Ratios and Proportions Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 6

Question 101Question →

A shipyard fabricator welds $30$ inches of steel seam every $15$ minutes. Operating at this exact constant rate, how many feet of steel seam will the fabricator weld in a $6$ -hour shift?

15

60

720

8,640

Show answer & explanation

Answer: 60

Answer

60

The correct answer is derived by first converting the 6-hour shift to 360 minutes. Setting up the direct proportion yields a total of 720 inches (

30 \times 360 \div 15

). Finally, dividing by 12 correctly converts the length into 60 feet.

Step-by-Step Solution

1

Harmonize the time units by converting the 6-hour shift into minutes.

6 \text{ hours} \times 60 \text{ minutes/hour} = 360 \text{ minutes}

The given rate is in minutes, so the total operational time must also be in minutes to set up a valid proportion.

2

Set up a direct proportion or find the unit rate of welding.

\frac{30 \text{ inches}}{15 \text{ minutes}} = 2 \text{ inches per minute}

Determining the rate per minute simplifies the calculation for the entire shift.

3

Calculate the total length of seam welded in inches.

2 \text{ inches/minute} \times 360 \text{ minutes} = 720 \text{ inches}

Applying the constant rate to the total operational time yields the total production in the initial unit.

4

Convert the total inches into the requested unit of feet.

720 \text{ inches} \div 12 \text{ inches/foot} = 60 \text{ feet}

The question explicitly asks for the final answer in feet, requiring a unit conversion.

Key Concept

Direct proportions with unit conversion

Question 102Question →

An engine transmission system uses three interconnected gears: Gear A, Gear B, and Gear C. The ratio of the number of teeth on Gear A to the number of teeth on Gear B is $4:3$ . The ratio of the number of teeth on Gear B to the number of teeth on Gear C is $5:2$ . If Gear C has exactly $18$ teeth, what is the total number of teeth on all three gears combined?

Show answer & explanation

Answer: 123

Answer

123

The correct answer is found by linking the two separate proportions using their common element, Gear B. By converting the A:B ratio (

4:3

) and the B:C ratio (

5:2

) to share a common value for B (

15

), we get a unified, continuous ratio for A:B:C of

20:15:6

. The problem states that Gear C has

18

teeth. Since Gear C represents

6

parts in our combined ratio, each part must equal

3

teeth (

18 \div 6 = 3

). The total number of parts across all three gears is

41

(

20 + 15 + 6

). Multiplying the

41

total parts by

3

teeth per part results in a total of

123

teeth for the entire system.

Step-by-Step Solution

1

Align the separate ratios based on the common element.

Gear A : Gear B =

4:3

and Gear B : Gear C =

5:2

.

To compare all three gears simultaneously, Gear B must have the same value in both ratios.

2

Find a common multiple for the ratio numbers representing Gear B (

3

and

5

).

The least common multiple of

3

and

5

is

15

.

Creating a common denominator allows us to link the two independent proportions.

3

Scale both ratios so Gear B is represented by

15

parts.

Ratio A:B becomes

(4 \times 5):(3 \times 5) = 20:15

. Ratio B:C becomes

(5 \times 3):(2 \times 3) = 15:6

.

Multiplying both sides of a ratio by the same number maintains the proportion.

4

Combine into a single three-part ratio and determine total parts.

Combined ratio A:B:C is

20:15:6

. Total ratio parts =

20 + 15 + 6 = 41

.

This single ratio precisely maps the relationships between all three gears in the system.

5

Calculate the multiplier using Gear C's known teeth count.

6

parts =

18

teeth. Therefore,

1

part =

18 / 6 = 3

teeth.

This establishes the real-world value of a single ratio unit in this specific transmission system.

6

Calculate the total number of teeth on all gears combined.

41

total parts

\times 3

teeth per part =

123

teeth.

Multiplying the total number of parts by the value of a single part gives the final aggregate quantity.

Key Concept

Combining overlapping ratios by calculating the least common multiple of the shared term, then scaling the combined ratio to determine unknown totals.

Question 103Question →

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 104Question →

A combat engineering squad fills $150$ sandbags in $40$ minutes. Operating at this exact constant rate, how many sandbags can the squad fill in $4$ hours?

Show answer & explanation

Answer: 900

Answer

900

Because the production rate is constant, the number of sandbags filled is directly proportional to the time spent. Harmonizing the time units by converting

4

hours into

240

minutes allows you to correctly scale up the initial ratio of

150

sandbags per

40

minutes.

Step-by-Step Solution

1

Convert the total working time from hours to minutes.

4 \text{ hours} \times 60 \text{ minutes/hour} = 240 \text{ minutes}

The given rate is in minutes (

150

sandbags per

40

minutes), so the target time must also be in minutes to set up an accurate proportion.

2

Set up a direct proportion.

\frac{150 \text{ sandbags}}{40 \text{ minutes}} = \frac{x \text{ sandbags}}{240 \text{ minutes}}

Since the squad works at a constant rate, the ratio of sandbags filled to time spent must remain equal.

3

Solve the proportion for

x

.

x = 900

The denominator

40

must be multiplied by

6

to get

240

. To keep the fraction equivalent, multiply the numerator

150

by

6

, yielding

900

.

Key Concept

Direct Proportions with Unit Harmonization

Estimated Time:1m 0s

Question 105Question →

Match each military and logistics operational scenario to the correct final mathematical ratio. Read the conditions carefully, as some require calculating a new ratio after a change occurs.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: Scenario 1 maps to 5:14, Scenario 2 maps to 2:7, Scenario 3 maps to 15:16, and Scenario 4 maps to 9:25.

Each scenario involves distinct ratio logic: Scenario 1 requires combining parts to find a whole. Scenarios 2 and 4 require breaking a total into parts using an initial ratio, subtracting from one part, and writing a new simplified ratio. Scenario 3 requires using part-to-part logic to find an unknown part given a specific base, then comparing the two results.

Step-by-Step Solution

1

Calculate the ratios for the Naval Strike Group (Scenario 1).

The total proportion is

2

(cruisers) +

5

(destroyers) +

7

(frigates) =

14

. The ratio of destroyers to total ships is

5:14

.

You must establish the total baseline by summing the parts before finding the part-to-whole ratio.

2

Determine the new personnel ratio (Scenario 2).

Initially, officers =

18

and enlisted =

42

. Subtracting

6

officers leaves

12

. The new ratio is

12:42

, simplified to

2:7

.

Changes to one subset of a population require determining the original absolute values first.

3

Compare the fuel additive amounts (Scenario 3).

Correct additive is

\frac{3}{8} \times 40 = 15

. Mistaken additive is

\frac{4}{10} \times 40 = 16

. The ratio is

15:16

.

Scaling both ratios up to the common base fuel amount of

40

reveals the exact additive quantities to compare.

4

Adjust the convoy vehicles ratio (Scenario 4).

Initially, cargo =

25

and escort =

10

. Subtracting

1

escort leaves

9

. The required ratio of escort to cargo is

9:25

.

Calculating initial absolute quantities is necessary before applying the subtraction of

1

vehicle and interpreting the final specific order requested.

Key Concept

Applying part-to-part and part-to-whole ratio relationships to calculate absolute values, applying condition changes, and then re-evaluating and simplifying the final ratios.

Alternative Method

For Scenario 2, you can also calculate the fractional drop without finding absolute enlisted numbers. The officer count drops from

18

to

12

, which is exactly

\frac{2}{3}

of the original amount. Thus, the new ratio must simply be

\frac{2}{3}

of the original officer portion (

3

) against the same enlisted portion (

7

), resulting directly in a

2:7

ratio.

Estimated Time:2m 30s

Question 106Question →

A defense contractor manufactures three types of vehicle armor panels: Type A, Type B, and Type C. The ratio of Type A panels to Type B panels produced is $2:3$ . The ratio of Type B panels to Type C panels produced is $4:5$ . If the contractor produces a total of $1,400$ panels across all three types, how many Type C panels are produced?

320

500

600

1,000

Show answer & explanation

Answer: 600

Answer

600

To find the number of Type C panels, the two separate ratios must first be combined into one continuous ratio. Since Type B is in both ratios (

3

parts in the first,

4

parts in the second), we find a common multiple, which is

12

. Scaling the ratios gives Type A : Type B =

8:12

and Type B : Type C =

12:15

. Combined, the ratio is

8:12:15

. Adding these gives

35

total parts. Dividing the

1,400

total panels by

35

reveals each part equals

40

panels. Because Type C represents

15

of those parts, we multiply

15

by

40

to get

600

Type C panels.

Step-by-Step Solution

1

Identify the overlapping category (Type B) in both given ratios and find the least common multiple for its values (

3

and

4

).

The least common multiple of

3

and

4

is

12

.

To compare all three panel types simultaneously, they must be expressed within a single, unified ratio sequence.

2

Scale both original ratios so that Type B is represented by

12

in both.

Multiply the first ratio (

2:3

) by

4

to get

8:12

. Multiply the second ratio (

4:5

) by

3

to get

12:15

.

Multiplying both sides of a ratio by the same number keeps the proportions mathematically equivalent.

3

Combine the scaled ratios into a single three-part ratio and calculate the total number of parts.

The unified ratio of Type A to Type B to Type C is

8:12:15

. The total number of parts is

8 + 12 + 15 = 35

.

The total number of parts represents the entire production batch in ratio units.

4

Divide the total actual number of panels by the total ratio parts to find the value of one part, then calculate the total for Type C.

1,400 \div 35 = 40

panels per part. Type C has

15

parts, so

15 \times 40 = 600

Type C panels.

This scales the ratio back up to the real-world quantities provided in the problem stem.

Key Concept

Combining Overlapping Ratios

Estimated Time:1m 30s

Question 107Question →

A combat engineering team lays $180$ feet of portable runway surface every $20$ minutes. If they continue working at this exact same constant rate, how many yards of runway surface can they lay during a standard $5$ -hour shift?

4

15

900

2,700

Show answer & explanation

Answer: 900

Answer

900

First, find the unit rate of runway laid per minute by dividing the given distance by the given time (

180 \text{ feet} \div 20 \text{ minutes} = 9 \text{ feet per minute}

). Next, convert the total shift time from hours to minutes (

5 \text{ hours} \times 60 \text{ minutes/hour} = 300 \text{ minutes}

). Multiply the unit rate by the total minutes to find the total distance in feet (

9 \text{ feet/minute} \times 300 \text{ minutes} = 2,700 \text{ feet}

). Finally, convert feet to yards by dividing by

3

(

2,700 \text{ feet} \div 3 \text{ feet/yard} = 900 \text{ yards}

).

Step-by-Step Solution

1

Determine the unit rate of construction in feet per minute.

180 \text{ feet} \div 20 \text{ minutes} = 9 \text{ feet per minute}

Finding the unit rate allows for easy calculation of distance over any given amount of time.

2

Convert the total shift time from hours to minutes.

5 \text{ hours} \times 60 \text{ minutes/hour} = 300 \text{ minutes}

The unit rate is in feet per minute, so the total time must also be in minutes to maintain consistent units.

3

Calculate the total distance constructed in feet.

9 \text{ feet/minute} \times 300 \text{ minutes} = 2,700 \text{ feet}

Multiplying the rate by the total time gives the total amount of surface laid.

4

Convert the total distance from feet to yards.

2,700 \text{ feet} \div 3 \text{ feet/yard} = 900 \text{ yards}

The question specifically asks for the final answer in yards, requiring division by

3

.

Key Concept

Direct proportions with multiple unit conversions

Alternative Method

Instead of breaking the problem down into steps, you can set up a single dimensional analysis equation:

\frac{180 \text{ ft}}{20 \text{ min}} \times \frac{60 \text{ min}}{1 \text{ hr}} \times \frac{5 \text{ hr}}{1} \times \frac{1 \text{ yd}}{3 \text{ ft}} = 900 \text{ yd}

.

Estimated Time:2m 0s

Question 108Question →

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 109Question →

Fill in the blank with the correct numerical value.

Fill in the blanks below

A logistics detachment inspects

30

cargo containers in

45

minutes. Operating at this constant rate, it will take the detachment hours to inspect

160

cargo containers.

Show answer & explanation

Answer

4

Since the detachment operates at a constant rate, this is a direct proportion problem. They inspect

30

containers in

45

minutes. Using the proportion

\frac{30}{45} = \frac{160}{x}

, we find it takes

240

minutes to inspect

160

containers. Dividing

240

minutes by

60

minutes per hour gives the final answer of

4

hours.

Step-by-Step Solution

1

Set up a direct proportion relating the number of containers to the time in minutes.

\frac{30}{45} = \frac{160}{x}

Because the inspection rate is constant, the ratio of containers to time remains the same.

2

Solve the proportion for

x

to find the total time required in minutes.

30x = 160 \times 45 \Rightarrow 30x = 7200 \Rightarrow x = 240 \text{ minutes}

Cross-multiplying and isolating

x

determines the unknown time in the same units (minutes) used in the initial ratio.

3

Convert the total time from minutes to hours.

\frac{240 \text{ minutes}}{60 \text{ minutes/hour}} = 4 \text{ hours}

The question specifically asks for the final answer in hours, requiring a unit conversion.

Key Concept

Direct proportions with unit conversion

Question 110Question →

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 111Question →

During a facility expansion, a landscaping crew uses $2.5$ pounds of specialized grass seed to cover a $400$ -square-foot section of terrain. Operating at this exact constant rate, how many ounces of grass seed will be required to cover a $3,200$ -square-foot sector?

320

20

3,200

1.25

Show answer & explanation

Answer: $320$

Answer

The correct answer is

320

.

The correct answer is found by setting up a direct proportion. Since

3,200

square feet is exactly

8

times larger than

400

square feet, the amount of seed required must also be

8

times the original amount. Multiplying

2.5

pounds by

8

gives

20

pounds of seed. Finally, converting

20

pounds into ounces by multiplying by

16

(since

1

pound =

16

ounces) yields exactly

320

ounces.

Step-by-Step Solution

1

Determine the scale factor by dividing the target area by the initial area.

3,200 \div 400 = 8

This establishes how many times larger the new sector is compared to the original section.

2

Multiply the initial amount of grass seed by the scale factor to find the total pounds needed.

2.5 \times 8 = 20

pounds

Because this is a direct proportion, the amount of seed scales by the exact same factor as the area.

3

Convert the total weight from pounds to ounces.

20 \times 16 = 320

ounces

The question specifically asks for the final answer in ounces, and there are 16 ounces in 1 pound.

Key Concept

Direct Proportions and Unit Conversion

Alternative Method

Convert the initial seed amount to ounces first:

2.5

pounds =

40

ounces. Then, set up the proportion:

\frac{40}{400} = \frac{x}{3,200}

. Solving for

x

gives

x = 40 \times 8 = 320

.

Estimated Time:1m 15s

Question 112Question →

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 113Question →

Match each military personnel or equipment scenario to its correctly simplified part-to-part ratio.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches simplify each starting ratio by finding the greatest common divisor for both terms in the specified order.

Each scenario describes a straightforward part-to-part relationship. By arranging the quantities in the requested order and dividing both numbers by their greatest common divisor, we arrive at the simplest equivalent ratio for each match.

Step-by-Step Solution

1

Identify the initial quantities in the specific order requested by each scenario.

20:5 for trucks to command vehicles; 6:18 for medics to infantry; 10:15 for water to food; 12:8 for officers to enlisted.

Establishing the correct order is strictly required to prevent inverted ratios.

2

Determine the greatest common divisor (GCD) for each pair of numbers.

The GCD for 20 and 5 is 5. The GCD for 6 and 18 is 6. The GCD for 10 and 15 is 5. The GCD for 12 and 8 is 4.

Dividing by the GCD reduces the ratio to its lowest, simplest form.

3

Divide both terms of each raw ratio by their respective GCD.

20:5 becomes 4:1. 6:18 becomes 1:3. 10:15 becomes 2:3. 12:8 becomes 3:2.

This calculation yields the final simplified part-to-part ratio for each scenario, which directly pairs with the right-hand items.

Key Concept

Simplifying part-to-part ratios

Question 114Question →

A military supply convoy consumes $24$ gallons of diesel fuel to travel $140$ miles. Assuming the fuel consumption rate remains constant, how many gallons of diesel fuel are required for the convoy to travel a distance of $315$ miles?

Show answer & explanation

Answer: 54

Answer

54

By setting up a direct proportion between the distance traveled and the fuel consumed (

\frac{140}{24} = \frac{315}{x}

) and cross-multiplying, we solve for

x

and determine that the convoy requires exactly

54

gallons of fuel.

Step-by-Step Solution

1

Set up a direct proportion relating distance to fuel consumption.

\frac{140}{24} = \frac{315}{x}

Since the fuel consumption rate is constant, the ratio of miles traveled to gallons consumed remains the same.

2

Cross-multiply the proportion.

140x = 24 \times 315

Cross-multiplication converts the proportion into a linear equation to solve for the unknown variable

x

.

3

Simplify the right side of the equation.

140x = 7560

Multiplying the known values prepares the equation for the final isolation step.

4

Divide by

140

to solve for

x

.

x = \frac{7560}{140} = 54

Dividing both sides by

140

isolates

x

, revealing the required gallons of fuel.

Key Concept

Direct Proportions

Estimated Time:1m 0s

Question 115Question →

An architectural blueprint for a new warehouse uses a scale where $\frac{1}{4} \text{ inch} = 3 \text{ feet}$ . If a storage room has an area of $5 \text{ square inches}$ on the blueprint, what is the actual area of the storage room in square feet?

15

45

60

720

Show answer & explanation

Answer: 720

Answer

720

To find the actual area from a scale drawing, you must first convert the given scale to a unit linear scale. Multiplying

\frac{1}{4} \text{ inch} = 3 \text{ feet}

by

4

shows that

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

. Because we are dealing with area (a 2D measurement), we must square this linear relationship:

1 \text{ square inch} = 12^2 \text{ square feet} = 144 \text{ square feet}

. Finally, multiplying the drawing's area of

5 \text{ square inches}

by

144 \text{ square feet per square inch}

yields

720 \text{ square feet}

.

Step-by-Step Solution

1

Find the linear scale rate for a full inch.

Since

\frac{1}{4} \text{ inch} = 3 \text{ feet}

, multiply both sides by

4

to get

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

.

Establishing a base unit rate of

1 \text{ inch}

makes area calculations straightforward.

2

Convert the linear scale into an area scale factor.

Square both sides of the linear scale:

(1 \text{ inch})^2 = (12 \text{ feet})^2

, which simplifies to

1 \text{ square inch} = 144 \text{ square feet}

.

Area is a two-dimensional measurement, so the one-dimensional (linear) scale factor must be squared.

3

Calculate the actual area of the storage room.

Multiply the blueprint area by the area scale factor:

5 \times 144 = 720 \text{ square feet}

.

This scales the drawing's area up to the actual physical area.

Key Concept

When using scale drawings, the scale factor for area is the square of the linear scale factor.

Question 116Question →

A blueprint of a machine part uses a scale of $1$ inch = $2.5$ centimeters. If the part is drawn as $8$ inches long on the blueprint, what is the actual length of the part in centimeters?

20

3.2

2

200

Show answer & explanation

Answer: 20

Answer

20

The scale establishes a direct proportion where every

1

inch on the blueprint represents

2.5

centimeters in reality. To find the actual length corresponding to an

8

-inch drawing, you must multiply the drawing length by the scale factor:

8 \times 2.5 = 20

centimeters.

Step-by-Step Solution

1

Identify the given scale and the drawing measurement.

The scale is

1

inch =

2.5

cm, and the drawing length is

8

inches.

Establishing the relationship between the drawing and the actual object is required to set up the calculation.

2

Multiply the drawing length by the scale factor to find the actual length.

8 \times 2.5 = 20

cm.

Since every

1

inch represents

2.5

cm,

8

inches will represent

8

times that amount.

Key Concept

Scale Drawings

Question 117Question →

A military transport truck travels $180$ miles on $20$ gallons of diesel fuel. A newer model travels $220$ miles on $22$ gallons of diesel fuel. The logistics team needs to select a truck for a supply route that is $360$ miles one-way. If the selected truck must complete a full round trip along this route, how many fewer gallons of diesel will the newer model use compared to the older model for the entire journey?

8

4

80

720

Show answer & explanation

Answer: 8

Answer

8

The correct answer is found by first calculating the unit rate for each truck: the older truck gets

9

mpg (

180 \div 20

) and the newer truck gets

10

mpg (

220 \div 22

). The total distance for a round trip is

720

miles (

360 \times 2

). For this journey, the older truck requires

80

gallons (

720 \div 9

) and the newer truck requires

72

gallons (

720 \div 10

). The difference is

8

gallons.

Step-by-Step Solution

1

Calculate the fuel efficiency (unit rate) for the older truck.

180 \div 20 = 9

miles per gallon (mpg).

Finding the unit rate allows us to calculate fuel consumption for any given distance.

2

Calculate the fuel efficiency for the newer model.

220 \div 22 = 10

miles per gallon (mpg).

This establishes the unit rate for the second vehicle.

3

Determine the total distance for the round trip.

360 \times 2 = 720

miles.

The route is

360

miles one-way, so a full round trip is double that distance.

4

Calculate the total fuel required for each truck for the round trip.

Older truck:

720 \div 9 = 80

gallons. Newer truck:

720 \div 10 = 72

gallons.

Divides the total distance by each truck's mpg to find the fuel needed.

5

Find the difference in fuel consumption.

80 - 72 = 8

gallons.

Subtracting the newer truck's fuel from the older truck's fuel gives the final savings.

Key Concept

Calculating and applying unit rates across multi-step word problems involving round trips.

Question 118Question →

A military cargo plane consumes $400$ gallons of fuel to fly $150$ miles. At this rate, how many gallons of fuel will the plane consume to fly a distance of $600$ miles?

100

320

1,600

16,000

Show answer & explanation

Answer: 1,600

Answer

1,600

The relationship between distance and fuel consumption is a direct proportion. Because the new distance (

600

miles) is exactly

4

times the original distance (

150

miles), the fuel needed is also

4

times the original amount. Multiplying

400

gallons by

4

yields

1,600

gallons.

Step-by-Step Solution

1

Determine the scale factor between the two distances.

600 \div 150 = 4

Since the rate of fuel consumption is constant, finding how many times greater the new distance is allows us to scale the fuel proportionally.

2

Multiply the original fuel amount by the scale factor.

400 \times 4 = 1,600

Direct proportion dictates that if the distance is multiplied by 4, the fuel required must also be multiplied by 4.

Key Concept

Direct Proportions

Question 119Question →

A military logistics planner is calculating fuel requirements for a supply convoy consisting of $9$ identical heavy transport trucks. The planned route on a tactical map measures $10$ inches, and the map's scale is $1 \text{ inch} = 20 \text{ miles}$ . Field tests show that a group of $3$ of these trucks operating together consumes $5$ gallons of diesel fuel for every $2$ miles traveled. Assuming all trucks maintain this exact rate of consumption, what is the total amount of diesel fuel, in gallons, required for the entire convoy of $9$ trucks to complete the route?

15,000

240

75

1,500

Show answer & explanation

Answer: 1,500

Answer

The total amount of diesel fuel required is 1,500 gallons.

The correct calculation involves three proportional steps: converting the map distance to actual distance (10 × 20 = 200 miles), finding the fuel rate per mile for the 3-truck group (5 / 2 = 2.5 gal/mi), and scaling this rate for the 9-truck convoy (which is 3 groups of 3). Multiplying the 3 groups by the 2.5 gal/mi rate and the 200-mile distance yields exactly 1,500 gallons.

Step-by-Step Solution

1

Calculate the actual route distance using the map scale.

10 inches × 20 miles/inch = 200 miles.

The map distance must be converted to actual miles using the given scale to determine the true travel distance.

2

Determine the fuel consumption rate per mile for a group of 3 trucks.

5 gallons ÷ 2 miles = 2.5 gallons per mile.

Finding the unit rate (gallons per single mile) simplifies the calculation for the total distance.

3

Calculate the total fuel required for the entire convoy of 9 trucks.

(9 trucks ÷ 3 trucks/group) × 2.5 gallons/mile × 200 miles = 3 × 2.5 × 200 = 1,500 gallons.

The rate is given for 3 trucks, so we find how many groups of 3 are in the convoy (3 groups), multiply by the group rate, and then by the total distance.

Key Concept

Direct proportions in multi-step scenarios involving map scales and grouped rates.

Question 120Question →

A machine stamps $45$ metal tags in $3$ minutes. At this constant rate, how many tags will the machine stamp in $1$ hour?

15

4

900

90

Show answer & explanation

Answer: 900

Answer

The correct answer is 900 tags.

The correct answer is derived by first harmonizing the time units, converting 1 hour to 60 minutes. Next, find the unit rate by dividing 45 tags by 3 minutes, which equals 15 tags per minute. Finally, multiply the 15 tags per minute by the 60 minutes in an hour to get 900 total tags.

Step-by-Step Solution

1

Convert the target time frame into the same units used in the given rate.

1 hour = 60 minutes

Units must match before applying a proportion or unit rate.

2

Calculate the unit rate by dividing the number of tags by the number of minutes.

45 tags / 3 minutes = 15 tags per minute

The unit rate tells you exactly how much is produced in one single unit of time.

3

Multiply the unit rate by the total converted time.

15 tags/minute * 60 minutes = 900 tags

This scales the unit rate up to the total production over the full hour.

Key Concept

Direct Proportions

Estimated Time:1m 0s

Ratios and Proportions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution