# What's the teacher's fractional addition algorithm?

One day my tenth-grade math teacher walked into our classroom and said, "Class, today to start off the class I've got an addition problem for you. Give me any two positive proper fractions and I'll add them up."

Some of us were a bit bewildered at that, but one of the students decided to bite. "1/2 plus 1/2", he said.

We were a bit surprised. "Shouldn't that be 1, teacher?"

"Alright, what about 1/3 + 2/3"?

"That's 4/11", said the teacher.

"That's 3/11."

The teacher wrote our questions and his answers on the board along with a few more:

1/2 + 1/2 = 11/20
1/3 + 2/3 = 4/11
2/9 + 7/9 = 3/11
3/20 + 4/20 = 1/8
1/9 + 1/9 = 1/9
1/30 + 5/6 = 1/12


Can you figure out what addition algorithm the teacher was using?

• If you want the results of any more additions, just ask. – Joe Z. Jun 12 '15 at 4:53
• How much is 11/13 + 11/13? – leoll2 Jun 12 '15 at 6:53
• I want to know some points about your representation system (which your question currently does not mention) : Is AB/AC=B/C ? Is A+B=B+A ? If A+B1=C & A+B2=C, then is B1=B2 ? Is (A+B)+C=A+(B+C) ? Is RightHandSide the "exact output" or simplified by removing common factors ? – Prem Jun 12 '15 at 10:34
• @JoeZ. [unrelated] I took the liberty of decoding your display picture's GB values to ascii characters as I have been wondering about it. This is what I got: "∙►☻ tπZ╫@U─cVê┐" ...very interesting secret. – Mark N Jun 12 '15 at 17:46
• @MarkN: It's no secret. That's just the infamous AACS 09F9 key. – Joe Z. Jun 13 '15 at 0:44

Rule:

Write both numbers on the left side as decimal, create the resulting number by picking digits in alternating order.

Calculations:

Left first, right second:

1) 0.5 + 0.5 = 0.55
2) 0.3333 + 0.6666 = 0.36363636
3) 0.2222 + 0.7777 = 0.27272727
4) 0.15 + 0.2 = 0.125
5) 0.1111 + 0.1111 = 0.11111111
6) 0.0333 + 0.8333 = 0.08333333

• This is the same answer I came up with, didn't notice tho that 4 and 7 were inversed so I thought it wasn't the right answer :P – Wouter Jun 12 '15 at 10:54
• It seems like the simpler fraction out of left and right was chosen in each case, which might explain the discrepancy of 4 and 7. – isaacg Jun 12 '15 at 11:01
• Looking over it again, number 7 doesn't quite match. Keep on looking, there might still be a better answer that correctly explains it all. Then again, I can't seem to get the numbers typed correctly myself, even after two edits, so maybe the author fell for the same trap. :-) – Moghwyn Jun 12 '15 at 11:06
• The additional rule could be the even vs. odd digit after the decimal point. – Luke Jun 12 '15 at 13:54
• +1 , I think there is no need for looking further. Discrepancies must be due typos by OP. It is unlikely that some other algorithm can get the same output. – Prem Jun 12 '15 at 15:32