All practice questions

Question 241Question →

A footbridge spans a $20$ -foot wide creek, resting on a concrete pillar at each end (Left Pillar and Right Pillar). A person weighing $160$ pounds stands on the bridge, exactly $5$ feet from the Right Pillar. If we ignore the weight of the bridge itself, how many pounds of the person's weight are supported by the Left Pillar?

Show answer & explanation

Answer: 40

Answer

40

The weight supported by the Left Pillar is inversely proportional to the load's distance from it. Since the person is

15

feet from the Left Pillar and

5

feet from the Right Pillar on a

20

-foot bridge, they are

3

times closer to the Right Pillar. Therefore, the Right Pillar carries

3/4

of the weight (

120

lbs) and the Left Pillar carries

1/4

of the weight (

40

lbs).

Step-by-Step Solution

1

Determine the relevant distances along the bridge.

Total span =

20

ft. Distance to Right Pillar =

5

ft. Distance to Left Pillar =

15

ft.

To calculate the load on one support, we need the total distance and the distance to the opposite support.

2

Set up the load distribution calculation for the Left Pillar.

Load on Left Pillar = Total Weight

\times

(Distance to Right Pillar / Total Span)

The proportion of weight supported by one end of a beam is equal to the distance from the load to the opposite end, divided by the total length.

3

Calculate the supported weight using the values.

160 \times (5 / 20) = 160 \times (1 / 4) = 40

pounds.

Solving the equation gives the exact weight carried by the left support.

Key Concept

Load Distribution and Balancing

Alternative Method

You can also use the concept of torque (moments) in static equilibrium. Taking the Right Pillar as the pivot point: (Force of Left Pillar

\times

20

ft) = (

160

lbs

\times

5

ft). Therefore,

20 \times F_{left} = 800

, which means

F_{left} = 40

lbs.

Estimated Time:45s

Question 242Question →

A single-axle utility trailer has its axle located $8$ feet behind the hitch. The empty trailer weighs $600$ pounds, and its center of gravity is $5$ feet behind the hitch. A machine weighing $1,200$ pounds is loaded onto the trailer so that its center of gravity is $6$ feet behind the hitch. When the trailer is parked horizontally, what is the downward force, in pounds, exerted on the hitch?

Show answer & explanation

Answer: 525

Answer

525

The correct answer is derived by treating the trailer as a lever in static equilibrium. Taking the sum of torques around the axle, the total rotational force from the empty trailer and the loaded machine is

4200

lb-ft. Dividing this by the

8

-foot lever arm to the hitch yields exactly

525

pounds of force.

Step-by-Step Solution

1

Identify a pivot point and determine all relevant distances.

Using the axle as the pivot point. Distance from axle to trailer CG is 3 ft (

8 - 5

). Distance from axle to machine CG is 2 ft (

8 - 6

). Distance from axle to hitch is 8 ft.

Establishing a pivot point allows us to use the principle of moments (torques) to find the unknown reaction force.

2

Calculate the total counter-clockwise torque around the axle.

Trailer torque =

600 \text{ lbs} \times 3 \text{ ft} = 1800 \text{ lb-ft}

. Machine torque =

1200 \text{ lbs} \times 2 \text{ ft} = 2400 \text{ lb-ft}

. Total torque =

4200 \text{ lb-ft}

.

The weight of the trailer and the machine both push down, attempting to rotate the system around the axle.

3

Set up the static equilibrium equation.

4200 \text{ lb-ft} = F_{hitch} \times 8 \text{ ft}

.

For the trailer to remain stationary, the clockwise torque provided by the hitch must perfectly balance the total counter-clockwise torque.

4

Solve for the force at the hitch (

F_{hitch}

).

F_{hitch} = \frac{4200}{8} = 525 \text{ lbs}

.

Dividing the total torque by the distance to the hitch gives the required balancing force.

Key Concept

Static Equilibrium and Torque Distribution

Alternative Method

You can also solve this by setting the hitch as the pivot point. The downward torque would be

(600 \times 5) + (1200 \times 6) = 3000 + 7200 = 10200

lb-ft. The upward force from the axle balances this:

F_{axle} \times 8 = 10200

, so

F_{axle} = 1275

lbs. Since the total weight is

1800

lbs (

600 + 1200

), the remaining force supported by the hitch is

1800 - 1275 = 525

lbs.

Estimated Time:2m 0s

Question 243Question →

A mechanic is inspecting a steering mechanism. If the steering wheel has a radius of $14 \text{ inches}$ and the attached steering column (axle) has a radius of $2 \text{ inches}$ , what is the ideal mechanical advantage of this wheel and axle system?

Show answer & explanation

Answer: 7

Answer

7

The ideal mechanical advantage (IMA) is the ratio of the distance the effort moves to the distance the load moves. In a wheel and axle, this is proportional to their radii. Dividing the wheel's radius (

14 \text{ inches}

) by the axle's radius (

2 \text{ inches}

) gives an IMA of

7

.

Step-by-Step Solution

1

Identify the relevant formula

\text{IMA} = \frac{R}{r}

The ideal mechanical advantage of a wheel and axle is the ratio of the effort radius (wheel) to the load radius (axle).

2

Substitute the known values

\text{IMA} = \frac{14}{2}

The wheel radius

R

is

14 \text{ inches}

and the axle radius

r

is

2 \text{ inches}

.

3

Perform the division

7

Dividing

14

by

2

yields an ideal mechanical advantage of

7

.

Key Concept

For a wheel and axle system where the input force is applied to the wheel, the ideal mechanical advantage is calculated by dividing the radius of the wheel by the radius of the axle.

Question 244Question →

A backup breathing canister has a volume of $50 \text{ in}^3$ and stores gas at an absolute pressure of $2000 \text{ psi}$ . During a transfer process, the gas is allowed to expand into a larger holding tank, reaching a total final volume of $250 \text{ in}^3$ . Assuming the temperature remains constant, what is the new absolute pressure of the gas in $\text{psi}$ ?

Show answer & explanation

Answer: 400

Answer

400

According to Boyle's Law, the pressure and volume of a gas are inversely proportional when the temperature is constant (

P_1V_1 = P_2V_2

). Since the volume increased by a factor of 5 (from 50 to 250 cubic inches), the pressure must decrease by a factor of 5. Dividing the initial 2000 psi by 5 gives a final pressure of 400 psi.

Step-by-Step Solution

1

Identify the initial and final states of the gas.

P_1 = 2000 \text{ psi}

,

V_1 = 50 \text{ in}^3

, and

V_2 = 250 \text{ in}^3

.

Organizing the known variables is the first step in solving any gas law problem.

2

Apply Boyle's Law to relate pressure and volume.

Equation to use:

P_1 V_1 = P_2 V_2

.

Boyle's Law states that for a fixed mass of gas at constant temperature, pressure and volume are inversely proportional.

3

Substitute the known values into the equation and solve for the final pressure (

P_2

).

2000 \times 50 = P_2 \times 250

, which simplifies to

100000 = 250 P_2

. Dividing both sides by 250 gives

P_2 = 400

.

This mathematical calculation yields the new absolute pressure after the gas has expanded.

Key Concept

Boyle's Law (Inverse relationship between pressure and volume)

Alternative Method

Instead of full algebraic substitution, recognize the proportionality: the volume increases by a factor of 5 (

250 / 50 = 5

). Because pressure and volume are inversely proportional, the pressure must be divided by 5. Thus,

2000 \text{ psi} / 5 = 400 \text{ psi}

.

Estimated Time:45s

Question 245Question →

A salvage crane is lifting a solid brass propeller from the bottom of a freshwater lake. The propeller has a true weight of $1,200 \text{ N}$ in the air. When it is completely submerged in the water, it displaces $35 \text{ liters}$ of water.

If $1 \text{ liter}$ of water weighs exactly $10 \text{ N}$ , what is the apparent weight of the propeller, in Newtons, while it is still completely underwater?

Show answer & explanation

Answer: 850

Answer

850

The propeller experiences an upward buoyant force equal to the weight of the water it displaces. Since it displaces

35 \text{ liters}

and each liter weighs

10 \text{ N}

, the buoyant force is

350 \text{ N}

. The apparent weight is the true weight (

1,200 \text{ N}

) minus this buoyant force (

350 \text{ N}

), resulting in

850 \text{ N}

.

Step-by-Step Solution

1

Calculate the weight of the displaced water to find the buoyant force.

Buoyant force =

350 \text{ N}

According to Archimedes' Principle, the buoyant force on an object is equal to the weight of the fluid it displaces. Here,

35 \text{ liters} \times 10 \text{ N/liter} = 350 \text{ N}

.

2

Subtract the buoyant force from the true weight to find the apparent weight.

Apparent weight =

850 \text{ N}

The apparent weight of a submerged object is its actual weight in air minus the upward buoyant force exerted by the fluid (

1,200 \text{ N} - 350 \text{ N} = 850 \text{ N}

).

Key Concept

Archimedes' principle states that the buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object. The apparent weight of a submerged object is its true weight minus the buoyant force.

Estimated Time:1m 0s

Question 246Question →

A mechanic checks a heavy-duty engine's coolant temperature gauge, which displays a reading of $35^\circ\text{C}$ . What is this temperature reading in degrees Fahrenheit?

Show answer & explanation

Answer: 95

Answer

95

The correct answer is derived by multiplying the given Celsius value by 9/5 to adjust for the difference in degree size, and then adding 32 to account for the difference in the freezing point baselines of the two scales.

Step-by-Step Solution

1

Identify the proper conversion formula.

F = \frac{9}{5}C + 32

This is the mathematical relationship used to convert temperatures from the Celsius scale to the Fahrenheit scale.

2

Substitute the given Celsius value into the formula.

F = \frac{9}{5}(35) + 32

The gauge reading is exactly

35^\circ\text{C}

.

3

Calculate the proportional factor.

63 + 32

Dividing 35 by 5 yields 7, and multiplying 7 by 9 yields 63.

4

Add the baseline offset to find the final temperature.

95

The Fahrenheit scale offset is 32 degrees above the freezing point equivalent.

Key Concept

Celsius to Fahrenheit Conversion

Question 247Question →

During a testing sequence, a heavy torpedo with a mass of $1,500 \text{ kg}$ is fired from a submarine tube. The launch mechanism applies a forward propulsive force of $45,000 \text{ N}$ . If the water resistance creates a constant opposing drag force of $9,000 \text{ N}$ , what is the initial acceleration of the torpedo?

Show answer & explanation

Answer: 24

Answer

24

To find the acceleration, we first determine the net force acting on the torpedo. The forward propulsive force is

45,000 \text{ N}

, but water resistance pushes back with

9,000 \text{ N}

. Subtracting the opposing force gives a net force of

36,000 \text{ N}

. Using Newton's Second Law (

F = ma

), we rearrange the formula to solve for acceleration (

a = \frac{F}{m}

). Dividing the net force (

36,000 \text{ N}

) by the mass (

1,500 \text{ kg}

) results in an acceleration of

24 \text{ m/s}^2

.

Step-by-Step Solution

1

Calculate the net force (

F_{net}

) acting on the torpedo.

F_{net} = 45,000 \text{ N} - 9,000 \text{ N} = 36,000 \text{ N}

The torpedo experiences forces in opposite directions, so the drag force must be subtracted from the forward thrust to find the total effective force.

2

Calculate the acceleration (

a

) using Newton's Second Law.

a = \frac{F_{net}}{m} = \frac{36,000 \text{ N}}{1,500 \text{ kg}} = 24 \text{ m/s}^2

Newton's Second Law states that acceleration equals the net force divided by the mass.

Key Concept

Newton's Second Law (F=ma) and Net Force

Question 248Question →

During a hydro-engineering project, a completely sealed hollow steel sphere is used as an underwater sensor housing. The sphere has an external volume of $2.5 \text{ m}^3$ and a dry weight of $35,000 \text{ N}$ in the air. When completely submerged in a testing pool where the water's weight density is $10,000 \text{ N/m}^3$ , what is the net downward force (apparent weight) of the housing in Newtons?

Show answer & explanation

Answer: 10000

Answer

10000

The net downward force, or apparent weight, is found by subtracting the upward buoyant force from the object's true dry weight. Since the object displaces

2.5 \text{ m}^3

of water with a weight density of

10,000 \text{ N/m}^3

, the buoyant force is

25,000 \text{ N}

. Subtracting this upward force from the

35,000 \text{ N}

downward dry weight leaves a net downward force of

10,000 \text{ N}

.

Step-by-Step Solution

1

Calculate the buoyant force acting on the submerged housing.

25,000 N

According to Archimedes' principle, the buoyant force equals the weight of the fluid displaced by the object's external volume (

2.5 \text{ m}^3 \times 10,000 \text{ N/m}^3 = 25,000 \text{ N}

).

2

Calculate the apparent weight (net downward force).

10,000 N

The apparent weight is the actual weight in air minus the upward buoyant force (

35,000 \text{ N} - 25,000 \text{ N} = 10,000 \text{ N}

).

Key Concept

Archimedes' Principle and Apparent Weight

Question 249Question →

A heavy-duty diving bell is being tested in a deep saltwater facility. At a depth of $10\text{ meters}$ , the hydrostatic pressure exerted on its exterior is measured to be $100\text{ kilopascals (kPa)}$ . If the diving bell is lowered further to a depth of $30\text{ meters}$ , what will the new hydrostatic pressure be in $\text{kPa}$ , assuming the fluid density remains constant?

Show answer & explanation

Answer: 300

Answer

The new hydrostatic pressure will be 300 kPa.

Because hydrostatic pressure is directly proportional to depth in a fluid of constant density, tripling the depth from 10 meters to 30 meters results in exactly three times the pressure. Thus, 100 kPa multiplied by 3 equals 300 kPa.

Step-by-Step Solution

1

Identify the relationship between hydrostatic pressure and depth.

Pressure is directly proportional to depth.

This is a fundamental principle of hydrostatic pressure for a fluid with constant density.

2

Calculate the ratio of the new depth to the initial depth.

30\text{ m} / 10\text{ m} = 3

This ratio tells us how much the pressure will increase.

3

Calculate the new pressure.

100\text{ kPa} \times 3 = 300\text{ kPa}

Applying the direct proportion to find the final pressure value.

Key Concept

Hydrostatic pressure increases in direct proportion to the depth of the fluid, assuming fluid density remains constant.

Question 250Question →

A hand-cranked emergency power generator uses a two-stage compound gear train to increase the rotational speed of its rotor. The hand crank is connected directly to the driving gear, Gear A, which has $60$ teeth. Gear A meshes with Gear B, which has $15$ teeth. Gear B shares a solid intermediate shaft with Gear C, which has $40$ teeth. Finally, Gear C meshes with Gear D on the generator's rotor shaft, which has $10$ teeth.

If a user turns the hand crank at a steady rate of $30\text{ rpm}$ , what is the resulting rotational speed of the generator rotor in $\text{rpm}$ ?

Show answer & explanation

Answer: 480

Answer

480

The correct answer is 480 because the total speed multiplication of a compound gear train is the product of the individual gear ratios of each stage. Stage 1 has a ratio of 60:15 (a 4:1 multiplier). Stage 2 has a ratio of 40:10 (a 4:1 multiplier). The total multiplier is 4 * 4 = 16. The input speed of 30 rpm multiplied by 16 gives a final output speed of 480 rpm.

Step-by-Step Solution

1

Calculate the speed multiplication factor of the first gear pair.

Factor of 4

Gear A (60 teeth) drives Gear B (15 teeth). The ratio of teeth is 60/15 = 4. Because a larger gear is driving a smaller gear, the rotational speed increases by this factor.

2

Calculate the rotational speed of the intermediate shaft.

120 rpm

Multiply the input speed (30 rpm) by the first stage factor (4). 30 * 4 = 120 rpm. Both Gear B and Gear C spin at this exact rate since they are mounted on the same solid shaft.

3

Calculate the speed multiplication factor of the second gear pair.

Factor of 4

Gear C (40 teeth) drives Gear D (10 teeth). The ratio is 40/10 = 4. Again, a larger gear is driving a smaller one, increasing the speed.

4

Calculate the final rotational speed of the rotor.

480 rpm

Multiply the intermediate shaft speed (120 rpm) by the second stage factor (4). 120 * 4 = 480 rpm.

Key Concept

Calculating output speed in a compound gear train

Question 251Question →

A factory conveyor belt transports manufactured parts a distance of $45 \text{ meters}$ across the production floor in exactly $15 \text{ seconds}$ . What is the average speed of the conveyor belt in meters per second?

Show answer & explanation

Answer: 3

Answer

The average speed of the conveyor belt is

3 \text{ m/s}

.

The average speed is correctly calculated by dividing the total distance traveled (

45 \text{ meters}

) by the total time taken (

15 \text{ seconds}

), which equals

3 \text{ m/s}

.

Step-by-Step Solution

1

Identify the given values for distance and time.

Distance (

d

) =

45 \text{ meters}

, Time (

t

) =

15 \text{ seconds}

.

These variables are required to calculate average speed.

2

Apply the average speed formula.

v = \frac{d}{t} = \frac{45}{15}

.

Average speed is defined as the total distance traveled divided by the elapsed time.

3

Calculate the final speed.

v = 3

.

To find the numerical value of the speed in meters per second.

Key Concept

Average Speed

Question 252Question →

A uniform window-washing scaffold is $20$ feet long and weighs $180$ pounds. It is supported horizontally by two vertical cables, one attached at the extreme left end and one at the extreme right end. A worker weighing $220$ pounds stands exactly $5$ feet from the left cable. If the system is in static equilibrium, what is the tension, in pounds, in the right cable?

Show answer & explanation

Answer: 145

Answer

145

The system is in equilibrium, so the sum of the moments about the left support must be zero. The downward weights create a clockwise moment of

2900 \text{ lb-ft}

(

180 \text{ lbs} \times 10 \text{ ft}

plus

220 \text{ lbs} \times 5 \text{ ft}

). To balance this, the right cable, located

20

feet away, must provide an equal counter-clockwise moment. Dividing

2900

by

20

yields a tension of

145

pounds.

Step-by-Step Solution

1

Select a pivot point to calculate moments (torques).

The left end of the scaffold is chosen as the pivot point.

Choosing the left end eliminates the unknown tension of the left cable from the equation, allowing direct calculation of the right cable's tension.

2

Calculate the total clockwise moment caused by the downward weights.

Scaffold moment =

180 \text{ lbs} \times 10 \text{ ft} = 1800 \text{ lb-ft}

. Worker moment =

220 \text{ lbs} \times 5 \text{ ft} = 1100 \text{ lb-ft}

. Total clockwise moment =

2900 \text{ lb-ft}

.

The scaffold's weight acts at its center (

10

ft), and the worker's weight acts at

5

ft from the pivot. Moment equals force multiplied by distance.

3

Equate the clockwise moment to the counter-clockwise moment and solve.

T_R \times 20 \text{ ft} = 2900 \text{ lb-ft}

, which gives

T_R = 145 \text{ lbs}

.

For the scaffold to be in static equilibrium, the sum of all moments about any pivot point must be zero.

Key Concept

Rotational equilibrium and calculation of moments (torque) to find support reaction forces.

Question 253Question →

A naval response boat is towing a disabled speedboat through choppy waters. The disabled speedboat has a mass of $4,000 \text{ kg}$ . The response boat's winch maintains a constant forward tension of $14,000 \text{ N}$ on the tow cable. Meanwhile, the water and wind create a combined resistive drag force of $4,400 \text{ N}$ acting against the speedboat.

What is the forward acceleration of the disabled speedboat in $\text{m/s}^2$ ?

Show answer & explanation

Answer: 2.4

Answer

2.4

The acceleration is determined by first finding the net force (

14,000 \text{ N} - 4,400 \text{ N} = 9,600 \text{ N}

) and then dividing it by the mass (

4,000 \text{ kg}

) according to Newton's Second Law (

a = F_{net}/m

), resulting in

2.4 \text{ m/s}^2

.

Step-by-Step Solution

1

Identify all horizontal forces acting on the speedboat.

Forward tension force =

14,000 \text{ N}

, Backward resistive drag force =

4,400 \text{ N}

.

Newton's Second Law requires the total net force, not just the applied force, to find the resulting acceleration.

2

Calculate the net horizontal force (

F_{net}

).

F_{net} = 14,000 \text{ N} - 4,400 \text{ N} = 9,600 \text{ N}

.

The drag force acts in the opposite direction of the tow cable, so it must be subtracted.

3

Use Newton's Second Law to solve for acceleration.

a = \frac{F_{net}}{m} = \frac{9,600 \text{ N}}{4,000 \text{ kg}} = 2.4 \text{ m/s}^2

.

Acceleration is directly proportional to net force and inversely proportional to mass.

Key Concept

Newton's Second Law (

F=ma

) requires first calculating the net force by combining all opposing forces (like applied tension and drag friction) before dividing by the mass to find the resulting acceleration.

Question 254Question →

A naval architect is reviewing the weight distribution of a new patrol boat. The design specifies that the aft storage compartment can safely hold a maximum of $\frac{7}{40}$ of the vessel's total displacement. What percentage of the total displacement does this represent?

Show answer & explanation

Answer: 17.5

Answer

17.5

To find the percentage equivalent of a fraction, divide the top number (numerator) by the bottom number (denominator) to get a decimal. Dividing 7 by 40 results in 0.175. Next, multiply this decimal by 100 to convert it into a percentage, which yields 17.5%.

Step-by-Step Solution

1

Divide the numerator by the denominator.

7 \div 40 = 0.175

A fraction represents division. Performing this division converts the fraction into a decimal.

2

Multiply the decimal by 100.

0.175 \times 100 = 17.5

To convert any decimal representation to a percentage, it must be multiplied by 100.

Key Concept

Converting a fraction to a percentage

Question 255Question →

A quartermaster issues $1,050$ uniform shirts to new recruits over the course of a $3$ -hour and $30$ -minute processing block. Assuming a constant pace, what is the unit rate of shirts issued per minute?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5 shirts per minute.

To find the unit rate in shirts per minute, the total time must first be converted entirely into minutes.

3

hours and

30

minutes is equal to

210

minutes. Dividing the total number of shirts (

1,050

) by the total number of minutes (

210

) results in exactly

5

shirts per minute.

Step-by-Step Solution

1

Convert the given time block into a single unit (minutes) to align with the question's requested rate.

3

hours and

30

minutes

= (3 \times 60) + 30 = 180 + 30 = 210

minutes.

The unit rate asked for is per minute, so the total time elapsed must be expressed in minutes.

2

Calculate the unit rate by dividing the total quantity by the total time in minutes.

1,050 \text{ shirts} \div 210 \text{ minutes} = 5 \text{ shirts per minute}

.

A unit rate compares a quantity to one unit of another quantity. Dividing the total items by the total minutes yields the number of items per single minute.

Key Concept

Calculating unit rates involving time conversion.

Question 256Question →

An electronics store is liquidating its marine equipment inventory. A high-frequency boat radar system is marked down by $20\%$ from its original price. During a special weekend clearance event, an additional $10\%$ is taken off the already discounted price. If the final sale price of the radar system is $\$360$ , what was the original price of the system in dollars?

Show answer & explanation

Answer: 500

Answer

The original price of the radar system was $500.

The correct answer is found by setting up an equation with successive multipliers for each discount phase:

x \times 0.80 \times 0.90 = 360

. Simplifying the left side yields

0.72x = 360

, and dividing by

0.72

correctly identifies the original price as $500.

Step-by-Step Solution

1

Set up an expression for the price after the first discount.

0.80x

A

20\%

discount means the buyer pays

100\% - 20\% = 80\%

of the original price

x

.

2

Set up an expression for the price after the second discount.

0.72x

An additional

10\%

discount means the buyer pays

90\%

of the already discounted price. Multiplying

0.90 \times 0.80x

gives

0.72x

.

3

Set the final expression equal to the sale price and solve for the original price.

0.72x = 360 \Rightarrow x = 500

This isolates the original price,

x

, finding the initial value before any markdowns.

Key Concept

Calculating successive discounts and setting up an equation to find the original pre-sale price.

Question 257Question →

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)

Show answer & explanation

Answer: 67.5

Answer

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.

Step-by-Step Solution

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.

Key Concept

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

Question 258Question →

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?

Show answer & explanation

Answer: 3

Answer

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.

Step-by-Step Solution

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.

Key Concept

Calculating and comparing unit rates involving US customary capacity conversions.

Question 259Question →

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?

Show answer & explanation

Answer: 1080

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Successive/Chain Discounts

Question 260Question →

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?

Show answer & explanation

Answer: 800

Answer

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.

Step-by-Step Solution

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.

Key Concept

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

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