Tüm alıştırma soruları

Soru 3301Soru →

The word ARDUOUS most nearly means which of the following?

passionate

strenuous

effortless

dangerous

Cevabı ve açıklamayı göster

Cevap: strenuous

Cevap

strenuous

The word 'arduous' means demanding significant effort, labor, or endurance. The option 'strenuous' is the most accurate synonym because it specifically characterizes tasks requiring great exertion.

Adım Adım Çözüm

1

Define the target word

The adjective 'arduous' describes an activity, journey, or task that involves great labor, difficulty, or exhausting physical effort.

Isolating the precise definition allows for accurate identification of its closest direct synonym.

2

Evaluate the choices for semantic equivalence

'Strenuous' directly matches the definition of requiring or characterized by intense exertion and difficulty.

Matching core meanings confirms the correct synonym while distinguishing it from false friends and antonyms.

Anahtar Kavram

Direct Synonym Identification for Isolated Words

Tahmini Süre:30s

Soru 3302Soru →

A military logistics center is distributing its annual fuel reserve among different divisions:

- The Aviation division receives $\frac{3}{8}$ of the total reserve.
- The Ground Transport division receives $22.5\%$ of the total reserve.
- A Naval detachment receives a portion equivalent to the decimal $0.125$ of the total reserve.
- The remainder of the fuel is stored in the emergency depot.

Which of the following fractions correctly represents the ratio of the amount of fuel stored in the emergency depot to the amount of fuel allocated to the Aviation division?

\frac{11}{15}

\frac{11}{40}

\frac{7}{5}

\frac{31}{30}

Cevabı ve açıklamayı göster

Cevap: $\frac{11}{15}$

Cevap

The correct ratio is

\frac{11}{15}

.

To find the ratio, all given quantities must first be converted into a common format, such as decimals. The Aviation division's

\frac{3}{8}

is equivalent to

0.375

. The Ground Transport's

22.5\%

is equivalent to

0.225

. The Naval detachment is already given as

0.125

. Adding these distributed portions together gives

0.375 + 0.225 + 0.125 = 0.725

. Because the total fuel reserve represents

1

whole (

100\%

), the remainder stored in the emergency depot is

1 - 0.725 = 0.275

. The question asks for the ratio of the emergency depot (

0.275

) to the Aviation division (

0.375

). This creates the ratio

\frac{0.275}{0.375}

, which can be rewritten as

\frac{275}{375}

. Dividing both the numerator and the denominator by their greatest common divisor of

25

simplifies the fraction to

\frac{11}{15}

.

Adım Adım Çözüm

1

Convert all the distributed fuel allocations into a single format (decimals) to allow for easy addition.

Aviation =

0.375

; Ground =

0.225

; Naval =

0.125

.

Fractions, decimals, and percentages must be in the same format before they can be summed.

2

Sum the converted allocations to find the total distributed portion of the reserve.

0.375 + 0.225 + 0.125 = 0.725

This establishes what portion of the whole (

1.0

) has already been assigned.

3

Subtract the total distributed portion from the whole (

1.0

) to find the portion stored in the emergency depot.

1.0 - 0.725 = 0.275

The emergency depot contains whatever is left over from the original reserve.

4

Set up the requested ratio (Emergency Depot to Aviation Division) using their decimal values.

\frac{0.275}{0.375}

The stem asks for the ratio of the emergency depot to the Aviation division.

5

Simplify the decimal ratio into a proper fraction in its lowest terms.

\frac{275}{375} = \frac{11}{15}

Multiplying numerator and denominator by

1000

removes the decimals, and dividing both by their greatest common divisor (

25

) yields the simplest fraction.

Anahtar Kavram

To compare quantities given in mixed formats (fractions, decimals, and percentages), they must first be converted into a common format. After finding unknown quantities using a consistent format, their relative sizes can be expressed as a ratio and simplified into a final fraction.

Tahmini Süre:2m 0s

Soru 3303Soru →

Sarah's family goes out for dinner, and the total food bill is $60.00. They decide to leave a 15% tip for the server. How much money, in dollars, do they leave for the tip?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

The tip amount is found by calculating 15% of

60.00. By multiplying

60.00

by

0.15

, the result is

9.00$, which represents the tip amount in dollars.

Adım Adım Çözüm

1

Convert the tip percentage into a decimal.

15\% = 0.15

Percentages must be converted to decimals to perform multiplication.

2

Multiply the total bill by the decimal tip rate.

60.00 \times 0.15 = 9.00

The tip amount is a percentage of the total bill.

Anahtar Kavram

Calculating a percentage of a given monetary amount.

Soru 3304Soru →

A military engineering battalion is scheduled to pave a new airstrip requiring a total of $840$ tons of asphalt. During the first week of operations, the battalion completes $\frac{1}{3}$ of the total project. During the second week, bad weather slows progress, and they complete only $\frac{3}{4}$ of the remaining project. The asphalt needed to finish the rest of the airstrip will be delivered in dump trucks. If each dump truck has a capacity of $8\frac{3}{4}$ tons and must be fully loaded, how many dump trucks are required to deliver the exact remaining amount of asphalt?

48

64

16

96

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

The correct answer is found by tracking the remaining asphalt after each week. After week one,

\frac{2}{3}

of the

840

tons remain, which is

560

tons. In week two,

\frac{3}{4}

of that

560

tons is used, leaving

\frac{1}{4}

of

560

, which equals

140

tons. Finally, dividing the remaining

140

tons by the truck capacity of

8\frac{3}{4}

(or

\frac{35}{4}

) tons yields exactly

16

trucks.

Adım Adım Çözüm

1

Calculate the amount of asphalt remaining after the first week.

840 \times (1 - \frac{1}{3}) = 840 \times \frac{2}{3} = 560

tons

The first week completes a fraction of the total, so we subtract that fraction from the whole to find what is left.

2

Calculate the amount of asphalt remaining after the second week.

560 \times (1 - \frac{3}{4}) = 560 \times \frac{1}{4} = 140

tons

The second week completes a fraction of the remaining amount, not the total. We find the new remainder by multiplying the previous remainder by the uncompleted fraction.

3

Divide the final remaining amount of asphalt by the capacity of one dump truck.

140 \div 8\frac{3}{4} = 140 \div \frac{35}{4} = 140 \times \frac{4}{35} = 16

trucks

To find the number of trucks needed, divide the total remaining payload by the payload capacity per truck.

Anahtar Kavram

Multiplying fractions to find a part of a remainder, and dividing a whole number by a mixed number.

Soru 3305Soru →

A carpenter is purchasing a new circular saw. The regular retail price of the saw is $\$240.00$ . If the hardware store is offering a $25\%$ discount during a holiday sale, what is the final sale price of the circular saw in dollars?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

The final sale price is determined by subtracting the discount amount from the original price. The discount is calculated as

25\%

of

\$240.00

, which equals

\$60.00

. Subtracting

\$60.00

from

\$240.00

gives a final sale price of

\$180.00

.

Adım Adım Çözüm

1

Convert the percentage to a decimal to calculate the discount amount.

25\% = 0.25

Percentages must be converted to decimals or fractions to perform arithmetic calculations.

2

Multiply the regular price by the decimal to find the total discount.

\$240.00 \times 0.25 = \$60.00

The discount is

25\%

of the original price.

3

Subtract the discount from the original price to find the final sale price.

\$240.00 - \$60.00 = \$180.00

The final sale price is the original price reduced by the discount amount.

Anahtar Kavram

Calculating percentage discounts and subtracting them from the original price.

Soru 3306Soru →

Complete the statement below by calculating the missing numerical value. Enter only the number.

Aşağıdaki boşlukları doldurun

A military satellite uplink transmits

1.5

gigabytes of reconnaissance data every

40

seconds. Operating at this constant rate, it will take minutes to transmit a total of

18

gigabytes of data.

Cevabı ve açıklamayı göster

Cevap

8

By setting up a proportion, we find that transmitting

18

gigabytes takes

480

seconds. Dividing

480

seconds by

60

gives exactly

8

minutes.

Adım Adım Çözüm

1

Set up a direct proportion comparing data volume to time.

\frac{1.5}{40} = \frac{18}{x}

Because the transmission rate is constant, the ratio of gigabytes to seconds remains the same.

2

Solve for the unknown time

x

in seconds by cross-multiplying.

1.5x = 40 \times 18 \rightarrow 1.5x = 720 \rightarrow x = 480

seconds

Cross-multiplication isolates the variable, giving the total time required in seconds.

3

Convert the total seconds into minutes.

480 \div 60 = 8

minutes

The question specifically asks for the time in minutes, so the time in seconds must be divided by the number of seconds in a minute (

60

).

Anahtar Kavram

Direct proportions with unit conversions

Tahmini Süre:1m 30s

Soru 3307Soru →

A neutral isotope of zinc has a nucleus containing $36$ neutrons and has an overall mass number of $66$ . During a chemical reaction, this atom forms a $\text{Zn}^{2+}$ cation by losing $2$ electrons. Calculate the total number of electrons present in the $\text{Zn}^{2+}$ ion.

Cevabı ve açıklamayı göster

Cevap: 28

Cevap

28 electrons

The mass number (

66

) minus the number of neutrons (

36

) gives an atomic number of

30

, which corresponds to

30

protons. A neutral atom has

30

electrons, and losing

2

electrons to become

\text{Zn}^{2+}

leaves exactly

28

electrons.

Adım Adım Çözüm

1

Calculate the number of protons (atomic number) in the zinc atom

30 protons

The mass number equals the sum of protons and neutrons (

A = Z + N

), so

Z = 66 - 36 = 30

.

2

Calculate the remaining electrons in the

\text{Zn}^{2+}

cation

28 electrons

A neutral atom has an equal number of protons and electrons (30). A

+2

charge indicates the loss of 2 electrons (

30 - 2 = 28

).

Anahtar Kavram

Subatomic particle counts in isotopes and monatomic ions

Tahmini Süre:1m 0s

Soru 3308Soru →

A group of friends goes out for dinner. The total bill comes to $74.40, which includes a 6% sales tax and an 18% tip. If both the tax and the tip were calculated based on the original cost of the food, what was the cost of the food before the tax and tip were added?

Aşağıdaki boşlukları doldurun

The original cost of the food was $.

Cevabı ve açıklamayı göster

Cevap

The original cost of the food was $60.00.

By adding the 6% sales tax and the 18% tip, we find that a total of 24% was added to the original cost. This means the final bill of

74.40 is equal to 124% (or 1.24 times) the original food cost. Dividing

74.40 by 1.24 gives the correct original cost of $60.00.

Adım Adım Çözüm

1

Determine the combined percentage rate applied to the original cost.

Combined percentage = 6% tax + 18% tip = 24%.

Since both the tax and the tip are based on the original food cost, their percentages can be added together.

2

Express the total bill as a percentage of the original cost.

The total bill represents 100% of the original cost plus the 24% in additions, which equals 124% of the original cost.

The final amount includes the original base amount (100%) and the additional charges.

3

Set up an equation and solve for the original cost.

Let x be the original cost. 1.24x = $74.40. Dividing both sides by 1.24 gives x = 60.

Converting 124% to a decimal (1.24) and dividing the final amount by this multiplier isolates the original pre-tax cost.

Anahtar Kavram

Calculating an original base price by dividing the final total by a growth factor multiplier (1 + combined percentage rate).

Soru 3309Soru →

A stable isotope of silicon has $14$ protons and a total mass number of $30$ . How many neutrons are located inside its nucleus?

14

44

16

30

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

The nucleus contains 16 neutrons.

The number of neutrons in an atom is found by subtracting the atomic number (number of protons) from the mass number. For this silicon isotope with a mass number of

30

and

14

protons, the calculation is

30 - 14 = 16

neutrons.

Adım Adım Çözüm

1

Identify the given nuclear parameters.

Mass number (

A

) =

30

; number of protons / atomic number (

Z

) =

14

.

The mass number represents the sum of protons and neutrons in the nucleus, while the number of protons defines the atomic number.

2

Apply the relationship

N = A - Z

to determine the number of neutrons.

N = 30 - 14 = 16

.

Subtracting the number of protons from the total mass number yields the number of neutrons.

Anahtar Kavram

Calculating Subatomic Particles in an Atom

Tahmini Süre:45s

Soru 3310Soru →

A supply clerk is distributing a $20$ -pound bulk bag of coffee to different departments. If each department receives a standard ration of $1 \frac{1}{4}$ pounds of coffee, how many departments can be supplied from this single bag?

4

16

25

5

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

The correct answer is 16. To find how many equal portions can be made from a total amount, division must be used. Dividing the total

20

pounds by the ration size of

1 \frac{1}{4}

(which is equivalent to the improper fraction

\frac{5}{4}

) pounds yields

20 \times \frac{4}{5}

. Multiplying this out gives

\frac{80}{5}

, which simplifies exactly to

16

.

Adım Adım Çözüm

1

Identify the operation needed by recognizing that a total amount is being divided into equal smaller portions.

20 \div 1 \frac{1}{4}

Dividing the total weight by the weight per ration gives the total number of rations (departments) that can be supplied.

2

Convert the mixed number to an improper fraction for easier calculation.

1 \frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4}

Mathematical operations with fractions require mixed numbers to be in improper fraction form before multiplying or dividing.

3

Divide the whole number by the improper fraction by multiplying by its reciprocal.

20 \div \frac{5}{4} = 20 \times \frac{4}{5} = \frac{80}{5} = 16

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

Anahtar Kavram

Dividing a whole number by a mixed number in a real-world distribution context.

Soru 3311Soru →

A naval vessel's desalination plant produces fresh water at a rate directly proportional to its operational time. The plant produces $1,500$ liters of fresh water in $40$ minutes. The vessel's main storage tank has a total capacity of $24$ cubic meters and is currently $25\%$ full.

Assuming $1$ cubic meter equals $1,000$ liters, determine the correct numerical value to answer the question: How many continuous hours must the desalination plant operate to completely fill the remainder of the storage tank?

Aşağıdaki boşlukları doldurun

The plant must operate for hours.

Cevabı ve açıklamayı göster

Cevap

8

The correct answer is 8 hours. First, convert the tank capacity to liters:

24 \text{ m}^3 = 24,000 \text{ L}

. Since the tank is

25\%

full, it needs

75\%

of its capacity to be filled, which is

24,000 \times 0.75 = 18,000 \text{ L}

. Using the direct proportion

\frac{40}{1,500} = \frac{x}{18,000}

, we find

x = 480

minutes. Finally, converting minutes to hours by dividing by

60

gives exactly

8

hours.

Adım Adım Çözüm

1

Calculate the total tank capacity in liters.

24 \times 1,000 = 24,000

liters.

To harmonize the tank's volume units with the desalination plant's production rate, which is given in liters.

2

Determine the volume of water needed to fill the remainder of the tank.

24,000 \times (1 - 0.25) = 18,000

liters.

The tank is already

25\%

full, meaning only the remaining

75\%

needs to be produced.

3

Set up and solve a direct proportion to find the required time in minutes.

\frac{40 \text{ min}}{1,500 \text{ L}} = \frac{x \text{ min}}{18,000 \text{ L}} \Rightarrow x = \frac{40 \times 18,000}{1,500} = 480

minutes.

Because water production is directly proportional to operational time.

4

Convert the required time from minutes to hours.

480 \div 60 = 8

hours.

The question specifically asks for the operational time in continuous hours.

Anahtar Kavram

Solving multi-step direct proportions involving unit harmonization and percentages.

Soru 3312Soru →

Marcus is comparing two personal bank loan options to finance home repairs. Option X offers a $\$4,000$ loan at an annual simple interest rate of $6\%$ for $18$ months. Option Y offers a $\$5,000$ loan at an annual simple interest rate of $4\%$ for $9$ months. How much more interest will Marcus pay with Option X compared to Option Y?

\$210

\$252

\$2,100

\$2,520

Cevabı ve açıklamayı göster

Cevap: $\$210$

Cevap

\$210

The correct answer accurately converts both time periods into years and applies the simple interest formula (

I = P \times R \times T

) to find the interest for both loans before calculating their difference. Option X generates

\$360

in interest (

4000 \times 0.06 \times 1.5

), and Option Y generates

\$150

in interest (

5000 \times 0.04 \times 0.75

). The difference is

\$360 - \$150 = \$210

.

Adım Adım Çözüm

1

Convert the loan durations from months to years by dividing by 12.

Option X time:

18 \div 12 = 1.5

years. Option Y time:

9 \div 12 = 0.75

years.

The simple interest formula requires the time to be in the same units as the annual interest rate.

2

Calculate the total simple interest for Option X using

I = P \times R \times T

.

I_X = \$4,000 \times 0.06 \times 1.5 = \$360

This determines the exact cost of borrowing for the first option.

3

Calculate the total simple interest for Option Y using

I = P \times R \times T

.

I_Y = \$5,000 \times 0.04 \times 0.75 = \$150

This determines the exact cost of borrowing for the second option.

4

Subtract the interest of Option Y from the interest of Option X.

\$360 - \$150 = \$210

The question asks for the difference in interest paid between the two options.

Anahtar Kavram

Calculating and comparing simple interest across different principals, rates, and fractional year time periods.

Soru 3313Soru →

A naval shipyard receives a shipment of $945$ tons of specialized steel plating. The shipyard superintendent allocates $\frac{4}{9}$ of this shipment specifically for submarine hull repairs. If each submarine repair module requires exactly $17 \frac{1}{2}$ tons of steel plating, how many complete repair modules can be outfitted from the allocated steel?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

The correct answer is found by first calculating the portion of the steel available for repairs (

945 \times \frac{4}{9} = 420

tons), and then dividing that available amount by the requirement per module (

420 \div 17 \frac{1}{2} = 420 \times \frac{2}{35} = 24

complete modules).

Adım Adım Çözüm

1

Determine the total amount of steel allocated for submarine repairs.

420 tons

The problem states that only

\frac{4}{9}

of the

945

ton shipment is allocated for this specific purpose, requiring multiplication:

945 \times \frac{4}{9} = 105 \times 4 = 420

.

2

Convert the mixed number representing the steel required per module into an improper fraction.

\frac{35}{2}

tons

Converting

17 \frac{1}{2}

to an improper fraction (

\frac{17 \times 2 + 1}{2} = \frac{35}{2}

) is necessary to accurately divide the total allocated steel.

3

Calculate the number of complete modules by dividing the allocated steel by the required steel per module.

24 modules

Dividing the total allocated amount (

420

tons) by the amount needed per module (

\frac{35}{2}

tons) gives the number of modules:

420 \div \frac{35}{2} = 420 \times \frac{2}{35} = 12 \times 2 = 24

.

Anahtar Kavram

Applying fraction multiplication to find a portion of a whole, followed by fraction division to determine how many equal groups can be formed.

Soru 3314Soru →

Calculate the total consumption based on the given rate and time, and enter the correct numerical value in the blank.

Aşağıdaki boşlukları doldurun

A naval ship consumes

\frac{3}{4}

of a barrel of engine oil every hour of standard cruising. At this rate, the ship will consume barrels of engine oil during an

8

-hour patrol.

Cevabı ve açıklamayı göster

Cevap

6

The problem provides an hourly consumption rate of

\frac{3}{4}

of a barrel. To find the total amount used over an

8

-hour period, multiply the rate by the time:

\frac{3}{4} \times 8

. Treating

8

as

\frac{8}{1}

gives

\frac{3 \times 8}{4 \times 1} = \frac{24}{4}

. Dividing

24

by

4

results in exactly

6

barrels.

Adım Adım Çözüm

1

Identify the hourly consumption rate and the total duration.

Rate:

\frac{3}{4}

barrel per hour, Time:

8

hours

Extracting the given numbers sets up the necessary calculation for finding the total consumption.

2

Set up the multiplication expression to find the total amount.

\frac{3}{4} \times 8

Total consumption is calculated by multiplying the rate of consumption by the total time.

3

Multiply the fraction by the whole number and simplify.

\frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = 6

Converting the whole number to a fraction allows for direct multiplication of numerators and denominators to reach the final whole quantity.

Anahtar Kavram

Multiplying a fraction by a whole number in a real-world rate context.

Tahmini Süre:45s

Soru 3315Soru →

Wave interactions and behaviors produce distinct physical effects in sound and light. Match each wave phenomenon to its corresponding real-world observation.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Doppler effect matches the shift in pitch of a moving siren; Resonance matches the shattering of a glass at its natural frequency; Total internal reflection matches light confinement in fiber optics; Diffraction matches radio waves bending around hills and structures.

Each wave phenomenon maps directly to its underlying principle: relative motion alters perceived pitch via the Doppler effect; matching natural frequencies drives maximum vibrational amplitude via resonance; optical containment in fiber cores relies on total internal reflection; wave propagation around physical obstacles occurs through diffraction.

Adım Adım Çözüm

1

Analyze the effect of source motion on perceived sound pitch.

Compression and expansion of wavefronts due to motion produces the Doppler effect, matching the approaching and receding siren.

Perceived frequency changes result directly from relative velocity between the source and observer.

2

Identify the behavior where matching frequencies produce maximum vibration amplitude.

Energy transfer at an object's natural frequency defines resonance, matching the shattering crystal glass.

When forcing frequency matches natural frequency, energy accumulates constructively to cause large mechanical stress.

3

Examine the mechanism allowing light transmission without leakage in fiber optics.

Complete reflection at the core-cladding boundary beyond the critical angle is total internal reflection, matching fiber-optic light pulses.

Light cannot refract out when traveling from high to low refractive index at incident angles exceeding the critical threshold.

4

Determine how long-wavelength waves navigate physical terrain barriers.

The bending of waves around large structures without a change in medium is diffraction, matching obstructed radio wave reception.

Waves naturally spread into geometric shadow zones behind obstacles comparable to their wavelength.

Anahtar Kavram

Wave behaviors and boundary interactions

Tahmini Süre:1m 15s

Soru 3316Soru →

A contractor is purchasing supplies for a project. He buys a power drill originally priced at $\$150.00$ and a set of drill bits originally priced at $\$50.00$ . The drill is on sale for $30\%$ off, but the drill bits are sold at full price. If an $8\%$ sales tax is applied to the entire purchase after the discount is taken, what is the final total amount paid?

\$171.00

\$156.24

\$167.40

\$151.20

Cevabı ve açıklamayı göster

Cevap: $\$167.40$

Cevap

\$167.40

The correct amount is found by first calculating the

30\%

discount on the

\$150.00

drill, which is a

\$45.00

reduction, making the drill

\$105.00

. Adding the full-price

\$50.00

drill bits brings the subtotal to

\$155.00

. Finally, calculating the

8\%

sales tax on the

\$155.00

subtotal gives

\$12.40

. Adding the tax to the subtotal yields the final amount of

\$167.40

.

Adım Adım Çözüm

1

Calculate the discount amount for the power drill.

\$150.00 \times 0.30 = \$45.00

To determine how much will be reduced from the drill's original price.

2

Determine the sale price of the power drill.

\$150.00 - \$45.00 = \$105.00

To find the actual cost of the drill before tax.

3

Calculate the pre-tax subtotal for all purchased items.

\$105.00 \text{ (drill)} + \$50.00 \text{ (bits)} = \$155.00

The sales tax is applied to the combined cost of the discounted and full-priced items.

4

Calculate the sales tax on the subtotal.

\$155.00 \times 0.08 = \$12.40

An 8% tax applies to the entire pre-tax subtotal.

5

Calculate the final total amount paid.

\$155.00 + \$12.40 = \$167.40

The final total must include both the purchase subtotal and the applied sales tax.

Anahtar Kavram

Calculating multi-step financial transactions involving partial-order discounts and subsequent aggregate sales tax.

Soru 3317Soru →

A naval ship uses $400$ pounds of provisions to feed a specific crew for $5$ days. At this same constant rate, how many pounds of provisions are needed to feed the crew for $12$ days?

Cevabı ve açıklamayı göster

Cevap: 960

Cevap

960

By determining the unit rate of

80

pounds per day and multiplying it by

12

days, the direct proportionality rule is correctly applied to find the total of

960

pounds.

Adım Adım Çözüm

1

Calculate the unit rate of provisions per day.

400 \div 5 = 80

pounds per day.

Finding the amount used in one day allows scaling the proportion to any other number of days.

2

Multiply the unit rate by the new number of days.

80 \times 12 = 960

pounds.

This scales the daily rate up to the required

12

-day period to maintain the direct proportion.

Anahtar Kavram

Direct proportions using unit rates

Tahmini Süre:45s

Soru 3318Soru →

A military transport aircraft consumes $3,600$ pounds of aviation fuel during a $45$ -minute flight at cruising altitude. Operating at this same constant rate of fuel consumption, how many tons of fuel will the aircraft consume during a $6$ -hour flight at cruising altitude? ( $1 \text{ ton} = 2,000 \text{ pounds}$ )

8.1

14.4

24.0

10.8

Cevabı ve açıklamayı göster

Cevap: $14.4$

Cevap

The aircraft will consume

14.4

tons of fuel.

Fuel consumption is directly proportional to flight time. First, harmonize the time units by finding the scale factor between the two flights. A

6

-hour flight is

360

minutes. The proportion multiplier is

360 \div 45 = 8

. Multiplying the original

3,600

pounds by

8

gives

28,800

pounds for the longer flight. Finally, convert this weight to tons by dividing by

2,000

, resulting in

14.4

tons.

Adım Adım Çözüm

1

Calculate the scale factor between the target flight time and the initial flight time.

The target time is

6

hours, which equals

360

minutes (

6 \times 60

). The initial time is

45

minutes. The scale factor is

360 \div 45 = 8

.

Finding the multiplier makes it easy to scale up the fuel consumption proportionally.

2

Calculate the total fuel consumed in pounds for the

6

-hour flight.

3,600 \text{ pounds} \times 8 = 28,800 \text{ pounds}

.

Since fuel consumption is directly proportional to flight time, multiplying the initial fuel by the time scale factor gives the total fuel.

3

Convert the total fuel from pounds to tons.

28,800 \div 2,000 = 14.4 \text{ tons}

.

The question asks for the final answer in tons, and we are given that

1 \text{ ton} = 2,000 \text{ pounds}

.

Anahtar Kavram

Direct Proportions with Unit Conversions

Alternatif Yöntem

Instead of finding the scale factor, you can find the unit rate per minute:

3,600 \div 45 = 80

pounds per minute. Then, convert the

6

-hour flight to minutes (

6 \times 60 = 360

minutes). Multiply the rate by the total minutes (

80 \times 360 = 28,800

pounds), and finally convert to tons (

28,800 \div 2,000 = 14.4

tons).

Tahmini Süre:2m 0s

Soru 3319Soru →

A communications antenna emits radio waves that travel through uniform atmospheric air. If the transmitter is adjusted to emit radio waves at a higher frequency, how do the speed and wavelength of the transmitted waves change?

The speed increases, and the wavelength remains unchanged.

The speed increases, and the wavelength increases.

The speed remains constant, and the wavelength decreases.

The speed remains constant, and the wavelength increases.

Cevabı ve açıklamayı göster

Cevap: The speed remains constant, and the wavelength decreases.

Cevap

The speed remains constant, and the wavelength decreases.

In a uniform medium such as air, the speed of an electromagnetic wave is constant. According to the wave equation

v = f \lambda

, when the speed

v

is constant, frequency

f

and wavelength

\lambda

are inversely proportional. Therefore, increasing the frequency causes the wavelength to decrease while the speed remains unchanged.

Adım Adım Çözüm

1

Identify the factor that determines wave propagation speed.

The speed of electromagnetic radio waves is constant in a uniform medium (

v = c

in air).

Wave speed is an intrinsic property of the medium through which the wave travels, not the frequency of the source.

2

Apply the wave equation relating speed, frequency, and wavelength.

v = f \lambda

, which rearranges to

\lambda = \frac{v}{f}

.

This relationship mathematically couples frequency and wavelength for a given constant speed.

3

Determine the effect of increasing frequency on wavelength.

As frequency (

f

) increases while speed (

v

) remains fixed, wavelength (

\lambda

) must decrease proportionally.

Frequency and wavelength are inversely proportional when wave speed is held constant.

Anahtar Kavram

The speed of a wave is determined by the transmission medium, creating an inverse relationship between frequency and wavelength (

v = f \lambda

).

Tahmini Süre:1m 0s

Soru 3320Soru →

A commercial landscaping company is purchasing a new riding lawnmower. The dealership acquired the lawnmower at a wholesale cost of $\$1,200.00$ . To determine the retail price, the dealership applies a $30\%$ markup to the wholesale cost. During a spring promotional event, the dealership offers a $20\%$ discount off the retail price. What is the final sale price of the lawnmower?

\$312.00

\$1,248.00

\$1,320.00

\$1,540.00

Cevabı ve açıklamayı göster

Cevap: $\$1,248.00$

Cevap

The final sale price of the lawnmower is

\$1,248.00

.

The retail price is calculated by adding the

30\%

markup to the wholesale cost (

\$1,200.00 + \$360.00 = \$1,560.00

). The final sale price is then found by subtracting the

20\%

discount from this established retail price (

\$1,560.00 - \$312.00 = \$1,248.00

).

Adım Adım Çözüm

1

Calculate the markup amount based on the wholesale cost.

\$1,200.00 \times 0.30 = \$360.00

The dealership applies a

30\%

markup to the original wholesale cost to determine the retail price.

2

Calculate the retail price.

\$1,200.00 + \$360.00 = \$1,560.00

The retail price is the sum of the wholesale cost and the markup.

3

Calculate the discount amount based on the new retail price.

\$1,560.00 \times 0.20 = \$312.00

The promotional

20\%

discount must be calculated using the retail price, not the wholesale cost.

4

Calculate the final sale price.

\$1,560.00 - \$312.00 = \$1,248.00

The final sale price is determined by subtracting the discount amount from the retail price.

Anahtar Kavram

When applying consecutive percentage changes, such as a markup followed by a discount, each percentage must be applied to the newly updated base amount, not the original starting amount.

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