Scholarship Cutoffs and Set Inclusion
A district scholarship uses two screening rules on a 100-point mathematics score M and a 100-point physics score P. Rule A requires M at least 90 and P at least 85. Rule B requires M+P at least 180. Counsellor Chen has 40 applicants: 22 satisfy Rule A, 18 satisfy Rule B, and 9 satisfy both. A student is called priority if they satisfy at least one rule. The principal asks whether the two rules are interchangeable, how many applicants are priority, and how many satisfy neither. Chen also proposes Rule C: M at least 85 and P at least 90, claiming it is equivalent to Rule A for every possible score pair.
1. Test the implications A ⇒ B and B ⇒ A using feasible score pairs with 0 ≤ M,P ≤ 100. What does this imply about treating the rules as interchangeable?
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A ⇒ B fails. Choose (M,P)=(90,85). Rule A holds because 90≥90 and 85≥85. But M+P=175<180, so Rule B fails.
B ⇒ A fails. Choose (100,80). Then M+P=180, so Rule B holds, but P=80<85, so Rule A fails.
Therefore neither rule implies the other; they select different regions of the score square and cannot be treated as interchangeable.
2. Using the counts, find the number of priority applicants and the number satisfying neither. Is the data consistent?
Read the worked answer
By inclusion-exclusion, the number satisfying at least one rule is |A∪B|=22+18−9=31. Hence the number satisfying neither is 40−31=9.
The data are consistent because 9≤22 and 9≤18, and the union 31 does not exceed 40. So 31 applicants are priority and 9 satisfy neither.
3. Is Rule C equivalent to Rule A? If not, give a score pair satisfying exactly one. Also state whether satisfying Rule A is sufficient, necessary, or both for being priority.
Read the worked answer
Rule C is not equivalent to Rule A. The pair (95,87) satisfies A (M≥90, P≥85) but not C (P<90). The pair (86,95) satisfies C but not A (M<90). Thus each rule has a point the other misses.
Since priority means A or B, satisfying A is sufficient for priority. It is not necessary, because (100,80) satisfies B only and is still priority.