1973 AHSME Problems/Problem 5
Problem
Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),
those which are always true are
Solution
We can consider each statement independently and see which ones are true. The average of two numbers and is .
Statement I The above statement is not true for all , so Statement I is false.
Statement II
Since addition is commutative, the above is true for all , so Statement II is true.
Statement III
The above statement is not true for all , so Statement III is false.
Statement IV
The left side and the right side are equivalent expressions, so Statement IV is true.
Statement V
Let be the identity element. If such exists, then the following is true:
However, since is a variable, but must be a constant, fixed number, there is no identity element for the binary operation of averaging. Therefore, Statement V is false.
Statements II and IV are true, so the answer is .
See Also
1973 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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