All practice questions

Question 181Question →

An aviation maintenance depot uses two automated riveting machines to assemble fuselage panels. Machine A can install $240$ rivets in $4$ hours. Machine B can install $30$ rivets every $20$ minutes. If both machines operate simultaneously, how many hours will it take them to install a total of $6,300$ rivets?

30

42

63

70

Show answer & explanation

Answer: 42

Answer

The correct answer is 42 hours.

To solve this problem, both machine rates must first be converted to the same time unit (hours). Machine A operates at

60

rivets per hour (

240 \div 4

). Machine B operates at

90

rivets per hour (since there are three

20

-minute periods in an hour,

30 \times 3 = 90

). Together, they install

150

rivets per hour (

60 + 90

). Dividing the total requirement of

6,300

rivets by the combined rate of

150

rivets per hour yields exactly

42

hours.

Step-by-Step Solution

1

Calculate the unit rate for Machine A in rivets per hour.

240 \text{ rivets} \div 4 \text{ hours} = 60 \text{ rivets per hour}

To combine the rates of both machines, they must be in the same unit of measure (rivets per hour).

2

Calculate the unit rate for Machine B in rivets per hour.

30 \text{ rivets} \div 20 \text{ minutes} = 1.5 \text{ rivets per minute}

.

1.5 \times 60 = 90 \text{ rivets per hour}

Machine B's rate is given per

20

minutes, so it must be scaled up to an hourly rate by multiplying by

60

minutes.

3

Calculate the combined hourly rate of both machines.

60 + 90 = 150 \text{ rivets per hour}

The machines operate simultaneously, so their individual hourly rates are added together.

4

Determine the total time required to install

6,300

rivets.

6,300 \text{ rivets} \div 150 \text{ rivets per hour} = 42 \text{ hours}

Dividing the total work required by the combined hourly rate gives the total time needed.

Key Concept

Unit Rates and Multi-Step Rate Applications

Question 182Question →

A military dining facility prepares a standard meal using a ratio of $4$ pounds of beef for every $3$ pounds of rice. If the facility uses $24$ pounds of rice for a meal, how many pounds of beef will be needed?

Show answer & explanation

Answer: 32

Answer

32

Because the required ratio of beef to rice is

4:3

, every

3

pounds of rice requires

4

pounds of beef. Since

24

pounds of rice is

8

times the base amount (

24 \div 3 = 8

), the amount of beef must also be scaled by

8

. Multiplying

4

pounds by

8

yields exactly

32

pounds of beef.

Step-by-Step Solution

1

Identify the given ratio and the known actual quantity.

The ratio of beef to rice is

4:3

. The actual amount of rice used is

24

pounds.

Establishing the known values is the first step in solving a proportion.

2

Determine the scale factor between the base ratio and the actual amounts.

24 \div 3 = 8

. The scale factor is

8

.

Since the ratio amount for rice is

3

and the actual amount is

24

, multiplying by

8

scales the ratio up to the actual meal size.

3

Apply the scale factor to the beef portion of the ratio to find the missing quantity.

4 \times 8 = 32

.

To maintain the exact proportional relationship, both parts of the ratio must be multiplied by the same scale factor.

Key Concept

Solving for a missing quantity in a part-to-part ratio using a scale factor.

Estimated Time:1m 0s

Question 183Question →

A military supply depot has a stockpile of $26 \frac{2}{3}$ gallons of synthetic transmission fluid. If each transport vehicle requires exactly $1 \frac{1}{3}$ gallons of this fluid for a complete routine service, how many vehicles can be fully serviced using the entire stockpile?

20

28

35 \frac{5}{9}

25 \frac{1}{3}

Show answer & explanation

Answer: 20

Answer

20

To determine how many vehicles can be serviced, the total amount of fluid (

26 \frac{2}{3}

gallons) must be divided by the amount required per vehicle (

1 \frac{1}{3}

gallons). First, convert both amounts to improper fractions:

\frac{80}{3} \div \frac{4}{3}

. Next, multiply by the reciprocal of the second fraction:

\frac{80}{3} \times \frac{3}{4}

. The 3s in the numerator and denominator cancel out, leaving

80 \div 4

, which equals exactly 20.

Step-by-Step Solution

1

Convert both mixed numbers to improper fractions.

26 \frac{2}{3} = \frac{80}{3}

and

1 \frac{1}{3} = \frac{4}{3}

.

Mixed numbers must be written as improper fractions before they can be multiplied or divided.

2

Set up the division equation.

\frac{80}{3} \div \frac{4}{3}

To find how many times the vehicle requirement fits into the total stockpile, divide the total fluid by the fluid needed per vehicle.

3

Multiply by the reciprocal of the divisor to solve.

\frac{80}{3} \times \frac{3}{4} = \frac{240}{12} = 20

.

Dividing by a fraction is mathematically identical to multiplying by its reciprocal. The 3s cross-cancel, leaving 80 divided by 4.

Key Concept

Dividing mixed numbers by converting them to improper fractions and multiplying by the reciprocal.

Estimated Time:1m 30s

Question 184Question →

A submarine leaves port with enough provisions to feed its standard crew of $120$ personnel for $60$ days. Before departure, $30$ additional mission specialists are assigned to the submarine. If the daily ration per person remains constant, how many days will the provisions last the newly expanded crew?

Show answer & explanation

Answer: 48

Answer

48

Because the total amount of food remains constant, an increase in personnel causes a proportional decrease in the number of days the food will last. The product of personnel and days is constant:

120 \times 60 = 150 \times x

. Solving for

x

gives

7200 / 150 = 48

days.

Step-by-Step Solution

1

Calculate the total amount of provisions available.

120 \text{ personnel} \times 60 \text{ days} = 7200 \text{ person-days}

In an inverse proportion involving resources, the product of the number of consumers and the time the resource lasts represents the constant total resource.

2

Calculate the total number of personnel after the addition.

120 + 30 = 150 \text{ personnel}

The problem states that 30 additional specialists were assigned, so the new crew size is the sum of the original and additional personnel.

3

Calculate the new duration the provisions will last.

7200 \text{ person-days} \div 150 \text{ personnel} = 48 \text{ days}

Dividing the total constant resource by the new number of consumers yields the new time duration.

Key Concept

Inverse Proportions

Estimated Time:1m 0s

Question 185Question →

A mechanic purchases a broken bicycle for $\$40$ at a garage sale. He then buys new tires and a chain for a total of $\$25$ . After repairing the bicycle, he sells it for $\$110$ . What is his total profit, in dollars?

Show answer & explanation

Answer: 45

Answer

45

The total cost is the sum of the purchase price (

\$40

) and the repair parts (

\$25

), which is

\$65

. The profit is found by subtracting this total cost from the selling price (

\$110

), which gives

\$45

.

Step-by-Step Solution

1

Calculate the total cost.

\$40 + \$25 = \$65

The total cost includes the purchase price plus the cost of the repair parts.

2

Calculate the profit.

\$110 - \$65 = \$45

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

Key Concept

Profit is calculated by subtracting all associated costs (purchase price plus any overhead or repairs) from the final selling price.

Question 186Question →

A small business hires a contractor to install a new office security system. The initial quote for the work is $800. After negotiating, the business receives a 15% discount on the initial quote. The installation is subject to a 5% local service tax, which is applied to the discounted price. To show appreciation for completing the job ahead of schedule, the business adds a 10% tip calculated from the original$ 800 quote. What is the total amount the business pays in dollars?

Show answer & explanation

Answer: 794

Answer

794

To find the total amount paid, we must sequentially apply the rates to their correct base amounts. First, find the discounted price by subtracting 15% of

800 (

120) from the original quote, yielding

680. Next, calculate the 5% tax on this discounted price (

680 × 0.05 =

34). Then, calculate the 10% tip based on the original quote as specified in the scenario (

800 × 0.10 =

80). The final total is the sum of the discounted price, the tax amount, and the tip amount:

680 +

34 +

80 = $794.

Step-by-Step Solution

1

Calculate the discount amount

$120

The business received a 15% discount on the initial $800 quote.

2

Determine the discounted price

$680

Subtracting the

120 discount from the

800 initial quote gives the new base price for the tax.

3

Calculate the local service tax

$34

The 5% tax is applied to the discounted price of

680 (0.05 ×

680 = $34).

4

Calculate the tip amount

$80

The problem states the 10% tip is specifically based on the original

800 quote (0.10 ×

800 = $80).

5

Calculate the final total

$794

Add the discounted price (

680), the tax (

34), and the tip ($80) to find the total amount paid.

Key Concept

Calculating percentage discounts, taxes, and tips when applied to different base amounts.

Question 187Question →

During a field training exercise, a supply sergeant purchases 8 identical waterproof tarps. If the total cost for all 8 tarps is $\$64.40$ , what is the cost of a single tarp in dollars?

Show answer & explanation

Answer: 8.05

Answer

8.05

The correct answer is found by taking the total cost of all tarps and dividing it by the number of tarps. Dividing

\$64.40

by 8 yields exactly

\$8.05

per tarp.

Step-by-Step Solution

1

Set up the division problem

64.40 \div 8

To find the cost of a single item, divide the total cost by the total quantity.

2

Perform the division

8.05

8 goes into 64 exactly 8 times. Bring down the decimal point. 8 goes into 4 zero times. Bring down the 0 to make 40. 8 goes into 40 exactly 5 times. Thus,

64.40 \div 8 = 8.05

.

Key Concept

Dividing a decimal currency amount by a whole number to find a unit price.

Estimated Time:45s

Question 188Question →

A motor pool technician is mixing a cleaning solvent and water in a ratio of $2:5$ to fill a $35$ -gallon container. How many gallons of cleaning solvent are needed?

7

10

14

25

Show answer & explanation

Answer: 10

Answer

10 gallons

The total number of ratio parts is

2 + 5 = 7

. The total volume is

35

gallons, so each part represents

35 \div 7 = 5

gallons. Since the cleaning solvent corresponds to

2

parts, the amount of solvent needed is

2 \times 5 = 10

gallons.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

2 + 5 = 7

total parts

The ratio

2:5

means the total mixture is made up of

7

equal parts.

2

Calculate the volume of a single part.

35 \div 7 = 5

gallons per part

Dividing the total volume by the total number of parts gives the amount for one individual part.

3

Calculate the volume of the cleaning solvent.

2 \times 5 = 10

gallons

The cleaning solvent represents

2

parts of the total mixture.

Key Concept

Dividing a total quantity into proportional parts based on a given part-to-part ratio.

Question 189Question →

A military logistics unit is preparing a supply drop. The standard loadout requires a ratio of $2$ pallets of Meals Ready-to-Eat (MREs) for every $3$ pallets of bottled water. If the unit loads $30$ pallets of MREs, how many pallets of bottled water are required for the drop?

45

20

18

75

Show answer & explanation

Answer: $45$

Answer

The correct answer is

45

pallets of bottled water.

The correct answer is calculated by setting up a proportion based on the part-to-part ratio. The MREs represent

2

parts of the ratio, which equals

30

pallets. Therefore, one part equals

15

pallets. The water represents

3

parts, so

3 \times 15 = 45

pallets of water.

Step-by-Step Solution

1

Identify the given ratio and known quantity.

The ratio of MREs to water is

2:3

, and there are

30

pallets of MREs.

Establishing the relationship between the parts is necessary to find the unknown quantity.

2

Determine the multiplier for the ratio.

30 \div 2 = 15

Since the

2

parts in the ratio represent

30

actual pallets, dividing determines the value of one part.

3

Calculate the unknown quantity using the multiplier.

3 \times 15 = 45

Multiplying the water ratio (

3

) by the value of one part (

15

) gives the actual number of water pallets.

Key Concept

Basic part-to-part ratio calculation involving finding a missing quantity when one part is known.

Question 190Question →

A military supply depot transfers $40\%$ of its medical kit inventory to a forward operating base. The following week, the depot transfers $30\%$ of its remaining medical kits to a local training facility. If the depot currently has $1,260$ medical kits left in stock, what was the original number of medical kits in the inventory?

3,000

4,200

7,000

2,142

Show answer & explanation

Answer: $3,000$

Answer

The original number of medical kits in the inventory was

3,000

.

The correct approach requires tracking the percentages relative to the original inventory. After

40\%

of the kits are transferred,

60\%

remains. The second transfer removes

30\%

of this

60\%

, meaning

70\%

of the

60\%

stays in stock. Multiplying

0.60 \times 0.70

gives

0.42

, which means the final

1,260

kits represent exactly

42\%

of the original inventory. Dividing

1,260

by

0.42

yields the original starting total of

3,000

.

Step-by-Step Solution

1

Determine the percentage of inventory remaining after the first transfer.

100\% - 40\% = 60\%

of the original inventory remains.

The second transfer is based on this remaining amount, not the original total.

2

Calculate the percentage of the original inventory that remains after the second transfer.

The second transfer takes

30\%

of the remaining kits, meaning

70\%

of those kits stay at the depot.

0.70 \times 0.60 = 0.42

, so

42\%

of the original inventory is left.

To set up an equation, the final quantity must be related to a single percentage representing a portion of the original whole.

3

Solve for the original number of medical kits.

Let

x

be the original inventory.

0.42x = 1,260

. Dividing both sides by

0.42

gives

x = 1,260 \div 0.42 = 3,000

.

Dividing the final remaining quantity by its corresponding decimal percentage yields the original starting value.

Key Concept

Calculating successive percentages and working backward from a final quantity to find an original whole.

Question 191Question →

A local cafe owner purchases $50$ pounds of raw coffee beans for $\$12$ per pound. The owner pays a flat shipping fee of $\$40$ for the delivery. During the roasting process, the beans lose $20\%$ of their original weight. If the owner wants to make a $50\%$ profit on the total cost of the order, what must be the selling price per pound of the roasted beans?

\$24.00

\$19.20

\$32.00

\$8.00

Show answer & explanation

Answer: $\$24.00$

Answer

The required selling price per pound of the roasted beans is

\$24.00

.

To find the selling price per pound, we must first determine the total cost. The beans cost

50 \times \$12 = \$600

, plus

\$40

for shipping, making the total cost

\$640

. A

50\%

profit on

\$640

is

\$320

, so the target total revenue is

\$640 + \$320 = \$960

. Next, we must account for the

20\%

weight loss during roasting. The initial

50

pounds loses

10

pounds (

0.20 \times 50 = 10

), leaving

40

pounds of sellable roasted beans. Dividing the target revenue by the sellable weight (

\$960 \div 40

) gives exactly

\$24.00

per pound.

Step-by-Step Solution

1

Calculate the total cost of purchasing and shipping the raw coffee beans.

50 \text{ lbs} \times \$12/\text{lb} = \$600

. Adding the

\$40

shipping fee gives a total cost of

\$640

.

Profit must be calculated based on the complete overhead and initial investment.

2

Determine the target total revenue required to make a

50\%

profit.

50\%

of

\$640

is

\$320

. The total target revenue is

\$640 + \$320 = \$960

.

The final sales must cover both the initial cost and the desired profit margin.

3

Calculate the final weight of the beans after the roasting process.

The beans lose

20\%

of their weight.

20\%

of

50 \text{ lbs}

is

10 \text{ lbs}

. The remaining weight is

50 - 10 = 40 \text{ lbs}

.

The owner can only sell the roasted beans, so the selling price must be based on the final weight.

4

Calculate the required selling price per pound.

\$960 \div 40 \text{ lbs} = \$24.00 \text{ per pound}

.

Dividing the total target revenue by the total sellable pounds yields the unit price.

Key Concept

Calculating multi-step profit requires finding the total investment cost (including overhead), applying the profit percentage to find target revenue, and distributing that target revenue over the final sellable inventory.

Question 192Question →

A workshop exhaust fan is connected to a $120\text{ V}$ power supply and consumes $480\text{ W}$ of electric power during normal operation. What is the effective electrical resistance of the fan motor, in ohms ( $\Omega$ )?

Show answer & explanation

Answer: 30

Answer

The effective electrical resistance of the fan motor is

30\,\Omega

.

To find the resistance from voltage and power, determine the current using

I = \frac{P}{V} = \frac{480\text{ W}}{120\text{ V}} = 4\text{ A}

, then use Ohm's law

R = \frac{V}{I} = \frac{120\text{ V}}{4\text{ A}} = 30\,\Omega

. Alternatively, using

R = \frac{V^2}{P}

yields

\frac{120^2}{480} = \frac{14400}{480} = 30\,\Omega

.

Step-by-Step Solution

1

Calculate the electric current flowing through the fan motor using the electrical power formula

P = V \times I

.

I = \frac{P}{V} = \frac{480\text{ W}}{120\text{ V}} = 4\text{ A}

Electric power equals the product of voltage and current, so dividing total wattage by applied voltage determines circuit current.

2

Calculate the resistance using Ohm's law rearranged as

R = \frac{V}{I}

.

R = \frac{120\text{ V}}{4\text{ A}} = 30\,\Omega

Resistance represents the opposition to current flow and is found by dividing the applied voltage by the circuit current.

Key Concept

Relationship between electrical power (

P = IV

) and Ohm's law (

V = IR

)

Estimated Time:1m 0s

Question 193Question →

A small business owner finances the purchase of new office equipment with a $\$3,000$ loan from a local credit union. The loan term is $18$ months. At the end of the term, the owner repays a total of $\$3,360$ , which includes the original loan amount and all accumulated simple interest.

What is the annual simple interest rate (as a percentage) applied to this loan?

Show answer & explanation

Answer: 8

Answer

8

The correct annual simple interest rate is 8%. First, determine the total interest by subtracting the principal from the total repayment (

\$3,360 - \$3,000 = \$360

). Next, convert the 18-month term to years (

18 \div 12 = 1.5 \text{ years}

). Using the simple interest formula

I = PRT

, substitute the values:

360 = 3,000 \times R \times 1.5

. Simplifying this gives

360 = 4,500R

, which means

R = 360 \div 4,500 = 0.08

. Finally, convert

0.08

to a percentage to get

8\%

.

Step-by-Step Solution

1

Calculate the total simple interest earned over the loan term.

Total Interest (I) =

\$3,360 - \$3,000 = \$360

The total repayment amount (A) is the sum of the principal (P) and the interest (I). Therefore, subtracting the principal from the total repayment isolates the interest.

2

Convert the loan duration from months to years.

Time (T) =

18 \text{ months} = 1.5 \text{ years}

The standard simple interest formula

I = PRT

requires time (T) to be expressed in years when calculating an annual interest rate.

3

Apply the simple interest formula and solve for the annual rate (R).

360 = 3000 \times R \times 1.5

360 = 4500 \times R

R = 0.08

Substituting the known values (I, P, T) allows us to isolate and solve for the missing rate variable (R).

4

Convert the decimal rate into a percentage.

0.08 = 8\%

Interest rates are universally expressed as percentages in financial contexts.

Key Concept

Calculating an unknown annual interest rate using the simple interest formula I = PRT, which requires determining the total interest from a final balance and converting time from months to years.

Question 194Question →

An agricultural equipment salesperson earns a fixed monthly salary of $\$2,400$ . They also earn a $4.5\%$ commission on their first $\$50,000$ in sales, and a $6\%$ commission on any sales exceeding $\$50,000$ for the month. If the salesperson's total sales for this month are $\$82,000$ , what is their total gross income for the month?

\$4,170

\$4,545

\$6,570

\$7,320

Show answer & explanation

Answer: $\$6,570$

Answer

The correct total gross income is

\$6,570

.

The correct total gross income is found by adding the base salary (

\$2,400

) to the commission from the first tier (

4.5\%

of

\$50,000 = \$2,250

) and the commission from the second tier (

6\%

of the remaining

\$32,000 = \$1,920

).

Step-by-Step Solution

1

Calculate the commission earned on the first $50,000 of sales.

\$50,000 \times 0.045 = \$2,250

The salesperson earns a 4.5% commission on sales up to the $50,000 threshold.

2

Determine the amount of sales that exceeded the $50,000 threshold.

\$82,000 - \$50,000 = \$32,000

The higher 6% commission rate only applies to the sales amount above the initial threshold.

3

Calculate the commission earned on the exceeding sales.

\$32,000 \times 0.06 = \$1,920

Apply the 6% commission rate to the excess sales amount.

4

Calculate the total gross income by adding the base salary and both commission amounts.

\$2,400 + \$2,250 + \$1,920 = \$6,570

Gross income includes the fixed monthly base salary plus all commissions earned.

Key Concept

Calculating tiered commissions and adding them to a base salary to determine total gross income.

Question 195Question →

A retail salesperson earns a weekly base salary of $400 plus a 5% commission on their total sales. If the salesperson sells$ 2,000 worth of merchandise in a given week, what is their total gross pay for that week?

$100

$500

$1,400

$2,500

Show answer & explanation

Answer: $500

Answer

$500

The total gross pay is found by adding the base salary to the commission earned. The commission is 5% of

2,000, which is

100. Adding this to the base salary of

400 gives a total of

500.

Step-by-Step Solution

1

Calculate the commission earned on the total sales.

2,000 \times 0.05 =

100

The commission is a percentage of the total sales for the week.

2

Add the earned commission to the base salary to find the total gross pay.

400 +

100 = $500

Total gross pay equals the guaranteed base salary plus any commission earned.

Key Concept

Calculating total compensation using a base salary plus a percentage commission on sales.

Question 196Question →

A military mess hall receives a shipment of $131 \frac{1}{4}$ pounds of ground beef. The kitchen staff uses $\frac{3}{7}$ of the shipment to prepare lunch. The remaining ground beef is portioned into smaller storage bags that each hold $3 \frac{3}{4}$ pounds for dinner recipes. How many storage bags can be completely filled with the remaining ground beef?

Show answer & explanation

Answer: 20

Answer

20

First, the remaining fraction of beef is determined to be 4/7. Multiplying the total initial weight of 131 1/4 pounds (or 525/4) by 4/7 yields exactly 75 pounds of remaining ground beef. Finally, dividing these 75 pounds by the capacity of each bag, which is 3 3/4 pounds (or 15/4), gives exactly 20 completely filled storage bags.

Step-by-Step Solution

1

Calculate the fraction of the ground beef that remains after lunch.

1 - \frac{3}{7} = \frac{4}{7}

The kitchen staff used 3/7 of the shipment, so 4/7 of the shipment is left for the dinner recipes.

2

Convert the total shipment weight from a mixed number to an improper fraction.

131 \frac{1}{4} = \frac{525}{4}

Converting to an improper fraction makes it easier to multiply by the remaining fraction.

3

Multiply the total weight by the remaining fraction to find the remaining weight.

\frac{4}{7} \times \frac{525}{4} = \frac{525}{7} = 75

pounds

This calculation yields the exact weight of the ground beef available to be portioned into storage bags. Notice that the 4 in the numerator and denominator cancel out.

4

Convert the storage bag capacity from a mixed number to an improper fraction.

3 \frac{3}{4} = \frac{15}{4}

This prepares the divisor for the fraction division step.

5

Divide the remaining weight by the capacity of each storage bag.

75 \div \frac{15}{4} = 75 \times \frac{4}{15} = \frac{300}{15} = 20

bags

Dividing the total available weight by the weight per bag yields the exact number of bags that can be completely filled.

Key Concept

Fraction Multiplication and Division in Context

Estimated Time:1m 30s

Question 197Question →

In an industrial facility, three identical light fixtures are connected in parallel across a constant $120\text{ V}$ electrical line. If one of the light fixtures burns out and creates an open circuit in its branch, what will happen to the remaining light fixtures and the total current drawn from the line?

All remaining fixtures go out immediately, and the total current drawn from the line drops to zero.

The remaining fixtures stay lit at the same brightness, and the total current drawn from the line decreases.

The remaining fixtures glow noticeably brighter, and the total current drawn from the line increases.

The remaining fixtures become dimmer, while the total current drawn from the line remains unchanged.

Show answer & explanation

Answer: The remaining fixtures stay lit at the same brightness, and the total current drawn from the line decreases.

Answer

The remaining fixtures stay lit at the same brightness, and the total current drawn from the line decreases.

In a parallel circuit, each device is connected across the full source voltage independently. When one fixture burns out, the voltage across the remaining fixtures remains at

120\text{ V}

, so their current and brightness stay the same. However, because the total current is the sum of the branch currents (

I_{\text{total}} = I_1 + I_2 + I_3

), losing one branch reduces the total current supplied by the line from three branch currents down to two.

Step-by-Step Solution

1

Analyze voltage across individual branches in a parallel circuit

Each remaining fixture continues to experience the full

120\text{ V}

line voltage.

In a parallel circuit, every branch is connected directly across the power source, so removing or opening one branch does not alter the voltage applied to the others.

2

Determine current and power dissipation for each remaining fixture

Individual current

I = \frac{V}{R}

and power

P = \frac{V^2}{R}

remain constant, keeping brightness unchanged.

Since both the resistance of each remaining fixture and the supply voltage remain unchanged, the power output of each active lamp does not change.

3

Calculate the effect on total circuit current

Total current drops from

I_{\text{total}} = 3I_{\text{branch}}

to

I_{\text{total}} = 2I_{\text{branch}}

.

Total current in a parallel circuit is the sum of the individual branch currents (

I_{\text{total}} = I_1 + I_2 + I_3

). When one branch opens, its current drops to zero, reducing the total current drawn from the supply.

Key Concept

Parallel Circuit Branch Independence and Current Summation

Estimated Time:1m 0s

Question 198Question →

A supply sergeant initially distributes $2,400$ rounds of specialized training ammunition among three firing squads—Alpha, Bravo, and Charlie—in a ratio of $2:3:5$ , respectively. During the drill, Squad Bravo requests and is issued exactly $180$ additional rounds.

To maintain the exact original $2:3:5$ distribution ratio among the three squads, the sergeant must also issue additional rounds to Alpha and Charlie. How many total additional rounds must be issued to Squads Alpha and Charlie combined to perfectly restore the original ratio?

Show answer & explanation

Answer: 420

Answer

420

By determining that Bravo's 180 additional rounds correspond to its 3 parts in the ratio, we can establish that the proportional increase per part is 60 rounds (180 ÷ 3). Since Alpha and Charlie make up 7 parts combined (2 + 5), they require an additional 7 × 60 = 420 rounds to maintain the ratio.

Step-by-Step Solution

1

Calculate the initial number of rounds representing a single ratio part.

240 rounds per part.

The total of 2,400 rounds is distributed across 2 + 3 + 5 = 10 total parts.

2

Determine Squad Bravo's new total amount of ammunition.

900 rounds.

Bravo initially had 3 parts (3 × 240 = 720). Adding the 180 new rounds brings their total to 720 + 180 = 900 rounds.

3

Calculate the new value of a single ratio part required to maintain the 2:3:5 distribution.

300 rounds per part.

Bravo's new total of 900 rounds must represent its 3 parts in the ratio. Therefore, 900 ÷ 3 = 300.

4

Calculate the total additional rounds needed for Alpha and Charlie combined.

420 rounds.

The proportional increase per ratio part is 300 - 240 = 60 rounds. Alpha and Charlie represent 2 + 5 = 7 parts combined. 7 parts × 60 rounds = 420 rounds.

Key Concept

Dividing quantities based on a given ratio and recalculating proportional parts after a specific adjustment.

Alternative Method

Instead of calculating the initial and final totals for each squad, analyze the proportional change directly. Squad Bravo's allocation increases by 180 rounds. Since Bravo represents 3 parts of the ratio, the increase is 180 ÷ 3 = 60 rounds per ratio part. Squads Alpha and Charlie account for 2 + 5 = 7 ratio parts combined. Therefore, their combined additional requirement is simply 7 parts × 60 rounds/part = 420 rounds.

Estimated Time:2m 30s

Question 199Question →

A platoon leader is organizing a holiday banquet for their unit at a local restaurant. The initial food and beverage total is $\$400.00$ . The restaurant offers a $15\%$ military discount on this total. A $5\%$ local sales tax and a $20\%$ gratuity (tip) are then applied. If both the tax and the gratuity are calculated based on the discounted total, what is the final overall cost of the banquet?

\$440.00

\$425.00

\$481.25

\$428.40

Show answer & explanation

Answer: $\$425.00$

Answer

The final overall cost of the banquet is

\$425.00

.

The correct answer accurately determines the discount (

60), subtracts it to find the new subtotal (

340), calculates the tax (

17) and tip (

68) based on that new subtotal, and adds everything together (

340 +

17 +

68) to reach the total of

425.00.

Step-by-Step Solution

1

Calculate the dollar amount of the military discount.

15\%

of

\$400.00 = 0.15 \times \$400.00 = \$60.00

The discount must be determined before calculating the new taxable subtotal.

2

Subtract the discount from the initial total to find the discounted subtotal.

\$400.00 - \$60.00 = \$340.00

This establishes the new base amount on which the tax and gratuity will be calculated.

3

Calculate the sales tax on the discounted subtotal.

5\%

of

\$340.00 = 0.05 \times \$340.00 = \$17.00

The problem states the

5\%

tax is applied to the discounted total.

4

Calculate the gratuity (tip) on the discounted subtotal.

20\%

of

\$340.00 = 0.20 \times \$340.00 = \$68.00

The problem specifies the gratuity is also calculated based on the discounted total.

5

Add the discounted subtotal, tax, and gratuity to find the final cost.

\$340.00 + \$17.00 + \$68.00 = \$425.00

Summing the adjusted base cost with all applicable fees gives the total amount due.

Key Concept

Taxes and Tips

Question 200Question →

In the analysis of direct-current electrical circuits, fundamental physical quantities define how electrical energy is stored, transported, and dissipated. Match each electrical quantity on the left with its correct physical definition on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Electric Current matches 'The rate at which electric charge flows through a cross-sectional area of a conductor over time'; Voltage (Electric Potential Difference) matches 'The amount of electric potential energy delivered or work done per unit of electric charge'; Electrical Resistance matches 'The measure of a material's opposition to the passage of electric charge'; Electrical Power matches 'The rate at which electrical energy is transferred, converted, or dissipated per unit of time'.

Each term corresponds directly to its foundational physical definition in electromagnetism: Electric Current is the rate of charge flow (

I = \frac{\Delta Q}{\Delta t}

), Voltage is the work/energy per unit charge (

V = \frac{W}{Q}

), Resistance is the opposition to charge movement (

R = \frac{V}{I}

), and Electrical Power is the time rate of total energy transfer (

P = VI

).

Step-by-Step Solution

1

Determine the physical definition of Electric Current.

Electric current is the flow of electric charge per unit time (

I = \frac{\Delta Q}{\Delta t}

), which matches 'The rate at which electric charge flows through a cross-sectional area of a conductor over time'.

Current measures how many coulombs of charge pass through a point each second.

2

Determine the physical definition of Voltage (Electric Potential Difference).

Voltage is the potential energy difference per unit charge (

V = \frac{W}{Q}

), which matches 'The amount of electric potential energy delivered or work done per unit of electric charge'.

Voltage represents the electrical 'push' or work done to move a unit charge between two points.

3

Determine the physical definition of Electrical Resistance.

Resistance describes the opposition encountered by flowing charge (

R = \frac{V}{I}

), which matches 'The measure of a material's opposition to the passage of electric charge'.

Resistance impedes electric current and creates a potential drop across a circuit component.

4

Determine the physical definition of Electrical Power.

Electrical power is the rate of energy transformation or consumption (

P = \frac{\Delta E}{\Delta t} = VI

), which matches 'The rate at which electrical energy is transferred, converted, or dissipated per unit of time'.

Power quantifies how rapidly electrical potential energy is converted into thermal energy, light, or mechanical work.

Key Concept

Fundamental electrical circuit quantities, their standard physical definitions, and underlying unit relationships

Estimated Time:1m 0s

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