There are plenty of webpages and videos that go through the standard textbook motions for reducing fractions and finding common factors — prime factorization, factor trees, factor “U”s, and maybe a few others as well. And maybe you (the teacher), you (the parent), you (the student), or you (the self-study individual) have gone over these methods and just don’t get them. Or maybe you’re just looking for another way.
Look no further! Let’s start off with a cryptic example.
Example #1
How do you reduce $$\frac{60}{24}$$
Before I explain what’s going on, see if you can follow the pattern here (and notice I’m using arrows not equal signs)
$$\frac{60}{24} \rightarrow \frac{36}{24} \rightarrow \frac{12}{24} \rightarrow \frac{12}{12}$$
and now we know that \(12\) is a common factor between \(60\) and \(24\). Hooray! No factor trees required, no division required. Just some possibly strange subtraction. And so $$\frac{60}{24} = \frac{5}{2}$$ once we divide \(60\) and \(24\) both by their common factor of \(12\)
Let’s do another example.
Example #2
How do you reduce $$\frac{84}{119}$$
Looks tricky! Let’s try this again. See if you can pick up the pattern. It’s ok if it doesn’t make sense.
$$\frac{84}{119} \rightarrow \frac{84}{35} \rightarrow \frac{49}{35} \rightarrow \frac{14}{35} \rightarrow \frac{14}{21} \rightarrow \frac{14}{7} \rightarrow \frac{7}{7}$$
And our common factor is \(7\) which means that $$\frac{84}{119} = \frac{14}{17}$$ once we divide the original numerator and denominator by \(7\).
One more?
Example #3
How do you reduce $$\frac{7}{2}$$
You may already know by inspection that this fraction can’t be reduced, but let’s do the same thing we did in the above two examples.
$$\frac{7}{2} \rightarrow \frac{5}{2} \rightarrow \frac{3}{2} \rightarrow \frac{1}{2} \rightarrow \frac{1}{1}$$
And from this we see that the common factor is \(1\) which means that our original fraction was already in fully-reduced form.
How does this work?
The method is straightforward, keep subtracting the smaller term from the larger term and create a new fraction retaining the smaller term and the subtracted off larger term. Keep doing this until both terms are the same. Whatever that number, that’s our largest common factor.
What’s the intuition?
Let’s take \(60\) and \(24\). In the illustration below, if we keep subtracting 24 from 60, we’ll do it twice with 12 left over. Then, we want to see how many times the 12 goes into 24. It goes in twice with no remainder. And what this means is that we can “chop up” 60 and 24 by the same amount! And that amount is 12! And 12 is the biggest common factor. Now, any number that divides 12 with no remainder (less formal phrasing is “goes in evenly”), also divides 60 and 24. So, 2, 3, 4, and 6 all are common factors of 60 and 24 as well. But 12 is the largest one.
Click on the image below to see more details.
Can you see that all we are really doing is trying to find remainders between the two numbers? And once we find that there is no remainder we find our greatest common factor. In this way, we don’t have to first factor both our numbers and inspect for the greatest common factor. Rather we can just continue with this procedure of finding remainders.
What’s nice about this is that if you have a computer program, they may not always have a built-in “find the greatest common factor” function, but they do often have a way to find remainders.
Let’s take a quick look to see if we can find the greatest common factor between 24688642 and 1357997531. Looks like challenge!
Here’s how a computer program would do this.
24688642 1357997531
122221 24688642
0 122221
The greatest common factor is 122221
Let’s look at 84 and 119.
84 119
35 84
14 35
7 14
0 7
The greatest common factor is 7
What about two large numbers?
4159558231249136722906328 92237040066437027769835738064600023984384685186604
41264745900528713278268 4159558231249136722906328
33083641196265395079528 41264745900528713278268
8181104704263318198740 33083641196265395079528
359222379212122284568 8181104704263318198740
278212361596627938244 359222379212122284568
81010017615494346324 278212361596627938244
35182308750144899272 81010017615494346324
10645400115204547780 35182308750144899272
3246108404531255932 10645400115204547780
907074901610779984 3246108404531255932
524883699698915980 907074901610779984
382191201911864004 524883699698915980
142692497787051976 382191201911864004
96806206337760052 142692497787051976
45886291449291924 96806206337760052
5033623439176204 45886291449291924
583680496706088 5033623439176204
364179465527500 583680496706088
219501031178588 364179465527500
144678434348912 219501031178588
74822596829676 144678434348912
69855837519236 74822596829676
4966759310440 69855837519236
321207173076 4966759310440
148651714300 321207173076
23903744476 148651714300
5229247444 23903744476
2986754700 5229247444
2242492744 2986754700
744261956 2242492744
9706876 744261956
6539380 9706876
3167496 6539380
204388 3167496
101676 204388
1036 101676
148 1036
0 148
The greatest common factor is 148