All practice questions

Question 841Question →

A data set $S$ consists of $9$ positive integers arranged in non-decreasing order: $x_1 \le x_2 \le x_3 \le x_4 \le x_5 \le x_6 \le x_7 \le x_8 \le x_9$ . The arithmetic mean of set $S$ is $22$ , the median is $20$ , and the set has a unique mode of $15$ . If the range of set $S$ is $30$ , what is the maximum possible value of the largest element, $x_9$

31

40

45

50

52

Show answer & explanation

Answer: 45

Answer

The maximum possible value of the largest element is 45.

To maximize the largest value in a set with a fixed range of 30, we must maximize the smallest value because the maximum value equals the minimum value plus 30. Since 15 is the unique mode of the ordered set, 15 must be an element of the set, which restricts the minimum value to at most 15. Setting the smallest element to 15 allows the largest element to reach 15 + 30 = 45, which can be verified to satisfy all mean, median, and mode constraints.

Step-by-Step Solution

1

Relate total sum, range, and median to the elements of the set.

The sum of all 9 elements is

9 \times 22 = 198

. For a 9-element set ordered as

x_1 \le x_2 \le \dots \le x_9

, the median is the 5th element

x_5 = 20

. The range is

x_9 - x_1 = 30

, so

x_9 = x_1 + 30

.

Establishing the basic relationships among the statistical measures given in the problem.

2

Determine the upper bound for the smallest element

x_1

.

Because

15

is the unique mode of the set,

15

must be present in the set. Since the set is in non-decreasing order,

x_1 \le 15

.

If

x_1 > 15

, then no element in the set could equal

15

, violating the condition that

15

is the mode.

3

Maximize

x_9

using the relationship

x_9 = x_1 + 30

.

To maximize

x_9

,

x_1

must be as large as possible. The maximum possible value for

x_1

is

15

, which gives

x_9 = 15 + 30 = 45

.

Connecting the range equation to the upper bound on

x_1

.

4

Verify that a valid set exists for

x_1 = 15

and

x_9 = 45

.

Consider the set

\{15, 15, 15, 15, 20, 20, 25, 28, 45\}

. Sum

= 198

, Mean

= 22

, Median

= 20

(5th element), Mode

= 15

(frequency 4), Range

= 45 - 15 = 30

. All conditions are satisfied.

Ensuring the upper bound is achievable under all given statistical constraints.

Key Concept

Interplay between statistical measures (mean, median, mode, range) and set boundaries

Question 842Question →

The scores of a student on five quizzes are $72, 85, 90, 78,$ and $95$ . What is the positive difference between the median and the arithmetic mean of these five quiz scores?

Show answer & explanation

Answer: 1

Answer

The positive difference between the median and the arithmetic mean of the five quiz scores is

1

.

To find the arithmetic mean, sum all five scores (

72 + 85 + 90 + 78 + 95 = 420

) and divide by

5

, yielding

84

. To find the median, list the scores in ascending order (

72, 78, 85, 90, 95

); the 3rd score is

85

. The positive difference between the median (

85

) and the arithmetic mean (

84

) is

1

.

Step-by-Step Solution

1

Calculate the arithmetic mean of the given set of scores.

The sum of the scores is

72 + 78 + 85 + 90 + 95 = 420

. Dividing by 5 gives an arithmetic mean of

84

.

The mean is defined as the total sum of all values divided by the number of values.

2

Order the scores in ascending order to identify the median.

The ordered list is

72, 78, 85, 90, 95

. The middle term (the 3rd value) is

85

.

The median of a set with an odd number of elements is the middle value when arranged in numerical order.

3

Subtract the arithmetic mean from the median to find the positive difference.

85 - 84 = 1

.

The question asks for the positive difference between the median and the arithmetic mean.

Key Concept

Calculating and comparing the arithmetic mean and median of a finite numerical data set.

Estimated Time:1m 0s

Question 843Question →

A global logistics firm consists of three distinct divisions: Express, Freight, and Solutions. In a given fiscal year, the annual revenue per employee was $\$120,000$ for the Express division, $\$180,000$ for the Freight division, and $\$240,000$ for the Solutions division. The number of employees in the Freight division was $25\%$ greater than the number of employees in the Express division. If the overall weighted average revenue per employee across all three divisions combined was $\$180,000$ , and the total annual revenue generated by the Solutions division was $\$14,400,000$ , what was the total number of employees across all three divisions?

Show answer & explanation

Answer: 195

Answer

The total number of employees across all three divisions was 195.

By setting up the weighted average equation based on employee counts, we establish that the Solutions division headcount equals the Express division headcount (

L = E = 60

). Incorporating the Freight division headcount (

F = 1.25 \times 60 = 75

) gives a total of

60 + 75 + 60 = 195

employees.

Step-by-Step Solution

1

Relate the number of employees in the Freight and Express divisions

F = 1.25E

The Freight division employs 25% more people than the Express division.

2

Set up the weighted average formula and solve for the relationship between Express (

E

) and Solutions (

L

) headcount

L = E

Setting the overall average revenue per employee to

180,000 yields

34.5E + 24L = 40.5E + 18L

, which simplifies to

6L = 6E$.

3

Calculate the exact employee count of the Solutions division

L = 60

Dividing the total revenue of

\$14,400,000

by the revenue per employee of

\$240,000

gives 60 employees.

4

Determine employee counts for all divisions and calculate the total

Total employees =

60 + 75 + 60 = 195

Since

E = L = 60

and

F = 1.25(60) = 75

, the total employee count is

60 + 75 + 60 = 195

.

Key Concept

Weighted Average in Combined Sets

Question 844Question →

An athlete swims across a lake at a constant rate of $10$ miles per hour and swims back to the starting point along the exact same path at a constant rate of $15$ miles per hour. What is the athlete's average speed, in miles per hour, for the entire trip?

11.5

12

12.5

13

25

Show answer & explanation

Answer: $12$

Answer

12

miles per hour

To find the average speed for a round trip, divide the total distance by the total time. If the distance of one leg is

d

, the time for the first leg is

\frac{d}{10}

and for the second leg is

\frac{d}{15}

. Total distance is

2d

, and total time is

\frac{d}{10} + \frac{d}{15} = \frac{d}{6}

. Dividing total distance by total time yields

\frac{2d}{\frac{d}{6}} = 12

miles per hour.

Step-by-Step Solution

1

Assign a convenient distance for one leg of the trip.

Let the distance across the lake be

d = 30

miles (the least common multiple of

10

and

15

).

Choosing a concrete distance simplifies calculating travel times.

2

Calculate the time spent on each leg.

Outbound time

t_1 = \frac{30}{10} = 3

hours; return time

t_2 = \frac{30}{15} = 2

hours.

Time equals distance divided by rate.

3

Calculate the total distance and total time.

Total distance

= 30 + 30 = 60

miles; total time

= 3 + 2 = 5

hours.

The round trip consists of two equal legs.

4

Compute the average speed.

Average speed

= \frac{60}{5} = 12

miles per hour.

Average speed is defined as total distance divided by total time.

Key Concept

Average speed for a multi-leg trip is equal to the total distance divided by the total time, not the simple arithmetic mean of the individual speeds.

Estimated Time:1m 0s

Question 845Question →

What is the value of the expression $\frac{3^8 + 3^8 + 3^8}{3^4}$ ?

3^4

3^5

3^8

3^{12}

3^{20}

Show answer & explanation

Answer: $3^5$

Answer

3^5

The correct answer is

3^5

. Adding three identical values

3^8 + 3^8 + 3^8

is equivalent to

3 \times 3^8

, which equals

3^9

. Dividing

3^9

by

3^4

gives

3^{9-4} = 3^5

.

Step-by-Step Solution

1

Simplify the numerator by combining like terms

3^8 + 3^8 + 3^8 = 3 \cdot 3^8 = 3^1 \cdot 3^8 = 3^{1+8} = 3^9

Repeated addition of

3

identical terms is equivalent to multiplying one term by

3

.

2

Apply the quotient rule for exponents

\frac{3^9}{3^4} = 3^{9-4} = 3^5

When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Key Concept

Combining like exponential terms using multiplication rules and applying the quotient rule for exponents

Question 846Question →

A research laboratory's secure vault contains $N$ data drives, where $N > 4$ . Exactly $3$ of the drives contain encrypted security keys, while the remaining $N - 3$ drives contain standard operational data. If two data drives are selected at random one after another without replacement, the probability that at least one of the selected drives contains an encrypted security key is $\frac{8}{15}$ . What is the value of $N$ ?

8

9

10

12

15

Show answer & explanation

Answer: 10

Answer

10

The correct option is 10. The probability of choosing at least one encrypted security key drive in two draws without replacement is

1 - P(\text{no security key drives})

. Since there are

N - 3

standard drives out of

N

total drives,

P(\text{no security key drives}) = \frac{N - 3}{N} \times \frac{N - 4}{N - 1}

. Setting

1 - \frac{(N - 3)(N - 4)}{N(N - 1)} = \frac{8}{15}

gives

\frac{(N - 3)(N - 4)}{N(N - 1)} = \frac{7}{15}

. Solving

15(N^2 - 7N + 12) = 7(N^2 - N)

leads to

8N^2 - 98N + 180 = 0

, or

4N^2 - 49N + 90 = 0

, which factors as

(4N - 9)(N - 10) = 0

. Because

N

must be an integer,

N = 10

.

Step-by-Step Solution

1

Express the complementary probability event

The probability of at least one drive having an encrypted key is

1 - P(\text{both drives are standard data drives})

. Thus,

P(\text{both standard}) = 1 - \frac{8}{15} = \frac{7}{15}

.

Using the complementary probability rule is more efficient than calculating the individual probabilities of key-then-standard, standard-then-key, and key-then-key.

2

Set up the probability expression for dependent sequential selection without replacement

For the first draw, the probability of selecting a standard drive is

\frac{N - 3}{N}

. For the second draw, since one standard drive has been removed, the probability is

\frac{N - 4}{N - 1}

. Therefore,

\frac{(N - 3)(N - 4)}{N(N - 1)} = \frac{7}{15}

.

The events are dependent because the selection is made without replacement.

3

Solve the algebraic equation for N

Expanding both sides gives

15(N^2 - 7N + 12) = 7(N^2 - N)

, which simplifies to

15N^2 - 105N + 180 = 7N^2 - 7N \implies 8N^2 - 98N + 180 = 0

. Dividing by 2 yields

4N^2 - 49N + 90 = 0

. Factoring gives

(4N - 9)(N - 10) = 0

.

Since

N

must be a positive integer greater than 4,

N = 10

.

Key Concept

Dependent probability without replacement and complementary probability

Estimated Time:2m 0s

Question 847Question →

A ferry travels from Port Alpha to Port Beta across a bay, a distance of $30$ miles, at a constant speed of $15$ miles per hour. On the return trip from Port Beta to Port Alpha along the exact same route, the ferry travels at a constant speed of $30$ miles per hour. What is the ferry's average speed, in miles per hour, for the entire round trip?

Show answer & explanation

Answer: 20

Answer

The ferry's average speed for the entire round trip is

20

miles per hour.

Average speed is found by dividing the total distance traveled by the total time taken. The round trip consists of two 30-mile legs, making the total distance 60 miles. The outbound leg takes 2 hours (30 / 15), while the return leg takes 1 hour (30 / 30), giving a total time of 3 hours. Dividing 60 miles by 3 hours yields an average speed of 20 miles per hour.

Step-by-Step Solution

1

Calculate outbound trip duration

2 hours

Dividing distance by rate gives time:

30 / 15 = 2

hours.

2

Calculate return trip duration

1 hour

Dividing distance by rate gives time:

30 / 30 = 1

hour.

3

Calculate total distance and total time

60 miles in 3 hours

Total distance is

30 + 30 = 60

miles, and total time is

2 + 1 = 3

hours.

4

Calculate overall average speed

20 mph

Average speed is total distance divided by total time:

60 / 3 = 20

mph.

Key Concept

Average speed for any multi-leg or round-trip journey is defined as Total Distance divided by Total Time, rather than the simple arithmetic mean of the speeds.

Question 848Question →

A piece of industrial machinery valued at $\$5,000$ depreciates in value by $20\%$ during its first year of operation, and by $10\%$ of its reduced value during its second year. What is the value of the machinery, in dollars, at the end of the second year?

Show answer & explanation

Answer: 3600

Answer

3600 dollars

Applying the successive percent reductions sequentially gives

\$5,000 \times 0.80 = \$4,000

at the end of Year 1, and

\$4,000 \times 0.90 = \$3,600

at the end of Year 2.

Step-by-Step Solution

1

Calculate the value after Year 1 depreciation

$4,000

Depreciation of 20% on $5,000 leaves 80% of the original value.

2

Calculate the value after Year 2 depreciation

$3,600

Depreciation of 10% on the new base of $4,000 leaves 90% of the Year 1 value.

Key Concept

Successive Percent Change

Question 849Question →

For how many integer values of $k$ does the equation $|x - 2| + |x + 4| = kx + 1$ have no real solutions for $x$ ?

Show answer & explanation

Answer: 4

Answer

4

The equation has no real solutions when the line y = kx + 1 lies completely below the graph of f(x) = |x - 2| + |x + 4|. Analyzing the three piecewise regions of f(x) shows that f(x) = 6 on [-4, 2], with linear rays extending upward outside this interval. Setting the line to remain below f(x) forces -1.25 < k <= 2. The four integers in this range are -1, 0, 1, and 2.

Step-by-Step Solution

1

Decompose the sum of absolute values into a piecewise linear function

f(x) = -2x - 2 for x < -4; f(x) = 6 for -4 <= x < 2; f(x) = 2x + 2 for x >= 2

The critical points x = -4 and x = 2 split the real number line into three intervals where each absolute value expression maintains a constant sign.

2

Set up conditions for zero intersections with the line g(x) = kx + 1

g(x) must remain strictly below f(x) for all x

Any intersection point between y = f(x) and y = g(x) corresponds to a real solution of the equation.

3

Evaluate boundary points and slope constraints for each interval

From the middle interval and right ray: k <= 2. From the left ray: k > -1.25.

For x >= 2, the line slope k cannot exceed the ray slope of 2. For x <= -4, g(-4) < 6 requires -4k + 1 < 6, which yields k > -1.25.

4

Determine the allowable interval for k and count integer values

-1.25 < k <= 2, giving integer values k in {-1, 0, 1, 2}

The integer values strictly inside (-1.25, 2] are -1, 0, 1, and 2, making a total of 4 integer values.

Key Concept

Piecewise analysis of absolute value functions and linear line intersection conditions

Question 850Question →

A solar-powered desalination facility uses two purification units, Unit $X$ and Unit $Y$ , to process standard batches of seawater. Working alone at its constant rate, Unit $X$ can desalinate a full batch of seawater in $12$ hours. Working alone at its constant rate, Unit $Y$ can desalinate an identical batch of seawater in $18$ hours. Unit $X$ begins processing a batch alone and operates for $3$ hours. Then, Unit $Y$ is also turned on, and both units work together at their respective constant rates to complete the remaining portion of the batch. What is the total number of hours required to process the entire batch from the time Unit $X$ was first turned on?

5.4

7.2

8.4

14.25

25.5

Show answer & explanation

Answer: $8.4$

Answer

8.4 hours

Unit X completes 1/12 of the batch per hour, so in 3 hours it processes 3/12 = 1/4 of the batch. This leaves 3/4 of the batch unfinished. Working together, Unit X and Unit Y have a combined rate of 1/12 + 1/18 = 5/36 batch per hour. Dividing the remaining 3/4 batch by 5/36 gives 27/5 = 5.4 hours for the joint operation phase. Summing the initial 3 hours and the 5.4 hours yields a total processing time of 8.4 hours.

Step-by-Step Solution

1

Calculate individual hourly work rates for each unit

Unit X rate =

\frac{1}{12}

batch/hour, Unit Y rate =

\frac{1}{18}

batch/hour

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

2

Determine the fraction of work completed during the first 3 hours

Work completed by Unit X =

3 \times \frac{1}{12} = \frac{1}{4}

of the batch

Unit X operates alone for 3 hours at a rate of

\frac{1}{12}

batch per hour.

3

Calculate the remaining fraction of work to be done

Remaining work =

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

of the batch

Subtract the completed fraction from the total job equal to 1 whole batch.

4

Find the combined hourly work rate of both units

Combined rate =

\frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36}

batch/hour

When entities work together, their individual rates add up.

5

Calculate the time required for both units to complete the remaining work

Time together =

\frac{\frac{3}{4}}{\frac{5}{36}} = \frac{3}{4} \times \frac{36}{5} = \frac{27}{5} = 5.4

hours

Time equals remaining work divided by combined rate.

6

Add the initial single-unit operating time to the combined operating time

Total time =

3 + 5.4 = 8.4

hours

The question asks for the total elapsed time from when Unit X first started.

Key Concept

Work Rate and Combined Work

Question 851Question →

Tank X initially contains $50$ liters of a solvent solution that is $20\%$ chemical compound by volume. Tank Y initially contains $30$ liters of a solvent solution that is $70\%$ of the same chemical compound by volume. First, $v$ liters of the solution in Tank X are transferred to Tank Y and thoroughly mixed. Then, $v$ liters of the resulting mixture in Tank Y are transferred back to Tank X. If the final concentration of the chemical compound in Tank X is $32\%$ by volume, what is the value of $v$ ?

Show answer & explanation

Answer: 20

Answer

The value of

v

is 20.

By setting up the conservation of solute equation for Tank X, the initial

10

liters of compound minus the

0.20v

liters transferred to Tank Y, plus the return portion

v \cdot \frac{21 + 0.20v}{30 + v}

equals the final

16

liters of compound. Solving this relation yields

v = 20

.

Step-by-Step Solution

1

Calculate the initial volume of pure chemical compound in each tank.

Tank X initially contains

50 \times 0.20 = 10

liters of compound. Tank Y initially contains

30 \times 0.70 = 21

liters of compound.

Establishing initial solute quantities is essential for building the mass balance equations.

2

Determine the concentration of Tank Y after the first transfer of

v

liters from Tank X.

Tank Y contains

30 + v

total liters of solution and

21 + 0.20v

liters of compound. Its concentration becomes

C_Y = \frac{21 + 0.20v}{30 + v}

.

The solution transferred from Tank X carries a 20% concentration into Tank Y, altering Tank Y's volume and concentration.

3

Formulate the total compound equation for Tank X after transferring

v

liters back from Tank Y.

Remaining compound in X after first transfer =

10 - 0.20v

. Compound returned from Y =

v \cdot C_Y = v \cdot \frac{21 + 0.20v}{30 + v}

. Final compound in X =

50 \times 0.32 = 16

liters.

Tank X ends with its original total volume (50 liters) at a 32% concentration.

4

Solve the algebraic equation for

v

.

(10 - 0.20v) + \frac{21v + 0.20v^2}{30 + v} = 16

-0.20v + \frac{21v + 0.20v^2}{30 + v} = 6

Multiply by

(30 + v)

:

-0.20v(30 + v) + 21v + 0.20v^2 = 6(30 + v)

-6v - 0.20v^2 + 21v + 0.20v^2 = 180 + 6v

15v = 180 + 6v \implies 9v = 180 \implies v = 20

The quadratic terms cancel out cleanly, leaving a simple linear equation.

Key Concept

Two-Stage Transfer and Replacement Mixture Balance

Question 852Question →

A research team needs to select a subcommittee of 3 scientists from a department containing 7 scientists. How many different 3-member subcommittees can be selected?

Show answer & explanation

Answer: 35

Answer

35 different subcommittees can be selected.

The total number of ways to choose a committee of 3 members from a group of 7 without regard to order is given by the combination formula

C(7,3) = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

.

Step-by-Step Solution

1

Determine whether the selection depends on order.

Order does not matter since all 3 members of the subcommittee have equal roles.

When order does not matter in group selection, combinations (

nCr

) must be used rather than permutations (

nPr

).

2

Apply the combination formula

C(n,k) = \frac{n!}{k!(n-k)!}

with

n=7

and

k=3

.

C(7,3) = \frac{7 \times 6 \times 5}{3 \times 2 \times 1}

This counts the unique groups of 3 that can be chosen from a pool of 7.

3

Calculate the numerical result.

35

Dividing

7 \times 6 \times 5 = 210

by

3 \times 2 \times 1 = 6

yields 35.

Key Concept

Combinations and Group Selections

Question 853Question →

A security system requires each employee to create a 6-character access code consisting of 3 non-zero digits followed by 3 letters from the standard 26-letter English alphabet. The access code must satisfy the following restrictions:

1. The first digit must be odd.
2. No two adjacent digits in the code can be identical.
3. The first letter must be a vowel ( $A, E, I, O, U$ ), the second letter must be a consonant, and the third letter cannot be identical to the first letter.

How many distinct access codes can be created under these rules?

806,400

840,000

1,040,000

1,063,125

2,945

Show answer & explanation

Answer: $840,000$

Answer

840,000

The total number of valid codes is obtained by multiplying the number of choices at each stage. For the digits:

5

(odd non-zero)

\times 8

(non-zero digit

\neq

1st digit)

\times 8

(non-zero digit

\neq

2nd digit)

= 320

. For the letters:

5

(vowels)

\times 21

(consonants)

\times 25

(any letter

\neq

1st letter)

= 2,625

. The overall number of codes is

320 \times 2,625 = 840,000

.

Step-by-Step Solution

1

Calculate the number of choices for the first 3 digit positions.

5 choices for the 1st digit, 8 choices for the 2nd digit, and 8 choices for the 3rd digit.

Non-zero odd digits are

\{1, 3, 5, 7, 9\}

(5 choices). The 2nd digit can be any non-zero digit except the 1st digit (

9 - 1 = 8

choices). The 3rd digit can be any non-zero digit except the 2nd digit (

9 - 1 = 8

choices). Thus,

5 \times 8 \times 8 = 320

digit combinations.

2

Calculate the number of choices for the 3 letter positions.

5 choices for the 1st letter, 21 choices for the 2nd letter, and 25 choices for the 3rd letter.

There are 5 vowels and 21 consonants in the 26-letter alphabet. The 1st letter has 5 choices. The 2nd letter has 21 choices (consonant). The 3rd letter can be any letter except the 1st letter (

26 - 1 = 25

choices). Thus,

5 \times 21 \times 25 = 2,625

letter combinations.

3

Apply the Fundamental Counting Principle to find the total number of access codes.

320 \times 2,625 = 840,000

Since choosing the digits and choosing the letters are independent stages of creating a code, multiply the total number of digit combinations by the total number of letter combinations.

Key Concept

Fundamental Counting Principle with Positional Restrictions

Question 854Question →

In a certain training program, Class A has $15$ participants with an average score of $80$ on a final exam, and Class B has $25$ participants with an average score of $88$ on the same exam. What is the average score for all $40$ participants combined?

Show answer & explanation

Answer: 85

Answer

The combined average score for all 40 participants is 85.

To find the combined average of two groups of different sizes, sum the total values of both groups (

15 \times 80 + 25 \times 88 = 3,400

) and divide by the total number of participants (

15 + 25 = 40

). This gives a weighted average of

85

.

Step-by-Step Solution

1

Find total score of Class A

1,200 points

Multiply the number of participants in Class A by their average score.

2

Find total score of Class B

2,200 points

Multiply the number of participants in Class B by their average score.

3

Find the total combined score

3,400 points

Add the total points from Class A and Class B together.

4

Calculate the weighted average score

85

Divide the total combined score by the total number of participants (15 + 25 = 40).

Key Concept

Weighted Average of Combined Sets

Estimated Time:1m 0s

Question 855Question →

A financial consulting firm audited $500$ corporate investment portfolios for exposure to three alternative asset classes: Private Equity ( $P$ ), Venture Capital ( $V$ ), and Infrastructure ( $I$ ). Every audited portfolio contained at least one of the three asset classes.

The audit revealed the following:
- Exactly $270$ portfolios contained Private Equity.
- Exactly $250$ portfolios contained Venture Capital.
- Exactly $220$ portfolios contained Infrastructure.
- The ratio of the number of portfolios containing both Private Equity and Venture Capital to those containing both Venture Capital and Infrastructure to those containing both Private Equity and Infrastructure was $11 : 9 : 10$ , respectively.
- The number of portfolios containing all three asset classes was $60$ .

How many portfolios contained exactly two of the three asset classes?

Show answer & explanation

Answer: 120

Answer

The total number of portfolios containing exactly two of the three asset classes is 120.

Using the 3-set inclusion-exclusion formula, we establish that

500 = 270 + 250 + 220 - (11k + 9k + 10k) + 60

, which simplifies to

30k = 300

, so

k = 10

. The sum of the pairwise intersections is

110 + 90 + 100 = 300

. Since each pairwise intersection includes the 60 portfolios that contain all three asset classes, the number of portfolios containing exactly two asset classes is

(110-60) + (90-60) + (100-60) = 50 + 30 + 40 = 120

.

Step-by-Step Solution

1

Set up the 3-Set Inclusion-Exclusion Principle equation

\text{Total} = |P| + |V| + |I| - (|P \cap V| + |V \cap I| + |P \cap I|) + |P \cap V \cap I|

This formula accounts for all region overlaps without double-counting or triple-counting.

2

Define variables using the given ratio for pairwise intersections

|P \cap V| = 11k

,

|V \cap I| = 9k

, and

|P \cap I| = 10k

Expressing the pairwise intersection sizes with a common ratio constant

k

allows us to substitute them into a single-variable linear equation.

3

Substitute the known numerical values and solve for

k

500 = 270 + 250 + 220 - (11k + 9k + 10k) + 60 \implies 500 = 800 - 30k \implies k = 10

Using algebraic simplification to solve for

k

gives the scaling factor for the pairwise intersections.

4

Calculate the actual sizes of the pairwise intersections

|P \cap V| = 110

,

|V \cap I| = 90

, and

|P \cap I| = 100

; Sum of pairwise intersections =

300

Multiplying each ratio term by

k=10

gives the exact count of portfolios containing at least the respective pairs.

5

Calculate portfolios containing exactly two asset classes

(110 - 60) + (90 - 60) + (100 - 60) = 50 + 30 + 40 = 120

The triple intersection (

60

) must be subtracted from each pairwise overlap to isolate the regions corresponding to portfolios holding exactly two asset classes.

Key Concept

Three-Set Overlapping Venn Diagrams and Inclusion-Exclusion Principle

Estimated Time:2m 0s

Question 856Question →

For how many integer values of $k$ does the equation $|x - 4| + |x + 2| = k$ have at least one real solution $x$ satisfying $|x| \leq 5$ ?

5

6

7

8

12

Show answer & explanation

Answer: 7

Answer

7 integer values of k satisfy the equation.

The sum of absolute values

f(x) = |x - 4| + |x + 2|

represents the total distance from

x

to

4

and

-2

on the number line. On the interval

[-2, 4]

, this total distance is constant and equal to

6

, which is the minimum possible value. On the restricted domain

[-5, 5]

, the maximum distance occurs at the boundary point

x = -5

, where

f(-5) = |-9| + |-3| = 12

. Because

f(x)

is continuous, it takes every real value from

6

to

12

inclusive. The integer values that

k

can assume are

6, 7, 8, 9, 10, 11,

and

12

, yielding a total of 7 integer values.

Step-by-Step Solution

1

Define the piecewise absolute value function

f(x) = |x - 4| + |x + 2|

and analyze its critical points.

Critical points occur at

x = -2

and

x = 4

.

Absolute value expressions change definition where their inner terms equal zero.

2

Evaluate

f(x)

across the three intervals defining the domain

[-5, 5]

.

For

x < -2

,

f(x) = (4 - x) - (x + 2) = 2 - 2x

. For

-2 \leq x \leq 4

,

f(x) = (4 - x) + (x + 2) = 6

. For

x > 4

,

f(x) = (x - 4) + (x + 2) = 2x - 2

.

Removing absolute value bars according to domain regions gives explicit linear formulas.

3

Find the minimum and maximum values of

f(x)

on the domain

|x| \leq 5

(i.e.,

-5 \leq x \leq 5

).

Minimum value is

f(x) = 6

for

x \in [-2, 4]

. Maximum value occurs at

x = -5

, giving

f(-5) = 2 - 2(-5) = 12

. At

x = 5

,

f(5) = 2(5) - 2 = 8

. Range of

f(x)

is

[6, 12]

.

Continuous piecewise functions achieve extreme values at domain boundaries or constant minimum intervals.

4

Count the integer values of

k

in the closed interval

[6, 12]

.

The integers are

6, 7, 8, 9, 10, 11, 12

, giving

12 - 6 + 1 = 7

integers.

The number of integers in an inclusive range

[a, b]

is given by

b - a + 1

.

Key Concept

Absolute Value Equations and Piecewise Function Ranges

Estimated Time:2m 0s

Question 857Question →

How many integer values of $x$ satisfy the inequality $\big||x + 1| - 4\big| \leq 3 - |x|$ ?

2

3

4

5

7

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. The inequality requires

3 - |x| \geq 0

, restricting candidate solutions to integers in the range

[-3, 3]

. Testing each integer directly reveals that only

x = 0, 1, 2, 3

satisfy the inequality, yielding a total of 4 valid integers.

Step-by-Step Solution

1

Establish the domain restriction from the non-negativity of the absolute value.

Since the left-hand side

\big||x + 1| - 4\big| \geq 0

for all real numbers, the right-hand side must also be non-negative:

3 - |x| \geq 0 \implies |x| \leq 3 \implies -3 \leq x \leq 3

.

An absolute value expression cannot be less than a negative number.

2

Test non-negative integer candidates in the domain

[-3, 3]

.

For

x = 3

:

\big|4 - 4\big| = 0 \leq 3 - 3 = 0

(True);
For

x = 2

:

\big|3 - 4\big| = 1 \leq 3 - 2 = 1

(True);
For

x = 1

:

\big|2 - 4\big| = 2 \leq 3 - 1 = 2

(True);
For

x = 0

:

\big|1 - 4\big| = 3 \leq 3 - 0 = 3

(True).

Substituting non-negative integers verifies that

x \in \{0, 1, 2, 3\}

satisfy the inequality.

3

Test negative integer candidates in the domain

[-3, 3]

.

For

x = -1

:

\big|0 - 4\big| = 4 \leq 3 - 1 = 2

(False);
For

x = -2

:

\big|-1 - 4\big| = 5 \leq 3 - 2 = 1

(False);
For

x = -3

:

\big|-2 - 4\big| = 6 \leq 3 - 3 = 0

(False).

Evaluating negative integers shows no negative integer satisfies the inequality.

4

Count the total number of valid integer solutions.

The valid integer solutions are

x = 0, 1, 2, 3

, giving a total count of 4.

Combining all valid cases yields exactly 4 integer solutions.

Key Concept

Solving nested absolute value inequalities using domain constraints and piecewise sign analysis.

Question 858Question →

A boutique marketing agency charges a corporate client a fixed monthly retainer fee of $\$1,600$ plus a uniform monthly fee for each active digital ad campaign managed. The client has a total monthly ad management budget of $\$4,000$ . If the agency were to increase the fee per campaign by $\$40$ , the number of ad campaigns the client could afford under the same total budget would decrease by $2$ . What is the original monthly fee charged per ad campaign?

\$160

\$180

\$200

\$240

\$250

Show answer & explanation

Answer: $\$200$

Answer

The original monthly fee charged per ad campaign is

\$200

.

Subtracting the

\$1,600

retainer from the

\$4,000

total budget leaves

\$2,400

for campaign fees. Letting

p

represent the original price per campaign, the original number of campaigns is

\frac{2,400}{p}

and the new number of campaigns at price

p+40

is

\frac{2,400}{p+40}

. The condition that the client can afford

2

fewer campaigns yields the equation

\frac{2,400}{p} - \frac{2,400}{p+40} = 2

. Simplifying leads to

p^2 + 40p - 48,000 = 0

, which factors as

(p - 200)(p + 240) = 0

. Since price must be positive, the original fee per campaign is

\$200

.

Step-by-Step Solution

1

Determine the portion of the budget available for ad campaigns.

Variable Budget = Total Budget - Fixed Retainer =

\$4,000 - \$1,600 = \$2,400

.

The fixed retainer fee must be subtracted before calculating how many campaigns can be purchased.

2

Define variables and set up the equation for the number of campaigns.

Let

p

be the original fee per campaign. The original number of campaigns is

\frac{2,400}{p}

. With the fee increase, the new number of campaigns is

\frac{2,400}{p + 40}

. The difference is given as

2

, so

\frac{2,400}{p} - \frac{2,400}{p + 40} = 2

.

Model the real-world reduction in campaigns using algebraic expressions.

3

Simplify the equation and form a standard quadratic equation.

Divide the entire equation by

2

:

\frac{1,200}{p} - \frac{1,200}{p + 40} = 1

. Multiply by

p(p + 40)

:

1,200(p + 40) - 1,200p = p(p + 40) \implies 48,000 = p^2 + 40p \implies p^2 + 40p - 48,000 = 0

.

Clear fractions to transform the rational equation into quadratic form.

4

Solve the quadratic equation for

p

.

(p + 240)(p - 200) = 0

. Since price must be positive,

p = 200

.

Factoring gives two roots (

200

and

-240

); reject the negative root in physical pricing context.

Key Concept

Algebraic Word Problems and Equation Modeling

Estimated Time:2m 0s

Question 859Question →

For all real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = a^2 - b^2$ . The function $f$ is defined by $f(x) = x \star 3$ . If $f(x) = 4x - 12$ , what is the sum of all real values of $x$ that satisfy this equation?

4

3

-4

-3

12

Show answer & explanation

Answer: 4

Answer

The sum of all real values of x that satisfy the equation is 4.

Applying the custom operator definition gives f(x) = x^2 - 9. Setting this equal to 4x - 12 produces the quadratic equation x^2 - 4x + 3 = 0. Factoring yields (x - 1)(x - 3) = 0, which has solutions x = 1 and x = 3. The sum of these solutions is 1 + 3 = 4.

Step-by-Step Solution

1

Evaluate the custom operator expression f(x) = x * 3.

f(x) = x^2 - 3^2 = x^2 - 9

Applying the definition of the custom operator a * b = a^2 - b^2 with a = x and b = 3.

2

Set f(x) equal to 4x - 12 and rearrange into standard quadratic form.

x^2 - 9 = 4x - 12 => x^2 - 4x + 3 = 0

Equating the function expression to the given linear expression.

3

Factor the quadratic equation to find the roots.

(x - 1)(x - 3) = 0 => x = 1 or x = 3

Finding the values of x that satisfy the quadratic equation.

4

Calculate the sum of the roots.

1 + 3 = 4

Summing all real solutions to satisfy the prompt requirement.

Key Concept

Custom Operator Evaluation and Quadratic Equation Solving

Estimated Time:1m 30s

Question 860Question →

In a class of $40$ students, each student received a distinct score on a final examination. If a particular student's score is at the $85\text{th}$ percentile of the class scores, how many students scored lower than this student?

34

85

6

35

33

Show answer & explanation

Answer: $34$

Answer

34 students scored lower than this student.

The 85th percentile of a group of 40 students means that 85% of the class scored below this student. Calculating

85\%

of

40

yields

0.85 \times 40 = 34

. Therefore, exactly 34 students scored lower.

Step-by-Step Solution

1

Understand the definition of percentile rank.

Being at the

85\text{th}

percentile means that the student's score is strictly higher than

85\%

of the other scores in the dataset.

By definition, the

p\text{th}

percentile of a set of distinct values separates the lower

p\%

of the data from the rest.

2

Calculate

85\%

of the total number of students (

40

).

0.85 \times 40 = 34

.

Multiplying the percentage by the sample size gives the exact count of students below the percentile threshold.

Key Concept

Percentile Rank

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept