All practice questions

Question 321Question →

A base housing maintenance team is inspecting a total of $150$ residential units. If $30\%$ of these units require new HVAC filters, calculate the number of units that need new filters.

Fill in the blanks below

The maintenance team needs to replace the HVAC filters in units.

Show answer & explanation

Answer

45

To determine how many units require new filters, find

30\%

of

150

. First, convert

30\%

to its decimal form, which is

0.30

. Next, multiply the total number of units by this decimal:

150 \times 0.30 = 45

. Therefore, exactly

45

units need new filters.

Step-by-Step Solution

1

Convert the given percentage to a decimal equivalent.

30\% = 0.30

Percentages must be expressed as decimals (or fractions) to be used mathematically in multiplication.

2

Multiply the total number of units by the decimal.

150 \times 0.30 = 45

Multiplying the total quantity by the percentage representation yields the exact portion requested.

Key Concept

Calculating a specific percentage of a whole quantity by converting the percent to a decimal and multiplying.

Question 322Question →

What is the measure of one interior angle of a regular octagon?

45^\circ

135^\circ

180^\circ

225^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

The measure of one interior angle of a regular octagon is

135^\circ

.

To find a single interior angle of a regular polygon, use the formula

\frac{(n - 2) \times 180^\circ}{n}

. Substituting

n = 8

for an octagon gives

\frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ

.

Step-by-Step Solution

1

Identify the number of sides (

n

) for an octagon.

n = 8

An octagon is a polygon with 8 sides.

2

Calculate the total sum of the interior angles using the formula

(n - 2) \times 180^\circ

.

(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

This formula determines the combined measure of all interior angles for any polygon.

3

Divide the total sum by the number of sides to find the measure of a single angle.

1080^\circ \div 8 = 135^\circ

In a regular polygon, all interior angles are equal, so dividing the total sum by the number of angles gives the measure of one.

Key Concept

Interior Angles of Regular Polygons

Question 323Question →

A florist purchases $120$ wholesale roses for \ $50.00 and pays an additional \$30.00 for cold shipping. Upon delivery,$ 24 $of the roses are found to be damaged and cannot be sold. If the florist sells the remaining roses in bunches of one dozen ($ 12$ roses) for \$12.50 per bunch, what is the florist's profit percentage on the total investment?

20\%

25\%

100\%

125\%

Show answer & explanation

Answer: $25\%$

Answer

25\%

To find the profit percentage, first calculate the total investment: \

50.00 for the roses plus \$30.00 for shipping equals a total cost of \$80.00. Next, find the sellable inventory by subtracting the

24

damaged roses from the original

120

, leaving

96

roses. Since they are sold by the dozen, divide

96

by

12

to get

8

bunches. Selling

8

bunches at \$12.50 each generates \$100.00 in total revenue. Subtract the total cost from the revenue to find the profit: \$100.00 - \$80.00 = \$20.00. Finally, divide the profit by the total cost to find the percentage: \$20.00 \div \$80.00 = 0.25

, which is

25\%

.

Step-by-Step Solution

1

Calculate the total cost (investment).

\$50.00 + \$30.00 = \$80.00

Profit percentage must be calculated using the entire cost basis, including overhead like shipping.

2

Determine the number of sellable bunches.

120 - 24 = 96

sellable roses.

96 \div 12 = 8

bunches.

Damaged goods cannot be sold and must be subtracted from inventory before calculating revenue.

3

Calculate the total revenue.

$

8 \times \$12.50 = \$100.00

Multiply the number of sellable units by the price per unit.

4

Calculate the profit and the profit percentage.

Profit = \$100.00 - \$80.00 = \$20.00. Percentage = (\$20.00 \div \$80.00) \times 100\% = 25\%.

Profit is revenue minus cost. Profit percentage is the profit divided by the total cost.

Key Concept

Calculating percentage profit requires finding the comprehensive total cost, determining actual total revenue (accounting for lost inventory), subtracting cost from revenue to find the profit amount, and then dividing that profit strictly by the total original cost.

Estimated Time:1m 30s

Question 324Question →

The area of a rectangular board is given by the algebraic expression $4x^2 - 8x - 5$ . Which of the following shows the dimensions of the board expressed as a product of two binomials?

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

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

(4x + 1)(x - 5)

(2x - 1)(2x - 5)

Show answer & explanation

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

Answer

The correct dimensions are given by the factored form

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

.

To factor

4x^2 - 8x - 5

, find two values that multiply to

-20

(which is

4 \times -5

) and add to

-8

. These values are

-10

and

2

. Splitting the middle term gives

4x^2 - 10x + 2x - 5

. Factoring by grouping gives

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

, which then factors out to

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

.

Step-by-Step Solution

1

Identify the coefficients for the quadratic expression

ax^2 + bx + c

.

a = 4

,

b = -8

, and

c = -5

.

To factor a non-monic quadratic trinomial by grouping, we first need to determine the product of

a

and

c

.

2

Find two numbers that multiply to the product of

a

and

c

(

-20

) and add to

b

(

-8

).

The two numbers are

-10

and

2

.

These numbers allow us to split the middle term (

-8x

) into two parts that can be factored by grouping.

3

Rewrite the original expression by splitting the middle term.

4x^2 - 10x + 2x - 5

This sets up the expression for factoring by grouping.

4

Factor the greatest common factor (GCF) from the first two terms and the last two terms.

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

Extracting the GCF reveals a common binomial factor in both groups.

5

Factor out the common binomial factor.

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

This provides the completely factored form of the original area expression.

Key Concept

Factoring Non-Monic Quadratic Trinomials

Question 325Question →

Simplify the following algebraic expression completely. Assume all variables represent non-zero real numbers.

\left( \frac{-3 x^{-2} y^5}{z^{-3}} \right)^{-2} \times \left( \frac{6 x^4 z^{-1}}{y^{-3}} \right)^3

Which of the following represents the fully simplified expression?

1944 x^8 y^{19} z^3

24 x^3 y^3 z^7

\frac{24 x^{16}}{y z^9}

-\frac{18 x^{16}}{y z^9}

Show answer & explanation

Answer: $\frac{24 x^{16}}{y z^9}$

Answer

The fully simplified expression is

\frac{24 x^{16}}{y z^9}

.

The correct answer is found by first distributing the outer exponents to all terms inside the parentheses (including coefficients), which yields (1/9) * x^4 * y^(-10) * z^(-6) and 216 * x^12 * z^(-3) * y^9. Multiplying these expressions by combining the coefficients and adding the exponents of like variables gives 24 * x^16 * y^(-1) * z^(-9). Rewriting with positive exponents moves y and z to the denominator, resulting in the correct simplified fraction.

Step-by-Step Solution

1

Resolve negative exponents in the denominators of both fractions to bring the variables to the numerators.

\left( -3 x^{-2} y^5 z^3 \right)^{-2} \times \left( 6 x^4 z^{-1} y^3 \right)^3

Moving negative exponents from the denominator to the numerator simplifies the terms before applying the outer powers.

2

Apply the power of a product rule to the first expression, multiplying each inner exponent by the outer exponent of -2.

(-3)^{-2} x^{(-2)(-2)} y^{(5)(-2)} z^{(3)(-2)} = \frac{1}{9} x^4 y^{-10} z^{-6}

The power of a product rule states that (abc)^x = a^x b^x c^x, and the power of a power rule states that (a^x)^y = a^(xy).

3

Apply the power of a product rule to the second expression, multiplying each inner exponent by the outer exponent of 3.

6^3 x^{(4)(3)} z^{(-1)(3)} y^{(3)(3)} = 216 x^{12} z^{-3} y^9

Distributing the cube to the coefficient and each variable's exponent prepares the expression for multiplication.

4

Multiply the resulting expressions together by grouping coefficients and like bases.

\left( \frac{1}{9} \times 216 \right) \left( x^4 \cdot x^{12} \right) \left( y^{-10} \cdot y^9 \right) \left( z^{-6} \cdot z^{-3} \right)

The commutative property of multiplication allows us to group similar terms to apply the product of powers rule.

5

Simplify the groups by multiplying coefficients and adding the exponents of like bases.

24 x^{16} y^{-1} z^{-9}

The product of powers rule states that a^x * a^y = a^(x+y).

6

Rewrite the expression with only positive exponents.

\frac{24 x^{16}}{y z^9}

Standard simplified form requires all exponents to be positive, so terms with negative exponents are moved to the denominator.

Key Concept

Applying the power of a product, power of a power, quotient of powers, and negative exponent rules to simplify complex algebraic expressions.

Question 326Question →

A dispatch rider is transporting urgent documents between two military installations. The rider travels on a paved highway at a constant speed of $65\text{ mph}$ for $1\text{ hour and } 12\text{ minutes}$ . Upon reaching a rural sector, the rider continues on an unpaved route at a steady speed of $42\text{ mph}$ for $35\text{ minutes}$ . What is the total distance, in miles, the dispatch rider traveled?

Show answer & explanation

Answer: 102.5

Answer

102.5

To find the total distance, calculate the distance for each leg of the trip separately by ensuring time is expressed in hours. For the first leg,

1\text{ hour and } 12\text{ minutes}

is

1.2\text{ hours}

. Multiplying the

65\text{ mph}

rate by

1.2\text{ hours}

gives

78\text{ miles}

. For the second leg,

35\text{ minutes}

is

\frac{35}{60}

or

\frac{7}{12}\text{ of an hour}

. Multiplying the

42\text{ mph}

rate by

\frac{7}{12}\text{ hours}

gives

24.5\text{ miles}

. Adding these two distances together (

78 + 24.5

) gives a total of

102.5\text{ miles}

.

Step-by-Step Solution

1

Convert the highway travel time into hours.

1\text{ hour and } 12\text{ minutes} = 1 + \frac{12}{60}\text{ hours} = 1.2\text{ hours}

.

The speed is given in miles per hour, so time must be converted to hours to match units.

2

Calculate the distance traveled on the highway.

65\text{ mph} \times 1.2\text{ hours} = 78\text{ miles}

.

Applying the formula: Distance = Rate

\times

Time.

3

Convert the unpaved route travel time into hours.

35\text{ minutes} = \frac{35}{60}\text{ hours} = \frac{7}{12}\text{ hours}

.

Time must be in hours for consistency with the speed unit.

4

Calculate the distance traveled on the unpaved route.

42\text{ mph} \times \frac{7}{12}\text{ hours} = 24.5\text{ miles}

.

Applying the formula: Distance = Rate

\times

Time.

5

Calculate the total distance.

78\text{ miles} + 24.5\text{ miles} = 102.5\text{ miles}

.

The total journey distance is the sum of the individual distances from each leg.

Key Concept

Calculating total distance for a multi-leg journey by accurately converting fractional times to hours and applying the distance formula.

Question 327Question →

A supply truck initially carries $78 \frac{3}{4}$ tons of cargo. The driver unloads $\frac{1}{3}$ of the total cargo at the first outpost. At the second outpost, the driver unloads $\frac{3}{5}$ of the *remaining* cargo.

How many tons of cargo are left on the truck after both deliveries are made?

Fill in the blanks below

Remaining cargo: tons

Show answer & explanation

Answer

21 tons

To find the final amount efficiently, multiply the original total by the fractions representing what remains on the truck after each stop. After dropping off

\frac{1}{3}

, exactly

\frac{2}{3}

of the load remains. After dropping off

\frac{3}{5}

of that remainder, exactly

\frac{2}{5}

of it remains. We calculate the final remaining amount as:

78 \frac{3}{4} \times \frac{2}{3} \times \frac{2}{5}

. Converting to an improper fraction, this becomes

\frac{315}{4} \times \frac{2}{3} \times \frac{2}{5}

. Multiplying the two remaining fractions first gives

\frac{4}{15}

. Finally,

\frac{315}{4} \times \frac{4}{15} = \frac{315}{15} = 21

tons.

Step-by-Step Solution

1

Calculate the amount of cargo left after the first delivery.

52 \frac{1}{2}

tons remain.

Since

\frac{1}{3}

of the cargo is unloaded,

1 - \frac{1}{3} = \frac{2}{3}

of the original cargo remains. Multiplying the original amount

\frac{315}{4}

by

\frac{2}{3}

yields

\frac{105}{2}

or

52 \frac{1}{2}

tons.

2

Calculate the amount of cargo left after the second delivery.

21 tons remain.

The driver unloads

\frac{3}{5}

of the remaining cargo, meaning

1 - \frac{3}{5} = \frac{2}{5}

of it is still on the truck. Multiplying the remaining

\frac{105}{2}

tons by

\frac{2}{5}

gives exactly 21 tons.

Key Concept

Calculating successive fractional parts by determining the remainder at each step.

Question 328Question →

A cafeteria chef is purchasing black pepper. A $4$ -ounce jar of black pepper costs $\$8.00$ . A $1$ -pound container of the same black pepper costs $\$16.00$ . How much money is saved per ounce by purchasing the $1$ -pound container instead of the $4$ -ounce jar?

\$1.90

\$14.00

\$1.00

\$0.50

Show answer & explanation

Answer: \$1.00

Answer

The correct answer is

\$1.00

saved per ounce.

The unit price of the

4

-ounce jar is calculated by dividing its cost by its size:

\$8.00 \div 4 \text{ ounces} = \$2.00

per ounce. Next, convert the

1

-pound container to ounces, knowing that

1 \text{ pound} = 16 \text{ ounces}

. The unit price of the

1

-pound container is

\$16.00 \div 16 \text{ ounces} = \$1.00

per ounce. The savings per ounce is found by subtracting the lower unit price from the higher unit price:

\$2.00 - \$1.00 = \$1.00

.

Step-by-Step Solution

1

Calculate the price per ounce for the

4

-ounce jar.

\$8.00 \div 4 = \$2.00

per ounce

To compare prices, we need the unit price of the first option.

2

Convert the weight of the

1

-pound container into ounces.

$1 \text{ pound} = 16 \text{ ounces}

Unit prices must be in the same measurement unit (ounces) to be compared.

3

Calculate the price per ounce for the

1

-pound container.

\$16.00 \div 16 = \$1.00

per ounce

This establishes the unit price of the second option for comparison.

4

Subtract the lower unit price from the higher unit price to find the savings.

\$2.00 - \$1.00 = \$1.00

per ounce

The question asks for the exact amount of money saved per ounce.

Key Concept

Comparing unit prices requires converting different units of measurement into a common unit (like ounces) before dividing cost by quantity.

Estimated Time:45s

Question 329Question →

What is the completely factored form of the polynomial $3x^2 - 3x - 36$ ?

3(x - 4)(x + 3)

3(x + 4)(x - 3)

(3x - 12)(x + 3)

3(x - 6)(x + 2)

Show answer & explanation

Answer: $3(x - 4)(x + 3)$

Answer

The completely factored form is

3(x - 4)(x + 3)

.

The correct answer accurately extracts the greatest common factor of

3

first, and then properly factors the resulting quadratic

x^2 - x - 12

into

(x - 4)(x + 3)

.

Step-by-Step Solution

1

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

3x^2 - 3x - 36

.

3(x^2 - x - 12)

Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor.

2

Find two numbers that multiply to the constant term (

-12

) and add up to the coefficient of the middle term (

-1

).

The numbers are

-4

and

3

, because

-4 \times 3 = -12

and

-4 + 3 = -1

.

These numbers are required to construct the binomial factors of the quadratic expression.

3

Write the final completely factored expression by combining the GCF and the two binomial factors.

3(x - 4)(x + 3)

This represents the full decomposition of the original polynomial into its simplest multiplicative components.

Key Concept

Factoring out a greatest common factor (GCF) followed by factoring a basic quadratic trinomial.

Question 330Question →

A catering manager is purchasing a commercial blender. The regular price of the blender is $\$400$ . The store offers a $20\%$ discount. If a $5\%$ sales tax is applied to the discounted price, what is the final cost of the blender?

\$399

\$336

\$340

\$84

Show answer & explanation

Answer: $\$336$

Answer

The final cost is

\$336

.

To solve this problem, first determine the discount amount by multiplying the original price by the discount percentage:

\$400 \times 0.20 = \$80

. Subtract this from the original price to find the discounted sale price:

\$400 - \$80 = \$320

. Next, calculate the

5\%

sales tax on this new price:

\$320 \times 0.05 = \$16

. Finally, add the tax to the sale price to find the total cost:

\$320 + \$16 = \$336

.

Step-by-Step Solution

1

Calculate the discount amount.

\$400 \times 0.20 = \$80

To determine how much money is saved from the

20\%

discount.

2

Calculate the sale price before tax.

\$400 - \$80 = \$320

To find the price that is actually subject to the sales tax.

3

Calculate the sales tax amount.

\$320 \times 0.05 = \$16

To determine the additional amount owed for the

5\%

tax on the sale price.

4

Calculate the final total cost.

\$320 + \$16 = \$336

To find the total amount the manager will pay at checkout.

Key Concept

Calculating final sale prices by applying percentage discounts followed by sales tax.

Alternative Method

You can calculate the final price in a single step using multipliers. A

20\%

discount means you pay

80\%

of the price (

0.80

). A

5\%

tax means you pay

105\%

of that amount (

1.05

). Chaining these gives:

\$400 \times 0.80 \times 1.05 = \$336

.

Estimated Time:1m 0s

Question 331Question →

Match each polynomial expression with 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 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 332Question →

A disaster relief team is transporting supplies to a stricken area. They first drive their trucks on a paved highway at a constant speed of $52\text{ miles per hour}$ for $1\text{ hour and } 15\text{ minutes}$ . They then transition to rough off-road terrain, where they travel at a steady speed of $18\text{ miles per hour}$ for $40\text{ minutes}$ to reach their final destination. What is the total distance traveled by the team?

65\text{ miles}

67\text{ miles}

77\text{ miles}

4,620\text{ miles}

Show answer & explanation

Answer: $77\text{ miles}$

Answer

The correct total distance is

77\text{ miles}

.

The total distance is the sum of the distances from both legs of the trip. The first leg is 1 hour and 15 minutes (1.25 hours) at 52 mph, which equals 65 miles. The second leg is 40 minutes (2/3 of an hour) at 18 mph, which equals 12 miles. Adding these gives a total of 77 miles.

Step-by-Step Solution

1

Convert the travel time for the first leg into hours.

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

.

Speed is given in miles per hour, so time must be converted into hours to correctly calculate the distance.

2

Calculate the distance traveled during the first leg.

52\text{ mph} \times 1.25\text{ hours} = 65\text{ miles}

.

Distance equals rate multiplied by time (

D = R \times T

).

3

Convert the travel time for the second leg into hours.

40\text{ minutes} = \frac{40}{60}\text{ hours} = \frac{2}{3}\text{ hours}

.

Consistent units are required for the second leg's calculation.

4

Calculate the distance traveled during the second leg.

18\text{ mph} \times \frac{2}{3}\text{ hours} = 12\text{ miles}

.

Using the

D = R \times T

formula for the off-road section.

5

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

65\text{ miles} + 12\text{ miles} = 77\text{ miles}

.

The question asks for the total combined distance traveled by the team to reach their destination.

Key Concept

Calculating total distance for multi-leg journeys by applying the

D = R \times T

formula with necessary time unit conversions.

Estimated Time:1m 30s

Question 333Question →

A civil engineer is designing a rectangular water reservoir. The length of the reservoir is represented by the polynomial $(4x - 3)$ meters, and the cross-sectional area of the side wall is represented by $(2x^2 + 5x - 1)$ square meters. If the total volume of the reservoir is calculated by multiplying its length by its cross-sectional area, which of the following polynomials represents the total volume in cubic meters?

8x^3 + 14x^2 - 19x + 3

8x^3 + 14x^2 - 19x - 3

8x^3 + 20x^2 - 4x - 3

8x^3 + 16x^2 - 21x + 3

Show answer & explanation

Answer: $8x^3 + 14x^2 - 19x + 3$

Answer

The correct polynomial representing the volume is

8x^3 + 14x^2 - 19x + 3

.

The correct expression is found by properly distributing both terms of the binomial

(4x - 3)

to all three terms of the trinomial

(2x^2 + 5x - 1)

, maintaining correct signs, and accurately combining the resulting like terms.

Step-by-Step Solution

1

Set up the multiplication of the two polynomials.

V = (4x - 3)(2x^2 + 5x - 1)

The problem states that the volume is calculated by multiplying the length by the cross-sectional area.

2

Distribute the first term of the binomial,

4x

, to every term in the trinomial.

4x(2x^2) + 4x(5x) + 4x(-1) = 8x^3 + 20x^2 - 4x

The distributive property requires multiplying every term in the second polynomial by the first term of the first polynomial.

3

Distribute the second term of the binomial,

-3

, to every term in the trinomial.

-3(2x^2) - 3(5x) - 3(-1) = -6x^2 - 15x + 3

Carefully apply the negative sign to all terms during multiplication, noting that a negative times a negative yields a positive.

4

Combine the results and group the like terms.

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

Like terms (terms with the exact same variable configuration) must be grouped together to simplify the expression.

5

Perform the final addition and subtraction to simplify completely.

8x^3 + 14x^2 - 19x + 3

This yields the fully simplified polynomial in standard form.

Key Concept

Multiplying a binomial by a trinomial using the Distributive Property

Question 334Question →

A quartermaster is evaluating different vendors to purchase AA batteries. Calculate the price per battery for each option and arrange the vendors in order from the lowest unit price to the highest unit price.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Vendor Delta, Vendor Bravo, Vendor Alpha, Vendor Charlie

The correct sequence is found by evaluating the cost per single battery for each vendor. By dividing the total cost by the quantity provided, we find that Vendor Delta offers the lowest rate at

\$0.70

per battery. Vendor Bravo follows at

\$0.80

per battery, Vendor Alpha at

\$0.90

per battery, and Vendor Charlie is the most expensive at

\$1.25

per battery. Arranging these calculated unit prices from smallest to largest provides the correct order.

Step-by-Step Solution

1

Calculate the unit price for Vendor Alpha.

\$9.00 \div 10 = \$0.90

per battery.

To find the cost of a single battery, divide the total cost by the quantity.

2

Calculate the unit price for Vendor Bravo.

\$9.60 \div 12 = \$0.80

per battery.

To find the cost of a single battery, divide the total cost by the quantity.

3

Calculate the unit price for Vendor Charlie.

\$2.50 \div 2 = \$1.25

per battery.

To find the cost of a single battery, divide the total cost by the quantity.

4

Calculate the unit price for Vendor Delta.

\$21.00 \div 30 = \$0.70

per battery.

To find the cost of a single battery, divide the total cost by the quantity.

5

Order the calculated unit prices from lowest to highest.

\$0.70

(Delta) <

\$0.80

(Bravo) <

\$0.90

(Alpha) <

\$1.25

(Charlie)

This establishes the correct sequence from the most cost-effective to the least cost-effective option.

Key Concept

Unit price is calculated by dividing the total price by the number of units. Comparing these prices reveals the lowest cost per item.

Question 335Question →

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 336Question →

A Coast Guard cutter is navigating to a waypoint that is $135$ miles away. If the vessel travels at a steady rate of $30$ miles per hour, how much time will it take to reach the waypoint?

4

hours and

5

minutes

7

hours and

30

minutes

4

hours and

30

minutes

4

hours and

50

minutes

Show answer & explanation

Answer: $4$ hours and $30$ minutes

Answer

The correct answer is

4

hours and

30

minutes.

To find the travel time, divide the total distance by the speed:

135 \div 30 = 4.5

hours. Since

0.5

of an hour is exactly half of

60

minutes, it equals

30

minutes. Thus, the total travel time is

4

hours and

30

minutes.

Step-by-Step Solution

1

Set up the formula for finding time given distance and rate.

Time = \frac{135}{30}

The standard relationship between distance, rate, and time is

Time = \frac{Distance}{Rate}

.

2

Divide the total distance by the rate to find the time in hours.

135 \div 30 = 4.5

hours

Performing the division gives the total travel time in decimal hours.

3

Convert the decimal portion of the hour into minutes.

0.5 \text{ hours} \times 60 \text{ minutes/hour} = 30 \text{ minutes}

Since the options are presented in hours and minutes, the fractional hour must be converted using the factor of

60

minutes per hour.

Key Concept

Calculating Time given Rate and Distance

Estimated Time:45s

Question 337Question →

A combat engineering detachment is mixing a specialized rapid-setting concrete. The dry mix requires sand, gravel, and cement in a ratio of $4:5:2$ by weight. If the engineers need to prepare exactly $3,300$ pounds of the dry mix for a runway repair, how many pounds of gravel are required?

Show answer & explanation

Answer: 1500

Answer

1500

To find the correct amount of gravel, the total weight of

3,300

pounds is divided by the total number of parts in the ratio (

4 + 5 + 2 = 11

) to find the weight of one part (

300

pounds). The gravel's share is 5 parts, so multiplying

300

pounds by 5 yields exactly

1,500

pounds.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

4 + 5 + 2 = 11

total parts

To distribute the total weight proportionally, you first need to know how many equal parts make up the entire mixture.

2

Determine the weight of one single part.

3,300 \div 11 = 300

pounds

Dividing the total weight of the mix by the total number of parts gives the value of one part.

3

Calculate the specific weight required for the gravel.

300 \times 5 = 1,500

pounds

The ratio specifies 5 parts of gravel. Multiplying the weight of one part by 5 gives the total weight of the gravel.

Key Concept

Dividing a total quantity into given ratio parts.

Question 338Question →

An aviation maintenance squadron stocks a specific fastener kit containing standard bolts, lock washers, and retaining nuts in a ratio of $7:4:3$ . If a master storage bin contains a total of $840$ of these combined items in this exact ratio, how many lock washers are in the bin?

Show answer & explanation

Answer: 240

Answer

There are 240 lock washers in the bin.

The total number of items (

840

) is divided by the sum of all ratio parts (

14

) to find the value of a single share (

60

). This share is then multiplied by the specific ratio number corresponding to lock washers (

4

) to find the final quantity (

240

).

Step-by-Step Solution

1

Calculate the total number of parts in the ratio.

7 + 4 + 3 = 14

parts

To determine what fraction of the total bin corresponds to a single proportional unit.

2

Determine the physical number of items represented by one part.

840 \div 14 = 60

items

Dividing the total inventory by the total ratio parts gives the baseline value of a single share.

3

Calculate the specific number of lock washers.

4 \times 60 = 240

lock washers

The ratio specifies there are

4

parts of lock washers, and each part equals

60

items.

Key Concept

Allocating a total known quantity into specific amounts based on a multi-term part-to-part ratio.

Estimated Time:1m 0s

Question 339Question →

Which of the following represents the completely factored form of the algebraic expression $3x^2 - 27$ ?

3(x - 3)(x + 3)

3(x^2 - 9)

3(x - 3)(x - 3)

3x(x - 9)

Show answer & explanation

Answer: $3(x - 3)(x + 3)$

Answer

The completely factored form is

3(x - 3)(x + 3)

.

The expression is completely factored by first extracting the greatest common factor of 3 to get

3(x^2 - 9)

, and then factoring the resulting difference of squares into

(x - 3)(x + 3)

.

Step-by-Step Solution

1

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

3(x^2 - 9)

Both terms share a common factor of 3. Factoring this out simplifies the remaining polynomial.

2

Examine the expression inside the parentheses to see if it can be factored further.

x^2 - 9

is a difference of squares.

The expression

x^2 - 9

follows the pattern

a^2 - b^2

, where

a = x

and

b = 3

.

3

Factor the difference of squares using the standard algebraic pattern

(a - b)(a + b)

.

(x - 3)(x + 3)

Expanding

(x - 3)(x + 3)

yields

x^2 + 3x - 3x - 9

, which simplifies back to

x^2 - 9

.

4

Combine the GCF and the factored binomials to write the final completely factored form.

3(x - 3)(x + 3)

An expression is only completely factored when all of its factors are prime (cannot be factored further).

Key Concept

Factoring by finding a Greatest Common Factor (GCF) followed by factoring a difference of squares.

Question 340Question →

A regional training center is preparing for an incoming class of $600$ recruits. If $45\%$ of these recruits are assigned to the engineering track, how many recruits are in the engineering track?

270

330

27

45

Show answer & explanation

Answer: $270$

Answer

There are

270

recruits in the engineering track.

The correct answer accurately determines the portion of the total by multiplying the total number of recruits (

600

) by the percentage in decimal form (

0.45

), yielding

270

.

Step-by-Step Solution

1

Convert the percentage to a decimal or fraction.

45\% = 0.45

(or

\frac{45}{100}

)

Percentages must be converted into a workable numerical format before performing calculations.

2

Multiply the total number of recruits by the converted percentage.

600 \times 0.45 = 270

To find a specific percentage of a whole quantity, multiply the total by the percentage's decimal or fractional equivalent.

Key Concept

Calculating Percentages of Quantities

Answer

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