All practice questions

Question 401Question →

A triangular pennant has a base of $8$ inches and a height of $15$ inches. What is the area of the pennant, in square inches?

Show answer & explanation

Answer: 60

Answer

60

The correct area is found by multiplying the base by the height, and then dividing by two according to the standard triangle area formula.

Step-by-Step Solution

1

Identify the formula for the area of a triangle.

A = \frac{1}{2}bh

The object is described as a triangular pennant.

2

Substitute the given base and height into the formula.

A = \frac{1}{2}(8)(15)

The base is given as

8

inches and the height is

15

inches.

3

Calculate the final product.

60

Multiplying the terms:

\frac{1}{2} \times 8 = 4

, and

4 \times 15 = 60

.

Key Concept

Area of a Triangle

Estimated Time:45s

Question 402Question →

The sum of the interior angles of a convex polygon is exactly three times the sum of its exterior angles. How many sides does this polygon have?

Show answer & explanation

Answer: 8

Answer

8

Applying the core polygon properties, the interior angle sum is

180^\circ(n - 2)

and the exterior angle sum is

360^\circ

. Setting

180^\circ(n - 2) = 3(360^\circ)

and solving for

n

yields 8.

Step-by-Step Solution

1

Identify the interior angle sum formula.

Sum =

180^\circ(n - 2)

This formula provides the total sum of all interior angles for an

n

-sided polygon.

2

Identify the exterior angle sum property.

Sum =

360^\circ

A fundamental property of all convex polygons is that the sum of their exterior angles is exactly

360^\circ

.

3

Set up the algebraic equation.

180^\circ(n - 2) = 3 \times 360^\circ

The question states the interior sum is exactly three times the exterior sum.

4

Solve the equation for

n

.

n = 8

Dividing

1080^\circ

by

180^\circ

yields 6, and adding 2 gives a final side count of 8.

Key Concept

Polygon Angle Sum Theorems

Question 403Question →

A rectangular garden has an area given by the expression $4n^2 - 4n - 24$ . Which of the following represents the completely factored form of this area expression?

4(n - 3)(n + 2)

4(n + 3)(n - 2)

(4n - 12)(n + 2)

4(n - 6)(n + 1)

Show answer & explanation

Answer: $4(n - 3)(n + 2)$

Answer

The completely factored form is

4(n - 3)(n + 2)

.

To factor the expression completely, first remove the greatest common factor of 4 from all terms, which gives

4(n^2 - n - 6)

. Then, factor the trinomial inside the parentheses. We need two numbers that multiply to -6 and add to -1, which are -3 and 2. This gives the completely factored form

4(n - 3)(n + 2)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all terms in the polynomial

4n^2 - 4n - 24

.

4(n^2 - n - 6)

Factoring out the GCF simplifies the polynomial and is the required first step for complete factorization.

2

Factor the remaining quadratic trinomial

n^2 - n - 6

by finding two numbers that multiply to -6 and add to -1.

The numbers are -3 and 2, yielding

(n - 3)(n + 2)

.

This breaks down the trinomial into its constituent binomial factors.

3

Combine the GCF and the binomial factors to write the final expression.

4(n - 3)(n + 2)

This represents the full, completely factored form of the original area expression.

Key Concept

Factoring out a GCF followed by factoring a quadratic trinomial

Question 404Question →

A delivery truck travels $40$ miles during the first $1$ hour of its route. It then travels another $75$ miles in $1$ hour and $30$ minutes. What is the truck's overall average speed for the entire trip?

45 mph

46 mph

50 mph

115 mph

Show answer & explanation

Answer: 46 mph

Answer

46 mph

To find the overall average speed, divide the total distance by the total time. The total distance is

40 + 75 = 115

miles. The total time is

1

hour plus

1

hour and

30

minutes. Since

30

minutes is

0.5

hours, the total time is

1 + 1.5 = 2.5

hours. Dividing

115

miles by

2.5

hours gives an average speed of

46

mph.

Step-by-Step Solution

1

Calculate the total distance traveled.

Total distance =

40 + 75 = 115

miles.

Average speed requires knowing the total distance of the trip.

2

Calculate the total time in consistent units (hours).

Total time =

1

hour +

1.5

hours =

2.5

hours.

Average speed requires the total time. The

30

minutes must be converted to

0.5

hours to match the unit.

3

Divide the total distance by the total time to find the average speed.

115 \div 2.5 = 46

mph.

The standard formula for average speed is total distance divided by total time.

Key Concept

Average speed is found by dividing the total distance of a journey by the total time it took to complete it.

Estimated Time:1m 0s

Question 405Question →

A machinist earns a regular hourly rate of $\$26.00$ for a standard $40$ -hour workweek. Any hours worked over $40$ in a single week are paid at a time-and-a-half overtime rate. Additionally, working the night shift adds a flat $\$3.50$ per hour differential to their base rate before the overtime multiplier is applied. If the machinist worked $46$ hours last week, and all of those hours were on the night shift, what was their total gross pay in dollars?

Show answer & explanation

Answer: 1445.5

Answer

1445.50

The total gross pay is calculated by first establishing the adjusted base rate for the night shift, which is the regular rate plus the differential:

\$26.00 + \$3.50 = \$29.50

. The machinist worked

40

regular hours at this rate, earning

40 \times \$29.50 = \$1,180.00

. The

6

remaining hours are overtime, paid at

1.5

times the adjusted rate:

\$29.50 \times 1.5 = \$44.25

. Multiplying this by the

6

overtime hours gives

6 \times \$44.25 = \$265.50

. Adding the regular pay and overtime pay results in a total gross pay of

\$1,180.00 + \$265.50 = \$1,445.50

.

Step-by-Step Solution

1

Calculate the adjusted base hourly rate.

\$26.00 + \$3.50 = \$29.50

per hour

The night shift differential is added to the standard base rate before any other multipliers are applied.

2

Calculate the regular pay for the first 40 hours.

40 \times \$29.50 = \$1,180.00

The standard workweek is 40 hours, which is paid at the regular adjusted rate.

3

Calculate the overtime hourly rate.

\$29.50 \times 1.5 = \$44.25

per hour

Overtime is paid at time-and-a-half (1.5 times the regular rate), applying to the adjusted base rate.

4

Calculate the pay for the overtime hours.

6 \times \$44.25 = \$265.50

The machinist worked 46 hours total, so 6 of those hours are considered overtime.

5

Calculate the total gross pay.

\$1,180.00 + \$265.50 = \$1,445.50

Total gross pay is the sum of the regular shift pay and the overtime pay.

Key Concept

Calculating gross pay involving base rate adjustments, shift differentials, and overtime multipliers.

Question 406Question →

A refueling tanker transfers $450$ gallons of aviation fuel into an aircraft in $15$ minutes. What is the tanker's transfer rate in gallons per hour?

1,800

3,000

112.5

30

Show answer & explanation

Answer: $1,800$

Answer

The correct answer is

1,800

.

The correct transfer rate is found by determining how much fuel is pumped in one minute, which is

450 \div 15 = 30

gallons per minute. Since there are

60

minutes in an hour, multiply this rate by

60

to get

1,800

gallons per hour.

Step-by-Step Solution

1

Find the unit rate in gallons per minute.

450 \div 15 = 30

gallons per minute.

This establishes the base rate of fuel transfer before scaling up to an hourly rate.

2

Convert the per-minute rate to a per-hour rate.

30 \times 60 = 1,800

gallons per hour.

There are 60 minutes in an hour, so the minute rate must be multiplied by 60 to find the hourly rate.

Key Concept

Calculating unit rates and applying time conversions.

Estimated Time:1m 0s

Question 407Question →

A signal corps detachment has a continuous spool of tactical fiber optic cable that is $240$ meters long. A technician must cut the entire spool into three separate sections to establish command, perimeter, and relay links in a ratio of $2:3:5$ respectively. What is the length of the shortest section in centimeters?

48

4,800

6,000

7,200

Show answer & explanation

Answer: $4,800$

Answer

4,800

The correct answer is found by first converting

240

meters to

24,000

centimeters. Then, the ratio

2:3:5

is summed to find

10

total parts. The shortest section corresponds to

2

parts. Taking

2

out of

10

parts of

24,000

centimeters gives

\frac{2}{10} \times 24,000 = 4,800

centimeters.

Step-by-Step Solution

1

Convert the total length of the cable from meters to centimeters.

240 \text{ meters} \times 100 \text{ cm/meter} = 24,000 \text{ cm}

The total length is given in meters, but the final question asks for the answer in centimeters.

2

Determine the total number of parts in the ratio.

2 + 3 + 5 = 10 \text{ total parts}

To find the size of a single part, the ratio must be converted into a fraction of the whole.

3

Calculate the length of the shortest section.

24,000 \text{ cm} \times \frac{2}{10} = 4,800 \text{ cm}

The shortest section corresponds to the smallest number in the ratio (

2

). Multiplying the total length by this fraction gives the length of that section.

Key Concept

To divide a quantity in a given ratio, first sum the parts of the ratio to find the total number of parts, then multiply the total quantity by the fraction representing the desired share. Always ensure units match the final question.

Alternative Method

You could also calculate the length of a single part first:

240

meters divided by

10

total parts is

24

meters per part. The shortest section is

2

parts, making it

24 \times 2 = 48

meters. Finally, convert

48

meters to centimeters by multiplying by

100

, which gives

4,800

centimeters.

Estimated Time:1m 15s

Question 408Question →

When analyzing algebraic interactions, it is often necessary to determine the exact coefficient of a specific term. Find the fully expanded and simplified form of the polynomial product

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

. What is the numerical coefficient of the

a^2b

term?

Show answer & explanation

Answer: -39

Answer

-39

To find the coefficient of the

a^2b

term, first expand the squared binomial

(3a - 4b)^2

to get the trinomial

9a^2 - 24ab + 16b^2

. Next, multiply this trinomial by

(2a + b)

. The specific terms that result in

a^2b

are the product of

9a^2

and

b

(which yields

9a^2b

) and the product of

-24ab

and

2a

(which yields

-48a^2b

). Combining these like terms gives

9a^2b - 48a^2b = -39a^2b

. Therefore, the numerical coefficient is -39.

Step-by-Step Solution

1

Expand the squared binomial

(3a - 4b)^2

.

9a^2 - 24ab + 16b^2

Order of operations requires resolving the exponent before multiplying by the binomial

(2a + b)

.

2

Set up the multiplication of the resulting trinomial and the binomial.

(9a^2 - 24ab + 16b^2)(2a + b)

This prepares the expression for distributing every term in the trinomial to every term in the binomial.

3

Identify and calculate only the partial products that result in an

a^2b

term.

(9a^2)(b) = 9a^2b

and

(-24ab)(2a) = -48a^2b

This is a more efficient method than fully expanding the entire polynomial if only a specific coefficient is needed.

4

Combine the identified like terms.

9a^2b - 48a^2b = -39a^2b

Combining these terms gives the final, simplified coefficient for the

a^2b

term.

Key Concept

Multiplying polynomials, expanding binomial squares, and combining like terms.

Question 409Question →

A circular target has a diameter of $16$ inches. Which of the following represents the area of the target, in square inches?

64\pi

256\pi

16\pi

32\pi

Show answer & explanation

Answer: $64\pi$

Answer

64\pi

The correct answer accurately divides the diameter of

16

inches by

2

to find the radius of

8

inches. It then squares the radius (

8^2 = 64

) and multiplies by

\pi

to find the area of

64\pi

square inches.

Step-by-Step Solution

1

Find the radius of the circle.

r = 16 / 2 = 8

inches

The area formula requires the radius, which is exactly half of the diameter.

2

Apply the area formula for a circle (

A = \pi r^2

).

A = \pi(8)^2

Substituting the calculated radius into the standard area formula.

3

Simplify the expression.

A = 64\pi

square inches

Squaring the radius (

8 \times 8 = 64

) provides the final area in terms of

\pi

.

Key Concept

The area of a circle is found using the formula

A = \pi r^2

. When given the diameter, it must first be divided by 2 to find the radius.

Question 410Question →

A motor pool sergeant is requisitioning maintenance supplies for his unit's vehicles. He orders $12$ quarts of synthetic oil at $\$6.75$ per quart, $4$ oil filters at $\$14.50$ each, and $3$ cans of heavy-duty degreaser at $\$5.25$ each. If he pays for the entire order using a $\$200.00$ unit procurement voucher, what is the remaining balance on the voucher in dollars?

Show answer & explanation

Answer: 45.25

Answer

45.25

The correct answer is found by first calculating the total cost for each group of items, summing these costs to determine the total purchase amount of

154.75, and then subtracting this total from the

200.00 voucher to find the remaining balance of $45.25.

Step-by-Step Solution

1

Calculate the total cost of the synthetic oil.

\$81.00

Multiply the quantity (12) by the price per quart (

\$6.75

).

2

Calculate the total cost of the oil filters.

\$58.00

Multiply the quantity (4) by the price per filter (

\$14.50

).

3

Calculate the total cost of the degreaser.

\$15.75

Multiply the quantity (3) by the price per can (

\$5.25

).

4

Calculate the total purchase amount.

\$154.75

Add the individual totals together (

\$81.00 + \$58.00 + \$15.75

).

5

Calculate the remaining balance on the voucher.

\$45.25

Subtract the total purchase amount from the initial voucher value (

\$200.00 - \$154.75

).

Key Concept

Multi-step decimal arithmetic involving unit prices and making change.

Estimated Time:1m 30s

Question 411Question →

A supply convoy is scheduled to travel a total distance of $135$ miles to a forward operating base in exactly $3$ hours and $30$ minutes. For the first $90$ minutes, the convoy travels along a secured highway at an average rate of $50$ miles per hour. For the remainder of the route, what must be the convoy's average rate of speed, in miles per hour, to arrive exactly on time?

Show answer & explanation

Answer: 30

Answer

30

The convoy must cover the remaining

60

miles in the remaining

2.0

hours, which requires an average speed of

30

miles per hour.

Step-by-Step Solution

1

Convert the given travel times from minutes to hours.

3

hours and

30

minutes =

3.5

hours.

90

minutes =

1.5

hours.

Since the speed is given in miles per hour, all time values must be in hours.

2

Calculate the distance covered on the secured highway.

50 \text{ miles/hour} \times 1.5 \text{ hours} = 75 \text{ miles}

To find out how much of the total distance has already been traveled.

3

Calculate the remaining distance to the forward operating base.

135 \text{ miles} - 75 \text{ miles} = 60 \text{ miles}

This is the distance that must be covered during the off-road portion of the route.

4

Calculate the remaining travel time available.

3.5 \text{ hours} - 1.5 \text{ hours} = 2.0 \text{ hours}

This is the time limit for completing the rest of the journey.

5

Calculate the required average speed for the remaining distance.

60 \text{ miles} \div 2.0 \text{ hours} = 30 \text{ miles per hour}

Rate equals distance divided by time (

r = \frac{d}{t}

).

Key Concept

Calculating Rate and Time

Alternative Method

You could set up an algebraic equation for the total distance:

(50 \times 1.5) + (r \times 2.0) = 135

. Solving for

r

:

75 + 2r = 135

, so

2r = 60

, which means

r = 30

.

Estimated Time:1m 30s

Question 412Question →

Which of the following expressions represents the completely factored form of the polynomial $3k^3 - 15k^2 - 42k$ ?

3k(k + 7)(k - 2)

k(3k - 21)(k + 2)

3k(k - 7)(k - 2)

3k(k - 7)(k + 2)

Show answer & explanation

Answer: $3k(k - 7)(k + 2)$

Answer

The completely factored form is

3k(k - 7)(k + 2)

.

The polynomial is completely factored by first extracting the Greatest Common Factor,

3k

, which simplifies the expression to

3k(k^2 - 5k - 14)

. Next, the remaining trinomial is factored by finding two numbers that multiply to

-14

and add to

-5

. These numbers are

-7

and

2

. Combining these gives the final fully factored form of

3k(k - 7)(k + 2)

.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all three terms of the polynomial.

The GCF of

3k^3

,

-15k^2

, and

-42k

is

3k

. Factoring this out leaves

3k(k^2 - 5k - 14)

.

Factoring out the GCF simplifies the remaining polynomial and is the required first step for complete factorization.

2

Factor the remaining quadratic trinomial,

k^2 - 5k - 14

.

Find two integers that multiply to

-14

(the constant term) and add to

-5

(the coefficient of the middle term). These integers are

-7

and

2

. This yields the binomial factors

(k - 7)

and

(k + 2)

.

These binomials represent the linear factors of the quadratic expression.

3

Combine the GCF and the binomial factors to write the fully factored expression.

3k(k - 7)(k + 2)

This represents the original polynomial as a product of its simplest prime factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.

Estimated Time:1m 0s

Question 413Question →

Two parallel lines, $L_1$ and $L_2$ , are intersected by three distinct transversals. The first transversal creates a pair of alternate interior angles that measure $(4x - 3y)^\circ$ and $(2x + y)^\circ$ . The second transversal creates a pair of consecutive (same-side) interior angles that measure $(3x + y)^\circ$ and $(5y + 36)^\circ$ . The third transversal creates a pair of corresponding angles. One of these corresponding angles measures $(x + y + z)^\circ$ , and the angle forming a linear pair with it measures $108^\circ$ . What is the value of $z$ ?

Show answer & explanation

Answer: 36

Answer

36

By applying geometric theorems for transversals (alternate interior angles are equal, same-side interior angles sum to 180), a system of equations reveals x = 24 and y = 12. Using the linear pair property, the final angle expression (x + y + z) equals 72, which isolates z to exactly 36.

Step-by-Step Solution

1

Set up an equation using the alternate interior angles.

4x - 3y = 2x + y \Rightarrow 2x = 4y \Rightarrow x = 2y

Alternate interior angles formed by a transversal intersecting parallel lines are equal.

2

Set up an equation using the consecutive (same-side) interior angles.

(3x + y) + (5y + 36) = 180 \Rightarrow 3x + 6y + 36 = 180 \Rightarrow 3x + 6y = 144

Consecutive interior angles formed by a transversal intersecting parallel lines are supplementary (sum to 180 degrees).

3

Solve the system of equations by substituting

x = 2y

into the second equation.

3(2y) + 6y = 144 \Rightarrow 6y + 6y = 144 \Rightarrow 12y = 144 \Rightarrow y = 12

. Since

x = 2y

,

x = 24

.

Substitution efficiently solves a two-variable linear system to find the individual angle components.

4

Use the linear pair to determine the measure of the corresponding angle, then solve for

z

.

(x + y + z) = 180 - 108 = 72

. Substituting known values:

24 + 12 + z = 72 \Rightarrow 36 + z = 72 \Rightarrow z = 36

.

Angles forming a linear pair are supplementary. Finding the target angle allows substitution of

x

and

y

to isolate

z

.

Key Concept

Complex systems of transversal angle relationships (alternate interior, consecutive interior, linear pairs)

Estimated Time:3m 0s

Question 414Question →

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 415Question →

A convex hexagon has five interior angles that measure $120^\circ$ , $135^\circ$ , $115^\circ$ , $140^\circ$ , and $110^\circ$ . If the measure of the sixth interior angle is represented by the algebraic expression $(4x + 16)^\circ$ , what is the value of $x$ ?

5

21

29

111

Show answer & explanation

Answer: $21$

Answer

The correct answer is

21

.

The sum of the interior angles of a hexagon (

6

sides) is

(6 - 2) \times 180^\circ = 720^\circ

. The five given angles add up to

620^\circ

. This means the sixth angle must be

720^\circ - 620^\circ = 100^\circ

. Setting the algebraic expression

4x + 16

equal to

100

and solving for

x

by subtracting

16

and dividing by

4

gives

x = 21

.

Step-by-Step Solution

1

Calculate the sum of the interior angles of a convex hexagon.

The sum is

(6 - 2) \times 180^\circ = 720^\circ

.

The formula for the sum of interior angles is

(n - 2) \times 180^\circ

, where

n

is the number of sides. A hexagon has

6

sides.

2

Find the sum of the five given angles.

120^\circ + 135^\circ + 115^\circ + 140^\circ + 110^\circ = 620^\circ

.

Adding the known angles allows us to find the combined measure of the known portion of the polygon.

3

Calculate the measure of the sixth angle.

720^\circ - 620^\circ = 100^\circ

.

Subtracting the sum of the five known angles from the total sum gives the measure of the missing angle.

4

Set up and solve the equation for

x

.

4x + 16 = 100 \implies 4x = 84 \implies x = 21

.

The sixth angle is represented by the expression

(4x + 16)^\circ

, so setting it equal to

100

allows us to isolate and solve for

x

.

Key Concept

Polygon Interior Angles and Linear Equations

Question 416Question →

A retail boutique manager receives a fixed monthly salary of $\$3,200$ . In addition to this base pay, she earns a $3\%$ commission on her own personal sales and a $0.5\%$ override commission on the sales generated by her staff. Last month, her personal sales totaled $\$14,000$ , and her staff's sales totaled $\$38,000$ . What was the manager's total gross income for the month?

$4,760

$610

$5,520

$3,810

Show answer & explanation

Answer: $3,810

Answer

\$3,810

To find the total gross income, calculate the separate commissions and add them to the base salary. The personal commission is

3\%

of

\$14,000

(which is

\$420

) and the staff commission is

0.5\%

of

\$38,000

(which is

\$190

). Adding these variable amounts to the fixed

\$3,200

base salary yields a total of

\$3,810

.

Step-by-Step Solution

1

Calculate the commission earned from personal sales.

\$14,000 \times 0.03 = \$420

The manager earns a 3% commission on her $14,000 of personal sales.

2

Calculate the override commission earned from staff sales.

\$38,000 \times 0.005 = \$190

The manager earns a 0.5% commission on the $38,000 sold by her staff. Note that 0.5% is 0.005 in decimal form.

3

Add the base salary and both commission amounts to find the total gross income.

\$3,200 + \$420 + \$190 = \$3,810

Gross income is the sum of all fixed pay and variable earnings before taxes.

Key Concept

Calculating Gross Pay with Base Salary and Multiple Commission Rates

Question 417Question →

A communications platoon is distributing a shipment of $8,400$ replacement radio batteries among three forward operating bases (FOB Alpha, FOB Bravo, and FOB Charlie) in a ratio of $4:5:6$ , respectively. Based on this ratio, how many batteries will FOB Charlie receive?

Show answer & explanation

Answer: 3360

Answer

FOB Charlie will receive 3,360 batteries.

The total ratio parts sum to 15 (4 + 5 + 6). Dividing the total 8,400 batteries by 15 yields 560 batteries per part. FOB Charlie corresponds to the '6' in the ratio, so multiplying 560 by 6 gives the correct allocation of 3,360 batteries.

Step-by-Step Solution

1

Calculate the total number of ratio parts.

4 + 5 + 6 = 15

parts

To divide a total quantity in a specific ratio, you first need to determine the total number of equal parts that make up the whole.

2

Determine the amount of batteries in a single part.

8,400 \div 15 = 560

batteries per part

Dividing the total number of batteries by the total number of parts establishes the baseline value of one unit in the ratio.

3

Calculate FOB Charlie's share.

560 \times 6 = 3,360

batteries

Since FOB Charlie receives 6 parts of the ratio, multiplying the value of a single part by 6 gives their final allocation.

Key Concept

To divide a quantity into a given ratio, find the total number of parts by summing the ratio values, divide the total quantity by this sum to find the value of one part, and then multiply by the desired part's ratio value.

Question 418Question →

A passenger train travels $80$ miles in $2$ hours. It then switches to a regional track and travels an additional $30$ miles in $3$ hours. Calculate the train's overall average speed for the entire journey.

Fill in the blanks below

The train's overall average speed is miles per hour.

Show answer & explanation

Answer

22

To calculate the average speed for the entire journey, determine the total distance traveled (

80 + 30 = 110

miles) and the total time taken (

2 + 3 = 5

hours). Dividing the total distance by the total time (

110 \div 5

) gives an average speed of

22

miles per hour.

Step-by-Step Solution

1

Calculate the total distance traveled.

80 + 30 = 110 miles

Average speed requires the total distance for all segments of the trip combined.

2

Calculate the total time taken.

2 + 3 = 5 hours

Average speed requires the total time duration for all segments.

3

Divide the total distance by the total time.

110 / 5 = 22

The formula for overall average speed is Total Distance divided by Total Time.

Key Concept

Average speed is found by dividing the total distance by the total time, not by averaging the individual speeds of different segments.

Question 419Question →

A tactical communications array transmits $1,800$ megabytes of encrypted data in $4.5$ minutes. At this constant rate, how many megabytes of data does the array transmit in $18$ seconds?

Show answer & explanation

Answer: 120

Answer

120

The correct answer is derived by first establishing the transmission rate of

400

megabytes per minute (

1,800 \div 4.5

). Since the required timeframe is given in seconds, it must be converted to minutes (

18 \div 60 = 0.3

minutes). Multiplying the rate by the converted time gives

120

megabytes.

Step-by-Step Solution

1

Calculate the unit transmission rate in megabytes per minute.

1,800 \div 4.5 = 400

megabytes per minute

Finding the rate per minute allows for scaling the transmission amount to any other duration.

2

Convert the target time from seconds to minutes.

18 \div 60 = 0.3

minutes

The established unit rate is in minutes, so the target time must match this temporal unit before multiplying.

3

Multiply the unit rate by the converted target time.

400 \times 0.3 = 120

megabytes

Multiplying the rate (megabytes per minute) by the time (minutes) yields the total amount of data transmitted.

Key Concept

Calculating unit rates and applying time unit conversions (seconds to minutes) to solve for a specific quantity.

Question 420Question →

The perimeter of a square courtyard is $36$ meters. What is the area of the courtyard, in square meters?

Show answer & explanation

Answer: 81

Answer

81

A square has four sides of equal length. Given a perimeter of

36

meters, each side measures

36 \div 4 = 9

meters. The area of a square is its side length squared, which is

9^2 = 81

square meters.

Step-by-Step Solution

1

Find the length of one side of the square.

36 \div 4 = 9

meters.

The perimeter of a square is the sum of its four equal sides, so dividing the total perimeter by 4 gives the length of a single side.

2

Calculate the area of the square.

9 \times 9 = 81

square meters.

The area of a square is calculated by squaring the length of one of its sides (

A = s^2

).

Key Concept

Perimeter and Area of 2D Figures

Answer

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