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Question 201Question →

An air base has a fleet of 200200 transport vehicles. Of these vehicles, 65%65\% are diesel-powered, and the remaining vehicles are electric. Currently, 20%20\% of the diesel-powered vehicles and 40%40\% of the electric vehicles are out of service for routine maintenance. What is the total number of transport vehicles that are currently in service?

Show answer & explanation

Answer: 146

Answer

146
The total number of vehicles in service is found by determining the quantity of each vehicle type, calculating the out-of-service vehicles for each subgroup (26 diesel and 28 electric), and subtracting this total (54) from the overall fleet of 200, resulting in 146 active vehicles.

Step-by-Step Solution

1
Calculate the total number of diesel and electric vehicles in the fleet.
130 diesel vehicles and 70 electric vehicles.
You must establish the correct baseline quantity for each vehicle type before calculating the out-of-service percentages.
2
Calculate the number of vehicles out of service for each type.
26 diesel vehicles and 28 electric vehicles are out of service.
The maintenance rates apply to the specific subgroups, so you take 20% of 130 and 40% of 70.
3
Determine the total number of vehicles that are out of service.
54 vehicles out of service.
Adding the unavailable diesel and electric vehicles gives the total number of inactive vehicles.
4
Subtract the out-of-service vehicles from the total fleet to find the number in service.
146 vehicles in service.
The entire fleet is composed of either in-service or out-of-service vehicles. Subtracting the unavailable ones yields the active fleet.

Key Concept

Calculating successive percentages and determining parts of different subgroups.

Alternative Method

Instead of finding the out-of-service vehicles and subtracting, you can directly calculate the in-service vehicles. Diesel in service = 80% of 130 = 104. Electric in service = 60% of 70 = 42. Total in service = 104 + 42 = 146.
Estimated Time:1m 30s
Question 202Question →

Match each quantity relationship to its equivalent simplified ratio.

Click a left item, then click its matching right item

Items

1010 instructors for every 5050 recruits (ratio of instructors to recruits)
1212 utility trucks for every 88 transport vans (ratio of trucks to vans)
1515 gallons of fuel for every 55 quarts of oil (ratio of fuel to oil)
2020 standard rifles for every 1515 sidearms (ratio of rifles to sidearms)

Matches

Show answer & explanation

Answer

The unsimplified ratios match to their reduced forms as follows: instructors to recruits (10:5010:50) is 1:51:5, trucks to vans (12:812:8) is 3:23:2, fuel to oil (15:515:5) is 3:13:1, and rifles to sidearms (20:1520:15) is 4:34:3.
Each relationship is found by writing down the initial numerical counts in the order specified and reducing them to their lowest integer terms using their greatest common factors.

Step-by-Step Solution

1
Set up the initial part-to-part ratio for each scenario in the exact order requested by the text.
The raw ratios are 10:5010:50, 12:812:8, 15:515:5, and 20:1520:15.
Ratio order matters. Ensuring the first mentioned item is the first number in the ratio prevents inverted answers.
2
Simplify each ratio by finding the greatest common factor (GCF) for each pair and dividing both numbers by it.
10:5010:50 reduces to 1:51:5. 12:812:8 reduces to 3:23:2. 15:515:5 reduces to 3:13:1. 20:1520:15 reduces to 4:34:3.
A ratio is properly expressed when the two numbers have no common factors other than 11.

Key Concept

Basic Ratios
Question 203Question →

A soldier buys a pair of tactical boots priced at $50.00\$50.00 at a local surplus store. He uses a coupon for $10.00\$10.00 off his purchase. After the discount is subtracted, a 6%6\% sales tax is applied to the remaining balance. What is the final total cost of the boots?

Show answer & explanation

Answer: $42.40\$42.40

Answer

$42.40\$42.40
The correct final cost is found by first subtracting the $10.00\$10.00 discount from the $50.00\$50.00 original price, resulting in a $40.00\$40.00 subtotal. The 6%6\% sales tax is then calculated on this new subtotal ($40.00×0.06=$2.40\$40.00 \times 0.06 = \$2.40). Finally, adding the $2.40\$2.40 tax to the $40.00\$40.00 subtotal gives the correct total of $42.40\$42.40.

Step-by-Step Solution

1
Subtract the coupon discount from the original price to find the subtotal.
$50.00−$10.00=$40.00\$50.00 - \$10.00 = \$40.00
The discount must be applied to the original price before calculating the sales tax.
2
Calculate the 6%6\% sales tax on the new subtotal.
$40.00×0.06=$2.40\$40.00 \times 0.06 = \$2.40
The 6%6\% tax rate is converted to the decimal 0.060.06 and multiplied by the discounted price.
3
Add the sales tax to the subtotal to find the final cost.
$40.00+$2.40=$42.40\$40.00 + \$2.40 = \$42.40
The final total is the sum of the discounted price and the applicable sales tax.

Key Concept

Calculating sequential discounts and taxes using decimal multiplication
Question 204Question →

Determine the commission earned and fill in the blank.

Fill in the blanks below

A travel agent receives a 4%4\% commission on all vacation packages they book. If the agent books a luxury cruise package priced at $8,500\$8,500, their commission for this booking is $\$ .
Show answer & explanation

Answer

340
The travel agent's commission is calculated by taking 4%4\% of the total package price of $8,500\$8,500. Converting 4%4\% to 0.040.04 and multiplying by $8,500\$8,500 gives exactly $340\$340.

Step-by-Step Solution

1
Convert the commission percentage to a decimal.
4%=0.044\% = 0.04
Percentages must be converted to decimals before performing multiplication to find a part of a whole.
2
Multiply the total sales price by the decimal commission rate.
$8,500×0.04=$340\$8,500 \times 0.04 = \$340
A commission is calculated as a percentage of the total sales amount.

Key Concept

Calculating commission based on a fixed percentage of a total sale amount.
Question 205Question →

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

Fill in the blanks below

Sarah works as a heavy equipment sales representative. She earns a monthly base salary of $2,400\$2,400. She also earns a 4%4\% commission on her first $40,000\$40,000 in monthly sales, and a 7.5%7.5\% commission on all sales exceeding $40,000\$40,000. In March, Sarah's total gross pay was $6,100\$6,100.

The total value of her sales in March was $\$$.
Show answer & explanation

Answer

68,000
Subtracting the base salary ($2,400\$2,400) and the maximum first tier commission ($1,600\$1,600) from the total gross pay ($6,100\$6,100) leaves $2,100\$2,100. This remaining amount is the commission earned at the 7.5%7.5\% rate. Dividing $2,100\$2,100 by 0.0750.075 yields $28,000\$28,000, which is the sales amount over the $40,000\$40,000 threshold. Adding this to the initial $40,000\$40,000 threshold gives the total monthly sales of $68,000\$68,000.

Step-by-Step Solution

1
Calculate the commission earned from the first tier of sales.
$40,000×0.04=$1,600\$40,000 \times 0.04 = \$1,600
To determine the maximum commission Sarah can earn at the 4%4\% base rate.
2
Subtract the base salary and first tier commission from the total gross pay to find the remaining commission.
$6,100−$2,400−$1,600=$2,100\$6,100 - \$2,400 - \$1,600 = \$2,100
To isolate the portion of her earnings that came exclusively from the 7.5%7.5\% commission tier.
3
Calculate the sales volume required to generate the remaining tier 2 commission.
$2,100÷0.075=$28,000\$2,100 \div 0.075 = \$28,000
To find out how much she sold above the $40,000\$40,000 threshold.
4
Add the tier 2 sales to the tier 1 threshold to find the total sales.
$40,000+$28,000=$68,000\$40,000 + \$28,000 = \$68,000
To find the complete total value of her sales in March.

Key Concept

Solving for required total sales given a base salary and a tiered commission structure
Question 206Question →

To purchase a used vehicle, a buyer finances $12,000\$12,000 through a dealership. The financing agreement includes an annual simple interest rate of 6.5%6.5\%. If the buyer pays off the loan over a period of 4242 months, what is the total amount repaid at the end of the financing period?

Show answer & explanation

Answer: $14,730\$14,730

Answer

The total amount repaid is $14,730\$14,730.
The correct answer is derived by first converting the 4242-month term into 3.53.5 years. Applying the simple interest formula (I=PRTI = PRT), the total interest is calculated as $12,000×0.065×3.5=$2,730\$12,000 \times 0.065 \times 3.5 = \$2,730. Finally, the interest is added to the principal to find the total repayment: $12,000+$2,730=$14,730\$12,000 + \$2,730 = \$14,730.

Step-by-Step Solution

1
Convert the loan duration from months to years.
42 months=4212=3.5 years42 \text{ months} = \frac{42}{12} = 3.5 \text{ years}.
The interest rate given is annual, so the time variable must be expressed in years.
2
Calculate the total simple interest accrued using the formula I=PRTI = PRT.
I=$12,000×0.065×3.5=$2,730I = \$12,000 \times 0.065 \times 3.5 = \$2,730.
This determines the cost of borrowing the money over the specified time period.
3
Add the interest to the original principal to find the total repayment amount.
Total Amount = $12,000+$2,730=$14,730\$12,000 + \$2,730 = \$14,730.
The buyer must repay both the original loan amount and the accumulated interest.

Key Concept

Calculating the total maturity value using simple interest by correctly converting months to years and adding the accrued interest to the principal.
Estimated Time:1m 15s
Question 207Question →

A mess hall cook prepares a large batch of stew. The recipe requires a total of 6060 pounds of meat and potatoes. If the ratio of meat to potatoes is 3:23:2, how many pounds of meat are needed?

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Answer: 3636

Answer

3636
The correct answer is found by first adding the parts of the ratio together (3+2=53 + 2 = 5). Next, divide the total weight (6060 pounds) by 55 to find the weight of a single part (1212 pounds). Finally, multiply this unit weight by the number of parts designated for meat (3×12=363 \times 12 = 36 pounds).

Step-by-Step Solution

1
Find the total number of parts in the ratio.
3+2=53 + 2 = 5 total parts.
To distribute the total weight correctly, we first need to determine how many proportional parts make up the whole.
2
Determine the weight of a single part.
60÷5=1260 \div 5 = 12 pounds per part.
Dividing the total weight by the total number of parts gives the value of one unit in the ratio.
3
Calculate the total weight of the meat.
3×12=363 \times 12 = 36 pounds.
The ratio states there are 33 parts of meat. Multiplying the weight of one part by 33 gives the final amount of meat needed.

Key Concept

Dividing a total quantity into parts based on a given ratio.
Estimated Time:45s
Question 208Question →

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

Fill in the blanks below

The tax amount is $\$,thegratuityis, the gratuity is \$,andthefinaltotalcostis, and the final total cost is \$$.
Show answer & explanation

Answer

Tax: $60\$60, Gratuity: $144\$144, Total: $2,204\$2,204
To find the final total, the tax and gratuity must be calculated separately based on their respective costs. The 5%5\% tax on the $1,200\$1,200 room is $60\$60. The 18%18\% gratuity on the $800\$800 catering is $144\$144. Adding the original costs ($1,200+$800=$2,000\$1,200 + \$800 = \$2,000) to the tax ($60\$60) and gratuity ($144\$144) yields a final total of $2,204\$2,204.

Step-by-Step Solution

1
Calculate the local tax on the room rental.
$1,200×0.05=$60\$1,200 \times 0.05 = \$60
The 5%5\% tax only applies to the $1,200\$1,200 room rental cost.
2
Calculate the mandatory gratuity on the catering.
$800×0.18=$144\$800 \times 0.18 = \$144
The 18%18\% gratuity only applies to the $800\$800 catering cost.
3
Calculate the final total cost of the retreat.
$1,200+$800+$60+$144=$2,204\$1,200 + \$800 + \$60 + \$144 = \$2,204
The final invoice includes the original room and catering costs, plus the calculated tax and gratuity.

Key Concept

Calculating specific taxes and tips based on different principal amounts within a single transaction.
Question 209Question →

A military surplus store acquires a bulk shipment of sleeping bags for a total cost of $15,000\$15,000. To determine the regular retail value, the store manager applies a 40%40\% markup to the total cost. Several months later, the store sells the entire shipment at a 15%15\% discount off the regular retail value. What is the final selling price of the shipment?

Show answer & explanation

Answer: $17,850\$17,850

Answer

The final selling price of the shipment is $17,850\$17,850.
The final selling price is calculated by first finding the retail value after a 40% markup on the 15,000cost,whichresultsin15,000 cost, which results in 21,000. Then, a 15% discount is applied to the new 21,000base,subtracting21,000 base, subtracting 3,150 to reach the final price of $17,850.

Step-by-Step Solution

1
Calculate the markup amount based on the initial cost.
$15,000×0.40=$6,000\$15,000 \times 0.40 = \$6,000
The regular retail value is established by adding a 40% markup to the original cost.
2
Determine the regular retail value.
$15,000+$6,000=$21,000\$15,000 + \$6,000 = \$21,000
The total retail value serves as the new base amount for any subsequent discounts.
3
Calculate the discount amount during the clearance event.
$21,000×0.15=$3,150\$21,000 \times 0.15 = \$3,150
The 15% discount is applied to the current retail value, not the original cost.
4
Subtract the discount from the retail value to find the final price.
$21,000−$3,150=$17,850\$21,000 - \$3,150 = \$17,850
This yields the final selling price after both the percentage increase and decrease have been sequentially applied.

Key Concept

Successive Percentage Calculations
Question 210Question →

A quartermaster purchases 22 dozen identical tactical flashlights for a total cost of $720\$720. What is the unit price of a single flashlight?

Show answer & explanation

Answer: $30\$30

Answer

The correct unit price is $30\$30 per flashlight.
To find the unit price, the total cost must be divided by the total number of individual items. First, convert 22 dozen into a specific quantity by multiplying by 1212, yielding 2424 flashlights. Then, divide the total cost of $720\$720 by 2424, which equals $30\$30 per flashlight.

Step-by-Step Solution

1
Determine the total number of flashlights purchased.
2 dozen=2×12=24 flashlights2 \text{ dozen} = 2 \times 12 = 24 \text{ flashlights}
You must know the exact number of individual items before you can calculate the cost of a single item.
2
Divide the total cost by the total number of items to find the unit price.
$720÷24=$30\$720 \div 24 = \$30
Unit price is found by dividing the total amount spent by the total quantity received.

Key Concept

Unit Price Calculation
Estimated Time:45s
Question 211Question →

A small business owner takes out a $12,500\$12,500 personal loan to finance new equipment. The loan agreement specifies an annual simple interest rate of 6.4%6.4\%. If the owner repays the loan in full at the end of 4545 months, what is the total amount that must be repaid?

Show answer & explanation

Answer: $15,500\$15,500

Answer

$15,500\$15,500
The correct total amount to repay is found by first converting the 45-month term into years by dividing by 12, which gives 3.75 years. Next, the simple interest is calculated by multiplying the principal ($12,500\$12,500), the annual rate as a decimal (0.0640.064), and the time in years (3.753.75) to get $3,000\$3,000. Finally, adding this interest to the original principal yields the total repayment amount of $15,500\$15,500.

Step-by-Step Solution

1
Convert the loan duration from months to years.
45 months÷12 months/year=3.75 years45 \text{ months} \div 12 \text{ months/year} = 3.75 \text{ years}
The interest rate is an annual rate, so the time period in the formula must be expressed in years.
2
Calculate the total simple interest earned over the period.
I=P×r×t=$12,500×0.064×3.75=$3,000I = P \times r \times t = \$12,500 \times 0.064 \times 3.75 = \$3,000
This determines the additional cost of borrowing the money for the specified time.
3
Add the calculated interest to the original principal to find the total repayment amount.
A=$12,500+$3,000=$15,500A = \$12,500 + \$3,000 = \$15,500
The question asks for the total amount that must be repaid, which includes both the principal balance and the accumulated interest.

Key Concept

To find the total amount to be repaid on a loan, you must calculate the simple interest using the formula I=P×r×tI = P \times r \times t, ensuring the time tt is converted to years, and then add this interest to the principal PP.
Question 212Question →

A military maintenance crew is draining a water purification tank for routine cleaning. They manage to drain 2125\frac{21}{25} of the water before the primary pump requires a filter change. What percentage of the water was drained?

Show answer & explanation

Answer: 84

Answer

84
To convert 2125\frac{21}{25} to a percentage, multiply both the numerator and the denominator by 4 to reach a denominator of 100. This results in 84100\frac{84}{100}, which is exactly 84%.

Step-by-Step Solution

1
Set up the conversion from fraction to percentage.
2125\frac{21}{25}
This is the proportion of the tank that was drained and needs to be converted.
2
Find an equivalent fraction with a denominator of 100.
21×425×4=84100\frac{21 \times 4}{25 \times 4} = \frac{84}{100}
Percentages are calculated out of 100. Since 25 is a factor of 100, multiplying by 4 scales the fraction appropriately.
3
Convert the resulting fraction directly to a percentage.
84
A fraction with a denominator of 100 indicates the percentage directly.

Key Concept

Converting a fraction with a easily scalable denominator to a percentage.
Question 213Question →

An engineering battalion is paving a section of a new airfield. The total length of the section to be paved is 451245 \frac{1}{2} kilometers. If the paving crew completes exactly 2352 \frac{3}{5} kilometers per day, how many days will it take them to finish this section?

Show answer & explanation

Answer: 171217 \frac{1}{2}

Answer

It will take the engineering battalion 171217 \frac{1}{2} days to finish paving the section.
The correct answer accurately converts the mixed numbers to improper fractions, sets up the division of the total distance by the daily rate, multiplies by the reciprocal, and simplifies to find the total days.

Step-by-Step Solution

1
Convert the given mixed numbers into improper fractions.
4512=91245 \frac{1}{2} = \frac{91}{2} and 235=1352 \frac{3}{5} = \frac{13}{5}
Mathematical operations such as fraction multiplication and division require working with improper fractions rather than mixed numbers.
2
Set up the division equation.
912÷135\frac{91}{2} \div \frac{13}{5}
To find the total time required, the total distance to be paved must be divided by the distance the crew can pave per day.
3
Convert the division problem into a multiplication problem.
912×513\frac{91}{2} \times \frac{5}{13}
Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.
4
Multiply the fractions and simplify the result.
91×52×13=45526=352=1712\frac{91 \times 5}{2 \times 13} = \frac{455}{26} = \frac{35}{2} = 17 \frac{1}{2}
Cross-canceling 9191 and 1313 by their common factor of 1313 simplifies the multiplication to 7×5=357 \times 5 = 35 for the numerator and 2×1=22 \times 1 = 2 for the denominator. Converting 352\frac{35}{2} back yields the mixed number 171217 \frac{1}{2}.

Key Concept

Fraction Division in Applied Contexts
Question 214Question →

A logistics officer needs to distribute 150150 tactical backpacks between two infantry squads in a ratio of 2:32:3. How many backpacks will the squad with the smaller share receive?

Show answer & explanation

Answer: 60

Answer

60
To solve the problem, we first determine that the total ratio consists of 55 parts (2+32 + 3). Dividing the 150150 total backpacks by 55 parts gives us 3030 backpacks for every single part. Because the question asks for the squad with the smaller share, we multiply the 3030 backpacks by the smaller ratio number (22), which results in 6060 backpacks.

Step-by-Step Solution

1
Calculate the total number of parts in the ratio by adding the given values.
2+3=52 + 3 = 5 total parts.
This establishes the total number of equal shares the backpacks will be divided into.
2
Determine the number of backpacks in a single part by dividing the total amount by the total parts.
150÷5=30150 \div 5 = 30 backpacks per part.
Finding the unit value allows us to calculate any specific share in the proportion.
3
Calculate the smaller share by multiplying the unit value by the smaller number in the ratio.
30×2=6030 \times 2 = 60 backpacks.
The question asks for the smaller share, which corresponds to the 22 parts in the 2:32:3 ratio.

Key Concept

Dividing a total quantity into a given ratio
Question 215Question →

A medical equipment representative earns a base salary of $800 per week plus a tiered commission on weekly sales. The commission structure is as follows:

- 3% on the first 10,000ofsales−510,000 of sales - 5% on the next 15,000 of sales
- 7.5% on any sales exceeding $25,000

During a specific week, the representative's total sales are 32,000.Additionally,therepresentativereceivesa32,000. Additionally, the representative receives a 150 bonus for exceeding 30,000insales,butmustpay30,000 in sales, but must pay 275 out-of-pocket for weekly travel expenses which are deducted from their gross earnings. What are the representative's net earnings for the week?

Show answer & explanation

Answer: $2,250

Answer

$2,250
The net earnings are correctly found by evaluating the sales across three tiers: 3% on the first 10,000(10,000 ( 300), 5% on the next 15,000(15,000 ( 750), and 7.5% on the remaining 7,000(7,000 ( 525). Adding this 1,575commissiontothe1,575 commission to the 800 base salary and 150bonusyieldsgrossearningsof150 bonus yields gross earnings of 2,525. Subtracting the 275inexpensesresultsinafinalnetpayof275 in expenses results in a final net pay of 2,250.

Step-by-Step Solution

1
Calculate the commission on the first tier (0to0 to 10,000).
10,000×0.03=10,000 \times 0.03 = 300
The first $10,000 of sales earns a 3% commission.
2
Calculate the commission on the second tier (10,000to10,000 to 25,000).
15,000×0.05=15,000 \times 0.05 = 750
The next $15,000 of sales earns a 5% commission.
3
Calculate the commission on the third tier (sales exceeding $25,000).
Total sales (32,000)−32,000) - 25,000 = 7,000.7,000. 7,000 \times 0.075 = $525
The remaining sales amount above $25,000 earns a 7.5% commission.
4
Determine total commission.
300+300 + 750 + 525=525 = 1,575
Total commission is the sum of earnings from all three tiers.
5
Calculate net earnings.
800(base)+800 (base) + 1,575 (commission) + 150(bonus)−150 (bonus) - 275 (expenses) = $2,250
Net earnings equal base salary plus total commission plus bonuses minus deductions.

Key Concept

Calculating tiered commissions, bonuses, and deductions to determine net earnings.
Question 216Question →

A structural engineering team is mixing a specialized epoxy resin containing base, hardener, and silica filler in a strict ratio of 5:2:15:2:1 by weight. If the team uses exactly 3535 pounds of the base component, what is the total weight, in pounds, of the prepared epoxy mixture?

Show answer & explanation

Answer: 56

Answer

The total weight of the prepared epoxy mixture is 56 pounds.
The components are mixed in a ratio of 5:2:15:2:1, which combines to make a total of 88 parts (5+2+1=85 + 2 + 1 = 8). We know the base component accounts for 55 of those parts and weighs 3535 pounds. This means a single part weighs 77 pounds (35÷5=735 \div 5 = 7). To find the total mixture weight, multiply the total number of parts (88) by the weight of a single part (77), resulting in 5656 pounds.

Step-by-Step Solution

1
Calculate the total number of parts in the given ratio.
5+2+1=85 + 2 + 1 = 8 total parts.
To establish the relationship between the individual components and the entire mixture.
2
Find the weight equivalent of one ratio part using the known base component.
35÷5=735 \div 5 = 7 pounds per part.
The base component represents 55 parts of the ratio and weighs 3535 pounds, so dividing its weight by its parts gives the unit multiplier.
3
Calculate the total weight of the mixture.
8×7=568 \times 7 = 56 pounds.
Multiplying the total number of ratio parts by the weight of a single part yields the overall total weight.

Key Concept

Solving part-to-whole ratio problems by finding the value of a single unit part.

Alternative Method

Calculate the individual weights of the missing components first. Hardener = 2×7=142 \times 7 = 14 pounds. Silica = 1×7=71 \times 7 = 7 pounds. Then, add these to the given base weight: 35+14+7=5635 + 14 + 7 = 56 pounds.
Estimated Time:1m 0s
Question 217Question →

Match each non-monic quadratic polynomial to its completely factored form. Pay close attention to the signs in each binomial factor.

Click a left item, then click its matching right item

Items

2m2−11m+152m^2 - 11m + 15
2m2+m−152m^2 + m - 15
2m2+11m+152m^2 + 11m + 15
2m2−m−152m^2 - m - 15

Matches

Show answer & explanation

Answer

The correct pairings are: 2m2−11m+152m^2 - 11m + 15 with (2m−5)(m−3)(2m - 5)(m - 3); 2m2+m−152m^2 + m - 15 with (2m−5)(m+3)(2m - 5)(m + 3); 2m2+11m+152m^2 + 11m + 15 with (2m+5)(m+3)(2m + 5)(m + 3); and 2m2−m−152m^2 - m - 15 with (2m+5)(m−3)(2m + 5)(m - 3).
Each polynomial requires finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient. The correct pairings map the expanded polynomials directly to their unique, fully factored equivalents.

Step-by-Step Solution

1
Analyze the signs of the constants in each polynomial.
Polynomials with a positive constant term (+15) must have binomial factors with identical signs (both positive or both negative). Polynomials with a negative constant term (-15) must have binomial factors with opposite signs.
The product of the constant terms in the binomials must equal the constant term of the polynomial.
2
Factor 2m2−11m+152m^2 - 11m + 15.
Since the constant is positive and the middle term is negative, both factors must be negative. 2m2−6m−5m+15=2m(m−3)−5(m−3)=(2m−5)(m−3)2m^2 - 6m - 5m + 15 = 2m(m - 3) - 5(m - 3) = (2m - 5)(m - 3).
Applying factoring by grouping.
3
Factor 2m2+m−152m^2 + m - 15.
The constant is negative, so signs differ. The middle term is positive, so the larger product in FOIL must be positive. 2m2+6m−5m−15=2m(m+3)−5(m+3)=(2m−5)(m+3)2m^2 + 6m - 5m - 15 = 2m(m + 3) - 5(m + 3) = (2m - 5)(m + 3).
Applying factoring by grouping.
4
Factor 2m2+11m+152m^2 + 11m + 15.
Since both the constant and middle term are positive, both factors must be positive. 2m2+6m+5m+15=2m(m+3)+5(m+3)=(2m+5)(m+3)2m^2 + 6m + 5m + 15 = 2m(m + 3) + 5(m + 3) = (2m + 5)(m + 3).
Applying factoring by grouping.
5
Factor 2m2−m−152m^2 - m - 15.
The constant is negative, so signs differ. The middle term is negative, so the larger product in FOIL must be negative. 2m2−6m+5m−15=2m(m−3)+5(m−3)=(2m+5)(m−3)2m^2 - 6m + 5m - 15 = 2m(m - 3) + 5(m - 3) = (2m + 5)(m - 3).
Applying factoring by grouping.

Key Concept

Factoring Non-Monic Trinomials and Sign Analysis
Estimated Time:2m 30s
Question 218Question →

A car rental agent earns a base weekly salary of 500.Inaddition,theyearna5500. In addition, they earn a 5% commission on any rental upgrades they sell. If the agent sells 800 worth of upgrades in one week, what is their total weekly pay?

Show answer & explanation

Answer: $540

Answer

$540
The total weekly pay is found by adding the base salary and the commission. The commission is 5% of the 800inupgradesales,whichequals800 in upgrade sales, which equals 40. Adding the 40commissiontothe40 commission to the 500 base salary results in a total of $540.

Step-by-Step Solution

1
Calculate the commission earned from the upgrade sales.
$800×0.05=$40\$800 \times 0.05 = \$40
The agent earns 5% on all rental upgrades sold.
2
Add the commission to the base salary to find the total weekly pay.
$500+$40=$540\$500 + \$40 = \$540
Total pay is the sum of the guaranteed base salary and the variable commission.

Key Concept

Calculating total earnings by adding a percentage-based commission to a fixed base salary.
Question 219Question →

A food truck operator buys 3030 pounds of beef at $8\$8 per pound and spends an additional $60\$60 on packaging and spices. From these supplies, the operator prepares 8080 meals and sells them for $6\$6 each. What is the operator's net profit as a percentage of their total cost?

Show answer & explanation

Answer: 60%60\%

Answer

The operator's net profit as a percentage of their total cost is 60%60\%.
The total cost is the sum of the beef ($240\$240) and the packaging/spices ($60\$60), which equals $300\$300. The total revenue is 8080 meals sold at $6\$6 each, equaling $480\$480. The net profit is the revenue minus the total cost ($480−$300=$180\$480 - \$300 = \$180). To find the profit percentage based on the total cost, divide the net profit by the total cost and multiply by 100100: ($180÷$300)×100=60%(\$180 \div \$300) \times 100 = 60\%.

Step-by-Step Solution

1
Calculate the total cost of all supplies.
30 pounds×$8/pound=$24030 \text{ pounds} \times \$8/\text{pound} = \$240. Adding the overhead: $240+$60=$300\$240 + \$60 = \$300 total cost.
The total cost must include both the direct material cost and any additional expenses.
2
Calculate the total revenue from selling the meals.
80 meals×$6/meal=$48080 \text{ meals} \times \$6/\text{meal} = \$480 total revenue.
Total revenue is the amount generated from sales before any costs are deducted.
3
Determine the net profit.
$480 (revenue)−$300 (total cost)=$180 net profit\$480 \text{ (revenue)} - \$300 \text{ (total cost)} = \$180 \text{ net profit}.
Net profit is the amount of money remaining after subtracting all costs from the revenue.
4
Calculate the net profit as a percentage of the total cost.
($180÷$300)×100=60%(\$180 \div \$300) \times 100 = 60\%.
The question asks for the profit percentage based on the total cost.

Key Concept

Calculating net profit by subtracting total expenses from total revenue, and expressing it as a percentage of the total cost.
Question 220Question →

A patrol unit drives at a steady speed of 20 mph20\text{ mph} for 33 hours along a desert trail. They then enter a paved highway and drive at 50 mph50\text{ mph} for 22 hours. What is the total distance traveled by the patrol unit?

Show answer & explanation

Answer: 160; 160 miles; 160 mi; 160miles; 160.0

Answer

160 miles
To find the total distance, you must calculate the distance for each part of the journey separately. For the first leg, the distance is 20 mph×3 hours=60 miles20\text{ mph} \times 3\text{ hours} = 60\text{ miles}. For the second leg, the distance is 50 mph×2 hours=100 miles50\text{ mph} \times 2\text{ hours} = 100\text{ miles}. Adding these two distances together (60+10060 + 100) gives a total distance of 160160 miles.

Step-by-Step Solution

1
Calculate the distance traveled during the first segment on the desert trail.
Distance = Rate × Time = 20 mph × 3 hours = 60 miles
The distance formula D=R×TD = R \times T is used to find the distance for each constant-speed leg of the trip.
2
Calculate the distance traveled during the second segment on the paved highway.
Distance = Rate × Time = 50 mph × 2 hours = 100 miles
Applying the same formula to the second leg provides the remaining distance.
3
Add the distances from both segments to find the total distance traveled.
60 miles + 100 miles = 160 miles
The total distance of a multi-leg journey is the sum of the distances of each individual segment.

Key Concept

Calculating total distance for a multi-leg journey using the formula Distance = Rate × Time (D=R×TD = R \times T).
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