Statistics and Data Analysis Practice Questions — GMAT

Question 1Question →

In a corporate evaluation of $200$ employees, performance scores range from $0$ to $100$ . An employee with a score of $84$ scored strictly higher than $150$ employees and strictly lower than $42$ employees, while the remaining employees received a score of exactly $84$ . If the percentile rank of a score is defined as the percentage of all scores strictly below it plus half the percentage of all scores equal to it, what is the percentile rank of a score of $84$ ?

75th percentile

77th percentile

79th percentile

84th percentile

88th percentile

Show answer & explanation

Answer: 77th percentile

Answer

77th percentile

The correct answer represents the relative position of a score of 84. Since 150 employees scored lower and 8 employees scored equal to 84, applying the percentile formula yields ((150 + 4) / 200) * 100 = 77th percentile.

Step-by-Step Solution

1

Determine the number of employees scoring exactly 84.

Number of employees scoring 84 =

200 - 150 - 42 = 8

.

The total group size equals the sum of students scoring below 84, above 84, and equal to 84.

2

Calculate the effective count of scores at or below 84 under the given percentile definition.

Effective count =

150 + 0.5 \times 8 = 154

.

The definition specifies adding the count of strictly lower scores to half the count of tied scores.

3

Convert the effective count to a percentile rank.

Percentile rank =

\frac{154}{200} \times 100 = 77\text{th percentile}

.

Dividing by the total number of scores (

200

) and multiplying by

100

converts the count into a percentage.

Key Concept

Percentile Rank Calculation

Estimated Time:1m 30s

Question 2Question →

In a survey of $120$ corporate executives, $70$ executives read Magazine A, $50$ read Magazine B, and $25$ read both Magazine A and Magazine B. How many of the executives surveyed read neither Magazine A nor Magazine B?

0

25

45

50

95

Show answer & explanation

Answer: 25

Answer

25 executives surveyed read neither Magazine A nor Magazine B.

By the inclusion-exclusion principle, the total number of executives reading at least one magazine is given by

\text{Set A} + \text{Set B} - \text{Both} = 70 + 50 - 25 = 95

. Since

120

executives were surveyed in total, the number of executives who read neither magazine is

120 - 95 = 25

.

Step-by-Step Solution

1

Apply the Inclusion-Exclusion Principle for two overlapping sets to find the number of executives reading at least one magazine.

\text{Total(At least one)} = \text{Magazine A} + \text{Magazine B} - \text{Both} = 70 + 50 - 25 = 95

.

Directly adding the counts for Magazine A and Magazine B counts executives who read both twice, so the intersection must be subtracted once.

2

Subtract the number of executives who read at least one magazine from the total number of executives surveyed.

\text{Neither} = \text{Total Surveyed} - \text{Total(At least one)} = 120 - 95 = 25

.

The universe of surveyed executives consists of those who read at least one magazine and those who read neither.

Key Concept

Overlapping Sets (Double-Matrix / Two-Set Venn Diagram)

Question 3Question →

A company recorded the number of daily sales over a 7-day week. The numbers of daily sales, when arranged in ascending order, are $12, 15, 18, x, 22, y, 30$ . If the median of the 7 daily sales figures is $20$ and the arithmetic mean is $21$ , what is the value of $y$ ?

Show answer & explanation

Answer: 30

Answer

The value of

y

is 30.

For an ordered set of 7 numbers, the median is the 4th term, which means

x = 20

. The total sum of the set is found by multiplying the number of terms by the mean:

7 \times 21 = 147

. Adding all known terms (

12 + 15 + 18 + 20 + 22 + 30

) gives 117. Subtracting 117 from 147 yields

y = 30

.

Step-by-Step Solution

1

Determine the value of

x

using the median definition.

x = 20

Since the 7 numbers are given in ascending order, the median is the middle (4th) term.

2

Calculate the total sum of all 7 numbers.

Total sum =

147

The sum of elements in a set equals the number of elements multiplied by the arithmetic mean (

7 \times 21 = 147

).

3

Sum all known numbers and set up an equation for

y

.

117 + y = 147

12 + 15 + 18 + 20 + 22 + 30 = 117

.

4

Solve for

y

.

y = 30

Subtract 117 from 147 to isolate

y

.

Key Concept

Using properties of median and arithmetic mean to find missing terms in an ordered data set.

Question 4Question →

A dataset consists of five integers: $2, 5, 8, 11,$ and $x$ . The variance of the dataset is $10$ . If the median of the dataset is $8$ , what is the value of $x$ ?

Show answer & explanation

Answer: 9

Answer

The value of

x

is 9.

By setting up the equation for variance in terms of

x

, we obtain the quadratic equation

x^2 - 13x + 36 = 0

, which yields

x = 4

or

x = 9

. Arranging the set in ascending order for

x = 9

gives

\{2, 5, 8, 9, 11\}

, where the middle number (median) is 8, satisfying all conditions.

Step-by-Step Solution

1

Calculate the mean of the dataset in terms of

x

.

The mean is

\mu = \frac{26 + x}{5}

.

The mean of a dataset is the sum of all elements divided by the number of elements.

2

Set up the variance equation using the definition of population variance.

Variance

\sigma^2 = \frac{(2-\mu)^2 + (5-\mu)^2 + (8-\mu)^2 + (11-\mu)^2 + (x-\mu)^2}{5} = 10

.

Variance measures the average squared deviation from the mean.

3

Substitute

\mu = \frac{26+x}{5}

, clear denominators, expand the algebraic expression, and set up the quadratic equation.

x^2 - 13x + 36 = 0

.

Multiplying through by 25 and simplifying the quadratic terms yields a standard quadratic form.

4

Solve the quadratic equation for

x

.

x = 4

or

x = 9

.

Factoring

(x-4)(x-9) = 0

gives two potential solutions.

5

Evaluate the median condition for both potential values of

x

.

For

x = 9

, the ordered dataset is

\{2, 5, 8, 9, 11\}

, which has a median of 8.

For

x = 4

, the ordered set is

\{2, 4, 5, 8, 11\}

with median 5, which fails the given condition.

Key Concept

Variance calculation and dataset order statistics

Question 5Question →

A class of 20 students took a 10-point mathematics quiz. The frequency table below records the quiz scores achieved by 18 of the students:

Score	Frequency
6	3
7	5
8	4
9	4
10	2

The scores of the remaining 2 students were recorded later. If the score of every student is an integer from 0 to 10, inclusive, and adding the 2 missing scores causes the median score of the entire class of 20 students to be 8 and the arithmetic mean score to be an integer, what is the score of the higher-scoring student among the 2 remaining students?

6

7

8

9

10

Show answer & explanation

Answer: 10

Answer

10

The sum of the 18 known scores is 141. For the overall mean of 20 scores to be an integer, the total sum of all 20 scores must be a multiple of 20. Since each score is at most 10, the maximum possible total sum is

141 + 10 + 10 = 161

. The only multiple of 20 between 141 and 161 is 160, requiring the sum of the two missing scores to be

160 - 141 = 19

. The only valid integer scores bounded by 10 that sum to 19 are 9 and 10. Adding scores of 9 and 10 places the 10th and 11th ordered values at 8, giving a median of 8. Thus, the higher missing score is 10.

Step-by-Step Solution

1

Calculate the total sum and count of the 18 known student scores.

Known count =

3 + 5 + 4 + 4 + 2 = 18

students. Known sum =

(6 \times 3) + (7 \times 5) + (8 \times 4) + (9 \times 4) + (10 \times 2) = 18 + 35 + 32 + 36 + 20 = 141

.

Establishing baseline sum and count is essential before analyzing missing values.

2

Set up the equation for the total sum of all 20 student scores and apply the integer mean condition.

Let the missing scores be

a

and

b

with

0 \le a \le b \le 10

. The total sum for 20 students is

S_{20} = 141 + a + b

. The arithmetic mean is

\frac{141 + a + b}{20}

.

Since the mean must be an integer,

141 + a + b

must be a multiple of 20.

3

Determine the required sum of the two missing scores

a + b

.

Since

0 \le a \le 10

and

0 \le b \le 10

, we have

0 \le a + b \le 20

. The range for

S_{20}

is

[141, 161]

. The only multiple of 20 in this range is 160. Thus,

141 + a + b = 160 \implies a + b = 19

.

160 is the unique multiple of 20 reachable given score bounds.

4

Find the unique integer pair

(a, b)

satisfying

a + b = 19

with

a, b \le 10

.

Since

a \le 10

and

b \le 10

, the only integer solution with

a \le b

is

a = 9

and

b = 10

.

No other pair of integers between 0 and 10 sums to 19.

5

Verify that adding scores 9 and 10 maintains a median score of 8.

With 9 and 10 added, the frequencies are: Score 6 (3), Score 7 (5), Score 8 (4), Score 9 (5), Score 10 (3). The 10th and 11th values in order are both 8, so the median is

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

.

Confirms the median constraint is fully satisfied.

Key Concept

Properties of Weighted Means and Median Constraints in Frequency Tables

Estimated Time:2m 0s

Question 6Question →

In a survey of $200$ software developers regarding their proficiency in three programming languages—Python, Java, and C++:

- $110$ developers are proficient in Python.
- $95$ developers are proficient in Java.
- $85$ developers are proficient in C++.
- $20$ developers are proficient in all three languages.
- $30$ developers are proficient in none of the three languages.

If the total number of developers proficient in exactly two of these languages is twice the number of developers proficient in Python and Java only, how many developers are proficient in C++ only?

15

25

35

40

45

Show answer & explanation

Answer: $25$

Answer

The number of developers proficient in C++ only is 25.

The correct answer is 25. Subtracting the 30 developers proficient in none of the languages leaves 170 developers who are proficient in at least one language. Using the 3-set overlap equations, the number of developers proficient in exactly two languages is calculated to be 80. Since the number proficient in Python and Java only is half of this (40), the sum of the remaining two double-overlap regions (Java & C++ only, and Python & C++ only) must also equal 40. Subtracting this combined double-overlap count of 40 and the triple-overlap count of 20 from the total C++ count of 85 leaves 25 developers proficient in C++ only.

Step-by-Step Solution

1

Calculate the total number of developers proficient in at least one language.

\text{Total proficient} = 200 - 30 = 170

Developers proficient in none of the three languages must be excluded from the universe of set elements.

2

Apply the 3-set Venn Diagram formula relating total unique elements, individual set sums, exactly-two regions, and all-three regions.

\text{Total} = E_1 + E_2 + E_3

, where

E_1

is exactly one,

E_2

is exactly two, and

E_3

is exactly three. Also,

|P| + |J| + |C| = E_1 + 2E_2 + 3E_3

.

Sum of individual sets:

110 + 95 + 85 = 290

.
So,

290 = E_1 + 2E_2 + 3(20) \implies E_1 + 2E_2 = 230

.

From total elements:

170 = E_1 + E_2 + 20 \implies E_1 + E_2 = 150

.

Subtracting the two equations gives

E_2 = 230 - 150 = 80

.

Each element in

E_2

is counted twice in the sum of individual sets, and each element in

E_3

is counted three times.

3

Determine the breakdown of the double-overlap regions.

Given that

E_2 = 2 \times (\text{Python and Java only})

, we have

\text{Python and Java only} = \frac{80}{2} = 40

.

The remaining double-overlap regions involving C++ are

(\text{Java and C++ only}) + (\text{Python and C++ only}) = E_2 - 40 = 80 - 40 = 40

.

The total number of developers proficient in exactly two languages consists of three disjoint groups: Python/Java only, Java/C++ only, and Python/C++ only.

4

Calculate the number of developers proficient in C++ only.

|C| = (\text{C++ only}) + (\text{Java and C++ only}) + (\text{Python and C++ only}) + (\text{All 3}) 85 = (\text{C++ only}) + 40 + 20 85 = (\text{C++ only}) + 60 \text{C++ only} = 25

Subtract all overlapping subsets containing C++ from the total proficiency count of set C.

Key Concept

Three-Set Overlapping Venn Diagrams and Subset Decompositions

Estimated Time:2m 30s

Question 7Question →

At an investment firm with $200$ portfolio managers, $110$ managers invest in Equities, $95$ invest in Fixed Income, and $80$ invest in Commodities. Furthermore, $45$ managers invest in both Equities and Fixed Income, $35$ invest in both Equities and Commodities, and $30$ invest in both Fixed Income and Commodities. If $15$ managers do not invest in any of these three asset classes, how many portfolio managers invest in all three asset classes?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10 portfolio managers.

Subtracting the 15 managers who do not invest in any asset class from the total 200 yields 185 managers in at least one asset class. Applying the 3-set inclusion-exclusion formula

185 = (110 + 95 + 80) - (45 + 35 + 30) + x

simplifies to

185 = 175 + x

, which gives

x = 10

for the number of managers investing in all three asset classes.

Step-by-Step Solution

1

Determine the total number of portfolio managers investing in at least one asset class.

185 managers

Subtracting the 15 managers who invest in none of the three asset classes from the total of 200 gives the size of the union of the three sets:

200 - 15 = 185

.

2

Set up the Inclusion-Exclusion formula for three overlapping sets.

|E \cup F \cup C| = |E| + |F| + |C| - (|E \cap F| + |E \cap C| + |F \cap C|) + |E \cap F \cap C|

To avoid overcounting elements present in multiple set intersections, pairwise intersections are subtracted and the triple intersection is added back.

3

Substitute known values into the equation and solve for the unknown triple intersection

x

.

x = 10

185 = 110 + 95 + 80 - (45 + 35 + 30) + x \implies 185 = 285 - 110 + x \implies 185 = 175 + x \implies x = 10

.

Key Concept

Three-Set Inclusion-Exclusion Principle

Question 8Question →

A logistics company maintains a fleet consisting of three types of delivery vehicles: Vans, Medium Trucks, and Heavy Trucks. The ratio of the number of Vans to Medium Trucks is $3 : 2$ , and the ratio of the number of Medium Trucks to Heavy Trucks is $4 : 1$ . If the average fuel efficiency is $24$ miles per gallon for the Vans and $15$ miles per gallon for the Medium Trucks, and the average fuel efficiency for the entire combined fleet is $19$ miles per gallon, what is the average fuel efficiency, in miles per gallon, of the Heavy Trucks?

Show answer & explanation

Answer: 5

Answer

The average fuel efficiency of the Heavy Trucks is 5 miles per gallon.

Combining the given vehicle ratios gives a unified proportion of 6 Vans to 4 Medium Trucks to 1 Heavy Truck (total 11 units). The total weighted efficiency sum is

6(24) + 4(15) + 1(x) = 144 + 60 + x = 204 + x

. Setting this equal to the combined fleet total of

11 \times 19 = 209

yields

x = 5

miles per gallon.

Step-by-Step Solution

1

Unify the two relative ratios into a single compound ratio for all three vehicle groups.

Vans : Medium Trucks : Heavy Trucks = 6 : 4 : 1, giving a total of 11 ratio units.

Both individual ratios share Medium Trucks as a common element (

2

units in the first ratio and

4

units in the second). Multiplying the first ratio by

2

aligns the Medium Truck count.

2

Set up the weighted mean equation using group weights and individual group averages.

Weighted Sum = 6(24) + 4(15) + 1(x) = 204 + x, where total elements = 11.

The sum of (group quantity × group mean) divided by total quantity equals the combined average.

3

Equate to total fleet mean and solve for the unknown average fuel efficiency x.

204 + x = 11 × 19 = 209, so x = 5 miles per gallon.

Multiplying total items (11) by total mean (19) gives total combined miles per gallon units (209).

Key Concept

Weighted Average of Multiple Combined Subgroups

Question 9Question →

A technology company conducted a survey of its $200$ software engineers regarding their proficiency in three programming languages: Python, Java, and C++. The survey revealed the following data:
- $110$ engineers are proficient in Python.
- $85$ engineers are proficient in Java.
- $75$ engineers are proficient in C++.
- $40$ engineers are proficient in both Python and Java.
- $35$ engineers are proficient in both Java and C++.
- $30$ engineers are proficient in both Python and C++.
- $15$ engineers are proficient in all three programming languages.

How many of the surveyed engineers are proficient in none of these three programming languages?

15

20

25

35

50

Show answer & explanation

Answer: 20

Answer

The number of software engineers proficient in none of the three programming languages is 20.

Using the 3-set inclusion-exclusion formula, the total number of engineers proficient in at least one programming language is

110 + 85 + 75 - (40 + 35 + 30) + 15 = 180

. Subtracting this from the total sample size of

200

yields

200 - 180 = 20

engineers proficient in none of the three languages.

Step-by-Step Solution

1

State the Principle of Inclusion-Exclusion formula for three sets.

N(\text{At least one}) = N(P) + N(J) + N(C) - [N(P \cap J) + N(J \cap C) + N(P \cap C)] + N(P \cap J \cap C)

To find the total number of engineers in at least one category without double-counting overlapping groups.

2

Substitute the given values into the formula.

N(\text{At least one}) = 110 + 85 + 75 - (40 + 35 + 30) + 15 = 270 - 105 + 15 = 180

Calculate the total number of unique engineers who know at least one language.

3

Subtract the number of engineers proficient in at least one language from the total population.

N(\text{None}) = 200 - 180 = 20

The total population consists of engineers proficient in at least one language plus those proficient in none.

Key Concept

Three-Set Overlapping Venn Diagrams and the Principle of Inclusion-Exclusion

Estimated Time:2m 0s

Question 10Question →

In a medical study involving $300$ clinical trial participants, researchers recorded the occurrence of three common side effects: fatigue, nausea, and insomnia. A total of $160$ participants reported fatigue, $140$ reported nausea, and $120$ reported insomnia. Additionally, $60$ participants reported both fatigue and nausea, $50$ reported both fatigue and insomnia, and $55$ reported both nausea and insomnia. If $30$ participants reported none of these three side effects, how many participants reported exactly two of these side effects?

Show answer & explanation

Answer: 120

Answer

120 participants reported exactly two of the side effects.

To calculate the number of participants with exactly two side effects, first find the total number of participants experiencing at least one side effect (

300 - 30 = 270

). Next, using the inclusion-exclusion formula

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, we find that

x = 15

participants experienced all three side effects. Finally, subtracting

15

from each pairwise overlap gives the counts for exactly two side effects:

45

(Fatigue & Nausea only),

35

(Fatigue & Insomnia only), and

40

(Nausea & Insomnia only). Summing these regions yields

45 + 35 + 40 = 120

.

Step-by-Step Solution

1

Calculate the total number of participants who experienced at least one side effect

270 participants

Subtracting the 30 participants who reported no side effects from the total study size of 300 gives

|F \cup N \cup I| = 300 - 30 = 270

.

2

Determine the number of participants who experienced all three side effects

15 participants

Applying inclusion-exclusion gives

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, simplifying to

270 = 255 + x

, so

x = 15

.

3

Calculate the sum of participants experiencing exactly two side effects

120 participants

The sum of pairwise intersections (

60 + 50 + 55 = 165

) counts individuals with all three side effects three times. Subtracting

3 \times 15 = 45

from

165

yields

120

.

Key Concept

Three-Set Overlapping Sets and Inclusion-Exclusion Principle

Estimated Time:2m 0s

Question 11Question →

In a biotechnology laboratory with $200$ research scientists, a survey was conducted regarding the use of three analytical instruments: Mass Spectrometers, NMR Spectrometers, and X-ray Diffractometers. The survey revealed that $95$ scientists use Mass Spectrometers, $80$ use NMR Spectrometers, and $75$ use X-ray Diffractometers. Exactly $20$ scientists use only Mass Spectrometers and NMR Spectrometers, exactly $15$ use only NMR Spectrometers and X-ray Diffractometers, and exactly $10$ use only Mass Spectrometers and X-ray Diffractometers. If $15$ scientists use all three instruments, how many scientists do not use any of these three instruments?

10

15

25

35

50

Show answer & explanation

Answer: $25$

Answer

25 scientists do not use any of the three instruments.

The total number of scientists using at least one instrument is calculated by summing all disjoint regions: the scientists using only one instrument (

50 + 30 + 35 = 115

), the scientists using exactly two instruments (

20 + 15 + 10 = 45

), and the scientists using all three instruments (

15

). This yields

115 + 45 + 15 = 175

scientists who use at least one instrument. Subtracting this from the total sample of

200

scientists gives

200 - 175 = 25

scientists who use none of the three instruments.

Step-by-Step Solution

1

Identify the number of scientists in each distinct overlapping region

Regions with exactly two instruments are given as 20, 15, and 10. The region with all three instruments is 15.

To find the number of single-instrument users without double-counting, all overlapping regions must be isolated.

2

Calculate the number of scientists using only one instrument

Mass Spectrometers only:

95 - (20 + 10 + 15) = 50

. NMR Spectrometers only:

80 - (20 + 15 + 15) = 30

. X-ray Diffractometers only:

75 - (10 + 15 + 15) = 35

.

Subtracting all shared regions from each set's total yields the exclusive users of that single instrument.

3

Sum all mutually exclusive regions to find the total number of scientists using at least one instrument

Total using at least one = (Only 1 instrument) + (Only 2 instruments) + (All 3 instruments) =

(50 + 30 + 35) + (20 + 15 + 10) + 15 = 115 + 45 + 15 = 175

.

The sum of disjoint regions equals the cardinality of the union of all three sets.

4

Subtract the number of scientists using at least one instrument from the total surveyed scientists

Scientists using none =

200 - 175 = 25

.

The set of scientists using none of the instruments is the complement of the union set.

Key Concept

3-Set Venn Diagram Inclusion-Exclusion Principle

Alternative Method

Using the 3-Set Inclusion-Exclusion formula: Total = |A| + |B| + |C| - (|A∩B| + |B∩C| + |A∩C|) + |A∩B∩C| + Neither. Note that |A∩B| = (only A and B) + |A∩B∩C| = 20 + 15 = 35, |B∩C| = 15 + 15 = 30, and |A∩C| = 10 + 15 = 25. Thus, 200 = 95 + 80 + 75 - (35 + 30 + 25) + 15 + Neither => 200 = 250 - 90 + 15 + Neither => 200 = 175 + Neither => Neither = 25.

Estimated Time:2m 0s

Question 12Question →

A set of $7$ positive integers has an arithmetic mean of $16$ , a median of $15$ , a unique mode of $12$ , and a range of $14$ . What is the maximum possible value of the largest integer in this set?

Show answer & explanation

Answer: 26

Answer

The maximum possible value of the largest integer in the set is 26.

The total sum of the 7 positive integers is

7 \times 16 = 112

. Ordering the terms as

a \le b \le c \le d \le e \le f \le g

, the median constraint gives

d = 15

. The range constraint gives

g - a = 14

, or

g = a + 14

. To maximize

g

, we need to maximize

a

. Because 12 is the unique mode of the set, 12 must appear at least twice. Since

d = 15

, the number 12 can only occupy positions

a, b,

or

c

, which means

a \le 12

. Setting

a = 12

gives the maximum value

g = 12 + 14 = 26

. A valid set achieving this is

\{12, 12, 12, 15, 15, 20, 26\}

, which sums to 112 and meets all statistical constraints.

Step-by-Step Solution

1

Determine the sum of the set

Sum = 112

The mean of 7 positive integers is 16, so the sum is 7 times 16.

2

Apply median and range constraints

d = 15 and a = g - 14

In a sorted set of 7 integers, the 4th element is the median (15), and range is the difference between the largest element g and smallest element a.

3

Bound the smallest element using the mode constraint

a <= 12, so max g = 12 + 14 = 26

Since 12 is the unique mode, it must occur at least twice. Because elements are sorted and median is 15, 12 must be among the first three terms, so the smallest element a cannot exceed 12.

4

Verify existence of a valid set with g = 26

Set {12, 12, 12, 15, 15, 20, 26} satisfies all conditions

The sum is 112, median is 15, mode is 12 (appears 3 times), and range is 26 - 12 = 14.

Key Concept

Extremal problems involving mean, median, mode, and range constraints

Question 13Question →

A artisanal coffee shop creates a custom blend using three grades of coffee beans: Grade P costing $10 per pound, Grade Q costing$ 15 per pound, and Grade R costing $25 per pound. The weight ratio of Grade P to Grade Q in the mixture is$ 3 : 2 $. If the weighted average cost of the total mixture is$ 18 per pound, what fraction of the total mixture, by weight, consists of Grade R?

\frac{5}{12}

\frac{11}{25}

\frac{6}{13}

\frac{6}{7}

\frac{7}{13}

Show answer & explanation

Answer: $\frac{6}{13}$

Answer

The fraction of the total mixture, by weight, that consists of Grade R is

\frac{6}{13}

.

First, find the weighted average price of Grades P and Q together using their given weight ratio of

3 : 2

:

\frac{3(10) + 2(15)}{3 + 2} = \frac{60}{5} = \$12

per pound. Next, treat the entire mixture as a combination of two parts: the (P+Q) group at

12 per pound and Grade R at

25 per pound. Let

W_{PQ}

be the weight of the (P+Q) group and

W_R

be the weight of Grade R. Setting up the weighted average gives

12 W_{PQ} + 25 W_R = 18(W_{PQ} + W_R)

, which simplifies to

7 W_R = 6 W_{PQ}

, or

W_R = \frac{6}{7} W_{PQ}

. The fraction of the total mixture represented by Grade R is

\frac{W_R}{W_{PQ} + W_R} = \frac{6/7 W_{PQ}}{W_{PQ} + 6/7 W_{PQ}} = \frac{6}{13}

.

Step-by-Step Solution

1

Calculate the average cost per pound of the combined P and Q mixture.

Average cost of P and Q mixture =

\frac{3(\$10) + 2(\$15)}{3 + 2} = \frac{30 + 30}{5} = \$12

per pound.

Since Grade P and Grade Q are present in a weight ratio of

3 : 2

, their combined per-pound cost must be weighted by these relative amounts.

2

Set up a weighted average equation for the combined mixture of (P + Q) and Grade R.

Let

W_{PQ}

be the weight of the P and Q combined mixture, and

W_R

be the weight of Grade R. The total cost equation is

12 W_{PQ} + 25 W_R = 18 (W_{PQ} + W_R)

.

The entire mixture consists of the combined P+Q component costing

12/lb and Grade R costing

25/lb, yielding a final weighted average of $18/lb.

3

Solve for the ratio of

W_R

to

W_{PQ}

.

12 W_{PQ} + 25 W_R = 18 W_{PQ} + 18 W_R \implies 7 W_R = 6 W_{PQ} \implies \frac{W_R}{W_{PQ}} = \frac{6}{7}

.

Expanding and rearranging terms isolates the relationship between the weight of Grade R and the combined weight of Grades P and Q.

4

Convert the part-to-part ratio into the part-to-whole fraction for Grade R.

\text{Fraction of Grade R} = \frac{W_R}{W_{PQ} + W_R} = \frac{6}{7 + 6} = \frac{6}{13}.

The question asks for the fraction of the total mixture, which requires dividing the weight of Grade R by the total combined weight of all three grades.

Key Concept

Weighted Average of Sub-groups and Combined Sets

Estimated Time:2m 0s

Question 14Question →

Data set $S$ consists of the four integers $4, 8, 12,$ and $16$ . A fifth integer $k$ , which is greater than the mean of data set $S$ , is added to the set to form data set $T = \{4, 8, 12, 16, k\}$ . If the standard deviation of data set $T$ is equal to the standard deviation of data set $S$ , what is the value of $k$ ?

Show answer & explanation

Answer: 15

Answer

The value of

k

is 15.

The variance of the original set

S = \{4, 8, 12, 16\}

is 20 with a mean of 10. Adding

k

creates a 5-element set

T

with mean

\frac{40+k}{5}

. Setting the variance of set

T

equal to 20 yields the quadratic equation

k^2 - 20k + 75 = 0

, which factors as

(k-5)(k-15) = 0

. Since

k

must be greater than 10, the correct answer is 15.

Step-by-Step Solution

1

Calculate the mean and variance of data set

S

.

Mean

\mu_S = 10

and variance

\sigma_S^2 = 20

.

Determining the baseline mean and variance for the initial dataset of 4 elements.

2

Express the mean and variance of data set

T = \{4, 8, 12, 16, k\}

algebraically in terms of

k

.

Mean

\mu_T = \frac{40+k}{5}

and variance

\sigma_T^2 = \frac{480 + k^2 - 5\left(\frac{40+k}{5}\right)^2}{5}

.

Formulating the standard deviation equation for the expanded dataset.

3

Equate

\sigma_T^2

to

\sigma_S^2 = 20

and solve for

k

.

The resulting quadratic equation

k^2 - 20k + 75 = 0

yields solutions

k = 5

and

k = 15

.

Finding all potential values of

k

that leave the dataset's standard deviation unchanged.

4

Apply the condition that

k

must be strictly greater than the mean of data set

S

(

\mu_S = 10

).

k = 15

.

Rejecting

k = 5

because it is less than 10.

Key Concept

Standard deviation measures the average distance of data points from the mean. Adding a value to a dataset shifts the mean and alters variance; finding when the standard deviation remains unchanged requires solving the quadratic relationship generated by the addition of the new element.

Question 15Question →

A survey asked consumers to rate a new service on a scale from $1$ to $5$ . The table below shows the frequency of each rating received, where $k$ is a positive integer:

Rating	Frequency
1	4
2	6
3	7
4	$k$
5	5

If the median rating of all responses is equal to the arithmetic mean of all responses, what is the value of $k$ ?

2

3

4

5

6

Show answer & explanation

Answer: $4$

Answer

4

The option stating

4

is correct because with

k = 4

, the total number of responses is

22 + 4 = 26

and the sum of all ratings is

62 + 4(4) = 78

. The mean is

\frac{78}{26} = 3

. The median of

26

ordered values is the average of the 13th and 14th values, both of which are

3

. Thus, the mean equals the median.

Step-by-Step Solution

1

Calculate total number of data points and total sum in terms of

k

Total responses

N = 4 + 6 + 7 + k + 5 = 22 + k

. Total sum

= 1(4) + 2(6) + 3(7) + 4(k) + 5(5) = 4 + 12 + 21 + 4k + 25 = 62 + 4k

.

The mean of a frequency distribution is the total weighted sum divided by the total frequency.

2

Determine the median rating

Ratings

1, 2, 3

account for the first

4 + 6 + 7 = 17

responses. For small positive integer values of

k

, the middle position(s) fall within the rating of

3

, so the median rating is

3

.

Since more than half of the initial

22

data points are

3

or below (specifically positions 11 through 17 are 3s), the median is

3

for

k \le 12

.

3

Equate the arithmetic mean to the median and solve for

k

\frac{62 + 4k}{22 + k} = 3 \implies 62 + 4k = 3(22 + k) \implies 62 + 4k = 66 + 3k \implies k = 4.

Setting the mean formula equal to the median value of

3

allows solving for

k

algebraically.

Key Concept

Weighted Mean and Frequency Distribution Median

Estimated Time:1m 30s

Question 16Question →

A set $S$ consists of $6$ distinct positive integers. The arithmetic mean of the integers in $S$ is $15$ , and the median of $S$ is $14$ . If the largest integer in $S$ is $28$ , what is the greatest possible value of the second-largest integer in $S$ ?

24

25

27

28

31

Show answer & explanation

Answer: $27$

Answer

The greatest possible value of the second-largest integer in the set is

27

.

The correct option is

27

. With

6

distinct positive integers

x_1 < x_2 < x_3 < x_4 < x_5 < x_6

, the total sum is

6 \times 15 = 90

. The median gives

x_3 + x_4 = 28

. Given

x_6 = 28

, we have

x_1 + x_2 + x_5 = 34

. Since all integers are distinct and

28

is the largest,

x_5

must be strictly less than

28

, making

27

the maximum integer bound. Setting

x_5 = 27

allows

x_1 = 1, x_2 = 6, x_3 = 13, x_4 = 15, x_5 = 27, x_6 = 28

, which satisfies every requirement of the problem.

Step-by-Step Solution

1

Determine the total sum of the

6

integers.

Sum =

6 \times 15 = 90

.

Since the mean of

6

integers is

15

, the sum of all elements equals the count multiplied by the mean.

2

Express the median condition algebraically.

Let the ordered set be

x_1 < x_2 < x_3 < x_4 < x_5 < x_6

. Then

\frac{x_3 + x_4}{2} = 14 \implies x_3 + x_4 = 28

.

For an even number of terms (

n=6

), the median is the average of the 3rd and 4th terms.

3

Substitute known values into the sum equation.

x_1 + x_2 + (x_3 + x_4) + x_5 + x_6 = 90 \implies x_1 + x_2 + 28 + x_5 + 28 = 90 \implies x_1 + x_2 + x_5 = 34

.

We know

x_3 + x_4 = 28

and

x_6 = 28

(the largest integer).

4

Apply constraints to maximize

x_5

.

Since

x_6 = 28

is the largest element and all integers are distinct,

x_5 < 28

, so

x_5 \le 27

. Testing

x_5 = 27

yields

x_1 + x_2 = 34 - 27 = 7

.

To verify

x_5 = 27

is achievable, choose distinct positive integers

x_1 = 1, x_2 = 6, x_3 = 13, x_4 = 15, x_5 = 27, x_6 = 28

, which satisfies all conditions.

Key Concept

Mean and Median Properties in Constrained Sets

Question 17Question →

The numbers of packages delivered by a courier on seven consecutive days are listed below:

5, 13, 2, 10, 15, 7, 4

What is the median number of packages delivered per day over this seven-day period?

5

7

8

10

13

Show answer & explanation

Answer: $7$

Answer

The median number of packages delivered per day is

7

.

To find the median of a dataset, the values must first be arranged in ascending order:

2, 4, 5, 7, 10, 13, 15

. Since there are

7

numbers, the median is the

4

th number in the ordered list, which is

7

.

Step-by-Step Solution

1

Arrange the data values in ascending order.

The ordered set is

2, 4, 5, 7, 10, 13, 15

.

Finding the median requires ordering the data sequentially first.

2

Identify the middle term of the ordered set containing

7

values.

The middle term is the

\frac{7+1}{2} = 4

th value, which is

7

.

For an odd number of data points

N

, the median is the

\frac{N+1}{2}

th element.

Key Concept

Median of a finite data set

Estimated Time:45s

Question 18Question →

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 19Question →

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 20Question →

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

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