
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 earthshattering 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 "twodigit 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 "twodigit 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
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 



