All practice questions

Match each quantity relationship to its equivalent simplified ratio.

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Answer

The unsimplified ratios match to their reduced forms as follows: instructors to recruits (

10:50

) is

1:5

, trucks to vans (

12:8

) is

3:2

, fuel to oil (

15:5

) is

3:1

, and rifles to sidearms (

20:15

) is

4:3

.

Each relationship is found by writing down the initial numerical counts in the order specified and reducing them to their lowest integer terms using their greatest common factors.

Step-by-Step Solution

1

Set up the initial part-to-part ratio for each scenario in the exact order requested by the text.

The raw ratios are

10:50

,

12:8

,

15:5

, and

20:15

.

Ratio order matters. Ensuring the first mentioned item is the first number in the ratio prevents inverted answers.

2

Simplify each ratio by finding the greatest common factor (GCF) for each pair and dividing both numbers by it.

10:50

reduces to

1:5

.

12:8

reduces to

3:2

.

15:5

reduces to

3:1

.

20:15

reduces to

4:3

.

A ratio is properly expressed when the two numbers have no common factors other than

1

.

Key Concept

Basic Ratios

Question 42Question →

Match each non-monic quadratic polynomial to its completely factored form. Pay close attention to the signs in each binomial factor.

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Answer

The correct pairings are:

2m^2 - 11m + 15

with

(2m - 5)(m - 3)

;

2m^2 + m - 15

with

(2m - 5)(m + 3)

;

2m^2 + 11m + 15

with

(2m + 5)(m + 3)

; and

2m^2 - m - 15

with

(2m + 5)(m - 3)

.

Each polynomial requires finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient. The correct pairings map the expanded polynomials directly to their unique, fully factored equivalents.

Step-by-Step Solution

1

Analyze the signs of the constants in each polynomial.

Polynomials with a positive constant term (+15) must have binomial factors with identical signs (both positive or both negative). Polynomials with a negative constant term (-15) must have binomial factors with opposite signs.

The product of the constant terms in the binomials must equal the constant term of the polynomial.

2

Factor

2m^2 - 11m + 15

.

Since the constant is positive and the middle term is negative, both factors must be negative.

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

.

Applying factoring by grouping.

3

Factor

2m^2 + m - 15

.

The constant is negative, so signs differ. The middle term is positive, so the larger product in FOIL must be positive.

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

.

Applying factoring by grouping.

4

Factor

2m^2 + 11m + 15

.

Since both the constant and middle term are positive, both factors must be positive.

2m^2 + 6m + 5m + 15 = 2m(m + 3) + 5(m + 3) = (2m + 5)(m + 3)

.

Applying factoring by grouping.

5

Factor

2m^2 - m - 15

.

The constant is negative, so signs differ. The middle term is negative, so the larger product in FOIL must be negative.

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

.

Applying factoring by grouping.

Key Concept

Factoring Non-Monic Trinomials and Sign Analysis

Estimated Time:2m 30s

Question 43Question →

Match each inverse proportion scenario to its correct calculated value.

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Answer

Each scenario represents an inverse proportion and must be matched with the value that maintains the constant product of the initial conditions.

Each match is mathematically determined by setting up an inverse proportion equation where the product of the initial states equals the product of the final states (

x_1 \cdot y_1 = x_2 \cdot y_2

). Dividing the initial product by the new given value provides the correct corresponding answer.

Step-by-Step Solution

1

Identify that each scenario represents an inverse relationship.

As one quantity increases (troops, speed, workers, gear teeth), the other quantity must decrease.

This determines that the product of the variables is constant (

x_1 \cdot y_1 = x_2 \cdot y_2

), rather than the ratio.

2

Calculate the constant product (

k

) for the initial conditions in each scenario.

The constants are

15,000

troop-days,

2,400\text{ miles}

,

96\text{ man-hours}

, and

7,200\text{ teeth-RPM}

.

The constant represents the total amount of work, distance, resources, or mechanical transfer that must remain unchanged.

3

Divide the constant product by the new given variable to find the unknown required value.

The resulting values are

20\text{ days}

,

4\text{ hours}

,

6\text{ hours}

, and

160\text{ RPM}

.

Solving the equation

x_1 \cdot y_1 = x_2 \cdot y_2

for the unknown variable

y_2

requires dividing the constant by

x_2

.

Key Concept

Inverse Proportions

Question 44Question →

Match each military vehicle's travel scenario to its correct average unit rate in miles per hour.

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Answer

The jeep matches to 40 mph, the helicopter to 150 mph, the truck to 55 mph, and the patrol boat to 36 mph.

Each unit rate is found by applying the formula for speed: Speed = Distance ÷ Time. Calculating this for each vehicle yields the correct matching pairs.

Step-by-Step Solution

1

Calculate the unit rate for the jeep.

120 / 3 = 40 miles per hour

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

2

Calculate the unit rate for the helicopter.

300 / 2 = 150 miles per hour

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

3

Calculate the unit rate for the supply truck.

220 / 4 = 55 miles per hour

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

4

Calculate the unit rate for the patrol boat.

180 / 5 = 36 miles per hour

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

Key Concept

A unit rate compares a quantity to one unit of another quantity, often calculated by dividing the total amount by the total time or number of items.

Question 45Question →

A logistics officer is reviewing purchase orders for safety equipment. Match each bulk purchase scenario to its correct unit price.

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Answer

The correct pairings are determined by dividing the total cost by the quantity for each scenario, yielding the unit rates of

5,

3,

6, and

4 respectively.

Each correct match is found by identifying the unit price using the formula: Total Cost ÷ Quantity.

Step-by-Step Solution

1

Calculate the unit price for the protective goggles.

60 / 12 =

5 per pair.

Unit price is total cost divided by quantity.

2

Calculate the unit price for the work gloves.

90 / 30 =

3 per pair.

Unit price is total cost divided by quantity.

3

Calculate the unit price for the heavy-duty flashlights.

48 / 8 =

6 per flashlight.

Unit price is total cost divided by quantity.

4

Calculate the unit price for the high-visibility vests.

100 / 25 =

4 per vest.

Unit price is total cost divided by quantity.

Key Concept

Calculating unit rates by dividing the total amount by the number of units.

Question 46Question →

A motor pool tracks the inventory of tactical vehicles and support vehicles assigned to different units. Match each unit's vehicle inventory to its equivalent simplified ratio of tactical vehicles to support vehicles.

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Answer

Alpha matches to

3:1

, Bravo matches to

3:4

, Charlie matches to

7:3

, and Delta matches to

7:5

.

Each correct pairing is identified by writing the initial inventory as a part-to-part ratio in the specified order (tactical to support), then reducing the numbers by dividing both terms by their greatest common divisor.

Step-by-Step Solution

1

Set up the initial ratio for Alpha Unit as tactical to support.

45:15

The question specifically asks for the ratio of tactical vehicles to support vehicles in that exact order.

2

Simplify the ratio for Alpha Unit by finding the greatest common divisor (GCD).

3:1

The GCD of

45

and

15

is

15

.

45 \div 15 = 3

and

15 \div 15 = 1

.

3

Set up and simplify the ratio for Bravo Unit (

24

to

32

).

3:4

The GCD of

24

and

32

is

8

.

24 \div 8 = 3

and

32 \div 8 = 4

.

4

Set up and simplify the ratio for Charlie Unit (

42

to

18

).

7:3

The GCD of

42

and

18

is

6

.

42 \div 6 = 7

and

18 \div 6 = 3

.

5

Set up and simplify the ratio for Delta Unit (

35

to

25

).

7:5

The GCD of

35

and

25

is

5

.

35 \div 5 = 7

and

25 \div 5 = 5

.

Key Concept

To find an equivalent simplified ratio, establish the correct part-to-part order and divide both quantities by their greatest common divisor.

Estimated Time:1m 30s

Question 47Question →

Connect the unsimplified exponential expressions in the left column to their corresponding reduced forms in the right column, assuming $x$ and $y$ are non-zero constants.

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Answer

Each unsimplified expression matches to a unique reduced form by carefully applying the power, product, and quotient rules of exponents.

Each expression is correctly simplified by systematically applying the product, quotient, and power rules of exponents, paying special attention to numerical coefficients and negative signs.

Step-by-Step Solution

1

Simplify the first expression by expanding the numerator's power and applying the quotient rule.

\frac{(2x^{-3}y^2)^3}{4x^{-5}y^{-1}} = \frac{8x^{-9}y^6}{4x^{-5}y^{-1}} = 2x^{-4}y^7 = \frac{2y^7}{x^4}

The power of a product rule applies exponents to coefficients, while the quotient rule requires subtracting denominator exponents from numerator exponents.

2

Simplify the second expression by resolving the negative outer exponent and multiplying the coefficients and variables.

(3x^4y^{-2})^{-2} \cdot (9x^0y^5) = (3^{-2}x^{-8}y^4) \cdot (9 \cdot 1 \cdot y^5) = \frac{1}{9}x^{-8}y^4 \cdot 9y^5 = x^{-8}y^9 = \frac{y^9}{x^8}

Applying negative exponents to the coefficient creates a fraction, and recognizing that

x^0 = 1

simplifies the second term before multiplication.

3

Simplify the third expression by resolving the fraction inside first, then applying the external negative exponent.

\left( \frac{15x^3y^{-4}}{5x^{-2}y^3} \right)^{-1} = (3x^5y^{-7})^{-1} = 3^{-1}x^{-5}y^7 = \frac{y^7}{3x^5}

Simplifying the internal expression first reduces algebraic errors before applying the reciprocal operation for the -1 exponent.

4

Simplify the fourth expression by multiplying the numerator terms, then dividing by the denominator.

\frac{6x^{-1}y^4 \cdot 2x^5y^{-2}}{3x^8y^{-3}} = \frac{12x^4y^2}{3x^8y^{-3}} = 4x^{4-8}y^{2-(-3)} = 4x^{-4}y^5 = \frac{4y^5}{x^4}

Combining like terms in the numerator through multiplication organizes the expression for a straightforward application of the quotient rule.

Key Concept

Operations with Variable Exponents

Question 48Question →

Match each travel scenario to the correct amount of time it takes to complete the journey.

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Answer

The delivery van takes 3 hours, the cargo ship takes 4 hours, the marathon runner takes 2 hours, and the commuter train takes 5 hours.

Each correct match is determined by taking the stated distance in miles and dividing it by the average speed in miles per hour (

t = \frac{d}{r}

). Because the difficulty is intentionally basic, all divisions result in clean, whole-number integer hours.

Step-by-Step Solution

1

Identify the given distance (

d

) and rate (

r

) for each scenario.

Van: (

150

mi,

50

mph), Ship: (

80

mi,

20

mph), Runner: (

16

mi,

8

mph), Train: (

300

mi,

60

mph).

These variables are required to apply the fundamental distance formula.

2

Apply the formula for calculating time:

t = \frac{d}{r}

.

Van:

\frac{150}{50} = 3

. Ship:

\frac{80}{20} = 4

. Runner:

\frac{16}{8} = 2

. Train:

\frac{300}{60} = 5

.

Dividing the total distance traveled by the constant rate yields the total elapsed time.

Key Concept

Calculating elapsed time using the distance-rate-time relationship (

t = \frac{d}{r}

).

Question 49Question →

Match each polynomial expression with its completely factored form.

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Answer

The correct matches pair each polynomial with its fully factored equivalent using difference of squares, greatest common factor, and basic trinomial factoring rules.

Each correct pairing applies fundamental algebraic factoring rules: recognizing special binomial products like the difference of squares, pulling out the greatest common factor (GCF) from terms, and finding appropriate numerical binomial factors for quadratic trinomials.

Step-by-Step Solution

1

Analyze each polynomial to determine the applicable factoring method.

Identified difference of squares, GCF, and two basic quadratic trinomials.

Different polynomial structures require specific factoring techniques.

2

Factor the difference of squares:

x^2 - 25

.

Matched with

(x - 5)(x + 5)

.

The pattern

a^2 - b^2

factors into

(a - b)(a + b)

.

3

Factor out the greatest common factor for

4x^2 + 8x

.

Matched with

4x(x + 2)

.

Both terms share a maximum common factor of

4x

.

4

Factor the trinomial

x^2 + 7x + 10

.

Matched with

(x + 2)(x + 5)

.

The numbers

2

and

5

multiply to positive

10

and add to positive

7

.

5

Factor the trinomial

x^2 - 3x - 10

.

Matched with

(x - 5)(x + 2)

.

The numbers

-5

and

2

multiply to negative

10

and add to negative

3

.

Key Concept

Factoring Polynomials

Question 50Question →

Match each unsimplified algebraic expression with its fully simplified equivalent form. Assume all variables represent non-zero real numbers.

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Answer

Each unsimplified expression matches exactly one fully simplified form by applying the product, quotient, and power rules of exponents.

The correct matches are found by systematically applying the fundamental properties of exponents: adding exponents for multiplication, subtracting for division, multiplying for powers of powers, and converting negative exponents to reciprocals.

Step-by-Step Solution

1

Simplify the product of powers expression: (3x^-2y^4)(5x^6y^-3)

15x^4y

Multiply numerical coefficients together. Add exponents for variables with the same base according to the product rule.

2

Simplify the quotient of powers expression: (30x^5y^-2) / (2x^9y^3)

15 / (x^4y^5)

Divide numerical coefficients. Subtract the denominator's exponent from the numerator's exponent. Convert negative exponents to positive by moving them to the denominator.

3

Simplify the power of a product expression: (-2x^3y^-1)^4

(16x^12) / y^4

Raise every factor inside the parentheses to the 4th power. Note that an even power of a negative coefficient yields a positive result.

4

Simplify the expression with an outer negative exponent: (3x^-2 / y^4)^-2

(x^4y^8) / 9

Multiply all inner exponents by the outer exponent -2. Then take the reciprocal of terms with negative exponents to rewrite them with positive exponents.

Key Concept

Operations with Variable Exponents

Question 51Question →

A quartermaster is organizing deployment cargo for four different platoons. Match each platoon's cargo inventory to the corresponding simplified ratio of the first listed item to the second listed item.

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Answer

Platoon Alpha matches to

3:1

, Platoon Bravo matches to

2:3

, Platoon Charlie matches to

3:5

, and Platoon Delta matches to

4:1

.

Each platoon's inventory forms a direct part-to-part ratio that can be simplified by dividing both quantities by their greatest common factor (GCF). Platoon Alpha's

45:15

reduces to

3:1

(GCF is

15

). Platoon Bravo's

24:36

reduces to

2:3

(GCF is

12

). Platoon Charlie's

60:100

reduces to

3:5

(GCF is

20

). Platoon Delta's

120:30

reduces to

4:1

(GCF is

30

).

Step-by-Step Solution

1

Determine the simplified ratio for Platoon Alpha (

45

rations to

15

water).

The greatest common factor is

15

. Dividing both numbers by

15

yields

3:1

.

Simplifying the ratio reveals the baseline relationship between the two quantities.

2

Determine the simplified ratio for Platoon Bravo (

24

batteries to

36

flares).

The greatest common factor is

12

. Dividing both numbers by

12

yields

2:3

.

Both terms must be reduced completely to find the correct matching equivalent.

3

Determine the simplified ratio for Platoon Charlie (

60

sandbags to

100

stakes).

The greatest common factor is

20

. Dividing both numbers by

20

yields

3:5

.

Identifying the highest common divisor ensures the ratio is in its simplest form.

4

Determine the simplified ratio for Platoon Delta (

120

standard rounds to

30

tracer rounds).

The greatest common factor is

30

. Dividing both numbers by

30

yields

4:1

.

Applying the same simplification principle resolves the final pairing.

Key Concept

Simplifying basic part-to-part ratios by identifying and dividing by the greatest common factor.

Estimated Time:1m 30s

Question 52Question →

Match each polygon or quadrilateral with its defining geometric property.

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Answer

Parallelogram matches with 'two pairs of parallel opposite sides', Trapezoid matches with 'exactly one pair of parallel sides', Rhombus matches with 'four sides of equal length', and Hexagon matches with 'exactly six sides'.

Each shape is matched to its fundamental geometric definition based on the number of sides, side lengths, and parallel properties. Knowing these core definitions is essential for classifying polygons and quadrilaterals correctly.

Step-by-Step Solution

1

Analyze the properties of a parallelogram.

It has two pairs of parallel opposite sides.

This is the fundamental geometric definition of a parallelogram.

2

Analyze the properties of a trapezoid.

It is defined by having exactly one pair of parallel sides.

This distinguishes a trapezoid from other quadrilaterals like parallelograms.

3

Analyze the properties of a rhombus.

It is a quadrilateral where all four sides are of equal length.

While it also has parallel sides, the equal side lengths are its distinguishing feature.

4

Identify the number of sides for a hexagon.

A hexagon is a polygon with exactly six sides.

The prefix 'hexa-' denotes six in geometry.

Key Concept

Basic Properties of Quadrilaterals and Polygons

Question 53Question →

Match each quadrilateral with its precise geometric defining properties.

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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 54Question →

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.

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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 55Question →

Pair each algebraic expression with its completely factored equivalent.

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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 56Question →

Match each polynomial expression to its completely factored form.

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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 57Question →

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.

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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 58Question →

Match each polygon description based on its angle properties to the correct geometric shape.

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Answer

Match the

540^\circ

sum to Pentagon, the

120^\circ

interior angle to Regular Hexagon, the

1080^\circ

sum to Octagon, and the equal interior/exterior sums to Quadrilateral.

Each match correctly applies the sum of interior angles formula,

S = (n-2) \times 180^\circ

, or relies on the fact that the sum of exterior angles for any convex polygon is exactly

360^\circ

. Solving for

n

reveals the specific polygon.

Step-by-Step Solution

1

Find the number of sides for a polygon with a

540^\circ

interior angle sum.

n = 5

We apply the formula

(n-2) \times 180 = 540

, which simplifies to

n-2 = 3

.

2

Determine the number of sides for a regular polygon with a

120^\circ

interior angle.

n = 6

The exterior angle is

180^\circ - 120^\circ = 60^\circ

. The number of sides is

360^\circ / 60^\circ = 6

.

3

Find the number of sides for a polygon with a

1080^\circ

interior angle sum.

n = 8

Applying the formula gives

(n-2) \times 180 = 1080

, which simplifies to

n-2 = 6

.

4

Identify the polygon where the interior angle sum equals the exterior angle sum.

n = 4

Since the exterior sum is always

360^\circ

, we set

(n-2) \times 180 = 360

, yielding

n-2 = 2

.

Key Concept

Polygon Interior and Exterior Angle Formulas

Estimated Time:1m 30s

Question 59Question →

Determine the unknown rate or time for each of the following military travel profiles by matching them to their correctly calculated values.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The supply truck travels at

52

mph, the aircraft takes

4

hours, the infantry squad marches at

3

mph, and the destroyer takes

4.5

hours.

Each match is correct because the standard distance relationships (

r = d/t

or

t = d/r

) are applied after standardizing all units. Knots are correctly identified as speed, and minutes are accurately converted into fractions of an hour.

Step-by-Step Solution

1

Identify the given variables and the missing variable for each travel profile.

Two scenarios give distance and time (requiring rate), while two give distance and rate (requiring time).

This determines whether to use the formula

r = d/t

or

t = d/r

.

2

Convert mixed time units into a single fractional or decimal unit of hours.

2

hours

30

minutes becomes

2.5

hours.

2

hours

40

minutes becomes

2 \frac{2}{3}

or

8/3

hours.

Rates expressed in mph require time to be represented entirely in hours before dividing.

3

Calculate the missing values using the appropriate distance formula.

Truck:

130 \div 2.5 = 52

. Aircraft:

1,200 \div 300 = 4

. Squad:

8 \div (8/3) = 3

. Destroyer:

135 \div 30 = 4.5

.

Applying the correct operational relationship yields the final matched answers.

Key Concept

Calculating Rate and Time with Unit Conversions

Question 60Question →

A shipyard damage control team is evaluating four auxiliary water pumps. Match each pump's performance test results to its calculated unit rate in gallons per minute (GPM).

Click a left item, then click its matching right item

Show answer & explanation

Answer

Pump Alpha matches

150

GPM, Pump Bravo matches

140

GPM, Pump Charlie matches

160

GPM, and Pump Delta matches

170

GPM.

Each pump's unit rate is determined by dividing the total gallons removed by the total number of minutes. This yields the uniform rate in gallons per minute (GPM).

Step-by-Step Solution

1

Calculate the unit rate for Pump Alpha.

1,800 \div 12 = 150

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

2

Calculate the unit rate for Pump Bravo.

3,500 \div 25 = 140

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

3

Calculate the unit rate for Pump Charlie.

4,800 \div 30 = 160

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

4

Calculate the unit rate for Pump Delta.

6,800 \div 40 = 170

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

Key Concept

Calculating unit rates by dividing a total quantity by the elapsed time to find the amount per single unit of time.

Answer

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