A Math Puzzle

Here’s a little brainteaser for you. What do these four fractions have in common?

$\dfrac{16}{64} \phantom{xxxx} \dfrac{19}{95} \phantom{xxxx} \dfrac{26}{65} \phantom{xxxx} \dfrac{49}{98}$

As it happens, these are the only four fractions where the top and bottom are both two-digit numbers that have this property. Feel free to leave suggestions in the comments. Good luck!

1. #1 Jason
January 4, 2014

Very clever! I’d rather not spoil it for everyone else, but I will say that anyone who teaches math has had a student or two who tries to work with fractions this way. Interesting to see that there are times when they would accidentally be right!

2. #2 DonM
January 4, 2014

I won’t ruin it either, but I am happy to say I got it quickly. I usually have to think long and hard, and then read someone else’s answer.

3. #3 Shecky R
January 4, 2014

I won’t ruin it either, but interesting that these are the ONLY 4 fractions where it works…there must be a simple way of proving that (other than brute force technique), but don’t immediately see it??

4. #4 Robert
California
January 4, 2014

$\dfrac{14266}{26695}$, too.

5. #5 Max
January 4, 2014

Why would it be “ruining it”? Anyone looking in the comments would be in search of the answer.

6. #6 Chuck
January 5, 2014

ok I will ruin it.

$\dfrac{10a+b}{10b+c} = \dfrac{a}{c}$

where a, b, c are single digit numbers.

7. #7 Jason Rosenhouse
January 5, 2014

I think we’re far enough down into the comments now that no one will see the solution by accident. The idea is that if you take a very naive approach to fraction reduction and just “cancel” any digit that appears in both the top and the bottom of the fraction, you will nonetheless get the right answer. So, if you cancel out the 6’s in 16/64, you get 1/4 which is correct. Likewise for the other fractions.

Incidentally, when I said those were the only four fractions that have this property I had in mind cancelling out the ones digit on top and the tens digit on the bottom. If you’re a little looser you can do some other things. For example, if you allow zeros at the end of the number you can do things like 20/30=2/3. And if you allow fractions that are equal to one then you can do things like 37/37=3/3=7/7. Somehow, though, those examples are not really in the spirit of things.

Chuck–

I took the liberty of formatting your comment for you.

Robert–

I think your example is not quite the same thing. If I cancel out the 6’s on the top and bottom it does not work. But it seems like it does work if I just cancel out the last two digits on the top and bottom. Interesting, but not quite the same thing.

8. #8 Robert
January 5, 2014

Hi Jason,

In my example, you cancel *all* the common digits. So cancel out “266” from top and bottom, and it works.

About 10 years ago I wrote a little Mathematica program to find these types of fractions where if you cancelled the same number from top and bottom, you get the same value. There’s lots! But, as you pointed out, there’s only the four you identified with 2 digits on both top and bottom.

9. #9 Jason Rosenhouse
January 5, 2014

Ah! Thanks for the clarification.

10. #10 stephenk
January 5, 2014

Robert.
Your example doesn’t work.
If you “cancel” the 266 the fraction is 14/95 approx 0.14
Using all digits is approx 14k/27k approx 0.5.

There must be a digit mistyped in there somewhere

(you confused my by throwing in that 5 digit example)

New comments have been disabled.