All practice questions

Question 1901Question →

A sealed industrial holding tank contains water with a pocket of compressed air trapped above the water surface. The total depth of the water in the tank is $8.0\text{ m}$ . A pressure gauge located at the absolute bottom of the tank reads a gauge pressure of $130\text{ kPa}$ . A secondary discharge valve is located on the side of the tank, exactly $5.0\text{ m}$ above the tank's bottom. Assuming the density of water is $1,000\text{ kg/m}^3$ and the acceleration due to gravity is $g = 10\text{ m/s}^2$ , what is the gauge pressure, in kilopascals (kPa), at the secondary discharge valve?

Show answer & explanation

Answer: 80

Answer

80

The pressure at any point in a static fluid is equal to the pressure at a lower point minus the hydrostatic pressure of the fluid column between them. Since the valve is

5.0\text{ m}

above the bottom, the pressure there is less than the bottom pressure by

\rho g \Delta h = 1,000 \times 10 \times 5.0 = 50\text{ kPa}

. Subtracting this from the bottom pressure (

130\text{ kPa}

) gives the correct pressure of

80\text{ kPa}

. Alternatively, calculating the trapped air pressure (

130 - 80 = 50\text{ kPa}

) and adding the

3.0\text{ m}

water column above the valve (

30\text{ kPa}

) yields the same correct result.

Step-by-Step Solution

1

Determine the vertical distance between the pressure gauge at the bottom and the secondary valve.

\Delta h = 5.0\text{ m}

Hydrostatic pressure changes linearly with vertical height within a continuous fluid.

2

Calculate the hydrostatic pressure difference exerted by this

5.0\text{ m}

column of water.

\Delta P = \rho g \Delta h = 1,000 \times 10 \times 5.0 = 50,000\text{ Pa} = 50\text{ kPa}

This represents the amount of pressure lost as you move upward from the bottom to the valve.

3

Subtract the pressure difference from the gauge pressure at the bottom of the tank.

130\text{ kPa} - 50\text{ kPa} = 80\text{ kPa}

The pressure at the valve must be lower than the pressure at the bottom by exactly the weight of the water column between them.

Key Concept

Hydrostatic pressure varies directly with depth. The pressure difference between any two points in a continuous static fluid depends only on the vertical distance between them, regardless of surface air pressure.

Question 1902Question →

A rubber drive belt is slipping on a steel motor pulley because there is insufficient frictional resistance between the two surfaces. Which of the following adjustments will most effectively increase the frictional force to prevent the belt from slipping?

Using a narrower belt to decrease the contact area

Increasing the tension on the drive belt

Replacing the pulley with a heavier one of the same size

Applying a sudden rotational force to use the pulley's inertia

Show answer & explanation

Answer: Increasing the tension on the drive belt

Answer

Increasing the tension on the drive belt

The correct adjustment increases the belt's tension. Friction depends on two main factors: the materials involved (coefficient of friction) and how hard they are pressed together (normal force). In a mechanical drive system, increasing the belt tension pulls the belt tighter around the pulley, directly increasing the normal force pressing inward. This higher normal force generates a proportionally higher static friction, giving the belt better grip to prevent slipping.

Step-by-Step Solution

1

Identify the formula and components of mechanical friction.

The maximum static frictional force is determined by the equation

F_f = \mu F_N

, where

\mu

is the coefficient of friction and

F_N

is the normal force.

Understanding the mathematical relationship reveals which physical variables can be manipulated to change the friction.

2

Determine how normal force is generated in a belt and pulley system.

The normal force (the force pressing the belt perpendicularly into the face of the pulley) is generated by the tension pulling on the ends of the belt.

To increase

F_N

(and thereby increase

F_f

), we must increase the force that squeezes the two components together.

3

Evaluate the given modifications.

Increasing the tension directly increases the normal force. Changing the surface area or pulley mass does not affect the normal force between the mating surfaces.

This confirms that tightening the belt is the mechanically correct method to stop slipping.

Key Concept

Frictional force is directly proportional to the normal force pressing two surfaces together, and in a belt-drive system, this normal force is created by belt tension.

Estimated Time:1m 0s

Question 1903Question →

A warehouse automated sorting system applies a constant net horizontal force of $200 \text{ N}$ to a package with a mass of $50 \text{ kg}$ . What is the resulting acceleration of the package?

4 \text{ m/s}^2

0.25 \text{ m/s}^2

40 \text{ m/s}^2

10,000 \text{ m/s}^2

Show answer & explanation

Answer: $4 \text{ m/s}^2$

Answer

The resulting acceleration of the package is

4 \text{ m/s}^2

.

According to Newton's Second Law (

F = ma

), acceleration is calculated by dividing the applied net force by the object's mass. Dividing

200 \text{ N}

by

50 \text{ kg}

correctly results in

4 \text{ m/s}^2

.

Step-by-Step Solution

1

Identify the known variables provided in the problem.

Net Force (

F

) =

200 \text{ N}

; Mass (

m

) =

50 \text{ kg}

.

Correctly identifying the givens ensures the proper values are placed into the physics formula.

2

State Newton's Second Law of Motion and rearrange it to solve for acceleration.

The formula

F = ma

becomes

a = \frac{F}{m}

.

The question asks for the resulting acceleration, so the formula must isolate

a

.

3

Substitute the known values into the rearranged formula and calculate the result.

a = \frac{200}{50} = 4 \text{ m/s}^2

.

Executing the division yields the final acceleration value in standard metric units.

Key Concept

Newton's Second Law (

F = ma

) demonstrates that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Estimated Time:30s

Question 1904Question →

Three identical $10$ -foot planks are each placed across two sawhorses, one sawhorse at the left end (Support X) and one at the right end (Support Y). A single $60$ -pound bag of cement is placed on each plank at a different location. Rank the scenarios below based on the amount of downward force carried by Support X, from greatest to least.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The bag closest to Support X places the greatest load on it, followed by the centered bag, while the bag closest to Support Y places the least load on Support X.

Support X must carry the largest portion of the load when the weight is closest to it. As the weight moves further away across the plank, the amount of force supported by X decreases.

Step-by-Step Solution

1

Understand the principle of load distribution.

The closer a weight is to a given support, the more of that weight the support must carry.

This is a basic rule of mechanical balance and levers.

2

Evaluate the load on Support X for the bag placed

3

feet from it.

This bag is very close to Support X, so Support X carries more than half the weight.

The distance to X is smaller than the distance to Y.

3

Evaluate the load on Support X for the centered bag.

This bag is exactly

5

feet from both supports, so Support X carries exactly half the weight.

A centered load is shared evenly.

4

Evaluate the load on Support X for the bag placed

2

feet from Support Y.

This bag is

8

feet away from Support X, so Support Y carries almost all the weight, leaving very little for Support X.

The distance to X is much larger than the distance to Y.

Key Concept

The proportion of a load carried by a support increases as the center of gravity moves closer to that support.

Question 1905Question →

In order for a heavy freight locomotive to maintain a constant, steady speed along a straight, level track, the forward pulling force it exerts must be greater than the combined forces of friction and air resistance acting against it.

True

False

Show answer & explanation

Answer: False

Answer

False

The statement is false because maintaining a constant velocity requires all forces to be completely balanced (a net force of zero). The locomotive's forward pulling force must exactly equal the opposing forces of friction and air resistance, not exceed them. If the forward force were greater, the train would accelerate.

Step-by-Step Solution

1

Identify the locomotive's state of motion as described in the scenario.

The locomotive is traveling at a 'constant, steady speed along a straight, level track', which means it has a constant velocity (no change in speed or direction).

Understanding the state of motion is required to determine which of Newton's laws applies to the situation.

2

Apply Newton's First Law (Inertia) to the state of constant velocity.

An object traveling at a constant velocity must have a net force of exactly zero acting upon it.

Inertia ensures that an object will maintain its current velocity indefinitely unless acted upon by an unbalanced (net) force.

3

Determine the necessary relationship between the opposing forces to achieve a zero net force.

The forward pulling force must perfectly balance the backward forces of friction and air resistance.

If the forward force were greater than the opposing forces, there would be a net forward force, which would cause the locomotive to continuously accelerate rather than maintain a steady speed.

Key Concept

An object moving in a straight line at a constant speed has a net force of zero acting on it, meaning all applied driving forces perfectly balance all resisting forces.

Estimated Time:45s

Question 1906Question →

Match each mechanical system or scenario with the correct description of its convective heat transfer process.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Engine coolant matches forced convection into a liquid; the radiator blower matches forced convection into a gas; the smokestack matches natural convection of a heated gas; and the reversed ceiling fan matches forced convection overcoming gas buoyancy.

Each mechanical application relies on a specific interaction between a solid and a fluid, or manipulates fluid flow using either mechanical force or natural buoyancy. The pump moves a liquid medium, the blower moves a gaseous medium across solid fins, the smokestack utilizes density differences, and the ceiling fan directly opposes thermal rising.

Step-by-Step Solution

1

Determine if each scenario relies on a mechanical device (forced convection) or fluid density differences (natural convection).

The pumped coolant, the radiator blower, and the ceiling fan are forced convection. The unpowered smokestack is natural convection.

Categorizing the primary driving force narrows down the available matching options.

2

Identify the fluid medium and the direction of fluid movement for the forced convection scenarios.

Coolant is a liquid absorbing heat from a solid. Radiator air is a gas absorbing heat from a solid. The ceiling fan actively pushes warm air down against its natural tendency to rise.

Matching the specific fluid states (liquid vs. gas) and unique flow paths firmly links each remaining scenario to its correct definition.

Key Concept

Convection is 'forced' when a mechanical device (pump, fan) moves the fluid, and 'natural' when temperature-induced density changes cause the fluid to rise or fall.

Question 1907Question →

Water flows at a steady rate through a horizontal pipe that consists of four distinct sections with different diameters. According to Bernoulli's principle and the continuity equation, arrange the pipe sections in order of their static pressure, from the lowest static pressure to the highest static pressure.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from lowest to highest static pressure is: Section W (1-inch), Section X (2-inch), Section Y (3-inch), Section Z (4-inch).

According to the continuity equation, a fluid travels faster through a narrower section of pipe. According to Bernoulli's principle, an increase in fluid velocity results in a decrease in static pressure. Therefore, the pipe sections with the smallest diameters will have the highest velocities and the lowest static pressures, while the sections with the largest diameters will have the lowest velocities and the highest static pressures.

Step-by-Step Solution

1

Determine the relationship between the pipe's diameter and the fluid's velocity.

The continuity equation indicates that fluid must travel faster through narrower sections to maintain a steady flow rate. Therefore, the smallest diameter has the highest velocity, and the largest diameter has the lowest velocity.

Because the volume of fluid passing through any section per second must remain constant.

2

Determine the relationship between fluid velocity and static pressure.

Bernoulli's principle states that an increase in fluid velocity occurs simultaneously with a decrease in static pressure.

The total energy in a steady, horizontal fluid flow is constant; as kinetic energy (speed) increases, pressure potential energy must decrease.

3

Rank the sections based on these relationships.

Since smallest diameter = highest velocity = lowest pressure, the order from lowest to highest pressure corresponds exactly to the order from smallest to largest diameter.

This allows us to arrange the sections directly by their diameter sizes: 1-inch, 2-inch, 3-inch, and 4-inch.

Key Concept

Fluid Dynamics and Bernoulli's Principle

Question 1908Question →

A marine salvage team is recovering two solid objects resting on the ocean floor: a 100-pound lead weight and a 100-pound concrete block. Lead has a significantly higher density than concrete. While both objects are completely submerged, which of the following statements is correct regarding the buoyant forces acting on them?

Both objects experience the same buoyant force because they have identical weights.

The concrete block experiences a greater buoyant force because it displaces a larger volume of water.

The lead weight experiences a greater buoyant force because its higher density creates a stronger downward reaction.

The lead weight experiences a greater buoyant force because its concentrated mass exerts more pressure on the water.

Show answer & explanation

Answer: The concrete block experiences a greater buoyant force because it displaces a larger volume of water.

Answer

The concrete block experiences a greater buoyant force because it displaces a larger volume of water.

According to Archimedes' Principle, the upward buoyant force acting on a submerged object is exactly equal to the weight of the fluid it displaces. Because both objects weigh 100 pounds but concrete is less dense than lead, the concrete block must take up more space (larger volume). When completely submerged, the larger volume of the concrete block displaces a larger volume of water, resulting in a significantly greater upward buoyant force compared to the smaller, denser lead weight.

Step-by-Step Solution

1

Analyze the relationship between weight, density, and volume for both objects.

Both weigh 100 pounds, but lead is denser than concrete. Therefore, the 100-pound concrete block must have a larger physical volume than the 100-pound lead weight.

Density is mass per unit volume. For objects of equal mass, the less dense material takes up more space.

2

Apply Archimedes' Principle to completely submerged objects.

The buoyant force is equal to the weight of the displaced fluid. The volume of displaced fluid equals the volume of the submerged object.

Archimedes' Principle states that an object fully or partially immersed in a fluid is buoyed up by a force equal to the weight of the fluid that it displaces.

3

Compare the displaced fluid volumes and resulting buoyant forces.

Because the concrete block has a larger volume, it displaces more water than the lead weight. Consequently, the concrete block experiences a greater upward buoyant force.

Greater displaced volume directly results in a proportionally greater buoyant force, regardless of the object's interior weight.

Key Concept

Buoyancy and Archimedes' Principle

Question 1909Question →

A heavy transport truck is traveling East on a straight highway at a constant speed of $90 \text{ km/h}$ . The driver applies the brakes, producing a constant deceleration until the truck comes to a complete stop in exactly $10 \text{ seconds}$ . Which of the following correctly describes the truck's average acceleration vector during this braking period?

2.5 \text{ m/s}^2

West

2.5 \text{ m/s}^2

East

9.0 \text{ m/s}^2

West

9.0 \text{ m/s}^2

East

Show answer & explanation

Answer: $2.5 \text{ m/s}^2$ West

Answer

The correct acceleration vector is

2.5 \text{ m/s}^2

West.

To find the acceleration, the initial speed must first be converted from km/h to m/s by multiplying by

1000/3600

, resulting in

25 \text{ m/s}

. The acceleration magnitude is the change in velocity (

25 \text{ m/s}

) divided by the time (

10 \text{ s}

), giving

2.5 \text{ m/s}^2

. Because the truck is braking while moving East, the acceleration vector must oppose the motion, meaning it points West.

Step-by-Step Solution

1

Convert the initial velocity from km/h to m/s.

90 \text{ km/h} \times \frac{1000 \text{ m}}{3600 \text{ s}} = 25 \text{ m/s}

East.

Standard kinematic equations require consistent metric units, specifically meters and seconds, to yield an acceleration in

\text{m/s}^2

.

2

Calculate the magnitude of the average acceleration using the formula

a = \frac{v_f - v_i}{t}

.

a = \frac{0 \text{ m/s} - 25 \text{ m/s}}{10 \text{ s}} = -2.5 \text{ m/s}^2

.

The final velocity (

v_f

) is zero because the truck comes to a complete stop.

3

Determine the direction of the acceleration vector.

The acceleration is

2.5 \text{ m/s}^2

West.

Because the truck is slowing down, the acceleration vector must point in the opposite direction of its velocity (East). The negative sign in the calculation confirms this opposing direction.

Key Concept

Kinematic equations with unit conversions and the vector nature of acceleration

Estimated Time:1m 30s

Question 1910Question →

Consider the linear equation provided below:

-2(y - 8) + 3y = 4(y - 1) - y + 10

Which of the following numbers correctly satisfies this equation?

11

5

0

-11

Show answer & explanation

Answer: 5

Answer

The correct value that satisfies the equation is 5.

The correct answer is found by first distributing the values outside the parentheses, maintaining the positive sign when multiplying -2 by -8. This results in -2y + 16 + 3y = 4y - 4 - y + 10. Combining like terms simplifies the equation to y + 16 = 3y + 6. Finally, isolating the variable by subtracting y and 6 from both sides gives 10 = 2y, which divides cleanly to 5.

Step-by-Step Solution

1

Distribute the outside coefficients into their respective parentheses.

-2y + 16 + 3y = 4y - 4 - y + 10

Distribution eliminates the parentheses, ensuring careful attention to multiplying the negative coefficient (

-2 \times -8 = +16

).

2

Combine the like variable terms and constant terms on each side independently.

y + 16 = 3y + 6

Simplifying each side makes it easier to track and isolate the variable in the next steps.

3

Isolate the variable terms on one side by subtracting

y

from both sides.

16 = 2y + 6

Gathering all variable terms on a single side is required to solve for the unknown.

4

Isolate the constant terms by subtracting 6 from both sides, then divide by 2.

10 = 2y \Rightarrow y = 5

Using inverse operations isolates

y

to reveal its final numerical value.

Key Concept

Solving One-Variable Linear Equations

Question 1911Question →

Evaluate the following expression involving fractions:

-\frac{2}{3} + \frac{1}{4}

What is the value of the expression in simplest form?

-\frac{11}{12}

-\frac{1}{7}

-\frac{5}{12}

\frac{5}{12}

Show answer & explanation

Answer: $-\frac{5}{12}$

Answer

The correct value is

-\frac{5}{12}

.

The correct answer is found by first determining the least common denominator, which is 12. The expression is rewritten as

-\frac{8}{12} + \frac{3}{12}

. By combining the numerators (

-8 + 3

), we get

-5

, resulting in a final simplified fraction of

-\frac{5}{12}

.

Step-by-Step Solution

1

Find the least common denominator (LCD) for the fractions.

The LCD for 3 and 4 is 12.

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

2

Rewrite both fractions as equivalent fractions with a denominator of 12.

-\frac{2}{3} = -\frac{8}{12}

and

\frac{1}{4} = \frac{3}{12}

.

Multiply the numerator and denominator of the first fraction by 4, and the second fraction by 3.

3

Add the numerators while keeping the denominator the same, applying integer sign rules.

\frac{-8 + 3}{12} = -\frac{5}{12}

.

Adding a positive 3 to a negative 8 results in a negative 5.

Key Concept

Adding fractions with unlike denominators and applying integer sign rules.

Question 1912Question →

Determine the final numerical value of the expression $k^2 + 4j$ when substituting $k = -3$ and $j = -2$ .

-17

17

1

-26

Show answer & explanation

Answer: 1

Answer

The correct value of the expression is 1.

The expression correctly evaluates to 1 by substituting the variables with their respective negative values and strictly following the order of operations (calculating exponents, then multiplication, then addition).

Step-by-Step Solution

1

Substitute the given numerical values into the variables in the expression.

(-3)^2 + 4(-2)

To begin evaluating the algebraic expression, each variable must be replaced with its assigned value.

2

Evaluate the exponent portion of the expression.

9 + 4(-2)

According to the order of operations, exponents are calculated before multiplication or addition. Squaring a negative number results in a positive value.

3

Perform the multiplication.

9 - 8

Multiplication precedes addition in the order of operations. Multiplying a positive integer by a negative integer gives a negative result.

4

Calculate the final sum.

1

Combining the remaining terms provides the final evaluated result of the expression.

Key Concept

Evaluating Algebraic Expressions

Question 1913Question →

A drone is flying at an altitude of $45.5$ meters. It descends $12.75$ meters to take a photo, and then ascends $8.2$ meters. What is the new altitude of the drone?

40.95 meters

24.55 meters

33.57 meters

50.05 meters

Show answer & explanation

Answer: 40.95 meters

Answer

The new altitude of the drone is 40.95 meters.

The correct answer is found by sequentially subtracting the descent from the initial altitude (

45.5 - 12.75 = 32.75

) and then adding the ascent (

32.75 + 8.2 = 40.95

). Aligning the decimal points correctly ensures an accurate calculation.

Step-by-Step Solution

1

Identify the starting altitude and the mathematical operations for descending and ascending.

Starting altitude is

45.5

meters. Descending means subtracting (

12.75

), and ascending means adding (

8.2

).

Translating the real-world scenario into a mathematical expression ensures the correct operations are applied in the correct sequence.

2

Subtract the descent distance from the initial altitude.

45.5 - 12.75 = 32.75

This finds the drone's intermediate altitude after its initial descent.

3

Add the ascent distance to the intermediate altitude, making sure to align the decimal points.

32.75 + 8.20 = 40.95

This calculates the final altitude after the drone moves back up.

Key Concept

Decimal Operations

Question 1914Question →

If

\frac{3(p + 4)}{2} - 7 = 2p - 4

what is the value of

p

?

13

18

-2

6

Show answer & explanation

Answer: 6

Answer

6

The correct answer is found by first isolating the fractional term by adding 7 to both sides, then multiplying the entire equation by 2 to eliminate the denominator, distributing properly, and finally moving variables to one side and constants to the other to solve for p.

Step-by-Step Solution

1

Add 7 to both sides of the equation.

\frac{3(p + 4)}{2} = 2p + 3

This isolates the fractional term on the left side of the equation.

2

Multiply both sides by 2.

3(p + 4) = 4p + 6

This clears the denominator from the fraction.

3

Distribute the 3 on the left side.

3p + 12 = 4p + 6

This expands the parentheses so like terms can be combined.

4

Subtract 3p from both sides.

12 = p + 6

This groups all variable terms on the right side of the equation.

5

Subtract 6 from both sides.

p = 6

This isolates the variable p to find its final value.

Key Concept

Solving Linear Equations with Fractions

Question 1915Question →

If $x = -3$ , what is the value of the expression $8 - 3|2x - 7|$ ?

-31

5

47

65

Show answer & explanation

Answer: -31

Answer

-31

The expression is evaluated using the correct order of operations. First, the value inside the absolute value is simplified by substituting -3, yielding -13. The absolute value of -13 is positive 13. Finally, multiplying 13 by 3 gives 39, and subtracting 39 from 8 results in -31.

Step-by-Step Solution

1

Substitute -3 for x in the expression.

8 - 3|2(-3) - 7|

To evaluate the expression for the specific variable value.

2

Multiply the numbers inside the absolute value.

8 - 3|-6 - 7|

Following the order of operations, calculations grouped inside absolute value bars must be simplified first.

3

Subtract the remaining numbers inside the absolute value.

8 - 3|-13|

Continuing to simplify the contents within the absolute value grouping.

4

Evaluate the absolute value of -13.

8 - 3(13)

The absolute value of any non-zero number is its positive distance from zero.

5

Multiply 3 by 13.

8 - 39

According to the order of operations (PEMDAS), multiplication must be performed before subtraction.

6

Subtract 39 from 8.

-31

This is the final operation to find the total value of the expression.

Key Concept

Evaluating absolute value expressions with order of operations

Question 1916Question →

A carpenter is sorting four wooden planks based on their lengths to construct a custom frame. The lengths of the planks, in feet, are listed below. Sequence the plank lengths from shortest to longest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending sequence is

1\frac{1}{3}

feet,

1\frac{3}{8}

feet,

\frac{7}{5}

feet, and

\frac{17}{12}

feet.

To compare fractions with different denominators, they must be converted to a common denominator. The least common multiple of

3

,

8

,

5

, and

12

is

120

. Converting all values yields

\frac{160}{120}

,

\frac{165}{120}

,

\frac{168}{120}

, and

\frac{170}{120}

. Arranging these from smallest to largest gives the proper sequence of

1\frac{1}{3}

feet,

1\frac{3}{8}

feet,

\frac{7}{5}

feet, and

\frac{17}{12}

feet.

Step-by-Step Solution

1

Convert all mixed numbers to improper fractions.

1\frac{3}{8} = \frac{11}{8}

and

1\frac{1}{3} = \frac{4}{3}

.

Fractions must be in a consistent improper format to find a common denominator efficiently.

2

Find the least common multiple (LCM) for the denominators

8

,

12

,

3

, and

5

.

The LCM of

8

,

12

,

3

, and

5

is

120

.

A common denominator is required to directly compare the relative sizes of the fractions.

3

Convert each fraction to an equivalent fraction with a denominator of

120

.

\frac{4}{3} = \frac{160}{120}

,

\frac{11}{8} = \frac{165}{120}

,

\frac{7}{5} = \frac{168}{120}

, and

\frac{17}{12} = \frac{170}{120}

.

Normalizing the denominators isolates the variation to the numerators, making comparison straightforward.

4

Arrange the fractions by comparing their normalized numerators from smallest to largest.

160 < 165 < 168 < 170

, which corresponds to the original values

1\frac{1}{3} < 1\frac{3}{8} < \frac{7}{5} < \frac{17}{12}

.

The problem specifies sequencing the plank lengths from shortest to longest.

Key Concept

Ordering and Comparing Fractions

Alternative Method

Convert each fraction to its decimal equivalent:

1\frac{1}{3} \approx 1.333

,

1\frac{3}{8} = 1.375

,

\frac{7}{5} = 1.400

, and

\frac{17}{12} \approx 1.417

. The decimals can then be easily ordered from least to greatest.

Estimated Time:1m 0s

Question 1917Question →

Complete the evaluation of the algebraic expression by filling in the blanks with the correct numerical values.

Fill in the blanks below

Evaluate the algebraic expression

5x^2y - \frac{3z^3}{x}

for the given variables

x = -2

,

y = 3

, and

z = -2

.

Step 1: The numerical value of the first term,

5x^2y

, is .
Step 2: The numerical value of the second term (including the negative sign),

-\frac{3z^3}{x}

, is .
Step 3: The final evaluated value of the entire expression is .

Show answer & explanation

Answer

The first term evaluates to 60, the second term evaluates to -12, and the final combined expression value is 48.

The expression requires careful substitution and order of operations. First, squaring

x = -2

yields

4

, making the first term

5(4)(3) = 60

. Next, cubing

z = -2

gives

-8

, resulting in a numerator of

-24

. Dividing by

x = -2

gives

12

. Applying the subtraction sign makes the second term

-12

. Adding these components yields

60 - 12 = 48

.

Step-by-Step Solution

1

Substitute the given values into the first term

5x^2y

.

5(-2)^2(3)

Variables must be replaced with their assigned numerical values.

2

Evaluate the exponent and multiply to find the value of the first term.

5(4)(3) = 60

According to the order of operations, exponents are evaluated before multiplication.

3

Substitute the given values into the second term

-\frac{3z^3}{x}

.

-\frac{3(-2)^3}{-2}

Preparing the rational term for evaluation.

4

Evaluate the exponent in the numerator.

-\frac{3(-8)}{-2}

The cube of a negative number remains negative, so

(-2)^3 = -8

.

5

Multiply the numerator and simplify the fraction.

-\frac{-24}{-2} = - (12) = -12

Multiplying 3 by -8 gives -24. Dividing -24 by -2 gives positive 12. Applying the negative sign in front results in -12.

6

Combine the evaluated terms to find the final result.

60 - 12 = 48

Summing the intermediate values yields the final expression value.

Key Concept

Evaluating algebraic expressions involves correctly applying variable substitution, managing negative signs, and strictly following the order of operations.

Estimated Time:1m 0s

Question 1918Question →

Calculate the exact value of

n

that balances the following equation:

\frac{2}{3}(6n - 9) + 4 = 3(n - 2) - n

Show answer & explanation

Answer: -2

Answer

The correct value for n is -2.

The solution is found by correctly applying the distributive property to both sides of the equation, combining like terms to simplify the expression to

4n - 2 = 2n - 6

, and then isolating the variable

n

using inverse operations to get

n = -2

.

Step-by-Step Solution

1

Distribute the fraction on the left side of the equation.

4n - 6 + 4 = 3(n - 2) - n

To eliminate the parentheses on the left side and isolate the individual terms.

2

Distribute the coefficient on the right side of the equation.

4n - 6 + 4 = 3n - 6 - n

To expand the expressions before combining them.

3

Combine like terms on both sides.

4n - 2 = 2n - 6

To simplify the equation into the standard linear form ax + b = cx + d.

4

Group the variable terms on one side and the constants on the other.

2n = -4

By subtracting 2n and adding 2 to both sides, we isolate the variable term.

5

Divide by the coefficient of n.

n = -2

To solve for the isolated variable n.

Key Concept

Solving multi-step linear equations involving fractions, distribution, and combining like terms.

Question 1919Question →

What is the evaluated result of the algebraic expression $a^2 - 4ab + b^3$ when $a = -5$ and $b = -2$ ?

57

-7

-23

-57

Show answer & explanation

Answer: -23

Answer

The correct evaluated result is -23.

The correct answer is found by substituting

a = -5

and

b = -2

into the expression and strictly following the order of operations. Squaring

-5

results in

25

. The product of

-4(-5)(-2)

is

-40

. Cubing

-2

results in

-8

. Combining these gives

25 - 40 - 8

, which equals

-23

.

Step-by-Step Solution

1

Substitute the given values

a = -5

and

b = -2

into the expression.

(-5)^2 - 4(-5)(-2) + (-2)^3

Direct substitution is the first step to evaluating an algebraic expression.

2

Evaluate the exponential terms.

25 - 4(-5)(-2) + (-8)

According to the order of operations (PEMDAS), exponents must be calculated before multiplication. Squaring a negative number gives a positive result (

(-5)^2 = 25

), and cubing a negative number gives a negative result (

(-2)^3 = -8

).

3

Calculate the multiplication in the middle term.

25 - 40 - 8

Multiply the three factors:

-4 \times -5 \times -2

. Two negatives make a positive (

20

), and multiplying by the third negative makes it negative (

-40

).

4

Combine the simplified terms using addition and subtraction from left to right.

-15 - 8 = -23

Final arithmetic computation yields the fully evaluated numerical value.

Key Concept

Evaluating Algebraic Expressions

Estimated Time:1m 0s

Question 1920Question →

A quality control inspector is verifying the thickness of four different steel plates. Arrange the following plate thicknesses in order from thinnest to thickest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from thinnest to thickest is

1\frac{5}{16}

inches,

1\frac{11}{32}

inches,

1\frac{3}{8}

inches, and

\frac{23}{16}

inches.

To accurately compare these lengths, they must all be expressed with a common denominator. The lowest common denominator for 8, 16, and 32 is 32. Converting the values yields

1\frac{5}{16} = \frac{42}{32}

,

1\frac{11}{32} = \frac{43}{32}

,

1\frac{3}{8} = \frac{44}{32}

, and

\frac{23}{16} = \frac{46}{32}

. Ordering by the numerators from lowest to highest gives the sequence

1\frac{5}{16}

,

1\frac{11}{32}

,

1\frac{3}{8}

, and

\frac{23}{16}

.

Step-by-Step Solution

1

Convert all mixed numbers to improper fractions.

1\frac{3}{8} = \frac{11}{8}

;

1\frac{5}{16} = \frac{21}{16}

;

1\frac{11}{32} = \frac{43}{32}

. The fraction

\frac{23}{16}

is already in improper form.

Working with improper fractions simplifies the process of finding a common denominator.

2

Determine the lowest common denominator (LCD) for the fractions

\frac{11}{8}

,

\frac{23}{16}

,

\frac{21}{16}

, and

\frac{43}{32}

.

The LCD for 8, 16, and 32 is 32.

A common denominator is required to accurately compare the absolute sizes of the fractions.

3

Convert each fraction into an equivalent fraction with a denominator of 32.

\frac{11}{8} = \frac{44}{32}

;

\frac{23}{16} = \frac{46}{32}

;

\frac{21}{16} = \frac{42}{32}

;

\frac{43}{32}

remains

\frac{43}{32}

.

This normalization allows for direct comparison of the numerators to determine size.

4

Order the numerators from smallest to largest and map them back to their original forms.

The numerators in order are

42 < 43 < 44 < 46

. This corresponds to

\frac{42}{32}

(

1\frac{5}{16}

),

\frac{43}{32}

(

1\frac{11}{32}

),

\frac{44}{32}

(

1\frac{3}{8}

), and

\frac{46}{32}

(

\frac{23}{16}

).

When the denominators are identical, the magnitude of the fraction is directly proportional to its numerator.

Key Concept

Comparing and Ordering Fractions

Answer

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