All practice questions

Match each physical or chemical concept related to matter with its corresponding scientific description.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Density matches 'The measure of mass per unit volume of a substance', Deposition matches 'A physical phase transition where a gas converts directly into a solid', Thermal conductivity matches 'The rate at which heat energy is transferred through a material', and Oxidation matches 'A chemical change in which a substance reacts with oxygen to form a new compound'.

Each concept accurately pairs with its foundational physical or chemical definition: Density is mass per unit volume, deposition is the gas-to-solid phase change, thermal conductivity is the heat transfer capability, and oxidation is a chemical change involving reaction with oxygen.

Step-by-Step Solution

1

Identify the definitions of volumetric and mass relationships

Density is identified as the intensive property representing mass per unit volume (

D = \frac{m}{V}

).

This establishes the direct relationship between matter's mass and the volume it occupies.

2

Distinguish between phase changes and chemical changes

Deposition is identified as the direct gas-to-solid phase change, while oxidation is identified as a chemical reaction forming new substances.

Phase changes retain chemical identity, whereas oxidation alters the chemical formula and bonds.

3

Classify transport properties of matter

Thermal conductivity is identified as the property dictating heat flow rate through a substance.

Thermal conductivity describes the physical transport of thermal energy across a temperature gradient.

Key Concept

Physical Properties, Chemical Properties, and State Changes of Matter

Estimated Time:1m 0s

Question 3242Question →

A military dining facility begins the week with $30 \frac{1}{2}$ gallons of cooking oil. On Monday, the kitchen uses $12 \frac{3}{4}$ gallons of the oil. On Tuesday, the kitchen uses another $9 \frac{2}{3}$ gallons. How many gallons of cooking oil remain at the facility?

8 \frac{1}{12}

8 \frac{11}{14}

9 \frac{1}{12}

22 \frac{5}{12}

Show answer & explanation

Answer: $8 \frac{1}{12}$

Answer

8 \frac{1}{12}

The correct answer is found by first calculating the total oil used by finding a common denominator of 12 and regrouping (

12 \frac{3}{4} + 9 \frac{2}{3} = 22 \frac{5}{12}

gallons). Then, this combined amount is subtracted from the initial inventory using the same common denominator (

30 \frac{1}{2} - 22 \frac{5}{12} = 30 \frac{6}{12} - 22 \frac{5}{12} = 8 \frac{1}{12}

gallons).

Step-by-Step Solution

1

Find the total amount of oil used by adding the amounts from Monday and Tuesday.

12 \frac{3}{4} + 9 \frac{2}{3}

To find out how much oil is left, we must first know the total volume that was removed from the initial inventory.

2

Convert the fractions to a common denominator to add them.

The least common multiple of

4

and

3

is

12

. The fractions become

\frac{9}{12}

and

\frac{8}{12}

.

Fractions must have the same denominator before they can be added together.

3

Add the converted mixed numbers and regroup the improper fraction.

12 \frac{9}{12} + 9 \frac{8}{12} = 21 \frac{17}{12}

. Since

\frac{17}{12}

is

1 \frac{5}{12}

, the total used is

22 \frac{5}{12}

gallons.

Simplifying the combined total makes the final subtraction step accurate and easier to perform.

4

Subtract the total amount used from the initial inventory to find the remaining oil.

30 \frac{1}{2} - 22 \frac{5}{12}

Subtracting the part used from the whole gives the remaining balance.

5

Find a common denominator for the final subtraction.

The least common multiple of

2

and

12

is

12

. The initial amount becomes

30 \frac{6}{12}

.

Again, fractions require a common denominator for subtraction.

6

Perform the subtraction.

30 \frac{6}{12} - 22 \frac{5}{12} = 8 \frac{1}{12}

gallons.

This calculation yields the final volume of oil remaining in the facility.

Key Concept

Adding and subtracting mixed numbers with unlike denominators by finding the lowest common denominator and regrouping improper fractions.

Question 3243Question →

A paleontologist unearths a well-preserved fern fossil within a horizontal layer of shale situated directly between two distinct volcanic ash beds. Radiometric dating reveals that the underlying ash bed is 65 million years old, while the overlying ash bed is 60 million years old. Based on these stratigraphic relationships, which conclusion about the age of the fossil is most accurate?

The fossil is older than 65 million years because organisms buried in shale always predate underlying rock layers.

The fossil organism lived between 60 and 65 million years ago.

The fossil is exactly 60 million years old because it crystallized directly from cooling magma within the upper volcanic bed.

The fossil is younger than 60 million years because sedimentary rock can only form after all volcanic activity has ceased.

Show answer & explanation

Answer: The fossil organism lived between 60 and 65 million years ago.

Answer

The fossil organism lived between 60 and 65 million years ago.

According to the Law of Superposition in undisturbed strata, lower layers are older than upper layers. Because the fossil-bearing shale was deposited on top of the 65-million-year-old ash bed and was subsequently covered by the 60-million-year-old ash bed, the fern fossil is reliably bracketed to an age between 60 and 65 million years old.

Step-by-Step Solution

1

Apply the Law of Superposition to determine the relative sequence of the strata.

The lower volcanic ash is the oldest layer, the middle shale layer containing the fossil is intermediate in age, and the upper volcanic ash is the youngest layer.

In an undisturbed geological column, rock layers are deposited sequentially with older beds on the bottom and younger beds on top.

2

Bracket the age of the fossil-bearing shale using the numerical dates of the bounding volcanic ash beds.

The shale and its enclosed fossil must be younger than 65 million years and older than 60 million years.

The sedimentary layer was deposited after the lower ash layer settled but before the upper ash layer was laid down.

Key Concept

Radiometric age bracketing and the Law of Superposition

Estimated Time:1m 0s

Question 3244Question →

On a clear, sunny afternoon at the beach, the dry sand becomes hot enough to burn bare feet, while the adjacent ocean water remains relatively cool despite receiving identical amounts of solar radiation. Which property accounts for this difference in temperature?

Water reflects virtually all incoming solar radiation away from its surface.

Water has a significantly higher specific heat capacity than sand.

Sand undergoes rapid evaporative cooling when exposed to sunlight.

Sand converts absorbed radiant energy into potential energy rather than kinetic energy.

Show answer & explanation

Answer: Water has a significantly higher specific heat capacity than sand.

Answer

Water has a significantly higher specific heat capacity than sand.

Specific heat capacity is the amount of heat energy needed to raise the temperature of a given mass of a substance by one degree Celsius. Water has one of the highest specific heat capacities of common natural substances (around

4.184\text{ J}/(\text{g}\cdot^\circ\text{C})

), whereas dry sand has a relatively low specific heat capacity (around

0.80\text{ J}/(\text{g}\cdot^\circ\text{C})

). As a result, receiving the same amount of solar radiation causes sand to experience a rapid and dramatic temperature increase, while water exhibits only a modest temperature change.

Step-by-Step Solution

1

Analyze the thermal conditions described in the scenario

Both the sand and the water absorb equal radiant solar energy per unit area over the same duration, but the sand reaches a much higher temperature.

Establishing equal energy input isolates material-specific thermal properties as the cause of the temperature variance.

2

Apply the thermodynamic definition of specific heat capacity (

Q = mc\Delta T

)

Temperature change is inversely proportional to specific heat capacity (

\Delta T = \frac{Q}{mc}

).

Specific heat capacity (

c

) measures the amount of heat energy required to raise the temperature of one unit of mass by one degree Celsius.

3

Compare the specific heat capacities of water and sand

Water has a specific heat capacity of approximately

4.184\text{ J}/(\text{g}\cdot^\circ\text{C})

, whereas dry sand has a specific heat capacity of approximately

0.80\text{ J}/(\text{g}\cdot^\circ\text{C})

.

Because water requires roughly five times more energy per gram to raise its temperature by one degree, it warms up far more slowly than sand under the same sun exposure.

Key Concept

Specific Heat Capacity and Thermal Inertia

Estimated Time:1m 0s

Question 3245Question →

A portable work light operates on a simple direct-current circuit. When switched on, an electric current of $2\text{ A}$ passes through the bulb, which has a resistance of $3\,\Omega$ . According to Ohm's law ( $V = I \times R$ ), what is the voltage supplied by the battery?

1.5\text{ V}

5\text{ V}

6\text{ V}

0.67\text{ V}

Show answer & explanation

Answer: $6\text{ V}$

Answer

The voltage supplied by the battery is

6\text{ V}

.

According to Ohm's law, voltage (

V

) equals current (

I

) multiplied by resistance (

R

). Multiplying

2\text{ A}

by

3\,\Omega

gives

6\text{ V}

.

Step-by-Step Solution

1

Identify the given electrical quantities from the problem.

Current

I = 2\text{ A}

and resistance

R = 3\,\Omega

.

Ohm's law requires current and resistance to determine voltage.

2

Apply Ohm's law formula

V = I \times R

.

V = 2\text{ A} \times 3\,\Omega = 6\text{ V}

.

Multiplying electric current by internal resistance calculates the total potential difference (voltage).

Key Concept

Ohm's law states that the electrical potential difference (voltage) across a conductor is directly proportional to the current flowing through it, given by

V = I \times R

.

Estimated Time:40s

Question 3246Question →

A maintenance technician needs to identify an unknown metal sample by evaluating its characteristics in a testing laboratory. Which of the following procedures tests a chemical property of the metal rather than a physical property?

Heating the metal sample in a furnace to determine the temperature at which it melts

Submerging the metal sample in a graduated cylinder to measure its water displacement volume

Applying a drop of concentrated hydrochloric acid to observe whether hydrogen gas is produced

Weighing the metal sample on a spring scale to measure the gravitational force acting upon it

Show answer & explanation

Answer: Applying a drop of concentrated hydrochloric acid to observe whether hydrogen gas is produced

Answer

Applying a drop of concentrated hydrochloric acid to observe whether hydrogen gas is produced

Applying hydrochloric acid to observe gas production directly tests chemical reactivity, which is a chemical property because the metal undergoes a reaction to form new chemical substances (a metal salt and hydrogen gas).

Step-by-Step Solution

1

Define the difference between physical and chemical properties

Physical properties can be observed or measured without changing the chemical composition of the substance (e.g., melting point, density, mass, volume). Chemical properties describe how a substance interacts with other substances to form entirely new materials (e.g., reactivity with acid, flammability, oxidation).

Establishing the fundamental definitions allows proper classification of each lab procedure.

2

Evaluate the procedure testing reaction with acid

When acid is placed on the metal, the formation of hydrogen gas demonstrates a chemical reaction (

M + 2HCl \rightarrow MCl_2 + H_2

), altering the chemical bonds and forming new compounds.

A change in chemical composition indicates the measurement of a chemical property.

Key Concept

Distinction between physical properties (observable without altering composition) and chemical properties (observable only during a reaction that changes composition)

Estimated Time:1m 0s

Question 3247Question →

A local boutique acquires a designer winter coat from a distributor for a wholesale cost of $\$125.00$ . The store manager applies a $60\%$ markup to establish the regular retail price. Later in the season, the coat is moved to the clearance rack, receiving a $30\%$ discount off its regular retail price. A shopper buys the coat using a loyalty reward that takes an additional $10\%$ off the clearance price. If a $5\%$ sales tax is calculated on the final discounted price, what is the total amount the shopper pays at the register?

\$132.30

\$136.00

\$157.50

\$168.00

Show answer & explanation

Answer: $\$132.30$

Answer

The shopper pays a total of

\$132.30

at the register.

The total amount is found by applying each percentage change successively to the new amount from the previous step. First, the 60% markup sets the price at

200.00. The 30% clearance discount reduces this to

140.00. The 10% loyalty discount further reduces it to

126.00. Finally, a 5% tax on the

126.00 final price adds

6.30, resulting in a total of

132.30.

Step-by-Step Solution

1

Calculate the regular retail price by applying the

60\%

markup to the wholesale cost.

\$125.00 \times 1.60 = \$200.00

The markup increases the base cost to establish the starting retail price.

2

Calculate the clearance price by applying the

30\%

discount to the regular retail price.

\$200.00 \times (1 - 0.30) = \$140.00

The first discount is taken from the current retail price, not the wholesale cost.

3

Calculate the final item price by applying the

10\%

loyalty discount to the clearance price.

\$140.00 \times (1 - 0.10) = \$126.00

Successive discounts must be applied sequentially to the newly discounted price.

4

Calculate the total amount paid by adding the

5\%

sales tax to the final item price.

\$126.00 \times 1.05 = \$132.30

Sales tax is calculated strictly on the final discounted price of the item.

Key Concept

Calculating successive percentages for markups, discounts, and taxes sequentially based on the updated price at each step.

Question 3248Question →

A building contractor purchases a shipment of lumber with a regular retail price of $\$800.00$ . The supplier offers a $15\%$ trade discount for professional contractors. After the discount is applied, a $6\%$ sales tax is added to the purchase. What is the final total cost of the lumber?

\$728.00

\$832.10

\$1,088.00

\$720.80

Show answer & explanation

Answer: $\$720.80$

Answer

The correct answer is

\$720.80

.

The correct answer is found by taking the multi-step percentage changes in order. First, find the

15\%

discount on the regular price:

0.15 \times \$800.00 = \$120.00

. Subtract this from the original price to get the discounted price:

\$800.00 - \$120.00 = \$680.00

. Next, calculate the

6\%

sales tax on this new subtotal:

0.06 \times \$680.00 = \$40.80

. Finally, add the sales tax to the discounted price:

\$680.00 + \$40.80 = \$720.80

.

Step-by-Step Solution

1

Calculate the discount amount.

15\%

of

\$800.00

is

\$120.00

.

The trade discount is a percentage reduction based on the regular retail price.

2

Determine the discounted price.

\$800.00 - \$120.00 = \$680.00

.

Subtracting the discount from the original price gives the subtotal before tax.

3

Calculate the sales tax.

6\%

of

\$680.00

is

\$40.80

.

Sales tax is applied to the discounted price, not the original retail price.

4

Find the final total cost.

\$680.00 + \$40.80 = \$720.80

.

Adding the calculated sales tax to the discounted subtotal yields the final amount paid.

Key Concept

Calculating successive percentage changes (a discount followed by a tax) on a principal amount.

Question 3249Question →

A logistics convoy is assigned an $80$ -mile supply route. During the morning phase, the convoy travels $22 \frac{3}{8}$ miles. In the afternoon, they travel an additional $18 \frac{4}{5}$ miles before halting for a security check. After resuming, they cover another $14 \frac{1}{2}$ miles before nightfall. How many miles remain to complete the entire route? (Enter your answer as a decimal)

Show answer & explanation

Answer: 24.325

Answer

24.325

The total distance traveled is calculated by converting the fractional parts of the mixed numbers to a common denominator of 40. Adding

22 \frac{15}{40} + 18 \frac{32}{40} + 14 \frac{20}{40}

gives

55 \frac{27}{40}

miles. Subtracting this from the

80

-mile total route leaves

24 \frac{13}{40}

miles. Converting

\frac{13}{40}

to a decimal yields 0.325, giving a final answer of 24.325.

Step-by-Step Solution

1

Determine the least common denominator (LCD) for the fractional parts of the distances traveled.

The denominators are 8, 5, and 2. The LCD is 40.

Fractions must have a common denominator before they can be added together.

2

Convert each mixed number to have the common denominator of 40 and add them to find the total distance traveled.

22 \frac{15}{40} + 18 \frac{32}{40} + 14 \frac{20}{40} = 54 \frac{67}{40}

, which simplifies to

55 \frac{27}{40}

miles.

Adding the individual segments provides the total distance covered so far.

3

Subtract the total distance traveled from the overall route distance to find the remaining distance.

80 - 55 \frac{27}{40} = 79 \frac{40}{40} - 55 \frac{27}{40} = 24 \frac{13}{40}

miles.

Subtracting the part completed from the whole gives the remaining distance.

4

Convert the remaining distance into a decimal format.

\frac{13}{40} = 0.325

, making the final distance

24.325

miles.

The question specifically asks for the answer to be entered as a decimal.

Key Concept

Adding mixed numbers with unlike denominators and subtracting from a whole number.

Question 3250Question →

A light training aircraft with a mass of $750\text{ kg}$ accelerates from rest along a straight runway under a constant net forward force of $1{}500\text{ N}$ . What is the velocity of the aircraft, in meters per second ( $\text{m/s}$ ), after $8\text{ seconds}$ ?

Show answer & explanation

Answer: 16

Answer

16 m/s

According to Newton's second law (

F = ma

), a net force of

1{}500\text{ N}

applied to a mass of

750\text{ kg}

produces a constant acceleration of

a = \frac{1{}500}{750} = 2\text{ m/s}^2

. Starting from rest (

v_0 = 0

), the velocity after

8\text{ seconds}

is

v = at = 2\text{ m/s}^2 \times 8\text{ s} = 16\text{ m/s}

.

Step-by-Step Solution

1

Calculate the constant acceleration of the aircraft using Newton's second law of motion (

F_{\text{net}} = ma

).

a = \frac{1{}500\text{ N}}{750\text{ kg}} = 2\text{ m/s}^2

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

2

Calculate the final velocity after

8\text{ seconds}

using the kinematic equation for uniform acceleration (

v = v_0 + at

).

v = 0\text{ m/s} + (2\text{ m/s}^2)(8\text{ s}) = 16\text{ m/s}

Starting from rest, the final velocity equals the constant acceleration multiplied by elapsed time.

Key Concept

Newton's Second Law and Uniform Acceleration

Alternative Method

Apply the impulse-momentum theorem:

\text{Impulse} = F_{\text{net}} \times \Delta t = \Delta p = m \times v

. Solving for velocity gives

v = \frac{F_{\text{net}} \times \Delta t}{m} = \frac{1{}500\text{ N} \times 8\text{ s}}{750\text{ kg}} = \frac{12{}000}{750} = 16\text{ m/s}

.

Estimated Time:1m 0s

Question 3251Question →

A maintenance technician needs to cut a continuous length of electrical wire into smaller segments. If the technician has a piece of wire that is $45$ feet long and each segment must be $3 \frac{3}{4}$ feet long, how many full segments can be cut?

Show answer & explanation

Answer: 12

Answer

12

To find the total number of segments, the total length of wire (

45

feet) must be divided by the length of each individual segment (

3 \frac{3}{4}

feet). Converting

3 \frac{3}{4}

to the improper fraction

\frac{15}{4}

and multiplying

45

by its reciprocal (

\frac{4}{15}

) yields exactly

12

segments.

Step-by-Step Solution

1

Convert the mixed number to an improper fraction.

3 \frac{3}{4} = \frac{15}{4}

Division operations with mixed numbers require converting them to improper fractions first.

2

Set up the division equation.

45 \div \frac{15}{4}

To find how many smaller segments fit into the total length, you must divide the total wire length by the length of one individual segment.

3

Convert the division into multiplication.

45 \times \frac{4}{15}

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

4

Simplify the expression and calculate the final value.

3 \times 4 = 12

Cross-canceling

45

and

15

leaves a factor of

3

, which is then multiplied by

4

to reach the final whole number answer.

Key Concept

Dividing a whole quantity by a mixed number using improper fractions and reciprocals.

Question 3252Question →

Different materials transfer thermal energy by conduction at significantly different rates. Arrange the following substances in order of increasing thermal conductivity, starting from the poorest conductor (best thermal insulator) and ending with the best thermal conductor.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Trapped air, Liquid water, Iron, Copper

Thermal conductivity measures a material's ability to conduct heat. Gases like trapped air have widely dispersed molecules and are the poorest conductors. Liquids like water have closer molecules and conduct heat moderately better than gases. Solid metals conduct heat rapidly due to free electrons, with copper exhibiting superior conduction compared to iron.

Step-by-Step Solution

1

Identify the state of matter and thermal mechanism for each substance

Trapped air is a gas, liquid water is a non-metallic liquid, while iron and copper are solid metals.

Thermal conduction occurs through particle collisions and free electron movement, which depends heavily on state of matter and atomic bonding.

2

Rank the non-metals based on molecular density

Trapped air has the lowest thermal conductivity (

k \approx 0.026\text{ W/(m}\cdot\text{K)}

), followed by liquid water (

k \approx 0.6\text{ W/(m}\cdot\text{K)}

).

Gases have large intermolecular distances leading to rare collisions, whereas liquids are denser and transfer kinetic energy more readily.

3

Rank the metals based on electronic thermal transport

Iron has a thermal conductivity of approximately

80\text{ W/(m}\cdot\text{K)}

, whereas copper has a thermal conductivity of approximately

400\text{ W/(m}\cdot\text{K)}

.

Both metals conduct heat via free valence electrons, but copper's electronic structure allows significantly higher conductivity.

Key Concept

Thermal Conductivity of Materials

Estimated Time:1m 0s

Question 3253Question →

A rescue drone with a mass of $40\text{ kg}$ is ascending vertically. If the drone's motors generate a constant upward lift force of $520\text{ N}$ in an area where the acceleration due to gravity is $8\text{ m/s}^2$ , what is the net upward acceleration of the drone?

13\text{ m/s}^2

21\text{ m/s}^2

5\text{ m/s}^2

0.5\text{ m/s}^2

Show answer & explanation

Answer: $5\text{ m/s}^2$

Answer

5\text{ m/s}^2

The upward thrust of

520\text{ N}

is opposed by the downward gravitational force of

320\text{ N}

(

40\text{ kg} \times 8\text{ m/s}^2

). The resulting net upward force is

200\text{ N}

. Dividing this net force by the mass of

40\text{ kg}

gives an upward acceleration of

5\text{ m/s}^2

.

Step-by-Step Solution

1

Calculate the downward force of gravity (weight) acting on the drone.

W = m \times g = 40\text{ kg} \times 8\text{ m/s}^2 = 320\text{ N}

downward.

Gravity exerts a constant downward force equal to the product of mass and local gravitational acceleration.

2

Determine the net vertical force acting on the drone.

F_{\text{net}} = F_{\text{lift}} - W = 520\text{ N} - 320\text{ N} = 200\text{ N}

upward.

The upward lift and downward weight act in opposite directions along the vertical axis, so their magnitudes subtract.

3

Apply Newton's second law (

F_{\text{net}} = ma

) to solve for acceleration.

a = \frac{F_{\text{net}}}{m} = \frac{200\text{ N}}{40\text{ kg}} = 5\text{ m/s}^2

.

Acceleration equals the net unbalanced force divided by the total mass of the object.

Key Concept

Newton's Second Law and Net Force in Vertical Motion

Alternative Method

Calculate the upward acceleration produced by the motor thrust alone (

a_{\text{thrust}} = \frac{520\text{ N}}{40\text{ kg}} = 13\text{ m/s}^2

) and subtract the downward acceleration due to gravity (

8\text{ m/s}^2

), which gives

13\text{ m/s}^2 - 8\text{ m/s}^2 = 5\text{ m/s}^2

.

Estimated Time:1m 0s

Question 3254Question →

Solve the following word problem by calculating the missing value.

Fill in the blanks below

An engineering battalion can lay

40

feet of runway matting in

15

minutes. Operating at this constant rate, the battalion will lay feet of matting in

1.5

hours.

Show answer & explanation

Answer

240

Because

1.5

hours is equivalent to

90

minutes, the time period is exactly

6

times longer than the original

15

-minute period (

90 \div 15 = 6

). In a direct proportion, the amount of matting laid increases by the same factor, so

40 \text{ feet} \times 6 = 240 \text{ feet}

.

Step-by-Step Solution

1

Convert the target time frame into the same units used for the known rate.

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

Units of time must match to set up a valid direct proportion.

2

Set up a proportion relating feet of matting to minutes.

\frac{40 \text{ feet}}{15 \text{ minutes}} = \frac{x \text{ feet}}{90 \text{ minutes}}

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

3

Solve for the unknown value

x

.

x = \frac{40 \times 90}{15} = 40 \times 6 = 240

Isolating

x

provides the total feet of matting laid in

90

minutes.

Key Concept

Direct proportions with unit conversion

Question 3255Question →

A maintenance crew can paint $120$ square feet of a ship's hull in $2$ hours. Working at this same constant rate, how many square feet can the crew paint in $5$ hours?

Show answer & explanation

Answer: 300

Answer

300

Because the work happens at a constant rate, the area painted is directly proportional to the time spent working. Finding the unit rate (

60

square feet per hour) and multiplying it by the target time (

5

hours) yields the correct total of

300

square feet.

Step-by-Step Solution

1

Calculate the unit rate of painting.

60

square feet per hour

To find how much is painted in a different amount of time, we first need to know how much is painted in one hour. We do this by dividing the total area (

120

) by the total time (

2

).

2

Calculate the total area painted in the new timeframe.

300

square feet

Multiply the unit rate (

60

square feet per hour) by the new target time (

5

hours) to find the final proportional amount.

Key Concept

Direct Proportions

Question 3256Question →

A logistics specialist is calculating the total weight of cargo for a transport flight. The first crate weighs $45 \frac{1}{3}$ pounds, and the second crate weighs $32 \frac{1}{6}$ pounds. What is the total combined weight of both crates?

77 \frac{1}{2}

pounds

77 \frac{2}{9}

pounds

77 \frac{1}{3}

pounds

13 \frac{1}{6}

pounds

Show answer & explanation

Answer: $77 \frac{1}{2}$ pounds

Answer

The total combined weight is

77 \frac{1}{2}

pounds.

The correct answer is found by identifying a common denominator of 6 for the fractions. The fraction 1/3 becomes 2/6. Adding 2/6 and 1/6 equals 3/6, which reduces to 1/2. Adding the whole numbers 45 and 32 gives 77. The combined weight is 77 1/2 pounds.

Step-by-Step Solution

1

Set up the addition problem for the two mixed numbers.

45 \frac{1}{3} + 32 \frac{1}{6}

The question asks for the total combined weight, requiring addition.

2

Find a common denominator for the fractional parts.

The lowest common multiple of

3

and

6

is

6

. Convert

\frac{1}{3}

by multiplying both numerator and denominator by 2 to get

\frac{2}{6}

.

Fractions must have the same denominator before they can be added together.

3

Add the whole numbers and the fractions, then simplify.

Whole numbers:

45 + 32 = 77

. Fractions:

\frac{2}{6} + \frac{1}{6} = \frac{3}{6}

. Simplifying

\frac{3}{6}

yields

\frac{1}{2}

. The final result is

77 \frac{1}{2}

.

Combining the sums of the whole numbers and the simplified fraction gives the final total weight.

Key Concept

Adding Mixed Numbers with Unlike Denominators

Estimated Time:45s

Question 3257Question →

An office manager is purchasing a commercial printer with a regular retail price of $\$1,200.00$ . The store is offering a $20\%$ discount on all electronics. Using a corporate account, the manager receives an additional $10\%$ discount applied to the reduced sale price. Finally, a $5\%$ sales tax is added to the discounted total. What is the final total cost of the printer in dollars?

Show answer & explanation

Answer: 907.2

Answer

The final total cost of the printer is

\$907.20

.

The correct answer is derived by first taking

20\%

off the

\$1,200.00

base price to get

\$960.00

. Then, the

10\%

corporate discount is taken off the

\$960.00

, bringing the subtotal down to

\$864.00

. Lastly, a

5\%

tax on the

\$864.00

subtotal adds

\$43.20

, resulting in a final cost of

\$907.20

.

Step-by-Step Solution

1

Calculate the price after the initial

20\%

electronics discount.

\$1,200.00 - (\$1,200.00 \times 0.20) = \$1,200.00 - \$240.00 = \$960.00

The first discount is applied to the original retail price of the printer.

2

Calculate the price after the additional

10\%

corporate discount.

\$960.00 - (\$960.00 \times 0.10) = \$960.00 - \$96.00 = \$864.00

Successive discounts are calculated based on the previously reduced price, not the original price.

3

Calculate the

5\%

sales tax on the final discounted price and add it to the total.

\$864.00 \times 0.05 = \$43.20

. Total =

\$864.00 + \$43.20 = \$907.20

Sales tax is always applied to the final discounted sale price before checkout.

Key Concept

Successive discounts must be applied sequentially to the new reduced price at each step, and sales tax is computed on the final discounted price.

Question 3258Question →

A military transport unit has a fuel reserve of $420$ gallons. On the first day of a field exercise, the unit consumes $\frac{2}{7}$ of the total reserve. On the second day, they consume $\frac{3}{5}$ of the remaining fuel. If the fuel that is left after the second day is divided equally among $4$ vehicles for a final patrol, how many gallons of fuel will each vehicle receive?

12

30

61 \frac{1}{4}

120

Show answer & explanation

Answer: $30$

Answer

Each vehicle will receive

30

gallons of fuel.

To find the final amount per vehicle, calculate the remainder after each step. Day 1 consumes

120

gallons, leaving

300

gallons (

420 - 120

). Day 2 uses

\frac{3}{5}

of

300

, which is

180

gallons, leaving a final remainder of

120

gallons. Dividing these

120

gallons evenly among the

4

vehicles yields exactly

30

gallons per vehicle.

Step-by-Step Solution

1

Calculate the amount of fuel consumed on the first day and find the remaining balance.

First day consumption:

420 \times \frac{2}{7} = 120

gallons. Remaining fuel:

420 - 120 = 300

gallons.

The first day's usage is given as a fraction of the total initial reserve.

2

Calculate the amount of fuel consumed on the second day from the remaining balance, and find the new remainder.

Second day consumption:

300 \times \frac{3}{5} = 180

gallons. Remaining fuel:

300 - 180 = 120

gallons.

The problem explicitly states the second day's consumption is a fraction of the remaining fuel, not the original total.

3

Divide the final remaining fuel equally among the

4

vehicles.

120 \div 4 = 30

gallons per vehicle.

The total leftover fuel must be distributed evenly to find the specific amount allocated to each vehicle.

Key Concept

Applying sequential fraction multiplication to calculate remaining quantities in multi-step word problems.

Alternative Method

Use multiplication of the remaining fractions directly. After Day 1,

1 - \frac{2}{7} = \frac{5}{7}

of the fuel is left. After Day 2,

1 - \frac{3}{5} = \frac{2}{5}

of that remainder is left. The total fraction remaining of the original amount is

\frac{5}{7} \times \frac{2}{5} = \frac{2}{7}

. Multiply this by the original amount:

420 \times \frac{2}{7} = 120

gallons. Finally, divide by

4

to get

30

gallons per vehicle.

Estimated Time:1m 15s

Question 3259Question →

Electromagnetic principles are applied in many common electrical machines and tools. Match each device with its primary operational function.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Electric Generator matches with converting mechanical energy to electrical energy; Electric Motor matches with converting electrical energy to mechanical rotation; Transformer matches with adjusting alternating current voltage levels; Electromagnet matches with producing a controllable magnetic field from electric current.

Each electromagnetic device corresponds directly to its defined operational function: generators convert mechanical work into electricity, motors convert electricity into mechanical rotation, transformers step AC voltage up or down, and electromagnets create temporary magnetic fields via electric current.

Step-by-Step Solution

1

Identify the energy transformation in an electric generator.

Mechanical energy is converted into electrical energy via induction.

Generators are driven by mechanical power to produce current.

2

Identify the energy transformation in an electric motor.

Electrical energy is converted into mechanical motion.

Motors consume electricity to produce physical torque and rotation.

3

Identify the primary role of a transformer in power distribution.

Voltage levels of alternating current are increased or decreased.

Transformers rely on changing magnetic flux between primary and secondary coils to adjust voltage.

4

Identify the mechanism of an electromagnet.

Current flowing through a coil creates a magnetic field.

Moving electric charges naturally produce magnetic fields around the coil.

Key Concept

Electromagnetic Devices and Energy Transformations

Estimated Time:45s

Question 3260Question →

A commercial bakery makes a standard pastry dough using a ratio of $7$ parts flour to $3$ parts butter by weight. If a daily batch requires $42$ pounds of flour, how many pounds of butter are needed?

18

60

98

126

Show answer & explanation

Answer: $18$

Answer

18

The ratio of flour to butter is given as

7:3

. Setting up a direct proportion where

x

is the pounds of butter needed:

\frac{7}{3} = \frac{42}{x}

. Cross-multiplying gives

7x = 126

. Solving for

x

yields

x = 18

pounds of butter.

Step-by-Step Solution

1

Identify the given ratio and the known quantity.

The ratio of flour to butter is

7:3

. The known quantity of flour is

42

pounds.

To set up a proportion, we must align the correct parts of the ratio with the real-world amounts.

2

Set up a direct proportion equation.

\frac{7}{3} = \frac{42}{x}

The ratio of parts must equal the ratio of the actual weights.

3

Solve for the unknown value

x

.

7x = 3 \times 42 \rightarrow 7x = 126 \rightarrow x = 18

Cross-multiplying and dividing isolates the variable to find the required pounds of butter.

Key Concept

Direct Proportions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes