All practice questions

Which of the following represents the completely factored form of the algebraic expression $2x^2 - 4x - 30$ ?

2(x + 5)(x - 3)

2(x^2 - 2x - 15)

2(x - 5)(x + 3)

2(x - 5)(x - 3)

Show answer & explanation

Answer: $2(x - 5)(x + 3)$

Answer

The completely factored form is

2(x - 5)(x + 3)

.

To find the completely factored form, first factor out the greatest common factor (GCF), which is

2

. This gives

2(x^2 - 2x - 15)

. Next, factor the trinomial inside the parentheses. You need two numbers that multiply to

-15

and add up to

-2

. These numbers are

-5

and

3

. Substituting these into the binomial factors yields the final completely factored form:

2(x - 5)(x + 3)

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from all terms in the expression

2x^2 - 4x - 30

.

The GCF is

2

. Factoring it out leaves

2(x^2 - 2x - 15)

.

Identifying and extracting the GCF is always the first step in simplifying and completely factoring a polynomial.

2

Factor the remaining quadratic trinomial

x^2 - 2x - 15

by finding two integers that multiply to

-15

and add to

-2

.

The two integers are

-5

and

3

.

For a trinomial in the form

x^2 + bx + c

, the constants in the binomial factors must multiply to

c

and add to

b

.

3

Write the expression as the product of the GCF and the two binomial factors.

The fully factored expression is

2(x - 5)(x + 3)

.

This combines all factored components into their most simplified, prime form.

Key Concept

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

Question 302Question →

A maritime authority is reviewing a nautical chart to designate a restricted sailing zone. The chart uses a scale where $\frac{1}{2}\text{ inch}$ represents $30\text{ nautical miles}$ . If the restricted zone is drawn as a rectangle measuring $2\text{ inches}$ by $3\text{ inches}$ on the chart, what is the actual area of the restricted zone in square nautical miles?

360

1,350

21,600

2,160

Show answer & explanation

Answer: 21,600

Answer

21,600

The correct answer is found by first converting each chart dimension into its actual distance. Because a half inch represents 30 nautical miles, one full inch represents 60 nautical miles. Multiplying the chart's 2-inch and 3-inch dimensions by 60 yields actual dimensions of 120 and 180 nautical miles. Multiplying these true dimensions together gives a final area of 21,600 square nautical miles.

Step-by-Step Solution

1

Determine the linear scale for a full inch based on the provided fractional scale.

Since

1/2\text{ inch} = 30\text{ nautical miles}

, then

1\text{ inch} = 60\text{ nautical miles}

.

Establishing the unit rate for 1 inch simplifies the conversion of the chart dimensions.

2

Convert the chart dimensions to actual dimensions.

Width:

2\text{ inches} \times 60 = 120\text{ nautical miles}

. Length:

3\text{ inches} \times 60 = 180\text{ nautical miles}

.

Linear dimensions must be converted to actual lengths before computing the true area.

3

Calculate the actual area by multiplying the actual length and actual width.

120 \times 180 = 21,600\text{ square nautical miles}

.

The area of a rectangle is the product of its length and width.

Key Concept

Calculating true area from a scale drawing by converting linear dimensions first.

Question 303Question →

A boutique owner purchases $50$ plain canvas tote bags at a wholesale price of $\$8.00$ each. She pays a flat shipping fee of $\$100.00$ for the entire order and spends an additional $\$2.50$ per bag for custom screen printing. If she wants to make a $40\%$ profit on her total investment, what must be the selling price for a single custom tote bag?

\$12.90

\$13.00

\$15.70

\$17.50

Show answer & explanation

Answer: $\$17.50$

Answer

\$17.50

To find the selling price, the total cost per bag must first be established by combining the wholesale price, the per-bag shipping cost, and the printing fee (

\$8.00 + \$2.00 + \$2.50 = \$12.50

). Finding

40\%

of this comprehensive cost yields a

\$5.00

profit, resulting in a final selling price of

\$17.50

.

Step-by-Step Solution

1

Calculate the total cost investment per individual tote bag, including all overhead expenses.

Wholesale:

\$8.00

. Shipping per bag:

\$100.00 \div 50 = \$2.00

. Printing:

\$2.50

. Total Cost =

\$8.00 + \$2.00 + \$2.50 = \$12.50

.

Profit must be calculated on the total investment, which includes base cost, shipping, and modifications.

2

Determine the required profit amount per bag by calculating

40\%

of the total cost.

40\% \text{ of } \$12.50 = 0.40 \times 12.50 = \$5.00

.

The question asks for a

40\%

profit margin on the total cost.

3

Add the calculated profit amount to the total cost to find the final selling price.

\$12.50 \text{ (Total Cost)} + \$5.00 \text{ (Profit)} = \$17.50

.

The selling price is the sum of the total cost and the desired profit.

Key Concept

Calculating selling price based on total investment (cost plus overhead) and a desired profit percentage.

Estimated Time:1m 15s

Question 304Question →

An armored reconnaissance vehicle is conducting a route clearance mission. It travels at a steady speed of $32\text{ miles per hour}$ for $45\text{ minutes}$ . Then, due to rough terrain, its speed drops to $16\text{ miles per hour}$ for the next $1\text{ hour}$ and $15\text{ minutes}$ . What is the total distance the vehicle traveled, in miles?

Show answer & explanation

Answer: 44; 44 miles; 44 mi; forty-four; 44.0

Answer

44 miles

To find the total distance traveled, we evaluate each leg of the trip using the formula Distance = Rate × Time. The first leg is driven at 32 mph for 45 minutes (which is 0.75 hours). Distance = 32 × 0.75 = 24 miles. The second leg is driven at 16 mph for 1 hour and 15 minutes (which is 1.25 hours). Distance = 16 × 1.25 = 20 miles. The total distance is the sum of these two segments: 24 + 20 = 44 miles.

Step-by-Step Solution

1

Convert the travel time of the first leg from minutes to hours.

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

The speed is given in miles per hour, so the time must be in hours to calculate the distance properly.

2

Calculate the distance covered during the first leg.

32\text{ mph} \times 0.75\text{ hours} = 24\text{ miles}

Using the formula Distance = Rate × Time.

3

Convert the travel time of the second leg to hours.

1\text{ hour}

and

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

Time must be consistently in hours to match the rate of 16 mph.

4

Calculate the distance covered during the second leg.

16\text{ mph} \times 1.25\text{ hours} = 20\text{ miles}

Using the formula Distance = Rate × Time.

5

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

24\text{ miles} + 20\text{ miles} = 44\text{ miles}

The total distance is the sum of the individual segments.

Key Concept

Calculating distance across multiple journey segments requiring time unit conversions.

Question 305Question →

What is the completely factored form of the polynomial $3x^2 - 12x - 36$ ?

3(x - 6)(x + 2)

3(x + 6)(x - 2)

(3x + 6)(x - 6)

3(x - 6)(x - 2)

Show answer & explanation

Answer: $3(x - 6)(x + 2)$

Answer

3(x - 6)(x + 2)

To completely factor the polynomial, the greatest common factor must be extracted first. Factoring out the GCF of 3 yields

3(x^2 - 4x - 12)

. Next, factoring the trinomial

x^2 - 4x - 12

requires two integers that multiply to

-12

and sum to

-4

. These integers are

-6

and

2

. Therefore, the completely factored form is

3(x - 6)(x + 2)

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from all terms in the polynomial.

3(x^2 - 4x - 12)

Extracting the GCF simplifies the expression, making the quadratic trinomial easier to factor.

2

Find two numbers that multiply to the constant term (

-12

) and add to the coefficient of the middle term (

-4

).

The numbers are

-6

and

2

, because

(-6) \times 2 = -12

and

(-6) + 2 = -4

.

These values are required to correctly split the middle term and factor the quadratic trinomial.

3

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

3(x - 6)(x + 2)

This represents the polynomial broken down into its simplest, irreducible algebraic factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.

Question 306Question →

A graphic designer is purchasing a new drawing tablet. The regular price of the tablet is $\$200.00$ . If the electronics store is offering a $30\%$ discount on all tablets, what is the final sale price of the tablet?

\$140.00

\$170.00

\$60.00

\$194.00

Show answer & explanation

Answer: $\$140.00$

Answer

The final sale price of the drawing tablet is

\$140.00

.

To find the final sale price, first calculate the discount amount by multiplying the regular price by the discount percentage (

\$200.00 \times 0.30 = \$60.00

). Then, subtract this discount amount from the regular price to find the total cost (

\$200.00 - \$60.00 = \$140.00

).

Step-by-Step Solution

1

Convert the discount percentage into a decimal.

30\% = 0.30

To calculate the discount amount, the percentage must be in decimal form.

2

Multiply the regular price by the decimal to find the discount amount.

\$200.00 \times 0.30 = \$60.00

This determines exactly how much money will be taken off the regular price.

3

Subtract the discount amount from the regular price.

\$200.00 - \$60.00 = \$140.00

The final sale price is the original price minus the total savings.

Key Concept

Calculating a single-step percentage discount and subtracting it from the original price to find the final sale price.

Alternative Method

Instead of calculating the discount amount and subtracting, you can multiply the original price by the percentage you will actually pay. If the discount is

30\%

, you are paying

100\% - 30\% = 70\%

of the original price. Calculate

70\%

of

\$200.00

:

\$200.00 \times 0.70 = \$140.00

.

Estimated Time:45s

Question 307Question →

A supply clerk is comparing prices for standard issue notebooks. Vendor A sells a pack of $5$ notebooks for $15.00$ dollars. Vendor B sells a bulk box of $20$ notebooks for $48.00$ dollars. What is the savings per notebook, in dollars, if the clerk chooses Vendor B instead of Vendor A?

Show answer & explanation

Answer: 0.6

Answer

0.60

The unit price is found by dividing the total cost by the quantity. Vendor A charges

15.00 \div 5 = 3.00

dollars per notebook. Vendor B charges

48.00 \div 20 = 2.40

dollars per notebook. The savings per notebook is the difference:

3.00 - 2.40 = 0.60

dollars.

Step-by-Step Solution

1

Calculate Vendor A's unit price

3.00

dollars per notebook

To find the cost of a single item in a package, divide the total price by the number of items (

15.00 \div 5

).

2

Calculate Vendor B's unit price

2.40

dollars per notebook

Divide the bulk box total price by the number of notebooks it contains (

48.00 \div 20

).

3

Calculate the difference between the two unit prices

0.60

dollars

Subtracting Vendor B's unit price from Vendor A's unit price (

3.00 - 2.40

) gives the exact savings per notebook.

Key Concept

Unit price calculation and comparison

Question 308Question →

A logistics unit is transporting a total of $300$ supply crates. If $25\%$ of the crates contain medical equipment, how many medical crates are in the transport?

Show answer & explanation

Answer: 75

Answer

75

By converting

25\%

to the fraction

\frac{1}{4}

(or the decimal

0.25

) and multiplying it by the total of

300

supply crates, we determine that exactly

75

crates contain medical equipment.

Step-by-Step Solution

1

Convert the given percentage to a decimal or a simple fraction.

25\% = 0.25

or

\frac{1}{4}

Percentages must be converted into decimals or fractions before performing calculations with a base quantity.

2

Multiply the total quantity by the converted decimal or fraction to find the portion.

300 \times 0.25 = 75

Finding a percentage 'of' a number indicates multiplication.

Key Concept

To calculate a percentage of a quantity, convert the percent to a decimal or fraction and multiply it by the total amount.

Estimated Time:45s

Question 309Question →

A naval submarine travels a total distance of $144$ miles in $4$ hours and $30$ minutes. Assuming a constant rate of travel, which of the following is the average speed of the submarine in miles per hour?

3.2

32

36

648

Show answer & explanation

Answer: $32$

Answer

The average speed is

32

miles per hour.

To find the average speed, the total distance must be divided by the total time in hours. First, convert

4

hours and

30

minutes to

4.5

hours. Then, dividing

144

miles by

4.5

hours gives an average speed of

32

miles per hour.

Step-by-Step Solution

1

Convert the total travel time into a single unit of hours.

4

hours and

30

minutes equals

4.5

hours.

Since the requested speed is in miles per hour, the time must be expressed purely in hours before calculating.

2

Apply the formula for average speed:

Speed = Distance \div Time

.

Speed = 144 \div 4.5

Average speed is found by dividing the total distance traveled by the total time taken.

3

Perform the division.

1440 \div 45 = 32

Shifting the decimal point one place to the right for both the dividend and divisor makes the calculation easier, resulting in

32

miles per hour.

Key Concept

Calculating Rate given Distance and Time

Question 310Question →

Complete the factored form of the given quadratic polynomial by determining the missing constant.

What number correctly fills in the blank?

Fill in the blanks below

$n^2 + 11n + 24 = (n + 3)(n + )$

Show answer & explanation

Answer

8

The polynomial

n^2 + 11n + 24

factors into two binomials. We are given one factor,

(n + 3)

. To find the missing value in the second factor

(n + b)

, we can use the property that the product of the constant terms in the binomials must equal the constant term of the trinomial. Therefore,

3 \times b = 24

, which means

b = 8

. We verify this by adding the constants:

3 + 8 = 11

, which matches the coefficient of the middle term. Thus, the correct value for the blank is 8.

Step-by-Step Solution

1

Identify the relationship between the constant term of the polynomial and the constants in its binomial factors.

The product of the constants in the binomial factors must equal the constant term of the polynomial, 24.

When expanding

(n + a)(n + b)

, the constant term is calculated as

a \times b

.

2

Set up an equation to find the missing value.

3 \times \text{missing value} = 24

.

One of the binomial factors is given as

(n + 3)

, so one constant is 3.

3

Solve for the missing value and verify with the middle term.

The missing value is 8, since

24 \div 3 = 8

. Checking the middle term,

3 + 8 = 11

, which matches

11n

.

This confirms that 8 satisfies both the multiplication to 24 and the addition to 11 requirements for factoring the trinomial.

Key Concept

Factoring quadratic trinomials of the form x^2 + bx + c.

Question 311Question →

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

Click a left item, then click its matching right item

Show answer & explanation

Answer

Alpha matches to

3:1

, Bravo matches to

3:4

, Charlie matches to

7:3

, and Delta matches to

7:5

.

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

Step-by-Step Solution

1

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

45:15

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

2

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

3:1

The GCD of

45

and

15

is

15

.

45 \div 15 = 3

and

15 \div 15 = 1

.

3

Set up and simplify the ratio for Bravo Unit (

24

to

32

).

3:4

The GCD of

24

and

32

is

8

.

24 \div 8 = 3

and

32 \div 8 = 4

.

4

Set up and simplify the ratio for Charlie Unit (

42

to

18

).

7:3

The GCD of

42

and

18

is

6

.

42 \div 6 = 7

and

18 \div 6 = 3

.

5

Set up and simplify the ratio for Delta Unit (

35

to

25

).

7:5

The GCD of

35

and

25

is

5

.

35 \div 5 = 7

and

25 \div 5 = 5

.

Key Concept

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

Estimated Time:1m 30s

Question 312Question →

A naval destroyer is tasked with an interception mission. It travels in a straight line at a constant speed of $35\text{ knots}$ (where $1\text{ knot} = 1\text{ nautical mile per hour}$ ) for exactly $3\text{ hours and } 12\text{ minutes}$ . What is the total distance traveled by the destroyer?

109.2\text{ nautical miles}

112\text{ nautical miles}

1,120\text{ nautical miles}

6,720\text{ nautical miles}

Show answer & explanation

Answer: $112\text{ nautical miles}$

Answer

112\text{ nautical miles}

The correct distance is found by converting the travel time entirely into hours to match the rate. Since there are 60 minutes in an hour,

12\text{ minutes}

is equal to

0.2\text{ hours}

(

12 \div 60

). Adding this to the 3 hours gives a total travel time of

3.2\text{ hours}

. Multiplying the rate of

35\text{ knots}

by

3.2\text{ hours}

yields exactly

112\text{ nautical miles}

.

Step-by-Step Solution

1

Convert the given time into a single unit that matches the rate (hours).

12\text{ minutes} \div 60\text{ minutes/hour} = 0.2\text{ hours}

. Total time =

3.2\text{ hours}

.

Speed is given in knots (nautical miles per hour), so time must be expressed purely in hours to use the formula Distance = Rate x Time.

2

Multiply the rate of speed by the total time in hours.

35\text{ knots} \times 3.2\text{ hours} = 112\text{ nautical miles}

.

Applying the distance formula (d = r * t) yields the total distance traveled.

Key Concept

Calculating distance by converting mixed time units into a decimal fraction of an hour before multiplying by speed.

Question 313Question →

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

Click a left item, then click its matching right item

Show answer & explanation

Answer

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

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

Step-by-Step Solution

1

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

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

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

2

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

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

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

x^0 = 1

simplifies the second term before multiplication.

3

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

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

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

4

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

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

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

Key Concept

Operations with Variable Exponents

Question 314Question →

A structural engineer is analyzing a blueprint for a steel bridge truss. The blueprint uses a scale where $\frac{7}{8}\text{ inch}$ represents an actual length of $21\text{ feet}$ . If a particular support beam measures $6.5\text{ inches}$ on the blueprint, what is the actual length of this beam in feet?

Show answer & explanation

Answer: 156

Answer

156

The correct answer is found by establishing a proportion based on the scale. Since

\frac{7}{8}\text{ inch}

equals

21\text{ feet}

,

1\text{ inch}

must equal

21 \div \frac{7}{8}

, which is

24\text{ feet}

. Multiplying the blueprint length of

6.5\text{ inches}

by this unit rate of

24

gives an actual length of

156\text{ feet}

.

Step-by-Step Solution

1

Determine the unit scale factor in feet per inch.

21 \div \frac{7}{8} = 24\text{ feet/inch}

Finding the actual distance represented by a single inch creates a simple multiplier for any dimension on the blueprint.

2

Multiply the measurement on the blueprint by the unit scale factor.

6.5 \times 24 = 156\text{ feet}

Scaling up the drawing measurement by the unit rate yields the actual real-world length of the beam.

Key Concept

Direct proportions and unit rates in scale drawings

Alternative Method

You can set up a direct proportion:

\frac{\frac{7}{8}}{21} = \frac{6.5}{x}

. Cross-multiplying gives

\frac{7}{8}x = 136.5

. Multiplying both sides by

\frac{8}{7}

yields

x = 156

.

Estimated Time:1m 0s

Question 315Question →

A disaster response team has a stockpile of $93 \frac{3}{4}$ pounds of water purification powder. They must first reserve $11 \frac{1}{4}$ pounds of the powder for the main field hospital. The team then distributes the remaining powder evenly into smaller distribution kits. If each kit requires exactly $3 \frac{3}{4}$ pounds of powder, how many complete distribution kits can they fill?

25

27

22

28

Show answer & explanation

Answer: 22

Answer

The team can fill 22 complete distribution kits.

The correct answer is 22 because the problem requires two steps. First, find the amount of powder available for the kits by subtracting the reserved amount from the total stockpile:

93 \frac{3}{4} - 11 \frac{1}{4} = 82 \frac{2}{4} = 82 \frac{1}{2}

pounds. Second, determine how many kits can be filled by dividing the remaining powder by the amount required per kit:

82 \frac{1}{2} \div 3 \frac{3}{4}

. Convert to improper fractions to divide:

\frac{165}{2} \div \frac{15}{4}

. Multiply by the reciprocal of the divisor:

\frac{165}{2} \times \frac{4}{15}

. Cross-simplify and multiply to get

\frac{11}{1} \times \frac{2}{1} = 22

complete kits.

Step-by-Step Solution

1

Subtract the reserved powder from the total stockpile to find the remaining powder available for kits.

93 \frac{3}{4} - 11 \frac{1}{4} = 82 \frac{2}{4} = 82 \frac{1}{2}

pounds

The

11 \frac{1}{4}

pounds are set aside, so they cannot be used to fill the distribution kits.

2

Divide the remaining powder by the amount of powder required for a single kit.

82 \frac{1}{2} \div 3 \frac{3}{4} = \frac{165}{2} \div \frac{15}{4} = \frac{165}{2} \times \frac{4}{15} = 22

kits

Dividing the total available amount by the size of one batch determines how many complete batches can be formed.

Key Concept

Solving multi-step word problems involving fraction subtraction and division.

Question 316Question →

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

Click a left item, then click its matching right item

Show answer & explanation

Answer

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

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

t = \frac{d}{r}

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

Step-by-Step Solution

1

Identify the given distance (

d

) and rate (

r

) for each scenario.

Van: (

150

mi,

50

mph), Ship: (

80

mi,

20

mph), Runner: (

16

mi,

8

mph), Train: (

300

mi,

60

mph).

These variables are required to apply the fundamental distance formula.

2

Apply the formula for calculating time:

t = \frac{d}{r}

.

Van:

\frac{150}{50} = 3

. Ship:

\frac{80}{20} = 4

. Runner:

\frac{16}{8} = 2

. Train:

\frac{300}{60} = 5

.

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

Key Concept

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

t = \frac{d}{r}

).

Question 317Question →

Read the scenario below and fill in the missing numerical values for the discount amount and percentage.

Fill in the blanks below

A gym owner is purchasing a set of free weights. The original price of the weights is

\$500

. During a seasonal clearance sale, the gym owner purchases the set for

\$400

.

Based on these prices, the dollar amount of the discount is

\$

and the discount percentage is

\%

.

Show answer & explanation

Answer

The discount amount is $100 and the discount percentage is 20%.

The discount amount is the difference between the original price (

\$500

) and the sale price (

\$400

), which is

\$100

. The discount rate is the discount amount divided by the original price (

\$100 / \$500 = 0.20

), which equals

20\%

.

Step-by-Step Solution

1

Calculate the dollar amount of the discount.

Discount =

\$500

-

\$400

=

\$100

.

The discount amount is found by subtracting the sale price from the original price.

2

Calculate the discount percentage.

(\$100 / \$500) \times 100 = 0.20 \times 100 = 20\%

.

The discount rate is calculated by dividing the discount amount by the original price.

Key Concept

Calculating discount amounts and percentage rates based on original and sale prices.

Alternative Method

You can also determine the percentage the owner paid (

\$400 / \$500 = 0.80

, or

80\%

) and subtract that from

100\%

to find the discount percentage (

100\% - 80\% = 20\%

).

Estimated Time:45s

Question 318Question →

An aviation unit's weekly fuel allowance is $4,000$ gallons. If they have used $60\%$ of their allowance during training exercises, how many gallons of fuel have they used?

240

3,940

2,400

6,400

Show answer & explanation

Answer: $2,400$

Answer

2,400

To find how much fuel was used, we calculate

60\%

of the total

4,000

gallons. Converting

60\%

to a decimal gives us

0.60

. Multiplying

4,000

by

0.60

results in

2,400

gallons.

Step-by-Step Solution

1

Convert the percentage to a decimal or fraction.

60\% = 0.60

Percentages must be converted into a workable mathematical format like a decimal before multiplying.

2

Multiply the total weekly allowance by the decimal to find the amount of fuel used.

4,000 \times 0.60 = 2,400

gallons

Multiplying the total quantity by the percentage (as a decimal) yields the portion that represents that percentage.

Key Concept

To calculate a percentage of a quantity, convert the percentage to a decimal (or fraction) and multiply it by the total quantity.

Question 319Question →

Which of the following represents the fully simplified form of the algebraic expression below, assuming $x \neq 0$ and $y \neq 0$ ?

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

\frac{y^{10}}{4x^{10}}

\frac{4x^2}{y^6}

\frac{4x^{10}}{y^{10}}

4x^{12}y^8

Show answer & explanation

Answer: $\frac{4x^{10}}{y^{10}}$

Answer

The expression simplifies to

\frac{4x^{10}}{y^{10}}

.

The correct answer is derived by first applying the quotient rule to simplify the expression inside the parentheses, being careful to subtract negative exponents correctly (e.g.,

y^{4 - (-1)} = y^5

). Then, the negative exponent on the outside of the parentheses means taking the reciprocal of the fraction and squaring all terms, which applies to the coefficient (2 becomes 4) and the variables (

x^5

becomes

x^{10}

).

Step-by-Step Solution

1

Simplify the numerical coefficients inside the parentheses.

\frac{2}{4} = \frac{1}{2}

Simplifying the fraction before handling variables reduces computational complexity.

2

Apply the quotient rule for exponents to the variables

x

and

y

.

For

x

:

-3 - 2 = -5

. For

y

:

4 - (-1) = 4 + 1 = 5

. The expression inside the parentheses becomes

\frac{1}{2} x^{-5}y^5

or

\frac{y^5}{2x^5}

.

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

3

Rewrite the entire expression with the simplified inner term.

\left( \frac{y^5}{2x^5} \right)^{-2}

This establishes the simplified base expression before applying the final outer exponent.

4

Apply the negative exponent property to the fraction.

\left( \frac{2x^5}{y^5} \right)^2

A negative exponent on a fraction indicates the reciprocal of that fraction raised to the positive exponent.

5

Apply the power of a quotient and product rules.

\frac{2^2(x^5)^2}{(y^5)^2} = \frac{4x^{10}}{y^{10}}

When raising a power to a power, multiply the exponents, and ensure the outer exponent is applied to the numerical coefficient.

Key Concept

Operations with Variable Exponents

Estimated Time:2m 0s

Question 320Question →

A medical logistics unit prepares field trauma kits. The weight ratio of bandages to antiseptic solution in each bulk shipment is strictly maintained at $3:4$ . If a particular shipment contains $12$ pounds of bandages, what is the required weight of the antiseptic solution in ounces?

16

256

64

144

Show answer & explanation

Answer: 256

Answer

The correct weight of the antiseptic solution is 256 ounces.

The ratio of bandages to antiseptic is

3:4

. Since

12

pounds of bandages represents

3

parts of the ratio, each part is equal to

4

pounds (

12 \div 3 = 4

). The antiseptic represents

4

parts, so its weight is

4 \times 4 = 16

pounds. Finally, to find the weight in ounces, multiply by

16

(since there are

16

ounces in a standard pound), yielding

16 \times 16 = 256

ounces.

Step-by-Step Solution

1

Set up a proportion using the given ratio of bandages to antiseptic (

3:4

).

\frac{3}{4} = \frac{12}{x}

The weight of the items must maintain the specific part-to-part ratio given in the problem.

2

Solve for

x

to find the weight of the antiseptic in pounds.

3x = 48 \Rightarrow x = 16

pounds

Cross-multiplying and dividing isolates the unknown quantity of antiseptic.

3

Convert the weight of the antiseptic from pounds to ounces.

16 \text{ pounds} \times 16 \text{ ounces/pound} = 256 \text{ ounces}

The question explicitly requests the final answer in ounces.

Key Concept

Solving basic ratios and applying unit conversions.

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

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Key Concept

Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes