Check whether each of the statements below is true. Justify your reasoning. Use algebra, if necessary, to justify. (i) The average of two even numbers is even. (ii) The average of any two multiples of 5 will be a multiple of 5. (iii) The average of any 5 multiples of 5 will also be a multiple of 5.
False. Counter-example: The average of 2 and 4 is 3, which is odd. The average of two even numbers is always an integer, but not necessarily even.
False. Counter-example: The average of 5 and 10 is 7.5, which is not a multiple of 5. The average of two multiples of 5 is a multiple of 5 only when their sum is divisible by 10.
True. Five multiples of 5 (5a, 5b, 5c, 5d, 5e) have average = (5a + 5b + 5c + 5d + 5e)/5 = a + b + c + d + e, which is always an integer. The average equals 5 times an integer, so it is a multiple of 5.
Explanation
(i) Let even numbers be 2a and 2b. Average = a + b, which is an integer but may be odd or even. (ii) Two multiples of 5 (5a and 5b) have average 5(a+b)/2, which is a multiple of 5 only if (a+b) is even. (iii) The sum of 5 multiples of 5 is divisible by 5, and dividing by 5 gives an integer that is a multiple of 5.