Find all the proper fractions (in reduced form) whose denominators and numerators are single digits.
Then
put them in order from smallest to largest. For example, you would not include 3/9 because it is equivalent to 1/3.
People often ask where I come up with the Problems of the Month.
This particular problem arose from my daily walk with my wife.
We have a 4.5 mile path that we walk here in the city and that is nine half miles. Using my wife's pedometer
(
yes, the pedometer figured into another problem in 2010), we knew where several of the half-mile markers
were located. So, I could determine when we were 1/3 of the way, 1/2 of the way, 2/3 of the way, ... But on one walk when it
was 8 degrees out, I started telling my wife when we were 1/9 of the way, 1/6 of the way, and then I wondered about the fractions in between.
Hence this problem!
Solution to the Problem:
1/9, 1/8, 1/7, 1/6, 1/5, 2/9, 1/4, 2/7, 1/3. 3/8, 2/5, 3/7, 4/9, 1/2,
5/9, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 7/9, 4/5, 5/6, 6/7, 7/8, 8/9
First, I found the least common multiple of 2, 3, 4, 5, 6, 7, 8, and 9, which is 2,520.
Then, I converted each of the unit fractions with single digit denominators to equivalent fractions with 2520 as a
denominator and placed them on a number line:
1/9 = 280/2520
1/8 = 315/2520
1/7 = 360/2520
1/6 = 420/2520
1/5 = 504/2520
1/4 = 630/2520
1/3 = 840/2520
1/2 = 1260/2520
Then, I converted each of the remaining fractions to equivalent fractions with 2520 as a denominator and placed them on
the number line.