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 |

**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.

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 |