Tüm alıştırma soruları

Soru 1Soru →

A base quartermaster is reviewing supply orders for the mess hall. Arrange the following provisions by their calculated unit price per pound, from least expensive to most expensive.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from least expensive to most expensive unit price is: 10 pounds of rice, 4 pounds of salt, 5 pounds of flour, and 2 pounds of pepper.

To find the correct order, calculate the unit price for each item. Rice is

0.85 per pound, salt is

3.10 per pound, flour is

3.70 per pound, and pepper is

7.50 per pound. Arranging these unit prices from lowest to highest gives the sequence: rice, salt, flour, pepper.

Adım Adım Çözüm

1

Calculate the unit price of the rice.

8.50 ÷ 10 =

0.85 per pound

Dividing total cost by quantity gives the price per pound.

2

Calculate the unit price of the salt.

12.40 ÷ 4 =

3.10 per pound

Dividing total cost by quantity gives the price per pound.

3

Calculate the unit price of the flour.

18.50 ÷ 5 =

3.70 per pound

Dividing total cost by quantity gives the price per pound.

4

Calculate the unit price of the pepper.

15.00 ÷ 2 =

7.50 per pound

Dividing total cost by quantity gives the price per pound.

5

Compare all calculated unit prices and order them from least to greatest.

0.85 <

3.10 <

3.70 <

7.50

The question asks for the items to be arranged from least expensive to most expensive.

Anahtar Kavram

Calculating unit rates by dividing a decimal total by a whole number quantity.

Soru 2Soru →

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

Aşağıdaki boşlukları doldurun

The longer section of the wire will be feet long.

Cevabı ve açıklamayı göster

Cevap

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

The wire is divided into

7 + 3 = 10

total parts. Since the total length is

60

feet, each part is equal to

60 \div 10 = 6

feet. The longer section is

7

parts, so its length is

7 \times 6 = 42

feet.

Adım Adım Çözüm

1

Find the total number of parts in the ratio.

7 + 3 = 10

total parts.

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

2

Calculate the length of one single part.

60 \div 10 = 6

feet per part.

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

3

Determine the length of the longer section.

7 \times 6 = 42

feet.

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

Anahtar Kavram

To divide a quantity in a given ratio, find the total number of parts by adding the ratio terms, divide the total quantity by this sum to find the value of one part, and then multiply by the specific ratio term.

Soru 3Soru →

A medical equipment technician is compensated with both an hourly wage and a tiered commission. The technician earns a base rate of $24.00 per hour for a standard 40-hour workweek, and time-and-a-half for any hours worked beyond 40. Additionally, the technician earns a 6% commission on all service contracts sold up to$ 5,000 in a week, and an 8% commission on the portion of sales exceeding $5,000. If the technician worked 46 hours and sold$ 8,500 in service contracts last week, what were the technician's total gross earnings?

$1,856.00

$1,684.00

$1,234.00

$1,756.00

Cevabı ve açıklamayı göster

Cevap: $1,756.00

Cevap

The technician's total gross earnings for the week were $1,756.00.

The correct answer is found by calculating each component of the compensation structure and summing them. Regular pay is 40 hours at

24.00/hour (

960.00). Overtime pay is 6 hours at the time-and-a-half rate of

36.00/hour (

216.00). The tiered commission is calculated as 6% of the first

5,000 (

300.00) plus 8% of the remaining

3,500 in sales (

280.00). Adding these components together (

960.00 +

216.00 +

300.00 +

280.00) results in total gross earnings of $1,756.00.

Adım Adım Çözüm

1

Calculate the regular base pay.

40 hours \times

24.00/hour =

960.00

The technician is paid the base rate for the first 40 hours worked.

2

Calculate the overtime pay.

6 hours \times (

24.00 \times 1.5) = 6 \times

36.00 = $216.00

The 6 hours worked beyond the standard 40 hours are paid at time-and-a-half.

3

Calculate the first tier of commission.

5,000 \times 0.06 =

300.00

The technician earns a 6% commission on the first $5,000 of service contracts sold.

4

Calculate the second tier of commission.

(

8,500 -

5,000) \times 0.08 =

3,500 \times 0.08 =

280.00

The technician earns an 8% commission on any sales amount exceeding the initial $5,000.

5

Calculate total gross earnings.

960.00 +

216.00 +

300.00 +

280.00 = $1,756.00

Total gross earnings are the sum of regular pay, overtime pay, and all commission tiers.

Anahtar Kavram

Calculating gross earnings involving overtime premiums and multi-tiered commission structures.

Soru 4Soru →

A naval destroyer traveling at an average speed of $24\text{ knots}$ completes a specific patrol route in $3\text{ hours}$ and $20\text{ minutes}$ . If the captain is ordered to complete the identical route in $2\text{ hours}$ and $30\text{ minutes}$ , what average speed must the destroyer maintain?

18\text{ knots}

32\text{ knots}

16\text{ knots}

3.2\text{ knots}

Cevabı ve açıklamayı göster

Cevap: $32\text{ knots}$

Cevap

The destroyer must maintain an average speed of

32\text{ knots}

.

The distance of the patrol route is constant, meaning the product of the initial speed and time must equal the product of the new speed and time. Converting the travel times completely to minutes gives

200\text{ minutes}

and

150\text{ minutes}

. Using the inverse proportion formula,

24 \times 200 = 4800

. Dividing this constant by the required new time of

150\text{ minutes}

gives the necessary speed of

32\text{ knots}

.

Adım Adım Çözüm

1

Convert both time durations entirely into minutes to ensure consistent units.

3\text{ hours and } 20\text{ minutes} = 200\text{ minutes}

, and

2\text{ hours and } 30\text{ minutes} = 150\text{ minutes}

.

Calculations involving mixed time units require a single common unit before a proportion can be accurately applied.

2

Identify the relationship between speed and time for a constant distance.

Speed and time are inversely proportional, governed by the equation

S_1 \times T_1 = S_2 \times T_2

.

When the distance of a route remains the same, an increase in speed results in a proportional decrease in travel time.

3

Set up the inverse proportion equation and solve for the new speed (

S_2

).

24 \times 200 = S_2 \times 150 \implies 4800 = 150 \times S_2 \implies S_2 = 32

.

Isolating the unknown variable yields the required average speed to complete the route in the shorter timeframe.

Anahtar Kavram

Inverse Proportions in Speed and Time

Tahmini Süre:1m 30s

Soru 5Soru →

Consider the algebraic expression: $3x^2 + 6x - 24$ . When this expression is factored completely, which of the following represents the correct product?

3(x + 4)(x - 2)

3(x - 4)(x + 2)

(3x + 12)(x - 2)

3(x + 4)(x + 2)

Cevabı ve açıklamayı göster

Cevap: $3(x + 4)(x - 2)$

Cevap

The completely factored form is

3(x + 4)(x - 2)

.

To factor the expression completely, you must first factor out the greatest common factor (GCF) of 3 from all terms. This leaves

3(x^2 + 2x - 8)

. Next, you factor the quadratic trinomial inside the parentheses. You need two numbers that multiply to -8 and add to +2. Those numbers are +4 and -2. Therefore, combining the GCF and the binomials gives the completely factored form:

3(x + 4)(x - 2)

.

Adım Adım Çözüm

1

Identify and factor out the greatest common factor (GCF) from all three terms in the expression

3x^2 + 6x - 24

.

The GCF is 3. Factoring it out yields

3(x^2 + 2x - 8)

.

Factoring out the GCF simplifies the remaining trinomial, making it easier to factor and ensuring the final answer is factored completely.

2

Factor the remaining quadratic trinomial

x^2 + 2x - 8

by finding two integers that multiply to the constant term (-8) and add up to the middle coefficient (2).

The correct integers are 4 and -2 (since 4 * -2 = -8 and 4 + -2 = 2). This gives the binomial factors

(x + 4)(x - 2)

.

This process decomposes the monic quadratic trinomial into its constituent binomial factors.

3

Combine the GCF and the binomial factors to write the final factored expression.

The final product is

3(x + 4)(x - 2)

.

This represents the original expression as a product of its simplest prime factors.

Anahtar Kavram

Factoring Polynomials using GCF and Trinomial Decomposition

Soru 6Soru →

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

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

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

Aşağıdaki boşlukları doldurun

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

Cevabı ve açıklamayı göster

Cevap

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

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

3,056 gross -

1,456 wages =

1,600 commission). Then, subtract the

400 earned from the first

10,000 tier. This leaves

1,200 in commission earned on the remaining

20,000 in sales. Dividing

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

Adım Adım Çözüm

1

Calculate the base pay for standard hours.

$1,120

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

2

Calculate the overtime pay.

$336

The overtime rate is

28.00 × 1.5 =

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

42.00 × 8 =

336.

3

Calculate the total hourly wages.

$1,456

Add the base pay and overtime pay:

1,120 +

336 = $1,456.

4

Calculate the total commission earned.

$1,600

Subtract the total hourly wages from the total gross earnings:

3,056 -

1,456 = $1,600.

5

Calculate the tier 1 commission amount.

$400

The commission is 4% on the first

10,000 of sales:

10,000 × 0.04 = $400.

6

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

Tier 2 Commission:

1,200. Tier 2 Sales:

20,000.

Subtract tier 1 commission from total commission (

1,600 -

400 =

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

30,000 -

10,000 =

20,000).

7

Determine the tier 2 commission rate.

6%

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

1,200 ÷

20,000 = 0.06, which is 6%.

Anahtar Kavram

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

Soru 7Soru →

A sales representative earns a base monthly salary of $2,400. They also earn a commission based on their total monthly sales volume. The commission structure is as follows: -$ 4\% $on the first$ 15,000 of sales
- $6\%$ on the next $25,000 of sales -$ 8.5\% $on any sales exceeding$ 40,000

In addition, if their total monthly sales exceed $50,000, they receive a one-time performance bonus of$ 750. Last month, the representative had a total sales volume of $54,200. What was the representative's total gross income for the month?

Cevabı ve açıklamayı göster

Cevap: 6457

Cevap

6457

The correct gross income is found by calculating the commission for each tier separately (

600 for the first tier,

1,500 for the second tier, and

1,207 for the third tier), summing them to get a total commission of

3,307, and then adding the base salary of

2,400 and the

750 performance bonus (awarded since sales exceeded

50,000) for a final total of

6,457.

Adım Adım Çözüm

1

Calculate the commission for the first tier (

0 to

15,000).

15,000 \times 0.04 =

600

The representative earns a 4% commission on the first $15,000 of sales.

2

Calculate the commission for the second tier (the next $25,000).

25,000 \times 0.06 =

1,500

The 6% rate applies to the next

25,000 of sales (covering sales from

15,000 to $40,000).

3

Calculate the commission for the third tier (sales exceeding $40,000).

54,200 -

40,000 =

14,200.

14,200 \times 0.085 = $1,207

The 8.5% rate applies to any sales exceeding the $40,000 threshold.

4

Sum the total commission.

600 +

1,500 +

1,207 =

3,307

Adding the amounts from all three tiers gives the total commission earned.

5

Calculate the total gross income by adding the base salary, total commission, and the performance bonus.

2,400 +

3,307 +

750 =

6,457

Total sales (

54,200) exceeded the

50,000 threshold, so the $750 bonus is awarded. Gross income includes base pay, total commission, and the bonus.

Anahtar Kavram

Calculating tiered commissions, applying percentages to multiple thresholds, and combining various income streams.

Soru 8Soru →

A transport aircraft flies at a constant cruising speed of $360\text{ miles per hour}$ . If it maintains this speed, what is the total distance it travels in $40\text{ minutes}$ , in miles?

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

240

To find the distance, the time must first be converted to match the 'hours' unit in the speed rate. Forty minutes is equal to two-thirds of an hour. Multiplying the rate of

360\text{ miles per hour}

by

\frac{2}{3}\text{ of an hour}

yields a total distance of

240\text{ miles}

.

Adım Adım Çözüm

1

Convert the travel time from minutes to hours.

\frac{2}{3}\text{ hours}

The speed is given in miles per hour, so the time must be converted to hours to ensure the units cancel out correctly during multiplication.

2

Calculate the distance using the formula

Distance = Rate \times Time

.

240\text{ miles}

Multiplying the constant rate of

360\text{ mph}

by the travel time of

\frac{2}{3}\text{ hours}

yields the total distance traveled.

Anahtar Kavram

Calculating distance using a constant rate and performing a basic time unit conversion.

Alternatif Yöntem

You can also find the distance traveled per minute by dividing

360\text{ miles}

by

60\text{ minutes}

, which equals

6\text{ miles per minute}

. Then, multiply

6\text{ miles per minute}

by

40\text{ minutes}

to get

240\text{ miles}

.

Tahmini Süre:45s

Soru 9Soru →

A specialized reconnaissance team hikes through rough terrain at a steady pace of $3\text{ mph}$ for $1\text{ hour and } 40\text{ minutes}$ . They then rendezvous with an extraction helicopter that flies them to a base at an average speed of $150\text{ mph}$ for $24\text{ minutes}$ . What is the total distance covered by the team? (Provide your answer in miles, e.g., '50 miles')

Cevabı ve açıklamayı göster

Cevap: 65 miles; 65; 65 mi; 65mi; 65.0 miles; sixty-five miles

Cevap

65 miles

The correct answer is found by calculating the distance of each leg separately using the formula Distance = Rate × Time, ensuring all times are properly converted into hours, and then summing the two distances (5 miles + 60 miles = 65 miles).

Adım Adım Çözüm

1

Convert the hiking time into hours.

1 hour and 40 minutes = 1 + 40/60 hours = 5/3 hours.

Speed is given in miles per hour, so time must be in hours to properly use the distance formula.

2

Calculate the distance covered during the hike.

Distance = Rate × Time = 3 mph × (5/3) hours = 5 miles.

This finds the length of the first leg of the journey.

3

Convert the helicopter flight time into hours.

24 minutes = 24/60 hours = 2/5 hours (or 0.4 hours).

Again, the helicopter speed is in mph, requiring time to be in hours.

4

Calculate the distance covered by the helicopter.

Distance = Rate × Time = 150 mph × 0.4 hours = 60 miles.

This finds the length of the second leg of the journey.

5

Calculate the total distance covered.

5 miles + 60 miles = 65 miles.

The total distance is the sum of the individual leg distances.

Anahtar Kavram

Calculating distance for multiple legs of a journey with mixed time units.

Soru 10Soru →

A rare book dealer recorded the financial details for four different inventory items. Match each item's transaction history to its final profit or loss percentage.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The transactions match their respective percentages by dividing the net dollar profit or loss by the total initial cost.

Each correct pair accurately reflects the percentage change from the initial total investment. The total cost always serves as the base (denominator) for these percentage calculations.

Adım Adım Çözüm

1

Calculate the profit percentage for the first edition novel.

Profit is

\$150 - \$120 = \$30

. Percentage is

\frac{30}{120} = 25\%

.

To find the profit percentage, divide the absolute profit by the original purchase cost.

2

Calculate the total cost and loss percentage for the antique atlas.

Total cost is

\$300 + \$100 = \$400

. Loss is

\$400 - \$320 = \$80

. Percentage is

\frac{80}{400} = 20\%

.

Overhead and restoration costs must be added to the purchase price to determine the true initial cost.

3

Calculate the profit percentage for the signed biography.

Profit is

\$90 - \$60 = \$30

. Percentage is

\frac{30}{60} = 50\%

.

Dividing the absolute profit by the initial cost provides the correct percentage.

4

Calculate the loss percentage for the vintage encyclopedias.

Loss is

\$80 - \$72 = \$8

. Percentage is

\frac{8}{80} = 10\%

.

Dividing the dollar loss by the original cost yields the correct loss percentage.

Anahtar Kavram

Calculating profit and loss percentages by dividing the net financial change by the total initial cost, ensuring all overhead is included.

Soru 11Soru →

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

Aşağıdaki boşlukları doldurun

The platoon can ground a total of complete radio systems.

Cevabı ve açıklamayı göster

Cevap

15

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

42 \frac{1}{2}

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

2 \frac{5}{6}

meters). Converting both to improper fractions gives

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

. Multiplying by the reciprocal yields

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

, which simplifies to

5 \times 3 = 15

.

Adım Adım Çözüm

1

Convert both mixed numbers to improper fractions.

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

and

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

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

2

Set up the division equation.

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

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

3

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

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

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

4

Simplify by cross-canceling common factors and multiply.

Divide

85

by

17

to get

5

. Divide

6

by

2

to get

3

. Then,

5 \times 3 = 15

.

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

Anahtar Kavram

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

Soru 12Soru →

A base logistics officer needs to requisition 1,500 feet of paracord. Distributor A sells a 100-yard spool of paracord for $36.00. Distributor B sells the same paracord in a 50-foot roll for$ 5.50. If the officer purchases the entire 1,500 feet from the distributor with the lower overall cost, how much total money will be saved compared to purchasing from the more expensive distributor?

$163.50

$15.00

$375.00

$1,455.00

Cevabı ve açıklamayı göster

Cevap: $15.00

Cevap

The total savings is $15.00 when purchasing from Distributor B instead of Distributor A.

The correct answer is found by equalizing the units and calculating the total cost required to obtain 1,500 feet from each distributor. Distributor A provides 300 feet (100 yards) per

36.00 spool, so 5 spools are needed, costing

180.00. Distributor B provides 50 feet per

5.50 roll, so 30 rolls are needed, costing

165.00. The total savings is the difference:

180.00 -

165.00 = $15.00.

Adım Adım Çözüm

1

Calculate the total cost if buying from Distributor A.

100 yards × 3 feet/yard = 300 feet per spool. Number of spools needed = 1,500 feet ÷ 300 feet/spool = 5 spools. Total cost = 5 ×

36.00 =

180.00.

Units must be consistent (feet) to compare or fulfill the 1,500-foot requirement accurately.

2

Calculate the total cost if buying from Distributor B.

Number of rolls needed = 1,500 feet ÷ 50 feet/roll = 30 rolls. Total cost = 30 ×

5.50 =

165.00.

We must find the exact total cost for fulfilling the entire requisition of 1,500 feet with the second option.

3

Find the difference between the two total costs to determine savings.

180.00 -

165.00 = $15.00.

Subtracting the lower cost from the higher cost reveals the total amount saved.

Anahtar Kavram

Comparing unit prices across different bulk packages by establishing a common metric or total purchasing cost.

Soru 13Soru →

Find the greatest integer value of $x$ that makes the following inequality true:

\frac{2}{3}(6x - 9) - 4(2x + 1) > \frac{1}{4}(8x - 20) - 3x

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The greatest integer value that satisfies the inequality is -2.

Expanding both sides yields

-4x - 10 > -x - 5

. Isolating

x

on one side gives

-3x > 5

. Dividing by -3 requires flipping the inequality symbol, resulting in

x < -\frac{5}{3}

(or

x < -1.67

). The greatest integer strictly less than -1.67 is -2.

Adım Adım Çözüm

1

Distribute the constants and fractions into the parentheses on both sides.

4x - 6 - 8x - 4 > 2x - 5 - 3x

Applying the distributive property removes the parentheses so like terms can be combined.

2

Combine like terms on each side of the inequality.

-4x - 10 > -x - 5

Simplifying both expressions makes it easier to isolate the variable.

3

Add

x

to both sides and add

10

to both sides to separate the variable terms from the constant terms.

-3x > 5

This isolates the

x

term on one side of the inequality.

4

Divide both sides by -3 and flip the inequality sign.

x < -\frac{5}{3}

When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.

5

Identify the greatest integer less than

-\frac{5}{3}

.

Since

-\frac{5}{3} \approx -1.67

, the integers strictly less than this are

-2, -3, -4, \dots

The greatest of these is

-2

.

The question specifically asks for the greatest integer value that makes the inequality true.

Anahtar Kavram

Solving complex linear inequalities involves applying the distributive property, combining like terms, and reversing the inequality symbol when dividing or multiplying by a negative number.

Soru 14Soru →

A mechanical winch uses a gear system where the driving gear and the driven gear have a revolution ratio of $4:7$ . If the driving gear completes $124$ revolutions during a lifting operation, how many revolutions does the driven gear complete?

Cevabı ve açıklamayı göster

Cevap: 217

Cevap

217

The correct answer is found by setting up a direct proportion based on the 4:7 ratio. Since the driving gear's 4 parts scale up to 124 actual revolutions, the scale factor is 31 (124 ÷ 4). Multiplying the driven gear's 7 parts by this same scale factor yields 217 revolutions.

Adım Adım Çözüm

1

Identify the known ratio

Driving : Driven = 4 : 7

This establishes the proportional relationship between the two gears.

2

Set up a proportion using the known quantity

4 / 7 = 124 / x

We know the driving gear completes 124 revolutions, which corresponds to the '4' in the ratio.

3

Find the scale factor

124 ÷ 4 = 31

This tells us how many times greater the actual operation is compared to the base ratio.

4

Calculate the unknown quantity

7 × 31 = 217

Applying the same scale factor to the driven gear maintains the proportional relationship.

Anahtar Kavram

Solving for an unknown part in a part-to-part relationship using scale factors.

Soru 15Soru →

A bakery owner purchases a commercial oven for $\$2,400$ . She pays $\$250$ for freight shipping and $\$150$ for professional installation. If she later sells the oven for a $15\%$ loss on her total initial investment, what is the selling price in dollars?

Cevabı ve açıklamayı göster

Cevap: 2380

Cevap

2380

The correct answer is found by first determining the total investment (

\$2,400 + \$250 + \$150 = \$2,800

), then finding the

15\%

loss (

0.15 \times \$2,800 = \$420

), and finally subtracting that loss from the total investment to get the selling price (

\$2,800 - \$420 = \$2,380

).

Adım Adım Çözüm

1

Calculate the total initial investment.

\$2,400 + \$250 + \$150 = \$2,800

All overhead costs must be added to the purchase price to find the true total cost.

2

Calculate the dollar amount of the

15\%

loss.

0.15 \times \$2,800 = \$420

The loss is based on the total initial investment, not just the purchase price.

3

Determine the final selling price.

\$2,800 - \$420 = \$2,380

Subtracting the loss amount from the total investment gives the selling price.

Anahtar Kavram

Calculating total cost including overhead, and applying a percentage loss to find the final selling price.

Alternatif Yöntem

Instead of calculating the loss and subtracting it, you can multiply the total investment by

85\%

(since

100\% - 15\% = 85\%

).

0.85 \times \$2,800 = \$2,380

.

Tahmini Süre:1m 0s

Soru 16Soru →

A motor pool mechanic needs to buy spark plugs. A single spark plug costs $3.00. A box of one dozen spark plugs costs$ 24.00. How much money is saved per spark plug by purchasing the box instead of buying them individually?

$1.00

$2.00

$2.50

$12.00

Cevabı ve açıklamayı göster

Cevap: $1.00

Cevap

$1.00

Purchasing the box of one dozen means buying 12 spark plugs for

24.00, which results in a unit price of

2.00 each (

24.00 ÷ 12). Buying them individually costs

3.00 each. The difference per spark plug is

3.00 -

2.00 = $1.00.

Adım Adım Çözüm

1

Determine the total number of items in the box.

1 dozen = 12 spark plugs.

To find the cost of a single spark plug from the box, the total quantity is required.

2

Calculate the unit price of a spark plug from the box.

24.00 ÷ 12 =

2.00 per spark plug.

Dividing the total cost by the quantity yields the price per individual item.

3

Calculate the savings per spark plug.

3.00 -

2.00 = $1.00.

Subtracting the bulk unit price from the individual unit price gives the savings per item.

Anahtar Kavram

Unit Price Comparisons

Soru 17Soru →

A Navy maintenance crew is applying a new non-slip coating to a ship's deck. A single standard drum of the coating covers exactly $12 \frac{3}{4}$ square yards of deck surface. If the crew needs to coat a helicopter landing pad that requires exactly $5 \frac{1}{3}$ times the amount of coating contained in a single drum, what is the total area of the helicopter landing pad in square yards?

Cevabı ve açıklamayı göster

Cevap: 68

Cevap

68

To find the total area, you must multiply the area covered by one drum (

12 \frac{3}{4}

square yards) by the relative size of the landing pad (

5 \frac{1}{3}

). Converting to improper fractions gives

\frac{51}{4} \times \frac{16}{3}

. Cross-canceling the terms yields

17 \times 4 = 68

square yards.

Adım Adım Çözüm

1

Convert the mixed numbers

12 \frac{3}{4}

and

5 \frac{1}{3}

to improper fractions.

12 \frac{3}{4} = \frac{12 \times 4 + 3}{4} = \frac{51}{4}

and

5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{16}{3}

.

Multiplication of mixed numbers requires converting them to improper fractions first.

2

Multiply the two improper fractions to find the total area.

\frac{51}{4} \times \frac{16}{3}

The total area is the area covered by one drum multiplied by the number of drum-equivalents required.

3

Simplify and compute the final product.

\frac{51 \times 16}{4 \times 3} = \left(\frac{51}{3}\right) \times \left(\frac{16}{4}\right) = 17 \times 4 = 68

.

Simplifying the fractions by cross-canceling before multiplying makes the calculation easier and provides the final area.

Anahtar Kavram

Multiplying mixed numbers by converting them to improper fractions in an applied context.

Soru 18Soru →

A military motor pool contains a total of $250$ utility vehicles. A morning report indicates that exactly $20\%$ of these vehicles are currently dispatched for a training exercise. How many vehicles are dispatched for the exercise?

50

5

200

230

Cevabı ve açıklamayı göster

Cevap: $50$

Cevap

There are

50

vehicles dispatched for the exercise.

The correct answer accurately determines

20\%

of

250

by multiplying the total (

250

) by the decimal form of the percentage (

0.20

).

Adım Adım Çözüm

1

Convert the percentage to a decimal equivalent.

20\% = 0.20

Percentages must be expressed as decimals or fractions to perform arithmetic calculations.

2

Multiply the total number of vehicles by the decimal value.

250 \times 0.20 = 50

To calculate a percentage of a specific quantity, multiply the total amount by the percentage rate.

Anahtar Kavram

To find a percentage of a quantity, convert the percentage into a decimal or fraction and multiply it by the total amount.

Tahmini Süre:30s

Soru 19Soru →

Evaluate whether the following statement is true or false:

The only positive integer solutions to the linear inequality

-\frac{3}{5}\left(\frac{10}{3}x - 15\right) + 4(x - 2) \geq -\frac{5}{2}(-2x + 4) + 2

are 1, 2, and 3.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is True.

By successfully distributing the fractional negative coefficients and combining like terms, the inequality simplifies to

-3x \geq -9

. Dividing by

-3

requires reversing the inequality to

x \leq 3

. The positive integers in this solution set are exactly 1, 2, and 3, meaning the statement is factually correct.

Adım Adım Çözüm

1

Distribute the negative fractions and constants on the left side of the inequality.

(-2x + 9) + (4x - 8)

Multiplying

-\frac{3}{5}

by

\frac{10}{3}x

gives

-2x

, and

-\frac{3}{5}

by

-15

gives

+9

. Multiplying

4

by

(x-2)

gives

4x - 8

.

2

Distribute the negative fraction on the right side of the inequality.

(5x - 10) + 2

Multiplying

-\frac{5}{2}

by

-2x

gives

5x

, and

-\frac{5}{2}

by

4

gives

-10

.

3

Combine the like terms on both sides.

2x + 1 \geq 5x - 8

On the left side,

-2x + 4x = 2x

and

9 - 8 = 1

. On the right side,

-10 + 2 = -8

.

4

Isolate the variable

x

by subtracting

5x

and

1

from both sides.

-3x \geq -9

Grouping all variables on one side and constants on the other prepares the equation for the final division.

5

Divide both sides by

-3

and reverse the inequality symbol.

x \leq 3

Dividing an inequality by a negative number requires flipping the inequality direction. Since

x \leq 3

, the only integers greater than zero that satisfy this are 1, 2, and 3.

Anahtar Kavram

Solving complex linear inequalities by distributing negative fractions, combining like terms, and properly applying the inequality sign flip rule.

Alternatif Yöntem

You can verify the statement by plugging the boundary values into the original inequality. Substitute

x=3

(which evaluates to

7 \geq 7

, a true statement),

x=4

(which evaluates to

9 \geq 12

, a false statement), and recognize that since it is a linear relation, solutions are valid downwards. Zero is not a positive integer, leaving 1, 2, and 3 as the only valid positive integers.

Tahmini Süre:2m 30s

Soru 20Soru →

Match each landscaping supply transaction to its correct final profit or loss percentage.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The soil pallet transaction resulted in a 25% profit. The lawnmower transaction resulted in a 20% loss. The gardening tools transaction resulted in a 40% loss. The decorative stones transaction resulted in a 100% profit.

Each matching pair accurately reflects the transaction's net profit or loss divided by the total initial cost (including any overhead).

Adım Adım Çözüm

1

Calculate the percentage profit for the soil pallet.

Profit =

250 -

200 =

50. Percentage = (

50 / $200) * 100 = 25%. This matches the '25% profit' option.

Profit is the positive difference between the selling price and cost. The percentage is always calculated over the original cost.

2

Determine the total initial cost of the lawnmower including overhead, then find the percentage loss.

Total cost =

400 +

100 =

500. Loss =

500 -

400 =

100. Percentage = (

100 /

500) * 100 = 20%. This matches the '20% loss' option.

Overhead expenses (like repairs) must be added to the initial purchase price to find the true total cost basis before evaluating profit or loss.

3

Calculate the percentage loss for the gardening tools.

Loss =

150 -

90 =

60. Percentage = (

60 / $150) * 100 = 40%. This matches the '40% loss' option.

Loss is the negative difference between the selling price and cost, and the percentage is found by dividing that difference by the original cost.

4

Calculate the percentage profit for the decorative stones.

Profit =

600 -

300 =

300. Percentage = (

300 / $300) * 100 = 100%. This matches the '100% profit' option.

When an item is sold for exactly double its purchase price, the profit equals the original cost, which corresponds to a 100% return.

Anahtar Kavram

Calculating percentage profit and loss based on total cost

Alternatif Yöntem

For profit scenarios, you can divide the selling price directly by the cost price to get a multiplier. For the decorative stones:

600 /

300 = 2.0. A multiplier of 2.0 corresponds to a 100% increase (profit).

Tahmini Süre:1m 30s

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem