Tüm alıştırma soruları

Soru 821Soru →

Train X and Train Y depart simultaneously from Station A and Station B, respectively, heading toward each other along a straight, 360-mile track. Train X travels at a constant speed of 40 miles per hour for the first 2 hours, after which its speed increases by 50 percent. Train Y travels at a constant speed of 50 miles per hour for the first hour, stops for 30 minutes due to a signal delay, and then resumes its journey at a constant speed of 70 miles per hour. How many hours after their simultaneous departure will the two trains meet?

Cevabı ve açıklamayı göster

Cevap: 3.5

Cevap

3.5 hours

Evaluating distance piecewise up to

t = 2

hours shows Train X covers 80 miles and Train Y covers 85 miles, leaving 195 miles between them. After

t = 2

hours, their relative rate of convergence is

60 + 70 = 130

mph. Dividing 195 miles by 130 mph yields 1.5 additional hours, bringing the total time since departure to 3.5 hours.

Adım Adım Çözüm

1

Determine distance traveled by Train X in the first 2 hours and its new speed.

Train X distance =

40 \text{ mph} \times 2 \text{ hours} = 80 \text{ miles}

. New speed =

40 \times (1 + 0.50) = 60 \text{ mph}

.

Train X increases its speed by 50% after the 2-hour mark.

2

Determine distance traveled by Train Y during the first 2 hours.

In hour 1 (

t=0

to

t=1

):

50 \text{ mph} \times 1 \text{ h} = 50 \text{ miles}

. During delay (

t=1

to

t=1.5

):

0 \text{ miles}

. From

t=1.5

to

t=2.0

:

70 \text{ mph} \times 0.5 \text{ h} = 35 \text{ miles}

. Total distance =

50 + 0 + 35 = 85 \text{ miles}

.

Accounting for Train Y's initial leg, 30-minute delay, and new speed up to

t = 2

hours.

3

Find the remaining distance separating the two trains at

t = 2

hours.

Total distance covered by both trains =

80 + 85 = 165 \text{ miles}

. Remaining distance =

360 - 165 = 195 \text{ miles}

.

Subtracting total distance covered by both trains from the track length of 360 miles.

4

Calculate the time required to cover the remaining distance using relative speed.

Relative closing speed for

t > 2

is

60 + 70 = 130 \text{ mph}

. Additional time =

\frac{195}{130} = 1.5 \text{ hours}

.

When two objects move toward each other, their rates add together to find the closing rate.

5

Calculate total elapsed time from departure.

Total time =

2.0 + 1.5 = 3.5 \text{ hours}

.

Combining the initial 2-hour evaluation window with the additional 1.5 hours required to meet.

Anahtar Kavram

Relative speed and piecewise distance calculations for converging vehicles with variable speeds and delays.

Tahmini Süre:3m 0s

Soru 822Soru →

At a culinary institute, a cohort of $250$ professional chefs was evaluated on three specialized culinary skills: Molecular Gastronomy, Pastry Arts, and Artisanal Baking. Each chef in the cohort possessed proficiency in at least one of the three skills. According to the evaluation report:

- $150$ chefs were proficient in Molecular Gastronomy
- $140$ chefs were proficient in Pastry Arts
- $130$ chefs were proficient in Artisanal Baking
- The number of chefs proficient in all three skills was $30$

How many chefs in the cohort were proficient in EXACTLY ONE of the three skills?

80

90

110

140

170

Cevabı ve açıklamayı göster

Cevap: 110

Cevap

110 chefs were proficient in exactly one of the three skills.

The correct answer of 110 is derived by setting up two fundamental 3-set relationships. First, the total number of chefs is the sum of those proficient in exactly one skill (

x

), exactly two skills (

y

), and all three skills (

z = 30

), giving

x + y + 30 = 250

, so

x + y = 220

. Second, the sum of individual set sizes counts elements according to how many sets they belong to:

150 + 140 + 130 = x + 2y + 3(30)

, which simplifies to

x + 2y = 330

. Subtracting the first equation from the second yields

y = 110

, and substituting

y = 110

back into

x + y = 220

gives

x = 110

.

Adım Adım Çözüm

1

Define variables for the regions of the 3-set Venn diagram

Let

x

be the number of chefs proficient in exactly 1 skill,

y

be the number proficient in exactly 2 skills, and

z

be the number proficient in all 3 skills (

z = 30

).

Categorizing elements by the exact number of sets they belong to simplifies the set inclusion-exclusion equations.

2

Set up the total population equation

x + y + z = 250 \implies x + y + 30 = 250 \implies x + y = 220

Since every chef is proficient in at least one skill, the sum of all distinct regions equals the total number of chefs (

250

).

3

Set up the sum of individual set sizes equation

150 + 140 + 130 = x + 2y + 3z \implies 420 = x + 2y + 3(30) \implies x + 2y = 330

Summing the individual set totals counts elements in exactly one set once, elements in exactly two sets twice, and elements in all three sets three times.

4

Solve the system of two linear equations for

x

and

y

Subtracting Equation 1 from Equation 2 gives

(x + 2y) - (x + y) = 330 - 220 \implies y = 110

. Substituting

y = 110

back into Equation 1 gives

x + 110 = 220 \implies x = 110

.

Finding

x

directly answers the question regarding chefs proficient in exactly one skill.

Anahtar Kavram

Overlapping Sets (3-Set Inclusion-Exclusion and Region Partitioning)

Tahmini Süre:2m 0s

Soru 823Soru →

The table below categorizes 100 analysts at a consulting firm by their department and experience level:

Department	Junior (1–3 yrs)	Senior (4–7 yrs)	Lead (8+ yrs)	Total
Technology	14	16	10	40
Analytics	12	18	5	35
Operations	9	11	5	25
Total	35	45	20	100

If one analyst is selected at random from this group, what is the probability that the selected analyst works in the Analytics department or has Lead experience, but NOT both?

Cevabı ve açıklamayı göster

Cevap: 0.45

Cevap

0.45

To find the probability of selecting an analyst who is either in the Analytics department or at the Lead level, but not both, we count the analysts in Analytics who are not Leads (

12 + 18 = 30

) and the analysts at the Lead level who are not in Analytics (

10 + 5 = 15

). The total number of favorable outcomes is

30 + 15 = 45

. Dividing by the total pool of

100

analysts yields a probability of

\frac{45}{100} = 0.45

.

Adım Adım Çözüm

1

Determine the total size of the sample space.

Total analysts

N = 100

.

Basic probability requires dividing favorable outcomes by total possible outcomes.

2

Calculate the number of analysts satisfying 'Analytics, but NOT Lead'.

Junior Analytics (

12

) + Senior Analytics (

18

) =

30

.

Excludes the

5

Lead analysts in the Analytics department.

3

Calculate the number of analysts satisfying 'Lead, but NOT Analytics'.

Technology Lead (

10

) + Operations Lead (

5

) =

15

.

Excludes the

5

Lead analysts in the Analytics department.

4

Sum the non-overlapping favorable counts and compute probability.

Favorable outcomes

= 30 + 15 = 45

; Probability

= \frac{45}{100} = 0.45

.

Probability of a single event is defined as

\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}

.

Anahtar Kavram

Basic Single-Event Probability from Two-Way Tabular Data with Mutually Exclusive Set Conditions

Soru 824Soru →

The roots of the quadratic polynomial $P(x) = x^2 + bx + c$ are $r_1$ and $r_2$ , where $b$ and $c$ are constants. If $r_1 + 2$ and $r_2 + 2$ are the roots of the quadratic equation $x^2 - 10x + 21 = 0$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

c

is 5.

Factoring

x^2 - 10x + 21 = 0

yields roots

3

and

7

. Because these roots represent

r_1 + 2

and

r_2 + 2

, subtracting

2

from each root gives the original roots

r_1 = 1

and

r_2 = 5

. In the monic polynomial

P(x) = x^2 + bx + c

, the constant coefficient

c

equals the product of the roots

r_1 \times r_2 = 1 \times 5 = 5

.

Adım Adım Çözüm

1

Solve for the roots of the given equation

x^2 - 10x + 21 = 0

The roots are

3

and

7

.

Factoring the quadratic expression yields

(x - 3)(x - 7) = 0

, so

x = 3

or

x = 7

.

2

Determine the roots

r_1

and

r_2

of

P(x)

r_1 = 1

and

r_2 = 5

.

Since the roots of the second equation are shifted by

+2

, we set

r_1 + 2 = 3 \implies r_1 = 1

and

r_2 + 2 = 7 \implies r_2 = 5

.

3

Calculate the constant term

c

c = 5

.

By Vieta's formulas, for any monic quadratic polynomial

x^2 + bx + c

, the constant term

c

is the product of the roots

r_1 r_2 = 1 \times 5 = 5

.

Anahtar Kavram

Root transformation of quadratic equations and relationship between roots and coefficients via Vieta's formulas.

Soru 825Soru →

A financial firm consists of three departments: Analytics, Trading, and Research. The ratio of the number of analysts in the Analytics department to traders in the Trading department is $2:3$ . The average annual bonus is $\$90,000$ for an analyst in the Analytics department, $\$60,000$ for a trader in the Trading department, and $\$45,000$ for a researcher in the Research department. If the Research department has $40$ researchers and the average annual bonus across all employees in all three departments combined is $\$64,000$ , how many analysts are in the Analytics department?

19

38

40

57

95

Cevabı ve açıklamayı göster

Cevap: 38

Cevap

The Analytics department has 38 analysts.

Using the given ratio

2:3

, let the number of analysts be

2k

and traders be

3k

. The weighted average bonus for these two groups is

\frac{2k(90,000) + 3k(60,000)}{5k} = \$72,000

. Combining this sub-group with the 40 researchers earning

\$45,000

gives an overall average of

\frac{72,000(5k) + 45,000(40)}{5k + 40} = 64,000

. Solving

\frac{360k + 1800}{5k + 40} = 64

yields

360k + 1800 = 320k + 2560

, so

40k = 760

and

k = 19

. The number of analysts is

2(19) = 38

.

Adım Adım Çözüm

1

Express the number of employees in Analytics and Trading in terms of a variable

k

.

Let

N_{\text{Analytics}} = 2k

and

N_{\text{Trading}} = 3k

. The total number of employees in these two departments is

5k

.

The ratio of analysts to traders is given as

2:3

.

2

Calculate the combined average bonus for the Analytics and Trading departments.

Combined average =

\frac{2k(90,000) + 3k(60,000)}{5k} = \frac{180,000k + 180,000k}{5k} = \frac{360,000k}{5k} = \$72,000

.

Weighting each department's average bonus by its relative ratio simplifies the calculation.

3

Set up the weighted average equation combining all three departments.

\frac{72,000(5k) + 45,000(40)}{5k + 40} = 64,000

The total bonus pool divided by the total number of employees across all three departments equals the overall average bonus of

\$64,000

.

4

Solve for

k

and find the number of analysts.

Dividing all terms by

1,000

yields

\frac{360k + 1800}{5k + 40} = 64 \implies 360k + 1800 = 320k + 2560 \implies 40k = 760 \implies k = 19

. Therefore,

N_{\text{Analytics}} = 2k = 2(19) = 38

.

Linear algebraic simplification isolates

k

to compute the requested quantity.

Anahtar Kavram

Weighted Averages with Compound Ratios

Alternatif Yöntem

Use the scale-balance method for weighted averages: The combined average of Analytics and Trading is

\$72,000

. The Research average is

\$45,000

, and the target combined average is

\$64,000

. The distances from the target are

(72,000 - 64,000) = 8,000

and

(64,000 - 45,000) = 19,000

. The ratio of group sizes

(N_{\text{Analytics}+\text{Trading}} : N_{\text{Research}})

is inversely proportional to these distances:

5k : 40 = 19,000 : 8,000 = 19 : 8

. Cross-multiplying gives

40k = 760 \implies k = 19

, so analysts

= 2k = 38

.

Tahmini Süre:2m 0s

Soru 826Soru →

A quality control analyst randomly selects one component from a shipment containing components labeled with distinct integer batch numbers from $21$ to $100$ , inclusive. What is the probability that the batch number of the selected component is a prime number?

\frac{1}{5}

\frac{17}{80}

\frac{17}{79}

\frac{9}{40}

\frac{17}{63}

Cevabı ve açıklamayı göster

Cevap: $\frac{17}{80}$

Cevap

\frac{17}{80}

The total number of batch numbers from 21 through 100 inclusive is given by

100 - 21 + 1 = 80

. The prime numbers in this range are 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97, which count to 17 prime numbers. Dividing the 17 favorable outcomes by the 80 total outcomes gives a probability of

\frac{17}{80}

.

Adım Adım Çözüm

1

Determine the total number of components (the sample space size).

Total components

N = 100 - 21 + 1 = 80

.

For an inclusive range of integers from

a

to

b

, the number of elements is

b - a + 1

.

2

Identify all prime numbers between

21

and

100

, inclusive.

The prime numbers in this range are:

23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

. There are

17

prime numbers in total.

Each listed number has exactly two distinct positive divisors:

1

and itself.

3

Calculate the probability of selecting a prime batch number.

P(\text{Prime}) = \frac{\text{Number of Prime Batch Numbers}}{\text{Total Number of Batch Numbers}} = \frac{17}{80}

.

Probability of a single event is the ratio of favorable outcomes to total possible outcomes.

Anahtar Kavram

Basic Single-Event Probability

Soru 827Soru →

A commercial bakery operates two automated production lines, Line A and Line B, to fulfill a production order for $B$ boxes of goods:

- Line A operates at a constant rate of $r$ boxes per hour ( $r > 0$ ) for $t$ hours.
- Line B operates at a constant rate that is $25\%$ higher than Line A's rate and works for $2$ hours longer than Line A.
- Line B produces twice as many boxes as Line A during their respective operational periods.
- The total contract volume $B$ is the sum of the boxes produced by Line A and Line B.

Which of the following statements MUST be true regarding this algebraic model? (Select ALL that apply.)

Geçerli olan tümünü seçin

Line A operates for

t = \frac{10}{3}

hours (3 hours and 20 minutes).

The total contract volume

B

is equal to

10r

boxes.

The overall average production rate over the combined total operating time of both lines is

1.125r

boxes per hour.

If both lines were to work together simultaneously to complete the total order of

B

boxes, the required time would equal the sum of their individual operating times (

\frac{26}{3}

hours).

Dividing both sides of the equation

1.25r(t + 2) = 2rt

by

r

loses

r = 0

as a physically valid non-zero rate solution.

Cevabı ve açıklamayı göster

Cevap: Line A operates for $t = \frac{10}{3}$ hours (3 hours and 20 minutes).; The total contract volume $B$ is equal to $10r$ boxes.

Cevap

The correct statements are the one establishing Line A's operating time as 10/3 hours and the one establishing the total volume as 10r boxes.

The model yields

t = \frac{10}{3}

hours by equating

1.25r(t + 2) = 2rt

and dividing by

r

. Substituting

t = \frac{10}{3}

into the total output expression

B = rt + 1.25r(t + 2)

gives

B = \frac{10}{3}r + \frac{20}{3}r = 10r

. Thus, both the operating time statement for Line A and the total contract volume statement are mathematically true.

Adım Adım Çözüm

1

Formulate algebraic expressions for the output of each production line.

Line A output:

Q_A = r \cdot t

. Line B rate is

1.25r

and operating time is

t + 2

, so Line B output:

Q_B = 1.25r(t + 2)

.

Word problem translation requires modeling quantities produced as rate multiplied by time.

2

Set up the relational equation based on the condition that Line B produces twice as much as Line A.

1.25r(t + 2) = 2(r \cdot t) \implies 1.25r(t + 2) = 2rt

.

Translates the given relationship

Q_B = 2 Q_A

into a single variable equation for

t

.

3

Solve for

t

by dividing out non-zero

r

.

1.25(t + 2) = 2t \implies 1.25t + 2.5 = 2t \implies 0.75t = 2.5 \implies t = \frac{2.5}{0.75} = \frac{10}{3}

hours.

Since rate

r > 0

, dividing by

r

isolates

t

.

4

Calculate individual outputs and total contract volume

B

.

Q_A = r \cdot \frac{10}{3} = \frac{10}{3}r

.

Q_B = 2 Q_A = \frac{20}{3}r

. Total volume

B = Q_A + Q_B = \frac{10}{3}r + \frac{20}{3}r = 10r

.

Summing the output of both lines gives the exact total contract volume in terms of rate

r

.

Anahtar Kavram

Algebraic Modeling of Multi-Rate Work Systems

Soru 828Soru →

A container holds a solution of acid and water that is $50\%$ acid by volume. First, $20$ liters of the solution are removed and replaced with $20$ liters of pure water, resulting in a solution that is $40\%$ acid by volume. Next, $25$ liters of this $40\%$ solution are removed and replaced with $25$ liters of pure acid. What is the percentage of acid, by volume, in the final solution?

47.5\%

52.5\%

55\%

65\%

70\%

Cevabı ve açıklamayı göster

Cevap: $55\%$

Cevap

The final concentration of acid by volume is

55\%

.

The solution requiring

55\%

concentration is correct. From the first replacement step, we establish that the total container volume is

100

liters (since removing

20

liters of

50\%

solution leaves

0.50V - 10

liters of acid, and

\frac{0.50V - 10}{V} = 0.40

yields

V = 100

). Before the second replacement, the container holds

40

liters of acid. Removing

25

liters of this

40\%

solution removes

10

liters of acid, leaving

30

liters of acid in

75

liters of solution. Adding

25

liters of pure acid increases the acid amount to

55

liters in a total volume of

100

liters, producing a final concentration of

55\%

.

Adım Adım Çözüm

1

Determine the total volume of the container (

V

) using the first replacement step.

Initial acid volume is

0.50V

. Removing

20

liters of solution removes

0.50 \times 20 = 10

liters of acid. Replacing with

20

liters of pure water restores total volume to

V

with an acid content of

0.50V - 10

. Setting up the new concentration equation:

\frac{0.50V - 10}{V} = 0.40 \implies 0.50V - 10 = 0.40V \implies 0.10V = 10 \implies V = 100

liters.

Finding the fixed total volume of the container is required to calculate the exact amounts of acid removed and added in subsequent steps.

2

Calculate the amount of acid remaining after removing

25

liters of the

40\%

solution.

Before the second step, the container has

100

liters of

40\%

acid solution, which contains

0.40 \times 100 = 40

liters of acid. Removing

25

liters of this solution removes

0.40 \times 25 = 10

liters of acid. Acid remaining in the container

= 40 - 10 = 30

liters.

When solution is removed, both solute (acid) and solvent (water) are removed proportional to their concentration.

3

Calculate the final acid concentration after adding

25

liters of pure acid.

Adding

25

liters of pure (

100\%

) acid adds exactly

25

liters of acid. Total final acid

= 30 + 25 = 55

liters. Total final volume

= (100 - 25) + 25 = 100

liters. Final concentration

= \frac{55}{100} \times 100\% = 55\%

.

The new concentration is the total volume of acid divided by the total volume of the solution.

Anahtar Kavram

Multi-stage removal and replacement in mixture problems

Tahmini Süre:2m 0s

Soru 829Soru →

A technology firm consists of two divisions: Division X and Division Y. Division X has $20$ employees with an average monthly salary of $\$5{}000$ , while Division Y has $30$ employees with an average monthly salary of $\$6{}000$ . What is the average monthly salary of all $50$ employees combined?

\$5{}400

\$5{}500

\$5{}600

\$5{}700

\$5{}800

Cevabı ve açıklamayı göster

Cevap: $\$5{}600$

Cevap

The combined average monthly salary for all

50

employees is

\$5{}600

.

The total salary for Division X is

20 \times \$5{}000 = \$100{}000

and for Division Y is

30 \times \$6{}000 = \$180{}000

. Combining both gives a total payout of

\$280{}000

across

50

employees. Dividing

\$280{}000

by

50

yields an exact weighted average of

\$5{}600

.

Adım Adım Çözüm

1

Calculate total monthly salary expenditure for Division X

20 \times \$5{}000 = \$100{}000

Total value of a set equals the number of items times the average value per item.

2

Calculate total monthly salary expenditure for Division Y

30 \times \$6{}000 = \$180{}000

Total value of a set equals the number of items times the average value per item.

3

Sum total monthly salary expenditures and total number of employees

Total salary =

\$100{}000 + \$180{}000 = \$280{}000

; Total employees =

20 + 30 = 50

Combined set averages require combined total sum divided by combined total quantity.

4

Compute the combined weighted average salary

\frac{\$280{}000}{50} = \$5{}600

Weighted average formula:

\text{Weighted Average} = \frac{\text{Total Sum}}{\text{Total Count}}

.

Anahtar Kavram

Weighted Average of Combined Sets

Tahmini Süre:1m 0s

Soru 830Soru →

A software consulting firm calculates the total cost of a project using a fixed administrative fee plus a constant hourly rate for each hour worked. For a project requiring $H$ hours, the firm originally charged a total of $\$2,400$ , where the fixed administrative fee was equal to the total hourly charges. Under a new pricing model, the fixed administrative fee is reduced by $25\%$ , while the hourly rate is increased by $20\%$ . Under this new model, a project requiring $H + 15$ hours costs a total of $\$2,880$ . What is the value of $H$ ?

25

30

40

50

60

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The value of

H

is 40.

The original cost model breaks into

F = \$1,200

and

R \cdot H = \$1,200

. Reducing

F

by

25\%

gives

\$900

. Increasing

R

by

20\%

gives a new hourly charge of

1.20 R(H + 15) = 1.20 RH + 18 R

. Substituting

RH = 1,200

yields

1,440 + 18 R = 1,980

, solving to

R = 30

. Dividing

1,200

by

30

gives

H = 40

.

Adım Adım Çözüm

1

Formulate equations for the original pricing structure.

Let

F

be the fixed administrative fee and

R

be the original hourly rate. The total original cost is given by

F + R \cdot H = 2,400

. Since

F

equals the total hourly charges (

R \cdot H

), we have

F = R \cdot H = 1,200

.

Splitting the total cost of

\$2,400

into two equal parts establishes the exact baseline values for

F

and

R \cdot H

.

2

Formulate the equation under the new pricing model.

The new fixed fee is

0.75 \times 1,200 = 900

. The new hourly rate is

1.20 R

. The new total cost equation for

H + 15

hours is

900 + 1.20 R(H + 15) = 2,880

.

Applies the specified percentage changes to the fixed fee and hourly rate individually.

3

Solve for the hourly rate

R

.

Subtract

900

from both sides:

1.20 R(H + 15) = 1,980 \implies 1.20 RH + 18 R = 1,980

. Substituting

RH = 1,200

gives

1.20(1,200) + 18 R = 1,980 \implies 1,440 + 18 R = 1,980 \implies 18 R = 540 \implies R = 30

.

Expands the algebraic expression and uses the substitution

RH = 1,200

to isolate

R

.

4

Calculate the value of

H

.

Since

R \cdot H = 1,200

and

R = 30

, we find

H = \frac{1,200}{30} = 40

.

Divides the total hourly charge by the hourly rate to obtain the number of hours.

Anahtar Kavram

Linear Equation Modeling with Percentage Adjustments

Tahmini Süre:2m 0s

Soru 831Soru →

What is the numerical value of $\frac{2^8 \cdot 3^4}{6^4}$ ?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

Rewriting the base

6

as

2 \cdot 3

gives

6^4 = 2^4 \cdot 3^4

. Substituting this into the original expression yields

\frac{2^8 \cdot 3^4}{2^4 \cdot 3^4}

. Cancelling

3^4

leaves

\frac{2^8}{2^4} = 2^{8-4} = 2^4 = 16

.

Adım Adım Çözüm

1

Rewrite composite bases into prime factors

6^4 = (2 \cdot 3)^4 = 2^4 \cdot 3^4

Applying the power of a product rule

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

allows base matching.

2

Simplify the quotient using exponent subtraction

\frac{2^8 \cdot 3^4}{2^4 \cdot 3^4} = 2^{8-4} \cdot 3^{4-4} = 2^4 \cdot 3^0 = 16 \cdot 1 = 16

Applying the quotient rule

\frac{a^m}{a^n} = a^{m-n}

for powers with equal bases.

Anahtar Kavram

Exponent rules: Power of a Product

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

and Quotient of Powers

\frac{a^m}{a^n} = a^{m-n}

Soru 832Soru →

A craft brewery blends three types of hops—Cascade, Centennial, and Mosaic—in a master batch. Initially, the weights of Cascade, Centennial, and Mosaic hops are in the ratio $3 : 4 : 5$ , respectively. After $12$ kilograms of Cascade hops and $12$ kilograms of Centennial hops are added to the batch, the new ratio of Cascade to Centennial to Mosaic becomes $5 : 6 : 5$ . What was the initial weight, in kilograms, of Centennial hops in the master batch?

6

18

24

30

36

Cevabı ve açıklamayı göster

Cevap: $24$

Cevap

The initial weight of Centennial hops in the master batch was

24

kilograms.

Let the initial weights of Cascade, Centennial, and Mosaic hops be

3x

,

4x

, and

5x

kilograms, respectively. Adding

12

kg to Cascade and Centennial gives new weights of

3x + 12

kg and

4x + 12

kg, while Mosaic stays at

5x

kg. Equating the ratio of New Cascade to New Mosaic (

\frac{3x + 12}{5x}

) to the new ratio term (

\frac{5}{5} = 1

) yields

3x + 12 = 5x

, so

2x = 12

and

x = 6

. Therefore, the initial weight of Centennial hops was

4x = 4(6) = 24

kg.

Adım Adım Çözüm

1

Define initial quantities using a common ratio multiplier

x

.

Cascade =

3x

kg, Centennial =

4x

kg, Mosaic =

5x

kg.

An initial ratio of

3 : 4 : 5

means the actual weights are multiples of

x

.

2

Express the new quantities after adding the specified weights.

New Cascade =

3x + 12

, New Centennial =

4x + 12

, New Mosaic =

5x

.

12

kg of Cascade and

12

kg of Centennial were added, while Mosaic remained unchanged.

3

Set up a proportion using the unchanged quantity and the new ratio.

Since Mosaic quantity remains

5x

and its component in the new ratio

5 : 6 : 5

is

5

, the multiplier for the new ratio is also

x

.

Comparing New Cascade to New Mosaic:

\frac{3x + 12}{5x} = \frac{5}{5} = 1 \implies 3x + 12 = 5x \implies 2x = 12 \implies x = 6

.

4

Calculate the initial weight of Centennial hops.

Initial Centennial weight =

4x = 4 \times 6 = 24

kilograms.

Multiplying the ratio part for Centennial (

4

) by the multiplier (

x = 6

) gives the initial weight.

Anahtar Kavram

Ratio scaling and setting up algebraic equations for altered multi-part ratios.

Tahmini Süre:2m 0s

Soru 833Soru →

Five speakers—Alice, Bob, Carol, David, and Eva—are scheduled to give consecutive presentations at a conference. If Alice must present first, in how many different linear orders can all five speakers present?

120

48

24

16

12

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

The total number of different presentation orders is 24.

Since Alice's position is fixed as the first speaker, there is only

1

way to fill the first slot. The remaining

4

speakers can be arranged in the remaining

4

slots in

4! = 4 \times 3 \times 2 \times 1 = 24

ways. Thus, the total number of linear arrangements is

1 \times 24 = 24

.

Adım Adım Çözüm

1

Identify the fixed element and available positions.

Alice must fill the 1st position, leaving 1 choice for slot 1.

The question specifies that Alice must present first.

2

Calculate the number of ways to arrange the remaining speakers.

The remaining 4 speakers (Bob, Carol, David, Eva) can be arranged in

4! = 4 \times 3 \times 2 \times 1 = 24

ways.

Linear permutations of

n

distinct objects use

n!

.

3

Apply the Fundamental Counting Principle.

Total arrangements =

1 \times 24 = 24

.

Combine the independent choices for each position.

Anahtar Kavram

Permutations with Fixed Positions

Soru 834Soru →

If $x$ is a real number satisfying the equation $(x^2 - 2x)^2 - (x^2 - 2x) - 12 = 0$ , what is the product of all distinct real roots of the equation?

-12

-4

-3

2

12

Cevabı ve açıklamayı göster

Cevap: $-4$

Cevap

The product of all distinct real roots of the equation is

-4

.

Substituting

u = x^2 - 2x

simplifies the given equation to

u^2 - u - 12 = 0

, which factors as

(u - 4)(u + 3) = 0

. Setting

x^2 - 2x = 4

yields

x^2 - 2x - 4 = 0

, which has two real roots (since the discriminant is

20 > 0

) with product

-4

. Setting

x^2 - 2x = -3

yields

x^2 - 2x + 3 = 0

, which has a negative discriminant (

-8

) and therefore no real solutions. Thus, the product of all distinct real roots is

-4

.

Adım Adım Çözüm

1

Use algebraic substitution to simplify the disguised quadratic equation.

Let

u = x^2 - 2x

. Substituting

u

into

(x^2 - 2x)^2 - (x^2 - 2x) - 12 = 0

gives

u^2 - u - 12 = 0

.

Recognizing the repeated expression

x^2 - 2x

reduces a 4th-degree polynomial into a standard 2nd-degree quadratic equation in terms of

u

.

2

Solve for

u

by factoring.

u^2 - u - 12 = (u - 4)(u + 3) = 0

, so

u = 4

or

u = -3

.

Factoring provides the possible numerical values for the expression

x^2 - 2x

.

3

Analyze each case for

x

and check for real roots using the discriminant.

Case 1:

x^2 - 2x = 4 \implies x^2 - 2x - 4 = 0

. Discriminant

D_1 = (-2)^2 - 4(1)(-4) = 20 > 0

. Since

D_1 > 0

, this quadratic has 2 distinct real roots. By Vieta's formula, the product of these two real roots is

\frac{c}{a} = \frac{-4}{1} = -4

.
Case 2:

x^2 - 2x = -3 \implies x^2 - 2x + 3 = 0

. Discriminant

D_2 = (-2)^2 - 4(1)(3) = -8 < 0

. Since

D_2 < 0

, this quadratic has no real roots.

Only roots with a non-negative discriminant belong to the real number domain and contribute to the product of real roots.

4

Determine the product of all distinct real roots.

The total product of all distinct real roots is

-4

.

Since Case 1 produces the only real roots of the original equation, their product

-4

is the complete product of all real solutions.

Anahtar Kavram

Disguised Quadratic Equations and Discriminant Real Root Filtering

Soru 835Soru →

What is the real solution to the equation $|x - 5| = 2x + 1$ ?

x = \frac{4}{3}

x = -6

x = -6

and

x = \frac{4}{3}

x = -\frac{4}{3}

x = 6

Cevabı ve açıklamayı göster

Cevap: $x = \frac{4}{3}$

Cevap

x = \frac{4}{3}

The correct answer is

x = \frac{4}{3}

. Setting

x - 5 = -(2x + 1)

gives

x - 5 = -2x - 1

, which solves to

3x = 4

or

x = \frac{4}{3}

. Plugging

x = \frac{4}{3}

into both sides of the original equation yields

\frac{11}{3} = \frac{11}{3}

, satisfying the equation.

Adım Adım Çözüm

1

Set up the two cases for the absolute value equation

Case 1:

x - 5 = 2x + 1

; Case 2:

x - 5 = -(2x + 1)

By definition,

|A| = B

implies

A = B

or

A = -B

, provided

B \geq 0

.

2

Solve Case 1 algebraically

x - 5 = 2x + 1 \implies x = -6

Subtracting

x

and

1

from both sides isolates

x

.

3

Solve Case 2 algebraically

x - 5 = -2x - 1 \implies 3x = 4 \implies x = \frac{4}{3}

Distributing the negative sign and adding

2x

and

5

to both sides isolates

x

.

4

Check for extraneous solutions in the original equation

|x - 5| = 2x + 1

For

x = -6

:

|-6 - 5| = |-11| = 11

, but

2(-6) + 1 = -11 \neq 11

(extraneous). For

x = \frac{4}{3}

:

|\frac{4}{3} - 5| = |-\frac{11}{3}| = \frac{11}{3}

, and

2(\frac{4}{3}) + 1 = \frac{11}{3}

(valid).

An absolute value expression cannot equal a negative number.

Anahtar Kavram

Solving absolute value equations with a variable on the right-hand side requires checking for extraneous solutions.

Soru 836Soru →

A commuter drives from home to work at a constant speed of $45$ miles per hour and arrives $10$ minutes late. On the following day, driving along the exact same route at a constant speed of $60$ miles per hour, the commuter arrives $5$ minutes early. What is the distance, in miles, from the commuter's home to work?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The distance from the commuter's home to work is 45 miles.

Let

d

be the distance in miles between home and work. At

45

miles per hour, the time taken is

\frac{d}{45}

hours. At

60

miles per hour, the time taken is

\frac{d}{60}

hours. The difference between being

10

minutes late and

5

minutes early is

15

minutes, or

\frac{15}{60} = \frac{1}{4}

hour. Setting up the equation

\frac{d}{45} - \frac{d}{60} = \frac{1}{4}

and finding a common denominator of

180

gives

\frac{4d - 3d}{180} = \frac{1}{4}

, which simplifies to

\frac{d}{180} = \frac{1}{4}

. Multiplying both sides by

180

yields

d = 45

miles.

Adım Adım Çözüm

1

Calculate the difference in travel time between the two trips in hours.

The time difference is 10 minutes late - (-5 minutes early) = 15 minutes = 1/4 hour.

Since speeds are given in miles per hour, time must be expressed in hours.

2

Set up an equation relating the two travel times using the formula time = distance / speed.

d/45 - d/60 = 1/4, where d represents the distance in miles.

The slower speed (45 mph) takes 1/4 hour longer than the faster speed (60 mph).

3

Solve the equation for the distance variable d.

(4d - 3d)/180 = 1/4 => d/180 = 1/4 => d = 45.

Find a common denominator for 45 and 60, which is 180, and solve for d.

Anahtar Kavram

Rate, Time, and Distance Problems

Soru 837Soru →

At a commercial bank, an audit of $300$ business loan applications revealed that each application underwent at least one of three specialized risk evaluations: Credit Risk, Market Risk, and Operational Risk. Exactly $180$ applications underwent Credit Risk evaluation, $150$ underwent Market Risk evaluation, and $135$ underwent Operational Risk evaluation. If exactly $95$ applications underwent exactly two of the three risk evaluations, how many loan applications underwent all three risk evaluations?

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

35 loan applications underwent all three risk evaluations.

The total number of applications is

300

. The sum of individual evaluations is

180 + 150 + 135 = 465

. Using the Venn diagram region formula,

\text{Sum of individual sets} - \text{Total} = \text{Exactly } 2 + 2 \times (\text{Exactly } 3)

. Substituting the known values gives

465 - 300 = 95 + 2x

, which simplifies to

165 = 95 + 2x

. Solving for

x

gives

2x = 70

, so

x = 35

.

Adım Adım Çözüm

1

State the fundamental overlapping set region equation for 3 sets.

\text{Total} = \text{Exactly } 1 + \text{Exactly } 2 + \text{Exactly } 3 + \text{Neither}

Every application falls into exactly one of these non-overlapping region categories.

2

Calculate the sum of the individual set counts.

180 + 150 + 135 = 465

Summing individual set counts counts items in 1 set once, 2 sets twice, and 3 sets three times.

3

Formulate the algebraic identity linking individual sums to region totals.

\text{Sum of Individual Sets} = \text{Exactly } 1 + 2(\text{Exactly } 2) + 3(\text{Exactly } 3)

This accounts for the multiple counting of overlapping regions.

4

Subtract the Total equation from the Sum equation to eliminate the 'Exactly 1' term.

465 - 300 = \text{Exactly } 2 + 2(\text{Exactly } 3) \Rightarrow 165 = 95 + 2(\text{Exactly } 3)

Subtracting

(\text{Exactly } 1 + \text{Exactly } 2 + \text{Exactly } 3)

from

(\text{Exactly } 1 + 2\cdot\text{Exactly } 2 + 3\cdot\text{Exactly } 3)

leaves

1\cdot\text{Exactly } 2 + 2\cdot\text{Exactly } 3

.

5

Solve the linear equation for the 'Exactly 3' intersection.

2(\text{Exactly } 3) = 165 - 95 = 70 \Rightarrow \text{Exactly } 3 = 35

Dividing the remaining difference of 70 by 2 gives the number of applications in all three sets.

Anahtar Kavram

Three-Set Venn Diagram Region Decomposition

Soru 838Soru →

For all real numbers $x$ and $y$ , the custom binary operator $\odot$ is defined by $x \odot y = x^2 y - x y^2$ . The function $f$ is defined by $f(x) = x \odot 3$ . What is the value of $f(f(2))$ ?

Cevabı ve açıklamayı göster

Cevap: 162

Cevap

The value of

f(f(2))

is 162.

To evaluate the nested function

f(f(2))

, first determine

f(2)

using the definition

f(x) = x \odot 3

. Applying the definition of the custom operator

x \odot y = x^2 y - x y^2

with

x = 2

and

y = 3

yields

2 \odot 3 = (2)^2(3) - (2)(3)^2 = 12 - 18 = -6

. Then, evaluate

f(-6) = -6 \odot 3 = (-6)^2(3) - (-6)(3)^2 = 108 - (-54) = 162

.

Adım Adım Çözüm

1

Evaluate the inner function expression f(2)

f(2) = 2 \odot 3 = (2)^2(3) - (2)(3)^2 = 12 - 18 = -6

By definition, f(x) = x \odot 3. Substituting x = 2 gives 2 \odot 3, which applies the custom operator definition x^2 y - x y^2.

2

Evaluate the outer function expression f(f(2)) = f(-6)

f(-6) = -6 \odot 3 = (-6)^2(3) - (-6)(3)^2 = 36(3) - (-54) = 108 + 54 = 162

Substitute the inner result -6 into the function f(x).

Anahtar Kavram

Custom Binary Operators and Nested Function Evaluation

Soru 839Soru →

An agricultural facility uses two automated loading conveyors, Conveyor $A$ and Conveyor $B$ , to fill a grain storage silo. Working alone at its constant rate, Conveyor $A$ can fill the silo in 10 hours. Working alone at its constant rate, Conveyor $B$ can fill the silo in 15 hours. Conveyor $A$ begins filling the empty silo alone. After 4 hours, Conveyor $B$ is turned on, and both conveyors work together at their respective constant rates until the silo is completely full. How many total hours does it take, from the moment Conveyor $A$ starts, to completely fill the silo?

Cevabı ve açıklamayı göster

Cevap: 7.6

Cevap

The total time required to fill the silo is 7.6 hours.

Conveyor A fills 1/10 of the silo per hour and works alone for 4 hours, completing 2/5 of the total job. The remaining 3/5 of the job is completed by both conveyors working together at a combined rate of 1/10 + 1/15 = 1/6 silo per hour. Dividing 3/5 by 1/6 gives 3.6 hours for the second stage. Adding the initial 4 hours yields a total time of 7.6 hours.

Adım Adım Çözüm

1

Determine the individual hourly rates of Conveyor A and Conveyor B.

Conveyor A rate =

1/10

silo per hour; Conveyor B rate =

1/15

silo per hour.

Work rate is the reciprocal of the time required to complete the total job.

2

Calculate the fraction of the silo filled by Conveyor A working alone for 4 hours.

Work completed =

4 \times \frac{1}{10} = \frac{2}{5}

of the silo.

Work done equals rate multiplied by time.

3

Determine the remaining fraction of work to be completed.

Remaining work =

1 - \frac{2}{5} = \frac{3}{5}

of the silo.

Subtract the completed fraction from the total job (1).

4

Calculate the combined work rate when both conveyors operate together.

Combined rate =

\frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6}

silo per hour.

When working together, individual rates add up.

5

Determine the time required for both conveyors to finish the remaining work.

Time =

\frac{3/5}{1/6} = \frac{18}{5} = 3.6

hours.

Divide remaining work by the combined work rate.

6

Sum the time spent in both stages to find the total time.

Total time =

4 + 3.6 = 7.6

hours.

The total duration includes 4 hours of Conveyor A operating alone plus 3.6 hours of combined operation.

Anahtar Kavram

Work Rate and Combined Work

Soru 840Soru →

A merchant purchased a shipment of 200 identical items for a total cost of $8,000. The merchant marked up the cost price of each item by$ P $percent to establish its list price. During the first month of sales, 60 percent of the items were sold at the list price. During the second month, 50 percent of the remaining items were sold at a 20 percent discount off the list price. The remaining items were then sold during a clearance sale at a 50 percent discount off the list price. If the merchant made an overall net profit of 29 percent on the total purchase cost of the shipment, what is the value of$ P$?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The value of

P

is 50.

The unit cost of each item is

\frac{\$8,000}{200} = \$40

. To make an overall net profit of 29%, total revenue must equal

\$8,000 \times 1.29 = \$10,320

. Sales are divided into three tiers: 120 items sold at list price

L

, 40 items sold at

0.80L

, and 40 items sold at

0.50L

. Summing the revenue yields

120L + 32L + 20L = 172L

. Solving

172L = 10,320

gives

L = 60

. The percentage markup over cost price is

\frac{60 - 40}{40} \times 100 = 50\%

.

Adım Adım Çözüm

1

Calculate the unit cost price and required total revenue.

Unit cost

C = \frac{\$8,000}{200} = \$40

. Total target revenue

R = \$8,000 \times (1 + 0.29) = \$10,320

.

Determining the total required revenue sets the benchmark to solve for the list price.

2

Determine the quantity of items sold at each price tier.

Month 1 sales:

0.60 \times 200 = 120

items at list price

L

.
Month 2 sales:

0.50 \times (200 - 120) = 40

items at discounted price

0.80L

.
Clearance sales:

200 - 120 - 40 = 40

items at clearance price

0.50L

.

Categorizing quantities by price tier enables building the total revenue expression.

3

Express total revenue in terms of list price

L

and solve for

L

.

Total Revenue =

120L + 40(0.80L) + 40(0.50L) = 120L + 32L + 20L = 172L

.
Setting

172L = 10,320 \implies L = \frac{10,320}{172} = 60

.

Summing revenue from all tiers and equating to target revenue yields the list price per item.

4

Calculate the markup percentage

P

.

P = \frac{L - C}{C} \times 100 = \frac{60 - 40}{40} \times 100 = 50\%

.

Percentage markup is measured relative to the initial cost price per unit.

Anahtar Kavram

Multi-tiered inventory pricing and percentage markup calculation

Tahmini Süre:2m 30s

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