All practice questions

Question 501Question →

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 502Question →

A commercial distributor purchased a shipment of 50 identical medical imaging monitors for a total cost of $20,000. To set the original list price of each monitor, the distributor marked up the cost price per unit by 75 percent. The distributor sold 30 monitors at the original list price. To clear the remaining inventory, the distributor sold the rest of the monitors at a discount of$ d $percent off the original list price. If the distributor realized an overall net profit of 33 percent on the entire shipment of 50 monitors, what is the value of$ d$?

24

30

40

60

84

Show answer & explanation

Answer: 60

Answer

The value of

d

is 60.

The correct response accurately determines that the distributor needs

5,600 from the remaining 20 monitors to achieve an overall 33 percent profit on the

20,000 investment. This requires selling each remaining monitor for

280, which reflects a

420 price reduction from the

700 list price. Expressed as a percentage of the

700 list price, a $420 reduction is equal to 60 percent.

Step-by-Step Solution

1

Calculate the unit cost price and original list price per monitor

Unit cost =

\frac{\$20,000}{50} = \$400

. List price =

\$400 \times (1 + 0.75) = \$700

.

Establishing individual unit costs and list prices is necessary to determine total revenues.

2

Determine total revenue required for an overall net profit of 33 percent

Total Target Revenue =

\$20,000 \times (1 + 0.33) = \$26,600

.

An overall 33 percent profit means total revenue must equal 133 percent of the total shipment cost.

3

Calculate revenue from the first 30 monitors and the remaining required revenue

Revenue from 30 units =

30 \times \$700 = \$21,000

. Remaining revenue needed from 20 units =

\$26,600 - \$21,000 = \$5,600

.

Subtracting the revenue generated by full-price sales isolates the revenue needed from the discounted items.

4

Determine the selling price per discounted monitor and calculate the discount percentage

d

Discounted selling price =

\frac{\$5,600}{20} = \$280

. Discount amount =

\$700 - \$280 = \$420

. Discount percentage

d = \left(\frac{\$420}{\$700}\right) \times 100 = 60

.

The discount percentage is the ratio of the dollar discount to the original list price.

Key Concept

Profit, Loss, and Markup (Successive percentage changes with mixed pricing)

Question 503Question →

For all real numbers $x > 1$ , the expression $\sqrt{x + 2\sqrt{x - 1}} - \sqrt{x - 2\sqrt{x - 1}}$ is equal to $2$ .

True

False

Show answer & explanation

Answer: False

Answer

The statement is False because the expression equals

2\sqrt{x-1}

(which is less than

2

) whenever

1 < x < 2

.

The statement asserts that the identity holds for all

x > 1

. However, evaluating

\sqrt{x - 2\sqrt{x-1}}

yields

|\sqrt{x-1} - 1|

. When

1 < x < 2

,

\sqrt{x-1} < 1

, so

|\sqrt{x-1} - 1| = 1 - \sqrt{x-1}

. Subtraction gives

2\sqrt{x-1} \neq 2

for all

x

in

(1, 2)

. Thus, the statement is false.

Step-by-Step Solution

1

Rewrite the expressions under the outer radicals as perfect squares.

x \pm 2\sqrt{x-1} = (x-1) \pm 2\sqrt{x-1} + 1 = (\sqrt{x-1} \pm 1)^2

Recognize the quadratic pattern

a^2 \pm 2ab + b^2

where

a = \sqrt{x-1}

and

b = 1

.

2

Apply the radical identity

\sqrt{u^2} = |u|

to remove the outer square roots.

\sqrt{x + 2\sqrt{x-1}} = |\sqrt{x-1} + 1| = \sqrt{x-1} + 1

and

\sqrt{x - 2\sqrt{x-1}} = |\sqrt{x-1} - 1|

The principal square root of a squared quantity is non-negative, requiring absolute value bars.

3

Analyze the absolute value

|\sqrt{x-1} - 1|

across the specified domain

x > 1

.

For

x \ge 2

,

\sqrt{x-1} \ge 1 \implies |\sqrt{x-1} - 1| = \sqrt{x-1} - 1

. For

1 < x < 2

,

\sqrt{x-1} < 1 \implies |\sqrt{x-1} - 1| = 1 - \sqrt{x-1}

.

The sign of the expression inside the absolute value changes at

x = 2

.

4

Evaluate the full expression for

1 < x < 2

.

(\sqrt{x-1} + 1) - (1 - \sqrt{x-1}) = 2\sqrt{x-1}

Since

2\sqrt{x-1} < 2

when

1 < x < 2

, the statement does not hold for all real numbers

x > 1

.

Key Concept

Nested Radicals and the Principal Square Root Absolute Value Identity

\sqrt{u^2} = |u|

Question 504Question →

Traditionally, economists believed that increasing minimum wage rates would inevitably lead to higher unemployment among low-skilled workers. However, recent empirical studies of municipal labor markets challenge this classic view. Researchers analyzing cities that implemented incremental minimum wage increases observed no significant drop in overall employment rates. Instead, the higher wages reduced costly employee turnover and improved worker productivity, offsetting the increased labor costs for businesses. Thus, modest minimum wage increases can stabilize low-wage workforce markets without forcing widespread job cuts.

Which of the following best states the primary purpose of the passage?

To present recent empirical findings that challenge a traditional economic view regarding minimum wage increases

To explain how reduced employee turnover lowers operational costs for municipal businesses

To argue that classical economic theories regarding labor markets are completely invalid and useless

To demonstrate that minimum wage increases inevitably cause higher unemployment among low-skilled workers

To propose a national legislative framework for implementing municipal wage increases

Show answer & explanation

Answer: To present recent empirical findings that challenge a traditional economic view regarding minimum wage increases

Answer

To present recent empirical findings that challenge a traditional economic view regarding minimum wage increases

The correct answer accurately synthesizes the main goal of the passage: highlighting new empirical evidence that challenges an old economic premise.

Step-by-Step Solution

1

Analyze the structural organization of the passage.

The passage starts with a traditional economic assumption, shifts perspective using the transition word 'However', presents empirical evidence, and concludes with a main takeaway.

Tracking structural pivots isolates the author's core thesis from background information.

2

Summarize the primary purpose across the whole passage.

The main goal is to introduce recent empirical research that disputes the traditional belief about minimum wage and unemployment.

The primary purpose must encompass the whole passage rather than isolated details.

Key Concept

Identifying Primary Purpose and Main Idea

Question 505Question →

A commercial bank's loan portfolio consists of three categories of loans: small business loans, residential mortgages, and commercial real estate loans. Small business loans carry an average annual interest rate of $8.5\%$ and account for $30\%$ of the portfolio's total value. Residential mortgages carry an average annual interest rate of $5.0\%$ , and commercial real estate loans carry an average annual interest rate of $6.5\%$ . If the overall weighted average annual interest rate for the entire portfolio is $6.35\%$ , what percentage of the portfolio's total value is invested in commercial real estate loans?

15\%

20\%

35\%

40\%

50\%

Show answer & explanation

Answer: $20\%$

Answer

The commercial real estate loans account for

20\%

of the portfolio's total value.

The overall portfolio rate of

6.35\%

is a weighted average of small business loans (

8.5\%

rate,

30\%

weight), residential mortgages (

5.0\%

rate,

x\%

weight), and commercial real estate loans (

6.5\%

rate,

(70-x)\%

weight). Solving

0.30(8.5) + 0.05x + 0.065(70 - x) = 6.35

gives

x = 50\%

, which means commercial real estate loans constitute

70\% - 50\% = 20\%

of the total portfolio.

Step-by-Step Solution

1

Define variables for the unknown portfolio weight components.

Let

w_1 = 0.30

(small business loans),

w_2 = x

(residential mortgages), and

w_3 = 0.70 - x

(commercial real estate loans).

The sum of all weights across the three portfolio categories must equal

100\%

(

1.00

). Since small business loans constitute

30\%

, the remaining two categories must sum to

70\%

(

0.70

).

2

Set up the weighted average formula for the overall portfolio interest rate.

0.30(8.5) + x(5.0) + (0.70 - x)(6.5) = 6.35

The overall weighted average interest rate is the sum of each component's rate multiplied by its respective weight in the portfolio.

3

Expand and simplify the algebraic equation.

2.55 + 5.0x + 4.55 - 6.5x = 6.35 \implies 7.10 - 1.5x = 6.35

Multiplying out terms allows combining like terms to solve for

x

.

4

Solve for

x

to find the weight of residential mortgages, then find the weight of commercial real estate loans.

1.5x = 0.75 \implies x = 0.50

(

50\%

). Thus, commercial real estate weight

= 0.70 - 0.50 = 0.20

(

20\%

).

Subtracting

x = 50\%

from the total remaining

70\%

gives the exact share allocated to commercial real estate loans.

Key Concept

Weighted Averages in Multi-Component Portfolios

Question 506Question →

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 507Question →

A quality control analyst measured the shelf life, in days, of a sample of $80$ manufactured batteries, where each battery had a distinct shelf life. Battery $X$ had a shelf life at the $65\text{th}$ percentile of the sample. If $20$ additional batteries are subsequently tested and every one of these $20$ batteries has a shelf life strictly greater than Battery $X$ , what is the percentile rank of Battery $X$ 's shelf life in the combined sample of $100$ batteries?

52\text{nd}

percentile

45\text{th}

percentile

51\text{st}

percentile

60\text{th}

percentile

65\text{th}

percentile

Show answer & explanation

Answer: $52\text{nd}$ percentile

Answer

The

52\text{nd}

percentile

In the initial sample of

80

batteries, a shelf life at the

65\text{th}

percentile means that

65\%

of

80

batteries, or

52

batteries, have a shelf life less than or equal to Battery

X

. When

20

new batteries are added—all with shelf lives strictly greater than Battery

X

—the new total number of batteries is

100

, while the number of batteries with shelf life less than or equal to Battery

X

remains

52

. Thus, Battery

X

is at the

\frac{52}{100} \times 100\% = 52\text{nd}

percentile of the combined sample.

Step-by-Step Solution

1

Determine the number of batteries in the original sample with a shelf life less than or equal to Battery X

Since

65\%

of the original

80

batteries are at or below Battery

X

,

0.65 \times 80 = 52

batteries.

By definition, a percentile rank of

65

in a dataset of size

N

represents

65\%

of the data points at or below that score.

2

Calculate the total number of batteries in the combined dataset

80 + 20 = 100

batteries.

Adding

20

new batteries increases the sample size from

80

to

100

.

3

Determine the number of batteries at or below Battery X in the combined dataset

The count remains

52

batteries.

All

20

newly added batteries have shelf lives strictly greater than Battery

X

, so none of them fall at or below Battery

X

.

4

Calculate the new percentile rank of Battery X

\frac{52}{100} \times 100\% = 52\text{nd percentile}.

The percentile rank is the percentage of total data points in the combined sample that are less than or equal to Battery

X

.

Key Concept

Percentile rank represents the proportion of values in a dataset that are less than or equal to a given value. When a dataset expands by adding values strictly above a target value, the number of values at or below the target remains constant while the total sample size increases.

Estimated Time:1m 30s

Question 508Question →

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 509Question →

In a dataset of $N$ distinct test scores, score $S_1$ is at the 75th percentile and score $S_2$ is at the 40th percentile. A group of 60 new distinct scores is added to the dataset: 12 are strictly less than $S_2$ , 18 are strictly between $S_2$ and $S_1$ , and 30 are strictly greater than $S_1$ . If score $S_1$ is at the 60th percentile of the combined dataset, what is the percentile rank of score $S_2$ in the combined dataset?

16th percentile

25th percentile

28th percentile

40th percentile

46th percentile

Show answer & explanation

Answer: 28th percentile

Answer

28th percentile

In the original dataset of

N

scores,

0.75N

scores are below

S_1

and

0.40N

scores are below

S_2

. Adding 60 scores (12 below

S_2

, 18 between

S_2

and

S_1

, and 30 above

S_1

) increases the total dataset size to

N + 60

. The number of scores below

S_1

becomes

0.75N + 12 + 18 = 0.75N + 30

. Given that

S_1

is at the 60th percentile of the new dataset,

0.75N + 30 = 0.60(N + 60)

, which simplifies to

0.15N = 6

, giving

N = 40

. The original number of scores below

S_2

is

0.40 \times 40 = 16

. Adding the 12 new scores that are below

S_2

gives

16 + 12 = 28

scores below

S_2

in the combined dataset. Out of 100 total scores in the combined dataset, the percentile rank of

S_2

is

\frac{28}{100} \times 100\% = 28\%

, which corresponds to the 28th percentile.

Step-by-Step Solution

1

Set up expressions for the number of scores below

S_1

and

S_2

in the original dataset.

In the original dataset of

N

scores,

0.75N

scores are strictly less than

S_1

, and

0.40N

scores are strictly less than

S_2

.

By definition of percentile rank,

k\%

percentile means

k\%

of the dataset scores fall strictly below that value.

2

Determine the number of scores strictly below

S_1

in the combined dataset.

The total number of new scores added strictly below

S_1

is

12 + 18 = 30

. Thus, the total number of scores below

S_1

in the combined dataset is

0.75N + 30

, while the new total dataset size is

N + 60

.

Scores added below

S_2

and scores added between

S_2

and

S_1

are all strictly less than

S_1

.

3

Solve for

N

using the 60th percentile rank condition for

S_1

in the combined dataset.

0.75N + 30 = 0.60(N + 60) \implies 0.75N + 30 = 0.60N + 36 \implies 0.15N = 6 \implies N = 40

.

Setting the count of scores below

S_1

equal to

60\%

of the new total dataset size

N + 60

forms a single-variable linear equation.

4

Calculate the percentile rank of score

S_2

in the combined dataset.

Original scores below

S_2 = 0.40 \times 40 = 16

. Combined scores below

S_2 = 16 + 12 = 28

. Combined total dataset size =

40 + 60 = 100

. Percentile rank of

S_2 = \frac{28}{100} \times 100\% = 28\%

.

Dividing the total count of scores strictly below

S_2

in the combined set by the total combined dataset size yields the updated percentile rank.

Key Concept

Percentile Rank and Combined Sets

Question 510Question →

Consider the four values defined below based on arithmetic and geometric sequences. Arrange these four items in ascending order (from smallest numerical value to largest numerical value).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order from smallest to largest value is: (1) The 7th term of the geometric sequence (

192

), (2) The sum of the first 8 terms of the arithmetic sequence (

208

), (3) The sum of the infinite geometric series (

225

), and (4) The 50th term of the arithmetic sequence (

250

).

Evaluating each item yields: the 7th term of the geometric sequence equals 192, the sum of the first 8 terms of the arithmetic sequence equals 208, the sum of the infinite geometric series equals 225, and the 50th term of the arithmetic sequence equals 250. Placing these in ascending numerical order produces 192 < 208 < 225 < 250.

Step-by-Step Solution

1

Calculate the value of the 7th term of the geometric sequence

Using

g_7 = 3 \cdot 2^{7-1} = 3 \cdot 64 = 192

.

The

n\text{th}

term of a geometric sequence is given by

g_n = g_1 r^{n-1}

.

2

Calculate the sum of the first 8 terms of the arithmetic sequence

Using

S_8 = \frac{8}{2}[2(5) + (8-1)6] = 4[10 + 42] = 208

.

The sum of the first

n

terms of an arithmetic sequence is given by

S_n = \frac{n}{2}[2a_1 + (n-1)d]

.

3

Calculate the sum of the infinite geometric series

Using

S_\infty = \frac{150}{1 - 1/3} = \frac{150}{2/3} = 225

.

The sum of an infinite geometric series with

|r| < 1

is

S_\infty = \frac{a_1}{1-r}

.

4

Determine the common difference and the 50th term of the arithmetic sequence

Find

d = \frac{35 - 15}{7 - 3} = 5

, then

a_{50} = 15 + (50 - 3)5 = 15 + 235 = 250

.

Linear spacing between terms

a_k

and

a_m

yields

a_m - a_k = (m - k)d

.

5

Compare the calculated values to order them from smallest to largest

192 < 208 < 225 < 250

.

Arranging the quantities according to their numerical values gives the final sorted sequence.

Key Concept

Arithmetic and Geometric Sequences and Series

Question 511Question →

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 512Question →

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 513Question →

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 514Question →

A box contains $4$ red balls and $6$ blue balls. If two balls are randomly drawn from the box one after another without replacement, what is the probability that both balls drawn are red?

\frac{4}{25}

\frac{3}{25}

\frac{2}{15}

\frac{1}{3}

\frac{11}{15}

Show answer & explanation

Answer: $\frac{2}{15}$

Answer

The probability that both balls drawn are red is

\frac{2}{15}

.

The probability of selecting a red ball on the first draw is

\frac{4}{10}

. Since the ball is not replaced,

9

total balls remain in the box, of which

3

are red. The probability of selecting a red ball on the second draw given the first was red is

\frac{3}{9}

. By the multiplication rule for dependent events, the probability that both balls drawn are red is

\frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}

.

Step-by-Step Solution

1

Find the probability of drawing a red ball on the first pick.

P(\text{First Red}) = \frac{4}{10} = \frac{2}{5}

There are

4

red balls out of a total of

10

balls (

4 + 6 = 10

).

2

Find the conditional probability of drawing a red ball on the second pick after one red ball has been removed.

P(\text{Second Red} \mid \text{First Red}) = \frac{3}{9} = \frac{1}{3}

Because sampling is done without replacement,

1

red ball and

1

total ball are removed, leaving

3

red balls out of

9

remaining balls.

3

Multiply the probabilities of the dependent events to find the joint probability.

P(\text{Both Red}) = \frac{2}{5} \times \frac{1}{3} = \frac{2}{15}

For dependent events,

P(A \text{ and } B) = P(A) \times P(B \mid A)

.

Key Concept

Probability of Dependent Events (Sampling Without Replacement)

Question 515Question →

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 516Question →

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 517Question →

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 518Question →

Let $n = 2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. If the integer $\frac{n}{6}$ has exactly $32$ positive divisors and the integer $10n$ has exactly $90$ positive divisors, what is the value of $a + b + c$ ?

8

9

10

11

12

Show answer & explanation

Answer: 9

Answer

The value of

a + b + c

is

9

.

Writing the prime factorizations gives

\frac{n}{6} = 2^{a-1} \cdot 3^{b-1} \cdot 5^c

with

a \cdot b \cdot (c+1) = 32

, and

10n = 2^{a+1} \cdot 3^b \cdot 5^{c+1}

with

(a+2)(b+1)(c+2) = 90

. Solving for positive integers

a

,

b

, and

c

yields

a = 4

,

b = 2

, and

c = 3

. Thus, the sum

a + b + c = 9

.

Step-by-Step Solution

1

Express the prime factorization of

\frac{n}{6}

and write its total divisor count equation.

\frac{n}{6} = \frac{2^a \cdot 3^b \cdot 5^c}{2 \cdot 3} = 2^{a-1} \cdot 3^{b-1} \cdot 5^c

. The number of positive divisors is

a \cdot b \cdot (c + 1) = 32

.

Dividing

n

by

6 = 2 \cdot 3

decreases the exponents of

2

and

3

by

1

each. The formula for the total number of divisors of a number

p_1^{x} p_2^{y} p_3^{z}

is

(x+1)(y+1)(z+1)

.

2

Express the prime factorization of

10n

and write its total divisor count equation.

10n = (2 \cdot 5) \cdot (2^a \cdot 3^b \cdot 5^c) = 2^{a+1} \cdot 3^b \cdot 5^{c+1}

. The number of positive divisors is

(a + 2) \cdot (b + 1) \cdot (c + 2) = 90

.

Multiplying by

10 = 2 \cdot 5

increases the exponents of

2

and

5

by

1

each.

3

Solve the system of equations for the positive integer exponents

a

,

b

, and

c

.

From

a \cdot b \cdot (c + 1) = 32

, test integer factors of

32

. Setting

c + 1 = 4 \implies c = 3

, we get

a \cdot b = 8

. Substituting

c = 3

into

(a + 2)(b + 1)(c + 2) = 90

yields

(a + 2)(b + 1)(5) = 90 \implies (a + 2)(b + 1) = 18

. Testing pairs where

a \cdot b = 8

: if

a = 4

and

b = 2

, then

(4 + 2)(2 + 1) = 6 \cdot 3 = 18

, which satisfies both equations.

Because

a, b, c

are positive integers, factor analysis uniquely pinpoints

a = 4

,

b = 2

, and

c = 3

.

4

Calculate

a + b + c

.

a + b + c = 4 + 2 + 3 = 9

.

Summing the derived values of the positive integer exponents.

Key Concept

Prime Factorization and Total Number of Divisors

Estimated Time:2m 0s

Question 519Question →

If $m$ and $n$ are positive integers such that $3^m \cdot 5^n$ is a factor of $15!$ , what is the maximum possible value of $m + n$ ?

8

9

10

11

12

Show answer & explanation

Answer: 9

Answer

9

To find the maximum power of a prime

p

dividing

n!

, we sum

\lfloor n/p^k \rfloor

for all

k \ge 1

. For

p = 3

,

m = \lfloor 15/3 \rfloor + \lfloor 15/9 \rfloor = 5 + 1 = 6

. For

p = 5

,

n = \lfloor 15/5 \rfloor = 3

. Thus, the maximum value of

m + n

is

6 + 3 = 9

.

Step-by-Step Solution

1

Find the maximum exponent

m

of the prime factor 3 in

15!

using Legendre's formula.

m = \lfloor \frac{15}{3} \rfloor + \lfloor \frac{15}{3^2} \rfloor = 5 + 1 = 6

The prime factor 3 appears in multiples of 3 (3, 6, 9, 12, 15) and contributes an extra factor in 9 (

3^2

).

2

Find the maximum exponent

n

of the prime factor 5 in

15!

using Legendre's formula.

n = \lfloor \frac{15}{5} \rfloor = 3

The prime factor 5 appears in multiples of 5 (5, 10, 15).

3

Sum the maximum possible integer values of

m

and

n

.

m + n = 6 + 3 = 9

The maximum possible value of

m+n

is the sum of the maximum individual prime exponents.

Key Concept

Counting Prime Factors in a Factorial (Legendre's Formula)

Alternative Method

List out the prime factorizations of all numbers from 1 to 15: 3 contributing numbers are 3, 6 (

2 \cdot 3

), 9 (

3^2

), 12 (

2^2 \cdot 3

), 15 (

3 \cdot 5

). Total factors of 3 =

1 + 1 + 2 + 1 + 1 = 6

. 5 contributing numbers are 5, 10 (

2 \cdot 5

), 15 (

3 \cdot 5

). Total factors of 5 =

1 + 1 + 1 = 3

. Sum

6 + 3 = 9

.

Estimated Time:1m 30s

Question 520Question →

If $k$ is a positive integer such that $\frac{3^{k+2} + 3^{k+4}}{10} = 3^{2k-1}$ , what is the value of $k$ ?

1

2

3

4

5

Show answer & explanation

Answer: 3

Answer

The value of

k

is 3.

Factoring

3^{k+2}

from the numerator gives

3^{k+2}(1 + 3^2) = 3^{k+2}(10)

. Dividing by 10 simplifies the left side to

3^{k+2}

. Setting

3^{k+2} = 3^{2k-1}

requires

k + 2 = 2k - 1

, which yields

k = 3

.

Step-by-Step Solution

1

Factor out the common exponential term

3^{k+2}

from the numerator on the left-hand side.

3^{k+2} + 3^{k+4} = 3^{k+2}(1 + 3^2) = 3^{k+2}(1 + 9) = 10 \cdot 3^{k+2}

When adding terms with identical bases, factor out the term with the smallest exponent to simplify the sum.

2

Substitute the factored expression into the fraction and simplify.

\frac{10 \cdot 3^{k+2}}{10} = 3^{k+2}

Canceling the common factor of 10 in the numerator and denominator simplifies the left-hand side.

3

Set the simplified left-hand side equal to the right-hand side of the original equation.

3^{k+2} = 3^{2k-1}

Both sides now have the same base of 3.

4

Equate the exponents since the bases are equal and solve for

k

.

k + 2 = 2k - 1 \implies 2 + 1 = 2k - k \implies k = 3

For any non-zero, non-one base

b

,

b^x = b^y

implies

x = y

.

Key Concept

Factoring exponential expressions with identical bases and equating exponents.

Answer

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