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Question 3341Question →

A customer brings their vehicle in for repairs. The mechanic's itemized bill lists $200.00\$200.00 for labor and $120.00\$120.00 for parts. The customer uses a promotional code that provides a 15%15\% discount on the labor cost. A 5%5\% sales tax is applied only to the cost of the parts. If the customer adds a 20%20\% tip calculated on the original pre-discount labor cost, what is the total amount the customer pays?

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Answer: $336.00\$336.00

Answer

The total amount the customer pays is $336.00\$336.00.
The total consists of four separate components: the discounted labor, the parts, the tax, and the tip. First, the discounted labor is $200−(0.15×$200)=$170\$200 - (0.15 \times \$200) = \$170. The parts remain at $120\$120. Second, the 5%5\% sales tax is applied only to the parts, resulting in an additional 0.05×$120=$60.05 \times \$120 = \$6. Finally, the 20%20\% tip is calculated strictly on the original $200\$200 labor cost, yielding $40\$40. Summing these amounts yields $170+$120+$6+$40=$336.00\$170 + \$120 + \$6 + \$40 = \$336.00.

Step-by-Step Solution

1
Calculate the discounted labor cost.
$170.00\$170.00
A 15%15\% discount on the $200.00\$200.00 labor cost is 0.15×200=300.15 \times 200 = 30. Subtracting this from the original cost gives $200−30=$170\$200 - 30 = \$170.
2
Calculate the sales tax on the parts.
$6.00\$6.00
The 5%5\% tax is only applied to the $120.00\$120.00 parts cost. 0.05×120=60.05 \times 120 = 6.
3
Calculate the tip based on the original labor cost.
$40.00\$40.00
The problem states the 20%20\% tip applies to the original pre-discount labor cost ($200.00\$200.00). 0.20×200=400.20 \times 200 = 40.
4
Sum all final costs to find the total amount paid.
$336.00\$336.00
Add the discounted labor, parts, tax on parts, and tip: $170+$120+$6+$40=$336.00\$170 + \$120 + \$6 + \$40 = \$336.00.

Key Concept

Calculating percentages on distinct bases in multi-step financial transactions.
Estimated Time:2m 0s
Question 3342Question →

A customer pays for a car detailing service that costs $120.00\$120.00 before tax. If the local sales tax rate is 7%7\%, what is the total cost of the service including tax?

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Answer: $128.40\$128.40

Answer

The total cost of the service is $128.40\$128.40.
To find the total cost, first calculate the sales tax by converting 7%7\% to a decimal (0.070.07) and multiplying it by the original price of $120.00\$120.00, which equals $8.40\$8.40. Then, add this tax amount to the original price ($120.00+$8.40=$128.40\$120.00 + \$8.40 = \$128.40) to get the final total.

Step-by-Step Solution

1
Convert the sales tax percentage to a decimal.
7%=0.077\% = 0.07
Percentages must be expressed as decimals to perform mathematical operations.
2
Calculate the amount of sales tax.
$120.00×0.07=$8.40\$120.00 \times 0.07 = \$8.40
Multiplying the base price by the tax rate gives the total tax amount.
3
Add the tax amount to the original price to find the total cost.
$120.00+$8.40=$128.40\$120.00 + \$8.40 = \$128.40
The total cost includes the base price plus the applied tax.

Key Concept

Calculating sales tax and total cost
Question 3343Question →

A motor pool technician is preparing a specialized coolant mixture. A full batch requires 45\frac{4}{5} of a gallon of antifreeze. If the technician needs to prepare exactly 13\frac{1}{3} of a batch for a small all-terrain vehicle, how much antifreeze is required?

Show answer & explanation

Answer: 415\frac{4}{15}

Answer

The technician requires 415\frac{4}{15} of a gallon of antifreeze.
To calculate a fractional portion of an existing fraction, you must multiply the two values. Multiplying the numerators (4×14 \times 1) yields 44, and multiplying the denominators (5×35 \times 3) yields 1515. This makes the correct amount 415\frac{4}{15} of a gallon.

Step-by-Step Solution

1
Identify the operation required to find a fraction of a given quantity.
The problem asks for 13\frac{1}{3} of a 45\frac{4}{5} gallon batch, which requires multiplication.
The word 'of' in this context indicates multiplication to find a part of a total amount.
2
Multiply the two fractions.
45×13=4×15×3=415\frac{4}{5} \times \frac{1}{3} = \frac{4 \times 1}{5 \times 3} = \frac{4}{15}
To multiply fractions, multiply the numerators together and the denominators together.

Key Concept

Fraction Multiplication in Context
Question 3344Question →

A motor pool mechanic is comparing four bulk purchasing options for a specialized synthetic lubricant. To make the most cost-effective decision, they need to determine the actual cost per ounce for each option, accounting for any additional fees or discounts.

Arrange the following purchasing options in order from the lowest cost per ounce to the highest cost per ounce.

Drag items to arrange them in the correct order

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Answer

The correct sequence from lowest to highest cost per ounce is: Bravo Brand ($4.15\$4.15), Delta Brand ($4.25\$4.25), Alpha Brand ($4.30\$4.30), and Charlie Brand ($4.40\$4.40).
To accurately compare the purchasing options, each one must be reduced to its unit cost (price per ounce) after adjusting for all fees and rebates. Bravo Brand is the cheapest at $4.15\$4.15 per ounce. Delta Brand is next at $4.25\$4.25 per ounce. Alpha Brand follows at $4.30\$4.30 per ounce after factoring in its handling fee. Charlie Brand is the most expensive at $4.40\$4.40 per ounce even after applying the instant rebate.

Step-by-Step Solution

1
Calculate the unit cost for Alpha Brand.
$4.30\$4.30 per ounce.
First, find the total cost by adding the handling fee: $52.50+$1.25=$53.75\$52.50 + \$1.25 = \$53.75. Next, divide by the volume: $53.75÷12.5=$4.30\$53.75 \div 12.5 = \$4.30.
2
Calculate the unit cost for Bravo Brand.
$4.15\$4.15 per ounce.
Divide the total cost by the volume: $33.20÷8.0=$4.15\$33.20 \div 8.0 = \$4.15.
3
Calculate the unit cost for Charlie Brand.
$4.40\$4.40 per ounce.
First, find the total cost by subtracting the rebate: $69.00−$3.00=$66.00\$69.00 - \$3.00 = \$66.00. Next, divide by the volume: $66.00÷15.0=$4.40\$66.00 \div 15.0 = \$4.40.
4
Calculate the unit cost for Delta Brand.
$4.25\$4.25 per ounce.
Divide the total cost by the volume: $102.00÷24.0=$4.25\$102.00 \div 24.0 = \$4.25.
5
Order the calculated unit costs from lowest to highest.
Bravo Brand, Delta Brand, Alpha Brand, Charlie Brand.
Comparing the decimal values gives 4.15<4.25<4.30<4.404.15 < 4.25 < 4.30 < 4.40.

Key Concept

Calculating and comparing unit prices involving decimals, fees, and discounts.
Question 3345Question →

Read the scenario and determine the total number of pieces that can be cut. Enter your numerical answer in the blank.

Fill in the blanks below

A quartermaster has a 5050-yard spool of parachute cord. They need to cut the entire spool into smaller pieces that are each 1231 \frac{2}{3} yards long. They will be able to cut a total of pieces.
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Answer

30
To find how many smaller pieces can be cut from a larger total, you must divide the total length (5050 yards) by the length of each individual piece (1231 \frac{2}{3} yards). First, convert 1231 \frac{2}{3} to the improper fraction 53\frac{5}{3}. The problem then becomes 50÷5350 \div \frac{5}{3}. To divide by a fraction, multiply by its reciprocal: 50×3550 \times \frac{3}{5}. Multiply 5050 by 33 to get 150150, then divide by 55 to get 3030 pieces.

Step-by-Step Solution

1
Identify the total length of the cord and the length of each individual piece.
The total length is 5050 yards, and the length of each piece is 1231 \frac{2}{3} yards.
These values are required to set up the division expression.
2
Convert the mixed number to an improper fraction.
123=1×3+23=531 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}
Fraction division is performed using improper fractions rather than mixed numbers.
3
Set up the division problem by dividing the total length by the length of one piece.
50÷5350 \div \frac{5}{3}
Dividing the total quantity by the size of each individual part gives the total number of parts.
4
Multiply the total length by the reciprocal of the fraction.
50×35=1505=3050 \times \frac{3}{5} = \frac{150}{5} = 30
To divide by a fraction, you multiply by its reciprocal.

Key Concept

Fraction Division in Context
Question 3346Question →

Match each military and logistics operational scenario to the correct final mathematical ratio. Read the conditions carefully, as some require calculating a new ratio after a change occurs.

Click a left item, then click its matching right item

Items

A naval strike group contains cruisers, destroyers, and frigates. For every 22 cruisers, there are 55 destroyers. The number of frigates equals the combined number of cruisers and destroyers. Find the ratio of destroyers to the total number of ships.
A unit has 6060 personnel total, with an initial officer to enlisted ratio of 3:73:7. After 66 officers are transferred out (while enlisted numbers remain unchanged), find the new ratio of officers to enlisted personnel.
A fuel mixture correctly requires 33 parts additive per 88 parts base fuel. A technician mistakenly uses 44 parts additive per 1010 parts base fuel instead. If both scenarios use exactly 4040 gallons of base fuel, find the ratio of the correct additive amount to the mistakenly used additive amount.
A supply convoy's ratio of cargo trucks to escort vehicles is 5:25:2. The convoy has 3535 total vehicles. If 11 escort vehicle breaks down and leaves the convoy, find the new ratio of escort vehicles to cargo trucks.

Matches

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Answer

The correct matches are: Scenario 1 maps to 5:14, Scenario 2 maps to 2:7, Scenario 3 maps to 15:16, and Scenario 4 maps to 9:25.
Each scenario involves distinct ratio logic: Scenario 1 requires combining parts to find a whole. Scenarios 2 and 4 require breaking a total into parts using an initial ratio, subtracting from one part, and writing a new simplified ratio. Scenario 3 requires using part-to-part logic to find an unknown part given a specific base, then comparing the two results.

Step-by-Step Solution

1
Calculate the ratios for the Naval Strike Group (Scenario 1).
The total proportion is 22 (cruisers) + 55 (destroyers) + 77 (frigates) = 1414. The ratio of destroyers to total ships is 5:145:14.
You must establish the total baseline by summing the parts before finding the part-to-whole ratio.
2
Determine the new personnel ratio (Scenario 2).
Initially, officers = 1818 and enlisted = 4242. Subtracting 66 officers leaves 1212. The new ratio is 12:4212:42, simplified to 2:72:7.
Changes to one subset of a population require determining the original absolute values first.
3
Compare the fuel additive amounts (Scenario 3).
Correct additive is 38×40=15\frac{3}{8} \times 40 = 15. Mistaken additive is 410×40=16\frac{4}{10} \times 40 = 16. The ratio is 15:1615:16.
Scaling both ratios up to the common base fuel amount of 4040 reveals the exact additive quantities to compare.
4
Adjust the convoy vehicles ratio (Scenario 4).
Initially, cargo = 2525 and escort = 1010. Subtracting 11 escort leaves 99. The required ratio of escort to cargo is 9:259:25.
Calculating initial absolute quantities is necessary before applying the subtraction of 11 vehicle and interpreting the final specific order requested.

Key Concept

Applying part-to-part and part-to-whole ratio relationships to calculate absolute values, applying condition changes, and then re-evaluating and simplifying the final ratios.

Alternative Method

For Scenario 2, you can also calculate the fractional drop without finding absolute enlisted numbers. The officer count drops from 1818 to 1212, which is exactly 23\frac{2}{3} of the original amount. Thus, the new ratio must simply be 23\frac{2}{3} of the original officer portion (33) against the same enlisted portion (77), resulting directly in a 2:72:7 ratio.
Estimated Time:2m 30s
Question 3347Question →

A neutral phosphorus atom contains 1515 protons and 1616 neutrons. If this atom gains 33 electrons to form a stable phosphide ion, what is the mass number of the resulting ion?

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Answer: 31

Answer

The mass number of the resulting phosphide ion is 3131.
The mass number (AA) is defined as the total number of protons and neutrons located in the nucleus (15+16=3115 + 16 = 31). When an atom gains or loses electrons, it forms an ion with an electric charge, but the contents of the nucleus remain unchanged.

Step-by-Step Solution

1
Identify the components that determine the mass number of an atom or ion.
Mass number (AA) is calculated as the sum of protons (ZZ) and neutrons (NN).
Electrons have negligible mass relative to nucleons and reside outside the nucleus, so they are not included in the mass number.
2
Calculate the mass number using the given proton and neutron counts.
A=15+16=31A = 15 + 16 = 31
Adding 1515 protons and 1616 neutrons yields the total number of nucleons in the nucleus.
3
Evaluate the effect of gaining 33 electrons on the mass number.
The mass number remains 3131.
Gaining electrons changes only the net electric charge from 00 to −3-3, leaving the nucleus and mass number unaltered.

Key Concept

The mass number of an atom or ion is strictly the sum of its protons and neutrons (A=Z+NA = Z + N) and does not change during ionization.
Estimated Time:45s
Question 3348Question →

Sarah deposits $2,000\$2,000 into a savings account that earns a simple interest rate of 3%3\% per year. If she makes no additional deposits or withdrawals, how much total interest will the account have earned at the end of 4848 months?

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Answer: $240\$240

Answer

$240\$240
To find the simple interest, use the formula I=PRTI = PRT. First, convert the time from months to years by dividing 4848 by 1212, which equals 44 years. Then, multiply the principal ($2,000\$2,000), the annual decimal rate (0.030.03), and the time in years (44). This gives 2000×0.03×4=2402000 \times 0.03 \times 4 = 240. The total interest earned is $240\$240.

Step-by-Step Solution

1
Convert the given time from months into years.
48 months÷12 months/year=4 years48 \text{ months} \div 12 \text{ months/year} = 4 \text{ years}
The interest rate is annual (per year), so the time must also be expressed in years.
2
Convert the percentage interest rate into a decimal.
3%=0.033\% = 0.03
Percentages must be expressed as decimals to be used mathematically in the formula.
3
Calculate the total simple interest using the formula I=PRTI = PRT.
I=2000×0.03×4=240I = 2000 \times 0.03 \times 4 = 240
Multiplying the principal (PP), the rate (RR), and the time (TT) yields the total interest earned over the period.

Key Concept

Calculating simple interest over a multi-year period requiring unit conversion from months to years.
Question 3349Question →

An IT consultant bills a corporate client for a bulk order of software licenses. The wholesale cost of the licenses is $850.00\$850.00. The consultant applies a 40%40\% markup to the wholesale cost to determine the retail price. After a contract negotiation, the consultant offers the client a 15%15\% discount off the retail price. What is the final price the client pays for the software licenses?

Show answer & explanation

Answer: $1,011.50\$1,011.50

Answer

$1,011.50\$1,011.50
First, find the retail price by adding a 40%40\% markup to the $850.00\$850.00 wholesale cost. 40%40\% of $850.00\$850.00 is $340.00\$340.00, making the retail price $1,190.00\$1,190.00. Next, calculate the 15%15\% discount on the new retail price. 15%15\% of $1,190.00\$1,190.00 is $178.50\$178.50. Subtracting this discount from the retail price gives a final price of $1,011.50\$1,011.50.

Step-by-Step Solution

1
Calculate the markup amount.
$340.00\$340.00
The markup is 40%40\% of the wholesale cost of $850.00\$850.00 (0.40×8500.40 \times 850).
2
Calculate the retail price.
$1,190.00\$1,190.00
Add the markup amount to the wholesale cost (850.00+340.00850.00 + 340.00).
3
Calculate the discount amount.
$178.50\$178.50
The discount is 15%15\% of the new retail price, not the wholesale cost (0.15×11900.15 \times 1190).
4
Calculate the final price.
$1,011.50\$1,011.50
Subtract the discount amount from the retail price (1,190.00−178.501,190.00 - 178.50).

Key Concept

Calculating successive percentage changes requires applying the second percentage to the new intermediate value (the retail price), not the original value.
Question 3350Question →

A defense contractor manufactures three types of vehicle armor panels: Type A, Type B, and Type C. The ratio of Type A panels to Type B panels produced is 2:32:3. The ratio of Type B panels to Type C panels produced is 4:54:5. If the contractor produces a total of 1,4001,400 panels across all three types, how many Type C panels are produced?

Show answer & explanation

Answer: 600

Answer

600
To find the number of Type C panels, the two separate ratios must first be combined into one continuous ratio. Since Type B is in both ratios (33 parts in the first, 44 parts in the second), we find a common multiple, which is 1212. Scaling the ratios gives Type A : Type B = 8:128:12 and Type B : Type C = 12:1512:15. Combined, the ratio is 8:12:158:12:15. Adding these gives 3535 total parts. Dividing the 1,4001,400 total panels by 3535 reveals each part equals 4040 panels. Because Type C represents 1515 of those parts, we multiply 1515 by 4040 to get 600600 Type C panels.

Step-by-Step Solution

1
Identify the overlapping category (Type B) in both given ratios and find the least common multiple for its values (33 and 44).
The least common multiple of 33 and 44 is 1212.
To compare all three panel types simultaneously, they must be expressed within a single, unified ratio sequence.
2
Scale both original ratios so that Type B is represented by 1212 in both.
Multiply the first ratio (2:32:3) by 44 to get 8:128:12. Multiply the second ratio (4:54:5) by 33 to get 12:1512:15.
Multiplying both sides of a ratio by the same number keeps the proportions mathematically equivalent.
3
Combine the scaled ratios into a single three-part ratio and calculate the total number of parts.
The unified ratio of Type A to Type B to Type C is 8:12:158:12:15. The total number of parts is 8+12+15=358 + 12 + 15 = 35.
The total number of parts represents the entire production batch in ratio units.
4
Divide the total actual number of panels by the total ratio parts to find the value of one part, then calculate the total for Type C.
1,400÷35=401,400 \div 35 = 40 panels per part. Type C has 1515 parts, so 15×40=60015 \times 40 = 600 Type C panels.
This scales the ratio back up to the real-world quantities provided in the problem stem.

Key Concept

Combining Overlapping Ratios
Estimated Time:1m 30s
Question 3351Question →

A customer orders groceries through a delivery service. The subtotal of the groceries is $50.00\$50.00. The service charges a 10%10\% delivery fee and an 8%8\% sales tax, both calculated on the grocery subtotal. The customer then adds a 20%20\% tip, which is calculated on the amount after the delivery fee and tax are applied. What is the total final cost to the customer?

Show answer & explanation

Answer: 70.8

Answer

70.80
The final cost is found by first determining the intermediate amount after the fee and tax are added to the subtotal. The delivery fee is $5.00\$5.00 (10%10\% of $50\$50) and the tax is $4.00\$4.00 (8%8\% of $50\$50), bringing the intermediate total to $59.00\$59.00. The 20%20\% tip is calculated sequentially on this $59.00\$59.00 amount, which equals $11.80\$11.80. Adding the tip to the intermediate total yields the final cost of $70.80\$70.80.

Step-by-Step Solution

1
Calculate the delivery fee based on the subtotal.
$50.00×0.10=$5.00\$50.00 \times 0.10 = \$5.00
The problem states the 10%10\% delivery fee is calculated on the grocery subtotal.
2
Calculate the sales tax based on the subtotal.
$50.00×0.08=$4.00\$50.00 \times 0.08 = \$4.00
The problem states the 8%8\% sales tax is also calculated on the grocery subtotal.
3
Calculate the intermediate total before the tip.
$50.00+$5.00+$4.00=$59.00\$50.00 + \$5.00 + \$4.00 = \$59.00
The tip is based on the amount after the delivery fee and tax are applied, so we must find this intermediate total.
4
Calculate the tip based on the intermediate total.
$59.00×0.20=$11.80\$59.00 \times 0.20 = \$11.80
The 20%20\% tip is applied to the intermediate total of $59.00\$59.00.
5
Calculate the final cost to the customer.
$59.00+$11.80=$70.80\$59.00 + \$11.80 = \$70.80
The final cost is the sum of the intermediate total and the tip.

Key Concept

Calculating sequential percentages for taxes, fees, and tips on a base amount.
Question 3352Question →

A combat engineering team lays 180180 feet of portable runway surface every 2020 minutes. If they continue working at this exact same constant rate, how many yards of runway surface can they lay during a standard 55-hour shift?

Show answer & explanation

Answer: 900

Answer

900
First, find the unit rate of runway laid per minute by dividing the given distance by the given time (180 feet÷20 minutes=9 feet per minute180 \text{ feet} \div 20 \text{ minutes} = 9 \text{ feet per minute}). Next, convert the total shift time from hours to minutes (5 hours×60 minutes/hour=300 minutes5 \text{ hours} \times 60 \text{ minutes/hour} = 300 \text{ minutes}). Multiply the unit rate by the total minutes to find the total distance in feet (9 feet/minute×300 minutes=2,700 feet9 \text{ feet/minute} \times 300 \text{ minutes} = 2,700 \text{ feet}). Finally, convert feet to yards by dividing by 33 (2,700 feet÷3 feet/yard=900 yards2,700 \text{ feet} \div 3 \text{ feet/yard} = 900 \text{ yards}).

Step-by-Step Solution

1
Determine the unit rate of construction in feet per minute.
180 feet÷20 minutes=9 feet per minute180 \text{ feet} \div 20 \text{ minutes} = 9 \text{ feet per minute}
Finding the unit rate allows for easy calculation of distance over any given amount of time.
2
Convert the total shift time from hours to minutes.
5 hours×60 minutes/hour=300 minutes5 \text{ hours} \times 60 \text{ minutes/hour} = 300 \text{ minutes}
The unit rate is in feet per minute, so the total time must also be in minutes to maintain consistent units.
3
Calculate the total distance constructed in feet.
9 feet/minute×300 minutes=2,700 feet9 \text{ feet/minute} \times 300 \text{ minutes} = 2,700 \text{ feet}
Multiplying the rate by the total time gives the total amount of surface laid.
4
Convert the total distance from feet to yards.
2,700 feet÷3 feet/yard=900 yards2,700 \text{ feet} \div 3 \text{ feet/yard} = 900 \text{ yards}
The question specifically asks for the final answer in yards, requiring division by 33.

Key Concept

Direct proportions with multiple unit conversions

Alternative Method

Instead of breaking the problem down into steps, you can set up a single dimensional analysis equation: 180 ft20 min×60 min1 hr×5 hr1×1 yd3 ft=900 yd\frac{180 \text{ ft}}{20 \text{ min}} \times \frac{60 \text{ min}}{1 \text{ hr}} \times \frac{5 \text{ hr}}{1} \times \frac{1 \text{ yd}}{3 \text{ ft}} = 900 \text{ yd}.
Estimated Time:2m 0s
Question 3353Question →

Match each chemical species with its correct composition of protons, neutrons, and electrons.

Click a left item, then click its matching right item

Items

Neutral Carbon-12 atom (612C^{12}_{6}\text{C})
Sodium ion (1123Na+^{23}_{11}\text{Na}^{+})
Fluoride ion (919F−^{19}_{9}\text{F}^{-})
Neutral Neon-20 atom (1020Ne^{20}_{10}\text{Ne})

Matches

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Answer

Neutral Carbon-12 matches 6 protons, 6 neutrons, and 6 electrons; Sodium ion (+1) matches 11 protons, 12 neutrons, and 10 electrons; Fluoride ion (-1) matches 9 protons, 10 neutrons, and 10 electrons; Neutral Neon-20 matches 10 protons, 10 neutrons, and 10 electrons.
Each species is correctly matched by identifying protons from the atomic number (ZZ), computing neutrons by subtracting the atomic number from the mass number (A−ZA - Z), and adjusting the electron count according to the ionic charge (e−=Z−chargee^- = Z - \text{charge}).

Step-by-Step Solution

1
Determine the number of protons from the atomic number (ZZ) for each species.
Carbon has 66 protons, Sodium has 1111 protons, Fluorine has 99 protons, and Neon has 1010 protons.
The atomic number uniquely defines the identity of an element and equals the total proton count.
2
Calculate the number of neutrons using Neutrons=Mass Number (A)−Atomic Number (Z)\text{Neutrons} = \text{Mass Number } (A) - \text{Atomic Number } (Z).
Carbon-12 has 12−6=612 - 6 = 6 neutrons; Sodium-23 has 23−11=1223 - 11 = 12 neutrons; Fluorine-19 has 19−9=1019 - 9 = 10 neutrons; Neon-20 has 20−10=1020 - 10 = 10 neutrons.
Mass number is the sum of protons and neutrons in the nucleus.
3
Determine the electron count based on net charge using Electrons=Protons−Charge\text{Electrons} = \text{Protons} - \text{Charge}.
Neutral Carbon has 66 electrons; Na+\text{Na}^{+} has 11−(+1)=1011 - (+1) = 10 electrons; F−\text{F}^{-} has 9−(−1)=109 - (-1) = 10 electrons; neutral Neon has 1010 electrons.
Cations lose electrons to form positive charges, anions gain electrons to form negative charges, and neutral atoms have equal numbers of protons and electrons.

Key Concept

Subatomic particle calculations in neutral atoms and isoelectronic ions
Estimated Time:1m 0s
Question 3354Question →

A monochromatic beam of light travels through air and enters a transparent block of crown glass. As the light wave enters the optically denser glass medium, what happens to its speed, wavelength, and frequency?

Show answer & explanation

Answer: Its speed decreases and its wavelength decreases, while its frequency remains constant.

Answer

Its speed decreases and its wavelength decreases, while its frequency remains constant.
When light transitions from air into a medium with a higher refractive index such as glass, its propagation speed decreases (v=c/nv = c/n). The frequency of the oscillation is an intrinsic property established by the wave source and remains invariant at the interface. By the wave relationship v=fλv = f\lambda, a reduction in wave velocity at constant frequency results in a proportional reduction in wavelength.

Step-by-Step Solution

1
Determine the effect of the medium change on wave speed.
The speed of light decreases from c≈3.0×108 m/sc \approx 3.0 \times 10^8\text{ m/s} in air to approximately 2.0×108 m/s2.0 \times 10^8\text{ m/s} in crown glass (n≈1.52n \approx 1.52) based on v=cnv = \frac{c}{n}.
An increase in optical density (higher index of refraction) reduces the propagation speed of electromagnetic waves.
2
Determine the behavior of wave frequency across the medium boundary.
The frequency (ff) remains completely unchanged.
Wave frequency is set solely by the emitting source and is conserved across media boundaries because the number of wave cycles arriving per second must equal the number transmitted.
3
Apply the wave equation to determine the resulting change in wavelength.
The wavelength (λ\lambda) decreases proportionally with speed according to λ=vf\lambda = \frac{v}{f}.
With speed vv reduced and frequency ff constant, the spatial distance between consecutive crests must shorten.

Key Concept

Wave speed and wavelength changes during medium transition under constant frequency (v=fλv = f\lambda).
Estimated Time:45s
Question 3355Question →

A homeowner hires a landscaping service for an original quoted price of $450.00\$450.00. The company applies a 15%15\% promotional discount to this price. A local service tax of 8%8\% is then calculated based on the discounted amount. Finally, the homeowner adds a tip for the workers equal to 20%20\% of the original quoted price. What is the total amount, in dollars, that the homeowner pays?

Show answer & explanation

Answer: 503.1

Answer

503.10
The final amount is the sum of the discounted price, the tax calculated on the discounted price, and the tip calculated on the original price. Adding $382.50\$382.50, $30.60\$30.60, and $90.00\$90.00 yields a final total of $503.10\$503.10.

Step-by-Step Solution

1
Calculate the discount amount.
$67.50\$67.50
The discount is 15%15\% of the original $450.00\$450.00 price.
2
Determine the discounted subtotal.
$382.50\$382.50
Subtract the $67.50\$67.50 discount from the original $450.00\$450.00 price.
3
Calculate the service tax.
$30.60\$30.60
The 8%8\% tax is applied to the discounted subtotal of $382.50\$382.50.
4
Calculate the tip amount.
$90.00\$90.00
The 20%20\% tip is based on the original quoted price of $450.00\$450.00.
5
Calculate the final total cost.
$503.10\$503.10
Add the discounted subtotal ($382.50\$382.50), tax ($30.60\$30.60), and tip ($90.00\$90.00).

Key Concept

Calculating sequential percentage changes, including taxes on discounted amounts and tips on original amounts.
Question 3356Question →

A Navy Seabee is framing a new barracks and starts with a wooden board that is 163816 \frac{3}{8} feet long. For the first section of the frame, a piece measuring 4344 \frac{3}{4} feet is cut from the board. For the second section, another piece measuring 5565 \frac{5}{6} feet is cut. Assuming there is no waste from the saw cuts, what is the remaining length of the wooden board?

Show answer & explanation

Answer: 519245 \frac{19}{24}

Answer

The remaining length of the wooden board is 519245 \frac{19}{24} feet.
To find the remaining length, the lengths of the two cut pieces must first be added together. Finding a common denominator of 24, the cuts are 418244 \frac{18}{24} and 520245 \frac{20}{24}. Their sum is 938249 \frac{38}{24}, which regroups to 10142410 \frac{14}{24}. Next, subtract this total from the original board length: 16924−10142416 \frac{9}{24} - 10 \frac{14}{24}. Since 924\frac{9}{24} is smaller than 1424\frac{14}{24}, borrowing from the whole number is required: 153324−101424=5192415 \frac{33}{24} - 10 \frac{14}{24} = 5 \frac{19}{24}.

Step-by-Step Solution

1
Calculate the total length of the wooden board that was cut off.
434+556=41824+52024=93824=1014244 \frac{3}{4} + 5 \frac{5}{6} = 4 \frac{18}{24} + 5 \frac{20}{24} = 9 \frac{38}{24} = 10 \frac{14}{24} feet.
Adding the removed sections determines the total amount to subtract from the original board.
2
Set up the subtraction of the total cut length from the original length.
1638−101424=16924−10142416 \frac{3}{8} - 10 \frac{14}{24} = 16 \frac{9}{24} - 10 \frac{14}{24}
A common denominator is needed to subtract the fractions.
3
Borrow from the whole number to subtract the fractions.
153324−101424=5192415 \frac{33}{24} - 10 \frac{14}{24} = 5 \frac{19}{24} feet.
Since 924\frac{9}{24} is less than 1424\frac{14}{24}, borrowing 11 (or 2424\frac{24}{24}) from the whole number 1616 makes the subtraction possible.

Key Concept

Adding and subtracting mixed numbers with unlike denominators, including regrouping (borrowing).
Question 3357Question →

A military logistics officer must calculate operational timelines based on available resources. Connect each situation below to its correctly calculated duration. Assume all rates are constant and inversely proportional.

Click a left item, then click its matching right item

Items

A runway repair takes 14 hours when 18 engineers are assigned. Time required if 21 engineers are tasked with the exact same repair.
A patrol boat traveling at 36 knots completes a coastal route in 4.5 hours. Time required to complete the route at a reduced speed of 27 knots.
Fifteen identical extractor fans ventilate a bunker complex in 6 hours. Time required if only 10 of these fans are operational.
A forward base's fuel cache can power 16 standard generators for 45 hours. Time the cache will last if 24 generators are running simultaneously.

Matches

Show answer & explanation

Answer

The scenarios pair correctly with 12 hours, 6 hours, 9 hours, and 30 hours respectively based on their constant products.
Each scenario involves an inverse relationship where the product of the two given quantities forms a constant. Dividing this constant by the new condition yields the correctly paired time.

Step-by-Step Solution

1
Identify the relationship for each scenario.
All four scenarios represent inverse proportions, meaning the product of the two variables (e.g., workers and time, speed and time) remains constant (x1×y1=x2×y2x_1 \times y_1 = x_2 \times y_2).
When one variable increases (like adding more workers or driving faster), the time required decreases proportionally.
2
Calculate the constant product for each individual scenario.
Engineer-hours: 18×14=25218 \times 14 = 252. Distance: 36×4.5=16236 \times 4.5 = 162. Fan-hours: 15×6=9015 \times 6 = 90. Generator-hours: 16×45=72016 \times 45 = 720.
This establishes the total amount of work, distance, or resource available before the change in variables.
3
Divide each constant product by the new given variable to find the new time.
Runway: 252÷21=12252 \div 21 = 12. Boat: 162÷27=6162 \div 27 = 6. Fans: 90÷10=990 \div 10 = 9. Generators: 720÷24=30720 \div 24 = 30.
Applying the inverse proportion formula (y2=kx2y_2 = \frac{k}{x_2}) isolates the unknown time.

Key Concept

Inverse Proportions in Work, Speed, and Resource Rates
Question 3358Question →

Maria purchases a $1,200\$1,200 government bond that earns an annual simple interest rate of 4%4\%. How much total interest will the bond earn after exactly 66 months?

Show answer & explanation

Answer: $24\$24

Answer

$24\$24
The correct answer is found by taking the principal ($1,200\$1,200), multiplying it by the annual interest rate (0.040.04), and then multiplying by the time in years (0.50.5). This results in $1,200×0.04×0.5=$24\$1,200 \times 0.04 \times 0.5 = \$24.

Step-by-Step Solution

1
Identify the given values for principal, rate, and time.
Principal (P) = $1,200. Rate (R) = 4% or 0.04. Time (T) = 6 months.
These are the necessary components for the simple interest formula.
2
Convert the time period from months into years.
6 months ÷ 12 months/year = 0.5 years.
The annual interest rate requires the time to be expressed in years.
3
Calculate the interest using the formula I = PRT.
I = 1200 × 0.04 × 0.5 = 48 × 0.5 = 24.
This yields the total interest earned over the given time period.

Key Concept

Calculating simple interest over a fractional year
Question 3359Question →

Electromagnetic waves span a wide range of frequencies, each exhibiting distinct physical properties and practical applications across technology and medicine. Match each type of electromagnetic radiation with its primary characteristic or practical use.

Click a left item, then click its matching right item

Items

Infrared Radiation
Ultraviolet Radiation
X-rays
Radio Waves

Matches

Show answer & explanation

Answer

Infrared Radiation matches thermal energy and thermal imaging; Ultraviolet Radiation matches vitamin D synthesis and sunburns; X-rays match skeletal medical imaging through soft tissue penetration; Radio Waves match broadcast transmissions and long-wavelength telecommunications.
Each type of electromagnetic radiation is characterized by distinct wavelength, frequency, and photon energy bands. Radio waves have the longest wavelengths and lowest energy, suitable for broadcasting; infrared radiation transfers thermal heat; ultraviolet radiation induces photochemical reactions in skin; and X-rays have sufficient penetration power to create shadow radiographs through soft tissue.

Step-by-Step Solution

1
Identify the characteristic behavior and application of infrared radiation.
Infrared radiation is associated with molecular thermal vibration and heat emission, used in night-vision and thermal sensors.
Infrared photons are emitted by warm bodies and absorbed by matter as sensible heat.
2
Identify the biological effects and energy level of ultraviolet radiation.
Ultraviolet radiation carries higher photon energy than visible light, stimulating vitamin D synthesis in the skin but causing cellular damage leading to sunburns.
UV photon energy is sufficient to initiate chemical bond changes in biological tissue.
3
Identify the high-energy penetration characteristics of X-rays.
X-rays pass easily through soft tissues like muscle and skin but are blocked by dense bone structure.
High-frequency electromagnetic waves have high penetration power that is stopped predominantly by high-density matter.
4
Identify the fundamental wave traits of radio waves.
Radio waves possess the lowest frequencies and longest wavelengths, making them ideal for long-distance broadcast communication.
Long wavelengths diffract readily around terrain and experience low atmospheric absorption.

Key Concept

Electromagnetic Spectrum Characteristics and Applications
Estimated Time:1m 0s
Question 3360Question →

A furniture showroom acquires a dining table for a wholesale cost of $550.00\$550.00. The showroom applies an 80%80\% markup to set the original retail price. During a holiday clearance event, the table is discounted by 30%30\% from its retail price. What is the final sale price of the dining table?

Fill in the blanks below

The final sale price of the dining table is $\$$.
Show answer & explanation

Answer

693
The correct final sale price is found by first calculating the retail price and then applying the discount to that new price. An 80%80\% markup on the $550.00\$550.00 wholesale cost adds $440.00\$440.00, making the retail price $990.00\$990.00. A 30%30\% discount on the $990.00\$990.00 retail price is $297.00\$297.00. Subtracting this discount from the retail price gives the final sale price of $693.00\$693.00.

Step-by-Step Solution

1
Calculate the markup amount.
$550.00×0.80=$440.00\$550.00 \times 0.80 = \$440.00
To determine how much value is added to the wholesale cost to set the retail price.
2
Calculate the original retail price.
$550.00+$440.00=$990.00\$550.00 + \$440.00 = \$990.00
To establish the base price before any sale discounts are applied.
3
Calculate the discount amount.
$990.00×0.30=$297.00\$990.00 \times 0.30 = \$297.00
To find the amount subtracted from the retail price during the clearance event.
4
Calculate the final sale price.
$990.00−$297.00=$693.00\$990.00 - \$297.00 = \$693.00
To find the ultimate cost to the consumer after all price adjustments.

Key Concept

Calculating successive percentage changes by first applying a markup to a base cost, then applying a discount to the new resulting price.
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