Inverse Proportions Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB)

Question 1Question →

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}

Show answer & explanation

Answer: $32\text{ knots}$

Answer

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}

.

Step-by-Step Solution

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.

Key Concept

Inverse Proportions in Speed and Time

Estimated Time:1m 30s

Question 2Question →

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

Show answer & explanation

Answer: 81

Answer

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.

Step-by-Step Solution

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.

Key Concept

Inverse Proportions

Estimated Time:1m 30s

Question 3Question →

A submarine leaves port with enough provisions to feed its standard crew of $120$ personnel for $60$ days. Before departure, $30$ additional mission specialists are assigned to the submarine. If the daily ration per person remains constant, how many days will the provisions last the newly expanded crew?

Show answer & explanation

Answer: 48

Answer

48

Because the total amount of food remains constant, an increase in personnel causes a proportional decrease in the number of days the food will last. The product of personnel and days is constant:

120 \times 60 = 150 \times x

. Solving for

x

gives

7200 / 150 = 48

days.

Step-by-Step Solution

1

Calculate the total amount of provisions available.

120 \text{ personnel} \times 60 \text{ days} = 7200 \text{ person-days}

In an inverse proportion involving resources, the product of the number of consumers and the time the resource lasts represents the constant total resource.

2

Calculate the total number of personnel after the addition.

120 + 30 = 150 \text{ personnel}

The problem states that 30 additional specialists were assigned, so the new crew size is the sum of the original and additional personnel.

3

Calculate the new duration the provisions will last.

7200 \text{ person-days} \div 150 \text{ personnel} = 48 \text{ days}

Dividing the total constant resource by the new number of consumers yields the new time duration.

Key Concept

Inverse Proportions

Estimated Time:1m 0s

Question 4Question →

Match each inverse proportion scenario to its correct calculated value.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each scenario represents an inverse proportion and must be matched with the value that maintains the constant product of the initial conditions.

Each match is mathematically determined by setting up an inverse proportion equation where the product of the initial states equals the product of the final states (

x_1 \cdot y_1 = x_2 \cdot y_2

). Dividing the initial product by the new given value provides the correct corresponding answer.

Step-by-Step Solution

1

Identify that each scenario represents an inverse relationship.

As one quantity increases (troops, speed, workers, gear teeth), the other quantity must decrease.

This determines that the product of the variables is constant (

x_1 \cdot y_1 = x_2 \cdot y_2

), rather than the ratio.

2

Calculate the constant product (

k

) for the initial conditions in each scenario.

The constants are

15,000

troop-days,

2,400\text{ miles}

,

96\text{ man-hours}

, and

7,200\text{ teeth-RPM}

.

The constant represents the total amount of work, distance, resources, or mechanical transfer that must remain unchanged.

3

Divide the constant product by the new given variable to find the unknown required value.

The resulting values are

20\text{ days}

,

4\text{ hours}

,

6\text{ hours}

, and

160\text{ RPM}

.

Solving the equation

x_1 \cdot y_1 = x_2 \cdot y_2

for the unknown variable

y_2

requires dividing the constant by

x_2

.

Key Concept

Inverse Proportions

Question 5Question →

A naval shipyard assigns $15$ automated painting drones to coat the hull of a cruiser in exactly $2\text{ hours}$ . If $5$ of these drones are taken offline for maintenance before the job begins, how many minutes will it take the remaining drones to complete the coating?

80

360

180

3

Show answer & explanation

Answer: $180$

Answer

It will take 180 minutes for the remaining drones to complete the hull coating.

The correct answer is

180

. First, subtract the

5

offline drones from the initial

15

to find that

10

drones are actively working. Because the number of drones and the time required are inversely proportional, their product remains constant. The equation is

15 \times 2 = 10 \times T_2

, which solves to

T_2 = 3\text{ hours}

. Finally, converting

3\text{ hours}

to minutes yields

3 \times 60 = 180\text{ minutes}

.

Step-by-Step Solution

1

Determine the number of active drones.

15 - 5 = 10\text{ drones}

The 5 offline drones cannot work, so only the remaining drones will paint the hull.

2

Set up an inverse proportion equation.

15\text{ drones} \times 2\text{ hours} = 10\text{ drones} \times T_2

Work rate problems involve inverse proportions; as the number of workers decreases, the time required increases proportionally to keep the total work constant.

3

Solve for the new time (

T_2

) in hours.

30 = 10 \times T_2 \Rightarrow T_2 = 3\text{ hours}

Dividing the total required drone-hours (

30

) by the number of active drones (

10

) isolates the time.

4

Convert the required time into minutes.

3 \times 60 = 180\text{ minutes}

The question explicitly asks for the final answer in minutes, not hours.

Key Concept

Inverse Proportions

Question 6Question →

A reconnaissance drone squadron currently takes $14\text{ hours}$ to completely map a designated tactical sector when using $6$ identical drones simultaneously. If the task force commander requires the exact same sector to be mapped in $4\text{ hours}$ , how many total drones must be deployed?

Show answer & explanation

Answer: 21

Answer

21

drones

The scenario describes an inverse proportion because increasing the number of active drones decreases the time required to finish the mapping. The total mapping workload is a constant

84

drone-hours (

6\text{ drones} \times 14\text{ hours}

). To complete this exact same workload in a compressed timeframe of

4

hours, a total of

21

drones are needed (

84 \div 4

).

Step-by-Step Solution

1

Identify the relationship between the variables.

The number of drones and the time required are inversely proportional.

Completing a fixed amount of work faster requires proportionally more equipment.

2

Calculate the total amount of work required (the constant of variation).

6\text{ drones} \times 14\text{ hours} = 84\text{ drone-hours}

In an inverse proportion (

x \times y = k

), the product of the two variables remains constant.

3

Calculate the new number of drones required for the target time.

84\text{ drone-hours} \div 4\text{ hours} = 21\text{ drones}

Divide the total work constant by the new required time to solve for the missing variable.

Key Concept

Inverse proportions occur when one variable increases while another decreases such that their product remains constant (

x_1 \times y_1 = x_2 \times y_2

).

Question 7Question →

A military transport vehicle's winch system uses a gear train consisting of three gears. The driving gear (Gear A) has $60$ teeth and rotates at $150\text{ RPM}$ . It meshes with an idler gear (Gear B) which has $45$ teeth. Gear B meshes directly with the driven gear (Gear C). If Gear C must rotate at a speed of $5\text{ revolutions per second}$ to safely operate the winch, how many teeth must Gear C have?

30

120

300

1800

Show answer & explanation

Answer: 30

Answer

30 teeth

The number of teeth on a gear and its rotational speed are inversely proportional. In a gear train, an idler gear (Gear B) only changes the direction of rotation, not the mathematical ratio between the first and last gears (

T_A \times S_A = T_C \times S_C

). First, ensure the speed units match by converting Gear C's speed to RPM:

5\text{ rev/s} \times 60\text{ seconds/minute} = 300\text{ RPM}

. Then, equate the products for Gear A and Gear C:

60 \times 150 = T_C \times 300

. Solving for

T_C

gives

9000 \div 300 = 30

teeth.

Step-by-Step Solution

1

Identify the relationship in a gear train.

The product of teeth and rotational speed is constant across meshed gears (

T_1 \times S_1 = T_2 \times S_2

). The idler gear (Gear B) does not change the final relationship between Gear A and Gear C.

To establish the mathematical formula needed and eliminate unnecessary variables.

2

Convert Gear C's speed to the same unit as Gear A (RPM).

5\text{ revolutions/second} \times 60\text{ seconds/minute} = 300\text{ RPM}

.

Both gears must use the exact same rate units before an accurate proportion can be calculated.

3

Set up the inverse proportion equation connecting Gear A to Gear C.

60\text{ teeth} \times 150\text{ RPM} = T_C \times 300\text{ RPM}

.

To isolate the unknown variable based on the constant product rule of gears.

4

Solve the equation for the number of teeth on Gear C (

T_C

).

9000 = 300 \times T_C

, which simplifies to

T_C = 9000 \div 300 = 30\text{ teeth}

.

To find the final required dimension for the driven gear.

Key Concept

Inverse Proportions in Compound Mechanics

Alternative Method

You can solve this in two steps by finding the idler gear's speed first. Gear B's speed:

60 \times 150 = 45 \times S_B \rightarrow 9000 = 45 \times S_B \rightarrow S_B = 200\text{ RPM}

. Then relate Gear B to Gear C (converted to

300\text{ RPM}

):

45 \times 200 = T_C \times 300 \rightarrow 9000 = 300 \times T_C \rightarrow T_C = 30

.

Estimated Time:1m 45s

Question 8Question →

A military logistics officer must calculate operational timelines based on available resources. Connect each situation below to its correctly calculated duration. Assume all rates are constant and inversely proportional.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The scenarios pair correctly with 12 hours, 6 hours, 9 hours, and 30 hours respectively based on their constant products.

Each scenario involves an inverse relationship where the product of the two given quantities forms a constant. Dividing this constant by the new condition yields the correctly paired time.

Step-by-Step Solution

1

Identify the relationship for each scenario.

All four scenarios represent inverse proportions, meaning the product of the two variables (e.g., workers and time, speed and time) remains constant (

x_1 \times y_1 = x_2 \times y_2

).

When one variable increases (like adding more workers or driving faster), the time required decreases proportionally.

2

Calculate the constant product for each individual scenario.

Engineer-hours:

18 \times 14 = 252

. Distance:

36 \times 4.5 = 162

. Fan-hours:

15 \times 6 = 90

. Generator-hours:

16 \times 45 = 720

.

This establishes the total amount of work, distance, or resource available before the change in variables.

3

Divide each constant product by the new given variable to find the new time.

Runway:

252 \div 21 = 12

. Boat:

162 \div 27 = 6

. Fans:

90 \div 10 = 9

. Generators:

720 \div 24 = 30

.

Applying the inverse proportion formula (

y_2 = \frac{k}{x_2}

) isolates the unknown time.

Key Concept

Inverse Proportions in Work, Speed, and Resource Rates

Question 9Question →

A maintenance crew uses $4$ identical pumps to empty a large aviation fuel tank in $12$ hours. If the crew decides to use $6$ of these identical pumps instead, how many hours will it take to completely empty the tank?

Show answer & explanation

Answer: 8

Answer

8 hours

Because the number of pumps and time are inversely proportional, their product must remain constant. The initial scenario gives a product of 4 pumps × 12 hours = 48 pump-hours. To find the new time, divide the constant product of 48 by the new number of pumps (6). 48 ÷ 6 = 8 hours.

Step-by-Step Solution

1

Identify the mathematical relationship between the variables.

The number of pumps and the time required are inversely proportional.

When working at a constant rate, increasing the number of machines decreases the time needed to complete the same job.

2

Set up the inverse proportion equation.

4 \times 12 = 6 \times x

For inverse proportions, the product of the two variables remains constant (

x_1 y_1 = x_2 y_2

).

3

Calculate the total pump-hours and solve for the new time.

x = 48 \div 6 = 8

Dividing the constant product (48) by the new number of pumps (6) isolates the unknown time variable.

Key Concept

Inverse Proportions

Estimated Time:45s

Question 10Question →

A detail of $18$ recruits can sweep and mop a large warehouse in $20$ minutes. If the schedule allows $1.5$ hours for the task, how many recruits, working at the same constant rate, are required?

240

24

4

81

Show answer & explanation

Answer: $4$

Answer

The correct answer is

4

.

The scenario describes an inverse proportion because giving the detail more time to complete the task means fewer recruits are needed. By converting

1.5

hours to

90

minutes and applying the inverse proportion relationship (

18 \times 20 = x \times 90

), we find that exactly

4

recruits are required.

Step-by-Step Solution

1

Calculate the total workload required to complete the task.

W_1 \times T_1 = 18 \times 20 = 360

recruit-minutes.

In inverse proportion problems involving work, the product of the number of workers and their time represents a constant total amount of work.

2

Convert the new target time into the same unit (minutes).

1.5 \text{ hours} = 1.5 \times 60 = 90 \text{ minutes}

.

Units must be consistent across the equation to yield a valid result.

3

Divide the total workload by the new target time to find the required number of recruits.

360 \div 90 = 4

recruits.

Applying the inverse proportion formula

W_2 = \text{Total Work} \div T_2

reveals how many workers are needed for the longer duration.

Key Concept

Inverse Proportions

Alternative Method

You can also solve this by finding the total work in 'recruit-hours'. First, convert

20

minutes to

1/3

of an hour. The total work is

18 \text{ recruits} \times (1/3) \text{ hour} = 6 \text{ recruit-hours}

. Since the new time is

1.5

hours, the required number of recruits is

6 \div 1.5 = 4

recruits.

Estimated Time:1m 0s

Question 11Question →

Match each military work scenario with the correct time it would take to complete the task. Assume all personnel work at the same constant rate.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The 4 mechanics will take 3 hours, the 6 medics will take 4 hours, the 10 logisticians will take 2 hours, and the 6 engineers will take 5 hours.

Each scenario is an inverse proportion problem where the product of workers and time is a constant (total work). To find the new time, multiply the original workers by the original hours, then divide by the new number of workers.

Step-by-Step Solution

1

Calculate the constant total work for each scenario.

Scenario 1: 12 hours. Scenario 2: 24 hours. Scenario 3: 20 hours. Scenario 4: 30 hours.

In inverse proportions, the product of workers and time is a constant (representing the total work required).

2

Divide the total work by the new number of workers.

12/4 = 3 hours; 24/6 = 4 hours; 20/10 = 2 hours; 30/6 = 5 hours.

Since the total work remains constant, dividing it by the available workforce yields the new time.

Key Concept

Inverse Proportions

Inverse Proportions

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Common Mistakes

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Common Mistakes