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

# Solution Name Date
1 1/4 + 6/8 Daniel Lu 4/30/01
2 1/2 + 3/6 Konstantin Knop 8/9/01
3 1/3 + 4/6 Jacqueline Hu 10/24/01
4 3/4 + 2/8 Jacqueline Hu 10/24/01
5 1/2 + 4/8 Leonard Lee 11/4/01
6 2/4 + 3/6 Leonard Lee 11/4/01
7
8

Type II

# Solution Name Date
1 1/4 + 2/8 = 3/6 Konstantin Knop 8/9/01
2 3/9 + 1/6 = 2/4 Leonard Lee 11/4/01
3
4
5
6

Type III

# Solution Name Date
1 2/10 + 4/5 Konstantin Knop 8/9/01
2 2/16 + 7/8 Jacqueline Hu 10/24/01
3 2/14 + 6/7 Jacqueline Hu 10/24/01
4 8/14 + 3/7 Jacqueline Hu 10/24/01
5 5/10 + 4/8 Leonard Lee 11/4/01
6 4/12 + 6/9 Leonard Lee 11/4/01
7 5/20 + 6/8 Leonard Lee 11/4/01
8 7/21 + 6/9 Leonard Lee 11/4/01
9
10
11
12

Type IV

# Solution Name Date
1 13/26 + 4/8 Konstantin Knop 8/9/01
2 15/30 + 2/4 Leonard Lee 11/4/01
3 15/30 + 4/8 Leonard Lee 11/4/01
4 15/60 + 3/4 Leonard Lee 11/4/01
5 19/38 + 2/4 Leonard Lee 11/4/01
6 16/48 + 2/3 Leonard Lee 11/4/01
7
8
9
10

On August 9 and 10, 2001, Konstantin Knop, from St. Petersburg, Russia, sent in some solutions to our problems posed above. But he extended the concept to include more types. And he provided solutions as well.
So we now present his extension ideas with two samples of solutions for each one. Wouldn’t you like to join him and send in a solution or two of your own?

Here is Type V.

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

Type V

# Solution Name Date
1 34/102 + 6/9 Konstantin Knop 8/9/01
2 26/130 + 4/5 Konstantin Knop 8/9/01
3 35/140 + 6/8 Leonard Lee 11/4/01
4 72/108 + 3/9 Leonard Lee 11/4/01
5 53/106 + 2/4 Leonard Lee 11/4/01
6 78/156 + 2/4 Leonard Lee 11/4/01
7
8
9
10

Next is Type VI.

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

Type VI

# Solution Name Date
1 64/208 + 9/13 Konstantin Knop 8/9/01
2 85/136 + 9/24 Konstantin Knop 8/9/01
3
4
5
6

And now Type VII.

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

Type VII

# Solution Name Date
1 24/136 + 70/85 Konstantin Knop 8/9/01
2 96/324 + 57/81 Konstantin Knop 8/9/01
3 45/180 + 27/36 Leonard Lee 11/4/01
4
5
6

This is Type VIII.

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

Type VIII

# Solution Name Date
1 148/296 + 35/70 Konstantin Knop 8/9/01
2 204/867 + 39/51 Konstantin Knop 8/9/01
3
4
5
6

This is Type IX.

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

Type IX

# Solution Name Date
1 57/204 + 98/136 Konstantin Knop 8/9/01
2 59/236 + 78/104 Konstantin Knop 8/9/01
3
4
5
6

We like Type X.

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

Type X

# Solution Name Date
1 1278/6390 + 4/5 Konstantin Knop 8/10/01
2 1485/2970 + 3/6 Konstantin Knop 8/10/01
3
4
5
6

August 12, 2001…

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

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

Type XI

# Solution Name Date
1 1/4 + 2/8 + 3/6 Leonard Lee 11/4/01
2 3/9 + 1/6 + 2/4 Leonard Lee 11/4/01
3
4
5
6
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