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Is the AVERAGE of two rational numbers always a rational number?
AYes, because (a/b + c/d)/2 = (ad + bc)/(2bd), which is a ratio of integers — hence rational
BNo, the average can be irrational
COnly if both rationals are positive
DOnly if the denominators match
Answer & Solution
Correct answer: A. Yes, because (a/b + c/d)/2 = (ad + bc)/(2bd), which is a ratio of integers — hence rational
For rationals a/b and c/d (b, d ≠ 0): their sum a/b + c/d = (ad + bc)/(bd) is a ratio of integers, so rational. Dividing by 2 (also rational) gives (ad + bc)/(2bd) — still a ratio of integers, so rational. This guarantees a rational number ALWAYS lies between any two rationals, which is the foundation of density.
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