All practice questions

Question 1Question →

A specialized reconnaissance team hikes through rough terrain at a steady pace of $3\text{ mph}$ for $1\text{ hour and } 40\text{ minutes}$ . They then rendezvous with an extraction helicopter that flies them to a base at an average speed of $150\text{ mph}$ for $24\text{ minutes}$ . What is the total distance covered by the team? (Provide your answer in miles, e.g., '50 miles')

Show answer & explanation

Answer: 65 miles; 65; 65 mi; 65mi; 65.0 miles; sixty-five miles

Answer

65 miles

The correct answer is found by calculating the distance of each leg separately using the formula Distance = Rate × Time, ensuring all times are properly converted into hours, and then summing the two distances (5 miles + 60 miles = 65 miles).

Step-by-Step Solution

1

Convert the hiking time into hours.

1 hour and 40 minutes = 1 + 40/60 hours = 5/3 hours.

Speed is given in miles per hour, so time must be in hours to properly use the distance formula.

2

Calculate the distance covered during the hike.

Distance = Rate × Time = 3 mph × (5/3) hours = 5 miles.

This finds the length of the first leg of the journey.

3

Convert the helicopter flight time into hours.

24 minutes = 24/60 hours = 2/5 hours (or 0.4 hours).

Again, the helicopter speed is in mph, requiring time to be in hours.

4

Calculate the distance covered by the helicopter.

Distance = Rate × Time = 150 mph × 0.4 hours = 60 miles.

This finds the length of the second leg of the journey.

5

Calculate the total distance covered.

5 miles + 60 miles = 65 miles.

The total distance is the sum of the individual leg distances.

Key Concept

Calculating distance for multiple legs of a journey with mixed time units.

Question 2Question →

Assume

x

and

y

are non-zero real numbers. When the algebraic expression

\frac{(2x^{-3}y^4)^3}{4x^5y^{-2}}

is fully simplified to the form

cx^ay^b

, what is the integer value of

b

?

Show answer & explanation

Answer: 14

Answer

14

The expression simplifies to

2x^{-14}y^{14}

. Because the question asks for the exponent of

y

, which is denoted by

b

in the target expression

cx^ay^b

, the correct value is 14.

Step-by-Step Solution

1

Apply the power of a product rule to the numerator.

(2x^{-3}y^4)^3 = 2^3(x^{-3})^3(y^4)^3 = 8x^{-9}y^{12}

When an algebraic term with coefficients and variables is raised to a power, the outer exponent must be distributed to each individual factor inside the parentheses.

2

Set up the division by applying the quotient rule for each base.

\frac{8x^{-9}y^{12}}{4x^5y^{-2}} = \left(\frac{8}{4}\right) \cdot \left(\frac{x^{-9}}{x^5}\right) \cdot \left(\frac{y^{12}}{y^{-2}}\right)

Separating the expression by identical bases allows the independent application of the quotient rule for exponents.

3

Evaluate the exponent for the

y

term to find the value of

b

.

y^{12} \div y^{-2} = y^{12 - (-2)} = y^{12 + 2} = y^{14}

When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent. Care must be taken with the double negative.

4

Identify the requested integer value.

b = 14

The question specifically asks for the integer value of

b

in the final simplified form

cx^ay^b

, which represents the exponent of the

y

term.

Key Concept

Applying the power of a power and quotient of powers rules for variable exponents.

Question 3Question →

The total revenue from selling a batch of electronic components is modeled by the polynomial $15x^2 - 14x - 8$ . If the number of units sold is given by the expression $(5x + 2)$ , what binomial represents the price per unit?

Show answer & explanation

Answer: 3x-4; (3x-4); 3x - 4; (3x - 4)

Answer

3x - 4

The correct answer is derived by factoring the revenue polynomial

15x^2 - 14x - 8

. Because revenue equals the number of units multiplied by the price per unit, the missing price expression is the binomial that, when multiplied by

(5x + 2)

, yields the given polynomial. By factoring the expression, we find that

15x^2 - 14x - 8 = (5x + 2)(3x - 4)

, confirming

(3x - 4)

is the correct binomial.

Step-by-Step Solution

1

Set up the algebraic relationship based on the context.

Total Revenue = (Number of Units) × (Price per Unit), which translates to

15x^2 - 14x - 8 = (5x + 2) \times (\text{Price per Unit})

.

This establishes that the price per unit is the missing factor of the given polynomial.

2

Determine the factoring method for the trinomial

15x^2 - 14x - 8

.

Using the ac-method, multiply the leading coefficient (15) by the constant term (-8) to get

-120

.

This is the standard approach for factoring a trinomial where the leading coefficient is greater than 1.

3

Find two integers that multiply to

-120

and add to the middle coefficient,

-14

.

The required integers are

-20

and

6

, because

-20 \times 6 = -120

and

-20 + 6 = -14

.

These integers allow us to correctly split the middle term for factoring by grouping.

4

Rewrite the trinomial and factor by grouping.

15x^2 - 20x + 6x - 8 = 5x(3x - 4) + 2(3x - 4) = (5x + 2)(3x - 4)

.

Grouping extracts the greatest common factor from pairs of terms, revealing the two binomial factors.

5

Identify the missing factor.

Since one factor is given as

(5x + 2)

, the other factor representing the price is

(3x - 4)

.

This completes the solution, satisfying the initial equation.

Key Concept

Factoring Non-Monic Trinomials

Question 4Question →

During a land navigation exercise, an instructor maps out 150 possible waypoint coordinates. Of these waypoints, 45 are located in dense forest, 35 are in open terrain, 50 are near water features, and the remaining waypoints are on elevated ridges. If a cadet is randomly assigned one waypoint coordinate, what is the probability that it is located on an elevated ridge? (Express your answer as a simplified fraction)

Show answer & explanation

Answer: 2/15; 2 / 15

Answer

2/15

The correct probability is found by dividing the number of ridge waypoints (20) by the total number of waypoints (150). Simplifying the fraction 20/150 yields 2/15.

Step-by-Step Solution

1

Determine the total number of possible outcomes.

Total waypoints = 150.

The denominator for a single event probability calculation is the total number of items in the set.

2

Calculate the number of favorable outcomes (elevated ridge waypoints).

150 - (45 + 35 + 50) = 150 - 130 = 20 waypoints on elevated ridges.

You must subtract the sum of all known waypoint categories from the total to find the unknown remainder.

3

Set up the probability ratio and simplify the fraction.

20 / 150 simplifies to 2 / 15.

Probability is the number of favorable outcomes divided by total outcomes. ASVAB format requires fractions to be reduced to their lowest terms.

Key Concept

Single Event Probability

Question 5Question →

Determine the exact ordered pair $(x, y)$ that satisfies the system of linear equations shown below.

4x - 3y = 25

x + 2y = 9

Enter your answer in the format $(x, y)$ .

Show answer & explanation

Answer: (7, 1); (7,1); 7, 1; 7,1

Answer

The correct ordered pair is (7, 1).

The correct solution is (7, 1). To verify, substitute x = 7 and y = 1 into both original equations. For the first equation: 4(7) - 3(1) = 28 - 3 = 25. For the second equation: 7 + 2(1) = 7 + 2 = 9. Since both equations hold true simultaneously, (7, 1) is the accurate solution.

Step-by-Step Solution

1

Isolate the variable

x

in the second equation.

x = 9 - 2y

The second equation has an

x

with a coefficient of 1, making it the most straightforward variable to isolate for substitution without creating fractions.

2

Substitute the expression for

x

into the first equation.

4(9 - 2y) - 3y = 25

This creates a single equation containing only the variable

y

, which can be solved algebraically.

3

Distribute the 4 and combine like terms.

36 - 8y - 3y = 25 \Rightarrow 36 - 11y = 25

Simplifying the equation prepares it for isolating the

y

term.

4

Solve for

y

by subtracting 36 from both sides and dividing by -11.

-11y = -11 \Rightarrow y = 1

This determines the numerical value for the y-coordinate of the solution.

5

Substitute

y = 1

back into the isolated

x

equation from Step 1.

x = 9 - 2(1) \Rightarrow x = 9 - 2 = 7

This provides the corresponding x-coordinate to complete the ordered pair.

Key Concept

Solving Systems of Linear Equations by Substitution

Question 6Question →

A patrol unit drives at a steady speed of $20\text{ mph}$ for $3$ hours along a desert trail. They then enter a paved highway and drive at $50\text{ mph}$ for $2$ hours. What is the total distance traveled by the patrol unit?

Show answer & explanation

Answer: 160; 160 miles; 160 mi; 160miles; 160.0

Answer

160 miles

To find the total distance, you must calculate the distance for each part of the journey separately. For the first leg, the distance is

20\text{ mph} \times 3\text{ hours} = 60\text{ miles}

. For the second leg, the distance is

50\text{ mph} \times 2\text{ hours} = 100\text{ miles}

. Adding these two distances together (

60 + 100

) gives a total distance of

160

miles.

Step-by-Step Solution

1

Calculate the distance traveled during the first segment on the desert trail.

Distance = Rate × Time = 20 mph × 3 hours = 60 miles

The distance formula

D = R \times T

is used to find the distance for each constant-speed leg of the trip.

2

Calculate the distance traveled during the second segment on the paved highway.

Distance = Rate × Time = 50 mph × 2 hours = 100 miles

Applying the same formula to the second leg provides the remaining distance.

3

Add the distances from both segments to find the total distance traveled.

60 miles + 100 miles = 160 miles

The total distance of a multi-leg journey is the sum of the distances of each individual segment.

Key Concept

Calculating total distance for a multi-leg journey using the formula Distance = Rate × Time (

D = R \times T

).

Question 7Question →

A military supply convoy travels at a steady speed of $40\text{ miles per hour}$ for $2\text{ hours}$ . Then, it travels at a steady speed of $50\text{ miles per hour}$ for $1\text{ hour}$ . What is the total distance traveled by the convoy? Provide your answer including the unit (e.g., '130 miles').

Show answer & explanation

Answer: 130 miles; 130 mi; 130

Answer

130 miles

To find the total distance, you must calculate the distance for each segment separately using the formula Distance = Rate × Time, and then add them together. The first segment covers 80 miles (40 mph × 2 hours) and the second segment covers 50 miles (50 mph × 1 hour), resulting in a total distance of 130 miles.

Step-by-Step Solution

1

Calculate the distance for the first part of the trip.

40 \times 2 = 80\text{ miles}

Distance is equal to rate multiplied by time.

2

Calculate the distance for the second part of the trip.

50 \times 1 = 50\text{ miles}

Apply the distance formula to the second segment.

3

Add the distances from both parts of the trip to find the total distance.

80 + 50 = 130\text{ miles}

The total distance is the sum of the individual segment distances.

Key Concept

Calculating total distance for a multi-segment journey by applying the formula Distance = Rate × Time to each segment.

Estimated Time:45s

Question 8Question →

An armored reconnaissance vehicle is conducting a route clearance mission. It travels at a steady speed of $32\text{ miles per hour}$ for $45\text{ minutes}$ . Then, due to rough terrain, its speed drops to $16\text{ miles per hour}$ for the next $1\text{ hour}$ and $15\text{ minutes}$ . What is the total distance the vehicle traveled, in miles?

Show answer & explanation

Answer: 44; 44 miles; 44 mi; forty-four; 44.0

Answer

44 miles

To find the total distance traveled, we evaluate each leg of the trip using the formula Distance = Rate × Time. The first leg is driven at 32 mph for 45 minutes (which is 0.75 hours). Distance = 32 × 0.75 = 24 miles. The second leg is driven at 16 mph for 1 hour and 15 minutes (which is 1.25 hours). Distance = 16 × 1.25 = 20 miles. The total distance is the sum of these two segments: 24 + 20 = 44 miles.

Step-by-Step Solution

1

Convert the travel time of the first leg from minutes to hours.

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

The speed is given in miles per hour, so the time must be in hours to calculate the distance properly.

2

Calculate the distance covered during the first leg.

32\text{ mph} \times 0.75\text{ hours} = 24\text{ miles}

Using the formula Distance = Rate × Time.

3

Convert the travel time of the second leg to hours.

1\text{ hour}

and

15\text{ minutes} = 1 + \frac{15}{60}\text{ hours} = 1.25\text{ hours}

Time must be consistently in hours to match the rate of 16 mph.

4

Calculate the distance covered during the second leg.

16\text{ mph} \times 1.25\text{ hours} = 20\text{ miles}

Using the formula Distance = Rate × Time.

5

Add the distances of both legs to find the total distance.

24\text{ miles} + 20\text{ miles} = 44\text{ miles}

The total distance is the sum of the individual segments.

Key Concept

Calculating distance across multiple journey segments requiring time unit conversions.

Question 9Question →

Simplify the following expression completely. Assume $x$ and $y$ are non-zero real numbers.

\frac{(3x^2y^{-3})^{-2} (-3x^{-2}y^2)^3}{(x^{-4}y^3)^{-2}}

If the simplified expression is written in the form $Ax^B y^C$ , where $A$ , $B$ , and $C$ are integers, what is the value of $A + B + C$ ?

Show answer & explanation

Answer: -3

Answer

-3

The correct answer is found by systematically applying the laws of exponents. First, distribute the outer exponents to all factors inside the parentheses, being careful with signs and coefficients. The numerator simplifies to

(\frac{1}{9}x^{-4}y^6) \cdot (-27x^{-6}y^6) = -3x^{-10}y^{12}

. The denominator simplifies to

x^8y^{-6}

. Dividing the numerator by the denominator using the quotient rule yields

-3x^{-18}y^{18}

. Setting this equal to the target form

Ax^By^C

yields

A = -3

,

B = -18

, and

C = 18

. The sum

A + B + C

is

-3 - 18 + 18 = -3

.

Step-by-Step Solution

1

Apply the power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

(a^m)^n = a^{m \cdot n}

to the first factor in the numerator.

(3x^2y^{-3})^{-2} = 3^{-2} \cdot x^{2(-2)} \cdot y^{-3(-2)} = \frac{1}{9}x^{-4}y^6

To eliminate the outer negative exponent on the first term.

2

Apply the same exponent rules to the second factor in the numerator.

(-3x^{-2}y^2)^3 = (-3)^3 \cdot x^{-2(3)} \cdot y^{2(3)} = -27x^{-6}y^6

To eliminate the outer exponent on the second term.

3

Multiply the simplified factors in the numerator using the product rule

a^m \cdot a^n = a^{m+n}

.

(\frac{1}{9}x^{-4}y^6) \cdot (-27x^{-6}y^6) = (\frac{1}{9} \cdot -27) \cdot x^{-4-6} \cdot y^{6+6} = -3x^{-10}y^{12}

To combine the expressions in the numerator into a single term.

4

Apply the power rules to simplify the denominator.

(x^{-4}y^3)^{-2} = x^{-4(-2)} \cdot y^{3(-2)} = x^8y^{-6}

To eliminate the outer negative exponent in the denominator.

5

Divide the numerator by the denominator using the quotient rule

\frac{a^m}{a^n} = a^{m-n}

.

\frac{-3x^{-10}y^{12}}{x^8y^{-6}} = -3 \cdot x^{-10-8} \cdot y^{12-(-6)} = -3x^{-18}y^{18}

To reach the fully simplified algebraic form.

6

Identify the values of

A

,

B

, and

C

to calculate the required sum.

A = -3

,

B = -18

,

C = 18

. The sum is

-3 + (-18) + 18 = -3

.

To answer the specific question asked in the stem.

Key Concept

Applying multiple rules of exponents (power of a power, negative exponents, product and quotient rules) to simplify complex algebraic expressions.

Question 10Question →

A military cargo plane flies from Base Alpha to Base Bravo at a steady cruising speed of $320\text{ miles per hour}$ . This first leg of the flight takes $2\text{ hours and } 45\text{ minutes}$ . After refueling, the plane continues to Base Charlie, flying at $280\text{ miles per hour}$ for $1\text{ hour and } 30\text{ minutes}$ . What is the total distance traveled by the cargo plane from Base Alpha to Base Charlie? Include both the numeric value and unit in your answer (e.g., '500 miles').

Show answer & explanation

Answer: 1300 miles; 1,300 miles; 1300 mi; 1300; 1,300

Answer

1300 miles

The correct answer is found by first converting all travel times into decimal hours. The first leg is

2.75

hours at

320\text{ mph}

, covering

880\text{ miles}

. The second leg is

1.5

hours at

280\text{ mph}

, covering

420\text{ miles}

. Adding these two distances yields a total of

1300\text{ miles}

.

Step-by-Step Solution

1

Convert the flight times into decimal hours.

2\text{ hours and } 45\text{ minutes} = 2.75\text{ hours}

;

1\text{ hour and } 30\text{ minutes} = 1.5\text{ hours}

.

To calculate distance using miles per hour, the time units must be converted to hours.

2

Calculate the distance for the first leg from Base Alpha to Base Bravo.

320\text{ mph} \times 2.75\text{ hours} = 880\text{ miles}

.

Distance equals rate multiplied by time (

d = r \times t

).

3

Calculate the distance for the second leg from Base Bravo to Base Charlie.

280\text{ mph} \times 1.5\text{ hours} = 420\text{ miles}

.

The second leg must be calculated independently due to the change in both speed and travel time.

4

Add the distances of both legs to find the total distance traveled.

880\text{ miles} + 420\text{ miles} = 1300\text{ miles}

.

The total journey distance is the sum of the distances of all individual legs.

Key Concept

Calculating multi-leg total distance using

d = r \times t

with unit conversions.

Question 11Question →

The polynomial $x^2 + 6x - 16$ can be factored as $(x - 2)(x + k)$ . What is the value of the constant $k$ ?

Show answer & explanation

Answer: 8; +8; eight

Answer

8

The constant

k

must be 8 because in the factored form

(x - 2)(x + k)

, the product of -2 and

k

must equal the constant term of the polynomial, which is -16. Since

-2 \times 8 = -16

and the sum

-2 + 8 = 6

gives the correct middle coefficient, the value of

k

is exactly 8.

Step-by-Step Solution

1

Analyze the relationship between the expanded polynomial and its factored form.

The expanded form is

x^2 + 6x - 16

and the factored form is

(x - 2)(x + k)

.

This sets up the equation to solve for the missing constant

k

.

2

Use the property of multiplying binomials to find the constant term.

The product of the constants in the binomials must equal the constant term of the polynomial:

-2 \times k = -16

.

When expanding

(x + a)(x + b)

, the constant term is always equal to

a \times b

.

3

Solve the equation for

k

.

k = -16 \div (-2) = 8

.

Dividing both sides by -2 isolates

k

.

4

Verify the middle term coefficient with the found value of

k

.

-2 + 8 = 6

, which matches the middle term

6x

.

Checking the sum of the constants ensures the entire factorization is correct.

Key Concept

Factoring quadratic trinomials by finding two numbers that multiply to the constant term and add to the middle coefficient.

Question 12Question →

When the algebraic expression $5x^2 - 13x - 6$ is factored completely, it can be written in the form $(5x + a)(x - b)$ , where $a$ and $b$ are positive integers. What is the value of $a + b$ ?

Show answer & explanation

Answer: 5; five

Answer

5

Factoring the quadratic

5x^2 - 13x - 6

yields

(5x + 2)(x - 3)

. By comparing this to the target format

(5x + a)(x - b)

, we can identify that

a = 2

and

b = 3

. Adding these together gives the final answer of 5.

Step-by-Step Solution

1

Determine the factoring method for the non-monic quadratic expression

5x^2 - 13x - 6

.

Since the leading coefficient is 5, look for two numbers that multiply to

(5)(-6) = -30

and add to the middle coefficient,

-13

.

This sets up the factor-by-grouping method.

2

Find the two numbers.

The numbers are

-15

and

2

.

(-15) \times 2 = -30

and

-15 + 2 = -13

.

3

Rewrite the middle term using these two numbers and factor by grouping.

5x^2 - 15x + 2x - 6 = 5x(x - 3) + 2(x - 3) = (5x + 2)(x - 3)

.

Grouping common terms extracts the binomial factors.

4

Match the factored form to the given format

(5x + a)(x - b)

.

Comparing

(5x + 2)(x - 3)

to

(5x + a)(x - b)

reveals that

a = 2

and

b = 3

.

Both

a

and

b

correspond exactly to the positive integer components of the binomials.

5

Calculate the final required sum,

a + b

.

2 + 3 = 5

.

The question specifically asks for the sum of the parameters.

Key Concept

Factoring non-monic quadratic trinomials

Question 13Question →

The algebraic expression $5x^2 + 35x$ can be completely factored into the form $5x(x + c)$ . What is the value of the constant $c$ ?

Show answer & explanation

Answer: 7; +7; seven; Seven

Answer

7

By extracting the greatest common factor of

5x

from

5x^2 + 35x

, we divide both terms by

5x

. This leaves

(x + 7)

inside the parentheses. Comparing this to the given structure

5x(x + c)

, it is clear that

c

must be 7.

Step-by-Step Solution

1

Identify the greatest common factor (GCF) of the terms

5x^2

and

35x

.

The GCF is

5x

.

Both terms are divisible by 5 and share a minimum of one

x

.

2

Divide each term of the original expression by the GCF to find the binomial factor.

\frac{5x^2}{5x} = x

and

\frac{35x}{5x} = 7

.

Factoring out the GCF is the reverse of distributing it.

3

Write the factored form and compare it to the given form

5x(x + c)

.

The factored form is

5x(x + 7)

. Therefore, comparing it to

5x(x + c)

,

c = 7

.

Matching corresponding terms reveals the value of the unknown constant.

Key Concept

Factoring the Greatest Common Factor (GCF)

Question 14Question →

When the polynomial $5x^2 - 20$ is factored completely into the form $k(x - c)(x + c)$ , where $k$ and $c$ are positive integers, what is the value of the sum $k + c$ ?

Show answer & explanation

Answer: 7; seven

Answer

7

The polynomial

5x^2 - 20

can be factored completely by first pulling out the greatest common factor,

5

, leaving

5(x^2 - 4)

. The expression inside the parentheses is a difference of squares and factors into

(x - 2)(x + 2)

. This makes the completely factored form

5(x - 2)(x + 2)

. Comparing this to the form

k(x - c)(x + c)

, we see that

k = 5

and

c = 2

. Their sum is

5 + 2 = 7

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from the polynomial.

5x^2 - 20 = 5(x^2 - 4)

Both terms share a common numerical factor of

5

. Factoring it out simplifies the remaining polynomial.

2

Factor the remaining binomial as a difference of squares.

5(x^2 - 4) = 5(x - 2)(x + 2)

The binomial

x^2 - 4

follows the pattern

a^2 - b^2 = (a - b)(a + b)

, where

a = x

and

b = 2

.

3

Identify the values of

k

and

c

from the given form.

k = 5

and

c = 2

By matching

5(x - 2)(x + 2)

to the target form

k(x - c)(x + c)

, the constant multiplier is

5

and the positive integer in the binomial is

2

.

4

Calculate the sum of

k

and

c

.

5 + 2 = 7

The question asks for the final value of the sum

k + c

.

Key Concept

Factoring polynomials using the greatest common factor and the difference of squares.

Question 15Question →

Given the system of equations below, what is the value of $x$ ?

3x + 2y = 16

x - 2y = 0

Show answer & explanation

Answer: 4; x=4; x = 4; four

Answer

4

Because the coefficients of

y

in the two equations are opposites (

+2

and

-2

), the fastest approach is the elimination method. Adding the equations top-to-bottom yields

(3x + x) + (2y - 2y) = 16 + 0

, which simplifies directly to

4x = 16

. Dividing both sides by

4

gives

x = 4

.

Step-by-Step Solution

1

Add the two equations together.

4x = 16

The

2y

and

-2y

terms are opposites, so adding the equations eliminates the

y

variable entirely.

2

Divide both sides of the resulting equation by 4.

x = 4

This isolates the variable

x

to find its exact numerical value.

Key Concept

Solving systems of linear equations using the elimination method.

Question 16Question →

Two inspection drones take off from the same cellular tower at the same time and fly in opposite directions along a straight power line. Drone A flies at $15$ mph and Drone B flies at $25$ mph. How many miles apart are the two drones after $2$ hours?

Show answer & explanation

Answer: 80; 80 miles; 80 mi; eighty

Answer

The drones are 80 miles apart.

Since the drones travel in opposite directions, their relative speed is the sum of their individual speeds (

15 + 25 = 40

mph). Over the course of

2

hours, the distance between them becomes

40 \times 2 = 80

miles.

Step-by-Step Solution

1

Calculate the combined relative speed of both drones.

15 + 25 = 40

mph

Because the drones are flying in opposite directions, the distance between them increases at a rate equal to the sum of their individual speeds.

2

Multiply the relative speed by the time traveled.

40 \times 2 = 80

miles

Total distance is found by multiplying the relative speed by the total time (

D = r \times t

).

Key Concept

Calculating separation distance for objects moving in opposite directions using relative speed.

Estimated Time:45s

Question 17Question →

Simplify the following algebraic expression completely. Write your final answer in standard form (highest to lowest power):

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

Show answer & explanation

Answer: n^2 + 2n - 6; n^2+2n-6; 1n^2 + 2n - 6; n^2 + 2n -6; n^2+2n - 6; 1n^2+2n-6

Answer

n^2 + 2n - 6

The completely simplified form is determined by first distributing the multipliers into their respective parentheses. For the first group, -2(n^2 - 4n + 3) expands to -2n^2 + 8n - 6. For the second group, 3n(n - 2) expands to 3n^2 - 6n. Combining the expansions results in -2n^2 + 8n - 6 + 3n^2 - 6n. By adding the coefficients of like terms, the final standard form evaluates to n^2 + 2n - 6.

Step-by-Step Solution

1

Distribute -2 into the first polynomial, (n^2 - 4n + 3).

-2n^2 + 8n - 6

The distributive property requires multiplying the outside term by every term inside the parentheses. Care must be taken with the negative sign: -2 multiplied by -4n becomes positive 8n.

2

Distribute 3n into the second binomial, (n - 2).

3n^2 - 6n

Using the distributive property again, multiplying 3n by n gives 3n^2, and 3n multiplied by -2 gives -6n.

3

Group the like terms from both expansions together.

(-2n^2 + 3n^2) + (8n - 6n) - 6

Like terms are terms that share the exact same variables raised to the exact same power. Grouping them helps systematically combine the coefficients.

4

Simplify the grouped terms.

n^2 + 2n - 6

Adding the coefficients of the grouped terms: -2 + 3 = 1 for the n^2 terms, and 8 - 6 = 2 for the n terms.

Key Concept

Simplifying algebraic expressions via polynomial expansion and combining like terms.

Question 18Question →

A maintenance kit contains $12$ metric wrenches, $15$ standard wrenches, $9$ screwdrivers, and an unknown number of pliers. If the probability of randomly selecting a screwdriver from the entire kit is $\frac{1}{6}$ , what is the probability of randomly selecting a standard wrench? Express your answer as a fraction in simplest form.

Show answer & explanation

Answer: $5/18; \frac{5}{18}; 5 / 18$

Answer

5/18

By using the given probability of selecting a screwdriver (1/6) and the known count of screwdrivers (9), we determine that the total number of items in the kit is 54 (because 9 is 1/6 of 54). The probability of randomly selecting a standard wrench is then the number of standard wrenches (15) divided by the total (54). Simplifying the fraction 15/54 by dividing both numbers by 3 yields 5/18.

Step-by-Step Solution

1

Set up an equation to find the total number of items in the kit.

\frac{9}{\text{Total}} = \frac{1}{6}

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. Here, we know the probability and the exact count of screwdrivers.

2

Solve for the total number of items.

\text{Total} = 9 \times 6 = 54

Cross-multiplying isolates the total variable, revealing there are 54 items in the entire kit.

3

Calculate the probability of selecting a standard wrench.

\text{Probability} = \frac{15}{54}

There are 15 standard wrenches out of the 54 total items calculated.

4

Simplify the fraction.

\frac{15}{54} = \frac{5}{18}

Both 15 and 54 are divisible by 3. Dividing the numerator and denominator by 3 gives the simplest form.

Key Concept

Calculating probabilities by determining the total sample size from a known probability.

Question 19Question →

A logistics officer is inspecting a shipment of $120$ combat helmets. The shipment contains $45$ large helmets, $35$ medium helmets, $24$ small helmets, and the remaining helmets are extra-large. If the officer randomly selects one helmet for quality control, what is the probability that it is an extra-large helmet? Express your answer as a fraction in simplest form.

Show answer & explanation

Answer: 2/15; 2 / 15

Answer

2/15

The correct answer is derived by first finding the number of extra-large helmets. Subtracting the known sizes from the total yields

120 - 104 = 16

extra-large helmets. The probability of selecting an extra-large helmet is the ratio of extra-large helmets to the total inventory, which is

\frac{16}{120}

. Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor (

8

) yields the final probability of

\frac{2}{15}

.

Step-by-Step Solution

1

Calculate the total number of known helmets.

45 + 35 + 24 = 104

To find the number of extra-large helmets, we first need to determine how many helmets belong to the other given size categories.

2

Determine the number of extra-large helmets.

120 - 104 = 16

Subtracting the sum of the known sizes from the total number of helmets gives the remaining count, which represents the extra-large helmets.

3

Set up the probability fraction.

\frac{16}{120}

The probability of a single event is the number of favorable outcomes (

16

extra-large helmets) divided by the total number of possible outcomes (

120

total helmets).

4

Simplify the fraction.

\frac{2}{15}

Divide the numerator and denominator by their greatest common divisor, which is

8

(

16 \div 8 = 2

and

120 \div 8 = 15

).

Key Concept

Single Event Probability

Question 20Question →

A motor pool contains a total of 24 vehicles, consisting exclusively of transport trucks and utility vehicles. The number of transport trucks is 6 less than twice the number of utility vehicles. How many utility vehicles are in the motor pool?

Show answer & explanation

Answer: 10; ten; 10 utility vehicles

Answer

10

By defining

t

as transport trucks and

u

as utility vehicles, we can create the system

t + u = 24

and

t = 2u - 6

. Substituting the value of

t

into the first equation yields

(2u - 6) + u = 24

. Combining the terms gives

3u - 6 = 24

. Adding 6 to both sides results in

3u = 30

, and dividing by 3 shows there are

10

utility vehicles.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

t

be the number of transport trucks and

u

be the number of utility vehicles.

This allows us to translate the word problem into a system of algebraic equations.

2

Set up the system of equations based on the problem statement.

Equation 1:

t + u = 24

(total vehicles).
Equation 2:

t = 2u - 6

(transport trucks are 6 less than twice the utility vehicles).

These equations accurately represent the given conditions in mathematical terms.

3

Substitute the expression for

t

from Equation 2 into Equation 1.

(2u - 6) + u = 24

Substitution eliminates the variable

t

, leaving a single-variable equation that can be solved directly for

u

.

4

Combine like terms and solve for

u

.

3u - 6 = 24 \Rightarrow 3u = 30 \Rightarrow u = 10

Isolating

u

gives the exact number of utility vehicles requested by the problem.

Key Concept

Solving Systems of Linear Equations Using Substitution

Alternative Method

This can also be solved by elimination if we rearrange the second equation to

-t + 2u = 6

. Adding this to

t + u = 24

yields

3u = 30

, which gives

u = 10

.

Estimated Time:1m 0s

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