Tüm alıştırma soruları

Soru 701Soru →

How many distinct 8-digit positive integers can be formed by rearranging all of the digits $1, 1, 1, 2, 2, 3, 3,$ and $3$ such that the resulting integer is even and no two $1$ s are adjacent?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

40

To ensure the 8-digit number is even, the units (8th) digit must be 2. This leaves 7 positions to fill with three 1s, one 2, and three 3s. By arranging the 4 non-1 digits ({2, 3, 3, 3}) first, there are 4! / (1! 3!) = 4 distinct arrangements. Placing 4 digits creates 5 distinct gaps where the 1s can be placed without being adjacent. Choosing 3 gaps out of 5 for the three identical 1s gives C(5, 3) = 10 ways. Thus, the total number of valid integers is 4 * 10 = 40.

Adım Adım Çözüm

1

Fix the last digit to satisfy the even integer restriction

The 8th digit is fixed as 2, leaving 7 positions to fill with the remaining digits {1, 1, 1, 2, 3, 3, 3}.

An integer is even if and only if its units digit is even. The digit 2 is the only even digit in the set.

2

Calculate the arrangements of the non-restricted digits

4! / (1! * 3!) = 4 distinct arrangements.

Arranging the four non-1 digits ({2, 3, 3, 3}) first sets up the framework for placing the restricted 1s.

3

Place the three identical 1s into the gaps using the combination formula

C(5, 3) = 10 distinct gap selections.

The 4 arranged digits create 5 gaps. Selecting 3 distinct gaps ensures no two 1s are adjacent.

4

Multiply the arrangements of non-1 digits by the number of gap choices

4 * 10 = 40.

By the Fundamental Counting Principle, each non-1 arrangement can be combined with any valid gap placement.

Anahtar Kavram

Permutations with Indistinguishable Objects and Non-Adjacency Restrictions (Gap Method)

Tahmini Süre:2m 0s

Soru 702Soru →

If 2 is a root of the quadratic equation $x^2 - kx + 24 = 0$ , where $k$ is a constant, and the quadratic equation $x^2 - (k + 2)x + m = 0$ has exactly one real solution, what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 64

Cevap

The value of

m

is 64.

Substituting

x = 2

into

x^2 - kx + 24 = 0

gives

4 - 2k + 24 = 0

, which simplifies to

2k = 28

or

k = 14

. Substituting

k = 14

into

x^2 - (k + 2)x + m = 0

produces

x^2 - 16x + m = 0

. For a quadratic equation

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant

b^2 - 4ac

must equal 0. Therefore,

(-16)^2 - 4(1)(m) = 0

, which means

256 - 4m = 0

, giving

m = 64

.

Adım Adım Çözüm

1

Substitute the known root

x = 2

into the equation

x^2 - kx + 24 = 0

.

k = 14

Since

x = 2

is a root of the quadratic equation, evaluating the expression at

x = 2

must equal zero.

2

Substitute

k = 14

into the second quadratic equation

x^2 - (k + 2)x + m = 0

.

x^2 - 16x + m = 0

This determines the linear coefficient of the second quadratic equation.

3

Set the discriminant of

x^2 - 16x + m = 0

to zero.

m = 64

A quadratic equation

ax^2 + bx + c = 0

has exactly one real solution if and only if its discriminant

b^2 - 4ac

equals zero.

Anahtar Kavram

Solving quadratic equations via root substitution and applying the discriminant condition for repeated roots.

Tahmini Süre:1m 30s

Soru 703Soru →

A bookshelf is to be arranged using 3 identical Mathematics books, 2 identical Physics books, and 1 Chemistry book. In how many distinct ways can all 6 books be arranged in a single row such that the Chemistry book is not adjacent to any Mathematics book?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The single Chemistry book must not touch any of the 3 Mathematics books. This leaves three mutually exclusive possibilities for the Chemistry book: at the left end preceded by a Physics book (4 ways), at the right end followed by a Physics book (4 ways), or sandwiched between both Physics books as a single block (4 ways). Summing these gives 12 total valid arrangements.

Adım Adım Çözüm

1

Analyze the placement restriction on the Chemistry book

The Chemistry book (C) cannot be placed next to any Mathematics book (M). Thus, C can only be adjacent to Physics books (P) or placed at the boundary of the row touching a P.

Eliminating adjacency to M restricts C to touch only P or the ends of the shelf.

2

Count valid arrangements when C is at the far left (position 1)

The row begins with C-P. The remaining 4 spots must be filled with 3 identical M's and 1 P, yielding 4! / (3! 1!) = 4 distinct ways.

Position 1 has only one neighbor (position 2), which must be P.

3

Count valid arrangements when C is at the far right (position 6)

The row ends with P-C. The remaining 4 spots must be filled with 3 identical M's and 1 P, yielding 4! / (3! 1!) = 4 distinct ways.

Position 6 has only one neighbor (position 5), which must be P.

4

Count valid arrangements when C is in an interior position (positions 2 through 5)

C must be sandwiched between two P's, forming the block (P-C-P). Arranging this single block along with the 3 identical M's (total of 4 items) yields 4! / (3! 1!) = 4 distinct ways.

Any interior placement requires both adjacent neighbors of C to be P.

5

Sum the counts from all mutually exclusive cases

Total valid arrangements = 4 + 4 + 4 = 12.

The three cases cover all possible non-overlapping valid placements for C.

Anahtar Kavram

Counting permutations with identical elements and positional restrictions using case analysis and block formation.

Tahmini Süre:1m 30s

Soru 704Soru →

A commercial bakery has two industrial ovens, Oven A and Oven B. Working alone at its constant rate, Oven A can bake a full order of $1,800$ pastries in $6$ hours. Working alone at its constant rate, Oven B can bake the same order in $9$ hours. If Oven A begins baking the order alone and is joined by Oven B after $2$ hours, how many total hours will it take from the time Oven A starts until the full order of $1,800$ pastries is baked?

Cevabı ve açıklamayı göster

Cevap: 4.4

Cevap

The total time required to complete the entire order of pastries is 4.4 hours.

The total time of 4.4 hours is derived by determining Oven A's rate (1/6 job/hour) and Oven B's rate (1/9 job/hour). In the first 2 hours, Oven A completes 1/3 of the job, leaving 2/3. Operating together, their combined rate is 5/18 job/hour, which finishes the remaining 2/3 of the job in 2.4 hours. Summing 2 hours and 2.4 hours yields 4.4 total hours.

Adım Adım Çözüm

1

Determine the individual work rates per hour.

Oven A completes

\frac{1}{6}

of the job per hour; Oven B completes

\frac{1}{9}

of the job per hour.

Work rate is the reciprocal of the total time needed to complete one full job.

2

Calculate the fraction of work completed during the first 2 hours by Oven A.

Oven A completes

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

of the total job.

Work done equals rate multiplied by time spent working.

3

Find the remaining fraction of work to be done.

Remaining work is

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

of the job.

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

4

Calculate the combined rate of Oven A and Oven B working together.

Combined rate is

\frac{1}{6} + \frac{1}{9} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}

job per hour.

When entities work together, their individual rates add up.

5

Calculate the time spent by both ovens working together to complete the remaining work.

Time together is

\frac{2/3}{5/18} = \frac{12}{5} = 2.4

hours.

Time equals remaining work divided by the combined rate.

6

Calculate total time from the start.

Total time =

2 + 2.4 = 4.4

hours.

Add the initial single-agent work duration to the combined work duration.

Anahtar Kavram

Work Rate and Combined Work

Soru 705Soru →

For all real numbers $x$ except $0$ and $1$ , the function $f(x)$ satisfies the relation $f(x) + 2f\left(\frac{1}{1-x}\right) = 9x$ . What is the value of $f(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

f(2)

is

6

.

Evaluating the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x

at the values

x = 2

,

x = -1

, and

x = \frac{1}{2}

creates a system of three linear equations in terms of

f(2)

,

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this system yields

9f(2) = 54

, which simplifies directly to

f(2) = 6

.

Adım Adım Çözüm

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) \implies f(2) + 2f(-1) = 18

Applying the input

x = 2

creates an equation connecting

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) \implies f(-1) + 2f\left(\frac{1}{2}\right) = -9

Evaluating at the new input

x = -1

generates a second equation connecting

f(-1)

and

f\left(\frac{1}{2}\right)

.

3

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) \implies f\left(\frac{1}{2}\right) + 2f(2) = 4.5

Evaluating at

x = \frac{1}{2}

completes the cyclic chain by linking back to

f(2)

.

4

Solve the system of three linear equations for

f(2)

From equation 3,

f\left(\frac{1}{2}\right) = 4.5 - 2f(2)

. Substituting into equation 2 yields

f(-1) = 4f(2) - 18

. Substituting into equation 1 gives

f(2) + 2(4f(2) - 18) = 18 \implies 9f(2) = 54 \implies f(2) = 6

.

Eliminating

f\left(\frac{1}{2}\right)

and

f(-1)

isolated

f(2)

to find its exact numerical value.

Anahtar Kavram

Cyclic Functional Equations and Substitution Systems

Soru 706Soru →

If $2x^2 + 5x - 3 = 0$ and $x > 0$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

The value of

x

is 0.5.

Factoring the quadratic equation

2x^2 + 5x - 3 = 0

yields

(2x - 1)(x + 3) = 0

. Setting each linear factor equal to zero gives two possible solutions for

x

:

x = 0.5

and

x = -3

. Because the problem stipulates that

x > 0

, the negative solution is discarded, leaving

x = 0.5

.

Adım Adım Çözüm

1

Factor the quadratic expression

(2x - 1)(x + 3) = 0

Splitting the middle term

5x

into

6x - x

allows grouping to factor by grouping.

2

Find the roots of the equation

x = 0.5

or

x = -3

By the zero-product property, if the product of two factors is zero, at least one factor must be zero.

3

Apply the positivity constraint

x > 0

x = 0.5

The root

x = -3

violates the given condition that

x

must be strictly greater than zero.

Anahtar Kavram

Solving quadratic equations by factoring and applying domain constraints

Soru 707Soru →

A freight logistics warehouse initially stores Standard, Refrigerated, and Oversized cargo containers in the ratio $4 : 3 : 2$ , respectively. During a morning shift, $12$ Standard containers are shipped out and $6$ Refrigerated containers arrive at the warehouse, while no Oversized containers are moved. Following these changes, the ratio of Standard containers to Refrigerated containers in the warehouse becomes $1 : 1$ . What was the total number of containers (Standard, Refrigerated, and Oversized) initially stored in the warehouse?

18

72

126

156

162

Cevabı ve açıklamayı göster

Cevap: 162

Cevap

162

The correct answer 162 is obtained by representing the initial numbers of containers as

4x

,

3x

, and

2x

. Equating the modified quantities of Standard (

4x - 12

) and Refrigerated (

3x + 6

) yields

x = 18

. Summing all three parts gives

9x = 9 \times 18 = 162

.

Adım Adım Çözüm

1

Define initial quantities using a common variable based on the given three-part ratio.

Let the initial numbers of Standard, Refrigerated, and Oversized containers be

4x

,

3x

, and

2x

, respectively. The initial total count of containers is

4x + 3x + 2x = 9x

.

Ratios represent relative parts, so multiplying each term by a constant multiplier

x

yields the actual quantities.

2

Set up an equation reflecting the change in container quantities.

After shipping

12

Standard containers and adding

6

Refrigerated containers, the new quantities are

(4x - 12)

Standard containers and

(3x + 6)

Refrigerated containers. Since their new ratio is

1 : 1

, we write

4x - 12 = 3x + 6

.

A

1 : 1

ratio means the two quantities are equal.

3

Solve for the multiplier

x

.

4x - 3x = 6 + 12 \implies x = 18

.

Isolating

x

gives the ratio scaling factor.

4

Calculate the initial total number of containers.

Initial total

= 9x = 9 \times 18 = 162

.

The question asks for the total initial count across all three container categories (

4x + 3x + 2x = 9x

).

Anahtar Kavram

Altering linear ratios by solving for a common multiplier across multi-part ratio components.

Soru 708Soru →

For all real numbers $a$ and $b$ such that $a + b \neq 0$ , the custom operator $\star$ is defined by $a \star b = \frac{a - b}{a + b}$ . What is the value of $(3 \star 1) \star 2$ ?

-\frac{5}{3}

-\frac{3}{5}

0

\frac{3}{5}

\frac{5}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{3}{5}$

Cevap

-\frac{3}{5}

Evaluating the expression step-by-step according to parentheses yields

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

. Substituting

\frac{1}{2}

as the first input and

2

as the second input gives

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

.

Adım Adım Çözüm

1

Evaluate the inner custom operation inside parentheses:

3 \star 1

.

3 \star 1 = \frac{3 - 1}{3 + 1} = \frac{2}{4} = \frac{1}{2}

.

Follow the order of operations by resolving the grouped expression first using the definition

a \star b = \frac{a - b}{a + b}

with

a = 3

and

b = 1

.

2

Substitute the result

\frac{1}{2}

back into the main expression to compute

(\frac{1}{2}) \star 2

.

(\frac{1}{2}) \star 2 = \frac{\frac{1}{2} - 2}{\frac{1}{2} + 2}

.

Apply the definition of the custom operator again, where the left operand is

\frac{1}{2}

and the right operand is

2

.

3

Simplify the complex fraction.

\frac{\frac{1}{2} - \frac{4}{2}}{\frac{1}{2} + \frac{4}{2}} = \frac{-\frac{3}{2}}{\frac{5}{2}} = -\frac{3}{5}.

Combine the fractions in the numerator and denominator, then divide.

Anahtar Kavram

Custom Operators and Order of Operations

Tahmini Süre:1m 30s

Soru 709Soru →

A conference schedule consists of $6$ consecutive time slots. The organizers must schedule $3$ identical workshops on Artificial Intelligence, $2$ identical workshops on Cybersecurity, and $1$ keynote address on Data Privacy. If the $2$ Cybersecurity workshops cannot be scheduled in consecutive time slots, how many distinct presentation schedules are possible?

20

40

50

60

120

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

40 distinct presentation schedules are possible.

The total number of unrestricted ways to arrange the 6 events (3 identical AI, 2 identical Cybersecurity, 1 Data Privacy) is calculated using multiset permutations as

\frac{6!}{3! \times 2! \times 1!} = 60

. To find the number of ways where the two Cybersecurity workshops are NOT consecutive, we use complementary counting. By treating the two Cybersecurity workshops as one glued block, we arrange 5 items (3 identical AI, 1 Cybersecurity block, 1 Data Privacy), yielding

\frac{5!}{3! \times 1! \times 1!} = 20

forbidden arrangements. Subtracting the 20 forbidden arrangements from the 60 total arrangements gives 40 valid presentation schedules.

Adım Adım Çözüm

1

Calculate the total number of distinct schedules without restrictions.

Total arrangements =

\frac{6!}{3! \times 2! \times 1!} = \frac{720}{6 \times 2 \times 1} = 60

.

There are

6

total slots with

3

identical AI workshops,

2

identical Cybersecurity workshops, and

1

Data Privacy keynote.

2

Calculate the number of forbidden schedules where the

2

Cybersecurity workshops are in consecutive slots.

Forbidden arrangements =

\frac{5!}{3! \times 1! \times 1!} = \frac{120}{6} = 20

.

Treat the

2

identical Cybersecurity workshops as a single combined block. This leaves

5

items to arrange (

3

AI,

1

combined Cybersecurity block,

1

Data Privacy).

3

Subtract the forbidden arrangements from the total arrangements using complementary counting.

Valid schedules =

60 - 20 = 40

.

The number of valid restricted arrangements is total arrangements minus restricted consecutive arrangements.

Anahtar Kavram

Counting with Restrictions and Repetitions (Complementary Counting and Permutations of Multisets)

Tahmini Süre:2m 0s

Soru 710Soru →

A company creates 5-digit employee identification codes using all of the digits $1, 1, 2, 2,$ and $3$ . How many distinct 5-digit identification codes can be formed such that the two $2$ 's are not adjacent to each other?

12

18

24

30

60

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

To find the number of distinct 5-digit identification codes where the two 2's are not adjacent, use complementary counting. First, compute the total distinct arrangements of the digits

1, 1, 2, 2, 3

, which is

\frac{5!}{2! \times 2!} = 30

. Next, find the number of arrangements where the two 2's are adjacent by treating

(22)

as a single item. Arranging

(22), 1, 1, 3

yields

\frac{4!}{2!} = 12

arrangements. Subtracting the adjacent arrangements from the total gives

30 - 12 = 18

.

Adım Adım Çözüm

1

Calculate the total number of distinct 5-digit arrangements of the digits

1, 1, 2, 2, 3

without any restrictions.

Total arrangements =

\frac{5!}{2! \times 2! \times 1!} = \frac{120}{4} = 30

.

When arranging elements with repeated indistinguishable items, divide

n!

by the factorials of the counts of each repeated item.

2

Calculate the number of restricted (forbidden) arrangements where the two

2

's are adjacent.

Adjacent arrangements =

\frac{4!}{2! \times 1!} = \frac{24}{2} = 12

.

Treat the two adjacent

2

's as a single block

(22)

. We now arrange 4 items:

(22), 1, 1, 3

, where the digit

1

appears twice.

3

Subtract the forbidden arrangements from the total arrangements using complementary counting.

Non-adjacent arrangements =

30 - 12 = 18

.

Complementary counting gives the number of valid arrangements where the two

2

's are not adjacent.

Anahtar Kavram

Permutations with Repeated Indistinguishable Elements and Complementary Counting

Tahmini Süre:1m 30s

Soru 711Soru →

The quadratic equation $x^2 - 6x + a = 0$ has two distinct real roots $\alpha$ and $\gamma$ , and the quadratic equation $y^2 - by + 21 = 0$ has two distinct real roots $\beta$ and $\delta$ . If $\alpha < \beta < \gamma < \delta$ and the four roots form an arithmetic progression in that order, what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of

a + b

is 15.

By representing the four ordered roots as

r, r+d, r+2d, r+3d

, Vieta's formula for the sum of roots of the first equation yields

r + (r+2d) = 2(r+d) = 6

, which implies

r+d = 3

. Thus, the second root is

\beta = 3

. Using the product of roots for the second equation,

3\delta = 21

gives

\delta = 7

. The common difference is

d = (7-3)/2 = 2

, which gives the roots

1, 3, 5, 7

. Finally,

a = 1 \times 5 = 5

and

b = 3 + 7 = 10

, so

a + b = 15

.

Adım Adım Çözüm

1

Apply Vieta's formulas to both quadratic equations.

\alpha + \gamma = 6

,

\alpha\gamma = a

,

\beta + \delta = b

, and

\beta\delta = 21

.

Vieta's relations connect the coefficients of a quadratic polynomial to the sum and product of its roots.

2

Set up the arithmetic progression representation for the roots.

\alpha = r

,

\beta = r + d

,

\gamma = r + 2d

, and

\delta = r + 3d

, where

d > 0

.

The roots form an increasing arithmetic progression in the order

\alpha, \beta, \gamma, \delta

.

3

Substitute the expressions for

\alpha

and

\gamma

into

\alpha + \gamma = 6

.

r + (r + 2d) = 2r + 2d = 2(r + d) = 6 \implies r + d = 3

.

Combining terms simplifies the sum of the first and third terms of the arithmetic progression.

4

Identify the value of

\beta

and solve for

\delta

.

\beta = r + d = 3

, so

\beta\delta = 21 \implies 3\delta = 21 \implies \delta = 7

.

Since

\beta = r + d

, its value is directly determined as 3, allowing

\delta

to be solved from the product relation.

5

Calculate the common difference

d

and the first term

r

.

\delta - \beta = 2d = 7 - 3 = 4 \implies d = 2

, and

r = 3 - 2 = 1

.

The difference between the fourth and second terms of an AP is equal to

2d

.

6

Find

a

,

b

, and their sum

a + b

.

a = \alpha\gamma = 1 \times 5 = 5

,

b = \beta + \delta = 3 + 7 = 10

, so

a + b = 5 + 10 = 15

.

With all four roots determined (

1, 3, 5, 7

), the missing coefficients are calculated using Vieta's formulas.

Anahtar Kavram

Combining Vieta's Formulas with Arithmetic Progressions to Solve Quadratic Systems

Soru 712Soru →

At a corporate law firm, the ratio of Partners to Associates was initially $3 : 8$ , and the ratio of Associates to Paralegals was initially $4 : 5$ . During an internal restructuring, $15$ Associates were promoted to Partners, and $2$ Paralegals resigned. As a result of these two changes, the ratio of Partners to Paralegals became $5 : 8$ . Shortly thereafter, the firm conducted a hiring drive, recruiting additional Associates until the ratio of Associates to Paralegals became $7 : 6$ , while the numbers of Partners and Paralegals remained unchanged. What was the total number of staff members (Partners, Associates, and Paralegals) at the firm after the hiring drive?

Cevabı ve açıklamayı göster

Cevap: 134

Cevap

134

The solution unifies the initial ratios into 3 : 8 : 10, sets up the algebraic equations reflecting both promotions and resignations to find the multiplier k = 5, updates the staff counts, and applies the final ratio to find 56 Associates, yielding a final total of 134 staff members.

Adım Adım Çözüm

1

Unify initial two-variable ratios into a continuous three-part ratio.

Partners : Associates : Paralegals = 3 : 8 : 10

Associates is the common term. Scaling 4 : 5 by 2 gives 8 : 10, matching the 8 parts in the Partners-to-Associates ratio.

2

Formulate algebraic expressions for post-restructuring staff counts.

Partners = 3k + 15, Associates = 8k - 15, Paralegals = 10k - 2

Promotions transfer 15 from Associates to Partners, and 2 Paralegals leave.

3

Solve for the ratio multiplier k using the new Partner-to-Paralegal ratio.

k = 5

Setting (3k + 15) / (10k - 2) = 5 / 8 yields 24k + 120 = 50k - 10, so 26k = 130.

4

Calculate exact staff counts after restructuring.

Partners = 30, Associates = 25, Paralegals = 48

Substitute k = 5 into the expressions from Step 2.

5

Determine final Associate count and total staff after hiring drive.

Final Associates = 56; Total Staff = 134

Associates = (7/6) * 48 = 56. Summing all roles gives 30 + 56 + 48 = 134.

Anahtar Kavram

Multi-part ratio unification, internal transfer ratio alteration, and sequential proportion scaling

Soru 713Soru →

A research laboratory mixes Solution A and Solution B in a ratio of $3 : 5$ by volume to create a standard compound. If a technician needs to prepare $240$ milliliters of the compound, how many milliliters of Solution A are required?

48

milliliters

80

milliliters

90

milliliters

144

milliliters

150

milliliters

Cevabı ve açıklamayı göster

Cevap: $90$ milliliters

Cevap

Solution A requires

90

milliliters of liquid.

The total compound consists of

3 + 5 = 8

parts. The fraction of Solution A in the total mixture is

\frac{3}{8}

. Multiplying this fraction by the total volume of

240

milliliters yields

\frac{3}{8} \times 240 = 90

milliliters.

Adım Adım Çözüm

1

Calculate the total number of ratio parts

The ratio

3 : 5

yields

3 + 5 = 8

equal parts in total.

To find the value of one part, the total volume must be divided by the total sum of the ratio terms.

2

Determine the volume of a single ratio part

\frac{240 \text{ mL}}{8 \text{ parts}} = 30 \text{ mL per part}

.

Dividing the total volume by the total parts gives the multiplier per ratio unit.

3

Multiply the single part volume by Solution A's ratio portion

3 \times 30 \text{ mL} = 90 \text{ mL}

.

Solution A accounts for

3

of the

8

total parts.

Anahtar Kavram

Part-to-Whole Ratio Scaling

Tahmini Süre:1m 0s

Soru 714Soru →

Scanner Unit Alpha and Scanner Unit Beta, operating independently at their respective constant rates, can complete a document digitization project together in $12$ hours. Scanner Unit Alpha operates at a rate that is $50\%$ faster than that of Scanner Unit Beta. The project is carried out in three consecutive stages: first, Scanner Unit Alpha operates alone for $4$ hours; second, Scanner Unit Beta joins Alpha and both operate together until $70\%$ of the entire project is completed; finally, Scanner Unit Beta finishes the remaining portion of the project alone. How many total hours does it take to complete the entire digitization project?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

The total time required to complete the entire digitization project is 19 hours.

To find the total time needed, calculate the individual rates first. With a combined rate of

r_A + r_B = \frac{1}{12}

and

r_A = 1.5 r_B

, solving

2.5 r_B = \frac{1}{12}

yields

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

. During Stage 1 (

4

hours), Alpha completes

4 \times \frac{1}{20} = 0.20

of the job. In Stage 2, both units work together to bring completion from

20\%

to

70\%

(

0.50

work), taking

\frac{0.50}{1/12} = 6

hours. In Stage 3, Beta completes the remaining

0.30

work alone, taking

\frac{0.30}{1/30} = 9

hours. Summing all stage durations yields

4 + 6 + 9 = 19

hours.

Adım Adım Çözüm

1

Determine the individual work rates of Scanner Unit Alpha (

r_A

) and Scanner Unit Beta (

r_B

).

r_B = \frac{1}{30}

project per hour, and

r_A = \frac{1}{20}

project per hour.

Since their combined rate is

\frac{1}{12}

project per hour and

r_A = 1.5 r_B

, we solve

2.5 r_B = \frac{1}{12}

to find

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

.

2

Calculate the fraction of work completed during Stage 1.

Scanner Unit Alpha completes

0.20

(

20\%

) of the project in

4

hours.

Alpha works alone for

4

hours at a rate of

\frac{1}{20}

project per hour:

4 \times \frac{1}{20} = 0.20

.

3

Calculate the duration of Stage 2 where both units work together.

Stage 2 takes

6

hours.

The combined units must complete the portion from

20\%

to

70\%

, which represents

0.70 - 0.20 = 0.50

of the project. At a combined rate of

\frac{1}{12}

project per hour, the time required is

\frac{0.50}{1/12} = 6

hours.

4

Calculate the duration of Stage 3 where Scanner Unit Beta works alone.

Stage 3 takes

9

hours.

Beta must complete the remaining

30\%

(

0.30

) of the project alone. At a rate of

\frac{1}{30}

project per hour, the time required is

\frac{0.30}{1/30} = 9

hours.

5

Sum the durations of all three stages to determine total project time.

Total time =

19

hours.

Adding the duration of each stage:

4 \text{ hours} + 6 \text{ hours} + 9 \text{ hours} = 19 \text{ hours}

.

Anahtar Kavram

Work Rate and Combined Work

Soru 715Soru →

A box contains 20 cards numbered sequentially from 1 through 20, inclusive. If one card is drawn at random from the box, what is the probability that the number on the drawn card is a prime number? Express your answer as a decimal.

Cevabı ve açıklamayı göster

Cevap: 0.4

Cevap

0.4

The total number of possible outcomes when selecting one card from 20 is 20. The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, and 19, giving 8 favorable outcomes (remembering that 1 is not prime). The single-event probability is calculated by dividing the number of favorable outcomes by the total number of outcomes, yielding

8 / 20 = 0.4

.

Adım Adım Çözüm

1

Determine the total number of possible outcomes in the sample space.

The sample space consists of 20 equally likely outcomes (integers 1 through 20).

Calculating single-event probability requires establishing the size of the total outcome space

N

.

2

Count the number of prime numbers in the set

\{1, 2, \dots, 20\}

.

There are 8 prime numbers:

\{2, 3, 5, 7, 11, 13, 17, 19\}

.

By definition, a prime number is an integer greater than 1 with exactly two positive divisors: 1 and itself. Thus, 1 is excluded.

3

Compute the probability using

P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}

.

P = \frac{8}{20} = 0.4

.

Directly apply the basic single-event probability formula.

Anahtar Kavram

Basic Single-Event Probability and Prime Number Identification

Tahmini Süre:45s

Soru 716Soru →

At a research institute, the ratio of senior scientists to associate scientists to junior researchers was initially $2 : 3 : 5$ , respectively. During an organizational expansion, 8 new senior scientists were hired, and 4 junior researchers were promoted to associate scientists, while no other personnel changes occurred. If the new ratio of senior scientists to associate scientists became $4 : 5$ , what was the initial total number of researchers at the institute?

40

60

100

120

128

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

By assigning a multiplier

x

to the initial ratio

2 : 3 : 5

, the counts are

2x

senior scientists,

3x

associate scientists, and

5x

junior researchers, making the initial total

10x

. Adding 8 senior scientists gives

2x + 8

, and adding 4 promoted associate scientists gives

3x + 4

. Equating their ratio to

\frac{4}{5}

yields

\frac{2x + 8}{3x + 4} = \frac{4}{5}

, which simplifies to

2x = 24

, or

x = 12

. Multiplying

x = 12

by the total 10 parts gives the correct initial total of 120.

Adım Adım Çözüm

1

Define initial quantities using a common ratio multiplier

x

.

Senior scientists =

2x

, Associate scientists =

3x

, Junior researchers =

5x

. Initial total researchers =

2x + 3x + 5x = 10x

.

Expressing quantities in terms of

x

allows setting up algebraic equations based on personnel updates.

2

Apply the specified personnel changes to find the new counts of senior and associate scientists.

New Senior scientists =

2x + 8

. New Associate scientists =

3x + 4

(since 4 junior researchers were promoted to associate scientists).

Promoting 4 junior researchers increases the associate scientist count by 4.

3

Set up the proportion equation for the new ratio of senior scientists to associate scientists.

\frac{2x + 8}{3x + 4} = \frac{4}{5}

The problem states the new ratio between senior and associate scientists is

4 : 5

.

4

Cross-multiply and solve for

x

.

5(2x + 8) = 4(3x + 4) \implies 10x + 40 = 12x + 16 \implies 2x = 24 \implies x = 12.

Solving for

x

provides the multiplier required to compute the initial total.

5

Calculate the initial total number of researchers.

Initial total =

10x = 10 \times 12 = 120

.

The total number of initial researchers corresponds to

10x

parts.

Anahtar Kavram

Setting up and solving algebraic proportions involving multi-part ratios after internal transfers and external additions

Tahmini Süre:2m 0s

Soru 717Soru →

At a financial analytics firm, a group of 200 analysts were evaluated on their proficiency in three software tools: Options Analytics, Futures Trader, and Swaps Pricing. Exactly 15% of the analysts had no proficiency in any of the three tools. Among the remaining analysts, 110 were proficient in Options Analytics, 95 were proficient in Futures Trader, and 85 were proficient in Swaps Pricing. If exactly 20 analysts were proficient in all three tools, how many analysts were proficient in exactly one of the three tools?

10

30

50

70

100

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

70 analysts were proficient in exactly one of the three tools.

The total number of analysts is 200, and 15% (30 analysts) have no proficiency, leaving 170 analysts proficient in at least one tool. Applying the 3-set inclusion-exclusion formula

N(A \cup B \cup C) = N(A) + N(B) + N(C) - S_2 + N(A \cap B \cap C)

, we get

170 = 110 + 95 + 85 - S_2 + 20

, which yields

S_2 = 140

. Since

S_2

counts elements in exactly two sets once and elements in all three sets three times, the number of analysts proficient in exactly two tools is

140 - 3(20) = 80

. Finally, subtracting those proficient in exactly two tools (80) and all three tools (20) from the total proficient in at least one tool (170) gives

170 - 80 - 20 = 70

.

Adım Adım Çözüm

1

Calculate the total number of analysts proficient in at least one tool

\text{At least one} = 200 - (0.15 \times 200) = 200 - 30 = 170

Analysts who are not proficient in any tool must be excluded from the total group size to find the union of the three sets.

2

Apply the 3-set inclusion-exclusion formula to find the sum of pairwise intersections

170 = 110 + 95 + 85 - S_2 + 20 \implies 170 = 310 - S_2 \implies S_2 = 140

, where

S_2 = N(\text{Options} \cap \text{Futures}) + N(\text{Futures} \cap \text{Swaps}) + N(\text{Options} \cap \text{Swaps})

The standard inclusion-exclusion principle states that

N(A \cup B \cup C) = N(A) + N(B) + N(C) - S_2 + N(A \cap B \cap C)

.

3

Determine the number of analysts proficient in exactly two tools

\text{Exactly 2} = S_2 - 3 \times N(\text{All 3}) = 140 - 3(20) = 140 - 60 = 80

Each member of the triple intersection is counted 3 times in

S_2

. Subtracting

3 \times N(\text{All 3})

isolates the elements belonging to exactly two sets.

4

Calculate the number of analysts proficient in exactly one tool

\text{Exactly 1} = N(\text{At least 1}) - \text{Exactly 2} - N(\text{All 3}) = 170 - 80 - 20 = 70

The union of the three sets consists of elements proficient in exactly 1 tool, exactly 2 tools, and all 3 tools.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Subset Decomposition

Tahmini Süre:2m 0s

Soru 718Soru →

Three specialized synthesis columns—Alpha, Beta, and Gamma—are used in a pharmaceutical facility to purify batches of a chemical compound. Operating simultaneously at their respective constant rates, Column Alpha and Column Beta can process a standard batch in $8$ hours, while Column Beta and Column Gamma working together can process the exact same batch in $12$ hours.

Column Alpha begins processing a standard batch alone. After $4$ hours of operation, Column Alpha experiences a mechanical restriction that reduces its processing rate by $33\frac{1}{3}\%$ . At that exact moment, Column Gamma is brought online to assist Column Alpha. Working together, Column Alpha (at its reduced rate) and Column Gamma complete the remaining portion of the batch in $8.4$ hours.

How many hours would it take Column Beta, operating alone at its normal constant rate, to process an entire standard batch of the chemical compound?

15 hours

18 hours

20 hours

24 hours

30 hours

Cevabı ve açıklamayı göster

Cevap: 20 hours

Cevap

Column Beta would take 20 hours to complete an entire batch working alone.

The correct response of 20 hours is determined by establishing the rates of the three columns. With

a + b = 1/8

and

b + c = 1/12

, subtracting these equations gives

c = a - 1/24

. During the first 4 hours, Column Alpha completes

4a

of the batch. In the second stage, Column Alpha works at

2/3 a

while Column Gamma works at

c

, giving a joint rate of

5/3 a - 1/24

. Multiplying this combined rate by

8.4

hours and setting it equal to the remaining work

1 - 4a

yields

a = 3/40

. Substituting this back into

a + b = 1/8

yields

b = 1/20

, meaning Column Beta requires 20 hours to complete a batch alone.

Adım Adım Çözüm

1

Define individual work rates in batches per hour for Column Alpha (

a

), Column Beta (

b

), and Column Gamma (

c

).

From the given combined rates:

a + b = \frac{1}{8}

and

b + c = \frac{1}{12}

.

Combined work rates equal the sum of individual work rates.

2

Express Column Gamma's rate (

c

) in terms of Column Alpha's rate (

a

).

Subtracting the second equation from the first gives

(a + b) - (b + c) = \frac{1}{8} - \frac{1}{12} \implies a - c = \frac{1}{24} \implies c = a - \frac{1}{24}

.

Isolating one rate variable simplifies the multi-stage work equation.

3

Calculate the work completed during Stage 1 and express the remaining work.

Work completed in Stage 1 (

4

hours at rate

a

) is

4a

. The remaining work to be done is

1 - 4a

.

Total work equals

1

batch.

4

Formulate the combined rate for Stage 2 and set up the equation for the remaining work.

Alpha's reduced rate is

\left(1 - \frac{1}{3}\right)a = \frac{2}{3}a

. The combined rate with Gamma is

\frac{2}{3}a + c = \frac{2}{3}a + \left(a - \frac{1}{24}\right) = \frac{5}{3}a - \frac{1}{24}

. Stage 2 takes

8.4 = \frac{42}{5}

hours, so

\frac{42}{5}\left(\frac{5}{3}a - \frac{1}{24}\right) = 1 - 4a

.

Work completed in Stage 2 equals combined rate multiplied by time spent in Stage 2.

5

Solve the algebraic equation for

a

, then find

b

.

Expanding the equation:

14a - \frac{7}{20} = 1 - 4a \implies 18a = \frac{27}{20} \implies a = \frac{3}{40}

. Then

b = \frac{1}{8} - a = \frac{1}{8} - \frac{3}{40} = \frac{2}{40} = \frac{1}{20}

.

Since Beta's rate is

\frac{1}{20}

batch/hour, the time taken by Beta alone is

\frac{1}{1/20} = 20

hours.

Anahtar Kavram

Work Rate Equations and Multi-Stage Combined Work

Soru 719Soru →

For any real numbers $u$ and $v$ , the binary operator $\star$ is defined by $u \star v = 2u^2 - 3v$ . A function $f$ is defined by $f(x) = x \star 4$ , and a function $g$ is defined by $g(x) = 3x + 1$ . If $m$ is a real number such that $f(g(m)) = 38$ , what is the product of all possible values of $m$ ?

-\frac{8}{3}

-1

-\frac{16}{9}

\frac{4}{3}

\frac{8}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{8}{3}$

Cevap

The product of all possible values of

m

is

-\frac{8}{3}

.

First, evaluate

f(x) = x \star 4 = 2x^2 - 3(4) = 2x^2 - 12

. Next, substitute

g(m) = 3m + 1

into

f(x)

to get

f(g(m)) = 2(3m + 1)^2 - 12

. Setting this expression equal to

38

gives

2(3m + 1)^2 - 12 = 38

, which simplifies to

(3m + 1)^2 = 25

. Taking the square root gives two possible linear equations:

3m + 1 = 5

(which gives

m = \frac{4}{3}

) and

3m + 1 = -5

(which gives

m = -2

). Multiplying these two solutions yields

\left(\frac{4}{3}\right) \times (-2) = -\frac{8}{3}

. Thus, the option equal to

-\frac{8}{3}

is correct.

Adım Adım Çözüm

1

Evaluate the function

f(x)

using the definition of the custom operator

\star

.

f(x) = x \star 4 = 2x^2 - 3(4) = 2x^2 - 12

Substitute

u = x

and

v = 4

into the formula

u \star v = 2u^2 - 3v

.

2

Express the nested function

f(g(m))

in terms of

m

.

f(g(m)) = 2(g(m))^2 - 12 = 2(3m + 1)^2 - 12

Substitute

g(m) = 3m + 1

into

f(x)

.

3

Set

f(g(m))

equal to

38

and solve for

(3m + 1)^2

.

2(3m + 1)^2 - 12 = 38 \implies 2(3m + 1)^2 = 50 \implies (3m + 1)^2 = 25

Isolate the squared binomial term using basic algebraic manipulation.

4

Take the square root of both sides to find all possible values of

m

.

3m + 1 = 5 \implies m = \frac{4}{3}

or

3m + 1 = -5 \implies m = -2

A positive real number has both positive and negative square roots.

5

Calculate the product of the two solutions for

m

.

\left(\frac{4}{3}\right) \times (-2) = -\frac{8}{3}

Multiply the two roots together as requested by the question stem.

Anahtar Kavram

Evaluating custom binary operators and nested composite functions, solving quadratic equations, and finding products of roots.

Soru 720Soru →

Machine X and Machine Y, operating independently at their respective constant rates, can complete a bottling order together in $6$ hours. If Machine X works alone for $2$ hours and is then joined by Machine Y, both machines work together for an additional $4.8$ hours to finish the remaining part of the order. How many hours would it take Machine Y, operating alone at its constant rate, to complete the entire bottling order?

10

12

15

16

18

Cevabı ve açıklamayı göster

Cevap: $15$

Cevap

15

hours

The correct answer is

15

hours. Using the work equation

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

, the combined rate of Machine X and Machine Y is

\frac{1}{6}

. The work completed by both machines in the

4.8

-hour period is

4.8 \times \frac{1}{6} = 0.8

of the job. Since the remaining

0.2

of the job was completed by Machine X in

2

hours, Machine X's rate is

\frac{0.2}{2} = 0.1 = \frac{1}{10}

per hour. Subtracting Machine X's rate from the combined rate yields Machine Y's rate:

\frac{1}{6} - \frac{1}{10} = \frac{1}{15}

per hour. Therefore, Machine Y takes

15

hours operating alone to finish the entire bottling order.

Adım Adım Çözüm

1

Define rates for Machine X and Machine Y

Let Machine X's rate be

r_x

orders per hour and Machine Y's rate be

r_y

orders per hour. Their combined rate is

r_x + r_y = \frac{1}{6}

orders per hour.

Combined work rate is the reciprocal of the combined completion time of

6

hours.

2

Express the total work completed in two stages

Machine X works alone for

2

hours completing

2 r_x

of the order. Then both work together for

4.8

hours completing

4.8(r_x + r_y)

of the order. Thus,

2 r_x + 4.8(r_x + r_y) = 1

.

The sum of work completed in the two stages equals

1

full job.

3

Substitute the combined rate into the equation to find

r_x

Since

r_x + r_y = \frac{1}{6}

, we substitute:

2 r_x + 4.8\left(\frac{1}{6}\right) = 1 \implies 2 r_x + 0.8 = 1 \implies 2 r_x = 0.2 \implies r_x = 0.1 = \frac{1}{10}

.

Substituting the combined rate simplifies the equation to a single variable,

r_x

.

4

Calculate

r_y

and the time needed for Machine Y working alone

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

orders per hour. Time taken by Machine Y alone

= \frac{1}{r_y} = 15

hours.

Subtracting Machine X's rate from the combined rate yields Machine Y's rate, whose reciprocal gives the time to complete the job alone.

Anahtar Kavram

Work Rate and Combined Work

Tahmini Süre:2m 0s

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