The World of Trotter Math

Distinct Digit Fraction Sums

Observe the following fraction addition carefully:

                  1     6
                 --- + ---  =  1
                  4     8

On the left side of the equation there are four distinct digits — 1, 4, 6, and 8. While that may not look like earth-shattering news to some people, I think it looks nice.
Can you make up a similar example?
This means, can you find another equation of the form

                  a     c
                 --- + ---  =  1
                  b     d

where a, b, c, and d are distinct digits?
You know, it may not be as easy as it looks. This shall be called a Type I expression.

Now, how about another variation on that theme? Observe this structure…

                  a     c       e
                 --- + ---  =  ---
                  b     d       f

where a, b, c, d, e, and f are distinct digits, and e/f < 1.

Can you find a solution to that?

This shall be called Type II.

Want to go for more? Well, then look at this.

                  a      d
                 ---- + ---  =  1
                  bc     e

where “bc” represents a “two-digit number” (like 27 or 83), and not the algebraic multiplication of 2 values.

This shall be called Type III.

Hey, I’m not done yet. Try your luck, er skill, on this one.

                  ab     e
                 ---- + ---  =  1
                  cd     f

where again “ab” and “cd” represent “two-digit numbers” (like 14 or 65), and not the algebraic multiplication of 2 values.

This shall be called Type IV.

If you can show me an answer to any of these questions, send it to me by email and I will present it here on this page in the charts below.

Please note: that in order for your solution to even be considered for posting, you must write “DDFS” in the subject line of your email; otherwise I will merely ignore it and delete it. Thank you.

trottermath@gmail.com or ttrotter3@yahoo.com

For an important UPDATE, see below Chart IV…

Type I

Type II

Type III

Type IV

Here is Type V.

                  ab      e
                 ----- + ---  =  1
                  cde     f

Type V

Next is Type VI.

                  ab      f
                 ----- + ----  =  1
                  cde     gh

Type VI

And now Type VII.

                  ab      fg
                 ----- + ----  =  1
                  cde     hi

Type VII

This is Type VIII.

                  abc     gh
                 ----- + -----  =  1
                  def     ij

Type VIII

This is Type IX.

                  ab      fg
                 ----- + -----  =  1
                  cde     hij

Type IX

We like Type X.

                  abcd     i
                 ------ + ---  =  1
                  efgh     j

Type X

August 12, 2001…

Let’s continue our patterns. Here’s another variation on Type II.

                  a     c     e
                 --- + --- + ---  =  1
                  b     d     f

Type XI