Verbal Section Allocation and Time Optimization

You are designing a 15-minute verbal section (900 seconds total) with 19 questions across four subtypes. The intended distribution is 30% GMAT-style grammar, 30% TOEFL-style vocabulary-inference, 30% tense/voice, and 10% other.

Assume a fixed 60-second instruction overhead, leaving 840 seconds for attempting questions. You may skip questions.

Part 1: Integer Allocation Closest to Target Mix

• Targets (as counts): 30% of 19 ≈ 5.7 for each of grammar, vocabulary, tense; 10% of 19 ≈ 1.9 for other.
• Feasible set: nonnegative integers that sum to 19, with each subtype within ±2 of its target (i.e., grammar/vocab/tense ∈ [4, 7], other ∈ [0, 3]).
• Objective: minimize total squared percentage error from the 30/30/30/10 targets. Break any ties by preferring higher grammar counts.
• Report the chosen counts and the residual error for each subtype.

Part 2: Attempt Plan to Maximize Expected Correct

• Performance profile (per attempted question):
  • Grammar: 35 sec/question, 85% accuracy
  • Vocabulary: 55 sec/question, 75% accuracy
  • Tense/voice: 45 sec/question, 80% accuracy
  • Other: 60 sec/question, 70% accuracy
• Use your Part 1 counts as the maximum available per subtype.
• With a 15-minute limit (900 sec) and 60 sec overhead, choose how many to attempt in each subtype to maximize expected correct answers.
• Show your formulation (e.g., bounded knapsack or greedy by expected-correct-per-second), your final attempt plan, expected correct, and any leftover time.