All practice questions

Question 381Question →

A base quartermaster is purchasing different types of office supplies for a training facility. Calculate the total cost for each order and arrange them from the least expensive to the most expensive.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least expensive to most expensive is:

5

boxes of classification folders (

\$89.25

),

12

packs of highlighters (

\$89.40

),

6

packs of permanent markers (

\$89.70

), and

4

magnetic whiteboards (

\$89.80

).

The total cost for each order is found by multiplying the quantity by the price per item. The folders cost

\$89.25

(

5 \times \$17.85

). The highlighters cost

\$89.40

(

12 \times \$7.45

). The markers cost

\$89.70

(

6 \times \$14.95

). The whiteboards cost

\$89.80

(

4 \times \$22.45

). Ordering these decimal values from least to greatest gives the correct sequence.

Step-by-Step Solution

1

Calculate the total cost for the classification folders.

\$89.25

Total cost equals quantity multiplied by unit price (

5 \times \$17.85

).

2

Calculate the total cost for the highlighters.

\$89.40

Total cost equals quantity multiplied by unit price (

12 \times \$7.45

).

3

Calculate the total cost for the permanent markers.

\$89.70

Total cost equals quantity multiplied by unit price (

6 \times \$14.95

).

4

Calculate the total cost for the magnetic whiteboards.

\$89.80

Total cost equals quantity multiplied by unit price (

4 \times \$22.45

).

5

Compare the calculated total costs and arrange them from least to greatest.

\$89.25 < \$89.40 < \$89.70 < \$89.80

To fulfill the requirement of ordering from least expensive to most expensive.

Key Concept

Calculating total cost by multiplying unit price by quantity, and comparing decimal values.

Question 382Question →

Match each quadrilateral with its precise geometric defining properties.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Rhombus matches the parallelogram with perpendicular diagonals that bisect angles. Rectangle matches the parallelogram with four right angles. Kite matches the quadrilateral with consecutive congruent sides. Isosceles Trapezoid matches the quadrilateral with exactly one pair of parallel sides.

Each quadrilateral has unique defining traits involving their sides, angles, and diagonals. A rhombus requires perpendicular diagonals and parallel opposite sides. A rectangle requires right angles and congruent diagonals. A kite requires adjacent equal sides and perpendicular diagonals. An isosceles trapezoid requires exactly one pair of parallel sides and congruent diagonals.

Step-by-Step Solution

1

Analyze the unique properties of a Rhombus.

It is a parallelogram with all equal sides, which leads to perpendicular diagonals that bisect the opposite angles.

To distinguish it from other standard parallelograms like rectangles.

2

Analyze the unique properties of a Rectangle.

It has four right angles and, unlike a standard parallelogram, has diagonals that are identical in length.

To correctly identify the item describing congruent diagonals in a parallelogram.

3

Analyze the unique properties of a Kite.

It has adjacent sides that are equal (consecutive congruent sides), which forces the diagonals to meet at 90 degrees.

To match it with the correct non-parallelogram definition.

4

Analyze the unique properties of an Isosceles Trapezoid.

It has exactly one pair of parallel sides (bases) and the symmetry from equal non-parallel sides creates equal diagonals.

To differentiate it from general trapezoids and generic parallelograms.

Key Concept

Quadrilaterals and Polygons

Question 383Question →

An engineer needs to simplify a calculation involving the volume of a component, modeled by the polynomial $4y^3 + 12y^2 - 72y$ . What is the fully factored equivalent of this polynomial?

4y(y + 6)(y - 3)

4y(y - 6)(y + 3)

4y(y^2 + 3y - 18)

4y(y + 6)(y + 3)

Show answer & explanation

Answer: $4y(y + 6)(y - 3)$

Answer

4y(y + 6)(y - 3)

To fully factor the given expression, first extract the greatest common factor (GCF). All terms share a common factor of

4y

, yielding

4y(y^2 + 3y - 18)

. Next, factor the resulting quadratic trinomial

y^2 + 3y - 18

. The two numbers that multiply to

-18

and sum to

3

are

6

and

-3

. This gives the binomial factors

(y + 6)

and

(y - 3)

. Combining these with the GCF results in the completely factored form

4y(y + 6)(y - 3)

.

Step-by-Step Solution

1

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

4y^3 + 12y^2 - 72y

.

4y(y^2 + 3y - 18)

Factoring out the GCF simplifies the remaining polynomial, making it easier to factor completely.

2

Factor the remaining quadratic trinomial

y^2 + 3y - 18

by finding two integers that multiply to

-18

and add to

3

.

(y + 6)(y - 3)

The integers

6

and

-3

satisfy both conditions:

6 \times (-3) = -18

and

6 + (-3) = 3

.

3

Combine the GCF with the factored binomials to state the final expression.

4y(y + 6)(y - 3)

This represents the original polynomial decomposed entirely into its lowest degree factors.

Key Concept

Factoring Polynomials with a GCF

Estimated Time:1m 0s

Question 384Question →

A supply sergeant is distributing $560$ pairs of cold-weather boots between two infantry companies (Company Alpha and Company Bravo) in a ratio of $3:4$ . How many MORE pairs of boots does Company Bravo receive than Company Alpha?

140

80

280

320

Show answer & explanation

Answer: 80

Answer

80

The correct approach is to first find the total number of parts in the ratio (

3 + 4 = 7

). Dividing the total

560

boots by

7

reveals that each part represents

80

boots. Since Company Bravo has

4

parts and Company Alpha has

3

parts, the difference is exactly

1

part, which equals

80

boots.

Step-by-Step Solution

1

Calculate the total number of ratio parts.

3 + 4 = 7

total parts

To distribute the total amount according to the ratio, you first need to know how many equal parts the total is being divided into.

2

Determine the value of a single part.

560 \div 7 = 80

boots per part

Dividing the total number of boots by the total number of ratio parts gives the value of one equal share.

3

Find the difference in ratio parts between the two companies.

4 - 3 = 1

part difference

Instead of calculating both companies' totals and subtracting, it is more efficient to find the difference in their ratio shares directly.

4

Calculate the difference in the number of boots.

1 \times 80 = 80

boots

Multiplying the difference in parts by the value of one part gives the exact difference in the final item count.

Key Concept

Dividing a quantity in a given ratio involves finding the total number of parts, the value of one part, and then applying that unit value to answer the specific question asked.

Alternative Method

You can calculate each company's share individually first: Company Alpha receives

3 \times 80 = 240

boots. Company Bravo receives

4 \times 80 = 320

boots. Then, subtract the smaller share from the larger share:

320 - 240 = 80

boots.

Estimated Time:1m 0s

Question 385Question →

A land surveyor is measuring a four-sided plot of land. They record three of the interior angles as $78^\circ$ , $104^\circ$ , and $83^\circ$ . What is the measure of the fourth interior angle, in degrees?

Show answer & explanation

Answer: 95

Answer

The fourth interior angle measures

95^\circ

.

The correct answer is found by subtracting the sum of the three known angles (

265^\circ

) from the total interior angle sum of a quadrilateral (

360^\circ

), which yields

95^\circ

.

Step-by-Step Solution

1

Identify the total sum of interior angles for a quadrilateral.

360^\circ

A quadrilateral has 4 sides, and the sum of interior angles is calculated as

(n - 2) \times 180^\circ

. For

n = 4

, this is

2 \times 180^\circ = 360^\circ

.

2

Calculate the sum of the three given angles.

78^\circ + 104^\circ + 83^\circ = 265^\circ

To find the missing part, we must first determine the combined total of the known angles.

3

Subtract the sum of the given angles from the total quadrilateral angle sum.

360^\circ - 265^\circ = 95^\circ

The remaining amount represents the exact measure of the fourth interior angle.

Key Concept

The sum of the interior angles of any convex quadrilateral is always exactly

360^\circ

.

Question 386Question →

A naval supply ship travels $150$ nautical miles in $10$ hours, and then travels an additional $90$ nautical miles in $5$ hours. What is the ship's average speed for the entire trip, in knots (nautical miles per hour)?

Show answer & explanation

Answer: 16

Answer

16 knots

The overall average speed is calculated correctly by finding the total distance (

150 + 90 = 240

nautical miles) and dividing it by the total time (

10 + 5 = 15

hours). This calculation (

240 \div 15

) results in exactly

16

knots.

Step-by-Step Solution

1

Calculate the total distance traveled for the entire trip.

150 + 90 = 240

nautical miles

Finding average speed requires knowing the complete distance covered across all segments of the trip.

2

Calculate the total time taken for the entire trip.

10 + 5 = 15

hours

Finding average speed requires knowing the total elapsed time.

3

Divide the total distance by the total time.

240 \div 15 = 16

knots

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

Key Concept

Average speed is calculated by dividing the total distance traveled by the total time taken.

Question 387Question →

A logistics battalion is coordinating different transport missions. Match each vehicle's specific mission profile with its correct calculated travel time, average speed, or arrival time.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The logistics mission profiles match successfully by applying the distance formula and executing necessary time conversions (e.g.,

2.25

hours is

2

hours and

15

minutes, and

30

minutes is

0.5

hours).

Each match strictly follows the distance formula

d = r \times t

, correctly manipulating the variables and converting fractional hours to minutes or minutes to hours where required.

Step-by-Step Solution

1

Calculate the travel time for the supply truck.

Total travel time of

2

hours and

15

minutes

Applying

t = d \div r

(

90 \div 40 = 2.25

hours) and converting the decimal remainder to minutes (

0.25 \times 60 = 15

).

2

Calculate the average speed of the patrol boat.

Average speed of

30

mph

Applying

r = d \div t

(

135 \div 4.5 = 30

).

3

Calculate the arrival time of the armored transport.

Arrival time of 1100 hours

Applying

t = d \div r

(

165 \div 55 = 3

hours), then adding

3

hours to 0800 hours.

4

Calculate the average speed of the medevac helicopter.

Average speed of

150

mph

Converting

30

minutes to

0.5

hours, then applying

r = d \div t

(

75 \div 0.5 = 150

).

Key Concept

Applying the fundamental distance relationship (

d = r \times t

) to calculate unknown rates and times, while managing unit conversions and clock mathematics.

Estimated Time:2m 0s

Question 388Question →

A cybersecurity task force is composed of junior analysts and senior specialists in a ratio of $7:2$ . If the task force has a total of $126$ members, how many senior specialists are on the team?

28

36

98

14

Show answer & explanation

Answer: $28$

Answer

There are

28

senior specialists on the team.

The correct answer is found by first adding the components of the ratio (

7 + 2 = 9

) to represent the entire task force. Next, divide the total number of members (

126

) by

9

to determine that each ratio 'part' equals

14

members. Finally, multiply this unit value by the

2

parts that represent the senior specialists (

14 \times 2 = 28

).

Step-by-Step Solution

1

Determine the total number of parts in the given ratio.

7 + 2 = 9

total parts.

To find the value of a single part relative to the entire group, you must first know how many equal parts make up the whole.

2

Calculate the numerical value of one single part.

126 \div 9 = 14

members per part.

Dividing the total number of members by the total ratio parts gives the quantity represented by one unit.

3

Multiply the value of one part by the number of parts designated for senior specialists.

14 \times 2 = 28

senior specialists.

The ratio states there are

2

parts of senior specialists for every

7

parts of junior analysts.

Key Concept

Calculating a specific part from a total quantity using a part-to-part ratio.

Question 389Question →

Pair each algebraic expression with its completely factored equivalent.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are:

4y^2 - 16y

matches with

4y(y - 4)

;

y^2 - 16

matches with

(y - 4)(y + 4)

;

y^2 - 8y + 16

matches with

(y - 4)^2

; and

4y + 16

matches with

4(y + 4)

.

Each polynomial requires a distinct factoring method. Identifying the correct structure—such as GCF, difference of squares, or perfect square trinomial—leads directly to the completely factored form.

Step-by-Step Solution

1

Factor

4y^2 - 16y

4y(y - 4)

The terms share a greatest common factor of

4y

.

2

Factor

y^2 - 16

(y - 4)(y + 4)

It follows the difference of squares pattern

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

.

3

Factor

y^2 - 8y + 16

(y - 4)^2

It is a perfect square trinomial, as

(-4) \times (-4) = 16

and

(-4) + (-4) = -8

.

4

Factor

4y + 16

4(y + 4)

The terms share a greatest common factor of

4

.

Key Concept

Factoring polynomials using greatest common factors, difference of squares, and perfect square trinomial patterns.

Question 390Question →

Match each polynomial expression to its completely factored form.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair each polynomial with its mathematically equivalent fully factored expression.

Each polynomial requires identifying its specific factoring archetype: the difference of two squares, a standard quadratic trinomial, GCF extraction followed by recognizing a perfect square trinomial, or factoring a non-monic trinomial.

Step-by-Step Solution

1

Analyze the binomial

x^2 - 16

.

Matches to

(x - 4)(x + 4)

.

It represents a difference of perfect squares.

2

Analyze the trinomial

x^2 + 6x + 8

.

Matches to

(x + 2)(x + 4)

.

The constants 2 and 4 have a product of 8 and a sum of 6.

3

Analyze the polynomial

2x^2 - 12x + 18

.

Matches to

2(x - 3)^2

.

Factoring out the common factor of 2 leaves

x^2 - 6x + 9

, which is the square of

(x - 3)

.

4

Analyze the polynomial

3x^2 - 5x - 2

.

Matches to

(3x + 1)(x - 2)

.

Multiplying the binomials

(3x + 1)

and

(x - 2)

confirms it expands to the original expression.

Key Concept

Factoring different archetypes of polynomials, including the difference of squares, monic trinomials, non-monic trinomials, and extracting the Greatest Common Factor (GCF).

Question 391Question →

A rectangular room has a length of $12$ feet and a width of $10$ feet. What is the perimeter of the room?

22

feet

34

feet

44

feet

120

feet

Show answer & explanation

Answer: $44$ feet

Answer

44

feet

The perimeter is the total distance around the boundary of the rectangle. Since a rectangle has two equal lengths and two equal widths, the perimeter is

2 \times 12 + 2 \times 10 = 24 + 20 = 44

feet.

Step-by-Step Solution

1

Identify the formula for the perimeter of a rectangle.

P = 2 \times (\text{Length} + \text{Width})

or

P = 2L + 2W

The perimeter is the total distance around the outside of the shape, which includes two lengths and two widths.

2

Substitute the given dimensions into the formula.

P = 2 \times (12 + 10)

The length is given as

12

feet and the width is

10

feet.

3

Calculate the final perimeter.

P = 2 \times 22 = 44

Performing the addition inside the parentheses first, then multiplying by

2

yields the total distance.

Key Concept

The perimeter of a rectangle is the total distance around its edges, calculated as twice the sum of its length and width.

Estimated Time:45s

Question 392Question →

When the polynomial expression

(3x - 2y)(Ax^2 + 4xy - 5y^2)

is fully expanded and simplified, the coefficient of the

x^2y

term is

-4

. What is the value of the constant

A

?

2

8

-4

-8

Show answer & explanation

Answer: $8$

Answer

The value of the constant

A

is

8

.

To find the coefficient of the

x^2y

term, we do not need to expand the entire expression. We only need to find the specific multiplications that result in an

x^2y

term. Multiplying the

x

term from the first polynomial by the

xy

term from the second gives

3x \times 4xy = 12x^2y

. Multiplying the

y

term from the first polynomial by the

x^2

term from the second gives

-2y \times Ax^2 = -2Ax^2y

. Combining these gives a total

x^2y

coefficient of

(12 - 2A)

. Since the problem states this coefficient is

-4

, we set up the equation

12 - 2A = -4

. Subtracting

12

from both sides gives

-2A = -16

. Dividing by

-2

gives

A = 8

.

Step-by-Step Solution

1

Identify the products that result in an

x^2y

term when expanding the polynomials.

The relevant terms are

(3x)(4xy)

and

(-2y)(Ax^2)

.

Only these specific combinations of terms will produce the exact variable configuration

x^2y

.

2

Multiply these specific terms to find their coefficients.

(3x)(4xy) = 12x^2y

and

(-2y)(Ax^2) = -2Ax^2y

.

This isolates the parts of the expansion that contribute to the final

x^2y

coefficient.

3

Combine the coefficients and set them equal to the given total coefficient of

-4

.

12 - 2A = -4

The algebraic sum of the individual

x^2y

coefficients must equal the final simplified coefficient provided in the problem.

4

Solve the linear equation for

A

.

-2A = -4 - 12 \Rightarrow -2A = -16 \Rightarrow A = 8

Standard algebraic isolation of the variable

A

gives the final answer.

Key Concept

Identifying specific terms in polynomial multiplication and solving for an unknown coefficient.

Estimated Time:1m 30s

Question 393Question →

In a parallelogram, the measure of one interior angle is $30^\circ$ less than twice the measure of the consecutive interior angle. What is the measure of the larger angle?

110^\circ

50^\circ

130^\circ

230^\circ

Show answer & explanation

Answer: $110^\circ$

Answer

The measure of the larger angle is

110^\circ

.

The correct answer correctly identifies that consecutive angles in a parallelogram are supplementary. Setting up the equation

x + (2x - 30) = 180^\circ

and solving for

x

yields

70^\circ

. Substituting this back into the expression for the larger angle gives

2(70) - 30 = 110^\circ

.

Step-by-Step Solution

1

Identify the geometric property relating the angles.

Consecutive interior angles of a parallelogram are supplementary, meaning they add up to

180^\circ

.

This is a fundamental property of parallelograms formed by parallel lines cut by a transversal.

2

Translate the word problem into an algebraic equation.

Let the smaller angle be

x

. The larger angle is

2x - 30

. Therefore,

x + (2x - 30) = 180

.

This captures both the relationship described ('

30^\circ

less than twice') and the supplementary geometric property.

3

Solve the algebraic equation for

x

.

Combine terms:

3x - 30 = 180

. Add

30

to both sides:

3x = 210

. Divide by

3

:

x = 70^\circ

.

Isolating the variable determines the exact measure of the smaller angle.

4

Calculate the measure of the larger angle.

2(70) - 30 = 140 - 30 = 110^\circ

.

The question specifically asks for the measure of the larger angle, not the smaller one.

Key Concept

Properties of Parallelograms and Algebraic Translation

Alternative Method

You can solve this by testing the answer choices. If the larger angle is

110^\circ

, the smaller consecutive angle must be

180^\circ - 110^\circ = 70^\circ

. Check if

110^\circ

is

30^\circ

less than twice

70^\circ

:

2(70) - 30 = 140 - 30 = 110^\circ

. The values perfectly match the problem statement.

Estimated Time:1m 15s

Question 394Question →

To optimize a packaging design, a technician models the surface area using the algebraic expression $18xy^2 - 24xy - 10x$ . What is the fully factored equivalent of this expression?

2x(3y - 5)(3y + 1)

2x(3y + 5)(3y - 1)

x(6y - 10)(3y + 1)

2xy(9y - 17)

Show answer & explanation

Answer: $2x(3y - 5)(3y + 1)$

Answer

The fully factored equivalent is

2x(3y - 5)(3y + 1)

.

The expression is fully factored by first extracting the greatest common factor of

2x

, resulting in

2x(9y^2 - 12y - 5)

. Then, using the ac-method, the remaining non-monic quadratic trinomial is factored into

(3y - 5)(3y + 1)

, yielding the final correct product.

Step-by-Step Solution

1

Identify and extract the greatest common factor (GCF) from all three terms.

2x(9y^2 - 12y - 5)

Extracting the GCF simplifies the polynomial and prepares the remaining quadratic trinomial for factoring.

2

Identify the components of the remaining quadratic trinomial

ay^2 + by + c

.

Here,

a = 9

,

b = -12

, and

c = -5

.

This establishes the values needed to use the ac-method for factoring non-monic quadratics.

3

Find two numbers that multiply to

ac

and add to

b

.

ac = 9 \times -5 = -45

. The two numbers that multiply to

-45

and add to

-12

are

-15

and

3

.

These numbers allow the middle term to be split, enabling factoring by grouping.

4

Rewrite the middle term and factor the trinomial by grouping.

9y^2 - 15y + 3y - 5 = 3y(3y - 5) + 1(3y - 5) = (3y - 5)(3y + 1)

This completes the factorization of the trinomial portion of the expression.

5

Combine the originally extracted GCF with the factored binomials.

2x(3y - 5)(3y + 1)

This yields the complete and fully factored form of the original polynomial.

Key Concept

Factoring Polynomials with a GCF

Question 395Question →

A custom cabinetry builder is paid an hourly wage of $\$24$ for a standard $40$ -hour workweek. For any hours worked beyond $40$ in a single week, they receive "time-and-a-half" overtime pay. Last week, the builder worked a total of $52$ hours and received an additional $\$150$ material handling bonus. What were the builder's total gross earnings for the week?

\$1,542

\$1,398

\$2,022

\$582

Show answer & explanation

Answer: $\$1,542$

Answer

The total gross earnings are

\$1,542

.

The correct answer accurately separates the total hours into regular and overtime components, applies the respective rates to each portion, and includes the fixed material handling bonus to find the total gross earnings.

Step-by-Step Solution

1

Calculate the base pay for the standard workweek.

40 \text{ hours} \times \$24/\text{hour} = \$960

The standard hourly rate only applies to the first 40 hours worked in the week.

2

Determine the number of overtime hours and the overtime pay rate.

52 - 40 = 12 \text{ overtime hours}

. The overtime rate is

\$24 \times 1.5 = \$36/\text{hour}

.

Overtime is paid at time-and-a-half for any hours beyond the standard 40.

3

Calculate the total overtime pay.

12 \text{ hours} \times \$36/\text{hour} = \$432

This accounts for the compensation earned during the extra hours.

4

Sum all earnings to find the total gross pay.

\$960 \text{ (base)} + \$432 \text{ (overtime)} + \$150 \text{ (bonus)} = \$1,542

Total gross earnings must include regular pay, overtime pay, and any additional bonuses.

Key Concept

Wages, Salaries, and Commissions

Question 396Question →

A naval supply officer is allocating a $\$4,500$ monthly discretionary budget among three ship departments—Deck, Engineering, and Operations—in a ratio of $4:5:6$ , respectively. Calculate the specific funding amounts based on this ratio.

Fill in the blanks below

The Engineering department will receive $\$

, and the Operations department will receive

\$$ more than the Deck department.

Show answer & explanation

Answer

The Engineering department receives

\$1,500

, and the Operations department receives

\$600

more than the Deck department.

The total budget of

\$4,500

is divided into

15

equal parts (

4 + 5 + 6

). Each part is worth

\$300

(

\$4,500 \div 15

). The Engineering department receives

5

parts, which amounts to

\$1,500

(

5 \times \$300

). The Operations department receives

6

parts and the Deck department receives

4

parts. The difference between them is

2

parts (

6 - 4

), which amounts to

\$600

(

2 \times \$300

).

Step-by-Step Solution

1

Calculate the total number of parts in the ratio.

4 + 5 + 6 = 15

total parts.

The total budget must be divided by the sum of the ratio numbers to find the value of a single part.

2

Determine the monetary value of one part.

\$4,500 \div 15 = \$300

.

This establishes the base unit value that can be multiplied by each department's respective ratio number.

3

Calculate the amount allocated to the Engineering department (second value in the ratio).

5 \times \$300 = \$1,500

.

This provides the answer for the first blank.

4

Calculate the difference between the Operations department (6 parts) and the Deck department (4 parts).

Difference in parts:

6 - 4 = 2

parts. Monetary difference:

2 \times \$300 = \$600

.

This provides the answer for the second blank without needing to calculate the full amounts for both departments.

Key Concept

Dividing a quantity in a given three-part ratio by finding the sum of the parts and the value of a single unit.

Question 397Question →

An operations chief needs to verify the hourly fuel consumption for a deployed mechanized unit. Match each asset's usage report to its calculated unit rate in gallons per hour.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Combat engineer vehicle matches to 25 gallons per hour; Utility helicopter matches to 80 gallons per hour; Tactical field generator matches to 15 gallons per hour; Cargo transport truck matches to 12 gallons per hour.

Each asset's fuel consumption rate is found by setting up a ratio of total gallons used to total hours operated, and then dividing to find the amount for exactly one hour. Any time given in minutes must be converted to hours to match the requested unit of 'gallons per hour'.

Step-by-Step Solution

1

Calculate the unit rate for the combat engineer vehicle.

350 \div 14 = 25

gallons per hour.

Unit rate is found by dividing the total quantity (gallons) by the total time (hours).

2

Calculate the unit rate for the utility helicopter.

90

minutes is

1.5

hours.

120 \div 1.5 = 80

gallons per hour.

The rate requested is in gallons per hour, so minutes must first be converted to hours before dividing.

3

Calculate the unit rate for the tactical field generator.

45 \div 3 = 15

gallons per hour.

Unit rate is found by dividing the total quantity (gallons) by the total time (hours).

4

Calculate the unit rate for the cargo transport truck.

144 \div 12 = 12

gallons per hour.

Unit rate is found by dividing the total quantity (gallons) by the total time (hours).

Key Concept

Calculating unit rates by dividing a total quantity by a total elapsed time, ensuring units align correctly.

Question 398Question →

If $3x + 5y = 11$ and $4x + 9y = 3$ , what is the value of the expression $x + 2y$ ?

22

4

2

-2

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. To solve the system, use the elimination method by multiplying the equations to create opposite or matching coefficients. Multiplying the first equation by 4 yields

12x + 20y = 44

. Multiplying the second equation by 3 yields

12x + 27y = 9

. Subtracting the first equation from the second gives

7y = -35

, so

y = -5

. Substituting

y = -5

into the first original equation gives

3x - 25 = 11

, which simplifies to

3x = 36

, or

x = 12

. Finally, substituting these values into the expression

x + 2y

gives

12 + 2(-5) = 12 - 10 = 2

.

Step-by-Step Solution

1

Multiply the first equation by 4 and the second equation by 3 to prepare for elimination.

12x + 20y = 44

and

12x + 27y = 9

Creating matching coefficients for

x

allows us to eliminate it by subtracting the equations.

2

Subtract the first new equation from the second new equation.

(12x + 27y) - (12x + 20y) = 9 - 44 \Rightarrow 7y = -35

This eliminates

x

and leaves a simple equation with only

y

.

3

Divide by 7 to solve for

y

.

y = -5

Isolating

y

gives its numerical value.

4

Substitute

y = -5

back into the original first equation (

3x + 5y = 11

) and solve for

x

.

3x + 5(-5) = 11 \Rightarrow 3x - 25 = 11 \Rightarrow 3x = 36 \Rightarrow x = 12

This finds the corresponding value for

x

.

5

Substitute

x = 12

and

y = -5

into the requested expression

x + 2y

.

12 + 2(-5) = 12 - 10 = 2

The question asks for the value of this specific expression, not just the ordered pair.

Key Concept

Solving systems of linear equations using the elimination method and evaluating algebraic expressions.

Estimated Time:2m 0s

Question 399Question →

A culinary specialist is ordering fresh provisions for a field kitchen. He orders $18.5$ pounds of ground beef at $\$4.20$ per pound and $24$ pounds of potatoes at $\$0.75$ per pound. What is the total cost of these provisions?

\$77.70

\$95.70

\$78.45

\$25.77

Show answer & explanation

Answer: $\$95.70$

Answer

The total cost of the provisions is

\$95.70

.

The total cost is calculated by multiplying the quantity of each item by its respective unit price, and then adding the two subtotals together. The ground beef costs

18.5 \times \$4.20 = \$77.70

. The potatoes cost

24 \times \$0.75 = \$18.00

. Adding these amounts together gives a final total of

\$95.70

.

Step-by-Step Solution

1

Calculate the total cost of the ground beef by multiplying the weight by the price per pound.

18.5 \times \$4.20 = \$77.70

To find the subtotal for the first provision item.

2

Calculate the total cost of the potatoes by multiplying the weight by the price per pound.

24 \times \$0.75 = \$18.00

To find the subtotal for the second provision item.

3

Add the two subtotals together to find the final overall cost.

\$77.70 + \$18.00 = \$95.70

To determine the combined total cost of the purchase.

Key Concept

Multiplying decimals to find itemized costs and adding the resulting subtotals to find a final sum.

Question 400Question →

An engineering platoon is paving a temporary runway. If their paving machine operates at a constant rate of $20$ feet per minute, how many hours will it take to pave a continuous $3,000$ -foot section?

2.3

2.5

25

150

Show answer & explanation

Answer: $2.5$

Answer

The paving process will take 2.5 hours.

The correct answer is found by taking the total distance (

3,000

feet) and dividing it by the rate (

20

feet per minute) to get a total time of

150

minutes. Since there are

60

minutes in an hour, dividing

150

by

60

yields exactly

2.5

hours.

Step-by-Step Solution

1

Determine the required time in the given units (minutes).

Time =

3,000 \div 20 = 150

minutes.

Using the formula Time = Distance ÷ Rate finds the duration in the same unit of time used by the rate.

2

Convert the calculated time from minutes to hours.

150 \div 60 = 2.5

hours.

The question explicitly asks for the final answer in hours, so the time in minutes must be divided by 60.

Key Concept

Calculating elapsed time using the distance formula (

t = d/r

) and performing time unit conversions.

Estimated Time:1m 30s

Answer

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