Placing Fractions on A Number Line

Directions: Using the digits 0 to 9 at most one time each, place a digit in each box to create five fractions and place them all on a number line with the correct order and spacing.

Hint

What will be your benchmark fractions?
Which fraction denominators will be more challenging to place?

Answer

There are many correct answers. One answer involves using 0/6 to represent 0, 8/4 to represent 2, and then placing the other fractions accordingly, for example, 9/3, 7/2, and 1/5.

Source: Robert Kaplinsky

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Comparing Fractions

Directions: Using the numbers 1-9, each one time only, fill in the blanks to make …

16 comments

  1. It seems like you all have developed a very well-specified theory of open middle task design. I can see you guys spinning that template out into a number of very interesting tasks. What’s missing here, I think, is a sense of how students react to the template. Any plans to get these in front of students?

    • Currently I am working through a group of problems for 8th grade to use this upcoming school year. I am going to field test some of the K-1 problems on my son however.

    • My concern with your example is related to the standard. Third graders are limited in the numbers they use for the denominator. How can you present 0/9 when 9ths in not acceptable fractions for 3rd grade per common core?

      • Good catch Jad, I have altered the wording in the solution set to avoid that. Thank you for contributing to the discussion.

      • Jad, the standard states: “Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.” Where are you seeing that 9ths can’t be a denominator?

        • Here is the other standard that we use along with the standard that Jad mentioned: 3MA.C.19: recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and 8. (e.g., 1/2 = 2/4, 4/6 = 2/3); explain why the fractions are equivalent by using a visual fraction model. In our analyzing the standards book it mentions that 3rd grade students are only required to know fractions with the above mentioned denominators. This activity that you presented seems more of a challenging extension to fractions and not one the should realistically be taught in 3rd grade. If we are suppose to teach to this level, then someone needs to tell all of the 3rd grade teachers in the county and students will have this expectation.

  2. Robert Kaplinsky

    Good question. I actually have tried several out with students. It was a very humbling experience in terms of dealing with lack of perseverance and content knowledge. I have some great student work samples I’ve already scanned. I just haven’t gotten around to blogging about it but will put it back on my radar.

  3. I know I’m late to the party, but….My thought is if they don’t feel comfortable with 9 in the denominator they can use it in the numerator. I’m excited to try this with my third graders. We’ll see how it goes. If it causes too much trouble, maybe I’ll alter the numbers to choose from…excluding two of the choices perhaps. That would still leave a lot of room for some fun!

  4. Rudolf Österreicher

    There are 13,944 different marked number lines (unfortunately too many to list here) and unsurprisingly, all of them have the number 0 marked. 801 of these only contain proper fractions:
    0/1 < 2/3 < 4/5 < 6/7 < 8/9
    0/1 < 2/3 < 6/8 < 7/9 < 4/5
    0/1 < 2/4 < 3/5 < 6/7 < 8/9
    0/1 < 2/4 < 3/5 < 6/8 < 7/9
    0/1 < 2/4 < 3/5 < 6/9 < 7/8
    0/1 < 2/5 < 3/4 < 6/7 < 8/9
    0/1 < 2/5 < 3/6 < 4/7 < 8/9
    0/1 < 2/5 < 3/7 < 4/6 < 8/9
    0/1 < 2/5 < 3/7 < 4/8 < 6/9
    0/1 < 2/5 < 3/7 < 4/9 < 6/8
    0/1 < 2/5 < 4/9 < 3/6 < 7/8
    0/1 < 2/5 < 6/9 < 3/4 < 7/8
    0/1 < 2/6 < 3/7 < 4/5 < 8/9
    0/1 < 2/6 < 3/7 < 4/8 < 5/9
    0/1 < 2/6 < 3/7 < 4/9 < 5/8
    0/1 < 2/6 < 3/8 < 4/9 < 5/7
    0/1 < 2/6 < 3/8 < 5/9 < 4/7
    0/1 < 2/6 < 3/8 < 7/9 < 4/5
    0/1 < 2/6 < 4/7 < 3/5 < 8/9
    0/1 < 2/6 < 4/8 < 3/5 < 7/9
    0/1 < 2/6 < 4/9 < 3/5 < 7/8
    0/1 < 2/6 < 5/7 < 3/4 < 8/9
    0/1 < 2/6 < 5/8 < 3/4 < 7/9
    0/1 < 2/6 < 5/9 < 3/4 < 7/8
    0/1 < 2/7 < 3/4 < 5/6 < 8/9
    0/1 < 2/7 < 3/5 < 4/6 < 8/9
    0/1 < 2/7 < 3/6 < 4/5 < 8/9
    0/1 < 2/7 < 3/8 < 4/9 < 5/6
    0/1 < 2/7 < 3/8 < 5/9 < 4/6
    0/1 < 2/7 < 3/8 < 6/9 < 4/5
    0/1 < 2/7 < 3/9 < 4/8 < 5/6
    0/1 < 2/7 < 3/9 < 5/8 < 4/6
    0/1 < 2/7 < 3/9 < 6/8 < 4/5
    0/1 < 2/7 < 4/8 < 3/5 < 6/9
    0/1 < 2/7 < 4/9 < 3/5 < 6/8
    0/1 < 2/7 < 4/9 < 3/6 < 5/8
    0/1 < 2/7 < 5/8 < 6/9 < 3/4
    0/1 < 2/8 < 3/4 < 7/9 < 5/6
    0/1 < 2/8 < 3/5 < 4/6 < 7/9
    0/1 < 2/8 < 3/6 < 5/9 < 4/7
    0/1 < 2/8 < 3/6 < 7/9 < 4/5
    0/1 < 2/8 < 3/7 < 4/9 < 5/6
    0/1 < 2/8 < 3/7 < 5/9 < 4/6
    0/1 < 2/8 < 3/7 < 6/9 < 4/5
    0/1 < 2/8 < 3/9 < 4/5 < 6/7
    0/1 < 2/8 < 3/9 < 4/6 < 5/7
    0/1 < 2/8 < 3/9 < 4/7 < 5/6
    0/1 < 2/8 < 4/7 < 3/5 < 6/9
    0/1 < 2/8 < 4/9 < 3/5 < 6/7
    0/1 < 2/8 < 4/9 < 3/6 < 5/7
    0/1 < 2/8 < 5/9 < 3/4 < 6/7
    0/1 < 2/8 < 6/9 < 5/7 < 3/4
    0/1 < 2/9 < 3/4 < 5/6 < 7/8
    0/1 < 2/9 < 3/5 < 4/6 < 7/8
    0/1 < 2/9 < 3/6 < 4/5 < 7/8
    0/1 < 2/9 < 3/6 < 4/7 < 5/8
    0/1 < 2/9 < 3/7 < 4/8 < 5/6
    0/1 < 2/9 < 3/7 < 5/8 < 4/6
    0/1 < 2/9 < 3/7 < 6/8 < 4/5
    0/1 < 2/9 < 3/8 < 4/5 < 6/7
    0/1 < 2/9 < 3/8 < 4/6 < 5/7
    0/1 < 2/9 < 3/8 < 4/7 < 5/6
    0/1 < 2/9 < 4/7 < 3/5 < 6/8
    0/1 < 2/9 < 4/8 < 3/5 < 6/7
    0/1 < 2/9 < 5/8 < 3/4 < 6/7
    0/1 < 3/7 < 2/4 < 5/6 < 8/9
    0/1 < 3/7 < 2/4 < 5/8 < 6/9
    0/1 < 3/7 < 2/4 < 5/9 < 6/8
    0/1 < 3/8 < 2/4 < 5/9 < 6/7
    0/1 < 3/8 < 2/4 < 6/9 < 5/7
    0/1 < 3/8 < 2/4 < 7/9 < 5/6
    0/1 < 3/8 < 2/5 < 4/6 < 7/9
    0/1 < 3/8 < 2/5 < 4/7 < 6/9
    0/1 < 3/8 < 2/5 < 4/9 < 6/7
    0/1 < 3/9 < 2/4 < 5/6 < 7/8
    0/1 < 3/9 < 2/4 < 5/7 < 6/8
    0/1 < 3/9 < 2/4 < 5/8 < 6/7
    0/1 < 3/9 < 2/5 < 4/6 < 7/8
    0/1 < 3/9 < 2/5 < 4/7 < 6/8
    0/1 < 3/9 < 2/5 < 4/8 < 6/7
    0/1 < 4/7 < 2/3 < 5/6 < 8/9
    0/1 < 4/8 < 2/3 < 7/9 < 5/6
    0/1 < 4/8 < 5/9 < 2/3 < 6/7
    0/1 < 4/9 < 2/3 < 5/6 < 7/8
    0/1 < 4/9 < 2/3 < 5/7 < 6/8
    0/1 < 4/9 < 5/8 < 2/3 < 6/7
    0/1 < 5/9 < 4/7 < 2/3 < 6/8
    0/2 < 1/3 < 4/5 < 6/7 < 8/9
    0/2 < 1/3 < 4/6 < 5/7 < 8/9
    0/2 < 1/3 < 4/7 < 5/6 < 8/9
    0/2 < 1/3 < 4/7 < 5/8 < 6/9
    0/2 < 1/3 < 4/8 < 5/9 < 6/7
    0/2 < 1/3 < 4/8 < 6/9 < 5/7
    0/2 < 1/3 < 4/8 < 7/9 < 5/6
    0/2 < 1/3 < 4/9 < 5/6 < 7/8
    0/2 < 1/3 < 4/9 < 5/7 < 6/8
    0/2 < 1/3 < 4/9 < 5/8 < 6/7
    0/2 < 1/3 < 5/8 < 4/6 < 7/9
    0/2 < 1/3 < 5/9 < 4/6 < 7/8
    0/2 < 1/3 < 5/9 < 4/7 < 6/8
    0/2 < 1/3 < 6/8 < 7/9 < 4/5
    0/2 < 1/3 < 6/9 < 4/5 < 7/8
    0/2 < 1/4 < 3/5 < 6/7 < 8/9
    0/2 < 1/4 < 3/5 < 6/8 < 7/9
    0/2 < 1/4 < 3/5 < 6/9 < 7/8
    0/2 < 1/4 < 3/6 < 5/7 < 8/9
    0/2 < 1/4 < 3/6 < 5/8 < 7/9
    0/2 < 1/4 < 3/6 < 5/9 < 7/8
    0/2 < 1/4 < 3/7 < 5/6 < 8/9
    0/2 < 1/4 < 3/7 < 5/8 < 6/9
    0/2 < 1/4 < 3/7 < 5/9 < 6/8
    0/2 < 1/4 < 3/8 < 5/9 < 6/7
    0/2 < 1/4 < 3/8 < 6/9 < 5/7
    0/2 < 1/4 < 3/8 < 7/9 < 5/6
    0/2 < 1/4 < 3/9 < 5/6 < 7/8
    0/2 < 1/4 < 3/9 < 5/7 < 6/8
    0/2 < 1/4 < 3/9 < 5/8 < 6/7
    0/2 < 1/5 < 3/4 < 6/7 < 8/9
    0/2 < 1/5 < 3/6 < 4/7 < 8/9
    0/2 < 1/5 < 3/7 < 4/6 < 8/9
    0/2 < 1/5 < 3/7 < 4/8 < 6/9
    0/2 < 1/5 < 3/7 < 4/9 < 6/8
    0/2 < 1/5 < 3/8 < 4/6 < 7/9
    0/2 < 1/5 < 3/8 < 4/7 < 6/9
    0/2 < 1/5 < 3/8 < 4/9 < 6/7
    0/2 < 1/5 < 3/9 < 4/6 < 7/8
    0/2 < 1/5 < 3/9 < 4/7 < 6/8
    0/2 < 1/5 < 3/9 < 4/8 < 6/7
    0/2 < 1/5 < 4/9 < 3/6 < 7/8
    0/2 < 1/5 < 6/9 < 3/4 < 7/8
    0/2 < 1/6 < 3/7 < 4/5 < 8/9
    0/2 < 1/6 < 3/7 < 4/8 < 5/9
    0/2 < 1/6 < 3/7 < 4/9 < 5/8
    0/2 < 1/6 < 3/8 < 4/9 < 5/7
    0/2 < 1/6 < 3/8 < 5/9 < 4/7
    0/2 < 1/6 < 3/8 < 7/9 < 4/5
    0/2 < 1/6 < 3/9 < 4/5 < 7/8
    0/2 < 1/6 < 3/9 < 4/7 < 5/8
    0/2 < 1/6 < 3/9 < 4/8 < 5/7
    0/2 < 1/6 < 4/7 < 3/5 < 8/9
    0/2 < 1/6 < 4/8 < 3/5 < 7/9
    0/2 < 1/6 < 4/9 < 3/5 < 7/8
    0/2 < 1/6 < 5/7 < 3/4 < 8/9
    0/2 < 1/6 < 5/8 < 3/4 < 7/9
    0/2 < 1/6 < 5/9 < 3/4 < 7/8
    0/2 < 1/7 < 3/4 < 5/6 < 8/9
    0/2 < 1/7 < 3/5 < 4/6 < 8/9
    0/2 < 1/7 < 3/6 < 4/5 < 8/9
    0/2 < 1/7 < 3/8 < 4/9 < 5/6
    0/2 < 1/7 < 3/8 < 5/9 < 4/6
    0/2 < 1/7 < 3/8 < 6/9 < 4/5
    0/2 < 1/7 < 3/9 < 4/8 < 5/6
    0/2 < 1/7 < 3/9 < 5/8 < 4/6
    0/2 < 1/7 < 3/9 < 6/8 < 4/5
    0/2 < 1/7 < 4/8 < 3/5 < 6/9
    0/2 < 1/7 < 4/9 < 3/5 < 6/8
    0/2 < 1/7 < 4/9 < 3/6 < 5/8
    0/2 < 1/7 < 5/8 < 6/9 < 3/4
    0/2 < 1/8 < 3/4 < 7/9 < 5/6
    0/2 < 1/8 < 3/5 < 4/6 < 7/9
    0/2 < 1/8 < 3/6 < 5/9 < 4/7
    0/2 < 1/8 < 3/6 < 7/9 < 4/5
    0/2 < 1/8 < 3/7 < 4/9 < 5/6
    0/2 < 1/8 < 3/7 < 5/9 < 4/6
    0/2 < 1/8 < 3/7 < 6/9 < 4/5
    0/2 < 1/8 < 3/9 < 4/5 < 6/7
    0/2 < 1/8 < 3/9 < 4/6 < 5/7
    0/2 < 1/8 < 3/9 < 4/7 < 5/6
    0/2 < 1/8 < 4/7 < 3/5 < 6/9
    0/2 < 1/8 < 4/9 < 3/5 < 6/7
    0/2 < 1/8 < 4/9 < 3/6 < 5/7
    0/2 < 1/8 < 5/9 < 3/4 < 6/7
    0/2 < 1/8 < 6/9 < 5/7 < 3/4
    0/2 < 1/9 < 3/4 < 5/6 < 7/8
    0/2 < 1/9 < 3/5 < 4/6 < 7/8
    0/2 < 1/9 < 3/6 < 4/5 < 7/8
    0/2 < 1/9 < 3/6 < 4/7 < 5/8
    0/2 < 1/9 < 3/7 < 4/8 < 5/6
    0/2 < 1/9 < 3/7 < 5/8 < 4/6
    0/2 < 1/9 < 3/7 < 6/8 < 4/5
    0/2 < 1/9 < 3/8 < 4/5 < 6/7
    0/2 < 1/9 < 3/8 < 4/6 < 5/7
    0/2 < 1/9 < 3/8 < 4/7 < 5/6
    0/2 < 1/9 < 4/7 < 3/5 < 6/8
    0/2 < 1/9 < 4/8 < 3/5 < 6/7
    0/2 < 1/9 < 5/8 < 3/4 < 6/7
    0/3 < 1/2 < 4/5 < 6/7 < 8/9
    0/3 < 1/2 < 4/6 < 5/7 < 8/9
    0/3 < 1/2 < 4/7 < 5/6 < 8/9
    0/3 < 1/2 < 4/7 < 5/8 < 6/9
    0/3 < 1/2 < 5/8 < 4/6 < 7/9
    0/3 < 1/2 < 5/9 < 4/6 < 7/8
    0/3 < 1/2 < 5/9 < 4/7 < 6/8
    0/3 < 1/2 < 6/8 < 7/9 < 4/5
    0/3 < 1/2 < 6/9 < 4/5 < 7/8
    0/3 < 1/4 < 2/5 < 6/7 < 8/9
    0/3 < 1/4 < 2/5 < 6/8 < 7/9
    0/3 < 1/4 < 2/5 < 6/9 < 7/8
    0/3 < 1/4 < 2/6 < 5/7 < 8/9
    0/3 < 1/4 < 2/6 < 5/8 < 7/9
    0/3 < 1/4 < 2/6 < 5/9 < 7/8
    0/3 < 1/4 < 2/7 < 5/6 < 8/9
    0/3 < 1/4 < 2/7 < 5/8 < 6/9
    0/3 < 1/4 < 2/7 < 5/9 < 6/8
    0/3 < 1/5 < 2/4 < 6/7 < 8/9
    0/3 < 1/5 < 2/4 < 6/8 < 7/9
    0/3 < 1/5 < 2/4 < 6/9 < 7/8
    0/3 < 1/5 < 2/6 < 4/7 < 8/9
    0/3 < 1/5 < 2/6 < 4/8 < 7/9
    0/3 < 1/5 < 2/6 < 4/9 < 7/8
    0/3 < 1/5 < 2/7 < 4/6 < 8/9
    0/3 < 1/5 < 2/7 < 4/8 < 6/9
    0/3 < 1/5 < 2/7 < 4/9 < 6/8
    0/3 < 1/5 < 2/8 < 4/6 < 7/9
    0/3 < 1/5 < 2/8 < 4/7 < 6/9
    0/3 < 1/5 < 2/8 < 4/9 < 6/7
    0/3 < 1/5 < 2/9 < 4/6 < 7/8
    0/3 < 1/5 < 2/9 < 4/7 < 6/8
    0/3 < 1/5 < 2/9 < 4/8 < 6/7
    0/3 < 1/6 < 2/4 < 5/7 < 8/9
    0/3 < 1/6 < 2/4 < 5/8 < 7/9
    0/3 < 1/6 < 2/4 < 5/9 < 7/8
    0/3 < 1/6 < 2/5 < 4/7 < 8/9
    0/3 < 1/6 < 2/5 < 4/8 < 7/9
    0/3 < 1/6 < 2/5 < 4/9 < 7/8
    0/3 < 1/6 < 2/7 < 4/5 < 8/9
    0/3 < 1/6 < 2/7 < 4/8 < 5/9
    0/3 < 1/6 < 2/7 < 4/9 < 5/8
    0/3 < 1/6 < 2/8 < 4/9 < 5/7
    0/3 < 1/6 < 2/8 < 5/9 < 4/7
    0/3 < 1/6 < 2/8 < 7/9 < 4/5
    0/3 < 1/6 < 2/9 < 4/5 < 7/8
    0/3 < 1/6 < 2/9 < 4/7 < 5/8
    0/3 < 1/6 < 2/9 < 4/8 < 5/7
    0/3 < 1/7 < 2/4 < 5/6 < 8/9
    0/3 < 1/7 < 2/4 < 5/8 < 6/9
    0/3 < 1/7 < 2/4 < 5/9 < 6/8
    0/3 < 1/7 < 2/5 < 4/6 < 8/9
    0/3 < 1/7 < 2/5 < 4/8 < 6/9
    0/3 < 1/7 < 2/5 < 4/9 < 6/8
    0/3 < 1/7 < 2/6 < 4/5 < 8/9
    0/3 < 1/7 < 2/6 < 4/8 < 5/9
    0/3 < 1/7 < 2/6 < 4/9 < 5/8
    0/3 < 1/7 < 2/8 < 4/9 < 5/6
    0/3 < 1/7 < 2/8 < 5/9 < 4/6
    0/3 < 1/7 < 2/8 < 6/9 < 4/5
    0/3 < 1/7 < 2/9 < 4/8 < 5/6
    0/3 < 1/7 < 2/9 < 5/8 < 4/6
    0/3 < 1/7 < 2/9 < 6/8 < 4/5
    0/3 < 1/8 < 2/4 < 5/9 < 6/7
    0/3 < 1/8 < 2/4 < 6/9 < 5/7
    0/3 < 1/8 < 2/4 < 7/9 < 5/6
    0/3 < 1/8 < 2/5 < 4/6 < 7/9
    0/3 < 1/8 < 2/5 < 4/7 < 6/9
    0/3 < 1/8 < 2/5 < 4/9 < 6/7
    0/3 < 1/8 < 2/6 < 4/9 < 5/7
    0/3 < 1/8 < 2/6 < 5/9 < 4/7
    0/3 < 1/8 < 2/6 < 7/9 < 4/5
    0/3 < 1/8 < 2/7 < 4/9 < 5/6
    0/3 < 1/8 < 2/7 < 5/9 < 4/6
    0/3 < 1/8 < 2/7 < 6/9 < 4/5
    0/3 < 1/8 < 2/9 < 4/5 < 6/7
    0/3 < 1/8 < 2/9 < 4/6 < 5/7
    0/3 < 1/8 < 2/9 < 4/7 < 5/6
    0/3 < 1/9 < 2/4 < 5/6 < 7/8
    0/3 < 1/9 < 2/4 < 5/7 < 6/8
    0/3 < 1/9 < 2/4 < 5/8 < 6/7
    0/3 < 1/9 < 2/5 < 4/6 < 7/8
    0/3 < 1/9 < 2/5 < 4/7 < 6/8
    0/3 < 1/9 < 2/5 < 4/8 < 6/7
    0/3 < 1/9 < 2/6 < 4/5 < 7/8
    0/3 < 1/9 < 2/6 < 4/7 < 5/8
    0/3 < 1/9 < 2/6 < 4/8 < 5/7
    0/3 < 1/9 < 2/7 < 4/8 < 5/6
    0/3 < 1/9 < 2/7 < 5/8 < 4/6
    0/3 < 1/9 < 2/7 < 6/8 < 4/5
    0/3 < 1/9 < 2/8 < 4/5 < 6/7
    0/3 < 1/9 < 2/8 < 4/6 < 5/7
    0/3 < 1/9 < 2/8 < 4/7 < 5/6
    0/3 < 2/9 < 1/4 < 5/6 < 7/8
    0/3 < 2/9 < 1/4 < 5/7 < 6/8
    0/3 < 2/9 < 1/4 < 5/8 < 6/7
    0/3 < 4/9 < 1/2 < 5/6 < 7/8
    0/3 < 4/9 < 1/2 < 5/7 < 6/8
    0/3 < 4/9 < 1/2 < 5/8 < 6/7
    0/4 < 1/2 < 3/5 < 6/7 < 8/9
    0/4 < 1/2 < 3/5 < 6/8 < 7/9
    0/4 < 1/2 < 3/5 < 6/9 < 7/8
    0/4 < 1/3 < 2/5 < 6/7 < 8/9
    0/4 < 1/3 < 2/5 < 6/8 < 7/9
    0/4 < 1/3 < 2/5 < 6/9 < 7/8
    0/4 < 1/5 < 2/3 < 6/7 < 8/9
    0/4 < 1/5 < 2/3 < 6/8 < 7/9
    0/4 < 1/5 < 2/6 < 3/7 < 8/9
    0/4 < 1/5 < 2/6 < 3/8 < 7/9
    0/4 < 1/5 < 2/7 < 3/6 < 8/9
    0/4 < 1/5 < 2/7 < 3/8 < 6/9
    0/4 < 1/5 < 2/7 < 3/9 < 6/8
    0/4 < 1/5 < 2/8 < 3/6 < 7/9
    0/4 < 1/5 < 2/8 < 3/7 < 6/9
    0/4 < 1/5 < 2/8 < 3/9 < 6/7
    0/4 < 1/5 < 2/9 < 3/6 < 7/8
    0/4 < 1/5 < 2/9 < 3/7 < 6/8
    0/4 < 1/5 < 2/9 < 3/8 < 6/7
    0/4 < 1/6 < 2/3 < 5/7 < 8/9
    0/4 < 1/6 < 2/5 < 3/7 < 8/9
    0/4 < 1/6 < 2/7 < 3/5 < 8/9
    0/4 < 1/6 < 2/7 < 3/8 < 5/9
    0/4 < 1/6 < 2/7 < 3/9 < 5/8
    0/4 < 1/6 < 2/8 < 3/5 < 7/9
    0/4 < 1/6 < 2/8 < 3/7 < 5/9
    0/4 < 1/6 < 2/8 < 3/9 < 5/7
    0/4 < 1/6 < 2/9 < 3/5 < 7/8
    0/4 < 1/6 < 2/9 < 3/7 < 5/8
    0/4 < 1/6 < 2/9 < 3/8 < 5/7
    0/4 < 1/6 < 3/8 < 2/5 < 7/9
    0/4 < 1/6 < 3/9 < 2/5 < 7/8
    0/4 < 1/6 < 5/8 < 2/3 < 7/9
    0/4 < 1/6 < 5/9 < 2/3 < 7/8
    0/4 < 1/7 < 2/3 < 5/6 < 8/9
    0/4 < 1/7 < 2/5 < 3/6 < 8/9
    0/4 < 1/7 < 2/6 < 3/5 < 8/9
    0/4 < 1/7 < 2/6 < 3/8 < 5/9
    0/4 < 1/7 < 2/8 < 3/5 < 6/9
    0/4 < 1/7 < 2/8 < 3/6 < 5/9
    0/4 < 1/7 < 2/8 < 3/9 < 5/6
    0/4 < 1/7 < 2/9 < 3/5 < 6/8
    0/4 < 1/7 < 2/9 < 3/6 < 5/8
    0/4 < 1/7 < 2/9 < 3/8 < 5/6
    0/4 < 1/7 < 3/8 < 2/5 < 6/9
    0/4 < 1/7 < 3/9 < 2/5 < 6/8
    0/4 < 1/7 < 5/9 < 2/3 < 6/8
    0/4 < 1/8 < 2/3 < 7/9 < 5/6
    0/4 < 1/8 < 2/5 < 3/6 < 7/9
    0/4 < 1/8 < 2/5 < 3/7 < 6/9
    0/4 < 1/8 < 2/6 < 3/5 < 7/9
    0/4 < 1/8 < 2/6 < 3/7 < 5/9
    0/4 < 1/8 < 2/7 < 3/5 < 6/9
    0/4 < 1/8 < 2/7 < 3/6 < 5/9
    0/4 < 1/8 < 2/7 < 3/9 < 5/6
    0/4 < 1/8 < 2/9 < 3/5 < 6/7
    0/4 < 1/8 < 2/9 < 3/6 < 5/7
    0/4 < 1/8 < 2/9 < 3/7 < 5/6
    0/4 < 1/8 < 3/9 < 2/5 < 6/7
    0/4 < 1/8 < 5/9 < 2/3 < 6/7
    0/4 < 1/9 < 2/3 < 5/6 < 7/8
    0/4 < 1/9 < 2/3 < 5/7 < 6/8
    0/4 < 1/9 < 2/5 < 3/6 < 7/8
    0/4 < 1/9 < 2/5 < 3/7 < 6/8
    0/4 < 1/9 < 2/6 < 3/5 < 7/8
    0/4 < 1/9 < 2/6 < 3/7 < 5/8
    0/4 < 1/9 < 2/6 < 3/8 < 5/7
    0/4 < 1/9 < 2/7 < 3/5 < 6/8
    0/4 < 1/9 < 2/7 < 3/6 < 5/8
    0/4 < 1/9 < 2/7 < 3/8 < 5/6
    0/4 < 1/9 < 2/8 < 3/5 < 6/7
    0/4 < 1/9 < 2/8 < 3/6 < 5/7
    0/4 < 1/9 < 2/8 < 3/7 < 5/6
    0/4 < 1/9 < 3/8 < 2/5 < 6/7
    0/4 < 1/9 < 5/8 < 2/3 < 6/7
    0/4 < 2/7 < 1/3 < 5/6 < 8/9
    0/4 < 2/7 < 1/3 < 5/8 < 6/9
    0/4 < 2/7 < 1/3 < 5/9 < 6/8
    0/4 < 2/8 < 1/3 < 5/9 < 6/7
    0/4 < 2/8 < 1/3 < 6/9 < 5/7
    0/4 < 2/8 < 1/3 < 7/9 < 5/6
    0/4 < 2/9 < 1/3 < 5/6 < 7/8
    0/4 < 2/9 < 1/3 < 5/7 < 6/8
    0/4 < 2/9 < 1/3 < 5/8 < 6/7
    0/4 < 3/7 < 1/2 < 5/6 < 8/9
    0/4 < 3/7 < 1/2 < 5/8 < 6/9
    0/4 < 3/7 < 1/2 < 5/9 < 6/8
    0/4 < 3/8 < 1/2 < 5/9 < 6/7
    0/4 < 3/8 < 1/2 < 6/9 < 5/7
    0/4 < 3/8 < 1/2 < 7/9 < 5/6
    0/4 < 3/9 < 1/2 < 5/6 < 7/8
    0/4 < 3/9 < 1/2 < 5/7 < 6/8
    0/4 < 3/9 < 1/2 < 5/8 < 6/7
    0/5 < 1/2 < 3/4 < 6/7 < 8/9
    0/5 < 1/2 < 6/9 < 3/4 < 7/8
    0/5 < 1/3 < 2/4 < 6/7 < 8/9
    0/5 < 1/3 < 2/4 < 6/8 < 7/9
    0/5 < 1/3 < 2/4 < 6/9 < 7/8
    0/5 < 1/4 < 2/3 < 6/7 < 8/9
    0/5 < 1/4 < 2/3 < 6/8 < 7/9
    0/5 < 1/4 < 2/6 < 3/7 < 8/9
    0/5 < 1/4 < 2/6 < 3/8 < 7/9
    0/5 < 1/4 < 2/7 < 3/6 < 8/9
    0/5 < 1/4 < 2/7 < 3/8 < 6/9
    0/5 < 1/4 < 2/7 < 3/9 < 6/8
    0/5 < 1/6 < 2/7 < 3/4 < 8/9
    0/5 < 1/6 < 2/7 < 3/8 < 4/9
    0/5 < 1/6 < 2/7 < 3/9 < 4/8
    0/5 < 1/6 < 2/8 < 3/4 < 7/9
    0/5 < 1/6 < 2/8 < 3/7 < 4/9
    0/5 < 1/6 < 2/8 < 3/9 < 4/7
    0/5 < 1/6 < 2/9 < 3/4 < 7/8
    0/5 < 1/6 < 2/9 < 3/7 < 4/8
    0/5 < 1/6 < 2/9 < 3/8 < 4/7
    0/5 < 1/6 < 3/7 < 2/4 < 8/9
    0/5 < 1/6 < 3/8 < 2/4 < 7/9
    0/5 < 1/6 < 3/9 < 2/4 < 7/8
    0/5 < 1/6 < 4/7 < 2/3 < 8/9
    0/5 < 1/6 < 4/8 < 2/3 < 7/9
    0/5 < 1/6 < 4/9 < 2/3 < 7/8
    0/5 < 1/7 < 2/6 < 3/4 < 8/9
    0/5 < 1/7 < 2/6 < 3/8 < 4/9
    0/5 < 1/7 < 2/8 < 3/9 < 4/6
    0/5 < 1/7 < 2/8 < 4/9 < 3/6
    0/5 < 1/7 < 2/8 < 6/9 < 3/4
    0/5 < 1/7 < 2/9 < 3/8 < 4/6
    0/5 < 1/7 < 3/8 < 2/4 < 6/9
    0/5 < 1/7 < 3/9 < 2/4 < 6/8
    0/5 < 1/7 < 4/9 < 2/3 < 6/8
    0/5 < 1/8 < 2/6 < 3/4 < 7/9
    0/5 < 1/8 < 2/6 < 3/7 < 4/9
    0/5 < 1/8 < 2/7 < 3/9 < 4/6
    0/5 < 1/8 < 2/7 < 4/9 < 3/6
    0/5 < 1/8 < 2/7 < 6/9 < 3/4
    0/5 < 1/8 < 2/9 < 3/4 < 6/7
    0/5 < 1/8 < 2/9 < 3/6 < 4/7
    0/5 < 1/8 < 2/9 < 3/7 < 4/6
    0/5 < 1/8 < 3/7 < 2/4 < 6/9
    0/5 < 1/8 < 3/9 < 2/4 < 6/7
    0/5 < 1/8 < 4/9 < 2/3 < 6/7
    0/5 < 1/9 < 2/6 < 3/4 < 7/8
    0/5 < 1/9 < 2/6 < 3/7 < 4/8
    0/5 < 1/9 < 2/6 < 3/8 < 4/7
    0/5 < 1/9 < 2/7 < 3/8 < 4/6
    0/5 < 1/9 < 2/8 < 3/4 < 6/7
    0/5 < 1/9 < 2/8 < 3/6 < 4/7
    0/5 < 1/9 < 2/8 < 3/7 < 4/6
    0/5 < 1/9 < 3/7 < 2/4 < 6/8
    0/5 < 1/9 < 3/8 < 2/4 < 6/7
    0/5 < 1/9 < 4/7 < 2/3 < 6/8
    0/5 < 1/9 < 4/8 < 2/3 < 6/7
    0/5 < 2/7 < 1/3 < 4/6 < 8/9
    0/5 < 2/7 < 1/3 < 4/8 < 6/9
    0/5 < 2/7 < 1/3 < 4/9 < 6/8
    0/5 < 2/8 < 1/3 < 4/6 < 7/9
    0/5 < 2/8 < 1/3 < 4/7 < 6/9
    0/5 < 2/8 < 1/3 < 4/9 < 6/7
    0/5 < 2/9 < 1/3 < 4/6 < 7/8
    0/5 < 2/9 < 1/3 < 4/7 < 6/8
    0/5 < 2/9 < 1/3 < 4/8 < 6/7
    0/5 < 2/9 < 1/4 < 3/6 < 7/8
    0/5 < 2/9 < 1/4 < 3/7 < 6/8
    0/5 < 2/9 < 1/4 < 3/8 < 6/7
    0/5 < 3/7 < 1/2 < 4/6 < 8/9
    0/5 < 3/7 < 4/9 < 1/2 < 6/8
    0/5 < 3/8 < 1/2 < 4/6 < 7/9
    0/5 < 3/8 < 1/2 < 4/7 < 6/9
    0/5 < 3/8 < 4/9 < 1/2 < 6/7
    0/5 < 3/9 < 1/2 < 4/6 < 7/8
    0/5 < 3/9 < 1/2 < 4/7 < 6/8
    0/6 < 1/2 < 4/7 < 3/5 < 8/9
    0/6 < 1/2 < 5/7 < 3/4 < 8/9
    0/6 < 1/2 < 5/8 < 3/4 < 7/9
    0/6 < 1/2 < 5/9 < 3/4 < 7/8
    0/6 < 1/3 < 2/4 < 5/7 < 8/9
    0/6 < 1/3 < 2/4 < 5/8 < 7/9
    0/6 < 1/3 < 2/4 < 5/9 < 7/8
    0/6 < 1/3 < 2/5 < 4/7 < 8/9
    0/6 < 1/3 < 2/5 < 4/8 < 7/9
    0/6 < 1/3 < 2/5 < 4/9 < 7/8
    0/6 < 1/4 < 2/3 < 5/7 < 8/9
    0/6 < 1/4 < 2/5 < 3/7 < 8/9
    0/6 < 1/4 < 2/7 < 3/5 < 8/9
    0/6 < 1/4 < 2/7 < 3/8 < 5/9
    0/6 < 1/4 < 2/7 < 3/9 < 5/8
    0/6 < 1/4 < 3/8 < 2/5 < 7/9
    0/6 < 1/4 < 3/9 < 2/5 < 7/8
    0/6 < 1/4 < 5/8 < 2/3 < 7/9
    0/6 < 1/4 < 5/9 < 2/3 < 7/8
    0/6 < 1/5 < 2/7 < 3/4 < 8/9
    0/6 < 1/5 < 2/7 < 3/8 < 4/9
    0/6 < 1/5 < 2/7 < 3/9 < 4/8
    0/6 < 1/5 < 2/8 < 3/4 < 7/9
    0/6 < 1/5 < 2/8 < 3/7 < 4/9
    0/6 < 1/5 < 2/8 < 3/9 < 4/7
    0/6 < 1/5 < 2/9 < 3/4 < 7/8
    0/6 < 1/5 < 2/9 < 3/7 < 4/8
    0/6 < 1/5 < 2/9 < 3/8 < 4/7
    0/6 < 1/5 < 3/7 < 2/4 < 8/9
    0/6 < 1/5 < 3/8 < 2/4 < 7/9
    0/6 < 1/5 < 3/9 < 2/4 < 7/8
    0/6 < 1/5 < 4/7 < 2/3 < 8/9
    0/6 < 1/5 < 4/8 < 2/3 < 7/9
    0/6 < 1/5 < 4/9 < 2/3 < 7/8
    0/6 < 1/7 < 2/3 < 4/5 < 8/9
    0/6 < 1/7 < 2/4 < 3/5 < 8/9
    0/6 < 1/7 < 2/5 < 3/4 < 8/9
    0/6 < 1/7 < 2/8 < 3/9 < 4/5
    0/6 < 1/7 < 2/8 < 4/9 < 3/5
    0/6 < 1/7 < 2/8 < 5/9 < 3/4
    0/6 < 1/7 < 2/9 < 3/8 < 4/5
    0/6 < 1/7 < 2/9 < 4/8 < 3/5
    0/6 < 1/7 < 2/9 < 5/8 < 3/4
    0/6 < 1/7 < 3/8 < 2/4 < 5/9
    0/6 < 1/7 < 3/8 < 2/5 < 4/9
    0/6 < 1/7 < 3/9 < 2/4 < 5/8
    0/6 < 1/7 < 3/9 < 2/5 < 4/8
    0/6 < 1/7 < 4/8 < 5/9 < 2/3
    0/6 < 1/7 < 4/9 < 5/8 < 2/3
    0/6 < 1/8 < 2/3 < 7/9 < 4/5
    0/6 < 1/8 < 2/4 < 3/5 < 7/9
    0/6 < 1/8 < 2/5 < 3/4 < 7/9
    0/6 < 1/8 < 2/5 < 3/7 < 4/9
    0/6 < 1/8 < 2/7 < 3/9 < 4/5
    0/6 < 1/8 < 2/7 < 4/9 < 3/5
    0/6 < 1/8 < 2/7 < 5/9 < 3/4
    0/6 < 1/8 < 2/9 < 3/7 < 4/5
    0/6 < 1/8 < 2/9 < 4/7 < 3/5
    0/6 < 1/8 < 2/9 < 5/7 < 3/4
    0/6 < 1/8 < 3/7 < 2/4 < 5/9
    0/6 < 1/8 < 3/9 < 2/4 < 5/7
    0/6 < 1/8 < 3/9 < 2/5 < 4/7
    0/6 < 1/8 < 4/9 < 2/3 < 5/7
    0/6 < 1/8 < 5/9 < 4/7 < 2/3
    0/6 < 1/9 < 2/3 < 4/5 < 7/8
    0/6 < 1/9 < 2/4 < 3/5 < 7/8
    0/6 < 1/9 < 2/5 < 3/4 < 7/8
    0/6 < 1/9 < 2/5 < 3/7 < 4/8
    0/6 < 1/9 < 2/7 < 3/8 < 4/5
    0/6 < 1/9 < 2/7 < 4/8 < 3/5
    0/6 < 1/9 < 2/7 < 5/8 < 3/4
    0/6 < 1/9 < 2/8 < 3/7 < 4/5
    0/6 < 1/9 < 2/8 < 4/7 < 3/5
    0/6 < 1/9 < 2/8 < 5/7 < 3/4
    0/6 < 1/9 < 3/7 < 2/4 < 5/8
    0/6 < 1/9 < 3/8 < 2/4 < 5/7
    0/6 < 1/9 < 3/8 < 2/5 < 4/7
    0/6 < 1/9 < 4/7 < 5/8 < 2/3
    0/6 < 1/9 < 4/8 < 2/3 < 5/7
    0/6 < 2/7 < 1/3 < 4/5 < 8/9
    0/6 < 2/7 < 1/3 < 4/8 < 5/9
    0/6 < 2/7 < 1/3 < 4/9 < 5/8
    0/6 < 2/8 < 1/3 < 4/9 < 5/7
    0/6 < 2/8 < 1/3 < 5/9 < 4/7
    0/6 < 2/8 < 1/3 < 7/9 < 4/5
    0/6 < 2/9 < 1/3 < 4/5 < 7/8
    0/6 < 2/9 < 1/3 < 4/7 < 5/8
    0/6 < 2/9 < 1/3 < 4/8 < 5/7
    0/6 < 2/9 < 1/4 < 3/5 < 7/8
    0/6 < 2/9 < 1/4 < 3/7 < 5/8
    0/6 < 2/9 < 1/4 < 3/8 < 5/7
    0/6 < 3/7 < 1/2 < 4/5 < 8/9
    0/6 < 3/7 < 4/9 < 1/2 < 5/8
    0/6 < 3/8 < 1/2 < 5/9 < 4/7
    0/6 < 3/8 < 1/2 < 7/9 < 4/5
    0/6 < 3/8 < 4/9 < 1/2 < 5/7
    0/6 < 3/9 < 1/2 < 4/5 < 7/8
    0/6 < 3/9 < 1/2 < 4/7 < 5/8
    0/6 < 4/9 < 1/2 < 3/5 < 7/8
    0/7 < 1/2 < 3/4 < 5/6 < 8/9
    0/7 < 1/2 < 3/5 < 4/6 < 8/9
    0/7 < 1/2 < 5/8 < 6/9 < 3/4
    0/7 < 1/3 < 2/4 < 5/6 < 8/9
    0/7 < 1/3 < 2/4 < 5/8 < 6/9
    0/7 < 1/3 < 2/4 < 5/9 < 6/8
    0/7 < 1/3 < 2/5 < 4/6 < 8/9
    0/7 < 1/3 < 2/5 < 4/8 < 6/9
    0/7 < 1/3 < 2/5 < 4/9 < 6/8
    0/7 < 1/4 < 2/3 < 5/6 < 8/9
    0/7 < 1/4 < 2/5 < 3/6 < 8/9
    0/7 < 1/4 < 2/6 < 3/5 < 8/9
    0/7 < 1/4 < 2/6 < 3/8 < 5/9
    0/7 < 1/4 < 3/8 < 2/5 < 6/9
    0/7 < 1/4 < 3/9 < 2/5 < 6/8
    0/7 < 1/4 < 5/9 < 2/3 < 6/8
    0/7 < 1/5 < 2/6 < 3/4 < 8/9
    0/7 < 1/5 < 2/6 < 3/8 < 4/9
    0/7 < 1/5 < 2/8 < 3/9 < 4/6
    0/7 < 1/5 < 2/8 < 4/9 < 3/6
    0/7 < 1/5 < 2/8 < 6/9 < 3/4
    0/7 < 1/5 < 2/9 < 3/8 < 4/6
    0/7 < 1/5 < 3/8 < 2/4 < 6/9
    0/7 < 1/5 < 3/9 < 2/4 < 6/8
    0/7 < 1/5 < 4/9 < 2/3 < 6/8
    0/7 < 1/6 < 2/3 < 4/5 < 8/9
    0/7 < 1/6 < 2/4 < 3/5 < 8/9
    0/7 < 1/6 < 2/5 < 3/4 < 8/9
    0/7 < 1/6 < 2/8 < 3/9 < 4/5
    0/7 < 1/6 < 2/8 < 4/9 < 3/5
    0/7 < 1/6 < 2/8 < 5/9 < 3/4
    0/7 < 1/6 < 2/9 < 3/8 < 4/5
    0/7 < 1/6 < 2/9 < 4/8 < 3/5
    0/7 < 1/6 < 2/9 < 5/8 < 3/4
    0/7 < 1/6 < 3/8 < 2/4 < 5/9
    0/7 < 1/6 < 3/8 < 2/5 < 4/9
    0/7 < 1/6 < 3/9 < 2/4 < 5/8
    0/7 < 1/6 < 3/9 < 2/5 < 4/8
    0/7 < 1/6 < 4/8 < 5/9 < 2/3
    0/7 < 1/6 < 4/9 < 5/8 < 2/3
    0/7 < 1/8 < 2/4 < 3/5 < 6/9
    0/7 < 1/8 < 2/5 < 4/9 < 3/6
    0/7 < 1/8 < 2/5 < 6/9 < 3/4
    0/7 < 1/8 < 2/6 < 4/9 < 3/5
    0/7 < 1/8 < 2/6 < 5/9 < 3/4
    0/7 < 1/8 < 2/9 < 3/4 < 5/6
    0/7 < 1/8 < 2/9 < 3/5 < 4/6
    0/7 < 1/8 < 2/9 < 3/6 < 4/5
    0/7 < 1/8 < 3/9 < 2/4 < 5/6
    0/7 < 1/8 < 3/9 < 2/5 < 4/6
    0/7 < 1/8 < 4/9 < 2/3 < 5/6
    0/7 < 1/9 < 2/3 < 6/8 < 4/5
    0/7 < 1/9 < 2/4 < 3/5 < 6/8
    0/7 < 1/9 < 2/6 < 3/8 < 4/5
    0/7 < 1/9 < 2/6 < 4/8 < 3/5
    0/7 < 1/9 < 2/6 < 5/8 < 3/4
    0/7 < 1/9 < 2/8 < 3/4 < 5/6
    0/7 < 1/9 < 2/8 < 3/5 < 4/6
    0/7 < 1/9 < 2/8 < 3/6 < 4/5
    0/7 < 1/9 < 3/8 < 2/4 < 5/6
    0/7 < 1/9 < 3/8 < 2/5 < 4/6
    0/7 < 1/9 < 4/8 < 2/3 < 5/6
    0/7 < 2/8 < 1/3 < 4/9 < 5/6
    0/7 < 2/8 < 1/3 < 5/9 < 4/6
    0/7 < 2/8 < 1/3 < 6/9 < 4/5
    0/7 < 2/9 < 1/3 < 4/8 < 5/6
    0/7 < 2/9 < 1/3 < 5/8 < 4/6
    0/7 < 2/9 < 1/3 < 6/8 < 4/5
    0/7 < 2/9 < 1/4 < 3/5 < 6/8
    0/7 < 2/9 < 1/4 < 3/6 < 5/8
    0/7 < 2/9 < 1/4 < 3/8 < 5/6
    0/7 < 3/8 < 1/2 < 5/9 < 4/6
    0/7 < 3/8 < 1/2 < 6/9 < 4/5
    0/7 < 3/8 < 4/9 < 1/2 < 5/6
    0/7 < 3/9 < 1/2 < 5/8 < 4/6
    0/7 < 3/9 < 1/2 < 6/8 < 4/5
    0/7 < 4/9 < 1/2 < 3/5 < 6/8
    0/8 < 1/2 < 3/4 < 7/9 < 5/6
    0/8 < 1/2 < 3/5 < 4/6 < 7/9
    0/8 < 1/2 < 4/7 < 3/5 < 6/9
    0/8 < 1/2 < 5/9 < 3/4 < 6/7
    0/8 < 1/2 < 6/9 < 5/7 < 3/4
    0/8 < 1/3 < 2/4 < 5/9 < 6/7
    0/8 < 1/3 < 2/4 < 6/9 < 5/7
    0/8 < 1/3 < 2/4 < 7/9 < 5/6
    0/8 < 1/3 < 2/5 < 4/6 < 7/9
    0/8 < 1/3 < 2/5 < 4/7 < 6/9
    0/8 < 1/3 < 2/5 < 4/9 < 6/7
    0/8 < 1/4 < 2/3 < 7/9 < 5/6
    0/8 < 1/4 < 2/5 < 3/6 < 7/9
    0/8 < 1/4 < 2/5 < 3/7 < 6/9
    0/8 < 1/4 < 2/6 < 3/5 < 7/9
    0/8 < 1/4 < 2/6 < 3/7 < 5/9
    0/8 < 1/4 < 2/7 < 3/5 < 6/9
    0/8 < 1/4 < 2/7 < 3/6 < 5/9
    0/8 < 1/4 < 2/7 < 3/9 < 5/6
    0/8 < 1/4 < 3/9 < 2/5 < 6/7
    0/8 < 1/4 < 5/9 < 2/3 < 6/7
    0/8 < 1/5 < 2/6 < 3/4 < 7/9
    0/8 < 1/5 < 2/6 < 3/7 < 4/9
    0/8 < 1/5 < 2/7 < 3/9 < 4/6
    0/8 < 1/5 < 2/7 < 4/9 < 3/6
    0/8 < 1/5 < 2/7 < 6/9 < 3/4
    0/8 < 1/5 < 2/9 < 3/4 < 6/7
    0/8 < 1/5 < 2/9 < 3/6 < 4/7
    0/8 < 1/5 < 2/9 < 3/7 < 4/6
    0/8 < 1/5 < 3/7 < 2/4 < 6/9
    0/8 < 1/5 < 3/9 < 2/4 < 6/7
    0/8 < 1/5 < 4/9 < 2/3 < 6/7
    0/8 < 1/6 < 2/3 < 7/9 < 4/5
    0/8 < 1/6 < 2/4 < 3/5 < 7/9
    0/8 < 1/6 < 2/5 < 3/4 < 7/9
    0/8 < 1/6 < 2/5 < 3/7 < 4/9
    0/8 < 1/6 < 2/7 < 3/9 < 4/5
    0/8 < 1/6 < 2/7 < 4/9 < 3/5
    0/8 < 1/6 < 2/7 < 5/9 < 3/4
    0/8 < 1/6 < 2/9 < 3/7 < 4/5
    0/8 < 1/6 < 2/9 < 4/7 < 3/5
    0/8 < 1/6 < 2/9 < 5/7 < 3/4
    0/8 < 1/6 < 3/7 < 2/4 < 5/9
    0/8 < 1/6 < 3/9 < 2/4 < 5/7
    0/8 < 1/6 < 3/9 < 2/5 < 4/7
    0/8 < 1/6 < 4/9 < 2/3 < 5/7
    0/8 < 1/6 < 5/9 < 4/7 < 2/3
    0/8 < 1/7 < 2/4 < 3/5 < 6/9
    0/8 < 1/7 < 2/5 < 4/9 < 3/6
    0/8 < 1/7 < 2/5 < 6/9 < 3/4
    0/8 < 1/7 < 2/6 < 4/9 < 3/5
    0/8 < 1/7 < 2/6 < 5/9 < 3/4
    0/8 < 1/7 < 2/9 < 3/4 < 5/6
    0/8 < 1/7 < 2/9 < 3/5 < 4/6
    0/8 < 1/7 < 2/9 < 3/6 < 4/5
    0/8 < 1/7 < 3/9 < 2/4 < 5/6
    0/8 < 1/7 < 3/9 < 2/5 < 4/6
    0/8 < 1/7 < 4/9 < 2/3 < 5/6
    0/8 < 1/9 < 2/3 < 4/5 < 6/7
    0/8 < 1/9 < 2/4 < 3/5 < 6/7
    0/8 < 1/9 < 2/5 < 3/4 < 6/7
    0/8 < 1/9 < 2/5 < 3/6 < 4/7
    0/8 < 1/9 < 2/5 < 3/7 < 4/6
    0/8 < 1/9 < 2/6 < 3/7 < 4/5
    0/8 < 1/9 < 2/6 < 4/7 < 3/5
    0/8 < 1/9 < 2/6 < 5/7 < 3/4
    0/8 < 1/9 < 2/7 < 3/4 < 5/6
    0/8 < 1/9 < 2/7 < 3/5 < 4/6
    0/8 < 1/9 < 2/7 < 3/6 < 4/5
    0/8 < 1/9 < 3/7 < 2/4 < 5/6
    0/8 < 1/9 < 4/7 < 2/3 < 5/6
    0/8 < 2/7 < 1/3 < 4/9 < 5/6
    0/8 < 2/7 < 1/3 < 5/9 < 4/6
    0/8 < 2/7 < 1/3 < 6/9 < 4/5
    0/8 < 2/9 < 1/3 < 4/5 < 6/7
    0/8 < 2/9 < 1/3 < 4/6 < 5/7
    0/8 < 2/9 < 1/3 < 4/7 < 5/6
    0/8 < 2/9 < 1/4 < 3/5 < 6/7
    0/8 < 2/9 < 1/4 < 3/6 < 5/7
    0/8 < 2/9 < 1/4 < 3/7 < 5/6
    0/8 < 3/7 < 1/2 < 5/9 < 4/6
    0/8 < 3/7 < 1/2 < 6/9 < 4/5
    0/8 < 3/7 < 4/9 < 1/2 < 5/6
    0/8 < 3/9 < 1/2 < 4/5 < 6/7
    0/8 < 3/9 < 1/2 < 4/6 < 5/7
    0/8 < 3/9 < 1/2 < 4/7 < 5/6
    0/8 < 4/9 < 1/2 < 3/5 < 6/7
    0/9 < 1/2 < 3/4 < 5/6 < 7/8
    0/9 < 1/2 < 3/5 < 4/6 < 7/8
    0/9 < 1/2 < 4/7 < 3/5 < 6/8
    0/9 < 1/2 < 5/8 < 3/4 < 6/7
    0/9 < 1/3 < 2/4 < 5/6 < 7/8
    0/9 < 1/3 < 2/4 < 5/7 < 6/8
    0/9 < 1/3 < 2/4 < 5/8 < 6/7
    0/9 < 1/3 < 2/5 < 4/6 < 7/8
    0/9 < 1/3 < 2/5 < 4/7 < 6/8
    0/9 < 1/3 < 2/5 < 4/8 < 6/7
    0/9 < 1/4 < 2/3 < 5/6 < 7/8
    0/9 < 1/4 < 2/3 < 5/7 < 6/8
    0/9 < 1/4 < 2/5 < 3/6 < 7/8
    0/9 < 1/4 < 2/5 < 3/7 < 6/8
    0/9 < 1/4 < 2/6 < 3/5 < 7/8
    0/9 < 1/4 < 2/6 < 3/7 < 5/8
    0/9 < 1/4 < 2/6 < 3/8 < 5/7
    0/9 < 1/4 < 2/7 < 3/5 < 6/8
    0/9 < 1/4 < 2/7 < 3/6 < 5/8
    0/9 < 1/4 < 2/7 < 3/8 < 5/6
    0/9 < 1/4 < 3/8 < 2/5 < 6/7
    0/9 < 1/4 < 5/8 < 2/3 < 6/7
    0/9 < 1/5 < 2/6 < 3/4 < 7/8
    0/9 < 1/5 < 2/6 < 3/7 < 4/8
    0/9 < 1/5 < 2/6 < 3/8 < 4/7
    0/9 < 1/5 < 2/7 < 3/8 < 4/6
    0/9 < 1/5 < 2/8 < 3/4 < 6/7
    0/9 < 1/5 < 2/8 < 3/6 < 4/7
    0/9 < 1/5 < 2/8 < 3/7 < 4/6
    0/9 < 1/5 < 3/7 < 2/4 < 6/8
    0/9 < 1/5 < 3/8 < 2/4 < 6/7
    0/9 < 1/5 < 4/7 < 2/3 < 6/8
    0/9 < 1/5 < 4/8 < 2/3 < 6/7
    0/9 < 1/6 < 2/3 < 4/5 < 7/8
    0/9 < 1/6 < 2/4 < 3/5 < 7/8
    0/9 < 1/6 < 2/5 < 3/4 < 7/8
    0/9 < 1/6 < 2/5 < 3/7 < 4/8
    0/9 < 1/6 < 2/7 < 3/8 < 4/5
    0/9 < 1/6 < 2/7 < 4/8 < 3/5
    0/9 < 1/6 < 2/7 < 5/8 < 3/4
    0/9 < 1/6 < 2/8 < 3/7 < 4/5
    0/9 < 1/6 < 2/8 < 4/7 < 3/5
    0/9 < 1/6 < 2/8 < 5/7 < 3/4
    0/9 < 1/6 < 3/7 < 2/4 < 5/8
    0/9 < 1/6 < 3/8 < 2/4 < 5/7
    0/9 < 1/6 < 3/8 < 2/5 < 4/7
    0/9 < 1/6 < 4/7 < 5/8 < 2/3
    0/9 < 1/6 < 4/8 < 2/3 < 5/7
    0/9 < 1/7 < 2/3 < 6/8 < 4/5
    0/9 < 1/7 < 2/4 < 3/5 < 6/8
    0/9 < 1/7 < 2/6 < 3/8 < 4/5
    0/9 < 1/7 < 2/6 < 4/8 < 3/5
    0/9 < 1/7 < 2/6 < 5/8 < 3/4
    0/9 < 1/7 < 2/8 < 3/4 < 5/6
    0/9 < 1/7 < 2/8 < 3/5 < 4/6
    0/9 < 1/7 < 2/8 < 3/6 < 4/5
    0/9 < 1/7 < 3/8 < 2/4 < 5/6
    0/9 < 1/7 < 3/8 < 2/5 < 4/6
    0/9 < 1/7 < 4/8 < 2/3 < 5/6
    0/9 < 1/8 < 2/3 < 4/5 < 6/7
    0/9 < 1/8 < 2/4 < 3/5 < 6/7
    0/9 < 1/8 < 2/5 < 3/4 < 6/7
    0/9 < 1/8 < 2/5 < 3/6 < 4/7
    0/9 < 1/8 < 2/5 < 3/7 < 4/6
    0/9 < 1/8 < 2/6 < 3/7 < 4/5
    0/9 < 1/8 < 2/6 < 4/7 < 3/5
    0/9 < 1/8 < 2/6 < 5/7 < 3/4
    0/9 < 1/8 < 2/7 < 3/4 < 5/6
    0/9 < 1/8 < 2/7 < 3/5 < 4/6
    0/9 < 1/8 < 2/7 < 3/6 < 4/5
    0/9 < 1/8 < 3/7 < 2/4 < 5/6
    0/9 < 1/8 < 4/7 < 2/3 < 5/6
    0/9 < 2/7 < 1/3 < 4/8 < 5/6
    0/9 < 2/7 < 1/3 < 5/8 < 4/6
    0/9 < 2/7 < 1/3 < 6/8 < 4/5
    0/9 < 2/8 < 1/3 < 4/5 < 6/7
    0/9 < 2/8 < 1/3 < 4/6 < 5/7
    0/9 < 2/8 < 1/3 < 4/7 < 5/6
    0/9 < 3/7 < 1/2 < 5/8 < 4/6
    0/9 < 3/7 < 1/2 < 6/8 < 4/5
    0/9 < 3/8 < 1/2 < 4/5 < 6/7
    0/9 < 3/8 < 1/2 < 4/6 < 5/7
    0/9 < 3/8 < 1/2 < 4/7 < 5/6

    • Rudolf Österreicher

      As you can see, many of these 801 solutions with only proper fractions (204 to be exact) also only contain irreducible fractions:
      0/2 < 1/3 < 4/5 < 6/7 < 8/9
      0/2 < 1/3 < 4/7 < 5/6 < 8/9
      0/2 < 1/3 < 4/9 < 5/6 < 7/8
      0/2 < 1/3 < 4/9 < 5/8 < 6/7
      0/2 < 1/4 < 3/5 < 6/7 < 8/9
      0/2 < 1/4 < 3/7 < 5/6 < 8/9
      0/2 < 1/4 < 3/8 < 5/9 < 6/7
      0/2 < 1/4 < 3/8 < 7/9 < 5/6
      0/2 < 1/5 < 3/4 < 6/7 < 8/9
      0/2 < 1/5 < 3/8 < 4/9 < 6/7
      0/2 < 1/6 < 3/7 < 4/5 < 8/9
      0/2 < 1/6 < 3/7 < 4/9 < 5/8
      0/2 < 1/6 < 3/8 < 4/9 < 5/7
      0/2 < 1/6 < 3/8 < 5/9 < 4/7
      0/2 < 1/6 < 3/8 < 7/9 < 4/5
      0/2 < 1/6 < 4/7 < 3/5 < 8/9
      0/2 < 1/6 < 4/9 < 3/5 < 7/8
      0/2 < 1/6 < 5/7 < 3/4 < 8/9
      0/2 < 1/6 < 5/8 < 3/4 < 7/9
      0/2 < 1/6 < 5/9 < 3/4 < 7/8
      0/2 < 1/7 < 3/4 < 5/6 < 8/9
      0/2 < 1/7 < 3/8 < 4/9 < 5/6
      0/2 < 1/8 < 3/4 < 7/9 < 5/6
      0/2 < 1/8 < 3/7 < 4/9 < 5/6
      0/2 < 1/8 < 4/9 < 3/5 < 6/7
      0/2 < 1/8 < 5/9 < 3/4 < 6/7
      0/2 < 1/9 < 3/4 < 5/6 < 7/8
      0/2 < 1/9 < 3/8 < 4/5 < 6/7
      0/2 < 1/9 < 3/8 < 4/7 < 5/6
      0/2 < 1/9 < 5/8 < 3/4 < 6/7
      0/3 < 1/2 < 4/5 < 6/7 < 8/9
      0/3 < 1/2 < 4/7 < 5/6 < 8/9
      0/3 < 1/4 < 2/5 < 6/7 < 8/9
      0/3 < 1/4 < 2/7 < 5/6 < 8/9
      0/3 < 1/6 < 2/5 < 4/7 < 8/9
      0/3 < 1/6 < 2/5 < 4/9 < 7/8
      0/3 < 1/6 < 2/7 < 4/5 < 8/9
      0/3 < 1/6 < 2/7 < 4/9 < 5/8
      0/3 < 1/6 < 2/9 < 4/5 < 7/8
      0/3 < 1/6 < 2/9 < 4/7 < 5/8
      0/3 < 1/8 < 2/5 < 4/9 < 6/7
      0/3 < 1/8 < 2/7 < 4/9 < 5/6
      0/3 < 1/8 < 2/9 < 4/5 < 6/7
      0/3 < 1/8 < 2/9 < 4/7 < 5/6
      0/3 < 2/9 < 1/4 < 5/6 < 7/8
      0/3 < 2/9 < 1/4 < 5/8 < 6/7
      0/3 < 4/9 < 1/2 < 5/6 < 7/8
      0/3 < 4/9 < 1/2 < 5/8 < 6/7
      0/4 < 1/2 < 3/5 < 6/7 < 8/9
      0/4 < 1/3 < 2/5 < 6/7 < 8/9
      0/4 < 1/5 < 2/3 < 6/7 < 8/9
      0/4 < 1/5 < 2/9 < 3/8 < 6/7
      0/4 < 1/6 < 2/3 < 5/7 < 8/9
      0/4 < 1/6 < 2/5 < 3/7 < 8/9
      0/4 < 1/6 < 2/7 < 3/5 < 8/9
      0/4 < 1/6 < 2/7 < 3/8 < 5/9
      0/4 < 1/6 < 2/9 < 3/5 < 7/8
      0/4 < 1/6 < 2/9 < 3/7 < 5/8
      0/4 < 1/6 < 2/9 < 3/8 < 5/7
      0/4 < 1/6 < 3/8 < 2/5 < 7/9
      0/4 < 1/6 < 5/8 < 2/3 < 7/9
      0/4 < 1/6 < 5/9 < 2/3 < 7/8
      0/4 < 1/7 < 2/3 < 5/6 < 8/9
      0/4 < 1/7 < 2/9 < 3/8 < 5/6
      0/4 < 1/8 < 2/3 < 7/9 < 5/6
      0/4 < 1/8 < 2/9 < 3/5 < 6/7
      0/4 < 1/8 < 2/9 < 3/7 < 5/6
      0/4 < 1/8 < 5/9 < 2/3 < 6/7
      0/4 < 1/9 < 2/3 < 5/6 < 7/8
      0/4 < 1/9 < 2/7 < 3/8 < 5/6
      0/4 < 1/9 < 3/8 < 2/5 < 6/7
      0/4 < 1/9 < 5/8 < 2/3 < 6/7
      0/4 < 2/7 < 1/3 < 5/6 < 8/9
      0/4 < 2/9 < 1/3 < 5/6 < 7/8
      0/4 < 2/9 < 1/3 < 5/8 < 6/7
      0/4 < 3/7 < 1/2 < 5/6 < 8/9
      0/4 < 3/8 < 1/2 < 5/9 < 6/7
      0/4 < 3/8 < 1/2 < 7/9 < 5/6
      0/5 < 1/2 < 3/4 < 6/7 < 8/9
      0/5 < 1/4 < 2/3 < 6/7 < 8/9
      0/5 < 1/6 < 2/7 < 3/4 < 8/9
      0/5 < 1/6 < 2/7 < 3/8 < 4/9
      0/5 < 1/6 < 2/9 < 3/4 < 7/8
      0/5 < 1/6 < 2/9 < 3/8 < 4/7
      0/5 < 1/6 < 4/7 < 2/3 < 8/9
      0/5 < 1/6 < 4/9 < 2/3 < 7/8
      0/5 < 1/8 < 2/9 < 3/4 < 6/7
      0/5 < 1/8 < 4/9 < 2/3 < 6/7
      0/5 < 2/9 < 1/4 < 3/8 < 6/7
      0/5 < 3/8 < 4/9 < 1/2 < 6/7
      0/6 < 1/2 < 4/7 < 3/5 < 8/9
      0/6 < 1/2 < 5/7 < 3/4 < 8/9
      0/6 < 1/2 < 5/8 < 3/4 < 7/9
      0/6 < 1/2 < 5/9 < 3/4 < 7/8
      0/6 < 1/3 < 2/5 < 4/7 < 8/9
      0/6 < 1/3 < 2/5 < 4/9 < 7/8
      0/6 < 1/4 < 2/3 < 5/7 < 8/9
      0/6 < 1/4 < 2/5 < 3/7 < 8/9
      0/6 < 1/4 < 2/7 < 3/5 < 8/9
      0/6 < 1/4 < 2/7 < 3/8 < 5/9
      0/6 < 1/4 < 3/8 < 2/5 < 7/9
      0/6 < 1/4 < 5/8 < 2/3 < 7/9
      0/6 < 1/4 < 5/9 < 2/3 < 7/8
      0/6 < 1/5 < 2/7 < 3/4 < 8/9
      0/6 < 1/5 < 2/7 < 3/8 < 4/9
      0/6 < 1/5 < 2/9 < 3/4 < 7/8
      0/6 < 1/5 < 2/9 < 3/8 < 4/7
      0/6 < 1/5 < 4/7 < 2/3 < 8/9
      0/6 < 1/5 < 4/9 < 2/3 < 7/8
      0/6 < 1/7 < 2/3 < 4/5 < 8/9
      0/6 < 1/7 < 2/5 < 3/4 < 8/9
      0/6 < 1/7 < 2/9 < 3/8 < 4/5
      0/6 < 1/7 < 2/9 < 5/8 < 3/4
      0/6 < 1/7 < 3/8 < 2/5 < 4/9
      0/6 < 1/7 < 4/9 < 5/8 < 2/3
      0/6 < 1/8 < 2/3 < 7/9 < 4/5
      0/6 < 1/8 < 2/5 < 3/4 < 7/9
      0/6 < 1/8 < 2/5 < 3/7 < 4/9
      0/6 < 1/8 < 2/7 < 4/9 < 3/5
      0/6 < 1/8 < 2/7 < 5/9 < 3/4
      0/6 < 1/8 < 2/9 < 3/7 < 4/5
      0/6 < 1/8 < 2/9 < 4/7 < 3/5
      0/6 < 1/8 < 2/9 < 5/7 < 3/4
      0/6 < 1/8 < 4/9 < 2/3 < 5/7
      0/6 < 1/8 < 5/9 < 4/7 < 2/3
      0/6 < 1/9 < 2/3 < 4/5 < 7/8
      0/6 < 1/9 < 2/5 < 3/4 < 7/8
      0/6 < 1/9 < 2/7 < 3/8 < 4/5
      0/6 < 1/9 < 2/7 < 5/8 < 3/4
      0/6 < 1/9 < 3/8 < 2/5 < 4/7
      0/6 < 1/9 < 4/7 < 5/8 < 2/3
      0/6 < 2/7 < 1/3 < 4/5 < 8/9
      0/6 < 2/7 < 1/3 < 4/9 < 5/8
      0/6 < 2/9 < 1/3 < 4/5 < 7/8
      0/6 < 2/9 < 1/3 < 4/7 < 5/8
      0/6 < 2/9 < 1/4 < 3/5 < 7/8
      0/6 < 2/9 < 1/4 < 3/7 < 5/8
      0/6 < 2/9 < 1/4 < 3/8 < 5/7
      0/6 < 3/7 < 1/2 < 4/5 < 8/9
      0/6 < 3/7 < 4/9 < 1/2 < 5/8
      0/6 < 3/8 < 1/2 < 5/9 < 4/7
      0/6 < 3/8 < 1/2 < 7/9 < 4/5
      0/6 < 3/8 < 4/9 < 1/2 < 5/7
      0/6 < 4/9 < 1/2 < 3/5 < 7/8
      0/7 < 1/2 < 3/4 < 5/6 < 8/9
      0/7 < 1/4 < 2/3 < 5/6 < 8/9
      0/7 < 1/6 < 2/3 < 4/5 < 8/9
      0/7 < 1/6 < 2/5 < 3/4 < 8/9
      0/7 < 1/6 < 2/9 < 3/8 < 4/5
      0/7 < 1/6 < 2/9 < 5/8 < 3/4
      0/7 < 1/6 < 3/8 < 2/5 < 4/9
      0/7 < 1/6 < 4/9 < 5/8 < 2/3
      0/7 < 1/8 < 2/9 < 3/4 < 5/6
      0/7 < 1/8 < 4/9 < 2/3 < 5/6
      0/7 < 2/9 < 1/4 < 3/8 < 5/6
      0/7 < 3/8 < 4/9 < 1/2 < 5/6
      0/8 < 1/2 < 3/4 < 7/9 < 5/6
      0/8 < 1/2 < 5/9 < 3/4 < 6/7
      0/8 < 1/3 < 2/5 < 4/9 < 6/7
      0/8 < 1/4 < 2/3 < 7/9 < 5/6
      0/8 < 1/4 < 5/9 < 2/3 < 6/7
      0/8 < 1/5 < 2/9 < 3/4 < 6/7
      0/8 < 1/5 < 4/9 < 2/3 < 6/7
      0/8 < 1/6 < 2/3 < 7/9 < 4/5
      0/8 < 1/6 < 2/5 < 3/4 < 7/9
      0/8 < 1/6 < 2/5 < 3/7 < 4/9
      0/8 < 1/6 < 2/7 < 4/9 < 3/5
      0/8 < 1/6 < 2/7 < 5/9 < 3/4
      0/8 < 1/6 < 2/9 < 3/7 < 4/5
      0/8 < 1/6 < 2/9 < 4/7 < 3/5
      0/8 < 1/6 < 2/9 < 5/7 < 3/4
      0/8 < 1/6 < 4/9 < 2/3 < 5/7
      0/8 < 1/6 < 5/9 < 4/7 < 2/3
      0/8 < 1/7 < 2/9 < 3/4 < 5/6
      0/8 < 1/7 < 4/9 < 2/3 < 5/6
      0/8 < 1/9 < 2/3 < 4/5 < 6/7
      0/8 < 1/9 < 2/5 < 3/4 < 6/7
      0/8 < 1/9 < 2/7 < 3/4 < 5/6
      0/8 < 1/9 < 4/7 < 2/3 < 5/6
      0/8 < 2/7 < 1/3 < 4/9 < 5/6
      0/8 < 2/9 < 1/3 < 4/5 < 6/7
      0/8 < 2/9 < 1/3 < 4/7 < 5/6
      0/8 < 2/9 < 1/4 < 3/5 < 6/7
      0/8 < 2/9 < 1/4 < 3/7 < 5/6
      0/8 < 3/7 < 4/9 < 1/2 < 5/6
      0/8 < 4/9 < 1/2 < 3/5 < 6/7
      0/9 < 1/2 < 3/4 < 5/6 < 7/8
      0/9 < 1/2 < 5/8 < 3/4 < 6/7
      0/9 < 1/4 < 2/3 < 5/6 < 7/8
      0/9 < 1/4 < 2/7 < 3/8 < 5/6
      0/9 < 1/4 < 3/8 < 2/5 < 6/7
      0/9 < 1/4 < 5/8 < 2/3 < 6/7
      0/9 < 1/6 < 2/3 < 4/5 < 7/8
      0/9 < 1/6 < 2/5 < 3/4 < 7/8
      0/9 < 1/6 < 2/7 < 3/8 < 4/5
      0/9 < 1/6 < 2/7 < 5/8 < 3/4
      0/9 < 1/6 < 3/8 < 2/5 < 4/7
      0/9 < 1/6 < 4/7 < 5/8 < 2/3
      0/9 < 1/8 < 2/3 < 4/5 < 6/7
      0/9 < 1/8 < 2/5 < 3/4 < 6/7
      0/9 < 1/8 < 2/7 < 3/4 < 5/6
      0/9 < 1/8 < 4/7 < 2/3 < 5/6
      0/9 < 3/8 < 1/2 < 4/5 < 6/7
      0/9 < 3/8 < 1/2 < 4/7 < 5/6

      • Rudolf Österreicher

        I forgot to include the cases where two or more fractions are equal to each other:
        There are an additional 1,176 solutions, for a total of 15,120, of which 945 only contain proper fractions and

        Here are the missing 144 solutions with only proper fractions:
        0/5 < 1/7 < 2/6 = 3/9 < 4/8
        0/5 < 1/7 < 2/9 < 3/6 = 4/8
        0/5 < 1/8 < 2/6 = 3/9 < 4/7
        0/5 < 1/9 < 2/7 < 3/6 = 4/8
        0/7 < 1/5 < 2/6 = 3/9 < 4/8
        0/7 < 1/5 < 2/9 < 3/6 = 4/8
        0/7 < 1/8 < 2/6 = 3/9 < 4/5
        0/7 < 1/9 < 2/5 < 3/6 = 4/8
        0/8 < 1/5 < 2/6 = 3/9 < 4/7
        0/8 < 1/7 < 2/6 = 3/9 < 4/5
        0/9 < 1/5 < 2/7 < 3/6 = 4/8
        0/9 < 1/7 < 2/5 < 3/6 = 4/8
        0/4 < 1/7 < 2/6 = 3/9 < 5/8
        0/4 < 1/8 < 2/6 = 3/9 < 5/7
        0/6 < 1/4 = 2/8 < 3/7 < 5/9
        0/6 < 1/4 = 2/8 < 3/9 < 5/7
        0/7 < 1/4 < 2/6 = 3/9 < 5/8
        0/7 < 1/4 = 2/8 < 3/6 < 5/9
        0/7 < 1/4 = 2/8 < 3/9 < 5/6
        0/7 < 1/8 < 2/4 = 3/6 < 5/9
        0/7 < 1/9 < 2/4 = 3/6 < 5/8
        0/8 < 1/4 < 2/6 = 3/9 < 5/7
        0/8 < 1/7 < 2/4 = 3/6 < 5/9
        0/8 < 1/9 < 2/4 = 3/6 < 5/7
        0/9 < 1/4 = 2/8 < 3/6 < 5/7
        0/9 < 1/4 = 2/8 < 3/7 < 5/6
        0/9 < 1/7 < 2/4 = 3/6 < 5/8
        0/9 < 1/8 < 2/4 = 3/6 < 5/7
        0/5 < 1/4 = 2/8 < 3/7 < 6/9
        0/5 < 1/4 = 2/8 < 3/9 < 6/7
        0/5 < 1/7 < 2/9 < 3/4 = 6/8
        0/5 < 1/9 < 2/7 < 3/4 = 6/8
        0/7 < 1/4 = 2/8 < 3/5 < 6/9
        0/7 < 1/5 < 2/9 < 3/4 = 6/8
        0/7 < 1/9 < 2/5 < 3/4 = 6/8
        0/9 < 1/4 = 2/8 < 3/5 < 6/7
        0/9 < 1/5 < 2/7 < 3/4 = 6/8
        0/9 < 1/7 < 2/5 < 3/4 = 6/8
        0/4 < 1/5 < 2/6 = 3/9 < 7/8
        0/5 < 1/4 < 2/6 = 3/9 < 7/8
        0/5 < 1/4 = 2/8 < 3/6 < 7/9
        0/5 < 1/8 < 2/4 = 3/6 < 7/9
        0/5 < 1/9 < 2/4 = 3/6 < 7/8
        0/6 < 1/4 = 2/8 < 3/5 < 7/9
        0/8 < 1/5 < 2/4 = 3/6 < 7/9
        0/9 < 1/5 < 2/4 = 3/6 < 7/8
        0/5 < 1/7 < 2/4 = 3/6 < 8/9
        0/7 < 1/5 < 2/4 = 3/6 < 8/9
        0/7 < 1/3 = 2/6 < 4/8 < 5/9
        0/7 < 1/3 = 2/6 < 4/9 < 5/8
        0/8 < 1/3 = 2/6 < 4/9 < 5/7
        0/8 < 1/9 < 2/3 = 4/6 < 5/7
        0/9 < 1/3 = 2/6 < 4/7 < 5/8
        0/9 < 1/3 = 2/6 < 4/8 < 5/7
        0/9 < 1/8 < 2/3 = 4/6 < 5/7
        0/5 < 1/3 = 2/6 < 4/8 < 7/9
        0/5 < 1/3 = 2/6 < 4/9 < 7/8
        0/5 < 1/8 < 2/3 = 4/6 < 7/9
        0/5 < 1/9 < 2/3 = 4/6 < 7/8
        0/8 < 1/5 < 2/3 = 4/6 < 7/9
        0/9 < 1/3 = 2/6 < 4/5 < 7/8
        0/9 < 1/5 < 2/3 = 4/6 < 7/8
        0/5 < 1/3 = 2/6 < 4/7 < 8/9
        0/5 < 1/7 < 2/3 = 4/6 < 8/9
        0/7 < 1/3 = 2/6 < 4/5 < 8/9
        0/7 < 1/5 < 2/3 = 4/6 < 8/9
        0/8 < 1/3 = 2/6 < 5/9 < 4/7
        0/3 < 1/4 = 2/8 < 5/9 < 6/7
        0/4 < 1/3 = 2/6 < 5/8 < 7/9
        0/4 < 1/3 = 2/6 < 5/9 < 7/8
        0/4 < 1/3 = 2/6 < 5/7 < 8/9
        0/7 < 1/8 < 2/3 = 6/9 < 4/5
        0/8 < 1/7 < 2/3 = 6/9 < 4/5
        0/3 < 1/4 = 2/8 < 6/9 < 5/7
        0/4 < 1/8 < 2/3 = 6/9 < 5/7
        0/8 < 1/4 < 2/3 = 6/9 < 5/7
        0/4 < 1/5 < 2/3 = 6/9 < 7/8
        0/5 < 1/4 < 2/3 = 6/9 < 7/8
        0/8 < 1/3 = 2/6 < 7/9 < 4/5
        0/3 < 1/4 = 2/8 < 7/9 < 5/6
        0/2 < 1/7 < 3/6 = 4/8 < 5/9
        0/2 < 1/9 < 3/6 = 4/8 < 5/7
        0/7 < 1/2 = 3/6 = 4/8 < 5/9
        0/9 < 1/2 = 3/6 < 4/7 < 5/8
        0/9 < 1/2 = 3/6 = 4/8 < 5/7
        0/2 < 1/5 < 3/6 = 4/8 < 7/9
        0/5 < 1/2 = 3/6 = 4/8 < 7/9
        0/9 < 1/2 = 3/6 < 4/5 < 7/8
        0/5 < 1/2 = 3/6 < 4/7 < 8/9
        0/7 < 1/2 = 3/6 < 4/5 < 8/9
        0/8 < 1/2 = 3/6 < 5/9 < 4/7
        0/4 < 1/2 = 3/6 < 5/8 < 7/9
        0/4 < 1/2 = 3/6 < 5/9 < 7/8
        0/4 < 1/2 = 3/6 < 5/7 < 8/9
        0/2 < 1/5 < 3/4 = 6/8 < 7/9
        0/5 < 1/2 < 3/4 = 6/8 < 7/9
        0/8 < 1/2 = 3/6 < 7/9 < 4/5
        0/5 < 1/7 < 4/8 < 2/3 = 6/9
        0/5 < 1/8 < 4/7 < 2/3 = 6/9
        0/7 < 1/5 < 4/8 < 2/3 = 6/9
        0/8 < 1/5 < 4/7 < 2/3 = 6/9
        0/7 < 1/2 = 4/8 < 3/5 < 6/9
        0/9 < 1/2 = 4/8 < 3/5 < 6/7
        0/6 < 1/2 = 4/8 < 3/5 < 7/9
        0/3 < 1/2 = 4/8 < 5/9 < 6/7
        0/3 < 1/2 = 4/8 < 6/9 < 5/7
        0/3 < 1/2 = 4/8 < 7/9 < 5/6
        0/7 < 1/8 < 5/9 < 2/3 = 4/6
        0/7 < 1/9 < 5/8 < 2/3 = 4/6
        0/8 < 1/7 < 5/9 < 2/3 = 4/6
        0/9 < 1/7 < 5/8 < 2/3 = 4/6
        0/4 < 1/7 < 5/8 < 2/3 = 6/9
        0/7 < 1/4 < 5/8 < 2/3 = 6/9
        0/2 < 1/7 < 5/9 < 3/4 = 6/8
        0/2 < 1/9 < 5/7 < 3/4 = 6/8
        0/7 < 1/2 < 5/9 < 3/4 = 6/8
        0/9 < 1/2 < 5/7 < 3/4 = 6/8
        0/1 < 2/6 = 3/9 < 4/7 < 5/8
        0/1 < 2/6 = 3/9 < 4/8 < 5/7
        0/1 < 2/7 < 3/6 = 4/8 < 5/9
        0/1 < 2/9 < 3/6 = 4/8 < 5/7
        0/1 < 2/5 < 3/6 = 4/8 < 7/9
        0/1 < 2/6 = 3/9 < 4/5 < 7/8
        0/1 < 2/4 = 3/6 < 5/8 < 7/9
        0/1 < 2/4 = 3/6 < 5/9 < 7/8
        0/1 < 2/4 = 3/6 < 5/7 < 8/9
        0/1 < 2/5 < 3/4 = 6/8 < 7/9
        0/1 < 2/3 = 4/6 < 5/7 < 8/9
        0/1 < 2/7 < 5/9 < 3/4 = 6/8
        0/1 < 2/9 < 5/7 < 3/4 = 6/8
        0/1 < 2/3 = 6/9 < 4/5 < 7/8
        0/6 < 3/7 < 1/2 = 4/8 < 5/9
        0/6 < 3/9 < 1/2 = 4/8 < 5/7
        0/7 < 3/9 < 1/2 = 4/8 < 5/6
        0/9 < 3/7 < 1/2 = 4/8 < 5/6
        0/5 < 3/7 < 1/2 = 4/8 < 6/9
        0/5 < 3/9 < 1/2 = 4/8 < 6/7
        0/7 < 4/9 < 1/2 = 3/6 < 5/8
        0/8 < 4/9 < 1/2 = 3/6 < 5/7
        0/5 < 4/9 < 1/2 = 3/6 < 7/8
        0/1 < 4/8 < 2/3 = 6/9 < 5/7
        0/1 < 4/7 < 5/8 < 2/3 = 6/9
        0/1 < 5/8 < 2/3 = 4/6 < 7/9
        0/1 < 5/9 < 2/3 = 4/6 < 7/8

        For the irreducible fractions, in one of my previous comments I skipped the 12 solutions with a 1 in the denominator because a/1 can be written as a whole number. But so can 0/a = 0, which means it doesn't make sense to exclude those 12, so I'll list them now:
        0/1 < 2/3 < 4/5 < 6/7 < 8/9
        0/1 < 2/5 < 3/4 < 6/7 < 8/9
        0/1 < 2/7 < 3/4 < 5/6 < 8/9
        0/1 < 2/7 < 3/8 < 4/9 < 5/6
        0/1 < 2/9 < 3/4 < 5/6 < 7/8
        0/1 < 2/9 < 3/8 < 4/5 < 6/7
        0/1 < 2/9 < 3/8 < 4/7 < 5/6
        0/1 < 2/9 < 5/8 < 3/4 < 6/7
        0/1 < 3/8 < 2/5 < 4/9 < 6/7
        0/1 < 4/7 < 2/3 < 5/6 < 8/9
        0/1 < 4/9 < 2/3 < 5/6 < 7/8
        0/1 < 4/9 < 5/8 < 2/3 < 6/7

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