Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer (relatively) soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

The Secret of Egyptian Fractions

Alex made a poster of Egyptian-style fractions, from 1/2 to 9/10. Many of the fractions were easy. She knew that…

$\frac{5}{10} = \frac{4}{8} = \frac{3}{6} = \frac{2}{4} = \frac{1}{2}$

Therefore, as soon as she figured out one fraction, she had the answer to all of its equivalents.

She had the most trouble with the 7ths and 9ths. She tried converting these to other fractions that were easier to work with. For example, 28 has more factors than 7, making 28ths easier to break up into other fractions with one in the numerator.

Alex changed 2/7 into 28ths:

$\frac{2}{7} = \frac{8}{28} = \frac{7}{28} + \frac{1}{28} = \frac{1}{4} + \frac{1}{28}$

The number 36 also has plenty of factors, so Alex tried:

$\frac{7}{9} = \frac{28}{36} = \frac{18}{36} + \frac{9}{36} + \frac{1}{36} = \frac{1}{2} + \frac{1}{4} + \frac{1}{36}$

But then she realized that:

$\frac{7}{9} = \frac{6}{9} + \frac{1}{9} = \frac{2}{3} + \frac{1}{9}$

…which was so much easier that she felt dumb for not noticing it at once.

Alex found Egyptian equivalents for all of the fractions on her chart. How well did you do? Download the answer sheet and compare:

Remember that many other correct answers are possible. For example, Alex chose…

$\frac{8}{9} = \frac{2}{3} + \frac{1}{6} + \frac{1}{18}$

But this answer would be equally valid:

$\frac{8}{9} = \frac{1}{2} + \frac{1}{3} + \frac{1}{18}$

To Be Continued…

Read all the posts from the January/February 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.