All practice questions

Question 1Question →

A communications specialist is preparing a field antenna and needs to cut a $60$ -foot spool of copper wire into two sections in a ratio of $7:3$ .

Fill in the blanks below

The longer section of the wire will be feet long.

Show answer & explanation

Answer

The longer section of the wire will be 42 feet long.

The wire is divided into

7 + 3 = 10

total parts. Since the total length is

60

feet, each part is equal to

60 \div 10 = 6

feet. The longer section is

7

parts, so its length is

7 \times 6 = 42

feet.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

7 + 3 = 10

total parts.

The total length of the wire is divided into this many equal parts.

2

Calculate the length of one single part.

60 \div 10 = 6

feet per part.

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

3

Determine the length of the longer section.

7 \times 6 = 42

feet.

The longer section corresponds to the larger number in the ratio, which is 7 parts.

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 sum to find the value of one part, and then multiply by the specific ratio term.

Question 2Question →

A specialized equipment technician earns a base wage of $28.00 per hour for a standard 40-hour workweek, with any additional hours paid at time-and-a-half. The technician also earns a tiered commission on equipment sold: 4% on the first$ 10,000 of sales, and a higher commission rate on all sales beyond $10,000.

Last week, the technician worked 48 hours, sold $30,000 worth of equipment, and had total gross earnings of$ 3,056.

Calculate the commission rate for the sales exceeding $10,000.

Fill in the blanks below

The commission rate on sales exceeding $10,000 is %.

Show answer & explanation

Answer

The commission rate on sales exceeding $10,000 is 6%.

The correct answer is found by first isolating the total commission earned (

3,056 gross -

1,456 wages =

1,600 commission). Then, subtract the

400 earned from the first

10,000 tier. This leaves

1,200 in commission earned on the remaining

20,000 in sales. Dividing

1,200 by $20,000 yields a 0.06 or 6% commission rate.

Step-by-Step Solution

1

Calculate the base pay for standard hours.

$1,120

The first 40 hours are paid at the standard rate: 40 hours × $28.00/hour.

2

Calculate the overtime pay.

$336

The overtime rate is

28.00 × 1.5 =

42.00/hour. The technician worked 8 overtime hours (48 - 40).

42.00 × 8 =

336.

3

Calculate the total hourly wages.

$1,456

Add the base pay and overtime pay:

1,120 +

336 = $1,456.

4

Calculate the total commission earned.

$1,600

Subtract the total hourly wages from the total gross earnings:

3,056 -

1,456 = $1,600.

5

Calculate the tier 1 commission amount.

$400

The commission is 4% on the first

10,000 of sales:

10,000 × 0.04 = $400.

6

Calculate the tier 2 commission amount and tier 2 sales volume.

Tier 2 Commission:

1,200. Tier 2 Sales:

20,000.

Subtract tier 1 commission from total commission (

1,600 -

400 =

1,200). Subtract tier 1 sales from total sales (

30,000 -

10,000 =

20,000).

7

Determine the tier 2 commission rate.

6%

Divide the tier 2 commission by the tier 2 sales volume:

1,200 ÷

20,000 = 0.06, which is 6%.

Key Concept

Reverse engineering multi-tier compensation structures involving overtime wages and graduated commission rates.

Question 3Question →

A communications platoon is upgrading its field radio systems. The unit has a single continuous spool of copper grounding wire that measures $42 \frac{1}{2}$ meters in length. Each radio system requires exactly $2 \frac{5}{6}$ meters of this wire for a proper ground connection. Calculate the maximum number of complete radio systems the platoon can ground using this spool.

Fill in the blanks below

The platoon can ground a total of complete radio systems.

Show answer & explanation

Answer

15

To determine how many complete systems can be grounded, the total length of the wire (

42 \frac{1}{2}

meters) must be divided by the length required per system (

2 \frac{5}{6}

meters). Converting both to improper fractions gives

\frac{85}{2} \div \frac{17}{6}

. Multiplying by the reciprocal yields

\frac{85}{2} \times \frac{6}{17}

, which simplifies to

5 \times 3 = 15

.

Step-by-Step Solution

1

Convert both mixed numbers to improper fractions.

42 \frac{1}{2} = \frac{85}{2}

and

2 \frac{5}{6} = \frac{17}{6}

Multiplication and division of fractions require improper fractions rather than mixed numbers.

2

Set up the division equation.

\frac{85}{2} \div \frac{17}{6}

To find how many items of a specific size fit into a total quantity, you must divide the total by the size of one item.

3

Rewrite the division problem as multiplication by taking the reciprocal of the divisor.

\frac{85}{2} \times \frac{6}{17}

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

4

Simplify by cross-canceling common factors and multiply.

Divide

85

by

17

to get

5

. Divide

6

by

2

to get

3

. Then,

5 \times 3 = 15

.

Cross-canceling before multiplying keeps the numbers small and manageable without a calculator.

Key Concept

Dividing a total quantity by a fractional size using improper fractions and reciprocal multiplication.

Question 4Question →

Determine the missing constants in the completely factored form of the given polynomial.

Fill in the blanks below

The polynomial

5x^2 - 5x - 60

can be factored completely. When written in the form

a(x - b)(x + c)

, where

b

and

c

are positive integers, the value of

a

is , the value of

b

is , and the value of

c

is .

Show answer & explanation

Answer

The constants are a = 5, b = 4, and c = 3.

The correct values are found by first factoring out the greatest common factor of

5

, leaving the quadratic expression

x^2 - x - 12

. This quadratic expression factors into two binomials by finding two numbers that multiply to

-12

and add to

-1

, which are

-4

and

3

. This yields the fully factored form

5(x - 4)(x + 3)

. By matching this result to the required algebraic structure

a(x - b)(x + c)

, we can directly identify that

a = 5

,

b = 4

, and

c = 3

.

Step-by-Step Solution

1

Find the greatest common factor (GCF) of the terms in the polynomial

5x^2 - 5x - 60

.

The GCF is

5

. Factoring it out gives

5(x^2 - x - 12)

.

Extracting the GCF simplifies the trinomial and makes it easier to factor into binomials.

2

Factor the remaining quadratic trinomial

x^2 - x - 12

.

We look for two integers that multiply to

-12

and add to

-1

.

This is the standard procedure for factoring a monic quadratic trinomial of the form

x^2 + bx + c

.

3

Identify the proper integers and write the binomial factors.

The integers are

-4

and

3

. The factored trinomial is

(x - 4)(x + 3)

.

(-4) \times 3 = -12

and

-4 + 3 = -1

.

4

Combine the factors and match them to the target form

a(x - b)(x + c)

.

The completely factored expression is

5(x - 4)(x + 3)

. Matching this to

a(x - b)(x + c)

yields

a = 5

,

b = 4

, and

c = 3

.

This establishes the precise values corresponding to the blanks in the question.

Key Concept

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

Question 5Question →

Find the completely factored form of the given polynomial:

2x^2 + 2x - 40

Assume all missing values in the blanks are positive integers.

Fill in the blanks below

The completely factored expression is (x + )(x - ).

Show answer & explanation

Answer

The missing values are 2, 5, and 4. The completely factored expression is 2(x + 5)(x - 4).

Factoring out the greatest common factor of 2 leaves

x^2 + x - 20

. This trinomial factors into

(x + 5)(x - 4)

because 5 and -4 multiply to give -20 and add to give 1. Thus, the complete factored form is

2(x + 5)(x - 4)

, corresponding to the blank values 2, 5, and 4.

Step-by-Step Solution

1

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

2(x^2 + x - 20)

Identifying and extracting the GCF of 2 simplifies the remaining trinomial, making it easier to factor.

2

Factor the resulting quadratic trinomial

x^2 + x - 20

.

Find two numbers that multiply to -20 and add to 1 (the coefficient of x).

To factor a trinomial of the form

x^2 + bx + c

, you must find two factors of

c

that add up to

b

.

3

Identify the two numbers that satisfy the conditions.

The numbers are 5 and -4.

5 \times (-4) = -20

and

5 + (-4) = 1

.

4

Write the complete factorization by combining the GCF and the factored binomials.

2(x + 5)(x - 4)

This represents the original polynomial completely factored into its simplest terms.

Key Concept

Factoring Polynomials with a Greatest Common Factor

Question 6Question →

Read the compensation scenario below and fill in the blanks with the correct numerical values.

Fill in the blanks below

A salon stylist earns a weekly base salary of

450. She also receives a 20% commission on all chemical treatments and a 10% commission on retail product sales. In a single week, she bills

850 for chemical treatments and sells

240 in retail products. For this week, the stylist earned a total of

in commissions, which brings her total gross pay to $.

Show answer & explanation

Answer

The stylist earned

194 in total commissions, making her total gross pay

644.

The correct commission is found by multiplying each sales category by its respective percentage (

850 \times 0.20 = 170

and

240 \times 0.10 = 24

), summing them (

170 + 24 = 194

), and adding the result to the

450 base salary for a total of

644.

Step-by-Step Solution

1

Calculate the commission earned from chemical treatments.

850 \times 0.20 = 170

The stylist earns 20% commission on the $850 billed for these specific services.

2

Calculate the commission earned from retail product sales.

240 \times 0.10 = 24

The stylist earns 10% commission on the $240 worth of retail sales.

3

Determine the total commission earned for the week.

170 + 24 = 194

Total commission is the sum of the commissions from both categories.

4

Calculate the total gross pay.

450 + 194 = 644

Total gross pay is the base salary plus the total commission earned.

Key Concept

Calculating varied commission rates on different sales categories and combining them with a base salary to find gross earnings.

Estimated Time:1m 30s

Question 7Question →

Determine the total gross earnings for the month based on the given compensation structure.

Fill in the blanks below

During the month of October, a medical equipment consultant recorded total sales of

\$45,000

. His compensation package includes a guaranteed monthly base salary of

\$1,200

. On top of this base, he earns a

5\%

commission on the first

\$20,000

of his sales, and an

8\%

commission on all sales volume beyond that initial

\$20,000

. Based on these figures, the consultant's total gross pay for October is $\$$.

Show answer & explanation

Answer

The consultant's total gross pay is $4,200.

The total gross pay is found by isolating the sales into two tiers: the first

20,000 earns a 5% commission (

1,000) and the remaining

25,000 earns an 8% commission (

2,000). Adding these two commissions to the

1,200 base salary yields a total gross pay of

4,200.

Step-by-Step Solution

1

Calculate the commission earned on the first tier of sales.

\$20,000 \times 0.05 = \$1,000

The consultant earns a 5% commission on the initial $20,000 of sales.

2

Determine the amount of sales that qualify for the second commission tier.

\$45,000 - \$20,000 = \$25,000

The higher commission rate only applies to the sales volume exceeding the first $20,000.

3

Calculate the commission earned on the second tier of sales.

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

The consultant earns an 8% commission on the remaining $25,000 of sales.

4

Sum the base salary and both commission amounts to find the total gross pay.

\$1,200 + \$1,000 + \$2,000 = \$4,200

Total earnings are the sum of the guaranteed salary and all commissions earned.

Key Concept

Calculating total earnings involving a base salary and a multi-tiered commission structure.

Question 8Question →

Complete the statement regarding the electrical parameters of the DC circuit by calculating the missing resistance and current values.

Fill in the blanks below

An indicator lamp with a constant resistance draws

150\text{ mA}

when connected to a

24\text{ V}

direct current source. The internal resistance of the lamp is

\Omega

, and if the supply voltage is increased to

36\text{ V}

, the current drawn by the lamp will increase to

\text{mA}

.

Show answer & explanation

Answer

The internal resistance of the lamp is 160 Ω, and the new current at 36 V is 225 mA.

Converting

150\text{ mA}

to standard amperes yields

0.15\text{ A}

. Using Ohm's law (

R = V / I

), the internal resistance is

24\text{ V} / 0.15\text{ A} = 160\ \Omega

. When the voltage rises to

36\text{ V}

, applying

I = V / R

produces

36\text{ V} / 160\ \Omega = 0.225\text{ A}

, which corresponds to

225\text{ mA}

.

Step-by-Step Solution

1

Convert the nominal current from milliamperes to base amperes

I_1 = 150\text{ mA} = 0.15\text{ A}

Ohm's law requires base SI units (amperes, volts, ohms) for direct calculation.

2

Calculate the constant internal resistance using Ohm's law

R = \frac{V_1}{I_1} = \frac{24\text{ V}}{0.15\text{ A}} = 160\ \Omega

Resistance equals voltage divided by current.

3

Calculate the new current at the elevated voltage level

I_2 = \frac{V_2}{R} = \frac{36\text{ V}}{160\ \Omega} = 0.225\text{ A} = 225\text{ mA}

Current across a fixed resistance is directly proportional to applied voltage (

I = V / R

).

Key Concept

Ohm's Law (

V = I \times R

) and Direct Proportionality in DC Circuits

Estimated Time:45s

Question 9Question →

Calculate the missing financial values for the equipment purchase described below.

Fill in the blanks below

A veterinary clinic is purchasing a new digital ultrasound machine originally listed at

\$6,500

. The medical supplier is offering a

20\%

promotional discount. Because the clinic agrees to pick up the equipment themselves, they receive an additional

5\%

discount off the already reduced price.

The total discount amount the clinic receives is $\$

and the final price paid for the machine is

\$$.

Show answer & explanation

Answer

The total discount amount is

1,560 and the final price is

4,940.

The first discount of 20% saves

1,300, bringing the price down to

5,200. The second discount of 5% applies to this new

5,200 balance, saving an additional

260. Subtracting the second discount from

5,200 yields a final price of

4,940. The total amount saved combines both discounts:

1,300 +

260 = $1,560.

Step-by-Step Solution

1

Calculate the amount of the first discount.

$1,300

Multiply the original price ($6,500) by the first discount rate (20% or 0.20).

2

Calculate the price after the first discount.

$5,200

Subtract the first discount (

1,300) from the original price (

6,500).

3

Calculate the amount of the second discount.

$260

Multiply the reduced price ($5,200) by the additional discount rate (5% or 0.05).

4

Determine the final price.

$4,940

Subtract the second discount (

260) from the previously reduced price (

5,200).

5

Determine the total discount amount.

$1,560

Add the first discount (

1,300) and the second discount (

260).

Key Concept

Successive Discounts

Question 10Question →

Simplify the given algebraic expression by factoring it completely, then determine the missing positive integer values.

Fill in the blanks below

The completely factored form of the expression

2y^2 - 10y - 28

can be written in the format

2(y - a)(y + b)

, where

a

and

b

are positive integers.

The value of

a

is and the value of

b

is .

Show answer & explanation

Answer

The value of a is 7 and the value of b is 2.

Factoring out the GCF of 2 yields

2(y^2 - 5y - 14)

. The trinomial

y^2 - 5y - 14

factors into

(y - 7)(y + 2)

. Matching this with the given structure

2(y - a)(y + b)

shows that the positive integers are

a = 7

and

b = 2

.

Step-by-Step Solution

1

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

2(y^2 - 5y - 14)

Identifying and factoring out the GCF of 2 simplifies the polynomial, revealing a trinomial with a leading coefficient of 1, which is easier to factor.

2

Factor the resulting quadratic trinomial

y^2 - 5y - 14

.

The binomial factors are

(y - 7)

and

(y + 2)

.

To factor

y^2 - 5y - 14

, we need to find two numbers that multiply to the constant term (-14) and add up to the middle term's coefficient (-5). The numbers -7 and 2 satisfy both conditions (-7 * 2 = -14 and -7 + 2 = -5).

3

Write the complete factorization and match it to the provided template

2(y - a)(y + b)

.

The expression is

2(y - 7)(y + 2)

. Matching this to the template gives

a = 7

and

b = 2

.

Comparing the forms directly, the term subtracted from y is 7 (so a = 7) and the term added to y is 2 (so b = 2).

Key Concept

Factoring polynomials using the Greatest Common Factor (GCF) and factoring quadratic trinomials.

Question 11Question →

Determine the missing side length based on the geometric properties of a right triangle.

Fill in the blanks below

In a right triangle, the two shorter sides (legs) measure

9

cm and

12

cm. According to the Pythagorean theorem, the length of the hypotenuse is cm.

Show answer & explanation

Answer

15

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) to a right triangle with legs of

9

and

12

, we square both numbers to get

81

and

144

. Their sum is

225

. The hypotenuse is the square root of

225

, which is

15

.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The legs (let's call them

a

and

b

) measure

9

cm and

12

cm.

The Pythagorean theorem requires knowing the lengths of two sides to find the third.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, which becomes

9^2 + 12^2 = c^2

.

This formula relates the squares of the legs to the square of the hypotenuse (

c

).

3

Square the lengths of the two legs and add them together.

81 + 144 = 225

This calculation determines the value of

c^2

.

4

Take the square root of the sum to find the hypotenuse.

\sqrt{225} = 15

Since

c^2 = 225

, taking the square root gives the actual length of the hypotenuse (

c

).

Key Concept

Pythagorean Theorem

Question 12Question →

Determine the required dimensions for the construction project.

Fill in the blanks below

A construction crew is building a wooden ramp with a right triangular cross-section. The horizontal base of the ramp measures

24\text{ feet}

and it rises to a vertical height of

7\text{ feet}

. According to the Pythagorean Theorem, the length of the sloped surface is feet, and the total perimeter of the ramp's triangular profile is feet.

Show answer & explanation

Answer

The sloped surface is 25 feet, and the total perimeter is 56 feet.

The sloped surface corresponds to the hypotenuse of a right triangle with legs of

7

and

24

feet. Applying the Pythagorean Theorem,

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

. Taking the square root of

625

gives

25\text{ feet}

for the sloped surface. The total perimeter is the sum of the base, height, and sloped surface:

24 + 7 + 25 = 56\text{ feet}

.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The legs of the right triangle are

a = 7

and

b = 24

.

The horizontal base and vertical height form the two legs that intersect at a right angle.

2

Apply the Pythagorean Theorem to find the length of the hypotenuse.

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

. Therefore,

c = \sqrt{625} = 25

.

The sloped surface represents the hypotenuse (

c

) of the right triangle.

3

Calculate the perimeter of the triangle.

7 + 24 + 25 = 56

.

The perimeter is found by summing the lengths of all three sides of the triangular cross-section.

Key Concept

Using the Pythagorean Theorem to find the hypotenuse of a right triangle, and applying it to calculate the perimeter.

Question 13Question →

Read the scenario and perform the necessary calculations. Enter the exact numerical values in the blanks.

Fill in the blanks below

A customer purchases a premium spa package for

\$250.00

. The local municipality applies a

6\%

sales tax to the base price of all spa services. After the tax is added, the customer decides to leave a

20\%

tip based on the new, post-tax total on the receipt.

The amount of the tip is $\$

and the final total cost paid by the customer is

\$$.

Show answer & explanation

Answer

The amount of the tip is 53 (or 53.00) and the final total cost is 318 (or 318.00).

The correct amounts are found by sequentially applying the percentages according to the narrative. First, a

6\%

tax on the

\$250.00

base price adds

\$15.00

, resulting in a

\$265.00

subtotal. The

20\%

tip is then applied to this

\$265.00

subtotal, resulting in a

\$53.00

tip. Adding this tip to the subtotal gives the final cost of

\$318.00

.

Step-by-Step Solution

1

Calculate the sales tax amount on the base price.

\$15.00

Multiply the base price (

\$250.00

) by the tax rate (

0.06

).

2

Determine the post-tax total.

\$265.00

Add the tax amount (

\$15.00

) to the base price (

\$250.00

) to find the amount upon which the tip will be calculated.

3

Calculate the tip amount.

\$53.00

Multiply the post-tax total (

\$265.00

) by the tip rate (

0.20

).

4

Calculate the final total cost.

\$318.00

Add the tip amount (

\$53.00

) to the post-tax total (

\$265.00

).

Key Concept

Calculating sales tax on a base price and applying a tip percentage to the sequential post-tax total.

Question 14Question →

Determine the correct numerical values to complete the statement.

Fill in the blanks below

An aviation mechanic calibrates

8

flight instruments in

3

hours. If a squadron requires

20

instruments to be calibrated, and the mechanic works at the same constant rate, it will take hours and minutes to complete the task.

Show answer & explanation

Answer

It will take 7 hours and 30 minutes.

Because the work rate is constant, the number of instruments calibrated is directly proportional to the time taken. Setting up the proportion 8/3 = 20/x yields x = 7.5 hours. Converting the remaining 0.5 hours to minutes gives 30 minutes, resulting in a total required time of 7 hours and 30 minutes.

Step-by-Step Solution

1

Set up a direct proportion comparing the number of instruments to the time in hours.

\frac{8 \text{ instruments}}{3 \text{ hours}} = \frac{20 \text{ instruments}}{x \text{ hours}}

When a rate is constant, the ratio of work completed to time taken remains equivalent.

2

Cross-multiply the terms to create a linear equation.

8x = 3 \times 20

Cross-multiplying is the standard algebraic method to solve for an unknown in a proportional relationship.

3

Simplify the equation and solve for

x

.

8x = 60 \Rightarrow x = \frac{60}{8} = 7.5 \text{ hours}

Isolating

x

provides the total required time expressed in hours.

4

Convert the decimal portion of the hours into minutes.

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

The question specifically requires the final answer to be formatted in hours and minutes.

Key Concept

Direct proportions with unit conversion

Estimated Time:1m 30s

Question 15Question →

A naval base allocates its quarterly maintenance budget among structural repairs, mechanical overhauls, and routine inspections in the ratio of $5:8:2$ . The amount allocated for mechanical overhauls is $\$45,000$ more than the amount allocated for routine inspections. Based on this information, calculate the total quarterly maintenance budget.

Fill in the blanks below

The total quarterly maintenance budget is $\$.

Show answer & explanation

Answer

The total quarterly maintenance budget is $112,500.

The difference between the mechanical overhaul and routine inspection allocations represents 6 parts of the total ratio (

8 - 2

). By dividing the given dollar difference (

\$45,000

) by 6, we find that a single ratio part is worth

\$7,500

. Since the entire maintenance budget is composed of 15 parts in total (

5 + 8 + 2

), multiplying 15 by

\$7,500

gives the correct total budget of

\$112,500

.

Step-by-Step Solution

1

Identify the number of ratio parts assigned to each category.

Structural =

5

parts, Mechanical =

8

parts, Routine =

2

parts.

This establishes the proportional distribution of the entire budget.

2

Determine the difference in parts between mechanical overhauls and routine inspections.

8

parts -

2

parts =

6

parts.

The problem provides the dollar amount for the difference between these two specific allocations, so we must find the corresponding difference in ratio parts.

3

Calculate the monetary value of a single ratio part.

\$45,000 \div 6 = \$7,500

per part.

Dividing the total monetary difference by the difference in parts gives the unit value of one part.

4

Find the total number of parts in the ratio and calculate the final budget.

Total parts =

5 + 8 + 2 = 15

. Total budget =

15 \times \$7,500 = \$112,500

.

Multiplying the total number of parts by the value of a single part yields the overall budget.

Key Concept

Using the monetary difference between two shares in a ratio to determine the value of a single unit, which is then used to find the total amount.

Question 16Question →

A marine supply store purchases commercial GPS units at a wholesale price of $\$320.00$ each. The store applies a $75\%$ markup to establish the regular retail price. During a spring clearance event, the store offers a $20\%$ discount off the regular retail price. What is the final sale price of the GPS unit?

Fill in the blanks below

The final sale price of the GPS unit is $\$$.

Show answer & explanation

Answer

The final sale price is

\$448.00

.

The final sale price is calculated in two sequential steps. First, the

\$240.00

markup (

75\%

of the

\$320.00

wholesale cost) is added to establish a regular retail price of

\$560.00

. Second, the

20\%

discount is applied to the

\$560.00

retail price, which equals

\$112.00

. Subtracting this

\$112.00

discount from the

\$560.00

retail price yields a final sale price of

\$448.00

.

Step-by-Step Solution

1

Calculate the markup amount.

\$320.00 \times 0.75 = \$240.00

To find the dollar amount added to the wholesale cost.

2

Determine the regular retail price.

\$320.00 + \$240.00 = \$560.00

To establish the base price before the clearance discount is applied.

3

Calculate the discount amount.

\$560.00 \times 0.20 = \$112.00

To find the dollar amount subtracted during the clearance sale.

4

Find the final sale price.

\$560.00 - \$112.00 = \$448.00

To determine the final cost to the customer after applying the discount.

Key Concept

Calculating sequential percentage changes by applying a markup to a base price and a subsequent discount to the resulting retail price.

Question 17Question →

A field communications unit has a spool containing $30 \frac{1}{4}$ yards of specialized copper wire. They need to cut the wire into smaller segments that are each $1 \frac{3}{8}$ yards long to connect field radios. How many complete segments of wire can the unit cut?

Fill in the blanks below

The unit can cut complete segments of wire.

Show answer & explanation

Answer

22

The correct answer is 22 because you must divide the total amount of wire by the length of one segment. Converting both mixed numbers to improper fractions gives

\frac{121}{4}

and

\frac{11}{8}

. Dividing

\frac{121}{4}

by

\frac{11}{8}

is equivalent to multiplying

\frac{121}{4}

by

\frac{8}{11}

, which simplifies to exactly 22.

Step-by-Step Solution

1

Convert the total wire length from a mixed number to an improper fraction.

30 \frac{1}{4} = \frac{30 \times 4 + 1}{4} = \frac{121}{4}

yards.

Improper fractions are required to perform fraction division.

2

Convert the length of each segment from a mixed number to an improper fraction.

1 \frac{3}{8} = \frac{1 \times 8 + 3}{8} = \frac{11}{8}

yards.

Improper fractions are required to perform fraction division.

3

Set up the division problem to find the total number of segments.

\frac{121}{4} \div \frac{11}{8}

Dividing the total length by the length of a single segment gives the total number of segments.

4

Perform the division by multiplying the first fraction by the reciprocal of the second fraction.

\frac{121}{4} \times \frac{8}{11}

The rule for dividing fractions is to multiply by the reciprocal of the divisor.

5

Simplify the expression by cross-canceling and multiply.

\frac{121}{11} \times \frac{8}{4} = 11 \times 2 = 22

segments.

Simplifying before multiplying makes the calculation easier and yields the final whole number of segments.

Key Concept

Dividing mixed numbers in a real-world word problem context

Question 18Question →

Determine the commission earned and fill in the blank.

Fill in the blanks below

A travel agent receives a

4\%

commission on all vacation packages they book. If the agent books a luxury cruise package priced at

\$8,500

, their commission for this booking is

\$

.

Show answer & explanation

Answer

340

The travel agent's commission is calculated by taking

4\%

of the total package price of

\$8,500

. Converting

4\%

to

0.04

and multiplying by

\$8,500

gives exactly

\$340

.

Step-by-Step Solution

1

Convert the commission percentage to a decimal.

4\% = 0.04

Percentages must be converted to decimals before performing multiplication to find a part of a whole.

2

Multiply the total sales price by the decimal commission rate.

\$8,500 \times 0.04 = \$340

A commission is calculated as a percentage of the total sales amount.

Key Concept

Calculating commission based on a fixed percentage of a total sale amount.

Question 19Question →

Solve the following problem by entering the correct numerical value in the blank.

Fill in the blanks below

Sarah works as a heavy equipment sales representative. She earns a monthly base salary of

\$2,400

. She also earns a

4\%

commission on her first

\$40,000

in monthly sales, and a

7.5\%

commission on all sales exceeding

\$40,000

. In March, Sarah's total gross pay was

\$6,100

.

The total value of her sales in March was $\$$.

Show answer & explanation

Answer

68,000

Subtracting the base salary (

\$2,400

) and the maximum first tier commission (

\$1,600

) from the total gross pay (

\$6,100

) leaves

\$2,100

. This remaining amount is the commission earned at the

7.5\%

rate. Dividing

\$2,100

by

0.075

yields

\$28,000

, which is the sales amount over the

\$40,000

threshold. Adding this to the initial

\$40,000

threshold gives the total monthly sales of

\$68,000

.

Step-by-Step Solution

1

Calculate the commission earned from the first tier of sales.

\$40,000 \times 0.04 = \$1,600

To determine the maximum commission Sarah can earn at the

4\%

base rate.

2

Subtract the base salary and first tier commission from the total gross pay to find the remaining commission.

\$6,100 - \$2,400 - \$1,600 = \$2,100

To isolate the portion of her earnings that came exclusively from the

7.5\%

commission tier.

3

Calculate the sales volume required to generate the remaining tier 2 commission.

\$2,100 \div 0.075 = \$28,000

To find out how much she sold above the

\$40,000

threshold.

4

Add the tier 2 sales to the tier 1 threshold to find the total sales.

\$40,000 + \$28,000 = \$68,000

To find the complete total value of her sales in March.

Key Concept

Solving for required total sales given a base salary and a tiered commission structure

Question 20Question →

A corporate team books a conference center for a retreat. The venue charges $\$1,200$ for the room rental and $\$800$ for the catering. The contract specifies a $5\%$ local tax that applies only to the room rental, and an $18\%$ mandatory gratuity that applies only to the catering. Calculate the required values based on this information.

Fill in the blanks below

The tax amount is $\$

, the gratuity is

\$

, and the final total cost is

\$$.

Show answer & explanation

Answer

Tax:

\$60

, Gratuity:

\$144

, Total:

\$2,204

To find the final total, the tax and gratuity must be calculated separately based on their respective costs. The

5\%

tax on the

\$1,200

room is

\$60

. The

18\%

gratuity on the

\$800

catering is

\$144

. Adding the original costs (

\$1,200 + \$800 = \$2,000

) to the tax (

\$60

) and gratuity (

\$144

) yields a final total of

\$2,204

.

Step-by-Step Solution

1

Calculate the local tax on the room rental.

\$1,200 \times 0.05 = \$60

The

5\%

tax only applies to the

\$1,200

room rental cost.

2

Calculate the mandatory gratuity on the catering.

\$800 \times 0.18 = \$144

The

18\%

gratuity only applies to the

\$800

catering cost.

3

Calculate the final total cost of the retreat.

\$1,200 + \$800 + \$60 + \$144 = \$2,204

The final invoice includes the original room and catering costs, plus the calculated tax and gratuity.

Key Concept

Calculating specific taxes and tips based on different principal amounts within a single transaction.

Answer

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