All practice questions

Question 501Question →

A contractor purchases lumber for a deck project at a wholesale price of $\$600$ . To cover transportation and handling, the contractor applies a $15\%$ markup to the cost of the lumber. What is the total amount, in dollars, the contractor bills the client for the lumber?

Show answer & explanation

Answer: 690

Answer

690

To find the total amount billed, first calculate the

15\%

markup on the

\$600

cost. Multiplying

\$600

by

0.15

gives a markup of

\$90

. Since it is a markup, this value is added to the original wholesale price.

\$600 + \$90 = \$690

.

Step-by-Step Solution

1

Calculate the markup amount.

\$90

The contractor applies a

15\%

markup to the

\$600

wholesale cost.

0.15 \times 600 = 90

.

2

Add the markup to the wholesale cost.

\$690

The final amount billed to the client is the original cost plus the markup.

600 + 90 = 690

.

Key Concept

Calculating a percentage markup and adding it to the original cost to find the final price.

Question 502Question →

A solid rectangular block of an unknown metal has dimensions of $5\text{ cm}$ in length, $4\text{ cm}$ in width, and $2\text{ cm}$ in height. If the total mass of the block is $316\text{ g}$ , what is the density of the metal in $\text{g/cm}^3$ ?

Show answer & explanation

Answer: 7.9

Answer

The density of the metal block is

7.9\text{ g/cm}^3

.

Density measures mass per unit volume (

D = \frac{m}{V}

). The volume of the rectangular prism is calculated by multiplying its dimensions (

5\text{ cm} \times 4\text{ cm} \times 2\text{ cm} = 40\text{ cm}^3

). Dividing the mass (

316\text{ g}

) by this volume (

40\text{ cm}^3

) yields

7.9\text{ g/cm}^3

.

Step-by-Step Solution

1

Calculate the volume of the rectangular metal block from its dimensions.

V = 5\text{ cm} \times 4\text{ cm} \times 2\text{ cm} = 40\text{ cm}^3

To determine density, the total volume must first be calculated from the given geometric measurements.

2

Divide the mass of the block by its calculated volume.

\text{Density} = \frac{316\text{ g}}{40\text{ cm}^3} = 7.9\text{ g/cm}^3

Density is defined as mass per unit volume (

D = \frac{m}{V}

).

Key Concept

Density is a characteristic physical property of matter defined as mass divided by volume (

D = \frac{m}{V}

).

Estimated Time:1m 0s

Question 503Question →

A logistics company is purchasing diesel fuel for its fleet. The base price of diesel is $\$4.50$ per gallon. The company uses a corporate discount card that provides a $\$0.30$ per gallon reduction on the first $120$ gallons, and a $20\%$ discount off the base price for any additional gallons purchased beyond the first $120$ .

The company purchases a total of $250$ gallons of diesel.

After calculating the cost of the fuel with the discounts applied, a federal excise tax of $\$0.40$ per gallon is added for the entire purchase volume. Finally, a $5\%$ state sales tax is applied to the subtotal (the subtotal includes both the discounted fuel cost and the excise tax).

If the company pays for the entire transaction using a prepaid account that currently has a balance of $\$1,500.00$ , what will be the remaining balance in dollars after the purchase?

Show answer & explanation

Answer: 374.4

Answer

374.40

The correct answer is found by first calculating the tiered fuel costs:

\$504.00

for the first 120 gallons (at

\$4.20

/gal) and

\$468.00

for the remaining 130 gallons (at

\$3.60

/gal). Adding the

\$100.00

federal excise tax gives a subtotal of

\$1072.00

. The

5\%

state tax is calculated on this subtotal, adding

\$53.60

for a final cost of

\$1125.60

. Subtracting this from the

\$1500.00

account balance leaves

\$374.40

.

Step-by-Step Solution

1

Calculate the fuel cost for the first tier (120 gallons).

$504.00

The first 120 gallons receive a

0.30 flat discount. The price per gallon is

4.50 -

0.30 =

4.20. Multiplying 120 by

4.20 gives

504.00.

2

Calculate the fuel cost for the second tier (remaining gallons).

$468.00

There are 250 - 120 = 130 remaining gallons. A 20% discount on

4.50 gives a price of

3.60 per gallon. Multiplying 130 by

3.60 gives

468.00.

3

Find the total discounted fuel cost before taxes.

$972.00

Add the costs of both tiers together:

504.00 +

468.00 = $972.00.

4

Calculate the federal excise tax.

$100.00

The excise tax applies to all 250 gallons at

0.40 per gallon. Multiplying 250 by

0.40 gives $100.00.

5

Calculate the subtotal and the state sales tax.

$53.60

The subtotal is

972.00 (fuel) +

100.00 (excise) =

1072.00. The 5% state tax on this subtotal is

1072.00 * 0.05 = $53.60.

6

Find the final transaction cost.

$1125.60

Add the state sales tax to the subtotal:

1072.00 +

53.60 = $1125.60.

7

Calculate the remaining balance in the prepaid account.

$374.40

Subtract the total transaction cost from the starting balance:

1500.00 -

1125.60 = $374.40.

Key Concept

Solving multi-step financial word problems involving tiered discounts, dual tax application sequences, and balance deductions.

Estimated Time:3m 0s

Question 504Question →

An automotive dealership initially offers a customer a trade-in value of $\$4,500.00$ for their used vehicle. After a mechanical inspection reveals worn brake pads and a minor oil leak, the dealership decreases this initial offer by $12\%$ . To help close the deal on a new vehicle, the sales manager then approves a $5\%$ increase on the revised trade-in offer. What is the final trade-in value offered to the customer?

Show answer & explanation

Answer: 4158

Answer

4158

The correct answer is found by taking the initial amount of

4,500, decreasing it by 12% to find the intermediate value of

3,960, and then increasing that intermediate value by 5% to arrive at the final value of $4,158.

Step-by-Step Solution

1

Calculate the dollar amount of the 12% decrease.

\$4,500 \times 0.12 = \$540

The initial offer is reduced by 12% due to mechanical issues found during the inspection.

2

Determine the revised trade-in offer after the initial decrease.

\$4,500 - \$540 = \$3,960

Subtracting the penalty from the original amount establishes the new base value for the next percentage change.

3

Calculate the dollar amount of the 5% increase.

\$3,960 \times 0.05 = \$198

The manager's 5% increase applies specifically to the revised offer, not the original amount.

4

Calculate the final trade-in value.

\$3,960 + \$198 = \$4,158

Adding the approved increase to the revised offer provides the final amount offered to the customer.

Key Concept

Successive percentage changes

Question 505Question →

A military supply depot distributes a shipment of Meals Ready-to-Eat (MREs) among three battalions: Alpha, Bravo, and Charlie. Battalion Alpha receives $\frac{1}{3}$ of the total shipment. The remaining MREs are then divided between Battalion Bravo and Battalion Charlie in a ratio of $3 : 5$ respectively. If Battalion Charlie receives exactly $1,200$ more MREs than Battalion Bravo, what is the total number of MREs in the entire shipment?

Show answer & explanation

Answer: 7200

Answer

The total number of MREs in the entire shipment is 7,200.

The correct total shipment is found by reasoning backward from the difference. Bravo and Charlie split the remaining 2/3 of the shipment in a 3:5 ratio, meaning there are 8 total parts for that remainder. Charlie receives 2 more parts than Bravo (

5 - 3 = 2

). Since these 2 parts equal 1,200 MREs, 1 part equals 600 MREs. The entire remainder (all 8 parts) is therefore

8 \times 600 = 4,800

MREs. Since this 4,800 represents the 2/3 of the total shipment left after Alpha's portion, the entire original shipment must be

(4,800 \div 2) \times 3 = 7,200

MREs.

Step-by-Step Solution

1

Define a variable for the total shipment and calculate the remaining amount after Alpha's share is removed.

Let the total shipment be T. Alpha receives

\frac{1}{3}T

, so the remaining portion for Bravo and Charlie is

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

.

We must determine exactly what portion of the whole shipment is being divided between the remaining two battalions.

2

Determine the fractional share of the remainder that Bravo and Charlie each receive based on their

3:5

ratio.

The ratio

3:5

has

3 + 5 = 8

total parts. Bravo gets

\frac{3}{8}

of the remainder, and Charlie gets

\frac{5}{8}

of the remainder.

Ratios represent parts of a whole; in this step, the 'whole' is the remaining

\frac{2}{3}T

of the shipment.

3

Express Bravo's and Charlie's individual shares in terms of the total original shipment T.

Bravo's share is

\frac{3}{8} \times \frac{2}{3}T = \frac{1}{4}T

. Charlie's share is

\frac{5}{8} \times \frac{2}{3}T = \frac{5}{12}T

.

Converting their shares back to fractions of the original total allows us to set up an algebraic equation for the entire shipment.

4

Set up an equation using the known difference between Charlie's and Bravo's shares.

Charlie receives 1,200 more than Bravo, so:

\frac{5}{12}T - \frac{1}{4}T = 1200

.

This translates the word problem's condition into a solvable algebraic equation.

5

Solve the equation for T.

Convert

\frac{1}{4}T

to

\frac{3}{12}T

. The equation becomes

\frac{2}{12}T = 1200

, which simplifies to

\frac{1}{6}T = 1200

. Multiplying both sides by 6 yields

T = 7200

.

Isolating the variable gives the final total number of MREs.

Key Concept

Dividing a remaining quantity into ratio parts and utilizing the algebraic difference between parts to find an original whole.

Question 506Question →

A customer invests $\$1,500$ in a savings certificate that pays an annual simple interest rate of $4\%$ . How much total interest, in dollars, will the investment earn after exactly $3$ years?

Show answer & explanation

Answer: 180

Answer

The investment will earn 180 dollars in interest.

Using the simple interest formula

I = PRT

, substituting the given values yields

I = 1500 \times 0.04 \times 3 = 180

. The total interest earned is

\$180

.

Step-by-Step Solution

1

Identify the given values for the simple interest formula.

Principal (

P

) =

1,500

, Rate (

R

) =

4\%

(or

0.04

), Time (

T

) =

3

years.

These are the necessary components to calculate simple interest.

2

Multiply the principal by the annual interest rate to find the interest earned in one year.

1500 \times 0.04 = 60

This determines the annual interest amount.

3

Multiply the annual interest by the number of years.

60 \times 3 = 180

This gives the total simple interest over the entire time period.

Key Concept

Calculating simple interest using the formula

I = PRT

.

Question 507Question →

A passenger takes a taxi ride that costs $\$30.00$ on the meter. The passenger wants to give the driver a $20\%$ tip. What is the total amount the passenger will pay in dollars?

Show answer & explanation

Answer: 36

Answer

36

The tip is calculated by converting

20\%

to the decimal

0.20

and multiplying it by the

\$30.00

meter cost, resulting in a

\$6.00

tip. Adding this tip to the original

\$30.00

cost gives a final total payment of

\$36.00

.

Step-by-Step Solution

1

Calculate the tip amount by finding

20\%

of the meter cost.

\$30.00 \times 0.20 = \$6.00

The tip is a percentage calculated based on the original cost of the service.

2

Add the tip amount to the original meter cost to find the total amount paid.

\$30.00 + \$6.00 = \$36.00

The total payment equals the original service cost plus the added tip.

Key Concept

Calculating a percentage of a base amount and adding it to find the total cost.

Question 508Question →

A rectangular training pool at a military base is $40$ feet long and $25$ feet wide. It is initially filled with water to a uniform depth of $8$ feet. If $1,500$ cubic feet of water is pumped out of the pool, what is the new depth of the water, in feet?

Show answer & explanation

Answer: 6.5

Answer

6.5

The pool's base has an area of

40 \times 25 = 1,000

square feet. When

1,500

cubic feet of water is removed, the water level drops by

1,500 \div 1,000 = 1.5

feet. Subtracting this drop from the initial depth of

8

feet gives a new depth of

6.5

feet. Alternatively, finding the original volume (

8,000

cubic feet), subtracting the removed water (

6,500

cubic feet), and dividing by the area (

1,000

square feet) also yields

6.5

feet.

Step-by-Step Solution

1

Calculate the base area of the rectangular pool.

1,000

square feet

The base area is needed to determine how changes in volume affect the depth of the water. Area = length

\times

width =

40 \times 25 = 1,000

.

2

Determine the change in the water depth.

The depth drops by

1.5

feet.

Dividing the removed volume by the base area gives the vertical drop in water level.

1,500 \div 1,000 = 1.5

.

3

Calculate the new depth.

6.5

feet

Subtract the drop in water level from the initial depth.

8 - 1.5 = 6.5

.

Key Concept

Calculating changes in the depth of a rectangular volume when a specific amount of capacity is removed.

Question 509Question →

What is the minimum integer value of $x$ that satisfies the inequality shown below?

3(2x - 5) - 8x \leq 4(x - 1) - 23

Show answer & explanation

Answer: 2

Answer

2

Solving the inequality yields

x \geq 2

. Since

x

must be greater than or equal to 2, the minimum integer value that makes this statement true is exactly 2.

Step-by-Step Solution

1

Distribute the constants into the parentheses on both sides of the inequality.

6x - 15 - 8x \leq 4x - 4 - 23

This removes the parentheses, allowing us to combine like terms.

2

Combine the like terms on each side of the inequality.

-2x - 15 \leq 4x - 27

Simplifying each side makes it easier to isolate the variable.

3

Subtract

4x

from both sides to group the variable terms on the left side.

-6x - 15 \leq -27

We need all terms with

x

on one side of the inequality to solve for it.

4

Add

15

to both sides to isolate the term with the variable.

-6x \leq -12

This leaves only the

-6x

term on the left side.

5

Divide both sides by

-6

, remembering to flip the inequality symbol.

x \geq 2

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

6

Determine the minimum integer value that satisfies the simplified inequality.

2

The inequality states that

x

must be greater than or equal to

2

. The smallest integer that meets this condition is

2

.

Key Concept

Solving linear inequalities requiring the distributive property, combining like terms, and reversing the inequality symbol when dividing by a negative coefficient.

Question 510Question →

Find the numerical value of the rational expression

\frac{3a^2 - 4b}{c + 2}

using the parameter values

a = -2

,

b = -5

, and

c = -6

.

Show answer & explanation

Answer: -8

Answer

The final numerical value is -8.

The correct value is found by carefully substituting the negative integer values into the rational expression, strictly applying the order of operations (exponents, then multiplication, then addition/subtraction), and accurately tracking the signs at each step to get 32 in the numerator and -4 in the denominator, which yields -8.

Step-by-Step Solution

1

Substitute the given values into the expression.

\frac{3(-2)^2 - 4(-5)}{-6 + 2}

Replace variables

a

,

b

, and

c

with their assigned numerical equivalents to begin evaluation.

2

Evaluate the exponent in the numerator.

\frac{3(4) - 4(-5)}{-6 + 2}

According to the order of operations, exponents must be evaluated before multiplication. The square of a negative number is positive:

(-2)^2 = 4

.

3

Perform the multiplications in the numerator.

\frac{12 - (-20)}{-6 + 2}

Calculate the products:

3 \times 4 = 12

and

4 \times (-5) = -20

.

4

Simplify the subtraction in the numerator.

\frac{12 + 20}{-6 + 2} = \frac{32}{-6 + 2}

Subtracting a negative number is equivalent to adding a positive number:

12 - (-20) = 12 + 20 = 32

.

5

Evaluate the addition in the denominator.

\frac{32}{-4}

Combine the constant terms in the denominator:

-6 + 2 = -4

.

6

Divide the numerator by the denominator.

-8

A positive number divided by a negative number results in a negative quotient:

32 \div (-4) = -8

.

Key Concept

Evaluating Algebraic Expressions

Question 511Question →

A fitness center offers two membership plans. Plan A requires a one-time enrollment fee of $45$ dollars and costs $30$ dollars per month. Plan B requires a one-time enrollment fee of $105$ dollars and costs $20$ dollars per month. For how many months of membership will the total cost of both plans be exactly the same?

Show answer & explanation

Answer: 6

Answer

The plans will cost the exact same amount at 6 months.

Setting the cost of Plan A (

45 + 30m

) equal to the cost of Plan B (

105 + 20m

) creates the equation

45 + 30m = 105 + 20m

. By isolating the variable, we find that

10m = 60

, which simplifies to

m = 6

. Therefore, it takes exactly

6

months for both memberships to cost the same.

Step-by-Step Solution

1

Define the variable and formulate a one-variable linear equation.

45 + 30m = 105 + 20m

To determine when the total costs are equal, the cost expression for Plan A (

45

fee plus

30

per month) must be set equal to the cost expression for Plan B (

105

fee plus

20

per month).

2

Subtract

20m

from both sides of the equation.

45 + 10m = 105

This groups all terms containing the variable

m

on one side of the equation.

3

Subtract

45

from both sides of the equation.

10m = 60

This isolates the variable term by moving all constant values to the opposite side of the equation.

4

Divide both sides by

10

.

m = 6

Dividing by the coefficient of the variable provides the final numerical solution for the unknown.

Key Concept

Translating a real-world scenario into a one-variable linear equation and isolating the variable to solve for an unknown value.

Estimated Time:1m 30s

Question 512Question →

Find the value of $x$ that satisfies the equation:

\frac{x - 3}{4} + \frac{2x + 1}{3} = \frac{5x}{6}

Enter the exact value of $x$ .

Show answer & explanation

Answer: 5

Answer

5

By finding the least common denominator (12) and multiplying every term by it, the fractions are cleanly eliminated. After applying the distributive property and combining like terms (

11x - 5 = 10x

), isolating the variable

x

yields exactly 5.

Step-by-Step Solution

1

Multiply all terms by the least common denominator (12).

12 \left(\frac{x - 3}{4}\right) + 12 \left(\frac{2x + 1}{3}\right) = 12 \left(\frac{5x}{6}\right)

Multiplying by the LCD eliminates all fractions from the equation, simplifying further steps.

2

Simplify the resulting fractions.

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

Dividing 12 by each respective denominator leaves an integer multiplier for the numerators.

3

Distribute the multipliers across the terms in the parentheses.

3x - 9 + 8x + 4 = 10x

The distributive property is required to remove the parentheses.

4

Combine like terms on the left side of the equation.

11x - 5 = 10x

Grouping

x

terms (

3x + 8x

) and constant terms (

-9 + 4

) simplifies the expression.

5

Isolate the variable

x

.

x = 5

Subtracting

10x

from both sides gives

x - 5 = 0

. Adding

5

to both sides results in

x = 5

.

Key Concept

Solving linear equations involving rational expressions by multiplying by a least common denominator.

Estimated Time:1m 30s

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