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Difficulty: MediumOverlapping Sets, Statistics, and Data Distributions

A community library cataloged a collection of 120120 historical manuscripts. Each manuscript is written in either Latin, Ancient Greek, or both. How many of the manuscripts are written in both Latin and Ancient Greek?

(1) Exactly 8080 manuscripts are written in Latin.
(2) The number of manuscripts written only in Ancient Greek is twice the number of manuscripts written in both languages.

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Neither statement alone provides enough independent relationships to solve for the intersection. Statement (1) determines the count of manuscripts written exclusively in Ancient Greek (120−80=40120 - 80 = 40), while Statement (2) establishes a proportional link between manuscripts exclusively in Ancient Greek and those in both languages (2x2x). Combining both facts produces the linear equation 2x=402x = 40, which uniquely solves to x=20x = 20.

Step-by-Step Solution

1
Rephrase the question stem using overlapping set principles.
Let LL be the set of Latin manuscripts, GG be the set of Ancient Greek manuscripts, and xx be the number of manuscripts in both (L∩GL \cap G). Since every manuscript is in at least one set, Total = (Latin only)+(Greek only)+x=120(\text{Latin only}) + (\text{Greek only}) + x = 120. Also, Total = ∣L∣+(Greek only)=120|L| + (\text{Greek only}) = 120. We need to find the unique value of xx.
Formulating the set relationships establishes the exact algebraic system needed to evaluate sufficiency.
2
Evaluate Statement (1) independently.
Statement (1) gives ∣L∣=80|L| = 80. Substituting into the total formula gives 80+(Greek only)=120  ⟹  Greek only=4080 + (\text{Greek only}) = 120 \implies \text{Greek only} = 40. However, ∣L∣=(Latin only)+x=80|L| = (\text{Latin only}) + x = 80. The overlap xx can range anywhere from 00 to 8080.
Statement (1) does not provide enough information to isolate xx from Latin only.
3
Evaluate Statement (2) independently.
Statement (2) states that Greek only=2x\text{Greek only} = 2x. Substituting into the total equation yields (Latin only)+2x+x=(Latin only)+3x=120(\text{Latin only}) + 2x + x = (\text{Latin only}) + 3x = 120. Multiple integer solutions exist for xx (e.g., if Latin only=90\text{Latin only} = 90, x=10x = 10; if Latin only=60\text{Latin only} = 60, x=20x = 20).
Statement (2) presents one equation with two unknown variables.
4
Evaluate Statements (1) and (2) together.
From Statement (1), Greek only=120−80=40\text{Greek only} = 120 - 80 = 40. From Statement (2), Greek only=2x\text{Greek only} = 2x. Therefore, 2x=40  ⟹  x=202x = 40 \implies x = 20. This gives a single, unique answer.
Combining both statements provides two independent linear equations, uniquely solving for xx.

Key Concept

Overlapping Sets (Two-Group Venn Diagram) Data Sufficiency
Estimated Time:2m 0s
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Topics

Overlapping Sets, Statistics, and Data Distributions

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