All practice questions

Question 2041Question →

Match each practical material or component to its primary thermal property or mechanism.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Engine coolant pairs with high specific heat capacity; Aluminum radiator fins pair with high thermal conductivity and low specific heat; Fiberglass batting pairs with trapping pockets of air; Spacecraft ceramic tiles pair with extremely low thermal conductivity.

Each material is chosen for mechanical applications based on distinct thermal properties. Water's high specific heat makes it an ideal coolant. Aluminum's high conductivity allows it to act as an effective heat sink and radiator. Fiberglass relies on the low conductivity of trapped air for home insulation. Ceramics provide robust thermal resistance for extreme environments.

Step-by-Step Solution

1

Analyze the thermal requirements for engine coolant.

Coolant must absorb a lot of heat without vaporizing, which requires a high specific heat capacity.

This corresponds to the property of absorbing large amounts of thermal energy with minimal temperature increase.

2

Analyze the function of aluminum radiator fins.

Fins are designed to pull heat away from the coolant and release it into the air quickly.

This requires high thermal conductivity to move the heat, and low specific heat so the aluminum itself doesn't hold onto the thermal energy.

3

Analyze how fiberglass batting insulates buildings.

Fiberglass is mostly empty space filled with air.

Air is a poor thermal conductor. By trapping air in small pockets, fiberglass prevents convective currents and minimizes conductive heat transfer.

4

Analyze the purpose of spacecraft ceramic tiles.

Tiles must protect the spacecraft interior from the extreme heat of atmospheric re-entry.

Ceramics have incredibly low thermal conductivity, meaning heat struggles to pass through them, effectively shielding underlying structures.

Key Concept

Different applications require specific thermal properties, such as high heat capacity for absorbing energy, high conductivity for dissipating energy, or low conductivity for insulation.

Question 2042Question →

A test sled has a weight of $2,940 \text{ N}$ (where $g = 9.8 \text{ m/s}^2$ ). During a braking test, a friction track applies a constant drag force of $588 \text{ N}$ . If it takes exactly $10 \text{ seconds}$ for the sled to come to a complete halt, what was its initial velocity?

1.96 \text{ m/s}

2 \text{ m/s}

19.6 \text{ m/s}

50 \text{ m/s}

Show answer & explanation

Answer: $19.6 \text{ m/s}$

Answer

19.6 \text{ m/s}

The correct answer applies the impulse-momentum theorem. First, the mass is found by dividing the weight by gravity (

2,940 \text{ N} / 9.8 \text{ m/s}^2 = 300 \text{ kg}

). Then, the impulse is calculated as the braking force multiplied by time (

588 \text{ N} \times 10 \text{ s} = 5,880 \text{ N}\cdot\text{s}

). Because impulse equals the change in momentum (

m \times \Delta v

), dividing the impulse by the mass yields the sled's initial velocity (

5,880 \text{ N}\cdot\text{s} / 300 \text{ kg} = 19.6 \text{ m/s}

).

Step-by-Step Solution

1

Calculate the mass of the sled from its given weight.

m = 2,940 \text{ N} / 9.8 \text{ m/s}^2 = 300 \text{ kg}

Momentum and kinematic equations require mass (in kg), not weight (in N).

2

Calculate the total impulse applied by the friction track.

\text{Impulse} = F \times t = 588 \text{ N} \times 10 \text{ s} = 5,880 \text{ N}\cdot\text{s}

Impulse is defined as the product of the average force and the time duration over which it acts.

3

Relate the impulse to the change in velocity.

5,880 = 300 \times \Delta v

, which solves to

\Delta v = 19.6 \text{ m/s}

.

The impulse-momentum theorem states that impulse equals the change in momentum (

\Delta p = m \times \Delta v

). Since the sled stops, the change in velocity equals its initial velocity.

Key Concept

Impulse and Momentum Conservation

Alternative Method

You can also solve this using Newton's Second Law and basic kinematics. First, find the deceleration:

a = F / m = 588 / 300 = 1.96 \text{ m/s}^2

. Then, use the kinematic equation

v_f = v_i - a \times t

. Since

v_f = 0

,

v_i = a \times t = 1.96 \times 10 = 19.6 \text{ m/s}

.

Estimated Time:1m 30s

Question 2043Question →

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 2044Question →

A commercial box truck has a wheelbase (the distance between the front and rear axles) of $15$ feet. The empty truck weighs $6,000$ pounds, with its center of gravity located $6$ feet behind the front axle. A $3,000$ -pound industrial generator is loaded onto the truck and positioned directly above the rear axle. What is the total load supported by the rear axle in this configuration?

4,500

pounds

5,400

pounds

6,000

pounds

6,600

pounds

Show answer & explanation

Answer: $5,400$ pounds

Answer

5,400

pounds

To find the load on the rear axle, we must evaluate the contributions of the empty truck and the generator separately. For the empty truck, the center of gravity is

6

feet from the front axle, meaning it is closer to the front. Taking moments around the front axle, the rear axle supports

6/15

of the

6,000

-pound truck, which equals

2,400

pounds. Because the

3,000

-pound generator is placed directly over the rear axle, its entire weight is added to that support. Adding both contributions together yields

2,400 + 3,000 = 5,400

pounds.

Step-by-Step Solution

1

Calculate the portion of the empty truck's weight supported by the rear axle.

Using the front axle as a pivot, the moment equation is: Rear Axle Force ×

15

ft =

6,000

lbs ×

6

ft. Solving for the force gives

36,000 / 15 = 2,400

lbs.

The principle of moments dictates that the reaction force at a support is proportional to the distance of the load from the opposite support.

2

Determine the load added by the generator to the rear axle.

Since the

3,000

-lb generator is placed directly over the rear axle,

100\%

of its weight rests on that axle (

3,000

lbs).

Loads placed directly on top of a support do not transfer any moment to the other supports.

3

Calculate the total weight supported by the rear axle.

2,400\text{ lbs} + 3,000\text{ lbs} = 5,400\text{ lbs}

.

The total load on a support is the sum of all individual load contributions.

Key Concept

Principle of Moments and Offset Load Distribution

Estimated Time:1m 15s

Question 2045Question →

A medical logistics unit processes $3,600$ standard first-aid kits in $45$ minutes. What is the unit rate of processing in kits per hour?

4,800

80

8,000

480

Show answer & explanation

Answer: 4,800

Answer

The unit rate is 4,800 kits per hour.

To find the hourly unit rate, the processing time must be converted from minutes to hours, or the per-minute rate must be calculated and multiplied by 60. Processing

3,600

kits in

45

minutes equals

80

kits per minute. Multiplying by

60

minutes per hour gives the correct rate of

4,800

kits per hour.

Step-by-Step Solution

1

Convert the processing time from minutes to hours.

45 \text{ minutes} = \frac{45}{60} \text{ hours} = 0.75 \text{ hours}

The question asks for the unit rate in kits per hour, so the time unit must match the requested rate.

2

Divide the total number of kits by the time in hours to find the unit rate.

\frac{3,600}{0.75} = 4,800 \text{ kits per hour}

A unit rate represents the ratio of the total quantity to the total time.

Key Concept

Calculating unit rates with time conversions

Question 2046Question →

A machinist is adjusting the feed rate on a lathe. The machine is currently operating at $0.18$ of its maximum safe speed. The technical manual specifies that for the selected alloy, the speed must be increased to exactly $\frac{21}{50}$ of the maximum safe speed. By what percentage of the maximum safe speed must the current speed be increased?

3%

4.2%

24%

42%

Show answer & explanation

Answer: 24%

Answer

24%

To find the required increase, both the current speed and the target speed must be in the same format. Converting the current speed to a percent yields 18% (0.18 * 100). Converting the target speed to a percent yields 42% (21/50 = 42/100 = 42%). The difference between the target speed and the current speed is 42% - 18% = 24%.

Step-by-Step Solution

1

Convert the current operating speed from a decimal to a percentage.

0.18 = 18\%

Converting all values to percentages makes them easy to compare and subtract.

2

Convert the target operating speed from a fraction to a percentage.

\frac{21}{50} = \frac{42}{100} = 42\%

The target speed must also be a percentage to find the precise difference.

3

Subtract the current percentage from the target percentage to find the required increase.

42\% - 18\% = 24\%

The difference between the two values reveals how much the speed needs to be increased.

Key Concept

Converting fractions and decimals to percentages to determine the difference between two values.

Question 2047Question →

A base quartermaster is evaluating the fuel efficiency of four different portable power generators. Match each generator's fuel consumption data to its correct unit rate in gallons per hour.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Generator W matches with

3.5

gallons per hour, Generator X matches with

3.75

gallons per hour, Generator Y matches with

6.5

gallons per hour, and Generator Z matches with

2.25

gallons per hour.

Each generator's fuel consumption is converted to an hourly rate by matching the time period into hours (e.g.,

2

days to

48

hours,

240

minutes to

4

hours) and then dividing the total gallons consumed by the total hours.

Step-by-Step Solution

1

Calculate the unit rate for Generator W.

14 \div 4 = 3.5

gallons per hour.

The rate is already given in hours, so divide the total gallons by the total hours.

2

Calculate the unit rate for Generator X by converting days to hours.

2

days =

48

hours.

180 \div 48 = 3.75

gallons per hour.

The unit rate must be in gallons per hour, so the time must first be converted from days to hours.

3

Calculate the unit rate for Generator Y.

26 \div 4 = 6.5

gallons per hour.

Divide the total volume of fuel by the given time in hours.

4

Calculate the unit rate for Generator Z by converting minutes to hours.

240

minutes =

4

hours.

9 \div 4 = 2.25

gallons per hour.

The unit rate must be in gallons per hour, so the time must first be converted from minutes to hours by dividing by

60

.

Key Concept

Calculating unit rates by ensuring equivalent units of measure before dividing the quantity by the time.

Question 2048Question →

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 2049Question →

A composite 2D figure is assembled from four disjoint polygon pieces. Before assembly, each piece must be mentally rotated clockwise from its starting orientation to a standard upright orientation.

- The Triangle's starting orientation is tilted 45° counter-clockwise from upright.
- The L-Shape's starting orientation is tilted 90° counter-clockwise from upright.
- The Trapezoid's starting orientation is completely inverted (180° from upright).
- The Rectangle's starting orientation is tilted 90° clockwise from upright.

Arrange the pieces in order from the SMALLEST clockwise rotation required to the LARGEST clockwise rotation required to align them for assembly.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from smallest to largest clockwise rotation is: Triangle (45°), L-Shape (90°), Trapezoid (180°), Rectangle (270°).

Based on a purely clockwise rotation requirement: the Triangle requires 45°, the L-Shape requires 90°, the Trapezoid requires 180°, and the Rectangle requires 270°. Ordering these degrees from smallest to largest results in Triangle, L-Shape, Trapezoid, Rectangle.

Step-by-Step Solution

1

Calculate the clockwise rotation for the Triangle.

The Triangle is tilted 45° counter-clockwise. Rotating it clockwise to the upright position requires 45°.

Moving clockwise from 45° left of vertical brings it to vertical.

2

Calculate the clockwise rotation for the L-Shape.

The L-Shape is tilted 90° counter-clockwise. Rotating it clockwise requires 90°.

Moving clockwise from 90° left of vertical brings it to vertical.

3

Calculate the clockwise rotation for the Trapezoid.

The Trapezoid is inverted (180°). Rotating it clockwise requires 180°.

180° is exactly halfway around.

4

Calculate the clockwise rotation for the Rectangle.

The Rectangle is tilted 90° clockwise. To reach the upright position moving only clockwise, it must complete the remaining 270°.

A full circle is 360°. Since it is already 90° clockwise, moving further clockwise to reach the top requires 360° - 90° = 270°.

5

Order the pieces based on rotation magnitude.

Triangle (45°), L-Shape (90°), Trapezoid (180°), Rectangle (270°).

Arranging from smallest to largest clockwise rotation as requested by the prompt.

Key Concept

Determining the magnitude of specific directional rotation (clockwise) to achieve a target spatial orientation.

Question 2050Question →

A square assembly component features a distinct triangular notch on its top edge. To evaluate how it fits into a larger system, you must mentally rotate this piece clockwise in continuous 90-degree increments. Arrange the resulting orientations in the chronological order they will appear during the rotation process.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence tracks the notch moving sequentially from the top edge to the right edge, then to the bottom edge, and finally to the left edge in a clockwise direction.

By following a continuous 90-degree clockwise rotational sequence, the defining notch moves in a standard circular progression: from its start at the top, to the right (first 90-degree turn), to the bottom (180-degree turn), and finally to the left (270-degree turn).

Step-by-Step Solution

1

Identify the baseline starting position of the shape.

The distinguishing feature (the notch) is located on the top edge (12 o'clock).

Establishing the initial orientation is necessary before applying rotational transformations.

2

Apply the first 90-degree clockwise rotation.

The notch rotates to the right edge (3 o'clock).

A 90-degree clockwise turn shifts the top face to the right.

3

Apply the second 90-degree clockwise rotation.

The notch rotates from the right edge to the bottom edge (6 o'clock).

Continuing clockwise, the right face shifts to the bottom.

4

Apply the third 90-degree clockwise rotation.

The notch rotates from the bottom edge to the left edge (9 o'clock).

The final 90-degree turn in this sequence shifts the bottom face to the left.

Key Concept

Tracking a distinct physical feature on a 2D shape as it undergoes incremental mental rotation.

Estimated Time:45s

Question 2051Question →

A supply depot crew loads $1,350$ cargo crates onto transport aircraft in $4.5$ hours. Assuming a constant work pace, how many crates does the crew load per minute?

3

50

300

5

Show answer & explanation

Answer: 5

Answer

5

The correct unit rate is found by first converting

4.5

hours to minutes. Since there are

60

minutes in an hour,

4.5 \times 60 = 270

minutes. Then, dividing the total quantity of

1,350

crates by the total time of

270

minutes yields exactly

5

crates per minute.

Step-by-Step Solution

1

Convert the total time from hours to minutes.

4.5 \text{ hours} \times 60 \text{ minutes/hour} = 270 \text{ minutes}

The question specifically asks for the unit rate in crates per minute, so the time must be in minutes.

2

Divide the total number of crates by the total time in minutes.

1,350 \text{ crates} \div 270 \text{ minutes} = 5 \text{ crates per minute}

The unit rate is found by dividing the total quantity by the total time.

Key Concept

Calculating unit rates requiring a preceding time unit conversion.

Question 2052Question →

Determine the total amount saved and the final pre-tax price for the successive discounts described below.

Fill in the blanks below

An auto parts store lists a set of all-terrain tires for a regular price of

\$800

. During a holiday event, the tires are marked down by

15\%

. A customer applies a military discount that takes an additional

10\%

off the already reduced sale price. The customer saves a total of $\$

from the regular price, resulting in a final pre-tax price of

\$$.

Show answer & explanation

Answer

The total savings is

\$188

, and the final price is

\$612

.

To find successive discounts, each percentage must be applied to the current reduced price. First, a

15\%

discount on

\$800

reduces the price by

\$120

, making it

\$680

. Then, a

10\%

discount is applied to the

\$680

sale price, which reduces it by another

\$68

. The final price is

\$680 - \$68 = \$612

. The total amount saved is the sum of both discounts,

\$120 + \$68 = \$188

.

Step-by-Step Solution

1

Calculate the first discount amount.

\$800 \times 0.15 = \$120

Find the dollar value of the initial

15\%

holiday mark-down.

2

Calculate the intermediate sale price.

\$800 - \$120 = \$680

Subtract the first discount from the regular price to find the new base price for the second discount.

3

Calculate the second discount amount.

\$680 \times 0.10 = \$68

Apply the

10\%

military discount to the intermediate sale price, not the original price.

4

Determine the total savings and final price.

Total savings:

\$120 + \$68 = \$188

. Final price:

\$800 - \$188 = \$612

.

Combine both discount amounts to find the total saved, and subtract from the original price to find the final price.

Key Concept

Calculating successive discounts requires applying the second percentage to the newly reduced price, rather than adding the percentages together.

Estimated Time:1m 30s

Question 2053Question →

A maintenance technician is preparing a chemical solvent for parts cleaning. The technical manual specifies that the active cleaning agent must make up $32.5\%$ of the total solution. What fraction of the total solution must be the active cleaning agent, expressed in simplest form?

\frac{13}{40}

\frac{13}{27}

\frac{13}{400}

\frac{40}{13}

Show answer & explanation

Answer: $\frac{13}{40}$

Answer

The correct fraction in its simplest form is

\frac{13}{40}

.

To convert

32.5\%

to a fraction, it is first written as

\frac{32.5}{100}

. Multiplying by

10

to remove the decimal results in

\frac{325}{1000}

. Dividing both numbers by

25

yields the simplest form,

\frac{13}{40}

.

Step-by-Step Solution

1

Convert the percentage to a fraction over 100.

32.5\% = \frac{32.5}{100}

A percentage represents a part out of 100.

2

Eliminate the decimal in the numerator by multiplying both the numerator and the denominator by 10.

\frac{32.5 \times 10}{100 \times 10} = \frac{325}{1000}

Fractions must be expressed using whole numbers before they can be simplified properly.

3

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 25.

325 \div 25 = 13

and

1000 \div 25 = 40

. The simplified fraction is

\frac{13}{40}

.

The question requires the final answer to be expressed in its simplest form.

Key Concept

Converting a percentage with a decimal to a fraction in simplest form requires creating a fraction out of 100, scaling the fraction to eliminate the decimal, and finding the greatest common divisor.

Question 2054Question →

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 2055Question →

A customer is purchasing a winter coat that has an original price of $\$140$ . The store offers a $25\%$ discount on the coat. After the discount is applied, a $6\%$ sales tax is added to the purchase. What is the final total cost of the winter coat?

\$113.40

\$121.90

\$111.30

\$105.63

Show answer & explanation

Answer: $\$111.30$

Answer

The final total cost of the winter coat is

\$111.30

.

The correct answer accurately calculates the discounted price first (

\$105

), and then correctly applies the

6\%

sales tax to that new, lower amount (

\$105 \times 1.06 = \$111.30

).

Step-by-Step Solution

1

Calculate the discount amount.

\$140 \times 0.25 = \$35

To find how much money will be taken off the original price, multiply the original price by the discount percentage.

2

Determine the discounted sale price.

\$140 - \$35 = \$105

Subtract the discount amount from the original price to find the price before tax.

3

Calculate the sales tax on the discounted price.

\$105 \times 0.06 = \$6.30

Sales tax is applied only to the final sale price, not the original sticker price.

4

Calculate the final total cost.

\$105 + \$6.30 = \$111.30

Add the sales tax amount to the discounted sale price to find the final amount paid at the register.

Key Concept

Discounts and Sale Prices

Question 2056Question →

A combat engineering commander is reviewing the progress of four specialized construction teams. The commander needs to prioritize resource drops, starting with the team that has completed the least amount of their assigned sector. Arrange the teams in order from the least portion of their sector completed to the most completed.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from least to most completed is: Team Charlie (

0.6

), Team Alpha (

\frac{5}{8}

), Team Delta (

\frac{13}{20}

), and Team Bravo (

66\%

).

By converting all values to percentages (

60\%

,

62.5\%

,

65\%

, and

66\%

), the relative sizes become immediately clear, allowing them to be ranked precisely from least to greatest.

Step-by-Step Solution

1

Identify the objective

We need to order four different numerical formats (fractions, decimals, percentages) from least to greatest.

Comparing mixed formats requires converting them all to a single, consistent format. Percentages are often the easiest format for direct comparison.

2

Convert Team Charlie's progress to a percentage

0.6 \times 100 = 60\%

Decimals are converted to percentages by multiplying by

100

or moving the decimal point two places to the right.

3

Convert Team Alpha's progress to a percentage

\frac{5}{8} = 5 \div 8 = 0.625

, which is

62.5\%

Fractions are converted to decimals by dividing the numerator by the denominator, and then to percentages by multiplying by

100

.

4

Convert Team Delta's progress to a percentage

\frac{13}{20} = \frac{13 \times 5}{20 \times 5} = \frac{65}{100} = 65\%

Scaling the fraction's denominator to

100

makes it easy to find the equivalent percentage without long division.

5

Compare and order all percentages from least to greatest

60\% < 62.5\% < 65\% < 66\%

This corresponds directly to Team Charlie, Team Alpha, Team Delta, and Team Bravo.

Key Concept

Converting fractions and decimals to a uniform format (percentages) in order to compare and rank their values.

Question 2057Question →

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 2058Question →

Calculate the total gross earnings based on the given compensation structure.

Fill in the blanks below

A medical supplies sales representative earns a weekly base salary of

\$600

. She also earns a

4\%

commission on her first

\$10,000

in sales for the week, and a

7\%

commission on any sales amount exceeding

\$10,000

. If she achieves

\$18,000

in total sales this week, her total gross earnings will be $\$$.

Show answer & explanation

Answer

1,560

To find the total gross earnings, we must add the base salary to the commissions earned from each tier. The base salary is

\$600

. The first commission tier is

4\%

of the first

\$10,000

, which is

\$400

. The second tier is

7\%

of the remaining

\$8,000

in sales (since

\$18,000 - \$10,000 = \$8,000

), which is

\$560

. Adding these three components together gives

\$600 + \$400 + \$560 = \$1,560

.

Step-by-Step Solution

1

Identify the fixed base salary.

The base salary is

\$600

.

This is a guaranteed amount earned regardless of the sales volume achieved.

2

Calculate the commission on the first

\$10,000

of sales.

0.04 \times \$10,000 = \$400

.

The representative earns a

4\%

commission on all sales up to the

\$10,000

threshold.

3

Determine the amount of sales that exceeds the

\$10,000

threshold.

\$18,000 - \$10,000 = \$8,000

.

The higher commission rate of

7\%

only applies to the portion of sales above

\$10,000

.

4

Calculate the commission on the sales exceeding

\$10,000

.

0.07 \times \$8,000 = \$560

.

This applies the

7\%

rate to the remaining

\$8,000

in sales.

5

Add the base salary and all commissions to find the total gross earnings.

\$600 + \$400 + \$560 = \$1,560

.

Total gross earnings are the sum of the fixed base salary and all variable commission tiers.

Key Concept

Calculating total gross earnings using a combination of a fixed base salary and a tiered commission structure.

Question 2059Question →

A damage control team reports that a leak has flooded $0.08$ of a storage compartment's total volume. Which of the following fractions represents the portion of the compartment that is NOT flooded?

\frac{4}{5}

\frac{2}{25}

\frac{23}{25}

\frac{1}{5}

Show answer & explanation

Answer: $\frac{23}{25}$

Answer

The fraction that represents the portion of the compartment that is not flooded is

\frac{23}{25}

.

The flooded portion is given as

0.08

. The portion that is not flooded is the total (

1.00

) minus the flooded portion, which is

0.92

. To convert the decimal

0.92

to a fraction, write it as

\frac{92}{100}

and simplify by dividing the numerator and denominator by

4

, resulting in

\frac{23}{25}

.

Step-by-Step Solution

1

Determine the decimal portion of the compartment that is not flooded.

1.00 - 0.08 = 0.92

The total volume is represented by

1.00

. The flooded portion must be subtracted to find the remaining unflooded volume.

2

Convert the resulting decimal into a fraction.

\frac{92}{100}

The decimal

0.92

represents ninety-two hundredths, which translates directly to a fraction with a denominator of

100

.

3

Simplify the fraction to its lowest terms.

\frac{23}{25}

Both

92

and

100

share a greatest common factor of

4

. Dividing the numerator and the denominator by

4

yields the simplified fraction.

Key Concept

Converting a decimal to a simplified fraction and finding the complement of a whole.

Question 2060Question →

A landscaping company is purchasing a bulk order of premium grass seed. The wholesale price of the order is $\$550$ . The supplier offers a $20\%$ commercial discount on the order. After the discount is applied, a $5\%$ sales tax is added. What is the final total cost of the grass seed order?

\$462.00

\$467.50

\$556.50

\$115.50

Show answer & explanation

Answer: $\$462.00$

Answer

The final total cost of the grass seed order is

\$462.00

.

The correct answer is found by completing the calculation in two distinct steps. First, apply the

20\%

discount to the original

\$550

price (

\$550 \times 0.80 = \$440

). Second, calculate the

5\%

sales tax on the new

\$440

balance and add it to the cost (

\$440 \times 1.05 = \$462.00

).

Step-by-Step Solution

1

Calculate the discount amount by finding

20\%

of

\$550

.

\$550 \times 0.20 = \$110

The commercial discount reduces the original wholesale price by a percentage.

2

Subtract the discount amount from the original price to find the sale price.

\$550 - \$110 = \$440

This establishes the actual price of the grass seed before sales tax is applied.

3

Calculate the

5\%

sales tax on the discounted sale price.

\$440 \times 0.05 = \$22

Sales tax must be applied to the final sale price, not the original sticker price.

4

Add the sales tax to the sale price to find the final total cost.

\$440 + \$22 = \$462.00

The final cost includes both the discounted price and the applicable tax.

Key Concept

Calculating multi-step financial arithmetic problems involving sequential discounts and sales taxes.

Answer

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