All practice questions

If $x$ is a real number satisfying the equation $x^2 + \frac{18}{x^2 + 3} = 8$ , what is the value of $x^2$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x^2

is 6.

Substituting

u = x^2

converts the equation to

u + \frac{18}{u+3} = 8

. Multiplying through by

(u+3)

yields

u^2 + 3u + 18 = 8u + 24

, which simplifies to the quadratic equation

u^2 - 5u - 6 = 0

. Factoring gives

(u-6)(u+1) = 0

, so

u = 6

or

u = -1

. Because

x

must be a real number,

x^2

must be non-negative, eliminating

u = -1

. Thus,

x^2 = 6

.

Step-by-Step Solution

1

Substitute a temporary variable

u = x^2

into the given equation

u + \frac{18}{u + 3} = 8

Using substitution simplifies the fourth-degree rational equation into a standard quadratic structure.

2

Clear the denominator by multiplying the entire equation by

(u + 3)

u(u + 3) + 18 = 8(u + 3)

, which expands to

u^2 + 3u + 18 = 8u + 24

Eliminating fractions allows rearranging terms into polynomial form.

3

Rearrange terms into standard quadratic form

au^2 + bu + c = 0

u^2 - 5u - 6 = 0

Moving all terms to one side prepares the expression for factoring.

4

Factor the quadratic polynomial into linear binomial factors

(u - 6)(u + 1) = 0

, giving

u = 6

or

u = -1

The roots of the transformed equation represent candidate values for

x^2

.

5

Apply the constraint that

x

is a real number to test candidate values

Since

x^2 \geq 0

for all real

x

,

u = -1

is rejected, leaving

u = x^2 = 6

Squaring any real number yields a non-negative result.

Key Concept

Solving rational equations reducible to quadratics via algebraic substitution and rejecting non-real roots.

Estimated Time:2m 0s

Question 122Question →

A container holds 60 cards, each marked with a distinct integer from 1 to 60, inclusive. If one card is selected at random, the probability that the integer on the selected card is divisible by 5, but divisible by neither 4 nor 6, can be expressed as a simplified fraction $\frac{a}{b}$ , where $a$ and $b$ are positive integers with no common factors greater than 1. What is the value of $a + b$ ?

Show answer & explanation

Answer: 17

Answer

The value of

a + b

is 17.

There are 60 equally likely outcomes. Out of 12 multiples of 5, exactly 4 integers (20, 30, 40, and 60) are divisible by 4 or 6. Thus, 12 - 4 = 8 integers satisfy the condition of being divisible by 5 but neither 4 nor 6. The single-event probability is 8/60 = 2/15. In simplest form, a = 2 and b = 15, yielding a + b = 17.

Step-by-Step Solution

1

Determine total number of possible outcomes.

Total outcomes

N = 60

.

Each of the 60 cards is equally likely to be selected.

2

Identify all multiples of 5 in the set

\{1, 2, \dots, 60\}

.

Multiples of 5:

\{5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60\}

, containing

\frac{60}{5} = 12

integers.

These represent all potential candidates before applying exclusion criteria.

3

Identify multiples of 5 that are also divisible by 4 or by 6.

Multiples of 5 divisible by 4:

\text{LCM}(5,4) = 20 \rightarrow \{20, 40, 60\}

(3 integers). Multiples of 5 divisible by 6:

\text{LCM}(5,6) = 30 \rightarrow \{30, 60\}

(2 integers). Overlap (divisible by both 4 and 6):

\text{LCM}(5,4,6) = 60 \rightarrow \{60\}

(1 integer).

An integer is divisible by both 5 and 4 if it is a multiple of 20, and by both 5 and 6 if it is a multiple of 30.

4

Apply the inclusion-exclusion principle to find the count of invalid multiples of 5.

Invalid multiples =

3 + 2 - 1 = 4

integers (

\{20, 30, 40, 60\}

). Favorable outcomes =

12 - 4 = 8

integers (

\{5, 10, 15, 25, 35, 45, 50, 55\}

).

The integer 60 is divisible by both 4 and 6 and must not be double-subtracted.

5

Calculate the probability, express it as a simplified fraction

\frac{a}{b}

, and compute

a + b

.

Probability

P = \frac{8}{60} = \frac{2}{15}

. Thus,

a = 2

,

b = 15

, and

a + b = 2 + 15 = 17

.

Dividing the numerator and denominator by 4 simplifies

\frac{8}{60}

to

\frac{2}{15}

, where

\text{gcd}(2, 15) = 1

.

Key Concept

Basic Single-Event Probability with Multiple Divisibility Constraints

Question 123Question →

A commercial bakery uses two automated dough mixing machines, Machine $A$ and Machine $B$ , to process large batches of dough. Working alone at its constant rate, Machine $A$ can process a full batch of dough in $12$ hours. Machine $B$ , working alone at its constant rate, can process a full batch of dough in $6$ hours. Machine $A$ starts processing a batch alone. After working for $3$ hours, Machine $B$ joins Machine $A$ , and both machines work together at their respective constant rates to finish the batch. What is the total number of hours required to process the entire batch of dough from start to finish?

Show answer & explanation

Answer: 6

Answer

The total number of hours required to process the entire batch of dough from start to finish is 6 hours.

Machine A processes 1/12 of the batch per hour. In the first 3 hours, Machine A finishes 3/12 = 1/4 of the batch, leaving 3/4 of the batch uncompleted. When Machine B joins, the combined processing rate is 1/12 + 1/6 = 1/4 batch per hour. Dividing the remaining 3/4 batch by the combined rate of 1/4 batch per hour yields 3 hours. Adding the initial 3 hours worked by Machine A alone gives a total elapsed time of 6 hours.

Step-by-Step Solution

1

Determine the individual work rates for Machine

A

and Machine

B

.

Machine

A

's rate is

\frac{1}{12}

batch/hour, and Machine

B

's rate is

\frac{1}{6}

batch/hour.

Work rate is the fraction of the job completed per hour (1 divided by total hours required alone).

2

Calculate the portion of the batch completed by Machine

A

operating alone during the first 3 hours.

Machine

A

completes

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

of the batch.

Work completed equals work rate multiplied by time worked.

3

Determine the remaining fraction of the batch to be completed.

Remaining work is

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

of the batch.

The entire job corresponds to 1 whole batch.

4

Calculate the combined work rate of Machine

A

and Machine

B

.

Combined rate

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

batch per hour.

When machines work together, their individual rates add up.

5

Calculate the time required for both machines working together to complete the remaining work.

Time together

= \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{3/4}{1/4} = 3

hours.

Time equals work divided by rate.

6

Calculate the total time required from start to finish.

Total time

= 3 \text{ hours (Machine A alone)} + 3 \text{ hours (both together)} = 6

hours.

The total time is the sum of time spent in each phase of the process.

Key Concept

Work Rate and Combined Work

Question 124Question →

An architectural firm initially allocates its annual project hours among three divisions—Residential, Commercial, and Urban Planning—in the ratio of $7 : 5 : 3$ , respectively. Mid-year, to meet new project demands, $300$ hours from Residential and $100$ hours from Commercial are transferred to Urban Planning. Following this transfer, the ratio of Commercial hours to Urban Planning hours becomes $1 : 1$ . What was the total number of project hours initially allocated across all three divisions?

Show answer & explanation

Answer: 3750

Answer

The total number of project hours initially allocated across all three divisions was 3,750 hours.

Representing the initial hours as

7x

,

5x

, and

3x

gives a total of

15x

hours. After the transfers, Commercial hours equal

5x - 100

and Urban Planning hours equal

3x + 300 + 100 = 3x + 400

. Setting these equal gives

5x - 100 = 3x + 400

, which simplifies to

2x = 500

, or

x = 250

. The initial total hours were

15 \times 250 = 3,750

.

Step-by-Step Solution

1

Define initial quantities using a multiplier variable

Let

x

represent the multiplier. Initial hours are Residential =

7x

, Commercial =

5x

, and Urban Planning =

3x

. Total initial hours =

15x

.

Ratios define the proportional relationship among the three divisions.

2

Determine updated hours after the transfers

Commercial hours become

5x - 100

. Urban Planning receives

300

hours from Residential and

100

hours from Commercial, becoming

3x + 300 + 100 = 3x + 400

.

Quantities added to Urban Planning must equal the sum of hours subtracted from the other two divisions.

3

Set up and solve the equation for the new ratio

Since Commercial and Urban Planning hours are now in a

1 : 1

ratio,

5x - 100 = 3x + 400 \implies 2x = 500 \implies x = 250

.

A

1 : 1

ratio means both quantities are equal.

4

Calculate the initial total project hours

Total initial hours =

15 \times 250 = 3,750

.

Substitute

x = 250

into the expression for total initial hours (

15x

).

Key Concept

Solving multi-part ratio word problems involving internal transfers by setting up algebraic equations with a single unknown multiplier.

Question 125Question →

A financial analyst chooses two investment assets at random, one after another without replacement, from a portfolio consisting of $3$ technology stocks and $2$ utility stocks. The probability of achieving a target return depends on the composition of the selected assets:
- If both selected assets are technology stocks, the probability of achieving the target return is $0.80$ .
- If exactly one selected asset is a technology stock, the probability of achieving the target return is $0.50$ .
- If neither selected asset is a technology stock, the probability of achieving the target return is $0.10$ .

What is the probability, expressed as a decimal, that the analyst achieves the target return?

Show answer & explanation

Answer: 0.55

Answer

The probability that the analyst achieves the target return is 0.55.

To find the overall probability of achieving the target return, calculate the probability of each mutually exclusive composition of assets selected without replacement, and weight each by its conditional probability of success. The probability of selecting two technology stocks is (3/5) * (2/4) = 0.30. The probability of selecting one technology stock and one utility stock in either order is (3/5)*(2/4) + (2/5)*(3/4) = 0.60. The probability of selecting two utility stocks is (2/5) * (1/4) = 0.10. Multiplying each by its respective success rate (0.80, 0.50, and 0.10) yields 0.24 + 0.30 + 0.01 = 0.55.

Step-by-Step Solution

1

Determine the probabilities of the three mutually exclusive selection scenarios without replacement.

P(TT) = 0.30, P(1T) = 0.60, P(UU) = 0.10.

Since selections are made without replacement, the outcome of the first draw affects the total count and available assets for the second draw.

2

Calculate the joint probability of achieving the target return for each scenario.

P(TT and Target) = 0.24, P(1T and Target) = 0.30, P(UU and Target) = 0.01.

Multiply the probability of drawing each combination by the conditional probability of success for that specific combination.

3

Sum the joint probabilities across all possible scenarios.

Total Probability = 0.24 + 0.30 + 0.01 = 0.55.

The scenarios are mutually exclusive and exhaustive, so their sum gives the total overall probability of achieving the target return.

Key Concept

Probability of Dependent Events and Law of Total Probability

Estimated Time:2m 0s

Question 126Question →

If the quadratic equation $3x^2 - kx + 48 = 0$ has equal real roots, where $k > 0$ , and the roots of the quadratic equation $x^2 + kx - m = 0$ differ by 10, what is the value of $m$ ?

Show answer & explanation

Answer: -119

Answer

The value of

m

is

-119

.

Setting the discriminant of

3x^2 - kx + 48 = 0

equal to zero gives

k^2 - 4(3)(48) = 0

, yielding

k = 24

since

k > 0

. Substituting

k = 24

into the second equation gives

x^2 + 24x - m = 0

. According to Vieta's formulas, the sum of roots is

r_1 + r_2 = -24

and the product of roots is

r_1 r_2 = -m

. Using the algebraic identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

with the given root difference

|r_1 - r_2| = 10

yields

10^2 = (-24)^2 - 4(-m)

, which simplifies to

100 = 576 + 4m

. Subtracting 576 from both sides gives

4m = -476

, so

m = -119

.

Step-by-Step Solution

1

Find

k

using the discriminant condition for equal roots of

3x^2 - kx + 48 = 0

.

k = 24

A quadratic equation

ax^2 + bx + c = 0

has equal real roots when its discriminant

b^2 - 4ac = 0

. Here,

(-k)^2 - 4(3)(48) = 0

, so

k^2 = 576

. Since

k > 0

,

k = 24

.

2

Set up the algebraic relationship for the root difference of

x^2 + 24x - m = 0

.

(r_1 - r_2)^2 = 100

Given that the difference between the roots

r_1

and

r_2

is 10,

|r_1 - r_2| = 10

, which implies

(r_1 - r_2)^2 = 100

.

3

Apply Vieta's formulas to express

(r_1 - r_2)^2

in terms of

m

and solve.

m = -119

By Vieta's formulas for

x^2 + 24x - m = 0

, the sum of roots is

r_1 + r_2 = -24

and the product of roots is

r_1 r_2 = -m

. Using the identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, substitute the known values:

100 = (-24)^2 - 4(-m) = 576 + 4m

. Solving

576 + 4m = 100

gives

4m = -476

, so

m = -119

.

Key Concept

Discriminant analysis and root difference identity via Vieta's formulas

Question 127Question →

An investment fund initially holds shares of Technology, Healthcare, and Energy stocks in the ratio $5 : 3 : 2$ , respectively. After the fund sells $30$ shares of Technology stock and purchases $30$ shares of Energy stock, the ratio of Technology, Healthcare, and Energy shares becomes $4 : 3 : 3$ . What was the total number of shares in the investment fund initially?

Show answer & explanation

Answer: 300

Answer

The initial total number of shares in the investment fund was 300.

The initial shares can be represented as

5x

,

3x

, and

2x

for a total of

10x

shares. After selling

30

Technology shares and buying

30

Energy shares, the Healthcare shares remain unchanged at

3x

. Since Healthcare represents

3

parts in both the initial and final ratios (

5:3:2

and

4:3:3

), the value of one ratio part remains constant at

x

. Therefore, the decrease in Technology shares from

5x

to

4x

corresponds directly to

1x = 30

shares. Multiplying

x = 30

by the initial total of

10

parts gives

300

shares.

Step-by-Step Solution

1

Define initial quantities using a common ratio multiplier

x

Technology =

5x

, Healthcare =

3x

, Energy =

2x

. Total initial shares =

10x

.

Representing ratio terms algebraically preserves their proportional relationships.

2

Account for changes in share counts

New Technology shares =

5x - 30

, Healthcare shares =

3x

, new Energy shares =

2x + 30

.

30 shares of Technology were sold and 30 shares of Energy were added.

3

Set up an equation comparing the new ratio of Technology to Healthcare

\frac{5x - 30}{3x} = \frac{4}{3}

The new ratio of Technology to Healthcare is given as

4 : 3

.

4

Solve for multiplier

x

3(5x - 30) = 12x \implies 15x - 90 = 12x \implies 3x = 90 \implies x = 30

Cross-multiplying eliminates fractions to solve for the unknown multiplier.

5

Calculate the initial total number of shares

10x = 10 \times 30 = 300

The initial total is the sum of the initial parts (

5x + 3x + 2x = 10x

).

Key Concept

Ratio Modification and Algebraic Scaling

Estimated Time:1m 30s

Question 128Question →

An analytics team tracked the monthly recurring revenue (in thousands of dollars) for 10 distinct client accounts of a SaaS company. The arithmetic mean of the monthly revenues across these 10 accounts was $\$45$ thousand, and the median monthly revenue was $\$38$ thousand. If the revenue for each account was a distinct positive integer (in thousands of dollars), what is the maximum possible monthly revenue, in thousands of dollars, generated by the single highest-performing account?

Show answer & explanation

Answer: 241

Answer

The maximum possible monthly revenue generated by the single highest-performing account is 241 thousand dollars.

The total sum of the 10 distinct positive integer revenues is

10 \times 45 = 450

thousand dollars. To maximize the largest term (

x_{10}

), the sum of the remaining 9 terms (

x_1

through

x_9

) must be minimized. The smallest 4 positive integers are

1, 2, 3, 4

(sum =

10

). Given a median of 38, the two central terms must satisfy

x_5 + x_6 = 76

. Because all values are distinct integers and

x_5 < x_6

, the smallest possible integer value for

x_6

is 39 (making

x_5 = 37

). To minimize

x_7, x_8, x_9

, they should be the smallest distinct integers greater than 39, which are

40, 41, 42

. Summing these minimal values gives

10 + 76 + 40 + 41 + 42 = 209

. Subtracting this sum from the total sum yields

450 - 209 = 241

thousand dollars.

Step-by-Step Solution

1

Calculate the total revenue across all 10 client accounts.

Total sum =

10 \times 45 = 450

thousand dollars.

The sum of a data set is equal to the number of elements multiplied by the arithmetic mean.

2

Use the median to determine the sum of the 5th and 6th terms.

Since

n = 10

, the median is

\frac{x_5 + x_6}{2} = 38

, so

x_5 + x_6 = 76

.

In an ordered set of an even number of elements, the median is the average of the two central terms.

3

Formulate the minimization strategy to maximize the largest term (

x_{10}

).

To maximize

x_{10} = 450 - (x_1 + x_2 + \dots + x_9)

, minimize the sum of the first 9 terms.

For a fixed sum, maximizing one element requires minimizing all other elements in the set.

4

Minimize the first four terms (

x_1, x_2, x_3, x_4

).

Assign

x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4

, which sum to

10

.

The revenues are distinct positive integers, so the smallest possible values are the first 4 positive integers.

5

Minimize

x_6

to minimize the subsequent terms (

x_7, x_8, x_9

).

Since

x_5 < x_6

and

x_5 + x_6 = 76

, the smallest integer value for

x_6

is

39

(which sets

x_5 = 37

). Then set

x_7 = 40, x_8 = 41, x_9 = 42

.

Minimizing

x_6

allows

x_7, x_8, x_9

to take the smallest possible distinct integer values greater than

x_6

.

6

Calculate the maximum value for

x_{10}

.

x_{10} = 450 - (10 + 76 + 40 + 41 + 42) = 450 - 209 = 241

.

Subtracting the minimal sum of the first 9 elements from the total sum gives the theoretical maximum for the 10th term.

Key Concept

Extremal optimization of distinct statistics data sets given mean and median constraints

Question 129Question →

A corporate training agency surveyed $180$ professionals to determine which of three skill workshops they attended: Leadership ( $L$ ), Negotiation ( $N$ ), and Communication ( $C$ ). The survey revealed that $90$ professionals attended Leadership, $80$ attended Negotiation, and $70$ attended Communication. Additionally, $35$ attended both Leadership and Negotiation, $25$ attended both Negotiation and Communication, $40$ attended both Leadership and Communication, and $20$ attended none of the three workshops. How many of the surveyed professionals attended exactly one workshop?

Show answer & explanation

Answer: 100

Answer

The number of professionals who attended exactly one workshop is 100.

By applying the principle of inclusion-exclusion for three overlapping sets, the number of professionals attending at least one workshop is

180 - 20 = 160

. Setting up the equation

160 = 90 + 80 + 70 - (35 + 25 + 40) + x

reveals that

x = 20

professionals attended all three workshops. Subtracting

20

from each pairwise intersection yields the counts for those attending exactly two workshops (

15

,

5

, and

20

, totaling

40

). Subtracting the exactly-two count (

40

) and the all-three count (

20

) from the total attending at least one (

160

) gives

100

professionals who attended exactly one workshop.

Step-by-Step Solution

1

Determine the number of professionals in the union of all three sets

|L \cup N \cup C| = 180 - 20 = 160

The total number of professionals who attended at least one workshop is equal to the total surveyed minus those who attended none.

2

Apply the principle of inclusion-exclusion for three sets to find the intersection of all three workshops

160 = 90 + 80 + 70 - (35 + 25 + 40) + |L \cap N \cap C| \implies |L \cap N \cap C| = 20

Summing individual set counts double-counts pairwise intersections and triple-counts the triple intersection, so we adjust using the standard three-set formula.

3

Calculate the number of professionals who attended exactly two workshops

(35 - 20) + (25 - 20) + (40 - 20) = 15 + 5 + 20 = 40

Each pairwise intersection includes those who attended all three workshops; subtracting the triple intersection leaves those in exactly two sets.

4

Calculate the number of professionals who attended exactly one workshop

\text{Exactly one} = 160 - 40 - 20 = 100

Subtracting the number of professionals who attended exactly two workshops and all three workshops from the total attending at least one leaves those attending exactly one workshop.

Key Concept

Overlapping Sets (Three-Set Inclusion-Exclusion Principle)

Estimated Time:2m 0s

Question 130Question →

A data set consists of $60$ distinct test scores arranged in ascending order. The $70\text{th}$ percentile of this data set is equal to the $42\text{nd}$ score. If $15$ new test scores, all strictly lower than the lowest score in the original data set, are added to form a new data set of $75$ scores, what is the percentile rank of the score that was the $70\text{th}$ percentile of the original data set?

Show answer & explanation

Answer: 76

Answer

76

The target score originally occupied position 42 in an ordered set of 60 values. When 15 values strictly smaller than all original values are introduced, they occupy the first 15 positions in the new ordered set. Consequently, the target score shifts to position 42 + 15 = 57 in the new set of 75 values. The percentile rank corresponds to the percentage of values at or below this position, calculated as (57 / 75) * 100% = 76%.

Step-by-Step Solution

1

Determine the rank position of the original 70th percentile score

The target score is at position 42 in the original sorted set of 60 scores.

The stem specifies that the 70th percentile corresponds to the 42nd score.

2

Calculate the target score's position in the expanded data set

The target score is now at position 57 in the new sorted set of 75 scores.

Since 15 scores smaller than the original minimum are prepended, every original score's rank position increases by 15. Thus, position 42 becomes position 42 + 15 = 57.

3

Compute the percentile rank of position 57 out of 75

76

The percentile rank is the percentage of values less than or equal to this score in the new set: (57 / 75) * 100% = 76%.

Key Concept

Percentile Rank and Rank Position Shifts

Question 131Question →

A private equity firm allocated its total investment capital between two tech startups, Company X and Company Y. Over the first year, the value of the investment in Company X increased by $60\%$ , while the value of the investment in Company Y increased by $20\%$ . Over the second year, the value of Company X decreased by $25\%$ from its Year 1 value, while the value of Company Y increased by $25\%$ from its Year 1 value. If the total combined value of the two investments at the end of the second year was $38\%$ greater than the original total investment capital allocated, what percent of the original total investment capital was allocated to Company X?

Show answer & explanation

Answer: 40

Answer

40%

Evaluating successive percent changes gives growth multipliers of

1.60 \times 0.75 = 1.20

for Company X and

1.20 \times 1.25 = 1.50

for Company Y. Setting the weighted sum of final values equal to

1.38(X + Y)

yields

1.20X + 1.50Y = 1.38X + 1.38Y

, which simplifies to

0.18X = 0.12Y

or

Y = 1.5X

. The proportion allocated to Company X is

\frac{X}{X + 1.5X} = \frac{1}{2.5} = 40\%

.

Step-by-Step Solution

1

Calculate the cumulative growth multiplier for Company X over the two-year period.

Company X's final value is

1.20X

, representing a net increase of

20\%

.

Successive percent changes are calculated by multiplying the growth factors:

(1 + 0.60) \times (1 - 0.25) = 1.60 \times 0.75 = 1.20

.

2

Calculate the cumulative growth multiplier for Company Y over the two-year period.

Company Y's final value is

1.50Y

, representing a net increase of

50\%

.

Successive percent changes are calculated by multiplying the growth factors:

(1 + 0.20) \times (1 + 0.25) = 1.20 \times 1.25 = 1.50

.

3

Set up an equation for the combined final portfolio value relative to initial capital.

1.20X + 1.50Y = 1.38(X + Y)

.

The problem states that the combined final value is

38\%

greater than the total initial capital

X + Y

.

4

Simplify the equation to express

Y

in terms of

X

.

0.12Y = 0.18X \implies Y = 1.5X

.

Expanding

1.38X + 1.38Y

and collecting like terms yields

1.50Y - 1.38Y = 1.38X - 1.20X

.

5

Determine Company X's proportion of the total initial allocation.

\frac{X}{X + 1.5X} = \frac{1}{2.5} = 40\%

.

The initial percentage allocated to Company X is

\frac{X}{X + Y} \times 100\%

.

Key Concept

Weighted Successive Percent Change

Question 132Question →

Set $S$ consists of consecutive integers. The sum of all positive integers in set $S$ is $105$ , and the sum of all integers in set $S$ is $-31$ . How many negative integers are contained in set $S$ ?

Show answer & explanation

Answer: 16

Answer

There are 16 negative integers in set

S

.

The correct response is 16. By setting up the sum of consecutive positive integers starting at 1, we determine that the set contains positive integers up to 14, which sum to 105. Subtracting 105 from the total set sum of -31 reveals that the negative integers must sum to -136. The consecutive negative integers -1, -2, ..., -p sum to -136 when p = 16, since 16 × 17 / 2 = 136.

Step-by-Step Solution

1

Find the maximum positive integer

k

in set

S

The largest positive integer in set

S

is

14

Because set

S

consists of consecutive integers, the positive integers are

1, 2, \dots, k

. The sum formula

\frac{k(k+1)}{2} = 105

leads to

k(k+1) = 210

. Factoring

210

into two consecutive integers gives

14 \times 15

, so

k = 14

.

2

Calculate the sum of all negative integers in set

S

The sum of all negative integers is

-136

The total sum of set

S

is the sum of its negative integers plus

0

plus the sum of its positive integers:

-31 = S_{\text{neg}} + 0 + 105 \implies S_{\text{neg}} = -136

.

3

Determine the count

p

of negative integers

The number of negative integers is

16

The negative integers are

-1, -2, \dots, -p

. Their sum is

-\frac{p(p+1)}{2} = -136 \implies p(p+1) = 272

. Solving

p(p+1) = 272

gives

p = 16

because

16 \times 17 = 272

.

Key Concept

Consecutive integer set properties and partitioning sets into positive and negative components using arithmetic series formulas.

Question 133Question →

A committee of $5$ members is to be formed from a pool of $6$ men and $5$ women. The committee must include at least $2$ men and at least $2$ women. However, two specific individuals in the pool, one man and one woman, refuse to serve on the committee together. How many different valid $5$ -member committees can be formed?

Show answer & explanation

Answer: 280

Answer

The correct answer is 280.

The correct answer is 280, found by taking the total possible committees satisfying the gender constraint (350) and subtracting the invalid committees containing both conflicting members (70).

Step-by-Step Solution

1

Calculate total combinations satisfying gender constraints without individual restriction

350 valid gender-balanced committees (200 from 3M/2W and 150 from 2M/3W)

Establishes the total baseline number of committee selections before removing forbidden pairings

2

Calculate forbidden combinations containing both specific conflicting individuals

70 forbidden combinations (40 with 3M/2W overall and 30 with 2M/3W overall)

Identifies committee selections that violate the condition that the two individuals cannot serve together

3

Subtract forbidden combinations from total baseline combinations

280 valid committee selections

Applying complementary counting (

350 - 70

) yields the exact number of allowable committees

Key Concept

Combinations with group constraints and complementary counting

Question 134Question →

A quality control engineer inspects a shipment of 25 solar panels, of which 9 are premium grade and 16 are standard grade. The engineer randomly selects 2 panels from the shipment, one after another without replacement. What is the probability, expressed as a decimal, that at least one of the two selected panels is premium grade?

Show answer & explanation

Answer: 0.6

Answer

The probability that at least one of the two selected panels is premium grade is 0.6.

Selection without replacement creates dependent events because the outcome of the first draw alters the pool available for the second draw. Out of 25 panels, 16 are standard grade. The probability that the first drawn panel is standard grade is 16/25. If the first panel is standard grade, 15 standard panels remain out of 24 total panels, giving a probability of 15/24 for the second draw. The probability of selecting two standard panels is (16/25) × (15/24) = 0.40. Using complementary probability, the probability that at least one panel is premium grade is 1 - 0.40 = 0.60.

Step-by-Step Solution

1

Determine the initial counts of total, premium, and standard grade panels.

Total panels = 25, premium grade panels = 9, standard grade panels = 16.

Establishing the sample space composition is necessary to calculate draw probabilities.

2

Calculate the probability that neither panel selected is premium grade (i.e., both are standard grade).

P(both standard) = (16 / 25) * (15 / 24) = (16 / 25) * (5 / 8) = 0.40.

Because selection is done without replacement, the total pool size and remaining standard panels each decrease by 1 for the second draw.

3

Apply complementary probability to determine the probability of selecting at least one premium panel.

P(at least one premium) = 1 - P(both standard) = 1 - 0.40 = 0.60.

The scenario of selecting at least one premium panel is the exact complementary event of selecting zero premium panels.

Key Concept

Dependent Events and Complementary Probability

Alternative Method

Sum the probabilities of mutually exclusive favorable outcomes: P(1st premium, 2nd standard) + P(1st standard, 2nd premium) + P(both premium) = (9/25)(16/24) + (16/25)(9/24) + (9/25)(8/24) = 0.24 + 0.24 + 0.12 = 0.60.

Estimated Time:1m 30s

Question 135Question →

A positive integer $N$ has the prime factorization $N = 3^a \cdot 5^b$ , where $a$ and $b$ are positive integers. If $N^2$ has exactly 35 positive integer divisors, what is the number of positive integer divisors of $N^3$ ?

Show answer & explanation

Answer: 70

Answer

70

Since

N = 3^a \cdot 5^b

,

N^2 = 3^{2a} \cdot 5^{2b}

. The number of positive divisors of

N^2

is

(2a+1)(2b+1) = 35

. The unique integer factor pair of 35 greater than 1 is

5 \times 7

, so the exponents

a

and

b

must be 2 and 3 (in some order). For

N^3 = 3^{3a} \cdot 5^{3b}

, the exponents are 6 and 9. Therefore, the number of positive divisors of

N^3

is

(6+1)(9+1) = 70

.

Step-by-Step Solution

1

Set up the formula for the number of positive divisors of

N^2

(2a + 1)(2b + 1) = 35

For an integer with prime factorization

p_1^{e_1} p_2^{e_2}

, the total number of divisors is

(e_1 + 1)(e_2 + 1)

.

2

Determine the positive integer values of

a

and

b

One exponent is 2 and the other exponent is 3

35 factors into

5 \times 7

. Solving

2a + 1 = 5

gives

a = 2

, and

2b + 1 = 7

gives

b = 3

.

3

Calculate the number of divisors of

N^3

(3(2) + 1)(3(3) + 1) = 7 \times 10 = 70

Exponents of

N^3

are

3a = 6

and

3b = 9

, so total divisors equal

(6 + 1)(9 + 1) = 70

.

Key Concept

Number of positive integer divisors from prime factorization

Estimated Time:1m 30s

Question 136Question →

How many integer values of $x$ satisfy both $|2x - 3| < 9$ and $|x + 1| \ge 3$ ?

Show answer & explanation

Answer: 4

Answer

There are 4 integer values of

x

that satisfy both inequalities.

Solving

|2x - 3| < 9

gives

-3 < x < 6

, so the set of possible integer values is

\{-2, -1, 0, 1, 2, 3, 4, 5\}

. Solving

|x + 1| \ge 3

gives

x \ge 2

or

x \le -4

. Taking the intersection of these two conditions gives

x \in \{2, 3, 4, 5\}

, for a total of 4 integer values.

Step-by-Step Solution

1

Unfold the first absolute value inequality

|2x - 3| < 9

-9 < 2x - 3 < 9, which simplifies to -3 < x < 6

An absolute value inequality of the form |A| < B is equivalent to -B < A < B.

2

Unfold the second absolute value inequality

|x + 1| \ge 3

x + 1 \ge 3 or x + 1 \le -3, which simplifies to x \ge 2 or x \le -4

An absolute value inequality of the form |A| >= B is equivalent to A >= B or A <= -B.

3

List integer candidates and find the intersection

Candidates from first condition: {-2, -1, 0, 1, 2, 3, 4, 5}. Applying second condition (x >= 2 or x <= -4) leaves {2, 3, 4, 5}

The solution must satisfy both conditions simultaneously.

Key Concept

Solving systems of absolute value inequalities for integer solutions

Estimated Time:1m 30s

Question 137Question →

If $k$ is the product of all real solutions to the equation $|x^2 - 7| = 3x - 3$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The product of all real solutions is 8.

The equation

|x^2 - 7| = 3x - 3

requires that

3x - 3 \ge 0

, which gives the restriction

x \ge 1

. Solving the two cases

x^2 - 7 = 3x - 3

and

x^2 - 7 = -(3x - 3)

produces the candidate roots

x = 4, -1, 2, -5

. Eliminating the negative candidate roots leaves

x = 2

and

x = 4

as the only valid real solutions. Their product is

2 \times 4 = 8

.

Step-by-Step Solution

1

Determine the domain constraint for valid solutions.

Since the absolute value expression

|x^2 - 7|

cannot be negative,

3x - 3 \ge 0

, which requires

x \ge 1

.

An absolute value quantity

|A|

is always non-negative, so any equation of the form

|A| = B

requires

B \ge 0

for real solutions.

2

Solve the positive case

x^2 - 7 = 3x - 3

and test for extraneous solutions.

x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0

, giving

x = 4

(valid, as

4 \ge 1

) and

x = -1

(extraneous, as

-1 < 1

).

Solutions must satisfy the non-negativity constraint of the right-hand side.

3

Solve the negative case

x^2 - 7 = -(3x - 3)

and test for extraneous solutions.

x^2 + 3x - 10 = 0 \implies (x + 5)(x - 2) = 0

, giving

x = 2

(valid, as

2 \ge 1

) and

x = -5

(extraneous, as

-5 < 1

).

Solutions must satisfy the non-negativity constraint of the right-hand side.

4

Compute the product of the valid real solutions.

k = 2 \times 4 = 8

.

The question asks for the product of all valid real solutions.

Key Concept

Absolute Value Equations with Variable Expressions and Extraneous Solution Elimination

Question 138Question →

Let $N = 2^4 \times 3^3 \times 5^2 \times 7$ . How many positive integer divisors of $N$ are even, divisible by 15, but not divisible by 9?

Show answer & explanation

Answer: 16

Answer

16

Any divisor of

N = 2^4 \times 3^3 \times 5^2 \times 7^1

takes the form

2^a \times 3^b \times 5^c \times 7^d

. The condition that the divisor is even requires

a \ge 1

, which gives 4 choices (

a \in \{1, 2, 3, 4\}

). The condition that the divisor is divisible by 15 requires

b \ge 1

and

c \ge 1

. The condition that it is not divisible by 9 requires

b < 2

. Together,

b \ge 1

and

b < 2

mean

b = 1

(1 choice). The condition

c \ge 1

allows

c \in \{1, 2\}

(2 choices). The exponent

d

can be 0 or 1 (2 choices). Multiplying these choices gives

4 \times 1 \times 2 \times 2 = 16

.

Step-by-Step Solution

1

Express the general prime factorization of a divisor

d = 2^a \times 3^b \times 5^c \times 7^d

with

0 \le a \le 4

,

0 \le b \le 3

,

0 \le c \le 2

,

0 \le d \le 1

Any divisor of

N

must consist only of the prime factors of

N

up to their respective maximum powers.

2

Apply the condition for even numbers

a \in \{1, 2, 3, 4\}

(4 choices)

An even integer must contain at least one factor of 2.

3

Apply the condition for divisibility by 15 and non-divisibility by 9

b = 1

(1 choice) and

c \in \{1, 2\}

(2 choices)

Divisibility by

15 = 3 \times 5

requires

b \ge 1

and

c \ge 1

. Non-divisibility by

9 = 3^2

requires

b < 2

. Thus

b

must be exactly 1.

4

Apply the condition for the exponent of 7

d \in \{0, 1\}

(2 choices)

There are no restrictions given for prime factor 7.

5

Calculate the product of choices

4 \times 1 \times 2 \times 2 = 16

By the fundamental counting principle, the total number of valid combinations of prime exponents is the product of the number of choices for each exponent.

Key Concept

Counting Divisors with Prime Factorization Restrictions

Question 139Question →

Set $S$ consists of a sequence of consecutive integers, ordered from least to greatest. If Set $S$ contains exactly $45$ integers and the sum of all the integers in Set $S$ is $405$ , what is the value of the smallest integer in Set $S$ ?

Show answer & explanation

Answer: -13

Answer

The smallest integer in Set

S

is

-13

.

The average (arithmetic mean) of the set is calculated by dividing the total sum (

405

) by the number of terms (

45

), which equals

9

. In an evenly spaced set with an odd number of elements, the mean is equal to the median (middle term). Because there are

45

terms, exactly

22

terms lie below the median. Subtracting

22

from

9

yields

-13

as the smallest integer in the set.

Step-by-Step Solution

1

Calculate the arithmetic mean and median of the set

Arithmetic mean = Median = 9

For an evenly spaced set, the arithmetic mean is equal to the median. Dividing the sum (

405

) by the number of terms (

45

) gives

9

.

2

Determine the number of terms preceding the median

22 terms precede the median

With

45

terms in total, the median is the

23\text{rd}

term, which leaves

\frac{45 - 1}{2} = 22

terms smaller than the median.

3

Calculate the smallest integer

Smallest integer =

-13

Subtracting

22

from the median gives

9 - 22 = -13

.

Key Concept

In any set of consecutive integers with

n

terms, the arithmetic mean equals the median. If

n

is odd, the smallest integer is given by

\text{Median} - \frac{n-1}{2}

.

Question 140Question →

Set $A$ consists of 5 consecutive even integers, and Set $B$ consists of 5 consecutive even integers. The smallest element of Set $B$ is 6 greater than the largest element of Set $A$ . If $S$ represents the standard deviation of the combined set of 10 integers formed by merging Set $A$ and Set $B$ , what is the value of $S^2$ ?

Show answer & explanation

Answer: 57

Answer

The value of

S^2

is 57.

Each subset of 5 consecutive even integers has an internal variance of 8. Because the mean of Set A is

x

and the mean of Set B is

x+14

, the combined mean is

x+7

. Each group mean is 7 units away from the overall mean. By the parallel axis theorem for variance, the combined variance is the within-group variance plus the squared distance from the combined mean:

8 + 7^2 = 57

.

Step-by-Step Solution

1

Define Set A algebraically and calculate its mean and variance.

Set

A = \{x-4, x-2, x, x+2, x+4\}

. Mean

\mu_A = x

, and variance

\sigma_A^2 = \frac{(-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2}{5} = \frac{40}{5} = 8

.

Symmetrically defining 5 consecutive even integers simplifies variance calculation.

2

Define Set B relative to Set A.

The largest element of Set

A

is

x+4

. The smallest element of Set

B

is

(x+4) + 6 = x+10

. Thus, Set

B = \{x+10, x+12, x+14, x+16, x+18\}

with mean

\mu_B = x+14

and variance

\sigma_B^2 = 8

.

Translating the word problem into exact algebraic representations for Set B.

3

Calculate the mean of the combined 10-element set.

The combined mean

\mu = \frac{5(x) + 5(x+14)}{10} = x+7

.

The combined mean is essential to evaluate overall variation.

4

Calculate the variance

S^2

of the combined set.

Using the combined variance formula

S^2 = \frac{1}{2}[\sigma_A^2 + (\mu_A - \mu)^2] + \frac{1}{2}[\sigma_B^2 + (\mu_B - \mu)^2] = \frac{1}{2}[8 + (-7)^2] + \frac{1}{2}[8 + 7^2] = 8 + 49 = 57

.

Combines internal set variances with the variance between group means.

Key Concept

Combined Variance and Standard Deviation of Sets

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Alternative Method

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Common Mistakes

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