All practice questions

Question 3321Question →

An electrician buys some wiring supplies that total $150.00 before tax. If the local sales tax rate is 8%, what is the total cost of the supplies including tax?

$12.00

$158.00

$162.00

$270.00

Show answer & explanation

Answer: $162.00

Answer

$162.00

The correct answer is found by first calculating the sales tax, which is 8% of the

150.00 subtotal. Multiplying

150.00 by 0.08 gives a tax amount of

12.00. Then, this tax amount is added to the original cost (

150.00 +

12.00) to find the final total of

162.00.

Step-by-Step Solution

1

Calculate the amount of sales tax.

\$150.00 \times 0.08 = \$12.00

To find the tax, multiply the pre-tax cost by the decimal form of the percentage (8% = 0.08).

2

Add the tax amount to the original cost.

\$150.00 + \$12.00 = \$162.00

The total cost includes both the base price and the applied sales tax.

Key Concept

Calculating Sales Tax and Total Cost

Question 3322Question →

Read the scenario carefully and calculate the final amount of fuel. Enter the exact numerical value in the blank.

Fill in the blanks below

A military logistics base begins a major operation with

12,000

gallons of fuel in reserve. During the first phase of the operation,

35\%

of the initial fuel supply is consumed. During the second phase,

45\%

of the *remaining* fuel is consumed. Following the second phase, a resupply convoy delivers an amount of fuel equal to

25\%

of the fuel that was consumed during the first phase. After the delivery, there are gallons of fuel in the reserve.

Show answer & explanation

Answer

5340

The correct answer accurately tracks the changing base values for each percentage calculation. First, find

35\%

of the initial

12,000

gallons (

4,200

used), leaving

7,800

gallons. Next, find

45\%

of the remaining

7,800

gallons (

3,510

used), leaving

4,290

gallons. Finally, calculate a resupply of

25\%

of the first phase's consumption (

25\%

of

4,200 = 1,050

). Adding this resupply gives

4,290 + 1,050 = 5,340

gallons.

Step-by-Step Solution

1

Calculate the fuel consumed in the first phase by multiplying the initial amount (

12,000

gallons) by

35\%

.

0.35 \times 12,000 = 4,200

gallons.

This determines the amount of fuel used during the initial stage based on the total starting quantity.

2

Subtract the consumed fuel from the initial amount.

12,000 - 4,200 = 7,800

gallons.

The second phase's percentage is based on the remaining fuel, not the original total.

3

Calculate the fuel consumed in the second phase by multiplying the remaining fuel (

7,800

gallons) by $45\%.

0.45 \times 7,800 = 3,510

gallons.

This finds the fuel used in the second stage based on the updated base value.

4

Subtract the second phase consumption from the fuel available before the second phase.

7,800 - 3,510 = 4,290

gallons.

This determines the total remaining fuel before the resupply.

5

Calculate the resupply amount by multiplying the fuel consumed in the first phase (

4,200

gallons) by $25\%.

0.25 \times 4,200 = 1,050

gallons.

The resupply is specifically defined as a percentage of a previously calculated quantity (Phase 1 consumption), not the remaining fuel.

6

Add the resupply amount to the fuel remaining after the second phase.

4,290 + 1,050 = 5,340

gallons.

This gives the final total of fuel in the reserve.

Key Concept

Calculating percentages of varying base quantities in multi-step word problems.

Question 3323Question →

A parachute rigger is inspecting and cutting suspension line for parachute maintenance. They start with a spool containing $25 \frac{1}{4}$ yards of nylon cord. They use $8 \frac{2}{3}$ yards to repair a main canopy and $6 \frac{5}{6}$ yards to repair a reserve parachute. How many yards of nylon cord remain on the spool?

15 \frac{1}{2}

14 \frac{7}{9}

9 \frac{3}{4}

10 \frac{1}{4}

Show answer & explanation

Answer: $9 \frac{3}{4}$

Answer

The correct answer is

9 \frac{3}{4}

yards.

To find the remaining cord, first determine the total amount used by adding

8 \frac{2}{3}

and

6 \frac{5}{6}

. Using a common denominator of

6

, this is

8 \frac{4}{6} + 6 \frac{5}{6} = 14 \frac{9}{6}

, which simplifies to

15 \frac{1}{2}

. Next, subtract this total from the initial amount:

25 \frac{1}{4} - 15 \frac{1}{2}

. Find a common denominator of

4

, making it

25 \frac{1}{4} - 15 \frac{2}{4}

. Since

\frac{1}{4}

is smaller than

\frac{2}{4}

, borrow

1

from the

25

to get

24 \frac{5}{4}

. Finally,

24 \frac{5}{4} - 15 \frac{2}{4} = 9 \frac{3}{4}

.

Step-by-Step Solution

1

Calculate the total amount of nylon cord used for both repairs by finding a common denominator.

8 \frac{2}{3} + 6 \frac{5}{6} = 8 \frac{4}{6} + 6 \frac{5}{6} = 14 \frac{9}{6}

To find the total amount removed from the spool, the two individual parts must be added together using a common denominator of 6.

2

Simplify the total amount used.

14 \frac{9}{6} = 15 \frac{3}{6} = 15 \frac{1}{2}

yards.

Converting the improper fraction part to a mixed number and reducing it to simplest form makes the next calculation easier.

3

Set up the subtraction to find the remaining cord, finding a common denominator for the total length and the used length.

25 \frac{1}{4} - 15 \frac{1}{2} = 25 \frac{1}{4} - 15 \frac{2}{4}

To find what is left, the used amount must be subtracted from the initial amount on the spool. A common denominator of 4 is required.

4

Regroup the initial mixed number to perform the subtraction.

24 \frac{5}{4} - 15 \frac{2}{4} = 9 \frac{3}{4}

yards.

Since

\frac{1}{4}

is smaller than

\frac{2}{4}

, one whole unit (

\frac{4}{4}

) must be borrowed from the whole number

25

to make the subtraction possible.

Key Concept

Adding and subtracting mixed numbers with unlike denominators, including necessary regrouping (borrowing) from whole numbers.

Question 3324Question →

Read the scenario below and determine the missing values.

Fill in the blanks below

Marcus buys a new winter coat with a price tag of

80.00. The store applies a 5% sales tax to all clothing purchases. The amount of sales tax added to his purchase is

, which brings the total cost of the coat to $.

Show answer & explanation

Answer

The sales tax amount is

4.00 and the total cost is

84.00.

First, the 5% tax on the

80.00 coat is calculated by multiplying 80 by 0.05, yielding

4.00 for the isolated tax amount. Then, this

4.00 tax is added to the initial

80.00 price, resulting in a final total of $84.00.

Step-by-Step Solution

1

Calculate the amount of sales tax.

80.00 \times 0.05 =

4.00

To find the tax amount, convert the percentage (5%) to a decimal (0.05) and multiply it by the original price.

2

Calculate the final total cost.

80.00 +

4.00 = $84.00

The final total is the sum of the original price and the calculated sales tax.

Key Concept

Calculating sales tax amounts and finding the final total cost.

Question 3325Question →

A corporate team of $8$ employees goes out to dinner. The original food subtotal for the group is $\$400.00$ . The team uses a coupon for $15\%$ off the food subtotal. The restaurant automatically adds an $18\%$ gratuity, which is calculated based on the original food subtotal before the discount. A $6\%$ sales tax is then applied only to the discounted food amount. The gratuity is not taxed. If the final total bill is split evenly among all $8$ employees, how much does each employee pay?

\$52.70

\$54.05

\$54.50

\$77.00

Show answer & explanation

Answer: $\$54.05$

Answer

Each employee pays

\$54.05

.

The problem requires calculating three components based on specific amounts. First, the discounted subtotal is found by subtracting

15\%

of

\$400

(

\$60

) to get

\$340

. Second, the

6\%

tax is applied to the

\$340

discounted subtotal, adding

\$20.40

. Third, the

18\%

gratuity is applied to the original

\$400

subtotal, adding

\$72.00

. The total bill is

\$340 + \$20.40 + \$72.00 = \$432.40

. Splitting this evenly among

8

employees gives

\$54.05

per person.

Step-by-Step Solution

1

Calculate the discount and the discounted food subtotal.

Discount =

0.15 \times \$400 = \$60

. Discounted subtotal =

\$400 - \$60 = \$340

.

The

15\%

coupon is applied to the original subtotal to find the amount subject to sales tax.

2

Calculate the sales tax.

Tax =

0.06 \times \$340 = \$20.40

.

The

6\%

sales tax is applied only to the discounted food amount.

3

Calculate the gratuity.

Gratuity =

0.18 \times \$400 = \$72.00

.

The gratuity must be calculated based on the original food subtotal (

\$400

), as explicitly stated in the problem.

4

Calculate the final total bill and split it evenly among the employees.

Total bill =

\$340 + \$20.40 + \$72.00 = \$432.40

. Per employee =

\$432.40 \div 8 = \$54.05

.

The final bill is the sum of the discounted food, tax, and gratuity. Dividing by

8

gives the individual cost.

Key Concept

Multi-step financial arithmetic involving independent bases for taxes and tips, followed by proportional division.

Question 3326Question →

Electromagnetic radiation travels through space as oscillating electric and magnetic fields, with each type characterized by its specific frequency, wavelength, and energy. Arrange the following regions of the electromagnetic spectrum in order of increasing photon energy, from the lowest energy to the highest energy.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from lowest to highest photon energy is Radio waves, Microwaves, Visible light, Ultraviolet radiation, and X-rays.

Photon energy is governed by the relation

E = hf = \frac{hc}{\lambda}

, where

h

is Planck's constant and

c

is the speed of light. Because energy increases directly with frequency and inversely with wavelength, radiation with the longest wavelengths (radio waves) carries the least energy per photon, followed in ascending order by microwaves, visible light, ultraviolet radiation, and X-rays.

Step-by-Step Solution

1

Recall the fundamental relationship between wave frequency and photon energy.

Photon energy is directly proportional to frequency as defined by Planck's equation,

E = hf

, and inversely proportional to wavelength,

E = \frac{hc}{\lambda}

.

Establishing that higher frequency and shorter wavelength correspond directly to higher photon energy.

2

Identify the positions of the given wave types across the continuous electromagnetic spectrum.

In order of increasing frequency: Radio waves (

< 10^9\text{ Hz}

)

\rightarrow

Microwaves (

10^9 - 10^{12}\text{ Hz}

)

\rightarrow

Visible light (

4 \times 10^{14} - 7.5 \times 10^{14}\text{ Hz}

)

\rightarrow

Ultraviolet radiation (

10^{15} - 10^{16}\text{ Hz}

)

\rightarrow

X-rays (

10^{16} - 10^{19}\text{ Hz}

).

Classifying each radiation type by its frequency band determines its relative energy.

3

Sequence the items from the lowest photon energy to the highest photon energy.

Radio waves

\rightarrow

Microwaves

\rightarrow

Visible light

\rightarrow

Ultraviolet radiation

\rightarrow

X-rays.

Arranging the items in ascending order of frequency matches the required ascending order of energy.

Key Concept

Photon energy across the electromagnetic spectrum is directly proportional to wave frequency (

E = hf

) and inversely proportional to wavelength (

E = hc/\lambda

).

Estimated Time:1m 0s

Question 3327Question →

An event planner organizes a retirement banquet. The standard cost for the food and beverage package is $500. The venue provides a 15% preferred client discount. A local sales tax of 8% is charged on the discounted price. In addition, an automatic gratuity of 20% is applied, which is also calculated based on the discounted price before tax. What is the total final cost of the banquet, in dollars?

Fill in the blanks below

Total final cost: $

Show answer & explanation

Answer

544

The final total is found by first applying the 15% discount to the original

500 cost, yielding a discounted base of

425. Both the 8% sales tax (

34) and the 20% gratuity (

85) are calculated using this

425 base. Adding the discounted base price, the tax, and the gratuity gives a final total of

544.

Step-by-Step Solution

1

Calculate the amount of the discount.

$75

The venue provides a 15% discount on the original $500 cost (0.15 × 500 = 75).

2

Determine the discounted price of the package.

$425

Subtract the discount amount from the original cost (500 - 75 = 425).

3

Calculate the sales tax.

$34

The 8% tax is charged on the $425 discounted price (0.08 × 425 = 34).

4

Calculate the gratuity.

$85

The 20% gratuity is also calculated based on the $425 discounted price (0.20 × 425 = 85).

5

Sum the components to find the total final cost.

$544

Add the discounted base price, the sales tax, and the gratuity together (425 + 34 + 85 = 544).

Key Concept

Calculating sequential percentages, including applying taxes and tips to a specific post-discount base amount.

Question 3328Question →

A shipyard fabricator welds $30$ inches of steel seam every $15$ minutes. Operating at this exact constant rate, how many feet of steel seam will the fabricator weld in a $6$ -hour shift?

15

60

720

8,640

Show answer & explanation

Answer: 60

Answer

60

The correct answer is derived by first converting the 6-hour shift to 360 minutes. Setting up the direct proportion yields a total of 720 inches (

30 \times 360 \div 15

). Finally, dividing by 12 correctly converts the length into 60 feet.

Step-by-Step Solution

1

Harmonize the time units by converting the 6-hour shift into minutes.

6 \text{ hours} \times 60 \text{ minutes/hour} = 360 \text{ minutes}

The given rate is in minutes, so the total operational time must also be in minutes to set up a valid proportion.

2

Set up a direct proportion or find the unit rate of welding.

\frac{30 \text{ inches}}{15 \text{ minutes}} = 2 \text{ inches per minute}

Determining the rate per minute simplifies the calculation for the entire shift.

3

Calculate the total length of seam welded in inches.

2 \text{ inches/minute} \times 360 \text{ minutes} = 720 \text{ inches}

Applying the constant rate to the total operational time yields the total production in the initial unit.

4

Convert the total inches into the requested unit of feet.

720 \text{ inches} \div 12 \text{ inches/foot} = 60 \text{ feet}

The question explicitly asks for the final answer in feet, requiring a unit conversion.

Key Concept

Direct proportions with unit conversion

Question 3329Question →

A family books a guided kayak tour that originally costs $\$250.00$ . They apply a promotional code that gives them a $20\%$ discount. After the discount is applied, an $8\%$ local booking tax is added. What is the final amount the family pays for the tour?

\$201.60

\$216.00

\$220.00

\$248.40

Show answer & explanation

Answer: $\$216.00$

Answer

\$216.00

The correct total is found by first reducing the original price by

20\%

(a

\$50.00

reduction, resulting in

\$200.00

). Then, an

8\%

tax is calculated on this new balance (

\$200.00 \times 0.08 = \$16.00

). Adding the tax to the discounted price yields

\$216.00

.

Step-by-Step Solution

1

Calculate the monetary value of the discount.

20\%

of

\$250.00 = 0.20 \times 250 = \$50.00

The discount amount must be determined before calculating the new adjusted price.

2

Subtract the discount from the original price.

\$250.00 - \$50.00 = \$200.00

This finds the discounted subtotal, which is the amount subject to taxation.

3

Calculate the tax amount on the discounted price.

8\%

of

\$200.00 = 0.08 \times 200 = \$16.00

Taxes are applied to the final selling price after discounts.

4

Add the tax to the discounted price to find the final total.

\$200.00 + \$16.00 = \$216.00

The customer pays the discounted price plus the applicable booking tax.

Key Concept

Calculating sequential percentages in financial transactions (discounts followed by taxes).

Question 3330Question →

An engine transmission system uses three interconnected gears: Gear A, Gear B, and Gear C. The ratio of the number of teeth on Gear A to the number of teeth on Gear B is $4:3$ . The ratio of the number of teeth on Gear B to the number of teeth on Gear C is $5:2$ . If Gear C has exactly $18$ teeth, what is the total number of teeth on all three gears combined?

Show answer & explanation

Answer: 123

Answer

123

The correct answer is found by linking the two separate proportions using their common element, Gear B. By converting the A:B ratio (

4:3

) and the B:C ratio (

5:2

) to share a common value for B (

15

), we get a unified, continuous ratio for A:B:C of

20:15:6

. The problem states that Gear C has

18

teeth. Since Gear C represents

6

parts in our combined ratio, each part must equal

3

teeth (

18 \div 6 = 3

). The total number of parts across all three gears is

41

(

20 + 15 + 6

). Multiplying the

41

total parts by

3

teeth per part results in a total of

123

teeth for the entire system.

Step-by-Step Solution

1

Align the separate ratios based on the common element.

Gear A : Gear B =

4:3

and Gear B : Gear C =

5:2

.

To compare all three gears simultaneously, Gear B must have the same value in both ratios.

2

Find a common multiple for the ratio numbers representing Gear B (

3

and

5

).

The least common multiple of

3

and

5

is

15

.

Creating a common denominator allows us to link the two independent proportions.

3

Scale both ratios so Gear B is represented by

15

parts.

Ratio A:B becomes

(4 \times 5):(3 \times 5) = 20:15

. Ratio B:C becomes

(5 \times 3):(2 \times 3) = 15:6

.

Multiplying both sides of a ratio by the same number maintains the proportion.

4

Combine into a single three-part ratio and determine total parts.

Combined ratio A:B:C is

20:15:6

. Total ratio parts =

20 + 15 + 6 = 41

.

This single ratio precisely maps the relationships between all three gears in the system.

5

Calculate the multiplier using Gear C's known teeth count.

6

parts =

18

teeth. Therefore,

1

part =

18 / 6 = 3

teeth.

This establishes the real-world value of a single ratio unit in this specific transmission system.

6

Calculate the total number of teeth on all gears combined.

41

total parts

\times 3

teeth per part =

123

teeth.

Multiplying the total number of parts by the value of a single part gives the final aggregate quantity.

Key Concept

Combining overlapping ratios by calculating the least common multiple of the shared term, then scaling the combined ratio to determine unknown totals.

Question 3331Question →

An acoustic engineer tests a noise-reduction device that emits a secondary sound wave exactly half a wavelength ( $180^\circ$ ) out of phase with an incoming noise wave of identical frequency and amplitude. What occurs when these two sound waves overlap in the air?

The waves undergo constructive interference, resulting in a sound wave with twice the original amplitude.

The waves undergo destructive interference, resulting in the cancellation of sound with zero net amplitude.

The waves undergo refraction, resulting in a permanent doubling of the wave's frequency.

The waves undergo dispersion, resulting in the sound separating into different component frequencies.

Show answer & explanation

Answer: The waves undergo destructive interference, resulting in the cancellation of sound with zero net amplitude.

Answer

The waves undergo destructive interference, resulting in the cancellation of sound with zero net amplitude.

When two waves with equal frequency and amplitude meet half a wavelength (

180^\circ

) out of phase, the crests of the first wave precisely coincide with the troughs of the second wave. According to the principle of superposition, their individual displacements subtract from each other, resulting in complete destructive interference and zero resultant amplitude.

Step-by-Step Solution

1

Analyze the phase relationship of the two sound waves.

The two waves have identical frequency and amplitude but are half a wavelength (

180^\circ

) out of phase.

Determining phase alignment indicates whether wave crests and troughs align constructively or destructively.

2

Apply the principle of wave superposition to calculate the net displacement.

Every positive peak (crest) of the primary wave perfectly aligns with an equal negative peak (trough) of the secondary wave, summing to zero net displacement (

A - A = 0

).

The principle of superposition states that overlapping waves combine algebraically at every point in space.

3

Identify the corresponding wave phenomenon.

Destructive interference.

Complete destructive interference occurs when opposing wave displacements of equal magnitude cancel each other out.

Key Concept

Destructive Wave Interference and Superposition

Estimated Time:1m 0s

Question 3332Question →

A supply sergeant is ordering office supplies for the base. He purchases $2.5$ cartons of printer paper at $\$14.40$ per carton and $2$ dozen pens at $\$0.85$ per pen. He applies a $\$6.40$ discount coupon to the entire purchase. After the discount is subtracted, an $8\%$ sales tax is added to the remaining balance. What is the final cost of the purchase?

\$54.00

\$54.51

\$33.80

\$50.40

Show answer & explanation

Answer: $\$54.00$

Answer

\$54.00

The correct answer is

\$54.00

. The paper costs

2.5 \times \$14.40 = \$36.00

. The pens cost

24 \times \$0.85 = \$20.40

. The total before the coupon is

\$36.00 + \$20.40 = \$56.40

. Subtracting the

\$6.40

coupon gives a subtotal of

\$50.00

. An

8\%

tax on

\$50.00

is

\$4.00

, making the final total

\$54.00

.

Step-by-Step Solution

1

Calculate the total cost of the printer paper.

2.5 \times \$14.40 = \$36.00

The unit price must be multiplied by the exact decimal quantity purchased.

2

Calculate the total cost of the pens.

2

dozen

= 24

pens.

24 \times \$0.85 = \$20.40

The quantity is given in dozens, which must be converted to individual units before multiplying by the unit price.

3

Find the total cost before the discount.

\$36.00 + \$20.40 = \$56.40

Add the individual totals together to find the gross purchase amount.

4

Subtract the discount coupon from the total.

\$56.40 - \$6.40 = \$50.00

The problem states the discount is applied to the total before sales tax is calculated.

5

Calculate the

8\%

sales tax on the discounted balance and add it to the total.

\$50.00 \times 0.08 = \$4.00

. Then,

\$50.00 + \$4.00 = \$54.00

Sales tax is applied as a percentage of the final discounted subtotal to find the final cost.

Key Concept

Multi-step decimal operations involving unit conversions, discounts, and percentage applications.

Question 3333Question →

An atom of bromine has an atomic number of $35$ and a mass number of $81$ . What is the total number of neutrons contained in its nucleus?

35

46

81

116

Show answer & explanation

Answer: $46$

Answer

The nucleus contains

46

neutrons.

The mass number of an atom is the total count of protons and neutrons in the nucleus (

A = \text{protons} + \text{neutrons}

). Given that the atomic number is

35

(protons =

35

) and the mass number is

81

, the number of neutrons is calculated as

81 - 35 = 46

.

Step-by-Step Solution

1

Identify the given values from the problem statement.

Atomic number (

Z

) =

35

(which equals the number of protons); Mass number (

A

) =

81

(total nucleons in the nucleus).

The atomic number defines the number of protons, and the mass number represents the sum of protons and neutrons.

2

Apply the relationship between mass number, protons, and neutrons:

\text{Mass Number } (A) = \text{Protons } (Z) + \text{Neutrons } (N)

.

N = A - Z

To find the unknown number of neutrons, rearrange the standard subatomic relationship.

3

Substitute the known integer values into the equation:

N = 81 - 35

.

N = 46

Subtracting

35

from

81

gives the total number of neutrons in the nucleus.

Key Concept

The mass number (

A

) of an atom equals the sum of its protons (

Z

) and neutrons (

N

). The number of neutrons is found by subtracting the atomic number from the mass number (

N = A - Z

).

Estimated Time:45s

Question 3334Question →

A military transport vehicle's winch system uses a gear train consisting of three gears. The driving gear (Gear A) has $60$ teeth and rotates at $150\text{ RPM}$ . It meshes with an idler gear (Gear B) which has $45$ teeth. Gear B meshes directly with the driven gear (Gear C). If Gear C must rotate at a speed of $5\text{ revolutions per second}$ to safely operate the winch, how many teeth must Gear C have?

30

120

300

1800

Show answer & explanation

Answer: 30

Answer

30 teeth

The number of teeth on a gear and its rotational speed are inversely proportional. In a gear train, an idler gear (Gear B) only changes the direction of rotation, not the mathematical ratio between the first and last gears (

T_A \times S_A = T_C \times S_C

). First, ensure the speed units match by converting Gear C's speed to RPM:

5\text{ rev/s} \times 60\text{ seconds/minute} = 300\text{ RPM}

. Then, equate the products for Gear A and Gear C:

60 \times 150 = T_C \times 300

. Solving for

T_C

gives

9000 \div 300 = 30

teeth.

Step-by-Step Solution

1

Identify the relationship in a gear train.

The product of teeth and rotational speed is constant across meshed gears (

T_1 \times S_1 = T_2 \times S_2

). The idler gear (Gear B) does not change the final relationship between Gear A and Gear C.

To establish the mathematical formula needed and eliminate unnecessary variables.

2

Convert Gear C's speed to the same unit as Gear A (RPM).

5\text{ revolutions/second} \times 60\text{ seconds/minute} = 300\text{ RPM}

.

Both gears must use the exact same rate units before an accurate proportion can be calculated.

3

Set up the inverse proportion equation connecting Gear A to Gear C.

60\text{ teeth} \times 150\text{ RPM} = T_C \times 300\text{ RPM}

.

To isolate the unknown variable based on the constant product rule of gears.

4

Solve the equation for the number of teeth on Gear C (

T_C

).

9000 = 300 \times T_C

, which simplifies to

T_C = 9000 \div 300 = 30\text{ teeth}

.

To find the final required dimension for the driven gear.

Key Concept

Inverse Proportions in Compound Mechanics

Alternative Method

You can solve this in two steps by finding the idler gear's speed first. Gear B's speed:

60 \times 150 = 45 \times S_B \rightarrow 9000 = 45 \times S_B \rightarrow S_B = 200\text{ RPM}

. Then relate Gear B to Gear C (converted to

300\text{ RPM}

):

45 \times 200 = T_C \times 300 \rightarrow 9000 = 300 \times T_C \rightarrow T_C = 30

.

Estimated Time:1m 45s

Question 3335Question →

A maintenance worker is painting transport vehicles. It takes $\frac{3}{5}$ of a gallon of paint to cover one vehicle. If the worker needs to paint $15$ identical vehicles, what is the total number of gallons of paint required?

Show answer & explanation

Answer: 9

Answer

9

The correct answer is found by taking the required amount of paint per vehicle (

\frac{3}{5}

) and multiplying it by the total number of vehicles (

15

). The calculation is

15 \times \frac{3}{5} = \frac{15 \times 3}{5} = \frac{45}{5} = 9

gallons.

Step-by-Step Solution

1

Set up the mathematical expression

15 \times \frac{3}{5}

To find the total amount of paint, multiply the amount of paint needed for a single vehicle by the total number of vehicles.

2

Calculate the final value

9

Multiply the whole number

15

by the numerator

3

to get

45

, then divide by the denominator

5

to get

9

.

Key Concept

Multiplying a whole number by a fraction to find a total quantity.

Question 3336Question →

Wave properties describe the spatial and temporal behavior of mechanical and electromagnetic oscillations. Which of the following correctly matches each wave parameter on the left with its standard physical definition on the right?

Click a left item, then click its matching right item

Show answer & explanation

Answer

Wavelength pairs with the spatial distance between two consecutive identical points on adjacent wave cycles; Frequency pairs with the number of complete wave cycles that pass a fixed reference point per second; Amplitude pairs with the maximum displacement of an oscillation from its central equilibrium position; and Wave speed pairs with the rate at which a wave disturbance propagates through a given medium.

Each fundamental wave term corresponds directly to its standard physical definition: wavelength is the distance between consecutive identical points on adjacent cycles; frequency is the number of cycles passing a reference point per second; amplitude is the maximum displacement from equilibrium; and wave speed is the propagation rate of the wave disturbance through a medium.

Step-by-Step Solution

1

Identify the spatial parameter describing the length of a full cycle

Wavelength (

\lambda

) is the physical distance between consecutive corresponding points, such as crest to crest or compression to compression.

This establishes the spatial periodicity of the wave.

2

Identify the temporal rate parameter measuring wave oscillations per unit time

Frequency (

f

) is the count of complete wave cycles passing a stationary point per unit time, measured in Hertz (

\text{Hz}

).

This establishes the temporal oscillation rate of the wave.

3

Identify the parameter measuring wave height or disturbance magnitude

Amplitude (

A

) is the maximum displacement of the medium or field from its central baseline equilibrium position.

This parameter indicates the energy carried by the wave, relating to loudness in sound and brightness in light.

4

Identify the propagation rate of the wave disturbance

Wave speed (

v

) is the distance a wave crest travels per unit time through a medium, determined by

v = f \lambda

.

This connects spatial wavelength and temporal frequency to linear propagation speed.

Key Concept

Fundamental wave parameters: wavelength, frequency, amplitude, and propagation speed

Estimated Time:1m 0s

Question 3337Question →

An aviation structural mechanic is cutting a piece of sheet metal to repair an aircraft fuselage. The initial piece of sheet metal is $45 \frac{1}{2}$ inches long. The mechanic cuts off one piece that is $14 \frac{3}{4}$ inches long and another piece that is $18 \frac{5}{8}$ inches long. Assuming no material is lost to the saw cut, how many inches of the sheet metal remain? (Enter your answer as a decimal).

Show answer & explanation

Answer: 12.125

Answer

The remaining piece of sheet metal is 12.125 inches long.

The remaining length is correctly found by determining the sum of the removed parts (

14 \frac{3}{4} + 18 \frac{5}{8} = 33 \frac{3}{8}

) and subtracting it from the whole (

45 \frac{1}{2} - 33 \frac{3}{8} = 12 \frac{1}{8}

), which equals 12.125 inches.

Step-by-Step Solution

1

Find a common denominator for the fractional parts to easily add and subtract the lengths.

The least common denominator for 2, 4, and 8 is 8. The initial length is

45 \frac{4}{8}

inches. The first cut is

14 \frac{6}{8}

inches, and the second cut is

18 \frac{5}{8}

inches.

Fractions must have the same denominator before they can be added or subtracted.

2

Calculate the total length of the sheet metal that was cut off.

14 \frac{6}{8} + 18 \frac{5}{8} = 32 \frac{11}{8}

, which simplifies to

33 \frac{3}{8}

inches.

Adding the cut pieces together gives the total amount of material removed.

3

Subtract the total cut length from the initial length to find the remaining amount.

45 \frac{4}{8} - 33 \frac{3}{8} = 12 \frac{1}{8}

inches.

Subtracting the removed material from the starting material leaves the remaining balance.

4

Convert the final mixed number to a decimal.

12 \frac{1}{8} = 12.125

The question requires a decimal format for the numeric entry.

Key Concept

Adding and subtracting mixed numbers with unlike denominators in a practical context.

Question 3338Question →

An ion with a $+3$ positive charge has $10$ electrons and a mass number of $27$ . How many neutrons are located inside the nucleus of this ion?

Show answer & explanation

Answer: 14

Answer

14 neutrons

For any atom or ion, the mass number equals the total count of protons and neutrons in the nucleus. Because this ion has a

+3

charge and

10

electrons, it must contain

13

protons (

10 + 3

). Subtracting

13

protons from the mass number of

27

gives exactly

14

neutrons.

Step-by-Step Solution

1

Calculate the atomic number (number of protons) using the ion's charge and electron count.

The atom has

13

protons.

A positive charge (

+3

) indicates that the atom has lost

3

electrons relative to its proton count. Therefore,

\text{Protons} = \text{Electrons} + \text{Charge} = 10 + 3 = 13

.

2

Calculate the number of neutrons using the mass number and the proton count.

The nucleus contains

14

neutrons.

Mass number is the sum of protons and neutrons (

A = Z + N

). Thus,

\text{Neutrons} = \text{Mass Number} - \text{Protons} = 27 - 13 = 14

.

Key Concept

Relationship between mass number, atomic number (protons), net ionic charge, and subatomic particle counts.

Estimated Time:1m 0s

Question 3339Question →

A combat engineering squad fills $150$ sandbags in $40$ minutes. Operating at this exact constant rate, how many sandbags can the squad fill in $4$ hours?

Show answer & explanation

Answer: 900

Answer

900

Because the production rate is constant, the number of sandbags filled is directly proportional to the time spent. Harmonizing the time units by converting

4

hours into

240

minutes allows you to correctly scale up the initial ratio of

150

sandbags per

40

minutes.

Step-by-Step Solution

1

Convert the total working time from hours to minutes.

4 \text{ hours} \times 60 \text{ minutes/hour} = 240 \text{ minutes}

The given rate is in minutes (

150

sandbags per

40

minutes), so the target time must also be in minutes to set up an accurate proportion.

2

Set up a direct proportion.

\frac{150 \text{ sandbags}}{40 \text{ minutes}} = \frac{x \text{ sandbags}}{240 \text{ minutes}}

Since the squad works at a constant rate, the ratio of sandbags filled to time spent must remain equal.

3

Solve the proportion for

x

.

x = 900

The denominator

40

must be multiplied by

6

to get

240

. To keep the fraction equivalent, multiply the numerator

150

by

6

, yielding

900

.

Key Concept

Direct Proportions with Unit Harmonization

Estimated Time:1m 0s

Question 3340Question →

A naval research vessel uses active sonar to determine the depth of the ocean floor. The ship's transducer transmits an acoustic pulse downward into seawater where sound travels at an average speed of $1{}500\text{ m/s}$ . If the reflected echo is recorded by the receiver exactly $4.0\text{ seconds}$ after transmission, what is the depth of the ocean floor?

6{}000\text{ meters}

750\text{ meters}

3{}000\text{ meters}

375\text{ meters}

Show answer & explanation

Answer: $3{}000\text{ meters}$

Answer

3{}000\text{ meters}

Sonar relies on reflected sound waves (echoes). Because the sound pulse must travel down to the seafloor and then back up to the ship's receiver, the

4.0\text{ seconds}

represents two-way travel time. The one-way travel time is

2.0\text{ seconds}

. Multiplying the speed of sound in seawater (

1{}500\text{ m/s}

) by the one-way time (

2.0\text{ s}

) gives an ocean depth of

3{}000\text{ meters}

.

Step-by-Step Solution

1

Calculate the total distance traveled by the sound wave during the round trip

\text{Total distance} = \text{speed} \times \text{total time} = 1{}500\text{ m/s} \times 4.0\text{ s} = 6{}000\text{ meters}

The acoustic pulse travels from the ship down to the ocean floor and reflects back up to the surface.

2

Divide the total round-trip distance by two to obtain the one-way depth

\text{Depth} = \frac{6{}000\text{ m}}{2} = 3{}000\text{ meters}

Sonar distance measurement involves two equal legs of travel (downward and upward).

Key Concept

Calculation of distance using sound wave reflection and two-way travel time

Estimated Time:1m 0s

Answer

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