If a is a positive whole number, is b also a positive whole number?
(1) The average (arithmetic mean) of a, b + 2, and a - 5 is a.
\frac{a+b+2+a-5}{3}=a so 2a+b-3=3a so b=a+3=integer+integer=integer and will be positive.
Sufficient.
(2) The average (arithmetic mean) of a and b is NOT a positive whole number.
\frac{a+b}{2}=\frac{4+5}{2}=4.5 and b is a whole number.
\frac{a+b}{2}=\frac{4+4.5}{2}=4.25 and b is NOT a whole number.
Not sufficient
(1) The average (arithmetic mean) of a, b + 2, and a - 5 is a.
\frac{a+b+2+a-5}{3}=a so 2a+b-3=3a so b=a+3=integer+integer=integer and will be positive.
Sufficient.
(2) The average (arithmetic mean) of a and b is NOT a positive whole number.
\frac{a+b}{2}=\frac{4+5}{2}=4.5 and b is a whole number.
\frac{a+b}{2}=\frac{4+4.5}{2}=4.25 and b is NOT a whole number.
Not sufficient