A Millbrook High School student reduced the following fraction by "canceling"
out the sixes:
He got lucky! Now you must find three other examples like this of lucky canceling without using the digit 0. The numbers must have different digits; i.e. 33/33 = 3/3 doesn't count. You should use two-digit or three-digit numbers.
Solution to the Problem:
I had only four or five examples of lucky canceling and I used them in algebra classes when simplifying rationals. But this week's Problem Solvers sent in many more examples for me to use in my classes! Thank you! This problem certainly generated lots of good comments! Below are some of them:
In addition to sending in three fractions that worked, Jeffrey Gaither made the following observations, "You should also add that you must use proper fractions. Otherwise, according to your problem, the answers could be 12/12, 13/13... etc. Also, what of fractions such as 21/126 where both the 2 and the 1 cancel out. Would that be 1/6, or 0/6 or what? If it is 1/6, then 21/126 and 25/125 are additional possible answers."
Good thinking -- I guess I would have had to accept any of those!
Julie (from Dallas) and Richard Johnson sent in two interesting examples:
(1) 166/664 which equals 16/64 (canceling out one 6)
and also equals 1/4 (canceling out two 6's)
(2) 199/995 which equals 19/95 (canceling out one 9)
and also equals 1/5 (canceling out two 9's).
Then Julie sent in additional examples...
1666/6664 = 1/4 as well as 1999/9995 = 1/5, and that if you continue by adding more 6's or 9's, it still
works: 19999999/99999995 = 1/5.
Mikael Wetterholm sent in an excellent analysis:
"I don't know the best method to solve this problem. I use a combination of
'thinking and testing'. Lets try to "reconstruct" the example 16/64=1/4 by
starting with the numerator 16. Then the following equation will help us to
find the correct fraction: 16/(60+x)=1/x The number x must 1) be a integer
and 2) be a one-digit number. We get x=4 and the fraction 16/64=1/4.
Maybe we can find other suitable fractions by start with a two-digit numerator and test it with an equation? I found 19 as a suitable numerator because 19/(90+x)=1/x gives x=5 and we get the following fraction 19/95=1/5. Another one is 26/65=2/5 because 26/(60+x)=2/x gives x=5 and 49/98 because 49/(90+x)=4/x gives x=8."
James Alarie sent in numerous examples:
"There are twelve proper, three-digit fractions which cancel last and
second digits:
139/695=13/65, 146/365=14/35, 149/298=14/28,
149/596=14/56, 179/895=17/85, 183/732=18/72,
186/465=18/45, 216/864=21/84, 346/865=34/85,
349/698=34/68, 386/965=38/95, 427/976=42/96
and nine which cancel second and first digits:
138/345=18/45, 162/648=12/48, 163/652=13/52,
168/672=18/72, 193/965=13/65, 197/985=17/85,
394/985=34/85, 491/982=41/82, 493/986=43/86"
John Funk also sent in over 20 solutions!
Correctly solved by:
| 1. Rich Murray | Ridgetown, Ontario, Canada |
| 2. Jeffrey Gaither | Winchester, Virginia |
| 3. Julie | Dallas, Texas |
| 4. Mikael Wetterholm | Danderyd, Sweden |
| 5. Richard Johnson | La Jolla, California |
| 6. John Funk | Ventura, California |
| 7. James Alarie | University of Michigan -- Flint Flint, Michigan |
| 8. Sang Lee | Harrisonburg, Virginia |
| 9. Kathryn Harris | Winchester, Virginia |
| 10. Bella Patel | Harrisonburg, Virginia |
| 11. Helna Patel | Harrisonburg, Virginia |
| 12. Dave Smith | Toledo, Ohio |