Reducing Fractions, the easy way. Or, 26/65 = 2/5

by Jonathan Wellons

I love fractions because they always cancel so nicely. It's well known that when you have an expression like xy/yz, you can cancel the common components. In this case, there's a y in the top and bottom, so you can just eliminate them both and reduce down to x/z.

Need a concrete example? Suppose I have 26/65. The 6's go away and we have 2/5. Check me on a calculator, 2/5 = 26/65 = 0.4.

Not convinced? Take 4784/7475. The 7's and 4's cancel and we get 48/75.

Another example picked at random: 6188/1820, here we can eliminate the 1 and an 8, to get it down to 68/20, which is a lot simpler. Or I'll take another one, say 4277/2730. Let's see, there's a 2 and 7, so we get 47/30. You might think this is some sort of trick, but Math really is this easy. Let's make up some more numbers to prove it: 138/1840, cancel the 1 and 8 and we get 3/40.


2006-09-16 14:31:46
I tried this with 32/35, and it doesn't work. I want my money back.

BTW, in your last example, you want to cancel 1 and 8

Jonathan Wellons
2006-09-16 14:34:50
Oh, good call. Fixed. With anything this easy, I get careless sometimes.
2006-09-16 15:02:51
What about 23/38?
Jon B
2006-09-16 15:17:59
In the case of numbers like 181/818 which 8 and 1 would be removed? is there a rule for removing always from the right or left or something?
2006-09-16 15:27:00
When I do 12/24, and remove the 2's, I get 1/4 (which is wrong). Is there some way to know it *isn't* going to work?
2006-09-16 17:40:18
This has to be a joke. 26/65 isn't xy/yz, it's x/y. xy/yz would be 2*6/6*5 which is reducible by eliminating the y in the numerator and denominator. According to your logic 22/23 reduces to 2/3, which it most certainly doesn't.
Jonathan Wellons
2006-09-16 17:43:28
Thank you Daniel. You're right on.

It's actually very challenging to find examples that work. I had to write a computer program that examined millions of fractions to find enough good ones to use in this entry.

2006-09-16 17:47:37
Ah, so it was a test. Can I get a cookie? :)
Jon B
2006-09-17 07:34:08
just maybe somebody has a little too much free time.

In justification of everyone (including myself) that were willing to believe there was something in it maths does have some very cool 'tricks' that aren't - however I think everyone highlighted flaws in the logic in their posts - maybe next time you can post some cool 'tricks' that work ;)

2006-09-18 10:46:03

Ah, yes. Deceptively simple. How easy it is to forget that in the example 26/65 the 6 in the numerator is in the "ones" position while the 6 in the denominator is in the "tens" position. I love these types of things... :)

Jonathan Wellons
2006-09-18 12:19:19
Jon B,

I was conflicted about posting it. In the end, I figured it was okay since trying even one example will show that it completely fails (unless you're phonomenally unlucky).

Too much time on my hands? I wouldn't have it any other way :)

2006-09-18 18:14:48
Oh boy, like we don't have enough crap on the net as it is! I just hope my boy doesn't google this while trying to do his homework!
Jonathan Wellons
2006-09-18 22:53:29
I wouldn't worry Abed, I don't think anyone will use this without testing it first.
2006-09-19 05:11:09
Cool! But don't tell my 16-year old daughter. She has enough trouble the old way ;-)
Josh Peters
2006-09-19 09:16:11
This post would have made for a refreshing change of pace come April 1.
Kurt Cagle
2006-09-20 10:32:07
This was the same method that the Indiana State Legislature used in 1880, legislating that pi was equivalent to 3 for all intents and purposes, and that as such 3 should be used.
2006-09-20 12:53:50
The four proper fractions with two-digit numerators and denominators are as follows:

16/64 = 1/4

19/95 = 1/5

26/65 = 2/5

49/98 = 4/8 ( = 1/2 )

These are pretty easy to find.

In an article for the Two-Year College Mathematics Journal, Ralph Boas refers to this as "anomalous cancellation".

Jonathan Wellons
2006-09-20 13:05:41

Cool! I googled Ralph Boas and found a little more info here:

Before I posted the entry I was really hoping to find a more comprehensive list (in base 10). I tried googling the ones I came up with but I had no luck. If anyone knows of such a list, please post a link to it.

2006-09-27 10:45:43
Very neat.
2007-01-04 14:50:02
what is 26 out of 3 and 12 5 16 5 and 4out of 3

2007-02-15 14:35:01
I've taken an 8th and 9th grade math test twice and failed. I know me and fractions and decimals have a fight every time.
2007-02-16 07:18:33
How do you know what numbers to council out?
Jonathan Wellons
2007-02-16 11:05:08
Dear Denise,

The idea behind the post is that you can cancel any numbers that are the same. But, as was revealed, the rule is completely made up.


2007-04-15 12:57:53
U SUCK!!! i was looking for an easy answer and do u know wat u did? u just confused me more but thanx for trying (NOT)
Jonathan Wellons
2007-04-15 13:12:10
Dear Anonymous,

Well, there were definitely pros and cons to posting this entry. I don't believe that the cons outweigh the pros, but I'd be happy to entertain the argument, if someone would care to make it.

2008-01-30 17:53:34
how dose 21/25 work?
Jonathan Wellons
2008-01-31 07:44:34
Dear j,

For 21/25, it turns out that you can cancel every digit other than 2. For instance, cancel all the 3's that appear in both the top and the bottom and it will give you the correct answer.

Let me know if you have any more questions,

2008-03-11 11:53:40
I hate fractions because I reducing them can be VERY HARD!

do you know a good website to get better at it.

Jonathan Wellons
2008-03-11 12:01:35
Dear hannah,

Post the fractions you would like to reduce here (or sample fractions if you don't have a fraction in particular but are just trying to learn), and we'll see if we can figure it out by applying the rules.


2008-07-20 07:04:03
I can under stand how simple and easy it works out, like when I tried it with 14/45, I got 1/5. But what I don't understand is how you get33/49 to be 1/7, and 27/22 to be 9/1. It doesn't work out that simple. Is the trick you're talking about only dealing with numbers that have a common number in it?
Jonathan Wellons
2008-07-21 15:24:27
Dear Kit,

I don't know where your last two examples come from -- I don't see that anyone has used them on this page. However, I'm glad that you're finding the rule works out for you!

Jonathan Wellons