Tüm alıştırma soruları

Read the problem carefully and fill in the blanks with the correct numerical values.

Aşağıdaki boşlukları doldurun

A military surplus store is selling tactical backpacks originally priced at

40. They are currently offering a 30% discount on these items. The amount of the discount is

and the new sale price is $.

Cevabı ve açıklamayı göster

Cevap

The discount amount is

12 and the new sale price is

28.

To solve this problem, first determine the dollar value of the discount. Convert 30% to a decimal (0.30) and multiply it by the original price (

40), which results in a discount amount of

12. To find the new sale price, subtract the

12 discount from the

40 original price, yielding a final sale price of $28.

Adım Adım Çözüm

1

Calculate the discount amount in dollars.

40 \times 0.30 =

12

The discount is 30% of the original price, so multiplying the original price by the decimal equivalent of 30% yields the discount amount.

2

Calculate the new sale price.

40 -

12 = $28

The sale price is found by subtracting the discount amount from the original price.

Anahtar Kavram

Calculating simple discounts and final sale prices

Soru 282Soru →

A property leasing agent earns a fixed weekly base salary of $\$800$ , plus a $5\%$ commission on the total dollar value of all leases signed that week. If the agent signed leases totaling $\$3,000$ last week, what was their total gross pay for the week?

\$150

\$815

\$950

\$840

Cevabı ve açıklamayı göster

Cevap: $\$950$

Cevap

The agent's total gross pay is

\$950

.

The total gross pay is the sum of the base salary and the commission. The commission is calculated as

5\%

of the

\$3,000

lease value, which is

\$150

. Adding this to the

\$800

base salary results in a total gross pay of

\$950

.

Adım Adım Çözüm

1

Calculate the commission amount by finding

5\%

of the total lease value.

0.05 \times \$3,000 = \$150

The commission is earned on the total value of the leases signed.

2

Add the calculated commission to the fixed base salary to find the total gross pay.

\$800 + \$150 = \$950

Total pay is the sum of the guaranteed base salary and any earned commission.

Anahtar Kavram

Calculating total earnings by combining a fixed base salary with a percentage-based commission.

Soru 283Soru →

A logistical transport convoy travels along a main supply route at a steady speed of $55\text{ miles per hour}$ . If the journey takes exactly $3\text{ hours and } 36\text{ minutes}$ , what is the total distance traveled in miles?

Cevabı ve açıklamayı göster

Cevap: 198

Cevap

The convoy traveled a total distance of 198 miles.

The correct total distance is determined by first converting the mixed travel time entirely into hours (

3.6\text{ hours}

) and then multiplying it by the constant speed (

55\text{ mph}

) to yield exactly

198\text{ miles}

.

Adım Adım Çözüm

1

Convert the travel time into a single unit (hours) to match the rate.

3\text{ hours and } 36\text{ minutes} = 3 + \frac{36}{60}\text{ hours} = 3.6\text{ hours}

.

The speed is given in miles per hour, so the time must be represented entirely in hours before multiplying.

2

Apply the distance formula: Distance = Rate

\times

Time.

Distance =

55 \times 3.6

This formula determines the total distance covered over a specific time frame at a constant speed.

3

Calculate the final distance.

198

Multiplying the constant rate of

55\text{ mph}

by the total duration of

3.6\text{ hours}

yields the total miles traveled.

Anahtar Kavram

Calculating distance using a constant rate and converting mixed time units.

Soru 284Soru →

Calculate the unknown value in the following financial scenario.

Aşağıdaki boşlukları doldurun

A facilities manager is purchasing a bulk supply of LED lighting fixtures from a wholesale distributor. The distributor applies a

15\%

volume discount to the regular total price. Because the manager pays the invoice within 10 days, an additional

5\%

early-payment discount is applied to the already reduced price. If the final price of the fixtures before tax is

\$3,230

, the regular total price of the order was $\$$.

Cevabı ve açıklamayı göster

Cevap

$4,000

To find the original price after successive discounts, you must work backwards from the final price. First, divide the final price by the complement of the second discount (

100\% - 5\% = 95\%

, or

0.95

) to find the intermediate price:

\$3,230 \div 0.95 = \$3,400

. Next, divide this intermediate price by the complement of the first discount (

100\% - 15\% = 85\%

, or

0.85

) to find the regular total price:

\$3,400 \div 0.85 = \$4,000

.

Adım Adım Çözüm

1

Calculate the intermediate price before the 5% early-payment discount.

\$3,230 \div 0.95 = \$3,400

The final price of

\$3,230

represents

95\%

(

100\% - 5\%

) of the price after the initial volume discount.

2

Calculate the original regular total price before the 15% volume discount.

\$3,400 \div 0.85 = \$4,000

The intermediate price of

\$3,400

represents

85\%

(

100\% - 15\%

) of the original regular total price.

Anahtar Kavram

Calculating original prices by reversing successive discounts using division.

Soru 285Soru →

A contractor is analyzing a zoning plan for a commercial parking lot. The plan's legend indicates a scale of $1\text{ inch} = 10\text{ yards}$ . If the parking lot covers an area of $4\text{ square inches}$ on the zoning plan, what is the actual area of the parking lot in square feet?

120

400

1,200

3,600

Cevabı ve açıklamayı göster

Cevap: $3,600$

Cevap

The actual area of the parking lot is

3,600

square feet.

The correct answer is found by first aligning the units: the scale of

1\text{ inch} = 10\text{ yards}

is equivalent to

1\text{ inch} = 30\text{ feet}

. Because we are dealing with area (a two-dimensional measurement), this linear scale must be squared, meaning

1\text{ square inch}

on the drawing represents

900\text{ square feet}

in reality (

30^2 = 900

). Finally, multiplying the drawing's area by this factor yields the actual area:

4 \times 900 = 3,600\text{ square feet}

.

Adım Adım Çözüm

1

Convert the linear scale from yards to feet.

10\text{ yards} = 30\text{ feet}

, so the scale is

1\text{ inch} = 30\text{ feet}

.

The final question asks for the area in square feet, so it is easiest to align the units of the scale factor first.

2

Determine the area scale factor by squaring the linear scale.

1\text{ square inch} = (30\text{ feet}) \times (30\text{ feet}) = 900\text{ square feet}

.

Because area represents two dimensions, the linear scale must be squared to find out how much actual area one square inch on the drawing represents.

3

Multiply the drawing area by the area scale factor.

4\text{ square inches} \times 900\text{ square feet per square inch} = 3,600\text{ square feet}

.

This scales the blueprint measurement up to the actual real-world area.

Anahtar Kavram

When using a scale drawing to find a real-world area, the linear scale factor must be squared. Furthermore, all units must be carefully tracked and converted to match the requested final unit.

Soru 286Soru →

A base supply depot has $1,200$ field rations in stock. If $15\%$ of these rations are designated for immediate deployment, how many field rations are designated for immediate deployment?

1,020

1,185

180

18

Cevabı ve açıklamayı göster

Cevap: $180$

Cevap

The correct number of field rations designated for immediate deployment is

180

.

The correct answer is determined by finding

15\%

of

1,200

, which is calculated by multiplying

1,200

by

0.15

to yield

180

.

Adım Adım Çözüm

1

Convert the percentage to a decimal.

15\% = 0.15

Percentages must be converted into decimals or fractions before performing calculations.

2

Multiply the total number of rations by the decimal equivalent of the percentage.

1,200 \times 0.15 = 180

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

Anahtar Kavram

Calculating Percentages of Quantities

Soru 287Soru →

A car wash attendant earns a base salary of $\$400$ per week, plus a $10\%$ commission on any detailing packages they sell. If the attendant sells $\$800$ worth of detailing packages in one week, what is their total gross pay for that week?

Aşağıdaki boşlukları doldurun

Total gross pay: $\$$

Cevabı ve açıklamayı göster

Cevap

480

The correct total gross pay is calculated by finding the commission (

10\%

of

\$800 = \$80

) and adding it to the weekly base salary (

\$400

). The sum is

\$400 + \$80 = \$480

.

Adım Adım Çözüm

1

Calculate the commission earned from the sale of detailing packages.

0.10 \times \$800 = \$80

The attendant earns a

10\%

commission on the total value of detailing packages sold.

2

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

\$400 + \$80 = \$480

Total earnings equal the base salary plus any additional commission earned.

Anahtar Kavram

Calculating total gross pay using a base salary plus a percentage-based commission.

Soru 288Soru →

A coast guard rescue helicopter flies directly toward a distressed vessel at a constant speed of $120\text{ miles per hour}$ . If the flight takes exactly $45\text{ minutes}$ , what is the total distance traveled by the helicopter?

90\text{ miles}

5,400\text{ miles}

54\text{ miles}

900\text{ miles}

Cevabı ve açıklamayı göster

Cevap: $90\text{ miles}$

Cevap

The total distance traveled by the helicopter is

90\text{ miles}

.

The correct answer accurately converts the

45\text{ minutes}

of flight time into

0.75\text{ hours}

, and then applies the distance formula (

D = R \times T

) to find

120 \times 0.75 = 90\text{ miles}

.

Adım Adım Çözüm

1

Convert the flight time from minutes to hours.

45\text{ minutes} = \frac{45}{60}\text{ hours} = 0.75\text{ hours}

.

The speed is given in miles per hour, so the time must also be in hours to calculate the distance correctly.

2

Calculate the distance using the formula

\text{Distance} = \text{Rate} \times \text{Time}

.

120\text{ miles per hour} \times 0.75\text{ hours} = 90\text{ miles}

.

Multiplying the constant speed by the total time in hours gives the total distance traveled.

Anahtar Kavram

To find distance, multiply the rate of speed by the time traveled, ensuring that the units of time match the units in the speed rate.

Soru 289Soru →

A logistics officer is reviewing purchase orders for safety equipment. Match each bulk purchase scenario to its correct unit price.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings are determined by dividing the total cost by the quantity for each scenario, yielding the unit rates of

5,

3,

6, and

4 respectively.

Each correct match is found by identifying the unit price using the formula: Total Cost ÷ Quantity.

Adım Adım Çözüm

1

Calculate the unit price for the protective goggles.

60 / 12 =

5 per pair.

Unit price is total cost divided by quantity.

2

Calculate the unit price for the work gloves.

90 / 30 =

3 per pair.

Unit price is total cost divided by quantity.

3

Calculate the unit price for the heavy-duty flashlights.

48 / 8 =

6 per flashlight.

Unit price is total cost divided by quantity.

4

Calculate the unit price for the high-visibility vests.

100 / 25 =

4 per vest.

Unit price is total cost divided by quantity.

Anahtar Kavram

Calculating unit rates by dividing the total amount by the number of units.

Soru 290Soru →

An outdoor gear shop is selling a camping tent with a regular price of $\$180.00$ . During a weekend clearance event, the tent is discounted by $15\%$ . What is the sale price of the tent?

\$165.00

\$27.00

\$153.00

\$177.30

Cevabı ve açıklamayı göster

Cevap: $\$153.00$

Cevap

\$153.00

The regular price of the tent is

\$180.00

. A

15\%

discount means saving

15\%

of the original price. Calculating

15\%

of

\$180.00

gives a

\$27.00

discount. Subtracting this discount from the regular price (

\$180.00 - \$27.00

) results in a final sale price of

\$153.00

.

Adım Adım Çözüm

1

Calculate the discount amount by finding

15\%

of

\$180.00

.

0.15 \times \$180.00 = \$27.00

To find the dollar amount of savings, multiply the original price by the discount percentage expressed as a decimal.

2

Subtract the discount amount from the regular price.

\$180.00 - \$27.00 = \$153.00

The final sale price is the original price reduced by the calculated discount amount.

Anahtar Kavram

Discounts and Sale Prices

Soru 291Soru →

A field generator consumes diesel fuel at a rate of $2 \frac{1}{4}$ gallons every $30$ minutes. If the fuel tank currently holds $33 \frac{3}{4}$ gallons, how many hours can the generator run before the tank is empty?

7 \frac{1}{2}

15

\frac{2}{15}

18

Cevabı ve açıklamayı göster

Cevap: $7 \frac{1}{2}$

Cevap

7 \frac{1}{2}

To find how long the generator can run, first align the time units. Since the generator uses

2 \frac{1}{4}

gallons every 30 minutes, it uses twice that amount in one hour, which is

4 \frac{1}{2}

gallons per hour. Then, divide the total fuel (

33 \frac{3}{4}

gallons) by the hourly rate (

4 \frac{1}{2}

gallons/hour). Converting to improper fractions, this is

\frac{135}{4} \div \frac{9}{2}

. Multiplying by the reciprocal gives

\frac{135}{4} \times \frac{2}{9} = \frac{270}{36}

, which simplifies to

7 \frac{1}{2}

hours.

Adım Adım Çözüm

1

Find the fuel consumption rate per hour.

2 \frac{1}{4} \times 2 = 4 \frac{1}{2}

gallons per hour

The rate is given for 30 minutes, which is half an hour. Multiply by 2 to find the hourly rate.

2

Convert the mixed numbers to improper fractions.

33 \frac{3}{4} = \frac{135}{4}

and

4 \frac{1}{2} = \frac{9}{2}

Fractions must be in improper form before dividing.

3

Divide the total fuel by the hourly rate to find the number of hours.

\frac{135}{4} \div \frac{9}{2} = \frac{135}{4} \times \frac{2}{9} = \frac{270}{36} = 7 \frac{18}{36} = 7 \frac{1}{2}

hours

Dividing total capacity by the rate of consumption gives the total time.

Anahtar Kavram

Fraction Division with Unit Conversion

Tahmini Süre:1m 30s

Soru 292Soru →

An architect is designing a community center and reviewing the floor plan. The floor plan uses a scale of $\frac{3}{4}\text{ inch} = 5\text{ feet}$ . If the main hall measures $4.5\text{ inches}$ long on the floor plan, what is the actual length of the main hall in feet?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

The correct answer is found by setting up a proportion or finding the unit rate. Since

\frac{3}{4}\text{ inch}

represents

5\text{ feet}

,

1\text{ inch}

represents

\frac{20}{3}\text{ feet}

. Multiplying the drawing measurement of

4.5\text{ inches}

by

\frac{20}{3}\text{ feet per inch}

yields exactly

30\text{ feet}

.

Adım Adım Çözüm

1

Find the actual distance represented by one inch.

1\text{ inch} = 5 \div \frac{3}{4} = 5 \times \frac{4}{3} = \frac{20}{3}\text{ feet}

Establishing a unit rate simplifies calculating the actual length for any given map measurement.

2

Multiply the unit rate by the blueprint measurement.

4.5 \times \frac{20}{3} = \frac{9}{2} \times \frac{20}{3} = \frac{180}{6} = 30\text{ feet}

Multiplying the map length by the actual distance per inch gives the total actual length.

Anahtar Kavram

Solving scale drawing problems using proportions or unit rates.

Soru 293Soru →

Read the scenario and calculate the correct numerical value to complete the statement.

Aşağıdaki boşlukları doldurun

A base supply warehouse holds

150

replacement tires for tactical vehicles. If a maintenance unit requisitions

20\%

of these tires for immediate use, the warehouse will issue tires to the unit.

Cevabı ve açıklamayı göster

Cevap

30

To find

20\%

of

150

, you must convert the percentage to a decimal (

0.20

) or a simple fraction (

\frac{1}{5}

). Then, multiply the total number of tires by this value:

150 \times 0.20 = 30

. Therefore,

30

tires will be issued to the maintenance unit.

Adım Adım Çözüm

1

Identify the total quantity and the percentage required.

Total tires =

150

, Percentage to issue =

20\%

These are the given values needed to find the partial amount.

2

Convert the percentage to a decimal or a fraction.

20\% = 0.20

(or

\frac{1}{5}

)

Percentages must be converted into a workable numeric format before calculating a portion.

3

Multiply the total quantity by the decimal or fraction to find the final amount.

150 \times 0.20 = 30

Multiplying the whole by the percentage rate gives the part of the whole.

Anahtar Kavram

Calculating a basic percentage of a whole quantity

Tahmini Süre:45s

Soru 294Soru →

A Coast Guard search and rescue helicopter departs its base and flies toward a distress signal at a steady speed of $140\text{ knots}$ for $45\text{ minutes}$ . Upon arriving at the scene, the helicopter hovers in place for $15\text{ minutes}$ to hoist a survivor. It then flies directly to a coastal hospital at a steady speed of $160\text{ knots}$ , arriving after $1\text{ hour and } 15\text{ minutes}$ . What is the total distance traveled by the helicopter during this mission?

305\text{ nautical miles}

300\text{ nautical miles}

247\text{ nautical miles}

200\text{ nautical miles}

Cevabı ve açıklamayı göster

Cevap: $305\text{ nautical miles}$

Cevap

305\text{ nautical miles}

The correct answer is found by determining the distance of each flight leg separately and adding them together. First, convert the times into hours:

45\text{ minutes}

is

0.75\text{ hours}

, and

1\text{ hour } 15\text{ minutes}

is

1.25\text{ hours}

. Then, apply the formula

\text{Distance} = \text{Rate} \times \text{Time}

. The first leg is

140\text{ knots} \times 0.75\text{ hours} = 105\text{ nautical miles}

. The hover adds

0

distance. The final leg is

160\text{ knots} \times 1.25\text{ hours} = 200\text{ nautical miles}

. Adding these segments together (

105 + 200

) yields a total distance of

305\text{ nautical miles}

.

Adım Adım Çözüm

1

Convert the travel time of the first leg from minutes to hours.

45\text{ minutes} = 45 / 60 = 0.75\text{ hours}

Speed is given in knots (nautical miles per hour), so the time must be represented in hours.

2

Calculate the distance traveled during the first leg.

140\text{ knots} \times 0.75\text{ hours} = 105\text{ nautical miles}

Distance is equal to rate multiplied by time.

3

Determine the distance traveled during the hover phase.

0\text{ nautical miles}

The helicopter is stationary while hovering, so no distance is covered.

4

Convert the travel time of the final leg to hours.

1\text{ hour and } 15\text{ minutes} = 1 + (15 / 60) = 1.25\text{ hours}

Consistent units are required to accurately apply the distance formula.

5

Calculate the distance traveled during the final leg.

160\text{ knots} \times 1.25\text{ hours} = 200\text{ nautical miles}

Using the distance formula again for the final phase of the journey.

6

Sum the distances from all phases of the mission.

105 + 0 + 200 = 305\text{ nautical miles}

The total mission distance is the sum of the distances covered in each individual segment.

Anahtar Kavram

Calculating distance across multiple segments by converting time to consistent units and applying the Distance = Rate × Time formula for each segment.

Soru 295Soru →

A squad of soldiers is conducting a foot march along a trail that is $12$ miles long. If they maintain a steady pace of $3$ miles per hour, how many hours will it take them to complete the march?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

Using the fundamental relationship

\text{Time} = \frac{\text{Distance}}{\text{Rate}}

, we divide the total distance of

12

miles by the steady pace of

3

miles per hour. This calculation (

12 \div 3

) results in exactly

4

hours.

Adım Adım Çözüm

1

Identify the given values for the distance and the rate of travel.

Distance =

12

miles, Rate =

3

miles per hour.

These values are required to calculate the total elapsed time.

2

Apply the appropriate formula to find the time.

\text{Time} = \frac{\text{Distance}}{\text{Rate}}

Time is equal to total distance divided by the speed.

3

Substitute the values and calculate the result.

\frac{12}{3} = 4

Dividing the distance by the rate gives the total time in hours.

Anahtar Kavram

Calculating elapsed time given distance and rate.

Tahmini Süre:45s

Soru 296Soru →

What is the complete solution set for the given inequality?

\frac{1}{2}(-8x - 10) - \frac{2}{3}(9x + 15) \leq -4(x - 2) + 2x + 1

x \geq -3

x \leq -3

x \geq -1

x \geq -2

Cevabı ve açıklamayı göster

Cevap: $x \geq -3$

Cevap

The correct solution is

x \geq -3

.

By systematically distributing the fractions and negative integers on both sides, we simplify the expression to

-10x - 15 \leq -2x + 9

. Moving the variable terms to the left and constants to the right results in

-8x \leq 24

. Finally, dividing by

-8

isolates

x

and requires flipping the inequality symbol, providing the correct solution.

Adım Adım Çözüm

1

Distribute the fractions into the terms inside the parentheses on the left side.

(-4x - 5) - (6x + 10) \leq -4(x - 2) + 2x + 1

To eliminate the parentheses and clear the fractions for easier manipulation.

2

Distribute the

-4

on the right side of the inequality.

-4x - 5 - 6x - 10 \leq -4x + 8 + 2x + 1

To expand all terms fully before attempting to combine them.

3

Combine the like terms on both the left and right sides independently.

-10x - 15 \leq -2x + 9

To simplify the inequality into a standard two-step format.

4

Add

2x

to both sides and add

15

to both sides to group variable terms and constant terms.

-8x \leq 24

To isolate the

x

term on one side of the inequality.

5

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq -3

Multiplying or dividing an inequality by a negative number requires flipping the inequality sign to maintain the mathematical truth.

Anahtar Kavram

Solving Complex Linear Inequalities

Soru 297Soru →

A military mess hall requires exactly $500$ pounds of flour. Supplier A sells flour in $25$ -pound bags for $\$14.00$ per bag. Supplier B sells the same flour in $40$ -ounce boxes for $\$1.50$ per box. How much total money, in dollars, is saved by purchasing the entire $500$ pounds from the more cost-effective supplier instead of the less cost-effective one?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The savings amount to 20 dollars.

Supplier A's total cost for 500 pounds is

280 (20 bags ×

14.00), and Supplier B's total cost is

300 (200 boxes ×

1.50). Purchasing from Supplier A saves $20.

Adım Adım Çözüm

1

Convert the weight of Supplier B's boxes from ounces to pounds.

40

ounces

\div 16

ounces/pound =

2.5

pounds per box.

Measurements must be in the same unit (pounds) to accurately calculate the required number of packages and compare costs.

2

Determine the number of packages required from each supplier to get exactly 500 pounds.

Supplier A:

500 \div 25 = 20

bags. Supplier B:

500 \div 2.5 = 200

boxes.

This establishes the exact purchasing volume needed for the final cost calculation.

3

Calculate the total cost for each supplier.

Supplier A:

20

bags

\times \$14.00

=

\$280

. Supplier B:

200

boxes

\times \$1.50

=

\$300

.

Total cost is the package count multiplied by the price per package.

4

Calculate the difference between the two total costs.

\$300 - \$280 = \$20

.

Subtracting the most cost-effective option from the alternative gives the total savings.

Anahtar Kavram

Calculating and comparing total costs across different measurement units by determining unit prices and required volumes.

Alternatif Yöntem

Instead of computing total costs, calculate the unit price per pound for both. Supplier A:

\$14.00 \div 25 = \$0.56

per pound. Supplier B:

40

oz is

2.5

lbs;

\$1.50 \div 2.5 = \$0.60

per pound. The difference is

\$0.04

per pound. Multiply this difference by the

500

pounds needed:

500 \times \$0.04 = \$20

.

Tahmini Süre:2m 30s

Soru 298Soru →

A naval maintenance crew is preparing an industrial cleaning solution for ship deck equipment. The standard procedure requires mixing chemical solvent and distilled water in a ratio of $3:2$ . If the crew needs to prepare a total of $150$ gallons of the cleaning solution, how many gallons of chemical solvent are required?

60

90

100

225

Cevabı ve açıklamayı göster

Cevap: $90$

Cevap

The required amount of chemical solvent is

90

gallons.

The correct answer accurately determines that the mixture consists of 5 total parts (3 parts solvent + 2 parts water). Dividing the total 150 gallons by 5 yields 30 gallons per part. Multiplying this by the 3 parts needed for the chemical solvent results in 90 gallons.

Adım Adım Çözüm

1

Determine the total number of parts in the ratio.

3 \text{ (solvent)} + 2 \text{ (water)} = 5 \text{ total parts}

To find the fractional share of the whole, you must first establish how many total proportional parts make up the complete mixture.

2

Calculate the volume of a single part by dividing the total volume by the total number of parts.

150 \text{ gallons} \div 5 \text{ parts} = 30 \text{ gallons per part}

This establishes the baseline unit value that can be scaled to find any individual component's volume.

3

Multiply the single part volume by the number of solvent parts required by the ratio.

30 \text{ gallons/part} \times 3 \text{ parts} = 90 \text{ gallons}

The ratio dictates there are 3 parts of solvent for every complete batch of the mixture.

Anahtar Kavram

Calculating a specific part from a total whole using a part-to-part ratio.

Tahmini Süre:1m 0s

Soru 299Soru →

A mechanic needs to buy a new set of socket wrenches. The regular price of the set is $\$150.00$ . If the store is offering a $30\%$ discount on all tools, what is the final sale price of the socket wrench set in dollars?

Cevabı ve açıklamayı göster

Cevap: 105

Cevap

The final sale price is

\$105.00

.

The correct answer is found by first determining the dollar value of the discount. A

30\%

discount on

\$150.00

is calculated as

150 \times 0.30

, which equals

\$45.00

. Subtracting this discount from the original price (

150 - 45

) gives the final sale price of

\$105.00

.

Adım Adım Çözüm

1

Calculate the discount amount.

\$150.00 \times 0.30 = \$45.00

To find out how much money is being taken off the regular price.

2

Subtract the discount amount from the regular price.

\$150.00 - \$45.00 = \$105.00

To determine the final sale price after the discount is applied.

Anahtar Kavram

Calculating a single-step discount and determining the final sale price.

Soru 300Soru →

Consider the algebraic expression below, where $x \neq 0$ and $y \neq 0$ . Which of the following is its fully simplified form?

\left( \frac{3x^{-2}y^3}{2xy^{-2}} \right)^{-2} \times \left( \frac{9x^2}{4y^4} \right)^{-1}

\frac{16}{81x^{12}y^{40}}

\frac{27x^4}{8y^6}

\frac{16x^4}{81y^6}

\frac{16y^2}{81}

Cevabı ve açıklamayı göster

Cevap: $\frac{16x^4}{81y^6}$

Cevap

The expression fully simplifies to

\frac{16x^4}{81y^6}

.

The correct expression is derived by sequentially simplifying the contents of the parentheses using the quotient rule, applying the outer negative exponents to both the numerical coefficients and variables, and then multiplying the resulting terms by adding the exponents of like bases.

Adım Adım Çözüm

1

Simplify the fraction inside the first set of parentheses using the quotient rule for exponents (

a^m / a^n = a^{m-n}

).

\frac{3x^{-2}y^3}{2xy^{-2}} = \frac{3}{2} x^{-2-1} y^{3-(-2)} = \frac{3}{2} x^{-3} y^5

Simplifying inside the parentheses first follows the order of operations and makes subsequent calculations more manageable.

2

Apply the outer negative exponent (

-2

) to all components of the first simplified term, including the numerical coefficient.

\left( \frac{3}{2} x^{-3} y^5 \right)^{-2} = \left( \frac{3}{2} \right)^{-2} (x^{-3})^{-2} (y^5)^{-2} = \frac{4}{9} x^6 y^{-10}

The power of a product rule states that

(ab)^n = a^n b^n

, and applying a negative power to a fraction flips it:

(a/b)^{-n} = (b/a)^n

.

3

Apply the outer negative exponent (

-1

) to the second term.

\left( \frac{9x^2}{4y^4} \right)^{-1} = \frac{4y^4}{9x^2} = \frac{4}{9} x^{-2} y^4

A negative exponent of

-1

simply takes the reciprocal (flips the fraction).

4

Multiply the two resulting expressions by grouping coefficients and using the product rule for exponents (

a^m \times a^n = a^{m+n}

).

\left( \frac{4}{9} x^6 y^{-10} \right) \times \left( \frac{4}{9} x^{-2} y^4 \right) = \left( \frac{4}{9} \times \frac{4}{9} \right) (x^6 \times x^{-2}) (y^{-10} \times y^4) = \frac{16}{81} x^4 y^{-6}

When multiplying terms with the same base, their exponents are added together.

5

Rewrite the final expression using only positive exponents.

\frac{16}{81} x^4 y^{-6} = \frac{16x^4}{81y^6}

Standard mathematical notation requires fully simplified algebraic expressions to be written without negative exponents.

Anahtar Kavram

Simplifying rational expressions using the laws of exponents, including the quotient rule, power rule, product rule, and handling negative exponents alongside numerical coefficients.

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