In a board election, three candidates P, Q, and R contest. How many votes did each receive? I. P received 17 more votes than Q and 103 more votes than R. II. The total number of votes cast was 1703.

Introduction / Context:
The task is to determine whether the provided statements allow unique vote counts for P, Q, and R. Data Sufficiency focuses on whether uniqueness is guaranteed, not on performing the full calculation unless needed.

Given Data / Assumptions:

• Let votes be p, q, r (nonnegative integers).
• Statement I: p = q + 17 and p = r + 103.
• Statement II: p + q + r = 1703.

Concept / Approach:
Combine linear relations to test if a consistent, integral solution exists. If the system is inconsistent or does not yield integers, the statements are not sufficient to answer “how many votes each received.”

Step-by-Step Solution:
From I: q = p − 17 and r = p − 103.Sum with II: p + (p − 17) + (p − 103) = 1703 ⇒ 3p − 120 = 1703 ⇒ 3p = 1823 ⇒ p = 1823 / 3, which is not an integer.Because vote counts must be integers, the two statements together are inconsistent and therefore cannot determine valid values.Individually, neither statement fixes unique values: I gives relations without a total; II gives a total without splits.

Verification / Alternative check:
Attempting integer rounding would violate the linear equalities, confirming insufficiency.

Why Other Options Are Wrong:
Neither I nor II alone suffices; together they still fail due to inconsistency with integer counts.

Common Pitfalls:
Assuming fractional votes are acceptable or overlooking the divisibility requirement.

Final Answer:
Both statements together are not sufficient (no consistent integer solution).