A seminar was attended by 100 professionals, each of whom speaks at least one of two langu… — GMAT

A seminar was attended by $100$ professionals, each of whom speaks at least one of two languages: Spanish or French. Exactly $60$ of the professionals speak Spanish, and exactly $50$ speak French. If all $100$ professionals took a language proficiency examination scored on a scale from $0$ to $100$ , is the average (arithmetic mean) score of all $100$ professionals greater than $75$ ?

(1) The average score of the professionals who speak only Spanish is $80$ , and the average score of the professionals who speak only French is $70$ .
(2) The average score of the professionals who speak both Spanish and French is $85$ .

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Cevap
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

The correct response identifies that both statements together are sufficient while neither statement alone is sufficient. By using the principle of inclusion-exclusion on the overlapping set sizes, we find that there are 50 professionals who speak only Spanish, 40 who speak only French, and 10 who speak both. Statement (1) alone is insufficient because the average score of the 10 dual-language professionals remains unknown, allowing the overall average to fall either above or below 75. Statement (2) alone is insufficient because it provides no score data for 90 of the 100 professionals. When both statements are combined, all three disjoint subgroup averages are known, giving an exact overall average of 76.5, which is definitively greater than 75.

Adım Adım Çözüm

Rephrase the question stem using overlapping set principles

Let

S

be Spanish speakers (

60

) and

F

be French speakers (

50

). Since every professional speaks at least one language,

|S \cup F| = 100

. By the inclusion-exclusion principle,

|S \cup F| = |S| + |F| - |S \cap F| \implies 100 = 60 + 50 - |S \cap F|

, so

|S \cap F| = 10

. The group partitions into: Spanish only =

60 - 10 = 50

, French only =

50 - 10 = 40

, and Both =

10

Deconstructing the total population into three mutually exclusive subgroups establishes the exact weights for computing the overall weighted average score.

Evaluate Statement (1) alone

Statement (1) provides average score for Spanish-only (

80

) and French-only (

70

). The total score sum is

50(80) + 40(70) + 10(A_{both}) = 6800 + 10(A_{both})

. The overall average is

68 + 0.1(A_{both})

. Depending on

A_{both}

(

0 \le A_{both} \le 100

), the overall average can range from

68

78

. For instance, if

A_{both} = 70

, overall average is

75

(not

>75

); if

A_{both} = 100

, overall average is

78

(

>75

). Thus, Statement (1) alone is NOT sufficient.

Since the score of the overlap group is unknown, the overall average cannot be uniquely tested against the threshold of 75.

Evaluate Statement (2) alone

Statement (2) provides

A_{both} = 85

, but gives no information about the average scores of the Spanish-only (

50

people) or French-only (

40

people) groups. Thus, the overall average score could be very low or very high. Statement (2) alone is NOT sufficient.

Without data on 90% of the population, statement (2) alone leaves the overall mean undetermined.

Evaluate Statements (1) and (2) together

Combining both statements gives:

50

people with average

80

40

people with average

70

, and

10

people with average

85

. Overall total score sum

= 50(80) + 40(70) + 10(85) = 4000 + 2800 + 850 = 7650

. Overall average

= 7650 / 100 = 76.5

. Since

76.5 > 75

, we get a definitive YES answer.

Having full weighted average data for all three disjoint components of the set yields a single, precise overall mean.

Anahtar Kavram

Weighted Average across Mutually Exclusive Partitions of Overlapping Sets

Tahmini Süre:2m 0s

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