Tüm alıştırma soruları

Soru 2061Soru →

A ship's culinary specialist is ordering bulk ground coffee. A $12$ -ounce bag of coffee costs $\$5.40$ . A $3$ -pound container of the same coffee costs $\$18.00$ . How much money is saved per pound by purchasing the $3$ -pound container instead of multiple $12$ -ounce bags?

\$1.20

\$1.95

\$3.60

\$5.55

Cevabı ve açıklamayı göster

Cevap: $\$1.20$

Cevap

\$1.20

The correct savings is found by determining the unit price of each option for exactly one pound and subtracting them. The

12

-ounce bag costs

\$0.45

per ounce, which equates to

\$7.20

per pound (

0.45 \times 16

). The

3

-pound container costs

\$6.00

per pound (

18.00 \div 3

). Subtracting the container's unit price from the bag's unit price yields a savings of

\$1.20

per pound.

Adım Adım Çözüm

1

Calculate the price per pound for the

3

-pound container.

\$18.00 \div 3 = \$6.00

per pound.

To find the baseline unit price of the bulk option for comparison.

2

Determine the cost per ounce of the

12

-ounce bag.

\$5.40 \div 12 = \$0.45

per ounce.

To find the base unit rate of the smaller bags before scaling up to a full pound.

3

Calculate the price per pound for the

12

-ounce bag.

\$0.45 \times 16 = \$7.20

per pound.

Because there are

16

ounces in a pound, multiplying the cost per ounce by

16

gives the comparable unit price.

4

Find the difference in price per pound.

\$7.20 - \$6.00 = \$1.20

.

Subtracting the unit price of the bulk container from the unit price of the bags determines the exact savings per pound.

Anahtar Kavram

Unit Price Comparisons across different measurement scales

Tahmini Süre:1m 30s

Soru 2062Soru →

A military patrol boat consumes 40 gallons of fuel per hour and travels at a constant speed of 20 miles per hour. If the boat completes a round trip to an island 60 miles away, how many gallons of fuel will it use in total?

6

60

120

240

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

The boat will use a total of 240 gallons of fuel.

First, find the total distance for the round trip, which is

120

miles (

60

miles each way). At a speed of

20

miles per hour, the entire trip takes

6

hours (

120 \div 20

). Since the boat burns

40

gallons per hour, multiply the time by the fuel rate to find the total fuel consumed:

6 \times 40 = 240

gallons.

Adım Adım Çözüm

1

Calculate the total distance for the round trip.

60 \times 2 = 120

miles

The boat must travel to the island and then return along the same route.

2

Calculate the total travel time in hours.

120 \div 20 = 6

hours

Time is equal to total distance divided by the constant speed.

3

Calculate the total fuel consumed.

6 \times 40 = 240

gallons

Total fuel used is the travel time multiplied by the hourly fuel consumption rate.

Anahtar Kavram

Calculating total fuel consumption for a round trip using speed, distance, and fuel rate.

Alternatif Yöntem

Find the fuel consumption per mile first:

40

gallons per hour

\div 20

miles per hour =

2

gallons per mile. Then, multiply this rate by the total round-trip distance:

120

miles

\times 2

gallons per mile =

240

gallons.

Tahmini Süre:45s

Soru 2063Soru →

You are visually tracing the orientation of a T-shaped connector as it is rotated counter-clockwise to fit into a matching slot. The long stem of the 'T' initially points to the right (3 o'clock position). Arrange the following positions of the T-stem in the chronological order they appear as the piece undergoes a continuous 270-degree counter-clockwise rotation.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct sequence tracks the counter-clockwise rotation of the stem from the right (3 o'clock): first it points up (12 o'clock), then left (9 o'clock), and finally down (6 o'clock).

Starting from the right (3 o'clock position), a counter-clockwise rotation moves the piece opposite to the hands of a clock. The first quarter-turn positions the stem pointing up (12 o'clock). The second quarter-turn positions it pointing left (9 o'clock). The third and final quarter-turn positions it pointing down (6 o'clock), successfully completing the 270-degree movement.

Adım Adım Çözüm

1

Identify the starting orientation and direction of rotation.

The stem starts at the 3 o'clock position and rotates counter-clockwise.

Establishing the baseline orientation is necessary before applying rotational transformations.

2

Determine the position after the first 90-degree rotation.

Moving counter-clockwise from 3 o'clock brings the stem to the 12 o'clock position.

A 90-degree counter-clockwise turn corresponds to moving backwards by a quarter-circle.

3

Track the subsequent 90-degree intervals.

From 12 o'clock, the next 90 degrees brings it to 9 o'clock, and the final 90 degrees brings it to 6 o'clock.

Continuing this pattern accurately completes the full 270-degree rotation.

Anahtar Kavram

Mental tracking of an object's orientation during continuous 2D counter-clockwise rotation.

Soru 2064Soru →

A cyber warfare technician detects that a network firewall has successfully blocked $\frac{27}{40}$ of the incoming botnet attack traffic. What percentage of the attack traffic was successfully blocked? (Enter the numerical value only)

Cevabı ve açıklamayı göster

Cevap: 67.5

Cevap

67.5

The fraction of blocked traffic is given as

\frac{27}{40}

. To find the percentage, first convert the fraction to a decimal by dividing the numerator by the denominator:

27 \div 40 = 0.675

. Next, multiply this decimal by

100

to convert it to a percentage:

0.675 \times 100 = 67.5

. Therefore,

67.5\%

of the traffic was blocked.

Adım Adım Çözüm

1

Identify the fraction of the attack traffic that was blocked.

The fraction is

\frac{27}{40}

.

This establishes the mathematical relationship between the blocked traffic and the total traffic.

2

Convert the fraction to a decimal.

27 \div 40 = 0.675

Dividing the numerator (

27

) by the denominator (

40

) provides the decimal representation of the fraction.

3

Convert the decimal to a percentage.

0.675 \times 100 = 67.5

Multiplying the decimal by

100

shifts the decimal point two places to the right, yielding the percentage.

Anahtar Kavram

Converting a fraction with a non-base-10 denominator into a percentage by first calculating its decimal equivalent and then multiplying by 100.

Soru 2065Soru →

An aviation logistics specialist is sourcing a specific synthetic lubricant for aircraft maintenance. A local supplier sells the lubricant in $1$ -quart bottles for $\$5.50$ each. A wholesale distributor sells the same lubricant in $5$ -gallon drums for $\$95.00$ per drum. How much money is saved per gallon by purchasing the $5$ -gallon drum instead of the individual $1$ -quart bottles?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

To find the savings per gallon, both purchase options must be converted to a per-gallon rate. Because there are 4 quarts in a US gallon, the 1-quart bottles effectively cost

\$5.50 \times 4 = \$22.00

per gallon. The 5-gallon drum costs

\$95.00 \div 5 = \$19.00

per gallon. The difference in price is

\$22.00 - \$19.00 = \$3.00

saved per gallon.

Adım Adım Çözüm

1

Calculate the price per gallon for the 1-quart bottles.

\$5.50 \times 4 = \$22.00

per gallon.

There are 4 quarts in a gallon, so the unit price must be scaled up to match the gallon comparison unit.

2

Calculate the price per gallon for the 5-gallon drum.

\$95.00 \div 5 = \$19.00

per gallon.

The total cost of the drum must be divided by its volume in gallons to find the unit price.

3

Calculate the difference between the two unit prices.

\$22.00 - \$19.00 = \$3.00

Subtracting the lower unit price from the higher unit price reveals the savings per gallon.

Anahtar Kavram

Calculating and comparing unit rates involving US customary capacity conversions.

Soru 2066Soru →

A straight fence is intersected by a diagonal property line. The two adjacent angles formed on the same side of the straight fence are supplementary. If the measure of the smaller angle is represented by $x^\circ$ and the measure of the larger angle is represented by $(x + 40)^\circ$ , what is the measure of the larger angle?

65^\circ

70^\circ

110^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $110^\circ$

Cevap

The measure of the larger angle is

110^\circ

.

Because the angles are supplementary, their sum must be exactly

180^\circ

. By creating the equation

x + (x + 40) = 180

, we combine the variables to get

2x + 40 = 180

. Subtracting 40 from both sides leaves

2x = 140

, making

x = 70

. Since the larger angle is defined as

(x + 40)^\circ

, adding 70 and 40 gives the correct measure of

110^\circ

.

Adım Adım Çözüm

1

Set up an equation based on the definition of supplementary angles.

x + (x + 40) = 180

Adjacent angles on a straight line are supplementary and must sum to 180 degrees.

2

Combine like terms and isolate the variable

x

.

2x + 40 = 180 \Rightarrow 2x = 140 \Rightarrow x = 70

To solve for

x

, first subtract 40 from both sides, then divide by 2.

3

Calculate the measure of the larger angle using the expression given in the problem.

(70) + 40 = 110

The question asks for the larger angle, which is

(x + 40)^\circ

, not just the value of

x

.

Anahtar Kavram

Supplementary Angles and Basic Algebra

Tahmini Süre:45s

Soru 2067Soru →

An asymmetric L-shaped bracket has its long arm pointing North and its short arm pointing West. The long arm features a semi-circular notch on its East-facing edge. To assemble a complex mechanism, this bracket must be mentally rotated clockwise to fit into various component slots. Arrange the following orientations of the bracket in order of the clockwise angular shift required to reach them from the initial position, starting from the smallest clockwise rotation and ending with the largest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct sequence of orientations based on their required clockwise rotation is: 90° (East-pointing long arm), 135° (South-East-pointing long arm), 225° (South-West-pointing long arm), and 270° (West-pointing long arm).

By tracking the long arm of the bracket, we can determine the exact angular shift. The initial position is North (0°). The required clockwise rotations are 90° (to East), 135° (to South-East), 225° (to South-West), and 270° (to West). Placing them in ascending order yields this exact sequence.

Adım Adım Çözüm

1

Analyze the initial reference orientation of the bracket.

The long arm is at 0° (North), the short arm is at 270° (West), and the notch is at 90° (East).

Establishing a baseline is necessary to calculate the angular shifts.

2

Calculate the rotation required for the orientation with the long arm pointing East.

Moving from North (0°) to East (90°) is a 90° clockwise rotation.

This determines the angular shift for the first described orientation.

3

Calculate the rotation required for the orientation with the long arm pointing South-East.

Moving from North (0°) to South-East (135°) is a 135° clockwise rotation.

This determines the angular shift for the second described orientation.

4

Calculate the rotation required for the orientation with the long arm pointing South-West.

Moving from North (0°) to South-West (225°) is a 225° clockwise rotation.

This determines the angular shift for the third described orientation.

5

Calculate the rotation required for the orientation with the long arm pointing West.

Moving from North (0°) to West (270°) is a 270° clockwise rotation.

This determines the angular shift for the fourth described orientation.

6

Arrange the determined angular shifts in ascending order.

The order is 90°, 135°, 225°, then 270°.

The question requests the order from the smallest to the largest clockwise rotation.

Anahtar Kavram

2D Mental Rotation

Tahmini Süre:1m 30s

Soru 2068Soru →

Match each vehicle trip scenario to the total number of gallons of fuel required to complete the journey.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The scout vehicle requires 5 gallons, the transport truck requires 12 gallons, the command SUV requires 16 gallons, and the hybrid commuter car requires 7 gallons.

The correct pairings are found by properly applying the formula: Gallons Consumed = Total Distance ÷ Fuel Efficiency (MPG). Performing this calculation for each scenario accurately identifies the fuel requirement.

Adım Adım Çözüm

1

Calculate the fuel required for the scout vehicle.

150 / 30 = 5 gallons.

Dividing the total travel distance by the vehicle's fuel efficiency (MPG) determines the total fuel consumed.

2

Calculate the fuel required for the transport truck.

240 / 20 = 12 gallons.

Dividing the total travel distance by the vehicle's fuel efficiency (MPG) determines the total fuel consumed.

3

Calculate the fuel required for the command SUV.

400 / 25 = 16 gallons.

Dividing the total travel distance by the vehicle's fuel efficiency (MPG) determines the total fuel consumed.

4

Calculate the fuel required for the hybrid commuter car.

350 / 50 = 7 gallons.

Dividing the total travel distance by the vehicle's fuel efficiency (MPG) determines the total fuel consumed.

Anahtar Kavram

Calculating total fuel consumption given total travel distance and fuel efficiency.

Soru 2069Soru →

A transport aircraft is flying a $1,200$ -nautical-mile resupply mission. It flies the first $600$ nautical miles at a cruising speed of $300$ knots. Due to headwind conditions, it flies the remaining $600$ nautical miles at a reduced speed of $240$ knots. What is the total flight time for this mission?

4 hours 30 minutes

4 hours 50 minutes

4 hours 27 minutes

2 hours 30 minutes

Cevabı ve açıklamayı göster

Cevap: 4 hours 30 minutes

Cevap

4 hours 30 minutes

The correct answer accurately finds the time for each leg of the journey by dividing distance by speed. The first leg takes 2 hours (

600 \div 300

) and the second leg takes 2.5 hours (

600 \div 240

). Combining these yields 4.5 hours, which is equal to 4 hours and 30 minutes.

Adım Adım Çözüm

1

Calculate the time taken for the first half of the mission.

600 \div 300 = 2

hours.

Time is calculated by dividing the distance by the rate for that specific segment.

2

Calculate the time taken for the second half of the mission.

600 \div 240 = 2.5

hours.

The rate changed, so this segment's time must be calculated independently using its specific distance and rate.

3

Add the times of both segments to find the total flight time in hours.

2 + 2.5 = 4.5

hours.

The total duration of the mission is the sum of the time spent on all individual segments.

4

Convert the fractional hour into minutes.

0.5 \text{ hours} \times 60 \text{ minutes/hour} = 30

minutes. Total time is 4 hours and 30 minutes.

Standard time format requires separating whole hours and converting the remaining fraction into minutes.

Anahtar Kavram

Calculating Rate and Time for Multi-leg Journeys

Tahmini Süre:1m 15s

Soru 2070Soru →

A military convoy is scheduled to travel a total distance of $1,500$ miles over a multi-day operation. If the convoy completes $32\%$ of the total distance on the first day, how many miles does the convoy have left to travel?

102

480

1,020

1,468

Cevabı ve açıklamayı göster

Cevap: 1,020

Cevap

1,020

The correct answer is found by determining that if 32 percent of the trip is complete, 68 percent remains. Calculating 68 percent of 1,500 miles gives exactly 1,020 miles. Alternatively, one could find that 32 percent of 1,500 is 480 miles, and then subtract 480 from 1,500 to get 1,020 miles.

Adım Adım Çözüm

1

Determine the percentage of the distance remaining to be traveled.

100\% - 32\% = 68\%

Since 32 percent of the distance is completed, the rest of the distance is the remaining percentage of the total.

2

Calculate the remaining distance by finding

68\%

of

1,500

miles.

1,500 \times 0.68 = 1,020

miles

Multiplying the total distance by the remaining decimal percentage gives the remaining physical distance.

Anahtar Kavram

Calculating Percentages of Quantities

Alternatif Yöntem

Find the completed distance first by calculating

32\%

of

1,500

(

0.32 \times 1,500 = 480

miles). Then subtract that distance from the total distance (

1,500 - 480 = 1,020

miles).

Tahmini Süre:1m 0s

Soru 2071Soru →

A carpenter is purchasing a commercial-grade table saw. The original price of the table saw is $\$850$ . The store offers a $20\%$ discount on all power tools. After the discount is applied, a $5\%$ sales tax is added to the purchase price. What is the final total amount the carpenter pays for the table saw?

\$722.50

\$871.50

\$714.00

\$1,020.00

Cevabı ve açıklamayı göster

Cevap: $\$714.00$

Cevap

\$714.00

To find the final price, first determine the sale price by taking

20\%

off the original

\$850

. Since

20\%

of

\$850

is

\$170

, the discounted price is

\$680

. Next, apply the

5\%

sales tax to the new sale price of

\$680

. The tax is

0.05 \times \$680 = \$34

. Adding the tax to the sale price gives the final total of

\$714.00

.

Adım Adım Çözüm

1

Calculate the discount amount.

\$170

The discount is

20\%

of the original price:

0.20 \times \$850 = \$170

.

2

Determine the discounted sale price.

\$680

Subtract the discount amount from the original price:

\$850 - \$170 = \$680

.

3

Calculate the sales tax amount on the sale price.

\$34

The tax is

5\%

of the discounted price:

0.05 \times \$680 = \$34

.

4

Calculate the final total amount.

\$714.00

Add the sales tax to the sale price:

\$680 + \$34 = \$714

.

Anahtar Kavram

Discounts and Sale Prices

Soru 2072Soru →

A construction battalion is procuring wood varnish for a barracks renovation. A supplier sells the varnish in a $5$ -gallon bucket for $\$120.00$ . Alternatively, the same varnish is sold in $1$ -quart cans for $\$8.00$ each. How much money is saved per gallon by purchasing the $5$ -gallon bucket instead of the $1$ -quart cans?

\$8.00

\$16.00

\$22.00

\$0.80

Cevabı ve açıklamayı göster

Cevap: \$8.00

Cevap

The

5

-gallon bucket saves

\$8.00

per gallon compared to the

1

-quart cans.

To accurately compare the costs, both options must be calculated as a price per gallon. The

5

-gallon bucket costs

\$120.00

, which breaks down to

\$120.00 \div 5 = \$24.00

per gallon. For the second option, because there are

4

quarts in

1

gallon, the price per gallon is

4 \times \$8.00 = \$32.00

. The savings per gallon is the difference between these two unit rates:

\$32.00 - \$24.00 = \$8.00

.

Adım Adım Çözüm

1

Calculate the unit price per gallon for the

5

-gallon bucket.

\$120.00 \div 5 = \$24.00

per gallon

This establishes the baseline unit price for the bulk option.

2

Convert the price of the

1

-quart cans into a price per gallon.

Since there are

4

quarts in a gallon,

4 \times \$8.00 = \$32.00

per gallon

Both unit prices must be in the same measurement (gallons) to make an accurate comparison.

3

Subtract the bucket's gallon price from the cans' gallon price to find the savings.

\$32.00 - \$24.00 = \$8.00

The question asks for the money saved per gallon by choosing the bucket.

Anahtar Kavram

Comparing unit prices requiring unit conversions

Soru 2073Soru →

A courier's motorcycle has a fuel efficiency of $40$ miles per gallon. The courier uses exactly $2.5$ gallons of fuel to complete a round trip from the dispatch center to a customer's location and back. What is the one-way distance between the dispatch center and the customer's location?

5

16

50

100

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The one-way distance is

50

miles.

To find the one-way distance, first determine the total distance traveled by multiplying the fuel efficiency by the amount of fuel used:

40 \text{ miles/gallon} \times 2.5 \text{ gallons} = 100 \text{ miles}

. Since this is a round trip, divide the total distance by

2

to find the one-way distance:

100 \div 2 = 50 \text{ miles}

.

Adım Adım Çözüm

1

Calculate the total distance traveled using the fuel efficiency and total fuel used.

40 \text{ miles/gallon} \times 2.5 \text{ gallons} = 100 \text{ miles}

Total distance is the product of the rate of fuel consumption (efficiency) and the amount of fuel consumed.

2

Calculate the one-way distance by dividing the total round trip distance in half.

100 \text{ miles} \div 2 = 50 \text{ miles}

A round trip consists of two equal segments (going and returning), so the one-way distance is half of the total distance.

Anahtar Kavram

Calculating distance from fuel efficiency and separating round trip distance into one-way legs.

Soru 2074Soru →

A bakery owner is purchasing a new commercial dough mixer. The original price of the mixer is $\$1,500$ . The restaurant supply store is running a promotional sale offering a $20\%$ discount on all equipment. Additionally, the owner receives an extra $10\%$ discount on the sale price for paying directly through a business bank transfer. What is the final sale price of the dough mixer in dollars?

Cevabı ve açıklamayı göster

Cevap: 1080

Cevap

The final sale price of the dough mixer is

\$1,080

.

To find the final price after successive discounts, each discount must be applied sequentially to the updated balance. First, applying the

20\%

discount to the original

\$1,500

price reduces it by

\$300

, resulting in an intermediate sale price of

\$1,200

. The subsequent

10\%

discount is then applied to this new

\$1,200

balance, taking off an additional

\$120

. Subtracting

\$120

from

\$1,200

yields a final price of

\$1,080

.

Adım Adım Çözüm

1

Calculate the amount of the first discount.

\$1,500 \times 0.20 = \$300

The promotional discount applies to the original base price.

2

Determine the intermediate sale price after the first discount.

\$1,500 - \$300 = \$1,200

Subtracting the discount from the original price gives the new price before further markdowns.

3

Calculate the amount of the second discount.

\$1,200 \times 0.10 = \$120

Successive discounts are calculated based on the new reduced sale price, not the original price.

4

Determine the final price after the second discount.

\$1,200 - \$120 = \$1,080

Subtracting the second discount from the intermediate price yields the final amount owed.

Anahtar Kavram

Successive/Chain Discounts

Soru 2075Soru →

Fill in the blank with the correct numeric value.

Aşağıdaki boşlukları doldurun

A military dispatch rider travels

75

miles on a highway in

1

hour and

30

minutes. He then travels an additional

25

miles on secondary roads in

1

hour. The rider's average speed for the entire trip is miles per hour.

Cevabı ve açıklamayı göster

Cevap

The average speed for the entire trip is 40 miles per hour.

The correct average speed is found by dividing the total distance (

100

miles) by the total travel time (

2.5

hours), which results in exactly

40

miles per hour.

Adım Adım Çözüm

1

Calculate the total distance traveled.

75 + 25 = 100

miles.

Average speed calculation requires the total distance of the trip.

2

Convert all time measurements to hours and calculate the total time.

1

hour and

30

minutes is equivalent to

1.5

hours. Total time

= 1.5 + 1 = 2.5

hours.

The requested rate is in miles per hour, so the total time must be consistently represented in hours.

3

Divide the total distance by the total time to find the average speed.

100 \div 2.5 = 40

miles per hour.

The fundamental formula for average speed is

\text{Total Distance} \div \text{Total Time}

.

Anahtar Kavram

Average speed is calculated by dividing the total distance traveled by the total time taken for the trip, not by taking the arithmetic mean of the individual segment speeds.

Soru 2076Soru →

A base cyber operations squadron is tasked with installing a critical security update on $2,500$ network terminals. If they have completed $68\%$ of the installations by the end of the day, how many network terminals still need to be updated?

Cevabı ve açıklamayı göster

Cevap: 800

Cevap

800 terminals

The total number of terminals is 2,500, which represents 100%. If 68% are updated, 32% remain. 32% of 2,500 is found by multiplying 0.32 by 2,500, which equals 800.

Adım Adım Çözüm

1

Calculate the percentage of terminals that still need to be updated.

100\% - 68\% = 32\%

Since 68% are already updated, the rest represent the remaining terminals.

2

Calculate 32% of the total 2,500 terminals.

0.32 \times 2,500 = 800

To find the actual number of remaining terminals, multiply the remaining percentage by the total number of terminals.

Anahtar Kavram

Calculating the remaining percentage of a total quantity in a multi-step word problem.

Tahmini Süre:1m 0s

Soru 2077Soru →

A field medical unit needs to resupply liquid surface disinfectant. The disinfectant is available in four different packaging options. Arrange the purchasing options in order from the lowest unit price (price per gallon) to the highest unit price.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from lowest to highest unit price is: the pack of eight 1-pint bottles (

\$23.04

/gal), the twin-pack of two 1-gallon jugs (

\$24.00

/gal), the 5-gallon drum (

\$25.60

/gal), and the case of four 1-quart bottles (

\$28.16

/gal).

By converting the capacity of each option to a common unit (gallons) and calculating the price per gallon, we can directly compare the unit costs. The

1

-pint bottles equate to

1

gallon for

\$23.04

. The

1

-gallon jugs cost

\$24.00

per gallon (

\$48.00 \div 2

). The

5

-gallon drum costs

\$25.60

per gallon (

\$128.00 \div 5

). The

1

-quart bottles equate to

1

gallon for

\$28.16

. Placing these calculated unit prices in ascending order provides the correct sequence.

Adım Adım Çözüm

1

Determine a common unit of measurement to compare all options.

The gallon is the most convenient common unit.

Converting all volumes to gallons allows for a direct dollar-to-dollar comparison of the unit prices.

2

Calculate the price per gallon for the

5

-gallon drum.

\$128.00 \div 5 \text{ gallons} = \$25.60

per gallon.

Dividing the total cost by the total volume yields the unit price.

3

Calculate the price per gallon for the case of

1

-quart bottles.

There are

4

quarts in a gallon, so four

1

-quart bottles equal exactly

1

gallon. The price is

\$28.16

per gallon.

Unit conversion is required before finding the price per gallon.

4

Calculate the price per gallon for the pack of

1

-pint bottles.

There are

2

pints in a quart and

4

quarts in a gallon, so there are

8

pints in a gallon. Eight

1

-pint bottles equal exactly

1

gallon. The price is

\$23.04

per gallon.

Unit conversion is required to establish the total volume in gallons.

5

Calculate the price per gallon for the twin-pack of

1

-gallon jugs.

\$48.00 \div 2 \text{ gallons} = \$24.00

per gallon.

Dividing the total cost by the total volume yields the unit price.

6

Compare the calculated prices per gallon and order them from lowest to highest.

\$23.04 < \$24.00 < \$25.60 < \$28.16

.

The question asks for the options to be arranged sequentially by increasing unit cost.

Anahtar Kavram

To compare the prices of items sold in different quantities and unit sizes, convert all volumes to a single common unit (such as gallons) and calculate the price per common unit for each option.

Soru 2078Soru →

Translate the following geometric relationships into algebraic equations and perform the first step of simplification. Pair each geometric description on the left with its corresponding simplified equation on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Match the complementary angles with

3x = 60

; the supplementary angles with

3x = 150

; the vertical angles with

x = 10

; and the adjacent angles on a straight line with

5x = 150

.

Each geometric term dictates a specific mathematical equality: 'complement' means the sum is

90^\circ

, 'supplement' and 'straight line' mean the sum is

180^\circ

, and 'vertical angles' means the measures are equal to each other. Applying these definitions and performing basic algebraic simplification yields the correct matches.

Adım Adım Çözüm

1

Formulate and simplify the equation for the complementary angles.

The setup is

3x + 30 = 90

, which simplifies to

3x = 60

.

Complementary angles always add up to

90^\circ

.

2

Formulate and simplify the equation for the supplementary angles.

The setup is

3x + 30 = 180

, which simplifies to

3x = 150

.

Supplementary angles always add up to

180^\circ

.

3

Formulate and simplify the equation for the vertical angles.

The setup is

3x = 30

, which simplifies to

x = 10

.

Vertical angles (angles opposite each other at an intersection) are geometrically equal.

4

Formulate and simplify the equation for the adjacent angles on a straight line.

The setup is

3x + (2x + 30) = 180

. Combining the

x

terms gives

5x + 30 = 180

, and subtracting 30 gives

5x = 150

.

Angles that form a linear pair on a straight line must sum to

180^\circ

.

Anahtar Kavram

Translating Geometric Angle Relationships into Algebraic Equations

Soru 2079Soru →

During a training exercise, an army jeep travels at a constant fuel consumption rate of 14 miles per gallon. If the jeep starts with exactly 8 gallons of fuel in its tank, what is the maximum number of miles the jeep can travel before running out of gas?

Cevabı ve açıklamayı göster

Cevap: 112

Cevap

112

The correct answer is found by multiplying the fuel efficiency rate (14 miles per gallon) by the amount of fuel in the tank (8 gallons). 14 * 8 equals 112 miles.

Adım Adım Çözüm

1

Identify the given values

Fuel efficiency = 14 miles per gallon, Fuel available = 8 gallons

These are the necessary components to calculate total distance.

2

Calculate the total distance

14 * 8 = 112 miles

Distance is the product of the rate of fuel consumption (miles per gallon) and the total amount of fuel (gallons).

Anahtar Kavram

Calculating maximum travel range based on fuel efficiency and fuel volume.

Soru 2080Soru →

You are assembling a square frame using four identical asymmetrical L-shaped brackets. Each bracket consists of a long arm and a short arm meeting at a 90-degree corner. When installed, the right-angle corner of each bracket sits exactly in one of the four corners of the square frame.

The installation requirements are:
- Top-Left Corner: The long arm extends from the corner along the top edge (pointing East), and the short arm extends along the left edge (pointing South).
- Top-Right Corner: The long arm extends from the corner along the right edge (pointing South), and the short arm extends along the top edge (pointing West).
- Bottom-Right Corner: The long arm extends from the corner along the bottom edge (pointing West), and the short arm extends along the right edge (pointing North).
- Bottom-Left Corner: The long arm extends from the corner along the left edge (pointing North), and the short arm extends along the bottom edge (pointing East).

A bracket on your workbench is currently oriented so that its long arm points North-East and its short arm points South-East.

Determine the exact 2D clockwise rotation needed to reorient this workbench bracket for each of the four installation corners. Arrange the corners in order from the LEAST clockwise rotation to the MOST clockwise rotation required (from 0° to 360°).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order based on clockwise rotation is: Top-Left Corner (45°), Top-Right Corner (135°), Bottom-Right Corner (225°), and Bottom-Left Corner (315°).

By mapping the long arm of the bracket to a compass rose, the starting position is established at 45° (North-East). The target positions for the long arm at each corner are East (90°), South (180°), West (270°), and North (360°). Subtracting the starting 45° from these targets yields clockwise rotations of 45°, 135°, 225°, and 315° respectively.

Adım Adım Çözüm

1

Identify the starting orientation of the workbench bracket on a standard compass rose.

The long arm points to North-East (45° clockwise from North) and the short arm points to South-East (135°).

Establishing a baseline angular coordinate allows for precise calculation of rotations.

2

Determine the required rotation for the Top-Left Corner.

Target long arm is East (90°). The rotation from 45° to 90° is a 45° clockwise rotation.

This represents the smallest clockwise adjustment needed among the four corners.

3

Determine the required rotation for the Top-Right Corner.

Target long arm is South (180°). The rotation from 45° to 180° is a 135° clockwise rotation.

Tracking a single reference point (the long arm) ensures consistent and accurate rotation calculations.

4

Determine the required rotation for the Bottom-Right Corner.

Target long arm is West (270°). The rotation from 45° to 270° is a 225° clockwise rotation.

Continuing clockwise through the compass points calculates the third position in the sequence.

5

Determine the required rotation for the Bottom-Left Corner.

Target long arm is North (360° or 0°). The rotation from 45° to 360° is a 315° clockwise rotation.

This represents the largest clockwise rotation, stopping just 45° short of a full circle.

Anahtar Kavram

2D Mental Rotation and Directional Tracking

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Sık Yapılan Hatalar

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