All practice questions

Consider the following algebraic expression:

(2x - 5y)^2(x + 2y)

If you were to fully expand this expression and combine all like terms, what would be the numerical coefficient of the $x^2y$ term?

Fill in the blanks below

The numerical coefficient of the

x^2y

term is .

Show answer & explanation

Answer

-12

The correct answer is derived by first expanding the squared binomial:

(2x - 5y)^2 = 4x^2 - 20xy + 25y^2

. Next, multiply this resulting trinomial by the remaining binomial:

(4x^2 - 20xy + 25y^2)(x + 2y)

. To find the coefficient of the

x^2y

term without expanding the entire expression, identify the term pairs whose product yields

x^2y

. These pairs are

(4x^2)(2y) = 8x^2y

and

(-20xy)(x) = -20x^2y

. Combining these like terms gives

8x^2y - 20x^2y = -12x^2y

. Thus, the numerical coefficient of the

x^2y

term is

-12

.

Step-by-Step Solution

1

Expand the squared binomial

(2x - 5y)^2

.

(2x)^2 - 2(2x)(5y) + (5y)^2 = 4x^2 - 20xy + 25y^2

Order of operations requires evaluating exponents before multiplying by the binomial

(x + 2y)

.

2

Set up the multiplication of the resulting trinomial and the remaining binomial.

(4x^2 - 20xy + 25y^2)(x + 2y)

Substitute the expanded form back into the original expression to continue the multiplication process.

3

Identify only the specific multiplications that will produce an

x^2y

term.

The term

4x^2

multiplied by

2y

, and the term

-20xy

multiplied by

x

.

To find the coefficient of a specific term, it is most efficient to calculate only the products that result in that exact variable combination.

4

Calculate and combine the identified products to find the final coefficient.

(4x^2)(2y) + (-20xy)(x) = 8x^2y - 20x^2y = -12x^2y

. The numerical coefficient is

-12

.

Combining these like terms gives the total

x^2y

term in the fully expanded polynomial.

Key Concept

Multiplying Polynomials

Question 222Question →

A military supply truck travels at a steady rate of $40\text{ mph}$ for exactly $45$ minutes along a designated supply route. What is the total distance traveled?

30\text{ miles}

18\text{ miles}

300\text{ miles}

1800\text{ miles}

Show answer & explanation

Answer: $30\text{ miles}$

Answer

30\text{ miles}

To find the correct distance, the time must first be converted to match the units of the rate. Dividing

45\text{ minutes}

by

60

gives

0.75\text{ hours}

. Multiplying the rate (

40\text{ mph}

) by the time (

0.75\text{ hours}

) gives exactly

30\text{ miles}

.

Step-by-Step Solution

1

Convert the travel time from minutes to hours.

45\text{ minutes} = \frac{45}{60}\text{ hours} = 0.75\text{ hours}

The speed is given in miles per hour (mph), so the time must be in hours to cancel out the units properly.

2

Calculate the total distance using the formula: Distance = Rate

\times

Time.

Distance

= 40\text{ mph} \times 0.75\text{ hours} = 30\text{ miles}

Multiplying the steady rate by the total time in hours yields the total distance.

Key Concept

Calculating Distance with Unit Conversion

Question 223Question →

Match each inverse proportion scenario to its correct calculated value.

Click a left item, then click its matching right item

Show answer & explanation

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 224Question →

Determine the missing principal amount based on the given simple interest details. Fill in the blank with the correct numeric value.

Fill in the blanks below

A homeowner paid

\$210

in simple interest on a short-term loan that lasted

14

months. If the loan had a

4\%

annual simple interest rate, the original principal amount borrowed was \$.

Show answer & explanation

Answer

The original principal amount borrowed was

\$4,500

.

The correct principal is found by rearranging the simple interest formula to

P = \frac{I}{R \times T}

. By converting the

14

months into

\frac{7}{6}

years and using the decimal rate of

0.04

, dividing the

\$210

interest by their product yields the original principal of

\$4,500

.

Step-by-Step Solution

1

Identify the known values: Interest (

I

), annual Rate (

R

), and Time (

T

) in months.

I = 210

,

R = 0.04

, and

T = 14

months

These are the given variables needed to set up the simple interest equation.

2

Convert the loan duration from months into years.

T = \frac{14}{12} = \frac{7}{6}

years

The simple interest formula requires time to be in years when the given interest rate is annual.

3

Substitute the values into the simple interest formula (

I = PRT

) to isolate Principal (

P

).

210 = P \times 0.04 \times \frac{7}{6}

This establishes the equation with only one unknown variable.

4

Calculate the product of the rate and time, then divide the total interest by this product.

P = \frac{210}{\frac{0.28}{6}} = 210 \times \frac{6}{0.28} = \frac{1260}{0.28} = 4500

Simplifying the expression yields the exact borrowed amount.

Key Concept

Solving for the principal using the simple interest formula

I = PRT

by converting time in months to a fraction of a year.

Question 225Question →

A naval supply ship pumps $3,600$ gallons of fresh water into a destroyer's holding tanks over the course of $4$ hours. At what constant rate, in gallons per hour, is the water being pumped?

Show answer & explanation

Answer: 900

Answer

900

To find the rate in gallons per hour, the total number of gallons (

3,600

) is divided by the total number of hours (

4

), resulting in

900

gallons per hour.

Step-by-Step Solution

1

Identify the total quantity and the total time given in the scenario.

Total quantity =

3,600

gallons. Total time =

4

hours.

A unit rate requires knowing both the total amount and the corresponding time frame.

2

Divide the total gallons by the total hours to find the unit rate.

3,600 \div 4 = 900

Dividing the total amount by the number of hours yields the amount per single hour.

Key Concept

Calculating a basic unit rate by dividing a total quantity by a total time.

Question 226Question →

A developer issues a short-term promissory note to cover construction costs. The note carries an annual simple interest rate of $8\%$ . After $9$ months, the developer repays a total of $\$42,400$ , which includes both the original principal borrowed and the interest earned. What was the original principal amount borrowed, in dollars?

Show answer & explanation

Answer: 40000

Answer

40000

The original principal is found by utilizing the total amount formula for simple interest,

A = P(1 + rt)

, and solving for

P

. By substituting

A = 42,400

,

r = 0.08

, and

t = 0.75

years (from

9

months), the equation simplifies to

1.06P = 42,400

, which yields a principal of $40,000.

Step-by-Step Solution

1

Identify the known variables from the word problem.

Total Amount (

A

) =

42,400

, Rate (

r

) =

8\% = 0.08

, Time (

t

) =

9

months.

Extracting the specific numbers and their corresponding variables is required before applying any formulas.

2

Convert the time duration from months into years.

t = \frac{9}{12} = 0.75

years.

Simple interest formulas require the time unit to match the rate unit, which is annual (

8\%

per year).

3

Apply the total amount simple interest formula and substitute the values.

42,400 = P(1 + 0.08 \times 0.75)

The total amount paid back (

42,400

) includes both the principal (

P

) and the interest (

Prt

), factored out as

P(1 + rt)

.

4

Simplify the equation.

42,400 = P(1 + 0.06) \Rightarrow 42,400 = 1.06P

Multiplying the rate by the time yields the total interest multiplier (

6\%

or

0.06

), adding it to

1

gives the total amount multiplier.

5

Solve for the principal (

P

).

P = \frac{42,400}{1.06} = 40,000

Isolating

P

gives the original amount borrowed. The division can be calculated mentally by noting that

424

is

4 \times 106

.

Key Concept

Solving for Principal in a Simple Interest Total Amount Scenario

Question 227Question →

A motor pool sergeant reviews a vehicle readiness report and notes that $\frac{3}{5}$ of the transport trucks are fully operational. What percentage of the transport trucks are fully operational?

60%

6%

37.5%

166.7%

Show answer & explanation

Answer: 60%

Answer

60%

The correct percentage is found by executing the division indicated by the fraction. Dividing the numerator (3) by the denominator (5) gives the decimal 0.6. Multiplying this decimal by 100 scales it to a percentage, giving 60%.

Step-by-Step Solution

1

Convert the given fraction to a decimal value by dividing the numerator by the denominator.

3 \div 5 = 0.6

Finding the decimal equivalent is the standard intermediate step for converting a fraction to a percentage.

2

Convert the resulting decimal to a percentage by multiplying it by 100.

0.6 \times 100 = 60\%

Percentages represent parts per hundred, so multiplying the decimal representation by 100 yields the correct percentage.

Key Concept

Converting fractions to percentages via division and decimal multiplication.

Alternative Method

Recognize that the denominator 5 is a factor of 100. Multiply both the numerator and the denominator by 20 to create an equivalent fraction over 100: (3 × 20) / (5 × 20) = 60 / 100, which is exactly 60%.

Estimated Time:45s

Question 228Question →

During a base renovation, a carpenter cuts a $5$ -foot wooden board into two pieces. If the lengths of the two pieces are in a ratio of $2:3$ , what is the length of the shorter piece in inches?

24

36

40

2

Show answer & explanation

Answer: $24$

Answer

The shorter piece is

24

inches long.

First, convert the

5

-foot board to inches by multiplying by

12

, which gives

60

inches. The ratio

2:3

means there are

2 + 3 = 5

total equal parts. Dividing the

60

-inch board by

5

gives

12

inches per part. The shorter piece is

2

parts, so

2 \times 12 = 24

inches.

Step-by-Step Solution

1

Convert the total length of the board from feet to inches.

5 \times 12 = 60

inches.

The question asks for the final answer in inches, so it is easiest to work in inches from the start.

2

Find the total number of parts in the ratio.

2 + 3 = 5

total parts.

A ratio of

2:3

means the whole board is divided into

5

equal segments.

3

Determine the length of one single part.

60 \div 5 = 12

inches per part.

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

4

Calculate the length of the shorter piece.

12 \times 2 = 24

inches.

The shorter piece represents

2

parts of the ratio.

Key Concept

To divide a quantity in a given ratio, find the total number of parts by adding the ratio terms, divide the total quantity by this number to find the value of one part, and multiply by the relevant ratio term. Always ensure units are consistent.

Question 229Question →

A naval shipyard receives a shipment of $4,500$ tons of high-grade steel for vessel construction. The shipyard allocates $36\%$ of this steel for hull plating. Of the remaining steel, $45\%$ is used for interior bulkheads. Calculate the exact number of tons of steel that will be placed into the strategic reserve.

Fill in the blanks below

The shipyard will place tons of steel into the strategic reserve.

Show answer & explanation

Answer

1,584

The problem requires calculating percentages sequentially based on updated totals. After taking

36\%

of the original

4,500

tons for the hull,

64\%

(

2,880

tons) remains. The next allocation of

45\%

for bulkheads is taken from this

2,880

tons, leaving

55\%

of it for the reserve. Multiplying

2,880

by

0.55

gives the final reserve amount of

1,584

tons.

Step-by-Step Solution

1

Calculate the percentage of the original steel shipment that remains after the hull plating allocation.

100\% - 36\% = 64\%

This establishes the base percentage for the next calculation without needing to compute the exact tonnage immediately.

2

Calculate the exact tonnage of the remaining steel.

4,500 \times 0.64 = 2,880

tons

The interior bulkheads are allocated from this remaining amount, not the original

4,500

tons.

3

Determine the percentage of the remaining steel that will go into the strategic reserve.

100\% - 45\% = 55\%

Since

45\%

of the remaining steel is used for bulkheads, the rest (

55\%

) goes to the reserve.

4

Calculate the final reserve tonnage based on the remaining steel.

2,880 \times 0.55 = 1,584

tons

This yields the final quantity of steel placed into the strategic reserve.

Key Concept

Calculating successive percentages of remaining quantities in multi-step scenarios.

Estimated Time:1m 30s

Question 230Question →

Read the scenario below and calculate the total number of complete IV bags the unit can prepare. Fill in the blank with the correct numerical value.

Fill in the blanks below

A field medical unit has a stockpile of

33 \frac{1}{3}

liters of a specialized saline solution. If each intravenous (IV) drip bag requires exactly

\frac{5}{9}

of a liter of the solution, the unit can prepare a total of complete IV bags.

Show answer & explanation

Answer

60

The unit needs to distribute the total volume of saline solution into smaller, equal-sized portions. This is an application of division. By converting the total volume into an improper fraction (

\frac{100}{3}

) and multiplying it by the reciprocal of the portion size (

\frac{9}{5}

), we determine that exactly 60 bags can be prepared.

Step-by-Step Solution

1

Convert the mixed number

33 \frac{1}{3}

to an improper fraction.

\frac{100}{3}

Improper fractions are required to perform multiplication or division with mixed numbers.

2

Set up the division expression to find how many

\frac{5}{9}

-liter bags can be made from

\frac{100}{3}

liters of solution.

\frac{100}{3} \div \frac{5}{9}

Dividing the total volume by the volume required per individual bag yields the total count of complete bags.

3

Multiply the total volume by the reciprocal of the volume per bag.

\frac{100}{3} \times \frac{9}{5} = 60

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal. Simplifying cross-terms (

100 \div 5 = 20

and

9 \div 3 = 3

) results in

20 \times 3 = 60

.

Key Concept

Dividing a mixed number by a proper fraction to determine a quantity of equal parts.

Question 231Question →

A furniture restorer buys an antique dining set for $\$250$ . The restorer spends $\$80$ on wood stain and varnish, and pays an upholsterer $\$170$ to repair the chair cushions. After the restoration is complete, the dining set is sold for $\$800$ . What is the restorer's profit as a percentage of their total cost?

Show answer & explanation

Answer: 60

Answer

60%

By summing the initial price and all repair costs, we find the total investment is

500. Subtracting this from the

800 selling price yields a

300 profit. The profit is then divided by the total cost (

300 / $500) to find the 60% profit.

Step-by-Step Solution

1

Calculate the total cost of the dining set project.

\$250 + \$80 + \$170 = \$500

The total cost must include the initial purchase price plus any additional expenses incurred before selling the item.

2

Calculate the total profit made on the sale.

\$800 - \$500 = \$300

Profit is the difference between the final selling price and the total cost.

3

Calculate the profit as a percentage of the total cost.

\frac{\$300}{\$500} \times 100 = 60\%

Percentage profit is found by dividing the dollar amount of profit by the original total cost and converting to a percentage.

Key Concept

Calculating profit percentage after accounting for overhead and multi-step production costs.

Estimated Time:1m 15s

Question 232Question →

A quartermaster is reviewing the completion status of four different supply warehouse inventories. To prioritize manpower, arrange the warehouses from the least completed inventory to the most completed inventory.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Warehouse Charlie, Warehouse Delta, Warehouse Alpha, Warehouse Bravo

By converting all values to percentages (

35\%

,

40\%

,

25\%

, and

30\%

), we can easily compare them. The lowest value is

25\%

(Charlie), followed by

30\%

(Delta), then

35\%

(Alpha), and the highest is

40\%

(Bravo).

Step-by-Step Solution

1

Convert all completion statuses to a common format, such as percentages, for direct comparison.

Warehouse Alpha =

35\%

, Warehouse Bravo =

40\%

.

Comparing numbers in different formats is difficult and prone to error.

2

Convert the fractions for Warehouse Charlie and Warehouse Delta to percentages.

Warehouse Charlie:

\frac{1}{4} = 25\%

. Warehouse Delta:

\frac{3}{10} = 30\%

.

Fractions with denominators of

4

and

10

can be easily converted to percentages.

3

Order the converted percentage values from least to greatest.

25\% < 30\% < 35\% < 40\%

.

The question asks for the order from least completed to most completed.

4

Match the ordered percentages back to their respective warehouses.

Charlie (

25\%

), Delta (

30\%

), Alpha (

35\%

), Bravo (

40\%

).

This provides the final required sequence.

Key Concept

Converting fractions and decimals to percentages to compare their values.

Estimated Time:45s

Question 233Question →

A delivery driver earns a base pay of $80.00 per day plus$ 3.50 for each package delivered. If the driver delivers 40 packages in one day, what is the driver's total gross pay for that day, in dollars?

Show answer & explanation

Answer: 220

Answer

220

The total gross pay is the sum of the fixed base pay (

80.00) and the variable delivery earnings (40 packages ×

3.50 per package =

140.00), which yields a total of

220.00.

Step-by-Step Solution

1

Calculate earnings from package deliveries

$140.00

The driver earns $3.50 for each of the 40 packages delivered (40 × 3.50).

2

Calculate total gross pay

$220.00

Add the base daily pay (

80.00) to the delivery earnings (

140.00) to find the total amount earned for the day.

Key Concept

Calculating total gross pay using a combination of base pay and piece-rate pay

Estimated Time:45s

Question 234Question →

A military detail fills $300$ sandbags in $150$ minutes. What is the constant rate at which the detail fills the sandbags, in sandbags per hour?

120

2

12

0.5

Show answer & explanation

Answer: 120

Answer

120

The unit rate is found by first converting the time into hours (

150

minutes is

2.5

hours) and then dividing the total quantity (

300

sandbags) by this time, resulting in

120

sandbags per hour.

Step-by-Step Solution

1

Convert the total time from minutes to hours.

150 \text{ minutes} \div 60 \text{ minutes/hour} = 2.5 \text{ hours}

The requested unit rate is in sandbags per hour, so the time must be expressed in hours.

2

Calculate the unit rate by dividing the total number of sandbags by the total time in hours.

300 \text{ sandbags} \div 2.5 \text{ hours} = 120 \text{ sandbags per hour}

A unit rate represents the quantity produced per single unit of time.

Key Concept

Calculating unit rates with time conversion

Question 235Question →

A homeowner secures a personal bank loan of $\$4,500$ to renovate their kitchen. The loan agreement specifies an annual simple interest rate of $8\%$ . If the homeowner repays the entire loan in exactly $9$ months, what is the total amount they must repay, including both principal and interest?

\$4,527

\$4,770

\$4,860

\$7,740

Show answer & explanation

Answer: $\$4,770$

Answer

The total repayment amount is

\$4,770

.

The correct answer accurately calculates the interest for exactly

9

months (

\$270

) and adds it to the original principal (

\$4,500

) to yield the total repayment of

\$4,770

.

Step-by-Step Solution

1

Convert the annual interest rate from a percentage to a decimal.

8\% = 0.08

Percentages must be expressed as decimals to be used in the simple interest formula.

2

Convert the loan term from months into years.

9 \text{ months} = \frac{9}{12} \text{ years} = 0.75 \text{ years}

Since the interest rate is annual, the time variable in the formula must also be expressed in years.

3

Calculate the total simple interest using the formula

I = PRT

.

I = \$4,500 \times 0.08 \times 0.75 = \$270

This formula determines the total cost of borrowing the principal over the specified time period.

4

Calculate the total repayment amount by adding the principal and the interest.

A = \$4,500 + \$270 = \$4,770

The total repayment requires paying back the original loan amount plus the accrued interest.

Key Concept

Simple Interest Applications

Question 236Question →

A naval submarine travels at a steady surface speed of $15\text{ knots}$ (nautical miles per hour) for $2\text{ hours and } 20\text{ minutes}$ before submerging. What is the total distance traveled during this time?

35\text{ nautical miles}

33\text{ nautical miles}

30\text{ nautical miles}

350\text{ nautical miles}

Show answer & explanation

Answer: $35\text{ nautical miles}$

Answer

35\text{ nautical miles}

The correct answer is found by converting the travel time of

2\text{ hours and } 20\text{ minutes}

into

2\frac{1}{3}\text{ hours}

, and then multiplying this time by the speed of

15\text{ knots}

. The calculation is

15 \times \frac{7}{3} = 35\text{ nautical miles}

.

Step-by-Step Solution

1

Convert the travel time into hours.

2\text{ hours and } 20\text{ minutes} = 2\frac{20}{60}\text{ hours} = 2\frac{1}{3}\text{ hours}

.

Speed is given in knots (nautical miles per hour), so time must be uniformly converted to hours to compute distance accurately.

2

Multiply the speed by the time to find the total distance.

\text{Distance} = 15 \times 2\frac{1}{3} = 15 \times \frac{7}{3} = 35\text{ nautical miles}

.

The formula for distance is

\text{Distance} = \text{Rate} \times \text{Time}

.

Key Concept

Calculating distance by converting minutes into a fraction of an hour and multiplying by the rate of speed.

Question 237Question →

Determine the total gross earnings based on the provided compensation structure.

Fill in the blanks below

A regional manager for a logistics company earns a base monthly salary of

\$3,200

. She receives a

5\%

commission on her first

\$40,000

in logistics contracts, and an

8\%

commission on any contract sales exceeding

\$40,000

within the month. If she secures

\$65,000

in contract sales this month, her total gross earnings will be $\$$.

Show answer & explanation

Answer

7200

The correct total is found by breaking the earnings into three parts: the

3,200 base salary, the 5% commission on the first

40,000 (

2,000), and the 8% commission on the remaining

25,000 (

2,000). Adding these together yields

7,200.

Step-by-Step Solution

1

Calculate the commission earned on the first tier of sales.

0.05 \times 40,000 = \$2,000

The manager earns a 5% commission on the first $40,000 in sales.

2

Determine the amount of sales that fall into the second commission tier.

65,000 - 40,000 = \$25,000

Sales exceeding the $40,000 threshold are subject to a different commission rate.

3

Calculate the commission earned on the second tier of sales.

0.08 \times 25,000 = \$2,000

The manager earns an 8% commission on the $25,000 that exceeded the initial threshold.

4

Sum the base salary and all commissions to find the total gross earnings.

3,200 + 2,000 + 2,000 = \$7,200

Total earnings equal the base salary plus the sum of all tiered commissions.

Key Concept

Calculating total compensation using a base salary combined with a tiered commission structure.

Question 238Question →

A naval shipyard assigns $15$ automated painting drones to coat the hull of a cruiser in exactly $2\text{ hours}$ . If $5$ of these drones are taken offline for maintenance before the job begins, how many minutes will it take the remaining drones to complete the coating?

80

360

180

3

Show answer & explanation

Answer: $180$

Answer

It will take 180 minutes for the remaining drones to complete the hull coating.

The correct answer is

180

. First, subtract the

5

offline drones from the initial

15

to find that

10

drones are actively working. Because the number of drones and the time required are inversely proportional, their product remains constant. The equation is

15 \times 2 = 10 \times T_2

, which solves to

T_2 = 3\text{ hours}

. Finally, converting

3\text{ hours}

to minutes yields

3 \times 60 = 180\text{ minutes}

.

Step-by-Step Solution

1

Determine the number of active drones.

15 - 5 = 10\text{ drones}

The 5 offline drones cannot work, so only the remaining drones will paint the hull.

2

Set up an inverse proportion equation.

15\text{ drones} \times 2\text{ hours} = 10\text{ drones} \times T_2

Work rate problems involve inverse proportions; as the number of workers decreases, the time required increases proportionally to keep the total work constant.

3

Solve for the new time (

T_2

) in hours.

30 = 10 \times T_2 \Rightarrow T_2 = 3\text{ hours}

Dividing the total required drone-hours (

30

) by the number of active drones (

10

) isolates the time.

4

Convert the required time into minutes.

3 \times 60 = 180\text{ minutes}

The question explicitly asks for the final answer in minutes, not hours.

Key Concept

Inverse Proportions

Question 239Question →

For the inequality below, which of the following expresses all possible values of $y$ ?

2(3y - 5) - 3(y - 4) \geq 6y - 16

y \leq 6

y \geq 6

y \leq -2

y \geq -2

Show answer & explanation

Answer: $y \leq 6$

Answer

The correct answer is

y \leq 6

.

The correct answer is derived by distributing the terms to get

6y - 10 - 3y + 12 \geq 6y - 16

, which simplifies to

3y + 2 \geq 6y - 16

. Subtracting

6y

from both sides and subtracting 2 from both sides yields

-3y \geq -18

. Finally, dividing by -3 requires flipping the inequality symbol, resulting in

y \leq 6

.

Step-by-Step Solution

1

Distribute the constants on the left side of the inequality.

6y - 10 - 3y + 12 \geq 6y - 16

To clear the parentheses. Note that multiplying -3 by -4 results in a positive 12.

2

Combine the like terms on the left side.

3y + 2 \geq 6y - 16

To simplify the expression before isolating the variable.

3

Move all variable terms to the left side and constant terms to the right side.

3y - 6y \geq -16 - 2

, which simplifies to

-3y \geq -18

To group the unknown variable on one side of the inequality.

4

Divide both sides by -3.

y \leq 6

To solve for y. Whenever multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.

Key Concept

Solving One-Variable Inequalities

Question 240Question →

A contractor purchases a bulk order of electrical wiring from a wholesale supply store. The store offers a $25\%$ discount on all orders over $\$500$ . Because the contractor is a loyal member, they receive an additional $10\%$ off the already discounted price. After both discounts are applied, an $8\%$ sales tax is added. If the original price of the electrical wiring order was $\$800$ , what is the final total amount the contractor pays?

\$561.60

\$583.20

\$604.00

\$826.20

Show answer & explanation

Answer: $\$583.20$

Answer

The final total amount the contractor pays is

\$583.20

.

The final amount must be calculated sequentially. First, apply the

25\%

discount to

\$800

, resulting in

\$600

. Then, apply the

10\%

loyal member discount to

\$600

, resulting in

\$540

. Finally, calculate

8\%

sales tax on the

\$540

final price (

\$540 \times 0.08 = \$43.20

) and add it to the price to get $\$583.20.

Step-by-Step Solution

1

Calculate the first

25\%

discount on the original

\$800

price.

\$800 \times 0.25 = \$200

discount. The new price is

\$800 - \$200 = \$600

.

The first discount is applied to the original wholesale price.

2

Calculate the second

10\%

loyal member discount on the new price of

\$600

.

\$600 \times 0.10 = \$60

discount. The new price is

\$600 - \$60 = \$540

.

Successive discounts are always applied to the previously discounted amount, not the original price.

3

Calculate the

8\%

sales tax on the final discounted price and add it to the total.

\$540 \times 0.08 = \$43.20

tax. The final total is

\$540 + \$43.20 = \$583.20

.

Sales tax is applied to the final amount after all discounts have been deducted.

Key Concept

Calculating successive discounts and applying sales tax to the final discounted price.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution