Tüm alıştırma soruları

Soru 861Soru →

Two automated document redaction pipelines, Pipeline $A$ and Pipeline $B$ , process digital legal archives. Working alone at its constant rate, Pipeline $A$ can redact a standard archive in $8$ hours. Working alone at its constant rate, Pipeline $B$ can redact the same standard archive in $12$ hours. Both pipelines begin processing a standard archive together. After $3$ hours of joint operation, Pipeline $A$ encounters a system error and stops. Pipeline $B$ continues working alone at its constant rate until the entire archive is redacted. How many total hours does it take from start to finish to complete the redaction of the archive?

Cevabı ve açıklamayı göster

Cevap: 7.5

Cevap

The total time required to redact the entire archive from start to finish is 7.5 hours.

To find the total time required, first determine the combined rate of Pipeline A and Pipeline B:

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

archive per hour. In the first 3 hours of joint work, they complete

3 \times \frac{5}{24} = \frac{5}{8}

of the job, leaving

1 - \frac{5}{8} = \frac{3}{8}

of the archive remaining. Pipeline B processes the remaining

\frac{3}{8}

at its individual rate of

\frac{1}{12}

per hour, taking

\frac{3/8}{1/12} = 4.5

hours. Adding the initial 3 hours to the 4.5 additional hours yields a total of 7.5 hours.

Adım Adım Çözüm

1

Determine the individual work rate for each pipeline.

Pipeline A's rate is

\frac{1}{8}

archive per hour; Pipeline B's rate is

\frac{1}{12}

archive per hour.

Work rate is the fraction of job completed per hour (

1 / \text{time required alone}

).

2

Calculate the combined work rate during joint operation.

Combined rate is

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

archive per hour.

When entities work concurrently, their rates are added together.

3

Determine the amount of work completed in the first 3 hours.

Work completed is

3 \times \frac{5}{24} = \frac{5}{8}

of the total archive.

Work done equals rate multiplied by time.

4

Find the fraction of the archive remaining to be redacted.

Remaining work is

1 - \frac{5}{8} = \frac{3}{8}

of the archive.

The full task represents 1 complete unit of work.

5

Calculate how long Pipeline B needs to finish the remaining work alone.

Solo time for Pipeline B is

\frac{3/8}{1/12} = 4.5

hours.

Time equals remaining work divided by Pipeline B's individual rate.

6

Sum the joint operation time and the solo operation time.

Total time is

3 + 4.5 = 7.5

hours.

The overall duration consists of two sequential periods: 3 hours working together plus 4.5 hours for Pipeline B working alone.

Anahtar Kavram

Combined Work Rates and Multi-Stage Work Scenarios

Soru 862Soru →

If $x$ and $y$ are non-zero real numbers such that $x^2 - 5xy + 6y^2 = 0$ , what is the sum of all possible values of $\frac{x^2 + y^2}{xy}$ ?

\frac{5}{2}

\frac{35}{6}

\frac{59}{6}

-\frac{5}{6}

5

Cevabı ve açıklamayı göster

Cevap: $\frac{35}{6}$

Cevap

The sum of all possible values of

\frac{x^2 + y^2}{xy}

is

\frac{35}{6}

.

Factoring

x^2 - 5xy + 6y^2 = 0

yields

(x - 2y)(x - 3y) = 0

. Thus,

\frac{x}{y} = 2

or

\frac{x}{y} = 3

. Simplifying

\frac{x^2 + y^2}{xy}

gives

\frac{x}{y} + \frac{y}{x}

. Substituting

2

yields

2 + \frac{1}{2} = \frac{5}{2}

, and substituting

3

yields

3 + \frac{1}{3} = \frac{10}{3}

. The sum of these values is

\frac{5}{2} + \frac{10}{3} = \frac{35}{6}

.

Adım Adım Çözüm

1

Factor the quadratic equation

x^2 - 5xy + 6y^2 = 0

in terms of

x

and

y

.

(x - 2y)(x - 3y) = 0

Factoring determines the proportional relationships between

x

and

y

.

2

Solve for the possible ratio values of

\frac{x}{y}

.

Either

x = 2y \implies \frac{x}{y} = 2

, or

x = 3y \implies \frac{x}{y} = 3

.

Since

y \neq 0

, dividing by

y

gives the linear root ratios.

3

Rewrite the target expression

\frac{x^2 + y^2}{xy}

as a sum of ratios.

\frac{x^2 + y^2}{xy} = \frac{x}{y} + \frac{y}{x}

Splitting the numerator simplifies substitution of

\frac{x}{y}

.

4

Evaluate the expression for each possible ratio.

Case 1 (

x/y = 2

):

2 + \frac{1}{2} = \frac{5}{2}

. Case 2 (

x/y = 3

):

3 + \frac{1}{3} = \frac{10}{3}

.

Each root ratio produces a distinct value for the target expression.

5

Sum the two values obtained.

\frac{5}{2} + \frac{10}{3} = \frac{15 + 20}{6} = \frac{35}{6}

The question asks for the sum of all possible values of the expression.

Anahtar Kavram

Homogeneous Quadratic Equations and Algebraic Factoring

Tahmini Süre:2m 0s

Soru 863Soru →

A motorboat travels downstream along a straight river from Port Alpha to Port Beta, a distance of $120$ miles, completing the downstream leg in exactly $3$ hours. At the moment the motorboat departs from Port Alpha, an unpowered raft is released from Port Alpha and drifts downstream driven solely by the river's constant current. Upon reaching Port Beta, the motorboat immediately turns around and travels upstream toward Port Alpha. If the motorboat meets the drifting raft at a point $30$ miles downstream from Port Alpha, and the motorboat's speed relative to the water remains constant throughout the journey, what is the motorboat's speed in still water, in miles per hour?

20

25

32

35

45

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

35 miles per hour

The correct answer is 35 miles per hour. Since the motorboat covers the 120-mile downstream distance in 3 hours, its downstream rate is 40 mph, which means the boat's speed in still water

v

plus the current speed

c

equals 40. The raft drifts 30 miles at speed

c

, requiring

\frac{30}{c}

hours. In that same total time, the boat spends 3 hours going downstream and then travels 90 miles upstream at speed

v - c = 40 - 2c

. Equating the two time expressions

\frac{30}{c} = 3 + \frac{90}{40-2c}

produces the quadratic equation

c^2 - 45c + 200 = 0

. The valid root is

c = 5

mph, yielding

v = 40 - 5 = 35

mph.

Adım Adım Çözüm

1

Determine the downstream speed of the motorboat and express the still-water speed

v

in terms of the current

c

.

The downstream speed is

\frac{120 \text{ miles}}{3 \text{ hours}} = 40

mph. Therefore,

v + c = 40

, which implies

v = 40 - c

mph and upstream speed

v - c = 40 - 2c

mph.

Downstream speed combines the motorboat's still-water speed and the river's current speed.

2

Formulate expressions for the time elapsed for both the raft and the motorboat until they meet.

The raft drifts

30

miles at speed

c

, so raft time

T = \frac{30}{c}

hours. The motorboat travels

120

miles downstream in

3

hours, and then

120 - 30 = 90

miles upstream at speed

(40 - 2c)

mph. So motorboat time

T = 3 + \frac{90}{40 - 2c}

hours.

Both vessels start at the exact same moment from Port Alpha, so their total travel times until they meet are equal.

3

Equate total times and solve the resulting algebraic equation for

c

.

Setting

\frac{30}{c} = 3 + \frac{90}{40 - 2c}

and dividing by

3

yields

\frac{10}{c} = 1 + \frac{15}{20 - c} = \frac{35 - c}{20 - c}

. Cross-multiplying gives

10(20 - c) = c(35 - c) \implies c^2 - 45c + 200 = 0

. Factoring gives

(c - 5)(c - 40) = 0

. Since

c < 40

,

c = 5

mph.

Solving the quadratic equation isolates the valid physical value for the current speed.

4

Calculate the motorboat's speed in still water,

v

.

v = 40 - 5 = 35

mph.

Subtracting the current speed from the downstream speed yields the speed in still water.

Anahtar Kavram

Relative speed in current and multi-leg journey time equilibrium

Soru 864Soru →

An integer $n$ is randomly selected from the set of all integers from $10$ to $59$ , inclusive. What is the probability that $n$ is a prime number whose units digit is $3$ ?

\frac{1}{10}

\frac{4}{49}

\frac{2}{25}

\frac{2}{23}

\frac{3}{50}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{25}$

Cevap

The probability that the selected integer is a prime number whose units digit is

3

is

\frac{2}{25}

.

The correct answer identifies that there are 50 total integers in the inclusive range from 10 to 59. Among the numbers ending in 3 (

13, 23, 33, 43, 53

), exactly four are prime (

13, 23, 43, 53

), while 33 is composite. Dividing 4 favorable outcomes by 50 total outcomes yields

\frac{4}{50}

, which simplifies to

\frac{2}{25}

.

Adım Adım Çözüm

1

Determine the total number of outcomes in the set.

The set contains

59 - 10 + 1 = 50

integers.

For an inclusive set of integers from

a

to

b

, the total number of integers is

b - a + 1

.

2

Identify all integers in the set with a units digit of

3

.

The candidate integers ending in 3 are

13, 23, 33, 43, 53

.

These are all two-digit numbers in the range

[10, 59]

ending with

3

.

3

Determine which candidate integers are prime numbers.

The prime numbers are

13, 23, 43,

and

53

(a total of 4 favorable outcomes). Note that

33 = 3 \times 11

, so it is composite.

A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself.

4

Calculate the single-event probability and simplify the fraction.

\text{Probability} = \frac{4}{50} = \frac{2}{25}

.

Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.

Anahtar Kavram

Basic Single-Event Probability

Tahmini Süre:1m 30s

Soru 865Soru →

A data set $S$ consists of $11$ positive integers. The set has an arithmetic mean of $15$ , a median of $14$ , and a unique mode of $18$ . If $M$ is the maximum possible value of an element in $S$ and $m$ is the minimum possible value of an element in $S$ , what is the maximum possible value of the range $M - m$ ?

68

71

72

73

74

Cevabı ve açıklamayı göster

Cevap: $73$

Cevap

The maximum possible value of the range

M - m

is

73

.

To maximize the range

M - m = x_{11} - x_1

, we minimize

x_1

and maximize

x_{11}

. Since elements are positive integers,

m = x_1 = 1

. The sum of all

11

elements is

11 \times 15 = 165

. With median

x_6 = 14

, the unique mode

18

must lie above the median. Setting the frequency of

18

to

3

allows other numbers to appear up to

2

times. Minimizing

x_1 \dots x_5

gives

1, 1, 2, 2, 3

(sum

9

). Minimizing

x_7 \dots x_{10}

by setting

x_7 = 14

and

x_8 = x_9 = x_{10} = 18

gives sum

68

. The sum of the first

10

terms is

9 + 14 + 68 = 91

, leaving

M = x_{11} = 165 - 91 = 74

. Thus, the maximum range is

74 - 1 = 73

.

Adım Adım Çözüm

1

Calculate the sum of all elements in the set.

Sum

= 11 \times 15 = 165

.

The arithmetic mean of

11

elements is

15

.

2

Identify the position of the median and order the elements.

Let elements be

x_1 \le x_2 \le \dots \le x_6 \le \dots \le x_{11}

, where

x_6 = 14

.

For an odd number of elements (

11

), the median is the 6th element.

3

Determine the mode frequency constraint to maximize

x_{11} - x_1

.

The unique mode is

18

. If

18

appears

3

times, the maximum frequency of any other value is

2

.

To maximize

x_{11}

, we minimize

x_1 \dots x_{10}

. Allowing the mode to appear

3

times permits other elements to appear up to

2

times.

4

Minimize the sum of the lower five elements

x_1, x_2, x_3, x_4, x_5

.

x_1 = 1, x_2 = 1, x_3 = 2, x_4 = 2, x_5 = 3

, giving sum

= 1 + 1 + 2 + 2 + 3 = 9

.

The smallest positive integers with maximum frequency

2

are

1, 1, 2, 2, 3

.

5

Minimize elements

x_7, x_8, x_9, x_{10}

.

x_7 = 14

(frequency

2

for

14

), and

x_8 = 18, x_9 = 18, x_{10} = 18

. Sum

= 14 + 18 + 18 + 18 = 68

.

Since

x_6 = 14

, setting

x_7 = 14

minimizes

x_7

while keeping the frequency of

14

at

2

, which is less than the mode frequency of

3

.

6

Calculate the maximum value

M = x_{11}

and the range

M - m

.

Sum of first 10 elements

= 9 + 14 + 68 = 91

. Thus

x_{11} = 165 - 91 = 74

. Range

= 74 - 1 = 73

.

Subtracting the sum of the first 10 elements from the total sum gives

M

, and subtracting

m = 1

yields the range.

Anahtar Kavram

Range, Mean, Median, and Mode Constraints in Data Sets

Soru 866Soru →

An industrial facility uses two automated assembly units, Unit $X$ and Unit $Y$ , to fabricate custom components. Working alone at its constant rate, Unit $X$ can complete a batch of components in $6$ hours, while Unit $Y$ can complete an identical batch working alone at its constant rate in $12$ hours. Unit $X$ begins working on a batch alone. After $2$ hours, Unit $Y$ joins Unit $X$ , and both units work together at their respective constant rates until the batch is complete. What is the total time, in hours, required to complete the batch from start to finish?

2\frac{2}{3}

3\frac{1}{2}

4\frac{2}{3}

6

8

Cevabı ve açıklamayı göster

Cevap: $4\frac{2}{3}$

Cevap

4\frac{2}{3}

hours

The correct answer is

4\frac{2}{3}

hours. Unit

X

works alone for

2

hours, completing

2 \times \frac{1}{6} = \frac{1}{3}

of the job. This leaves

\frac{2}{3}

of the job remaining. Working together, Unit

X

and Unit

Y

have a combined rate of

\frac{1}{6} + \frac{1}{12} = \frac{1}{4}

of the job per hour. The time spent working together is

\frac{2/3}{1/4} = \frac{8}{3}

hours. Adding the initial

2

hours gives a total duration of

2 + \frac{8}{3} = 4\frac{2}{3}

hours.

Adım Adım Çözüm

1

Calculate the individual work rates of Unit

X

and Unit

Y

.

Unit

X

's rate is

\frac{1}{6}

batch per hour, and Unit

Y

's rate is

\frac{1}{12}

batch per hour.

Work rate is defined as

\text{Rate} = \frac{1}{\text{Time}}

for completing one full job.

2

Determine the fraction of the batch completed by Unit

X

during its

2

hours of solo work.

Unit

X

completes

2 \times \frac{1}{6} = \frac{1}{3}

of the batch.

Work done equals

\text{Rate} \times \text{Time}

.

3

Find the remaining fraction of work to be completed.

The remaining work is

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

of the batch.

Subtract the completed fraction from the total job (

1

).

4

Calculate the combined work rate when both units operate together.

Combined rate =

\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}

batch per hour.

Simultaneous work rates add linearly.

5

Calculate the time needed for both units to finish the remaining work.

Time together =

\frac{2/3}{1/4} = \frac{2}{3} \times 4 = \frac{8}{3} = 2\frac{2}{3}

hours.

Time equals

\frac{\text{Remaining Work}}{\text{Combined Rate}}

.

6

Add the solo phase duration to the combined phase duration to get total elapsed time.

Total time =

2 + \frac{8}{3} = \frac{14}{3} = 4\frac{2}{3}

hours.

Total time is the sum of the times from each sequential stage.

Anahtar Kavram

Work Rate and Combined Work

Tahmini Süre:2m 0s

Soru 867Soru →

What is the sum of all real solutions to the equation $\sqrt{2x + 15} = x$ ?

5

2

-3

8

15

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The sum of all real solutions to the equation is 5.

Squaring both sides of

\sqrt{2x + 15} = x

yields

2x + 15 = x^2

, which rearranges to

x^2 - 2x - 15 = 0

. Factoring gives

(x - 5)(x + 3) = 0

, yielding candidate roots

x = 5

and

x = -3

. Substituting

x = 5

into the original equation gives

\sqrt{25} = 5

, which is true. Substituting

x = -3

gives

\sqrt{9} = -3

, which is false because the principal square root must be non-negative. Therefore,

x = 5

is the only valid real solution, and its sum is 5.

Adım Adım Çözüm

1

Square both sides of the equation to eliminate the radical.

2x + 15 = x^2

Squaring both sides removes the radical so the equation can be expressed in quadratic form.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 2x - 15 = 0

Setting the quadratic expression equal to zero enables solution by factoring.

3

Factor the quadratic polynomial.

(x - 5)(x + 3) = 0 \implies x = 5 \text{ or } x = -3

Factoring determines the candidate roots of the quadratic equation.

4

Check candidate roots in the original equation

\sqrt{2x + 15} = x

for extraneous solutions.

For

x = 5

:

\sqrt{2(5) + 15} = \sqrt{25} = 5

(Valid). For

x = -3

:

\sqrt{2(-3) + 15} = \sqrt{9} = 3 \neq -3

(Extraneous).

Squaring both sides of an equation can introduce extraneous roots that violate the non-negativity constraint of principal square roots.

5

Sum all valid real solutions.

Sum = 5

Since

x = 5

is the only valid solution, the sum of all solutions is 5.

Anahtar Kavram

Quadratic Equations and Extraneous Solutions in Radical Equations

Soru 868Soru →

An art gallery displays works in three mediums: oil paintings, sculptures, and watercolors. Initially, the ratio of oil paintings to sculptures to watercolors is $3 : 4 : 5$ , respectively. After the gallery acquires 12 additional oil paintings and sells 6 watercolors, the ratio of oil paintings to watercolors becomes $1 : 1$ . If no sculptures were acquired or sold, how many sculptures are currently on display at the gallery?

27

36

45

48

108

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

By defining the initial quantities of oil paintings, sculptures, and watercolors as

3x

,

4x

, and

5x

respectively, we update the modified amounts to

3x + 12

for oil paintings and

5x - 6

for watercolors. Equating these gives

3x + 12 = 5x - 6

, leading to

2x = 18

and

x = 9

. The number of sculptures, which remained unchanged, is

4x = 4 \times 9 = 36

.

Adım Adım Çözüm

1

Define variables based on the initial ratio

Let the multiplier be

x

. Initial quantities are Oil Paintings

= 3x

, Sculptures

= 4x

, and Watercolors

= 5x

.

Ratios expressed as

3 : 4 : 5

can be represented using a common multiplier

x

to find actual quantities.

2

Set up an equation incorporating the changes in inventory

New Oil Paintings

= 3x + 12

; New Watercolors

= 5x - 6

. The ratio between them is

1 : 1

, so

3x + 12 = 5x - 6

.

A

1 : 1

ratio means the updated quantity of oil paintings equals the updated quantity of watercolors.

3

Solve for the common multiplier

x

5x - 3x = 12 + 6 \implies 2x = 18 \implies x = 9

.

Linear algebraic manipulation isolates

x

.

4

Calculate the quantity of sculptures

Number of Sculptures

= 4x = 4 \times 9 = 36

.

Substitute the value of

x

back into the expression for sculptures.

Anahtar Kavram

Ratio modification and single-variable setting for multi-part ratios

Tahmini Süre:2m 0s

Soru 869Soru →

A logistics company operates two models of cargo trucks, Model X and Model Y, to transport freight between two distribution hubs. Model X consumes $10$ liters of fuel per $100$ kilometers driven in city traffic and $6$ liters of fuel per $100$ kilometers driven on open highway. Model Y consumes $14$ liters of fuel per $100$ kilometers in city traffic and $8$ liters per $100$ kilometers on open highway. On a trip between the two hubs along a route composed entirely of city traffic segments and open highway segments, Model X consumed a total of $42$ liters of fuel, and Model Y consumed a total of $58$ liters of fuel. What is the total length, in kilometers, of the open highway segments along this route?

Cevabı ve açıklamayı göster

Cevap: 200

Cevap

The total length of the open highway segments along the route is 200 kilometers.

By converting the per-100-kilometer fuel consumption into per-kilometer rates, we obtain two linear equations in two variables:

0.10C + 0.06H = 42

and

0.14C + 0.08H = 58

. Solving this system by elimination yields

H = 200

kilometers for the open highway distance.

Adım Adım Çözüm

1

Define variables for city traffic distance and open highway distance.

Let

C

equal the distance in km of city traffic segments, and

H

equal the distance in km of open highway segments.

Establishing explicit variables allows the translation of the word problem context into algebraic equations.

2

Convert consumption rates to liters per kilometer and formulate total fuel equations for both trucks.

Model X equation:

0.10C + 0.06H = 42

; Model Y equation:

0.14C + 0.08H = 58

.

Dividing the consumption per 100 km by 100 gives the unit rates per km for city and highway driving.

3

Simplify the system of linear equations by eliminating decimals and common factors.

First equation:

5C + 3H = 2100

; Second equation:

7C + 4H = 2900

.

Simplifying equations reduces computation errors during elimination.

4

Solve the system using elimination to isolate the highway distance variable

H

.

Multiplying the equations to match coefficients of

C

(

35C + 21H = 14700

and

35C + 20H = 14500

) and subtracting gives

H = 200

.

Eliminating

C

directly yields the target quantity

H

without needing additional substitution steps.

Anahtar Kavram

Formulating and Solving Systems of Linear Equations from Multi-Condition Word Problems

Tahmini Süre:2m 0s

Soru 870Soru →

A logistics company operates a fleet consisting of standard electric delivery vans and high-capacity electric delivery vans. A standard van consumes a fixed $2.5$ kilowatt-hours (kWh) of electricity per delivery route, while a high-capacity van consumes $50\%$ more electricity per route than a standard van. On a given day, the fleet completed a total of $40$ delivery routes and consumed a total of $130$ kWh of electricity. How many delivery routes were completed by high-capacity vans?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

The total number of delivery routes completed by high-capacity vans is 24.

By modeling the relationship using the system

S + H = 40

and

2.5S + 3.75H = 130

, substituting

S = 40 - H

gives

100 + 1.25H = 130

, which resolves to

H = 24

high-capacity van routes.

Adım Adım Çözüm

1

Calculate the power consumption per route for a high-capacity van.

High-capacity rate =

2.5 \times 1.5 = 3.75

kWh per route.

High-capacity vans consume

50\%

more energy than standard vans.

2

Formulate a system of linear equations representing total routes and total electricity used.

S + H = 40

and

2.5S + 3.75H = 130

, where

S

represents standard van routes and

H

represents high-capacity van routes.

The sum of routes equals 40 and the combined energy consumption equals 130 kWh.

3

Solve for the target variable

H

.

Substitute

S = 40 - H

into the energy equation:

2.5(40 - H) + 3.75H = 130 \implies 100 + 1.25H = 130 \implies 1.25H = 30 \implies H = 24

.

Substituting

S

yields a single-variable linear equation for

H

.

Anahtar Kavram

Algebraic Modeling of Systems of Linear Equations

Tahmini Süre:1m 30s

Soru 871Soru →

A dataset consists of five identical integers: $12, 12, 12, 12,$ and $12$ . What is the standard deviation of this dataset?

0

12

-12

144

2\sqrt{3}

Cevabı ve açıklamayı göster

Cevap: $0$

Cevap

0

The correct value is

0

. Standard deviation measures how spread out the numbers in a dataset are around the mean. Since every value in the set is

12

, the mean is

12

and every value lies exactly on the mean (distance of

0

). Thus, the standard deviation is

0

.

Adım Adım Çözüm

1

Calculate the arithmetic mean of the dataset

Mean

\mu = \frac{12 + 12 + 12 + 12 + 12}{5} = 12

The mean is needed to measure individual deviations.

2

Calculate the deviation of each data point from the mean

Deviations are

12 - 12 = 0

for all five elements

Standard deviation measures the average distance of data points from the mean.

3

Compute the standard deviation

Standard Deviation

SD = \sqrt{\frac{0^2 + 0^2 + 0^2 + 0^2 + 0^2}{5}} = 0

When all data points are identical, the spread around the mean is zero.

Anahtar Kavram

Standard deviation measures the dispersion or spread of a set of numbers around its mean. If all elements in a set are identical, there is no variation, so both the range and the standard deviation are equal to 0.

Soru 872Soru →

Eight executive team members—3 vice presidents, 3 directors, and 2 managers—are to stand in a single line for a company photograph. If no two vice presidents can stand next to each other, and the 2 managers must stand next to each other, how many different linear arrangements of the eight team members are possible?

Cevabı ve açıklamayı göster

Cevap: 2880

Cevap

2,880

To satisfy both constraints simultaneously, first treat the 2 managers as a single block, which has

2! = 2

internal arrangements. Next, arrange the non-vice-president units—consisting of the 3 distinct directors and the 1 manager block—giving 4 units total, which can be ordered in

4! = 24

ways. These 4 units create 5 distinct slot locations (at the ends and between adjacent units). To ensure no two vice presidents are adjacent, place the 3 distinct vice presidents into 3 of these 5 slots in

P(5, 3) = 60

ways. Multiplying these independent decisions yields

2 \times 24 \times 60 = 2,880

.

Adım Adım Çözüm

1

Group the elements that must remain adjacent and calculate internal permutations

The 2 managers form 1 block with

2! = 2

internal arrangements.

Since the 2 managers must stand next to each other, treating them as a single block ensures they are never separated.

2

Arrange all non-restricted base units in a line

The 3 directors and 1 manager block yield

4! = 24

linear arrangements.

Establishing the sequence of non-vice-president units creates the fixed slots into which the vice presidents will later be inserted.

3

Calculate the available slot arrangements for the separated elements

4 base units create 5 available slots. Permuting 3 vice presidents into 5 slots yields

P(5, 3) = 60

ways.

Placing at most one vice president per slot guarantees that no two vice presidents are placed adjacently.

4

Apply the Fundamental Counting Principle to determine total arrangements

Total arrangements =

2 \times 24 \times 60 = 2,880

.

The choices for internal block arrangement, base unit ordering, and slot placement are independent sequential events.

Anahtar Kavram

Linear Permutations with Simultaneous Grouping and Non-Adjacency Constraints

Soru 873Soru →

A board of directors consists of $7$ members. In how many different ways can a subcommittee of $3$ members be chosen from the board?

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

35 different subcommittees can be chosen.

The total number of ways to choose a subcommittee of 3 members from a group of 7 members when order does not matter is given by the combination formula

\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

.

Adım Adım Çözüm

1

Determine whether the selection requires permutations or combinations.

Since selecting members A, B, and C forms the exact same committee as selecting B, C, and A, the order of selection does not matter. Therefore, this is a combination problem.

Committees are unordered groups.

2

Calculate the number of combinations of 7 items taken 3 at a time using

\binom{7}{3} = \frac{7!}{3!(7-3)!}

.

\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

.

Simplify the factorial expression by canceling out common terms.

Anahtar Kavram

Combinations (Selection Without Regard to Order)

Soru 874Soru →

A laboratory container initially holds $120$ liters of a chemical solution that is $40\%$ acid by volume. A chemist removes $x$ liters of the solution and replaces it with an equal volume of pure acid. Next, the chemist removes $x$ liters of the resulting mixture and replaces it with an equal volume of pure water. If the final concentration of acid in the container is $40\%$ by volume, what is the value of $x$ ?

20

30

40

50

60

Cevabı ve açıklamayı göster

Cevap: $40$

Cevap

The value of

x

is

40

.

The initial amount of acid is

120 \times 0.40 = 48

liters. After replacing

x

liters with pure acid, the amount of acid becomes

48 - 0.40x + x = 48 + 0.60x

liters. Removing

x

liters of this intermediate solution removes a fraction

\frac{x}{120}

of its acid, leaving

(48 + 0.60x)\left(1 - \frac{x}{120}\right)

liters. Adding pure water does not add acid. Setting this equal to the final acid quantity of

48

liters yields the equation

(48 + 0.60x)(120 - x) = 5760

, which simplifies to

24x - 0.60x^2 = 0

. Solving for non-zero

x

gives

x = 40

.

Adım Adım Çözüm

1

Calculate initial acid quantity and acid quantity after the first replacement.

Initial acid =

120 \times 0.40 = 48

liters. Removing

x

liters of

40\%

solution removes

0.40x

liters of acid. Adding

x

liters of pure acid gives a new acid quantity of

48 - 0.40x + x = 48 + 0.60x

liters.

Pure acid contains

100\%

acid, so replacing

x

liters adds

1.00x

acid while removing

0.40x

acid.

2

Determine the acid quantity after the second replacement.

The solution concentration after Step 1 is

\frac{48 + 0.60x}{120}

. Removing

x

liters removes a fraction

\frac{x}{120}

of the solution, leaving a fraction

\left(1 - \frac{x}{120}\right)

. Replacing with pure water adds

0

acid, so final acid quantity is

(48 + 0.60x)\left(1 - \frac{x}{120}\right)

.

Pure water contributes

0\%

acid to the total solute count.

3

Set up and solve the equation for the final acid amount.

(48 + 0.60x)\left(\frac{120 - x}{120}\right) = 48 \implies (48 + 0.60x)(120 - x) = 5760

. Expanding gives

5760 - 48x + 72x - 0.60x^2 = 5760 \implies 24x - 0.60x^2 = 0

. Since

x > 0

,

0.60x = 24 \implies x = 40

.

The final concentration is given as

40\%

, which corresponds to

48

liters of acid in

120

liters total volume.

Anahtar Kavram

Multi-stage mixture dilution and replacement equations using solute mass balance.

Tahmini Süre:2m 0s

Soru 875Soru →

A dataset consists of 20 distinct numerical values arranged in increasing order. The lower quartile ( $Q_1$ ) of the dataset is equal to the average of the 5th and 6th values, and the upper quartile ( $Q_3$ ) is equal to the average of the 15th and 16th values. Given that the lower quartile $Q_1 = 42$ , the interquartile range ( $\text{IQR} = Q_3 - Q_1$ ) is 38, and the 15th value in the dataset is 74, what is the value of the 16th term?

Cevabı ve açıklamayı göster

Cevap: 86

Cevap

86

To find the 16th value, first determine the upper quartile

Q_3

using the interquartile range:

Q_3 = Q_1 + \text{IQR} = 42 + 38 = 80

. Since

Q_3

is the average of the 15th and 16th terms, set up the equation

\frac{74 + x_{16}}{2} = 80

. Multiplying by 2 gives

74 + x_{16} = 160

, so

x_{16} = 86

.

Adım Adım Çözüm

1

Calculate the upper quartile (

Q_3

) using the given lower quartile (

Q_1

) and interquartile range (IQR).

Q_3 = Q_1 + \text{IQR} = 42 + 38 = 80

By definition, the interquartile range is the difference between the upper and lower quartiles (

\text{IQR} = Q_3 - Q_1

).

2

Relate the upper quartile (

Q_3

) to the 15th and 16th values of the ordered dataset.

Q_3 = \frac{x_{15} + x_{16}}{2} = 80

For an ordered dataset of 20 elements,

Q_3

is the median of the upper half of the data (the 11th through 20th terms), which equals the arithmetic mean of the 15th and 16th terms.

3

Substitute the known value of the 15th term (

x_{15} = 74

) into the equation and solve for the 16th term (

x_{16}

).

\frac{74 + x_{16}}{2} = 80 \implies 74 + x_{16} = 160 \implies x_{16} = 86

Multiplying both sides by 2 gives 160, and subtracting 74 yields 86.

Anahtar Kavram

Interquartile Range and Quartile Calculations

Soru 876Soru →

Set $S$ consists of five distinct integers: $14, 22, 18, 9,$ and $31$ . What is the range of the numbers in Set $S$ ?

Cevabı ve açıklamayı göster

Cevap: 22

Cevap

The range of the numbers in Set

S

is

22

.

To find the range of a set of numbers, subtract the minimum value from the maximum value in the set. For Set

S = \{14, 22, 18, 9, 31\}

, the maximum value is

31

and the minimum value is

9

. Subtracting the minimum from the maximum gives

31 - 9 = 22

.

Adım Adım Çözüm

1

Identify the maximum and minimum elements in the given set.

The maximum element is

31

and the minimum element is

9

.

The range of a dataset is defined as the difference between its greatest and least values.

2

Subtract the minimum value from the maximum value.

31 - 9 = 22

.

Applying the formula

\text{Range} = \text{Maximum} - \text{Minimum}

yields the range of the set.

Anahtar Kavram

Range of a Numerical Data Set

Soru 877Soru →

In a corporate firm of $N$ employees, each employee earns a distinct annual salary. An employee earning a salary of $\$78,000$ is at the $60\text{th}$ percentile of all salaries in the firm. The firm subsequently hires $10$ new employees, each of whom earns an annual salary strictly less than $\$78,000$ . If $\$78,000$ is now at the $70\text{th}$ percentile of all salaries in the expanded firm, how many employees were originally in the firm?

30

40

50

70

100

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

The percentile rank of a score indicates the percentage of values in the set that are less than or equal to that score. Originally,

60\%

of

N

employees earned

\le \$78,000

, giving

0.60N

employees. Adding

10

employees who all earn under

\$78,000

brings the count of employees earning

\le \$78,000

to

0.60N + 10

, while the total workforce becomes

N + 10

. Since

\$78,000

is at the 70th percentile of the new distribution,

0.60N + 10 = 0.70(N + 10)

. Expanding and solving yields

0.60N + 10 = 0.70N + 7

, so

0.10N = 3

, which gives

N = 30

.

Adım Adım Çözüm

1

Define the initial number of employees earning at or below $78,000.

Initially,

0.60N

employees earn

\le \$78,000

.

By definition of percentile rank with distinct values, being at the 60th percentile of

N

scores means

60\%

of the total

N

salaries are less than or equal to

\$78,000

.

2

Determine the new count of employees and the updated count of employees earning at or below $78,000.

New total employees

= N + 10

; new count earning

\le \$78,000

is

0.60N + 10

.

All 10 newly hired employees earn salaries strictly less than

\$78,000

, increasing the count of salaries

\le \$78,000

by 10.

3

Set up an equation using the new percentile rank.

0.60N + 10 = 0.70(N + 10)

The salary

\$78,000

is now at the 70th percentile of the updated total group size of

(N + 10)

.

4

Solve the algebraic equation for

N

.

0.60N + 10 = 0.70N + 7 \implies 3 = 0.10N \implies N = 30

.

Subtracting

0.60N

and

7

from both sides gives

0.10N = 3

, which yields

N = 30

.

Anahtar Kavram

Percentile Rank in Expanding Data Sets

Soru 878Soru →

A software enterprise sells two annual licensing tiers, Tier 1 and Tier 2. In January, the company sold a combined total of $200$ licenses for the two tiers. The price of a Tier 1 license was $\$40$ and the price of a Tier 2 license was $\$60$ . In February, the price of a Tier 1 license increased by $25\%$ and the number of Tier 1 licenses sold increased by $50\%$ , while both the price and the number of Tier 2 licenses sold remained unchanged from January. If the total monthly revenue from these two tiers increased by $\$2,800$ in February compared to January, how many Tier 1 licenses were sold in January?

60

70

80

100

120

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80 Tier 1 licenses were sold in January.

The correct answer is 80. Modeling the sales volume of Tier 1 in January as

x

, Tier 1 generated

40x

dollars in January. In February, with a

25\%

price increase (to

\$50

) and a

50\%

volume increase (to

1.5x

), Tier 1 generated

50(1.5x) = 75x

dollars. Because Tier 2 price and sales volume remained constant, the total revenue increase of

\$2,800

is equal to the change in Tier 1 revenue:

75x - 40x = 35x = 2,800

, which gives

x = 80

.

Adım Adım Çözüm

1

Define variables for January sales volume and set up equations for each tier.

Let

x

be the number of Tier 1 licenses sold in January. The number of Tier 2 licenses sold in January is

200 - x

.

The total number of licenses sold in January was given as 200.

2

Calculate January revenue from Tier 1.

\text{Revenue}_{\text{Tier 1, Jan}} = 40x.

Tier 1 licenses cost

\$40

each in January.

3

Determine February price and quantity for Tier 1, and calculate February Tier 1 revenue.

\text{Price}_{\text{Tier 1, Feb}} = 40 \times 1.25 = \

50

; \text{Quantity}_{\text{Tier 1, Feb}} = 1.5x$. Thus, \text{Revenue}_{\text{Tier 1, Feb}} = 50(1.5x) = 75x.

Tier 1 price increased by

25\%

and volume increased by

50\%

.

4

Express the total revenue change between February and January.

Since Tier 2 price and volume did not change, net revenue change comes entirely from Tier 1:

75x - 40x = 35x

.

Unchanged terms cancel out when taking the difference (February Revenue minus January Revenue).

5

Solve for

x

using the given revenue increase of

\$2,800

.

35x = 2,800 \implies x = 80.

Dividing both sides by 35 gives the January quantity for Tier 1 licenses.

Anahtar Kavram

Algebraic Word Problems and Equation Modeling

Tahmini Süre:2m 0s

Soru 879Soru →

A dataset contains $20$ distinct test scores arranged in ascending order:

$10, 12, 15, 17, 20, 22, 25, 28, 30, 32, 35, 38, 40, 43, 45, 48, 50, 52, 55, 60$

If the $p\text{th}$ percentile of a dataset of $N$ values is defined as the value at position $k = \frac{p}{100} \times N$ when ordered from least to greatest, what is the $75\text{th}$ percentile of these test scores?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The 75th percentile of the given test scores is 45.

The 75th percentile corresponds to position k = (75/100) * 20 = 15 in the ordered list of 20 scores, which is equal to 45.

Adım Adım Çözüm

1

Calculate the position index

k

for the

75\text{th}

percentile.

k = \frac{75}{100} \times 20 = 15

The rank formula determines the 1-based index of the target percentile value in an ordered set of size

N = 20

.

2

Locate the

15\text{th}

score in the ordered dataset.

The

15\text{th}

score is

45

.

Counting from the lowest score (

10

at position 1), the

15\text{th}

score in the sequence is

45

.

Anahtar Kavram

Percentiles and Quartiles

Soru 880Soru →

An investor deposits a principal amount into a savings account that earns interest at a constant annual compound rate of $r\%$ . At the end of the first year, immediately after annual interest is credited, the investor withdraws $20\%$ of the interest earned during that first year, leaving the original principal and all remaining interest in the account. At the end of the second year, the interest earned in the second year alone is $10\%$ greater than the interest earned in the first year alone. What is the value of $r$ ?

10.0\%

12.5\%

15.0\%

25.0\%

37.5\%

Cevabı ve açıklamayı göster

Cevap: $12.5\%$

Cevap

12.5%

The option specifying 12.5% is correct. Let

P

be the initial principal. The interest earned in the first year is

I_1 = P r

. After withdrawing

20\%

of this interest,

80\%

of the interest remains in the account, making the new principal for Year 2 equal to

P + 0.8Pr = P(1 + 0.8r)

. The interest earned in Year 2 is

I_2 = P(1 + 0.8r)r

. Since Year 2 interest is

10\%

greater than Year 1 interest,

I_2 = 1.10 I_1

. Equating the two expressions gives

P(1 + 0.8r)r = 1.10 Pr

. Dividing both sides by

Pr

yields

1 + 0.8r = 1.10

, which simplifies to

0.8r = 0.10

, so

r = 0.125

or

12.5\%

.

Adım Adım Çözüm

1

Express the interest earned in Year 1 in terms of principal

P

and interest rate

r

.

Year 1 interest

I_1 = P \times r

.

Simple/compound interest earned over the first single compounding period on initial principal

P

is

P \times r

.

2

Determine the account balance at the start of Year 2 after the partial interest withdrawal.

Balance at start of Year 2 =

P + 0.80(P \times r) = P(1 + 0.8r)

.

The investor keeps the principal

P

and retains

80\%

of Year 1 interest (

I_1 - 0.20 I_1 = 0.80 I_1

).

3

Express the interest earned in Year 2 (

I_2

) and set up the equation

I_2 = 1.10 \times I_1

.

P(1 + 0.8r) \times r = 1.10 \times (P \times r)

.

Year 2 interest is calculated on the updated balance at rate

r

, and is given as

10\%

greater than Year 1 interest.

4

Simplify the equation to solve for

r

.

1 + 0.8r = 1.10 \implies 0.8r = 0.10 \implies r = \frac{0.10}{0.80} = 0.125 = 12.5\%

.

Divide both sides by the non-zero quantity

P \times r

, then solve linear equation for

r

.

Anahtar Kavram

Percent Change and Compound Interest Base Tracking

Tahmini Süre:2m 0s

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