## Solve Timed Logical and Quantitative Reasoning Cases Work through the seven short assessment cases below. For each reasoning case, use only the stated premises: do not turn “may,” “generally,” or “often” into a certainty. Show enough arithmetic to make every numerical answer auditable. ### Part 1 — Farm Relationships Facts: - Everyone who lives on the farm is related to X. - A is X's brother. - B lives on the farm. Classify each statement as `true`, `false`, or `cannot determine`: 1. B is related to X. 2. A lives on the farm. 3. X lives on the farm. #### What This Part Should Cover - Apply the farm-resident implication only in its stated direction. - Separate A's known family relationship from A's unknown residence. - Avoid assuming that X lives wherever X's relatives live. ```hint Follow each implication in one direction Write “lives on farm implies related to X” as an arrow before evaluating the three statements. ``` ### Part 2 — Juvenile Case Classification Assume the following material: - Juvenile offenses often end in release with a fine or release without a fine. - If a child is released without a fine, the court prohibits that child from seeking counseling. - Release without a fine is generally used for a first offense. - Ming is a first-time offender, and the court requires Ming to seek counseling. Classify “Ming was released without a fine” as `definitely true`, `probably true`, `cannot determine`, `probably false`, or `definitely false`. #### What This Part Should Cover - Distinguish the exact consequence of no-fine release from the merely typical first-offense pattern. - Reconcile the counseling requirement with the stated prohibition. - Explain why being a first-time offender alone does not prove the release type. ```hint Look for a contradiction, not a correlation Test the proposed release type against the required-counseling fact before using the weaker “generally” premise. ``` ### Part 3 — Salary Steps A position pays `13000` at step 1. Each move to the next step increases the previous step's salary by `10%`. What is the salary at step 4? #### What This Part Should Cover - Count how many increases occur between steps 1 and 4. - Apply the percentage to the preceding step rather than repeatedly to the starting step. - Give the exact result under the stated compounding rule. ```hint Count transitions There are three moves: step 1 to 2, step 2 to 3, and step 3 to 4. ``` ### Part 4 — Volunteer Funding Material: - If government funding increases, current volunteers will be paid. - Paying volunteers may attract more volunteers in the future. - Government funding has increased. Classify “There will be no volunteers next year” using the five labels from Part 2. #### What This Part Should Cover - Derive the certain consequence for current volunteers. - Treat “may attract” as a possibility, not a guarantee. - Decide whether the premises determine next year's volunteer count. ```hint Keep time periods separate The premises say what happens to current volunteers now but do not state who remains or joins next year. ``` ### Part 5 — Acquisition and Purchase Volume Material: - If Company A acquires a smaller company, it may change its name and temporarily lower prices. - If it temporarily lowers prices, purchase volume may increase. - Company A acquired a smaller company. Classify “Purchase volume increased” using the five labels from Part 2. #### What This Part Should Cover - Preserve both occurrences of “may.” - Avoid chaining possible events as though they were guaranteed implications. - State whether an observed acquisition determines an observed volume increase. ```hint Mark possibility words Underline each “may” before attempting to chain the two sentences. ``` ### Part 6 — Combined Work Rate Worker A completes `72` units in `6` hours. Worker B completes `108` units in `6` hours. If they work together at constant independent rates, how long do they need to complete `60` units? #### What This Part Should Cover - Convert both totals to units per hour. - Add rates, not completion times. - Divide the required work by the combined rate and include units. ```hint Normalize first Express each worker's output as units per hour before combining them. ``` ### Part 7 — Benefit Payment A four-person household has a basic standard of `420` and monthly income of `180`. Its countable income is the first `30` of income plus one third of all remaining income. The benefit equals `75%` of the basic standard minus countable income. Calculate the benefit. If the only numerical choice shown is `225`, explain whether to select it or `none of the above`. #### What This Part Should Cover - Split the income into the first `30` and the remaining amount. - Apply one third only to the remaining amount. - Compute the reduced standard before subtracting countable income. - Compare the derived result with the available choice rather than forcing a match. ```hint Keep the two reductions separate First calculate countable income; then calculate 75% of the standard; only then subtract. ``` ### What a Strong Answer Covers - Uses only the supplied implications and probability qualifiers. - Shows calculations with the correct number of percentage steps and consistent units. - Identifies when the premises are insufficient instead of guessing. - Flags an inconsistent answer choice when the stated formula yields a different result. ### Follow-up Questions 1. Which conclusions would change if “may” were replaced by “will” in Parts 4 and 5? 2. How would Part 3 differ if every raise were `10%` of the step-1 salary instead of the previous step? 3. Which single premise in Part 2 is decisive for rejecting no-fine release? 4. What rounding rule would need to be specified if Part 7 produced fractional cents?