Day 11, MAT115

But they didn't restrict themselves to "halving", as our next example shows. Divide 6 by 7:

The starred entries on the right add up to 6:

1. How would you divide 5 by 7?
2. How can we use Fraudini and the unit fraction table to get the same answer?

1. How would you divide 23 by 9?
2. How can we use Fraudini and the unit fraction table to get the same answer?

   Dominic Olivastro, "Ancient Puzzles", suggests a third reason why 
this use of unary fractions is good. Consider the problem Ahmes poses 
of dividing 3 loaves of bread between 5 people. We would answer "each
person gets 3/5-ths of a loaf". If we implemented our solution, we might
then cut 2 loaves into 3/5 | 2/5 pieces, with bread for 3 people; then
cut one of the smaller pieces in half, giving the other two people 
2/5 + 1/5 pieces. Mathematically acceptable, but try this with kids and
they will insist that it is not an even division. Some have larger pieces,
some have smaller. Ahmes would calculate 3/5 as :
   3/5 = ()3 + ()5 + ()15    [ = 1/3 + 1/5 + 1/15 ]
Now cut one loaf into fifths, cut two more into thirds, then take one of
the 1/3-rd pieces and cut it into 5-ths (for the 1/15-th pieces), and you
can now distribute everyone's 3/5-ths share in a way that _looks_ equal,
since they will have exactly the same size pieces. (And no, I don't want
to argue about the crust.)

(This illustrates that the priest would have used the unit fraction table, rather than our method -- which would have given us the answer 1/2+1/10 -- both answers are right! But I (and mathematicians in general) think that our shorter answer is better....)