Hints and solutions to
Problem of the day for Wednesday, January 8
A three digit number is selected at random from all three digit
numbers 100 - 999. What is the probability that the number is a perfect
square?
-
We have two things to count. First, how many 3-digit numbers are
there?
In other words, how many numbers from 100 to 999? The answer is
not
999 - 100,
but rather 1 + 999 - 100 = 900. To see this, try some small examples
(i.e., how many
numbers from 1 to 5, or 15 to 20 etc. Notice that you always have to add
1 to the
difference).
-
Second, we have to count how many squares between 100 and 999. Of
course, you don't
have to check every number to see if it is a perfect square. Rather, note
that
100 = 102, and the square root of 999 is a little bigger than
31. So the
biggest square before 999 is 312 = 961. So the number of
squares from
100 to 999 is 1 + 31 - 10 (there's that 1 again!) = 22.
-
That means 22 out of the 900 numbers from 100 to 999 are perfect
squares, so
the probability is
22/900 = 0.02444... = about 2.4 %
that a randomly
chosen 3-digit
number will be a perfect square.