilograms. Therefore, the weight of the sand needed is

150

kilograms.

Adım Adım Çözüm

1

Find the total weight of the mortar in the concrete mixture.

210

kilograms

The concrete mixture has a ratio of mortar to gravel of

7

to

3

by weight. This means mortar makes up

\frac{7}{7+3} = \frac{7}{10}

of the total concrete weight. Multiplying this fraction by the total weight of

300

kilograms gives

\frac{7}{10} \times 300 = 210

kilograms of mortar.

2

Find the weight of the sand within the mortar.

150

kilograms

The mortar is composed of cement and sand in a ratio of

2

to

5

by weight. Thus, sand makes up

\frac{5}{2+5} = \frac{5}{7}

of the mortar's weight. Multiplying this fraction by the mortar's weight of

210

kilograms yields

\frac{5}{7} \times 210 = 150

kilograms of sand.

Anahtar Kavram

Solving multi-step ratio problems by determining part-to-whole relationships and compounding ratios.

Soru 178Soru →

In a hydroponic farming system, a nutrient solution is created by mixing a liquid fertilizer concentrate with water in a ratio of $3$ to $50$ by volume. To fill a reservoir, a technician needs to prepare a total of $318$ liters of this nutrient solution. How many liters of the liquid fertilizer concentrate are required to prepare the solution?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

To find the amount of liquid fertilizer concentrate needed, we first express the ratio in terms of the total mixture. The ratio of concentrate to water is

3

to

50

, meaning there are

3

parts of concentrate for every

50

parts of water, making a total of

3 + 50 = 53

parts. Since the total volume of the solution is

318

liters, each part represents

\frac{318}{53} = 6

liters. The amount of concentrate required is

3

parts, which corresponds to

3 \times 6 = 18

liters.

Adım Adım Çözüm

1

Calculate the total parts represented in the ratio of liquid fertilizer concentrate to water.

The total number of parts is

3 + 50 = 53

.

Since the ratio of concentrate to water is

3

to

50

, the entire nutrient solution is divided into

53

equal parts.

2

Find the volume of solution that corresponds to one part.

One part is equal to

\frac{318}{53} = 6

liters.

Dividing the total volume of the nutrient solution (

318

liters) by the total number of parts (

53

) gives the volume of a single part.

3

Calculate the total volume of liquid fertilizer concentrate needed.

The volume of concentrate required is

3 \times 6 = 18

liters.

Since the concentrate accounts for

3

parts of the total mixture, multiplying the volume of one part by

3

gives the total amount of concentrate needed.

Anahtar Kavram

Solving part-to-whole ratio problems by determining the value of a single unit or part from a given total quantity.

Alternatif Yöntem

Let the volume of the fertilizer concentrate be

3x

and the volume of water be

50x

. The total volume of the nutrient solution is

3x + 50x = 53x

. Given that the total volume is

318

liters, we can set up the equation

53x = 318

. Solving for

x

gives

x = 6

. The volume of the concentrate is

3x = 3(6) = 18

liters.

Tahmini Süre:1m 15s

Soru 179Soru →

A chef prepares a spice mix by combining paprika, cumin, and coriander in a ratio of $8 : 5 : 2$ by weight. If the chef wants to make a total of $60$ ounces of the spice mix, how many ounces of cumin are needed?

15

20

30

37.5

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The correct answer is 20 ounces.

The correct answer is 20 ounces. To find the amount of cumin needed, first sum the parts of the ratio to find the total parts of the mixture:

8 + 5 + 2 = 15

parts. Since cumin represents 5 of these 15 parts, the fraction of the spice mix that is cumin is

\frac{5}{15}

, which simplifies to

\frac{1}{3}

. Multiplying this fraction by the total desired weight of the spice mix gives

\frac{1}{3} \times 60 = 20

ounces.

Adım Adım Çözüm

1

Calculate the total number of parts in the ratio.

8 + 5 + 2 = 15

parts.

Determining the total parts is necessary to find what fraction of the whole mixture is represented by cumin.

2

Determine the fractional portion of the mixture that is cumin.

The fraction is

\frac{5}{15}

, which simplifies to

\frac{1}{3}

.

This fraction represents the ratio of the cumin parts to the total parts of the spice mix.

3

Multiply the cumin fraction by the total weight of the mix to find the weight of cumin.

\frac{1}{3} \times 60 = 20

ounces.

Multiplying the proportion of cumin by the total desired weight of the spice mix yields the required ounces of cumin.

Anahtar Kavram

Solving part-to-whole ratio problems by finding the total number of parts and applying the fraction to the overall amount.

Tahmini Süre:1m 15s

Problem-Solving and Data Analysis

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