Tüm alıştırma soruları

Consider the following algebraic expression:

(2x - 5y)^2(x + 2y)

If you were to fully expand this expression and combine all like terms, what would be the numerical coefficient of the $x^2y$ term?

Aşağıdaki boşlukları doldurun

The numerical coefficient of the

x^2y

term is .

Cevabı ve açıklamayı göster

Cevap

-12

The correct answer is derived by first expanding the squared binomial:

(2x - 5y)^2 = 4x^2 - 20xy + 25y^2

. Next, multiply this resulting trinomial by the remaining binomial:

(4x^2 - 20xy + 25y^2)(x + 2y)

. To find the coefficient of the

x^2y

term without expanding the entire expression, identify the term pairs whose product yields

x^2y

. These pairs are

(4x^2)(2y) = 8x^2y

and

(-20xy)(x) = -20x^2y

. Combining these like terms gives

8x^2y - 20x^2y = -12x^2y

. Thus, the numerical coefficient of the

x^2y

term is

-12

.

Adım Adım Çözüm

1

Expand the squared binomial

(2x - 5y)^2

.

(2x)^2 - 2(2x)(5y) + (5y)^2 = 4x^2 - 20xy + 25y^2

Order of operations requires evaluating exponents before multiplying by the binomial

(x + 2y)

.

2

Set up the multiplication of the resulting trinomial and the remaining binomial.

(4x^2 - 20xy + 25y^2)(x + 2y)

Substitute the expanded form back into the original expression to continue the multiplication process.

3

Identify only the specific multiplications that will produce an

x^2y

term.

The term

4x^2

multiplied by

2y

, and the term

-20xy

multiplied by

x

.

To find the coefficient of a specific term, it is most efficient to calculate only the products that result in that exact variable combination.

4

Calculate and combine the identified products to find the final coefficient.

(4x^2)(2y) + (-20xy)(x) = 8x^2y - 20x^2y = -12x^2y

. The numerical coefficient is

-12

.

Combining these like terms gives the total

x^2y

term in the fully expanded polynomial.

Anahtar Kavram

Multiplying Polynomials

Soru 222Soru →

A military supply truck travels at a steady rate of $40\text{ mph}$ for exactly $45$ minutes along a designated supply route. What is the total distance traveled?

30\text{ miles}

18\text{ miles}

300\text{ miles}

1800\text{ miles}

Cevabı ve açıklamayı göster

Cevap: $30\text{ miles}$

Cevap

30\text{ miles}

To find the correct distance, the time must first be converted to match the units of the rate. Dividing

45\text{ minutes}

by

60

gives

0.75\text{ hours}

. Multiplying the rate (

40\text{ mph}

) by the time (

0.75\text{ hours}

) gives exactly

30\text{ miles}

.

Adım Adım Çözüm

1

Convert the travel 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 (mph), so the time must be in hours to cancel out the units properly.

2

Calculate the total distance using the formula: Distance = Rate

\times

Time.

Distance

= 40\text{ mph} \times 0.75\text{ hours} = 30\text{ miles}

Multiplying the steady rate by the total time in hours yields the total distance.

Anahtar Kavram

Calculating Distance with Unit Conversion

Soru 223Soru →

Match each inverse proportion scenario to its correct calculated value.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Each scenario represents an inverse proportion and must be matched with the value that maintains the constant product of the initial conditions.

Each match is mathematically determined by setting up an inverse proportion equation where the product of the initial states equals the product of the final states (

x_1 \cdot y_1 = x_2 \cdot y_2

). Dividing the initial product by the new given value provides the correct corresponding answer.

Adım Adım Çözüm

1

Identify that each scenario represents an inverse relationship.

As one quantity increases (troops, speed, workers, gear teeth), the other quantity must decrease.

This determines that the product of the variables is constant (

x_1 \cdot y_1 = x_2 \cdot y_2

), rather than the ratio.

2

Calculate the constant product (

k

) for the initial conditions in each scenario.

The constants are

15,000

troop-days,

2,400\text{ miles}

,

96\text{ man-hours}

, and

7,200\text{ teeth-RPM}

.

The constant represents the total amount of work, distance, resources, or mechanical transfer that must remain unchanged.

3

Divide the constant product by the new given variable to find the unknown required value.

The resulting values are

20\text{ days}

,

4\text{ hours}

,

6\text{ hours}

, and

160\text{ RPM}

.

Solving the equation

x_1 \cdot y_1 = x_2 \cdot y_2

for the unknown variable

y_2

requires dividing the constant by

x_2

.

Anahtar Kavram

Inverse Proportions

Soru 224Soru →

Determine the missing principal amount based on the given simple interest details. Fill in the blank with the correct numeric value.

Aşağıdaki boşlukları doldurun

A homeowner paid

\$210

in simple interest on a short-term loan that lasted

14

months. If the loan had a

4\%

annual simple interest rate, the original principal amount borrowed was \$.

Cevabı ve açıklamayı göster

Cevap

The original principal amount borrowed was

\$4,500

.

The correct principal is found by rearranging the simple interest formula to

P = \frac{I}{R \times T}

. By converting the

14

months into

\frac{7}{6}

years and using the decimal rate of

0.04

, dividing the

\$210

interest by their product yields the original principal of

\$4,500

.

Adım Adım Çözüm

1

Identify the known values: Interest (

I

), annual Rate (

R

), and Time (

T

) in months.

I = 210

,

R = 0.04

, and

T = 14

months

These are the given variables needed to set up the simple interest equation.

2

Convert the loan duration from months into years.

T = \frac{14}{12} = \frac{7}{6}

years

The simple interest formula requires time to be in years when the given interest rate is annual.

3

Substitute the values into the simple interest formula (

I = PRT

) to isolate Principal (

P

).

210 = P \times 0.04 \times \frac{7}{6}

This establishes the equation with only one unknown variable.

4

Calculate the product of the rate and time, then divide the total interest by this product.

P = \frac{210}{\frac{0.28}{6}} = 210 \times \frac{6}{0.28} = \frac{1260}{0.28} = 4500

Simplifying the expression yields the exact borrowed amount.

Anahtar Kavram

Solving for the principal using the simple interest formula

I = PRT

by converting time in months to a fraction of a year.

Soru 225Soru →

A naval supply ship pumps $3,600$ gallons of fresh water into a destroyer's holding tanks over the course of $4$ hours. At what constant rate, in gallons per hour, is the water being pumped?

Cevabı ve açıklamayı göster

Cevap: 900

Cevap

900

To find the rate in gallons per hour, the total number of gallons (

3,600

) is divided by the total number of hours (

4

), resulting in

900

gallons per hour.

Adım Adım Çözüm

1

Identify the total quantity and the total time given in the scenario.

Total quantity =

3,600

gallons. Total time =

4

hours.

A unit rate requires knowing both the total amount and the corresponding time frame.

2

Divide the total gallons by the total hours to find the unit rate.

3,600 \div 4 = 900

Dividing the total amount by the number of hours yields the amount per single hour.

Anahtar Kavram

Calculating a basic unit rate by dividing a total quantity by a total time.

Soru 226Soru →

A developer issues a short-term promissory note to cover construction costs. The note carries an annual simple interest rate of $8\%$ . After $9$ months, the developer repays a total of $\$42,400$ , which includes both the original principal borrowed and the interest earned. What was the original principal amount borrowed, in dollars?

Cevabı ve açıklamayı göster

Cevap: 40000

Cevap

40000

The original principal is found by utilizing the total amount formula for simple interest,

A = P(1 + rt)

, and solving for

P

. By substituting

A = 42,400

,

r = 0.08

, and

t = 0.75

years (from

9

months), the equation simplifies to

1.06P = 42,400

, which yields a principal of $40,000.

Adım Adım Çözüm

1

Identify the known variables from the word problem.

Total Amount (

A

) =

42,400

, Rate (

r

) =

8\% = 0.08

, Time (

t

) =

9

months.

Extracting the specific numbers and their corresponding variables is required before applying any formulas.

2

Convert the time duration from months into years.

t = \frac{9}{12} = 0.75

years.

Simple interest formulas require the time unit to match the rate unit, which is annual (

8\%

per year).

3

Apply the total amount simple interest formula and substitute the values.

42,400 = P(1 + 0.08 \times 0.75)

The total amount paid back (

42,400

) includes both the principal (

P

) and the interest (

Prt

), factored out as

P(1 + rt)

.

4

Simplify the equation.

42,400 = P(1 + 0.06) \Rightarrow 42,400 = 1.06P

Multiplying the rate by the time yields the total interest multiplier (

6\%

or

0.06

), adding it to

1

gives the total amount multiplier.

5

Solve for the principal (

P

).

P = \frac{42,400}{1.06} = 40,000

Isolating

P

gives the original amount borrowed. The division can be calculated mentally by noting that

424

is

4 \times 106

.

Anahtar Kavram

Solving for Principal in a Simple Interest Total Amount Scenario

Soru 227Soru →

A motor pool sergeant reviews a vehicle readiness report and notes that $\frac{3}{5}$ of the transport trucks are fully operational. What percentage of the transport trucks are fully operational?

60%

6%

37.5%

166.7%

Cevabı ve açıklamayı göster

Cevap: 60%

Cevap

60%

The correct percentage is found by executing the division indicated by the fraction. Dividing the numerator (3) by the denominator (5) gives the decimal 0.6. Multiplying this decimal by 100 scales it to a percentage, giving 60%.

Adım Adım Çözüm

1

Convert the given fraction to a decimal value by dividing the numerator by the denominator.

3 \div 5 = 0.6

Finding the decimal equivalent is the standard intermediate step for converting a fraction to a percentage.

2

Convert the resulting decimal to a percentage by multiplying it by 100.

0.6 \times 100 = 60\%

Percentages represent parts per hundred, so multiplying the decimal representation by 100 yields the correct percentage.

Anahtar Kavram

Converting fractions to percentages via division and decimal multiplication.

Alternatif Yöntem

Recognize that the denominator 5 is a factor of 100. Multiply both the numerator and the denominator by 20 to create an equivalent fraction over 100: (3 × 20) / (5 × 20) = 60 / 100, which is exactly 60%.

Tahmini Süre:45s

Soru 228Soru →

During a base renovation, a carpenter cuts a $5$ -foot wooden board into two pieces. If the lengths of the two pieces are in a ratio of $2:3$ , what is the length of the shorter piece in inches?

24

36

40

2

Cevabı ve açıklamayı göster

Cevap: $24$

Cevap

The shorter piece is

24

inches long.

First, convert the

5

-foot board to inches by multiplying by

12

, which gives

60

inches. The ratio

2:3

means there are

2 + 3 = 5

total equal parts. Dividing the

60

-inch board by

5

gives

12

inches per part. The shorter piece is

2

parts, so

2 \times 12 = 24

inches.

Adım Adım Çözüm

1

Convert the total length of the board from feet to inches.

5 \times 12 = 60

inches.

The question asks for the final answer in inches, so it is easiest to work in inches from the start.

2

Find the total number of parts in the ratio.

2 + 3 = 5

total parts.

A ratio of

2:3

means the whole board is divided into

5

equal segments.

3

Determine the length of one single part.

60 \div 5 = 12

inches per part.

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

4

Calculate the length of the shorter piece.

12 \times 2 = 24

inches.

The shorter piece represents

2

parts of the ratio.

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 number to find the value of one part, and multiply by the relevant ratio term. Always ensure units are consistent.

Soru 229Soru →

A naval shipyard receives a shipment of $4,500$ tons of high-grade steel for vessel construction. The shipyard allocates $36\%$ of this steel for hull plating. Of the remaining steel, $45\%$ is used for interior bulkheads. Calculate the exact number of tons of steel that will be placed into the strategic reserve.

Aşağıdaki boşlukları doldurun

The shipyard will place tons of steel into the strategic reserve.

Cevabı ve açıklamayı göster

Cevap

1,584

The problem requires calculating percentages sequentially based on updated totals. After taking

36\%

of the original

4,500

tons for the hull,

64\%

(

2,880

tons) remains. The next allocation of

45\%

for bulkheads is taken from this

2,880

tons, leaving

55\%

of it for the reserve. Multiplying

2,880

by

0.55

gives the final reserve amount of

1,584

tons.

Adım Adım Çözüm

1

Calculate the percentage of the original steel shipment that remains after the hull plating allocation.

100\% - 36\% = 64\%

This establishes the base percentage for the next calculation without needing to compute the exact tonnage immediately.

2

Calculate the exact tonnage of the remaining steel.

4,500 \times 0.64 = 2,880

tons

The interior bulkheads are allocated from this remaining amount, not the original

4,500

tons.

3

Determine the percentage of the remaining steel that will go into the strategic reserve.

100\% - 45\% = 55\%

Since

45\%

of the remaining steel is used for bulkheads, the rest (

55\%

) goes to the reserve.

4

Calculate the final reserve tonnage based on the remaining steel.

2,880 \times 0.55 = 1,584

tons

This yields the final quantity of steel placed into the strategic reserve.

Anahtar Kavram

Calculating successive percentages of remaining quantities in multi-step scenarios.

Tahmini Süre:1m 30s

Soru 230Soru →

Read the scenario below and calculate the total number of complete IV bags the unit can prepare. Fill in the blank with the correct numerical value.

Aşağıdaki boşlukları doldurun

A field medical unit has a stockpile of

33 \frac{1}{3}

liters of a specialized saline solution. If each intravenous (IV) drip bag requires exactly

\frac{5}{9}

of a liter of the solution, the unit can prepare a total of complete IV bags.

Cevabı ve açıklamayı göster

Cevap

60

The unit needs to distribute the total volume of saline solution into smaller, equal-sized portions. This is an application of division. By converting the total volume into an improper fraction (

\frac{100}{3}

) and multiplying it by the reciprocal of the portion size (

\frac{9}{5}

), we determine that exactly 60 bags can be prepared.

Adım Adım Çözüm

1

Convert the mixed number

33 \frac{1}{3}

to an improper fraction.

\frac{100}{3}

Improper fractions are required to perform multiplication or division with mixed numbers.

2

Set up the division expression to find how many

\frac{5}{9}

-liter bags can be made from

\frac{100}{3}

liters of solution.

\frac{100}{3} \div \frac{5}{9}

Dividing the total volume by the volume required per individual bag yields the total count of complete bags.

3

Multiply the total volume by the reciprocal of the volume per bag.

\frac{100}{3} \times \frac{9}{5} = 60

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal. Simplifying cross-terms (

100 \div 5 = 20

and

9 \div 3 = 3

) results in

20 \times 3 = 60

.

Anahtar Kavram

Dividing a mixed number by a proper fraction to determine a quantity of equal parts.

Soru 231Soru →

A furniture restorer buys an antique dining set for $\$250$ . The restorer spends $\$80$ on wood stain and varnish, and pays an upholsterer $\$170$ to repair the chair cushions. After the restoration is complete, the dining set is sold for $\$800$ . What is the restorer's profit as a percentage of their total cost?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60%

By summing the initial price and all repair costs, we find the total investment is

500. Subtracting this from the

800 selling price yields a

300 profit. The profit is then divided by the total cost (

300 / $500) to find the 60% profit.

Adım Adım Çözüm

1

Calculate the total cost of the dining set project.

\$250 + \$80 + \$170 = \$500

The total cost must include the initial purchase price plus any additional expenses incurred before selling the item.

2

Calculate the total profit made on the sale.

\$800 - \$500 = \$300

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

3

Calculate the profit as a percentage of the total cost.

\frac{\$300}{\$500} \times 100 = 60\%

Percentage profit is found by dividing the dollar amount of profit by the original total cost and converting to a percentage.

Anahtar Kavram

Calculating profit percentage after accounting for overhead and multi-step production costs.

Tahmini Süre:1m 15s

Soru 232Soru →

A quartermaster is reviewing the completion status of four different supply warehouse inventories. To prioritize manpower, arrange the warehouses from the least completed inventory to the most completed inventory.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

Warehouse Charlie, Warehouse Delta, Warehouse Alpha, Warehouse Bravo

By converting all values to percentages (

35\%

,

40\%

,

25\%

, and

30\%

), we can easily compare them. The lowest value is

25\%

(Charlie), followed by

30\%

(Delta), then

35\%

(Alpha), and the highest is

40\%

(Bravo).

Adım Adım Çözüm

1

Convert all completion statuses to a common format, such as percentages, for direct comparison.

Warehouse Alpha =

35\%

, Warehouse Bravo =

40\%

.

Comparing numbers in different formats is difficult and prone to error.

2

Convert the fractions for Warehouse Charlie and Warehouse Delta to percentages.

Warehouse Charlie:

\frac{1}{4} = 25\%

. Warehouse Delta:

\frac{3}{10} = 30\%

.

Fractions with denominators of

4

and

10

can be easily converted to percentages.

3

Order the converted percentage values from least to greatest.

25\% < 30\% < 35\% < 40\%

.

The question asks for the order from least completed to most completed.

4

Match the ordered percentages back to their respective warehouses.

Charlie (

25\%

), Delta (

30\%

), Alpha (

35\%

), Bravo (

40\%

).

This provides the final required sequence.

Anahtar Kavram

Converting fractions and decimals to percentages to compare their values.

Tahmini Süre:45s

Soru 233Soru →

A delivery driver earns a base pay of $80.00 per day plus$ 3.50 for each package delivered. If the driver delivers 40 packages in one day, what is the driver's total gross pay for that day, in dollars?

Cevabı ve açıklamayı göster

Cevap: 220

Cevap

220

The total gross pay is the sum of the fixed base pay (

80.00) and the variable delivery earnings (40 packages ×

3.50 per package =

140.00), which yields a total of

220.00.

Adım Adım Çözüm

1

Calculate earnings from package deliveries

$140.00

The driver earns $3.50 for each of the 40 packages delivered (40 × 3.50).

2

Calculate total gross pay

$220.00

Add the base daily pay (

80.00) to the delivery earnings (

140.00) to find the total amount earned for the day.

Anahtar Kavram

Calculating total gross pay using a combination of base pay and piece-rate pay

Tahmini Süre:45s

Soru 234Soru →

A military detail fills $300$ sandbags in $150$ minutes. What is the constant rate at which the detail fills the sandbags, in sandbags per hour?

120

2

12

0.5

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

The unit rate is found by first converting the time into hours (

150

minutes is

2.5

hours) and then dividing the total quantity (

300

sandbags) by this time, resulting in

120

sandbags per hour.

Adım Adım Çözüm

1

Convert the total time from minutes to hours.

150 \text{ minutes} \div 60 \text{ minutes/hour} = 2.5 \text{ hours}

The requested unit rate is in sandbags per hour, so the time must be expressed in hours.

2

Calculate the unit rate by dividing the total number of sandbags by the total time in hours.

300 \text{ sandbags} \div 2.5 \text{ hours} = 120 \text{ sandbags per hour}

A unit rate represents the quantity produced per single unit of time.

Anahtar Kavram

Calculating unit rates with time conversion

Soru 235Soru →

A homeowner secures a personal bank loan of $\$4,500$ to renovate their kitchen. The loan agreement specifies an annual simple interest rate of $8\%$ . If the homeowner repays the entire loan in exactly $9$ months, what is the total amount they must repay, including both principal and interest?

\$4,527

\$4,770

\$4,860

\$7,740

Cevabı ve açıklamayı göster

Cevap: $\$4,770$

Cevap

The total repayment amount is

\$4,770

.

The correct answer accurately calculates the interest for exactly

9

months (

\$270

) and adds it to the original principal (

\$4,500

) to yield the total repayment of

\$4,770

.

Adım Adım Çözüm

1

Convert the annual interest rate from a percentage to a decimal.

8\% = 0.08

Percentages must be expressed as decimals to be used in the simple interest formula.

2

Convert the loan term from months into years.

9 \text{ months} = \frac{9}{12} \text{ years} = 0.75 \text{ years}

Since the interest rate is annual, the time variable in the formula must also be expressed in years.

3

Calculate the total simple interest using the formula

I = PRT

.

I = \$4,500 \times 0.08 \times 0.75 = \$270

This formula determines the total cost of borrowing the principal over the specified time period.

4

Calculate the total repayment amount by adding the principal and the interest.

A = \$4,500 + \$270 = \$4,770

The total repayment requires paying back the original loan amount plus the accrued interest.

Anahtar Kavram

Simple Interest Applications

Soru 236Soru →

A naval submarine travels at a steady surface speed of $15\text{ knots}$ (nautical miles per hour) for $2\text{ hours and } 20\text{ minutes}$ before submerging. What is the total distance traveled during this time?

35\text{ nautical miles}

33\text{ nautical miles}

30\text{ nautical miles}

350\text{ nautical miles}

Cevabı ve açıklamayı göster

Cevap: $35\text{ nautical miles}$

Cevap

35\text{ nautical miles}

The correct answer is found by converting the travel time of

2\text{ hours and } 20\text{ minutes}

into

2\frac{1}{3}\text{ hours}

, and then multiplying this time by the speed of

15\text{ knots}

. The calculation is

15 \times \frac{7}{3} = 35\text{ nautical miles}

.

Adım Adım Çözüm

1

Convert the travel time into hours.

2\text{ hours and } 20\text{ minutes} = 2\frac{20}{60}\text{ hours} = 2\frac{1}{3}\text{ hours}

.

Speed is given in knots (nautical miles per hour), so time must be uniformly converted to hours to compute distance accurately.

2

Multiply the speed by the time to find the total distance.

\text{Distance} = 15 \times 2\frac{1}{3} = 15 \times \frac{7}{3} = 35\text{ nautical miles}

.

The formula for distance is

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

.

Anahtar Kavram

Calculating distance by converting minutes into a fraction of an hour and multiplying by the rate of speed.

Soru 237Soru →

Determine the total gross earnings based on the provided compensation structure.

Aşağıdaki boşlukları doldurun

A regional manager for a logistics company earns a base monthly salary of

\$3,200

. She receives a

5\%

commission on her first

\$40,000

in logistics contracts, and an

8\%

commission on any contract sales exceeding

\$40,000

within the month. If she secures

\$65,000

in contract sales this month, her total gross earnings will be $\$$.

Cevabı ve açıklamayı göster

Cevap

7200

The correct total is found by breaking the earnings into three parts: the

3,200 base salary, the 5% commission on the first

40,000 (

2,000), and the 8% commission on the remaining

25,000 (

2,000). Adding these together yields

7,200.

Adım Adım Çözüm

1

Calculate the commission earned on the first tier of sales.

0.05 \times 40,000 = \$2,000

The manager earns a 5% commission on the first $40,000 in sales.

2

Determine the amount of sales that fall into the second commission tier.

65,000 - 40,000 = \$25,000

Sales exceeding the $40,000 threshold are subject to a different commission rate.

3

Calculate the commission earned on the second tier of sales.

0.08 \times 25,000 = \$2,000

The manager earns an 8% commission on the $25,000 that exceeded the initial threshold.

4

Sum the base salary and all commissions to find the total gross earnings.

3,200 + 2,000 + 2,000 = \$7,200

Total earnings equal the base salary plus the sum of all tiered commissions.

Anahtar Kavram

Calculating total compensation using a base salary combined with a tiered commission structure.

Soru 238Soru →

A naval shipyard assigns $15$ automated painting drones to coat the hull of a cruiser in exactly $2\text{ hours}$ . If $5$ of these drones are taken offline for maintenance before the job begins, how many minutes will it take the remaining drones to complete the coating?

80

360

180

3

Cevabı ve açıklamayı göster

Cevap: $180$

Cevap

It will take 180 minutes for the remaining drones to complete the hull coating.

The correct answer is

180

. First, subtract the

5

offline drones from the initial

15

to find that

10

drones are actively working. Because the number of drones and the time required are inversely proportional, their product remains constant. The equation is

15 \times 2 = 10 \times T_2

, which solves to

T_2 = 3\text{ hours}

. Finally, converting

3\text{ hours}

to minutes yields

3 \times 60 = 180\text{ minutes}

.

Adım Adım Çözüm

1

Determine the number of active drones.

15 - 5 = 10\text{ drones}

The 5 offline drones cannot work, so only the remaining drones will paint the hull.

2

Set up an inverse proportion equation.

15\text{ drones} \times 2\text{ hours} = 10\text{ drones} \times T_2

Work rate problems involve inverse proportions; as the number of workers decreases, the time required increases proportionally to keep the total work constant.

3

Solve for the new time (

T_2

) in hours.

30 = 10 \times T_2 \Rightarrow T_2 = 3\text{ hours}

Dividing the total required drone-hours (

30

) by the number of active drones (

10

) isolates the time.

4

Convert the required time into minutes.

3 \times 60 = 180\text{ minutes}

The question explicitly asks for the final answer in minutes, not hours.

Anahtar Kavram

Inverse Proportions

Soru 239Soru →

For the inequality below, which of the following expresses all possible values of $y$ ?

2(3y - 5) - 3(y - 4) \geq 6y - 16

y \leq 6

y \geq 6

y \leq -2

y \geq -2

Cevabı ve açıklamayı göster

Cevap: $y \leq 6$

Cevap

The correct answer is

y \leq 6

.

The correct answer is derived by distributing the terms to get

6y - 10 - 3y + 12 \geq 6y - 16

, which simplifies to

3y + 2 \geq 6y - 16

. Subtracting

6y

from both sides and subtracting 2 from both sides yields

-3y \geq -18

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

y \leq 6

.

Adım Adım Çözüm

1

Distribute the constants on the left side of the inequality.

6y - 10 - 3y + 12 \geq 6y - 16

To clear the parentheses. Note that multiplying -3 by -4 results in a positive 12.

2

Combine the like terms on the left side.

3y + 2 \geq 6y - 16

To simplify the expression before isolating the variable.

3

Move all variable terms to the left side and constant terms to the right side.

3y - 6y \geq -16 - 2

, which simplifies to

-3y \geq -18

To group the unknown variable on one side of the inequality.

4

Divide both sides by -3.

y \leq 6

To solve for y. Whenever multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.

Anahtar Kavram

Solving One-Variable Inequalities

Soru 240Soru →

A contractor purchases a bulk order of electrical wiring from a wholesale supply store. The store offers a $25\%$ discount on all orders over $\$500$ . Because the contractor is a loyal member, they receive an additional $10\%$ off the already discounted price. After both discounts are applied, an $8\%$ sales tax is added. If the original price of the electrical wiring order was $\$800$ , what is the final total amount the contractor pays?

\$561.60

\$583.20

\$604.00

\$826.20

Cevabı ve açıklamayı göster

Cevap: $\$583.20$

Cevap

The final total amount the contractor pays is

\$583.20

.

The final amount must be calculated sequentially. First, apply the

25\%

discount to

\$800

, resulting in

\$600

. Then, apply the

10\%

loyal member discount to

\$600

, resulting in

\$540

. Finally, calculate

8\%

sales tax on the

\$540

final price (

\$540 \times 0.08 = \$43.20

) and add it to the price to get $\$583.20.

Adım Adım Çözüm

1

Calculate the first

25\%

discount on the original

\$800

price.

\$800 \times 0.25 = \$200

discount. The new price is

\$800 - \$200 = \$600

.

The first discount is applied to the original wholesale price.

2

Calculate the second

10\%

loyal member discount on the new price of

\$600

.

\$600 \times 0.10 = \$60

discount. The new price is

\$600 - \$60 = \$540

.

Successive discounts are always applied to the previously discounted amount, not the original price.

3

Calculate the

8\%

sales tax on the final discounted price and add it to the total.

\$540 \times 0.08 = \$43.20

tax. The final total is

\$540 + \$43.20 = \$583.20

.

Sales tax is applied to the final amount after all discounts have been deducted.

Anahtar Kavram

Calculating successive discounts and applying sales tax to the final discounted price.

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