All practice questions

Question 2081Question →

A surveyor is marking out a straight property boundary. Another reference line intersects this straight boundary, creating two adjacent angles. If the measure of the first angle is $(5x)^\circ$ and the measure of the second angle is $(4x + 81)^\circ$ , what is the value of $x$ ?

2

11

29

1

Show answer & explanation

Answer: 11

Answer

The value of x is 11.

Since the two adjacent angles form a straight line, they are supplementary and their measures must add up to 180 degrees. Setting up the equation

5x + 4x + 81 = 180

allows us to combine like terms to get

9x + 81 = 180

. Subtracting 81 from both sides gives

9x = 99

, and dividing by 9 results in

x = 11

.

Step-by-Step Solution

1

Identify the geometric relationship between the two angles.

The angles form a linear pair, meaning their sum is

180^\circ

.

Adjacent angles that form a straight line are always supplementary.

2

Set up an algebraic equation.

5x + (4x + 81) = 180

Substitute the given angle expressions into the supplementary angle relationship.

3

Combine the like terms on the left side of the equation.

9x + 81 = 180

Add the coefficients of the x terms (

5x + 4x = 9x

) while keeping the constant separate.

4

Isolate the variable term.

9x = 99

Subtract 81 from both sides of the equation to balance it.

5

Solve for x.

x = 11

Divide both sides by 9 to find the final value.

Key Concept

Supplementary Angles on a Straight Line

Question 2082Question →

An armory receives a shipment of $1,250$ tactical vests. During the initial inspection, $8\%$ of the vests are found to be defective and are removed from the inventory. If $20\%$ of the remaining functional vests are immediately issued to a deploying infantry company, how many vests does the company receive?

23

230

250

350

Show answer & explanation

Answer: 230

Answer

230

The problem requires a two-step percentage calculation on a changing base. First, find the number of defective vests by multiplying the original total by

8\%

(

1,250 \times 0.08 = 100

). Subtracting these from the total leaves

1,150

functional vests. The deploying company receives

20\%

of these remaining functional vests, which is calculated as

1,150 \times 0.20 = 230

vests.

Step-by-Step Solution

1

Calculate the number of defective vests.

1,250 \times 0.08 = 100

defective vests.

You must first determine how many vests are removed from the total inventory before calculating the issued amount.

2

Subtract the defective vests from the original shipment.

1,250 - 100 = 1,150

functional vests.

The

20\%

issue rate applies only to the remaining functional vests, so this new base amount must be established.

3

Calculate

20\%

of the remaining functional vests.

1,150 \times 0.20 = 230

vests.

This determines the final quantity of vests issued to the deploying infantry company.

Key Concept

Calculating sequential percentages on a changing base quantity

Estimated Time:1m 0s

Question 2083Question →

A local hardware store is purchasing a bulk wholesale order of industrial shelving. The distributor offers a $20\%$ volume discount on the order. Because the hardware store pays the invoice in full within 10 days, they receive an additional $5\%$ discount on the reduced price. If the hardware store pays $\$3,420$ for the shelving (before shipping and taxes), what was the original price of the bulk order?

\$3,445

\$4,275

\$4,500

\$4,560

Show answer & explanation

Answer: $\$4,500$

Answer

\$4,500

The correct answer is found by representing the unknown original price as a variable, applying the 20% discount (multiplying by 0.80), and then applying the 5% discount to that new amount (multiplying by 0.95). Setting the resulting expression (0.76 times the original price) equal to the final price of

3,420 and dividing yields an original price of

4,500.

Step-by-Step Solution

1

Set up an equation representing the first discount.

Let the original price be

P

. A

20\%

discount means the price becomes

100\% - 20\% = 80\%

of the original price, or

0.80P

.

The volume discount is applied to the starting original price.

2

Apply the second successive discount to the result of the first.

An additional

5\%

discount on the reduced price means the hardware store pays

100\% - 5\% = 95\%

of that reduced price. This gives

0.95 \times 0.80P = 0.76P

.

Successive discounts are calculated based on the new, already-discounted price, not the original price.

3

Solve for the original price

P

.

Set the expression equal to the final paid amount:

0.76P = 3420

. Divide both sides by

0.76

to find

P = \$4,500

.

This isolates

P

to determine the starting value before any discounts were applied.

Key Concept

Calculating the original price after successive discounts by working backward using algebraic representation.

Question 2084Question →

An architect is designing a rectangular layout where the area is represented by the algebraic expression $4y^2 + 12y - 40$ . To determine the possible dimensions of the layout in terms of $y$ , the architect needs to rewrite this expression in its completely factored form. Which of the following represents the completely factored expression?

4(y + 5)(y - 2)

4(y - 5)(y + 2)

(4y - 8)(y + 5)

4(y + 5)(y + 2)

Show answer & explanation

Answer: $4(y + 5)(y - 2)$

Answer

The completely factored expression is

4(y + 5)(y - 2)

.

The correct answer completely factors out the GCF of 4 from all terms and correctly factors the remaining trinomial

y^2 + 3y - 10

into

(y + 5)(y - 2)

.

Step-by-Step Solution

1

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

4y^2 + 12y - 40

.

4(y^2 + 3y - 10)

Factoring out the GCF simplifies the polynomial and makes the remaining trinomial easier to factor.

2

Find two integers that multiply to the constant term (-10) and add to the coefficient of the middle term (3).

The correct integers are 5 and -2.

These integers are required to construct the binomial factors of the simplified trinomial

y^2 + 3y - 10

.

3

Write the trinomial as a product of two binomials using the integers found, and combine them with the GCF.

4(y + 5)(y - 2)

This represents the original expression fully broken down into its prime polynomial factors.

Key Concept

Factoring Polynomials by Extracting the GCF and Grouping

Question 2085Question →

Angles $\angle P$ and $\angle Q$ are supplementary angles. The measure of $\angle P$ is given by the expression $(7x + 20)^\circ$ , and the measure of $\angle Q$ is given by $(3x - 10)^\circ$ . What is the measure of the angle that is complementary to $\angle Q$ ?

49^\circ

139^\circ

73^\circ

29^\circ

Show answer & explanation

Answer: $49^\circ$

Answer

The complementary angle measures

49^\circ

.

The sum of supplementary angles is

180^\circ

. Setting

(7x + 20) + (3x - 10) = 180

yields

10x + 10 = 180

, which simplifies to

10x = 170

, meaning

x = 17

. Substituting

x = 17

into the expression for

\angle Q

gives

3(17) - 10 = 41^\circ

. The complement of an angle sums to

90^\circ

, so the complementary angle is

90^\circ - 41^\circ = 49^\circ

.

Step-by-Step Solution

1

Set up an equation based on the definition of supplementary angles.

(7x + 20) + (3x - 10) = 180

Supplementary angles have a sum of

180^\circ

.

2

Combine like terms and solve for

x

.

10x + 10 = 180 \Rightarrow 10x = 170 \Rightarrow x = 17

This isolates the variable

x

to determine its numerical value.

3

Substitute the value of

x

into the expression for

\angle Q

.

\angle Q = 3(17) - 10 = 51 - 10 = 41^\circ

This finds the exact degree measure of

\angle Q

.

4

Calculate the complement of

\angle Q

.

90^\circ - 41^\circ = 49^\circ

Complementary angles have a sum of

90^\circ

.

Key Concept

Solving multi-step algebraic equations derived from the properties of supplementary and complementary angles.

Question 2086Question →

A maintenance supervisor is ordering floor sealant for a large building renovation. The sealant is available in two sizes: a $2$ -gallon jug priced at $\$18.00$ , and a $1$ -quart bottle priced at $\$3.25$ . If the renovation requires exactly $8$ gallons of sealant, how much money is saved in total by purchasing the $2$ -gallon jugs instead of the $1$ -quart bottles?

\$4.00

\$46.00

\$32.00

\$61.60

Show answer & explanation

Answer: $\$32.00$

Answer

\$32.00

To find the total savings, calculate the total cost for each option. For the

2

-gallon jugs, you need

4

jugs to reach

8

gallons (

4 \times \$18.00 = \$72.00

). For the

1

-quart bottles, first convert

8

gallons to quarts (

8 \times 4 = 32

quarts). You need

32

bottles (

32 \times \$3.25 = \$104.00

). The difference is

\$104.00 - \$72.00 = \$32.00

.

Step-by-Step Solution

1

Calculate the total cost of purchasing 8 gallons using the 2-gallon jugs.

\$72.00

The project needs 8 gallons, which requires 4 jugs (

8 \div 2

). At

\$18.00

per jug, the cost is

4 \times \$18.00 = \$72.00

.

2

Determine how many 1-quart bottles are needed to make 8 gallons.

32 bottles

There are 4 quarts in 1 gallon, so 8 gallons is equal to 32 quarts (

8 \times 4

).

3

Calculate the total cost of purchasing 32 quart bottles.

\$104.00

At

\$3.25

per bottle, the cost is

32 \times \$3.25 = \$104.00

.

4

Calculate the total savings between the two options.

\$32.00

Subtract the lower bulk cost from the higher individual cost:

\$104.00 - \$72.00 = \$32.00

.

Key Concept

Unit Price Comparisons

Question 2087Question →

A base landscaping crew needs to purchase concentrated herbicide. Vendor A sells a $5$ -gallon drum of the herbicide for $\$140.00$ . Vendor B sells the same herbicide in $2$ -quart bottles for $\$15.00$ each. What is the difference in price per gallon, in dollars, between the two vendors?

Show answer & explanation

Answer: 2

Answer

The difference in price per gallon between the two vendors is 2 dollars.

By converting both options to a common unit price of dollars per gallon, we find Vendor A charges

28.00 per gallon and Vendor B charges

30.00 per gallon. The difference between these unit prices is

30.00 -

28.00 = $2.00.

Step-by-Step Solution

1

Calculate the price per gallon for Vendor A.

\$140.00 \div 5 \text{ gallons} = \$28.00 \text{ per gallon}

To compare prices accurately, both must be expressed as a unit rate (price per gallon).

2

Convert the volume of Vendor B's bottles into gallons to find how many bottles equal one gallon.

Since there are

4 \text{ quarts}

in

1 \text{ gallon}

, a

2

-quart bottle is

0.5 \text{ gallons}

. Therefore, it takes

2

bottles to make

1 \text{ gallon}

.

Unit prices must be in the same measurement (gallons) for a valid comparison.

3

Calculate the price per gallon for Vendor B.

\$15.00 \times 2 \text{ bottles} = \$30.00 \text{ per gallon}

This establishes the unit rate for Vendor B.

4

Find the difference between the two unit prices.

\$30.00 - \$28.00 = \$2.00

The question asks for the numeric difference in price per gallon.

Key Concept

Comparing unit prices by converting to a common unit of measure.

Question 2088Question →

Read the scenario and calculate the final remaining amount. Enter only the numerical value in the blank.

Fill in the blanks below

A naval shipyard crew is assigned to apply a specialized anti-corrosion coating to

3,200

square feet of a submarine's hull. During the first week of the project, the crew successfully coats

45\%

of the total area. In the second week, due to bad weather, they only manage to coat

25\%

of the remaining uncoated area. Based on this progress, exactly square feet of the hull still need to be coated.

Show answer & explanation

Answer

1320

The correct calculation requires adjusting the base for the second percentage. First, find

45\%

of the original

3,200

square feet, which is

1,440

square feet. This leaves

1,760

square feet uncoated. The second week's progress is

25\%

of that remaining

1,760

square feet, equaling

440

square feet. Subtracting this from the

1,760

gives a final uncoated area of

1,320

square feet.

Step-by-Step Solution

1

Calculate the area coated during the first week.

3,200 \times 0.45 = 1,440

square feet.

To find the initial amount completed, multiply the total area by the first week's completion percentage.

2

Calculate the remaining uncoated area after the first week.

3,200 - 1,440 = 1,760

square feet.

Subtract the coated area from the total area to establish the new base amount for the second week's calculation.

3

Calculate the area coated during the second week.

1,760 \times 0.25 = 440

square feet.

Multiply the remaining uncoated area by the second week's completion percentage.

4

Calculate the final remaining uncoated area.

1,760 - 440 = 1,320

square feet.

Subtract the second week's coated area from the previously remaining area to find the final answer.

Key Concept

Calculating successive percentages by identifying the correct base quantity for each step.

Question 2089Question →

A square panel has a single alignment notch located on its top edge. The panel is continuously rotated flat on a table in consecutive 90-degree clockwise increments. In what chronological sequence will the alignment notch appear in the following positions, starting immediately after the first 90-degree rotation?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence tracks the notch moving clockwise in 90-degree increments: first to the right edge, then to the bottom edge, next to the left edge, and finally back to the top edge.

Starting from the top, a 90-degree clockwise rotation shifts the tracked feature to the right side (acting like a clock hand moving from 12 to 3). The next 90-degree turn moves it to the bottom (6 o'clock), followed by the left side (9 o'clock), and a final 90-degree turn completes the circle, bringing it back to the top (12 o'clock).

Step-by-Step Solution

1

Identify the starting position of the feature being tracked.

The alignment notch begins on the top edge of the square panel.

Establishing an initial reference point is necessary before applying any spatial transformations.

2

Apply the first 90-degree clockwise rotation mentally.

The top edge rotates to become the right edge, bringing the notch with it.

A 90-degree clockwise turn always shifts a feature on a square to the next adjacent side in a clockwise direction.

3

Track the feature through subsequent 90-degree clockwise rotations.

The notch moves to the bottom edge (at 180 degrees), then the left edge (at 270 degrees), and finally the top edge (at 360 degrees).

Continuing the rotation in the same direction continues to advance the feature to consecutive edges.

Key Concept

2D Mental Rotation and Feature Tracking

Estimated Time:30s

Question 2090Question →

Match each geometric angle relationship with the correct calculated value of $x$ .

Click a left item, then click its matching right item

Show answer & explanation

Answer

Match complementary to x = 19, vertical to x = 15, linear pair to x = 18, and alternate interior to x = 20.

Each geometric relationship dictates a distinct algebraic setup. Complementary angles sum to 90, vertical angles are set equal to each other, a linear pair sums to 180, and alternate interior angles are set equal to each other. Solving these linear equations gives the matching

x

values.

Step-by-Step Solution

1

Set up the equation for the complementary angles.

5x - 5 = 90

Complementary angles sum to

90^\circ

.

2

Solve for

x

in the complementary equation.

x = 19

Add 5 to both sides, then divide by 5.

3

Set up the equation for the vertical angles.

5x - 30 = 2x + 15

Vertical angles are congruent (equal in measure).

4

Solve for

x

in the vertical angles equation.

x = 15

Subtract 2x from both sides and add 30 to both sides, yielding 3x = 45.

5

Set up the equation for the linear pair.

8x + 36 = 180

Angles forming a linear pair are supplementary and sum to

180^\circ

.

6

Solve for

x

in the linear pair equation.

x = 18

Subtract 36 from both sides, then divide by 8.

7

Set up the equation for the alternate interior angles.

3x + 45 = 6x - 15

Alternate interior angles on parallel lines are congruent.

8

Solve for

x

in the alternate interior equation.

x = 20

Subtract 3x from both sides and add 15 to both sides, yielding 3x = 60.

Key Concept

Translating geometric angle relationships into algebraic equations to solve for unknown variables.

Question 2091Question →

A straight road intersects two parallel train tracks. One of the acute angles formed at an intersection measures $72^\circ$ . What is the measure of any of the obtuse angles formed by these parallel tracks and the intersecting road, in degrees?

Show answer & explanation

Answer: 108

Answer

108

When a transversal intersects parallel lines, the sum of any acute angle and any obtuse angle formed is always

180^\circ

. Subtracting

72^\circ

from

180^\circ

gives the correct obtuse angle of

108^\circ

.

Step-by-Step Solution

1

Identify the relationship between the angles.

The acute and obtuse angles are supplementary.

When a transversal intersects parallel lines, consecutive interior angles (or any adjacent pair of acute and obtuse angles) add up to

180^\circ

.

2

Calculate the measure of the obtuse angle.

180^\circ - 72^\circ = 108^\circ

Subtracting the given acute angle from

180^\circ

yields the supplementary obtuse angle.

Key Concept

Parallel Lines and Transversals

Estimated Time:30s

Question 2092Question →

Read the scenario below and fill in the missing financial values.

Fill in the blanks below

A musician is purchasing a new drum kit for

\$1,200

and a set of cymbals for

\$400

. The music store is running a promotion that offers a

15\%

discount on the total purchase if it exceeds

\$1,000

. After the discount is applied, a

6\%

state sales tax is added to the sale price.

Based on this information, the total amount saved from the discount is $\$

, and the final amount paid after tax is

\$$.

Show answer & explanation

Answer

The discount amount is

\$240

, and the final total paid is

\$1,441.60

.

The solution requires finding the total original price of

\$1,600

. The

15\%

discount on this amount is

\$240

, leaving a sale price of

\$1,360

. Finally, calculating a

6\%

sales tax on

\$1,360

adds

\$81.60

, bringing the final total to

\$1,441.60

.

Step-by-Step Solution

1

Calculate the total original price of the items.

\$1,200 + \$400 = \$1,600

To find the base amount before any discounts are applied.

2

Calculate the discount amount.

15\%

of

\$1,600 = 0.15 \times 1600 = \$240

Since the purchase exceeds

\$1,000

, the

15\%

discount applies to the total original price. This provides the first missing value.

3

Calculate the sale price before tax.

\$1,600 - \$240 = \$1,360

Subtracting the discount from the original price gives the taxable amount.

4

Calculate the sales tax.

6\%

of

\$1,360 = 0.06 \times 1360 = \$81.60

Sales tax is calculated based on the discounted sale price.

5

Calculate the final total paid.

\$1,360 + \$81.60 = \$1,441.60

Adding the sales tax to the sale price yields the final cost. This provides the second missing value.

Key Concept

Calculating a percentage discount on a total sum and then applying sales tax to the discounted price.

Alternative Method

You can calculate the final price in one continuous equation by multiplying the original price by the complement of the discount (

1 - 0.15 = 0.85

) to get the sale price, and then multiplying by the tax multiplier (

1 + 0.06 = 1.06

):

\$1,600 \times 0.85 \times 1.06 = \$1,441.60

.

Estimated Time:1m 30s

Question 2093Question →

A fleet manager is purchasing a replacement heavy-duty transmission. The original price of the transmission is $\$3,200$ . The supplier offers a $15\%$ commercial discount, and an additional $5\%$ discount off the reduced price if the invoice is paid within $10$ days. If the manager pays the invoice on the same day, what is the final price of the transmission before any taxes?

\$2,560

\$2,584

\$616

\$3,180

Show answer & explanation

Answer: $\$2,584$

Answer

The final price of the transmission is

\$2,584

.

The correct answer accurately calculates the successive discounts step by step. First, a

15\%

discount on

\$3,200

saves

\$480

, bringing the price to

\$2,720

. Then, a

5\%

discount on the new price of

\$2,720

saves

\$136

, resulting in a final price of

\$2,584

.

Step-by-Step Solution

1

Calculate the price after the first discount of

15\%

.

\$3,200 \times 0.15 = \$480

in savings. The new price is

\$3,200 - \$480 = \$2,720

.

The commercial discount is applied to the original price of the item.

2

Calculate the final price after the second discount of

5\%

.

\$2,720 \times 0.05 = \$136

in savings. The final price is

\$2,720 - \$136 = \$2,584

.

Successive (chain) discounts are always applied to the previously reduced price, not the original price.

Key Concept

Discounts and Sale Prices

Question 2094Question →

A military supply warehouse receives a shipment of $4,500$ replacement parts. During initial processing, $4\%$ of the parts are flagged for immediate deployment to forward operating bases. From the remaining parts, $25\%$ are placed in long-term storage, and the rest are distributed to active maintenance bays. How many replacement parts are distributed to the active maintenance bays?

2,025

3,240

3,195

1,080

Show answer & explanation

Answer: 3,240

Answer

3,240

The initial

4\%

reduction removes

180

parts (

4,500 \times 0.04

), leaving

4,320

. The subsequent

25\%

reduction is taken from this remaining base of

4,320

, removing another

1,080

parts. Subtracting this from

4,320

leaves

3,240

parts for the active maintenance bays.

Step-by-Step Solution

1

Calculate the number of parts flagged for immediate deployment.

4,500 \times 0.04 = 180

To find the

4\%

portion of the initial total shipment.

2

Determine the remaining number of parts after the first deployment.

4,500 - 180 = 4,320

The next percentage reduction applies only to the parts that remain in the warehouse.

3

Calculate the number of parts placed in long-term storage.

4,320 \times 0.25 = 1,080

To find the

25\%

of the newly calculated remaining pool.

4

Subtract the stored parts to find the final distributed amount.

4,320 - 1,080 = 3,240

To determine the final quantity distributed to the active maintenance bays.

Key Concept

Calculating sequential percentages on remaining quantities

Question 2095Question →

A motor pool sergeant is purchasing industrial degreaser for the vehicle maintenance bay. The facility requires exactly $8$ gallons of degreaser. A supply catalog offers the degreaser in two sizes: a $1$ -gallon jug for $\$14.50$ and a $32$ -ounce bottle for $\$4.00$ . If the sergeant purchases the required amount using the most cost-effective option, how much money will be saved in total compared to buying the more expensive option?

\$1.20

\$12.00

\$84.00

\$108.00

Show answer & explanation

Answer: $\$12.00$

Answer

\$12.00

The correct answer is found by first equalizing the volumes. There are 128 ounces in a gallon, so it takes four 32-ounce bottles to make a single gallon. Buying the degreaser in bottles costs 4 ×

4.00 =

16.00 per gallon. The 1-gallon jug costs

14.50, meaning the jug saves

1.50 per gallon compared to the bottles. Because the facility needs 8 gallons, the total savings is 8 ×

1.50 =

12.00.

Step-by-Step Solution

1

Determine the number of 32-ounce bottles needed to equal one gallon.

Since

1

gallon equals

128

ounces,

128 \div 32 = 4

bottles per gallon.

To accurately compare prices, both options must be evaluated at the same unit of volume.

2

Calculate the price per gallon for the 32-ounce bottle option.

4

bottles

\times \$4.00 = \$16.00

per gallon.

This establishes a direct unit price comparison against the 1-gallon jug.

3

Determine which option is more cost-effective and calculate the savings per gallon.

The jug costs

\$14.50

per gallon, which is

\$1.50

cheaper than the

\$16.00

per gallon cost of the bottles.

Finding the difference in unit price is necessary to calculate the total savings over the entire purchase.

4

Calculate the total savings for the required 8 gallons.

8

gallons

\times \$1.50 = \$12.00

total savings.

The facility requires 8 gallons, so the per-gallon savings must be multiplied by the total volume needed.

Key Concept

Comparing unit prices across different measurement units to find total purchase savings.

Question 2096Question →

A metal worker is joining two pieces of steel such that the two adjacent angles formed make a straight line. The measure of the first angle is $(2x + 10)^\circ$ and the measure of the second angle is $(3x)^\circ$ . What is the measure of the first angle?

78^\circ

42^\circ

58^\circ

34^\circ

Show answer & explanation

Answer: $78^\circ$

Answer

The measure of the first angle is

78^\circ

.

Angles that form a straight line are supplementary and add up to 180 degrees. Setting up the equation (2x + 10) + 3x = 180 yields 5x + 10 = 180. Solving for x gives x = 34. Substituting this value into the expression for the first angle gives 2(34) + 10 = 78 degrees.

Step-by-Step Solution

1

Set up an equation based on the properties of a straight line.

(2x + 10) + 3x = 180

Adjacent angles that form a straight line are supplementary, meaning their sum is exactly 180 degrees.

2

Combine like terms.

5x + 10 = 180

Adding the variable terms (2x and 3x) together simplifies the equation.

3

Solve for x.

x = 34

Subtract 10 from both sides to get 5x = 170, then divide by 5 to isolate x.

4

Substitute the value of x back into the expression for the first angle.

2(34) + 10 = 68 + 10 = 78

The question asks for the measure of the first angle, not just the value of x.

Key Concept

Supplementary Angles and Linear Equations

Question 2097Question →

Which of the following represents the completely factored form of the quadratic expression $3x^2 - 3x - 18$ ?

3(x - 3)(x + 2)

3(x + 3)(x - 2)

(3x - 9)(x + 2)

3(x - 6)(x + 1)

Show answer & explanation

Answer: $3(x - 3)(x + 2)$

Answer

The completely factored form is

3(x - 3)(x + 2)

.

The correct answer properly extracts the GCF of 3 first, leaving the trinomial

x^2 - x - 6

. This trinomial factors into

(x - 3)(x + 2)

because

-3

and

2

multiply to

-6

and add up to

-1

.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression.

3(x^2 - x - 6)

The coefficients

3, -3,

and

-18

are all divisible by

3

.

2

Find two numbers that multiply to the constant term (

-6

) and add to the coefficient of the middle term (

-1

).

The numbers are

-3

and

2

.

-3 \times 2 = -6

and

-3 + 2 = -1

.

3

Rewrite the quadratic trinomial as the product of two binomials using these numbers.

3(x - 3)(x + 2)

This represents the final, completely factored form of the original expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a quadratic trinomial.

Question 2098Question →

In quadrilateral $ABCD$ , $\angle B$ and $\angle D$ are right angles. The lengths of three sides are given as $AB = 7$ , $BC = 24$ , and $CD = 15$ . What is the length of side $AD$ ?

8

10

20

46

Show answer & explanation

Answer: 20

Answer

The length of side

AD

is 20.

Drawing the diagonal

AC

creates two right triangles that share a hypotenuse. The first triangle (

ABC

) has legs of

7

and

24

, giving a hypotenuse of

25

(

7^2 + 24^2 = 25^2

). In the second triangle (

ADC

), this

25

serves as the hypotenuse while

CD

is a leg of

15

. Solving for the remaining leg gives

20

(

25^2 - 15^2 = 400

, and

\sqrt{400} = 20

).

Step-by-Step Solution

1

Visualize or draw diagonal

AC

to split the quadrilateral into two right triangles.

Right

\triangle ABC

and right

\triangle ADC

are formed, sharing hypotenuse

AC

.

Since

\angle B

and

\angle D

are right angles, the segment connecting

A

and

C

must be the hypotenuse for both triangles.

2

Apply the Pythagorean theorem to

\triangle ABC

to find the length of

AC

.

AC^2 = 7^2 + 24^2 = 49 + 576 = 625

, which means

AC = 25

.

The lengths of both legs (

AB

and

BC

) are known, allowing calculation of the hypotenuse.

3

Apply the Pythagorean theorem to

\triangle ADC

using the known hypotenuse.

AD^2 + CD^2 = AC^2

, which becomes

AD^2 + 15^2 = 25^2

.

The hypotenuse

AC

is shared, and leg

CD

is given as 15, leaving

AD

as the only unknown variable.

4

Solve the equation for the missing leg

AD

.

AD^2 + 225 = 625 \Rightarrow AD^2 = 400 \Rightarrow AD = 20

.

Isolating

AD^2

and taking the square root provides the final length of the side.

Key Concept

Applying the Pythagorean theorem sequentially across adjacent right triangles that share a common hypotenuse.

Estimated Time:2m 0s

Question 2099Question →

An engineering battalion is tasked with laying $4,200$ tons of asphalt to repair a section of an airfield runway. By the end of the first week, they have completed $65\%$ of the total required amount. How many tons of asphalt do they still need to lay to complete the repair?

Show answer & explanation

Answer: 1470

Answer

1470

The correct answer is found by determining the remaining percentage of the total project. If

65\%

of the work is completed, then

100\% - 65\% = 35\%

remains. To find the exact tonnage, calculate

35\%

of

4,200

tons by converting the percentage to a decimal and multiplying:

4,200 \times 0.35 = 1,470

.

Step-by-Step Solution

1

Find the percentage of the asphalt that still needs to be laid.

100\% - 65\% = 35\%

The total project represents

100\%

. Subtracting the completed percentage gives the remaining percentage.

2

Calculate

35\%

of the total

4,200

tons.

4,200 \times 0.35 = 1,470

To find a percentage of a quantity, convert the percentage to a decimal and multiply it by the total quantity.

Key Concept

Calculating the remaining percentage of a whole quantity

Estimated Time:1m 0s

Question 2100Question →

A commercial photographer is purchasing a new studio lighting kit. The original retail price of the kit is $\$1,200$ . The camera supply store is currently offering a $25\%$ seasonal discount on all studio equipment. In addition, the photographer receives a $10\%$ professional discount, which is applied to the already reduced price. What is the final price of the lighting kit before sales tax?

Show answer & explanation

Answer: 810

Answer

The final price of the lighting kit is

\$810

.

The correct calculation requires applying the first discount to find a new subtotal, and then applying the second discount to that new subtotal. Applying

25\%

off

\$1,200

yields

\$900

. Applying the

10\%

professional discount to the new base of

\$900

reduces the price by another

\$90

, resulting in a final price of

\$810

.

Step-by-Step Solution

1

Calculate the first discount amount.

25\%

of

\$1,200

is

\$300

.

The seasonal discount applies to the original retail price.

2

Find the intermediate reduced price.

\$1,200 - \$300 = \$900

.

Subtracting the first discount gives the new base price before the second discount.

3

Calculate the second discount amount.

10\%

of

\$900

is

\$90

.

The professional discount applies to the already reduced price, not the original price.

4

Determine the final price.

\$900 - \$90 = \$810

.

Subtracting the second discount from the intermediate price gives the final amount paid.

Key Concept

Calculating successive discounts requires applying the second percentage to the new, already discounted price, rather than the original amount.

Answer

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