Ratios and Proportions Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB)

Soru 1Soru →

A communications specialist is preparing a field antenna and needs to cut a $60$ -foot spool of copper wire into two sections in a ratio of $7:3$ .

Aşağıdaki boşlukları doldurun

The longer section of the wire will be feet long.

Cevabı ve açıklamayı göster

Cevap

The longer section of the wire will be 42 feet long.

The wire is divided into

7 + 3 = 10

total parts. Since the total length is

60

feet, each part is equal to

60 \div 10 = 6

feet. The longer section is

7

parts, so its length is

7 \times 6 = 42

feet.

Adım Adım Çözüm

1

Find the total number of parts in the ratio.

7 + 3 = 10

total parts.

The total length of the wire is divided into this many equal parts.

2

Calculate the length of one single part.

60 \div 10 = 6

feet per part.

Dividing the total length by the total number of parts gives the value of one part.

3

Determine the length of the longer section.

7 \times 6 = 42

feet.

The longer section corresponds to the larger number in the ratio, which is 7 parts.

Anahtar Kavram

To divide a quantity in a given ratio, find the total number of parts by adding the ratio terms, divide the total quantity by this sum to find the value of one part, and then multiply by the specific ratio term.

Soru 2Soru →

A naval destroyer traveling at an average speed of $24\text{ knots}$ completes a specific patrol route in $3\text{ hours}$ and $20\text{ minutes}$ . If the captain is ordered to complete the identical route in $2\text{ hours}$ and $30\text{ minutes}$ , what average speed must the destroyer maintain?

18\text{ knots}

32\text{ knots}

16\text{ knots}

3.2\text{ knots}

Cevabı ve açıklamayı göster

Cevap: $32\text{ knots}$

Cevap

The destroyer must maintain an average speed of

32\text{ knots}

.

The distance of the patrol route is constant, meaning the product of the initial speed and time must equal the product of the new speed and time. Converting the travel times completely to minutes gives

200\text{ minutes}

and

150\text{ minutes}

. Using the inverse proportion formula,

24 \times 200 = 4800

. Dividing this constant by the required new time of

150\text{ minutes}

gives the necessary speed of

32\text{ knots}

.

Adım Adım Çözüm

1

Convert both time durations entirely into minutes to ensure consistent units.

3\text{ hours and } 20\text{ minutes} = 200\text{ minutes}

, and

2\text{ hours and } 30\text{ minutes} = 150\text{ minutes}

.

Calculations involving mixed time units require a single common unit before a proportion can be accurately applied.

2

Identify the relationship between speed and time for a constant distance.

Speed and time are inversely proportional, governed by the equation

S_1 \times T_1 = S_2 \times T_2

.

When the distance of a route remains the same, an increase in speed results in a proportional decrease in travel time.

3

Set up the inverse proportion equation and solve for the new speed (

S_2

).

24 \times 200 = S_2 \times 150 \implies 4800 = 150 \times S_2 \implies S_2 = 32

.

Isolating the unknown variable yields the required average speed to complete the route in the shorter timeframe.

Anahtar Kavram

Inverse Proportions in Speed and Time

Tahmini Süre:1m 30s

Soru 3Soru →

A mechanical winch uses a gear system where the driving gear and the driven gear have a revolution ratio of $4:7$ . If the driving gear completes $124$ revolutions during a lifting operation, how many revolutions does the driven gear complete?

Cevabı ve açıklamayı göster

Cevap: 217

Cevap

217

The correct answer is found by setting up a direct proportion based on the 4:7 ratio. Since the driving gear's 4 parts scale up to 124 actual revolutions, the scale factor is 31 (124 ÷ 4). Multiplying the driven gear's 7 parts by this same scale factor yields 217 revolutions.

Adım Adım Çözüm

1

Identify the known ratio

Driving : Driven = 4 : 7

This establishes the proportional relationship between the two gears.

2

Set up a proportion using the known quantity

4 / 7 = 124 / x

We know the driving gear completes 124 revolutions, which corresponds to the '4' in the ratio.

3

Find the scale factor

124 ÷ 4 = 31

This tells us how many times greater the actual operation is compared to the base ratio.

4

Calculate the unknown quantity

7 × 31 = 217

Applying the same scale factor to the driven gear maintains the proportional relationship.

Anahtar Kavram

Solving for an unknown part in a part-to-part relationship using scale factors.

Soru 4Soru →

A flight line supervisor is dividing $480$ hours of scheduled maintenance tasks between two mechanic crews in a ratio of $3:5$ . If the larger share of hours is assigned to the senior crew, how many maintenance hours does the senior crew receive?

160

180

288

300

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

The senior crew receives 300 maintenance hours.

To solve the problem, first find the total number of parts in the ratio:

3 + 5 = 8

parts. Next, divide the total number of hours by the total parts to find the value of one part:

480 \div 8 = 60

hours. Finally, multiply the value of one part by

5

(the larger share) to find the senior crew's allocation:

60 \times 5 = 300

hours.

Adım Adım Çözüm

1

Find the total number of parts in the ratio by adding the terms together.

3 + 5 = 8

total parts.

To distribute a total amount according to a ratio, you must first determine how many equal parts make up the whole.

2

Determine the value of a single part by dividing the total hours by the total number of parts.

480 \div 8 = 60

hours per part.

This establishes the baseline unit value that corresponds to '1' in the ratio.

3

Multiply the value of a single part by the larger number in the ratio to find the senior crew's share.

60 \times 5 = 300

hours.

The problem states the senior crew receives the larger share, which corresponds to the '5' in the

3:5

ratio.

Anahtar Kavram

To divide a total quantity into a given ratio, add the terms to find the total number of parts, divide the total quantity by this sum to find the value of one part, and then multiply by the desired ratio term.

Tahmini Süre:1m 0s

Soru 5Soru →

During a ship repair project, a naval shipyard assigns 24 welders and 8 pipefitters to complete the structural work. What is the ratio of welders to pipefitters on this project, expressed in its simplest form?

3:1

1:3

3:4

1:4

Cevabı ve açıklamayı göster

Cevap: 3:1

Cevap

3:1

The correct answer accurately extracts the given quantities (24 welders, 8 pipefitters), sets them up in the requested sequence (24 to 8), and successfully reduces the numbers to their simplest terms by dividing both by 8, yielding 3:1.

Adım Adım Çözüm

1

Identify the two quantities that need to be compared based on the question.

Welders = 24, Pipefitters = 8

The question asks for the specific ratio of welders to pipefitters.

2

Set up the initial ratio in the exact order requested by the problem.

24:8

The order of terms in a ratio matters; it must strictly match 'welders to pipefitters'.

3

Simplify the ratio by dividing both numbers by their greatest common divisor (GCD).

The GCD of 24 and 8 is 8. Therefore, (24 ÷ 8) : (8 ÷ 8) = 3:1

Ratios are conventionally expressed in their lowest possible integer terms.

Anahtar Kavram

Basic Ratios

Tahmini Süre:45s

Soru 6Soru →

On a regional map, a scale of $1\text{ inch} = 12\text{ miles}$ is used. If a straight road is $1.5\text{ inches}$ long on the map, what is the actual length of the road?

0.125\text{ miles}

8\text{ miles}

18\text{ miles}

180\text{ miles}

Cevabı ve açıklamayı göster

Cevap: $18\text{ miles}$

Cevap

The actual length of the road is

18\text{ miles}

.

The map uses a scale where each inch corresponds to

12

actual miles. To find the real distance, we multiply the map distance by the scale:

1.5 \times 12 = 18

. Therefore, the road is

18\text{ miles}

long in reality.

Adım Adım Çözüm

1

Identify the scale ratio and the given map measurement.

The scale is

1\text{ inch} = 12\text{ miles}

, and the map measurement is

1.5\text{ inches}

.

This establishes the relationship between the drawing size and the real-world size.

2

Set up the calculation to find the actual distance.

Actual distance =

1.5\text{ inches} \times 12\text{ miles/inch}

.

Since every inch represents

12\text{ miles}

, multiplying the number of map inches by

12

will yield the actual miles.

3

Perform the multiplication.

1.5 \times 12 = 18\text{ miles}

.

Completing the arithmetic gives the final actual length of the road.

Anahtar Kavram

To find actual dimensions from a map, multiply the measured distance on the map by the given scale factor.

Soru 7Soru →

A military logistics center uses two automated packaging machines to assemble field ration kits. Machine A can assemble $450$ kits in $1.5$ hours. Machine B can assemble $90$ kits every $15$ minutes. If both machines operate simultaneously at their respective constant rates, how many total ration kits can they assemble in $4$ hours?

2,640

1,224

1,560

165

Cevabı ve açıklamayı göster

Cevap: $2,640$

Cevap

They can assemble

2,640

ration kits in

4

hours.

The correct answer is found by standardizing the production rates into hourly unit rates, adding them together, and multiplying by the total time. Machine A produces at

300

kits/hr (

450 \div 1.5

). Machine B produces at

360

kits/hr (

90 \times 4

, since

15

minutes is a quarter of an hour). The combined hourly rate is

660

kits/hr. In

4

hours, they produce

660 \times 4 = 2,640

kits.

Adım Adım Çözüm

1

Calculate the hourly unit rate for Machine A.

450 \div 1.5 = 300

kits per hour.

To compare and combine rates, both machines must be evaluated using the same unit of time (hours).

2

Calculate the hourly unit rate for Machine B.

90

kits per

15

minutes is equal to

90 \times 4 = 360

kits per hour.

There are

60

minutes in an hour, which is exactly four

15

-minute segments (

60 \div 15 = 4

). Therefore, multiplying the

15

-minute production rate by

4

gives the hourly rate.

3

Calculate the combined hourly rate.

300 + 360 = 660

kits per hour.

Since both machines operate simultaneously, their individual hourly rates are added together to find the total combined rate.

4

Calculate the total kits assembled in

4

hours.

660 \times 4 = 2,640

kits.

Multiply the combined hourly unit rate by the total operating time to find the final production amount.

Anahtar Kavram

Unit Rates

Soru 8Soru →

A commercial 3D printer extrudes $1.2$ kilograms of plastic filament every $45$ minutes of operation. Assuming the printer continues to operate at this exact constant rate, how many grams of filament will it extrude during a $3$ -hour continuous printing session?

4.8

80

300

4800

Cevabı ve açıklamayı göster

Cevap: $4800$

Cevap

The printer will extrude 4800 grams of filament.

Solving this requires maintaining consistent units while scaling up the operation time. Converting the 3-hour print session into 180 minutes reveals that the printer operates for exactly 4 times the duration of the baseline 45-minute period (180 divided by 45 is 4). Because the relationship is a direct proportion, the mass of filament used also scales by a factor of 4. Multiplying 1.2 kilograms by 4 yields 4.8 kilograms. Finally, converting 4.8 kilograms into grams by multiplying by 1000 gives 4800 grams.

Adım Adım Çözüm

1

Convert the target time period from hours to minutes to match the units of the given rate.

3 \text{ hours} = 3 \times 60 = 180 \text{ minutes}

Units must be consistent across the proportion equation to find the correct multiplier.

2

Set up a direct proportion to find the amount of filament extruded in kilograms over the new time period.

\frac{1.2 \text{ kg}}{45 \text{ min}} = \frac{x \text{ kg}}{180 \text{ min}} \Rightarrow x = 1.2 \times \left(\frac{180}{45}\right) = 1.2 \times 4 = 4.8 \text{ kg}

The mass of filament extruded scales directly with the time the machine operates.

3

Convert the resulting mass from kilograms to grams.

4.8 \text{ kg} = 4.8 \times 1000 = 4800 \text{ grams}

The question specifically asks for the final answer in grams.

Anahtar Kavram

Direct Proportions with Unit Conversions

Soru 9Soru →

Match each military logistical operation to its correct unit rate. Pay close attention to the requested units of measurement and necessary conversions.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches are: Supply truck with 12 miles per gallon, Drainage system with 60 gallons per minute, Packaging line with 420 rations per hour, and Generator with 12.5 milliliters per minute.

Each scenario requires identifying the total quantity of the numerator (miles, gallons, rations, milliliters) and dividing it by the corresponding total of the denominator (gallons, minutes, hours). In every case, this necessitates at least one preliminary conversion or calculation step.

Adım Adım Çözüm

1

Calculate the fuel efficiency of the supply truck in miles per gallon.

Total distance =

45 \times 4 = 180

miles. Unit rate =

180 / 15 = 12

miles per gallon.

The fuel efficiency is found by dividing the total distance traveled by the total gallons of fuel consumed, requiring a preliminary distance calculation.

2

Calculate the pumping rate of the drainage system in gallons per minute.

Total time =

60 + 20 = 80

minutes. Unit rate =

4,800 / 80 = 60

gallons per minute.

The units requested are gallons per minute, so the total time must first be completely converted into minutes before dividing the volume.

3

Calculate the production rate of the packaging line in rations per hour.

Rate per minute =

315 / 45 = 7

rations/min. Rate per hour =

7 \times 60 = 420

rations per hour.

To find the hourly rate from a minute-based rate, first find the rate per minute, then scale it up by multiplying by the number of minutes in an hour (

60

).

4

Calculate the consumption rate of the generator in milliliters per minute.

Volume =

18 \times 1,000 = 18,000

mL. Time =

24 \times 60 = 1,440

minutes. Unit rate =

18,000 / 1,440 = 12.5

mL/min.

Both given quantities require unit conversions (liters to milliliters, and hours to minutes) before performing the final division to find the required rate.

Anahtar Kavram

Calculating unit rates involving multi-step operations and dimensional analysis (unit conversions).

Soru 10Soru →

A city planner is reviewing a map of a proposed municipal park. The map uses a scale where $\frac{1}{2}$ inch represents $30$ feet. If the park occupies an area of $15$ square inches on the map, what will be the actual area of the park in square yards?

300

6,000

18,000

54,000

Cevabı ve açıklamayı göster

Cevap: 6,000

Cevap

The actual area of the park is 6,000 square yards.

To solve this problem efficiently, first convert the linear scale to yards:

\frac{1}{2}\text{ inch} = 30\text{ feet}

means

1\text{ inch} = 60\text{ feet}

, which equals

20\text{ yards}

. Because we are dealing with area, we must square the linear scale to find the area scale factor:

1\text{ square inch} = 20^2\text{ square yards} = 400\text{ square yards}

. Finally, multiply the given map area by this factor:

15\text{ square inches} \times 400 = 6,000\text{ square yards}

.

Adım Adım Çözüm

1

Determine the linear scale factor for one full inch.

1\text{ inch} = 60\text{ feet}

The map scale is given as

\frac{1}{2}\text{ inch} = 30\text{ feet}

. Multiplying both sides by 2 gives the equivalent measurement for 1 inch.

2

Convert the linear scale into yards to match the requested final unit.

1\text{ inch} = 20\text{ yards}

Since there are 3 feet in a yard, dividing 60 feet by 3 yields 20 yards. This simplifies calculations by converting units early.

3

Calculate the area scale factor by squaring the linear scale.

1\text{ square inch} = 400\text{ square yards}

Area is a two-dimensional measurement. The linear scale must be squared (

20 \times 20 = 400

) to establish the relationship between map area and actual area.

4

Calculate the actual area of the park.

15 \times 400 = 6,000\text{ square yards}

Multiply the map area by the area scale factor to find the true dimensions.

Anahtar Kavram

Scale Factors for Area and Unit Conversions

Alternatif Yöntem

Instead of converting to yards early, you can find the actual area in square feet first. The area scale factor in feet is

60^2 = 3,600\text{ square feet per square inch}

. Multiply by the map area:

15 \times 3,600 = 54,000\text{ square feet}

. Then convert to square yards by dividing by 9:

54,000 \div 9 = 6,000\text{ square yards}

.

Tahmini Süre:1m 30s

Soru 11Soru →

A floor plan for a new office space is drawn using a scale of $1\text{ inch} = 6\text{ feet}$ . If the main conference room is $3.5\text{ inches}$ long on the floor plan, what is the actual length of the conference room in feet?

Cevabı ve açıklamayı göster

Cevap: 21

Cevap

21

The scale indicates that every

1\text{ inch}

on the drawing corresponds to

6\text{ feet}

in reality. Therefore, a length of

3.5\text{ inches}

on the drawing represents an actual length of

3.5 \times 6 = 21\text{ feet}

.

Adım Adım Çözüm

1

Identify the given scale and drawing measurement.

Scale:

1\text{ inch} = 6\text{ feet}

, Drawing length:

3.5\text{ inches}

.

These are the key values needed to calculate the actual length.

2

Multiply the drawing length by the scale factor.

3.5 \times 6 = 21

Since each inch represents

6\text{ feet}

, multiplying the number of inches by

6

gives the total actual feet.

Anahtar Kavram

Using a scale factor to convert a drawing measurement to an actual measurement.

Soru 12Soru →

Match each military supply and maintenance inventory count to its correctly simplified ratio.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches pair each scenario to its mathematically simplified ratio.

Each correct match is found by writing the quantities in the requested order and then dividing all terms by their greatest common factor.

Adım Adım Çözüm

1

Identify the quantities to be compared in the exact order requested by each scenario.

The numerical values are extracted in the correct sequence.

Ratios represent a specific relationship, and the order of the terms must match the order requested in the text.

2

Write the initial unsimplified ratios for all scenarios.

The initial ratios are

24:8

,

10:15

,

20:5

, and

10:5:15

.

This establishes the mathematical baseline before simplification.

3

Find the greatest common factor (GCF) for the numbers in each ratio and divide.

The simplified ratios are

3:1

,

2:3

,

4:1

, and

2:1:3

.

Ratios should always be expressed in their simplest terms.

Anahtar Kavram

Basic Ratios and Simplification

Soru 13Soru →

Determine the correct numerical values to complete the statement.

Aşağıdaki boşlukları doldurun

An aviation mechanic calibrates

8

flight instruments in

3

hours. If a squadron requires

20

instruments to be calibrated, and the mechanic works at the same constant rate, it will take hours and minutes to complete the task.

Cevabı ve açıklamayı göster

Cevap

It will take 7 hours and 30 minutes.

Because the work rate is constant, the number of instruments calibrated is directly proportional to the time taken. Setting up the proportion 8/3 = 20/x yields x = 7.5 hours. Converting the remaining 0.5 hours to minutes gives 30 minutes, resulting in a total required time of 7 hours and 30 minutes.

Adım Adım Çözüm

1

Set up a direct proportion comparing the number of instruments to the time in hours.

\frac{8 \text{ instruments}}{3 \text{ hours}} = \frac{20 \text{ instruments}}{x \text{ hours}}

When a rate is constant, the ratio of work completed to time taken remains equivalent.

2

Cross-multiply the terms to create a linear equation.

8x = 3 \times 20

Cross-multiplying is the standard algebraic method to solve for an unknown in a proportional relationship.

3

Simplify the equation and solve for

x

.

8x = 60 \Rightarrow x = \frac{60}{8} = 7.5 \text{ hours}

Isolating

x

provides the total required time expressed in hours.

4

Convert the decimal portion of the hours into minutes.

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

The question specifically requires the final answer to be formatted in hours and minutes.

Anahtar Kavram

Direct proportions with unit conversion

Tahmini Süre:1m 30s

Soru 14Soru →

A base commander is distributing a total of $240$ new combat helmets to two platoons in a ratio of $5:3$ . How many helmets will the platoon with the smaller share receive?

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

The platoon with the smaller share will receive 90 helmets.

To find the smaller share, first add the ratio parts together to find the total number of parts (

5 + 3 = 8

). Next, divide the total number of helmets by the total parts (

240 \div 8 = 30

) to find the value of one part. Finally, multiply the value of one part by the smaller ratio number (

30 \times 3 = 90

).

Adım Adım Çözüm

1

Find the total number of ratio parts

5 + 3 = 8

parts

To determine how many equal parts the total number of helmets is divided into.

2

Calculate the value of one part

240 \div 8 = 30

helmets per part

To find out how many helmets correspond to a single ratio unit.

3

Calculate the smaller share

3 \times 30 = 90

helmets

To find the number of helmets for the platoon receiving the smaller portion of the ratio.

Anahtar Kavram

Dividing a total quantity into a given ratio by finding the value of a single part.

Tahmini Süre:45s

Soru 15Soru →

A naval base allocates its quarterly maintenance budget among structural repairs, mechanical overhauls, and routine inspections in the ratio of $5:8:2$ . The amount allocated for mechanical overhauls is $\$45,000$ more than the amount allocated for routine inspections. Based on this information, calculate the total quarterly maintenance budget.

Aşağıdaki boşlukları doldurun

The total quarterly maintenance budget is $\$.

Cevabı ve açıklamayı göster

Cevap

The total quarterly maintenance budget is $112,500.

The difference between the mechanical overhaul and routine inspection allocations represents 6 parts of the total ratio (

8 - 2

). By dividing the given dollar difference (

\$45,000

) by 6, we find that a single ratio part is worth

\$7,500

. Since the entire maintenance budget is composed of 15 parts in total (

5 + 8 + 2

), multiplying 15 by

\$7,500

gives the correct total budget of

\$112,500

.

Adım Adım Çözüm

1

Identify the number of ratio parts assigned to each category.

Structural =

5

parts, Mechanical =

8

parts, Routine =

2

parts.

This establishes the proportional distribution of the entire budget.

2

Determine the difference in parts between mechanical overhauls and routine inspections.

8

parts -

2

parts =

6

parts.

The problem provides the dollar amount for the difference between these two specific allocations, so we must find the corresponding difference in ratio parts.

3

Calculate the monetary value of a single ratio part.

\$45,000 \div 6 = \$7,500

per part.

Dividing the total monetary difference by the difference in parts gives the unit value of one part.

4

Find the total number of parts in the ratio and calculate the final budget.

Total parts =

5 + 8 + 2 = 15

. Total budget =

15 \times \$7,500 = \$112,500

.

Multiplying the total number of parts by the value of a single part yields the overall budget.

Anahtar Kavram

Using the monetary difference between two shares in a ratio to determine the value of a single unit, which is then used to find the total amount.

Soru 16Soru →

A state map uses a scale where $1\text{ inch}$ represents $50\text{ miles}$ . If the straight-line distance between two cities on the map is $2.5\text{ inches}$ , what is the actual distance between the two cities?

20\text{ miles}

12.5\text{ miles}

125\text{ miles}

2.5\text{ miles}

Cevabı ve açıklamayı göster

Cevap: $125\text{ miles}$

Cevap

125\text{ miles}

The map scale indicates that every 1 inch on the map corresponds to 50 actual miles. To find the real-world distance for a measurement of 2.5 inches on the map, you must multiply the map measurement by the scale factor: 2.5 × 50 = 125 miles.

Adım Adım Çözüm

1

Set up a proportion or identify the scale factor from the problem description.

1\text{ inch} = 50\text{ miles}

This establishes how many real-world miles are represented by each inch on the map.

2

Multiply the measured map distance by the scale factor to find the actual distance.

2.5\text{ inches} \times 50\text{ miles/inch} = 125\text{ miles}

Since every inch represents 50 miles, 2.5 inches will represent 2.5 groups of 50 miles.

Anahtar Kavram

Using a linear scale factor to convert a map measurement to a real-world distance.

Soru 17Soru →

A technician spends $2$ hours diagnosing a communications system and $30$ minutes completing the repair. What is the ratio of the time spent diagnosing to the time spent repairing?

4:1

1:15

1:4

4:5

Cevabı ve açıklamayı göster

Cevap: $4:1$

Cevap

The correct ratio is

4:1

.

To find the correct ratio, both times must be expressed in the same unit. Converting

2

hours gives

120

minutes. The requested ratio of diagnosing time to repairing time is

120:30

. Dividing both numbers by their greatest common divisor (

30

) yields the simplest form,

4:1

.

Adım Adım Çözüm

1

Convert the diagnosing time to the same unit as the repair time.

2

hours =

2 \times 60

minutes =

120

minutes.

Ratios must compare quantities in the same units to be mathematically meaningful.

2

Set up the ratio of diagnosing time to repair time.

120

minutes :

30

minutes

The question specifically asks for the ratio of diagnosing time first, followed by repair time.

3

Simplify the ratio to its lowest terms.

120 : 30 = 4 : 1

Divide both parts of the ratio by their greatest common divisor, which is

30

.

Anahtar Kavram

To find a ratio between two real-world quantities, ensure they share the same units, express their relationship in the exact order requested, and simplify.

Soru 18Soru →

A unit of $36$ combat engineers can construct a perimeter defense system in $3$ hours and $45$ minutes. If the commanding officer orders the defense system to be completed in exactly $1$ hour and $40$ minutes, how many combat engineers in total must be assigned to the task, assuming all engineers work at the same constant rate?

16

32

81

89

Cevabı ve açıklamayı göster

Cevap: 81

Cevap

81

The problem describes an inverse proportion because decreasing the allotted time requires increasing the number of workers to accomplish the identical task. By converting all times to minutes, the total required work effort is calculated as

36

engineers

\times

225

minutes =

8,100

engineer-minutes. Dividing this constant work effort by the new strict time constraint of

100

minutes yields the total required personnel:

81

engineers.

Adım Adım Çözüm

1

Convert both time durations entirely into minutes to establish a single, consistent unit of measure.

3

hours and

45

minutes =

(3 \times 60) + 45 = 225

minutes.

1

hour and

40

minutes =

(1 \times 60) + 40 = 100

minutes.

Converting to minutes prevents errors associated with mixed units and base-60 time fractions.

2

Set up an inverse proportion equation, recognizing that the total amount of work (personnel multiplied by time) remains constant.

\text{Engineers}_1 \times \text{Time}_1 = \text{Engineers}_2 \times \text{Time}_2 \Rightarrow 36 \times 225 = x \times 100

When completing a fixed task, the number of workers and the time required are inversely proportional.

3

Solve the equation to find the total number of engineers required for the new timeframe.

8,100 = 100x \Rightarrow x = \frac{8,100}{100} = 81

engineers.

Dividing the total required engineer-minutes (

8,100

) by the available minutes (

100

) determines the necessary workforce.

Anahtar Kavram

Inverse Proportions

Tahmini Süre:1m 30s

Soru 19Soru →

An aviation maintenance team is preparing a specialized anti-icing fluid blend consisting of glycol, water, and an additive in the ratio of $8:3:1$ by volume. If the team needs to prepare a total of $60$ gallons of this fluid, how many more quarts of glycol are required than water?

25

100

150

160

Cevabı ve açıklamayı göster

Cevap: 100

Cevap

100

To solve this, first find the total number of parts in the ratio by adding

8 + 3 + 1 = 12

parts. Since the total volume is

60

gallons, each part represents

60 \div 12 = 5

gallons. Glycol accounts for

8

parts and water accounts for

3

parts, making a difference of

5

parts. The difference in gallons is

5 \times 5 = 25

gallons. Finally, convert gallons to quarts by multiplying by

4

(

1

gallon =

4

quarts), which gives

25 \times 4 = 100

quarts.

Adım Adım Çözüm

1

Determine the total number of parts in the ratio.

8 + 3 + 1 = 12

parts

To find the volume of a single part, the total parts must be known.

2

Calculate the volume of one part in gallons.

60 \div 12 = 5

gallons per part

Dividing the total volume by the total parts gives the value of one part.

3

Find the difference in volume between glycol and water in gallons.

Glycol is

8

parts and water is

3

parts. Difference =

5

parts.

5 \times 5 = 25

gallons.

The question asks for the exact difference between the two components.

4

Convert the difference from gallons to quarts.

25 \times 4 = 100

quarts

The question specifically asks for the final answer in quarts, and there are

4

quarts in a gallon.

Anahtar Kavram

Dividing a total amount into a multi-part ratio and converting between standard units of measurement.

Soru 20Soru →

An aviation maintenance depot uses two automated riveting machines to assemble fuselage panels. Machine A can install $240$ rivets in $4$ hours. Machine B can install $30$ rivets every $20$ minutes. If both machines operate simultaneously, how many hours will it take them to install a total of $6,300$ rivets?

30

42

63

70

Cevabı ve açıklamayı göster

Cevap: 42

Cevap

The correct answer is 42 hours.

To solve this problem, both machine rates must first be converted to the same time unit (hours). Machine A operates at

60

rivets per hour (

240 \div 4

). Machine B operates at

90

rivets per hour (since there are three

20

-minute periods in an hour,

30 \times 3 = 90

). Together, they install

150

rivets per hour (

60 + 90

). Dividing the total requirement of

6,300

rivets by the combined rate of

150

rivets per hour yields exactly

42

hours.

Adım Adım Çözüm

1

Calculate the unit rate for Machine A in rivets per hour.

240 \text{ rivets} \div 4 \text{ hours} = 60 \text{ rivets per hour}

To combine the rates of both machines, they must be in the same unit of measure (rivets per hour).

2

Calculate the unit rate for Machine B in rivets per hour.

30 \text{ rivets} \div 20 \text{ minutes} = 1.5 \text{ rivets per minute}

.

1.5 \times 60 = 90 \text{ rivets per hour}

Machine B's rate is given per

20

minutes, so it must be scaled up to an hourly rate by multiplying by

60

minutes.

3

Calculate the combined hourly rate of both machines.

60 + 90 = 150 \text{ rivets per hour}

The machines operate simultaneously, so their individual hourly rates are added together.

4

Determine the total time required to install

6,300

rivets.

6,300 \text{ rivets} \div 150 \text{ rivets per hour} = 42 \text{ hours}

Dividing the total work required by the combined hourly rate gives the total time needed.

Anahtar Kavram

Unit Rates and Multi-Step Rate Applications

Ratios and Proportions

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm