{
  "version": 1,
  "topic": "CE11",
  "title": "Why rounding at every step changes a cement worksheet",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Shared versus separately rounded batches",
    "premise": "Each independent interval requires 10.1 theoretical sacks. Compare one compatible shared batch with independently rounded batches.",
    "formula": "shared = ceil(10.1n); separate = n × ceil(10.1)",
    "variables": "n is a count of intervals. Separate rounding is appropriate only when batches are physically independent.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Number of intervals = 3",
        "explanation": "Each independent interval requires 10.1 theoretical sacks. Compare one compatible shared batch with independently rounded batches."
      },
      {
        "label": "Select the relation",
        "calculation": "shared = ceil(10.1n); separate = n × ceil(10.1)",
        "explanation": "n is a count of intervals. Separate rounding is appropriate only when batches are physically independent."
      },
      {
        "label": "Substitute the values",
        "calculation": "ceil(3 × 10.1) = 31; 3 × ceil(10.1) = 33; water difference = 2 × 6.3 = 12.6 US gal",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Shared whole sacks (sk) = 31; Separate whole sacks (sk) = 33; Water difference at 6.3 gal/sk (US gal) = 12.6",
        "explanation": "The difference comes from the number of rounding decisions, not a geometric change."
      }
    ],
    "result": "Shared whole sacks (sk) = 31; Separate whole sacks (sk) = 33; Water difference at 6.3 gal/sk (US gal) = 12.6",
    "interpretation": "The difference comes from the number of rounding decisions, not a geometric change.",
    "control": "Number of intervals",
    "columns": [
      "Number of intervals",
      "Shared whole sacks (sk)",
      "Separate whole sacks (sk)",
      "Water difference at 6.3 gal/sk (US gal)"
    ],
    "rows": [
      [
        1,
        11,
        11,
        0
      ],
      [
        2,
        21,
        22,
        6.3
      ],
      [
        3,
        31,
        33,
        12.6
      ],
      [
        4,
        41,
        44,
        18.9
      ],
      [
        5,
        51,
        55,
        25.2
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Number of intervals = 1. Shared whole sacks (sk): 11; Separate whole sacks (sk): 11; Water difference at 6.3 gal/sk (US gal): 0.",
      "Number of intervals = 2. Shared whole sacks (sk): 21; Separate whole sacks (sk): 22; Water difference at 6.3 gal/sk (US gal): 6.3.",
      "Number of intervals = 3. Shared whole sacks (sk): 31; Separate whole sacks (sk): 33; Water difference at 6.3 gal/sk (US gal): 12.6.",
      "Number of intervals = 4. Shared whole sacks (sk): 41; Separate whole sacks (sk): 44; Water difference at 6.3 gal/sk (US gal): 18.9.",
      "Number of intervals = 5. Shared whole sacks (sk): 51; Separate whole sacks (sk): 55; Water difference at 6.3 gal/sk (US gal): 25.2."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Shared whole sacks (sk) without rounding intermediate values.",
        "answer": "ceil(3 × 10.1) = 31; 3 × ceil(10.1) = 33; water difference = 2 × 6.3 = 12.6 US gal. Shared whole sacks (sk) = 31. The difference comes from the number of rounding decisions, not a geometric change."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Number of intervals changes from 1 to 5. Other controls remain as stated in the premise. Shared whole sacks (sk) changes from 11 to 51."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. n is a count of intervals. Separate rounding is appropriate only when batches are physically independent. The difference comes from the number of rounding decisions, not a geometric change."
      }
    ]
  },
  "recorded": null
}