Logic and data questions ask what a model or set of observations establishes. Define the cases and units before calculating. A numerical answer may be correct for an assumed model while the question requires noticing that the model’s assumption was never supplied.
Reason through the task
For sets, account for overlap so that shared members are not counted twice. For probability, identify outcomes and whether they are equally likely. For conditional statements, preserve the direction of implication: if every A is B, a B need not be A. For data, distinguish a mean from a total or a percentage of a particular group. Label the denominator in every proportion. These habits expose errors hidden by familiar-looking arithmetic.
Worked example and a tempting error
Original example: among thirty students, eighteen play an instrument, fourteen play a sport and eight do both. At least one activity includes 18 + 14 − 8 = 24 students, so six do neither. Adding eighteen and fourteen without subtracting the overlap gives thirty-two, impossible in a class of thirty. The overlap is not an extra group outside the two activities; each of its eight members has already appeared in both counts.
Try a different case
Transfer: all approved projects have budgets. A project has a budget. Must it be approved? No: having a budget is necessary under the stated rule, but may not be sufficient. An unapproved project can also have one. By contrast, a project without a budget cannot be approved if the rule holds. Drawing one circle inside another helps preserve that direction without confusing a rule with its converse.