Last time we examined the basic concept of equivalent fractions – the fact that different fractions can represent the same value. We saw that there will be one way to write a fraction that is “in lowest terms” – no other fraction with the same value will involve smaller numbers, and all the others can be obtained by multiplying the numerator and denominator by the same number. Now we’ll get to the matter of “reducing a fraction to lowest terms”, or “simplifying” it.
We’ll start with this 1998 question that gives us the big picture:
Tips for Simplifying Fractions I have no idea how to reduce or simplify a fraction. Could you please send me the solution to how to reduce or simplify a fraction (like 2 and 4/5 or 4/5)? It would help a lot. Please send me an example and how to do it.
I answered, starting with some encouragement:
Being able to reduce fractions is an important skill, so you're right to want to get it clear before you move on too far. I'm glad you've asked for help. I'll try to give you a quick summary of how to do it. If these examples aren't enough, don't hesitate to ask your teacher for some extra help. Sometimes all it takes is a chance to ask your questions privately and give a teacher a chance to point out one little step you might be missing. Or you could send us a specific problem and show us what you've done and where you're stuck, and we can help you figure out what you need.
A lot of students seem to get stuck around the time they learn fractions, and decide they’re bad at math because they struggle with the concepts. It’s important to get all the help you can – you’re not alone!
The two examples you gave actually don't need reducing. 2 4/5 is a mixed number, which is equal to the improper fraction 14/5, and there's nothing you can do to simplify it. And 4/5 is already as simple as it can get.
Sometimes this happens – the student’s examples are too simple to learn from! So we move on …
I started with the basics we looked at last time, about equivalent fractions:
The basic idea of reducing or simplifying fractions is that two fractions are the same if the numerator and denominator are multiplied by the same thing, because that's just a way of multiplying the fraction by one. For example: 4 8 - = -- 5 10 because: 4 4 2 4 * 2 8 - = - * - = ----- = -- 5 5 2 5 * 2 10
That’s the key idea behind writing a new fraction with the same value. But there’s more.
Dividing by the Greatest Common Factor (GCF)
Simplifying means find the simplest fraction that is equal to the one you're given. That's useful because it can save a lot of work if you have to do more with the fraction - you'll have smaller numbers to work with. You could say you're trying to find a small fraction (one with small numerator and denominator) hidden inside the given fraction. If you take my example above in reverse, you can see that if you are given 8/10, you have to recognize that 8 = 4*2 and 10 = 5*2, then divide both by 2.
So if you see that you can divide the numerator and denominator by the same number, do it! Notice that dividing each number by 2 is the same as removing a factor of 2 from both.
Let's take a harder example. If I'm given 36/54, I have to do something like this: 36 18*2 18 2 2 -- = ---- = -- * - = - 54 18*3 18 3 3 The hard part is how to find the Greatest Common Factor (or Divisor) of the two numbers, which is that 18 that appeared magically in what I just did. For some problems, you might just happen to see that 18, and you're almost done. I didn't. I'll show you what I actually did a little later. The important thing to realize is that you don't have to be a whiz at this to get it done.
This time I recognized that both numbers are multiples of 18, so I could write each as a multiple of 18 and removed the 18’s, or equivalently divide each number by 18. (How to find the things to divide by is the subject of most of the rest of this post, so don’t worry if you didn’t see that 18!)
Factoring completely as a product of prime numbers
Here’s one approach to finding factors to use: find all of them at once:
Some people will very carefully factor each number completely, turning each one into a product of prime numbers, and then match up any factors that appear in both numbers and cancel them: / / / 36 2*2*3*3 2*2*3*3 2 -- = ------- = --------- = --- 54 2*3*3*3 2 *3*3*3 3 / / / (In case that confuses you, canceling factors really means something like this: 2*2*3*3 2*2*3*3 2 2 3 3 1 1 2 ------- = --------- = - * - * - * - * - = 1 * 2 * 1 * 1 * - = - 2*3*3*3 2* 3*3*3 2 1 3 3 3 3 3 so that any factor on both the top and bottom turn into a 1.)
Here I typed a slash above each number I wanted to cancel; really we’d draw a slash through those numbers. Then I ignore them when I finally multiply the remaining factors together again.
For an example of how to find those long factorizations, see
Dividing by one factor at a time
This way of doing it makes the answer very neat, but it can be a little intimidating, because it takes some practice to completely factor a number. The fact is that you don't really have to do it that way. Often some factors jump out at you, but others hide better. You can just cancel out whatever common factors you do see, then go back to looking for more factors. In this example, I saw immediately that both numbers are even. I looked again and recognized that both are multiples of 6. So I divided each number by six, making the problem a lot simpler: 36 6*6 6 -- = --- = - 54 6*9 9 But I'm not finished yet. I look again, and I see another common factor, 3: 6 3*2 2 - = --- = - 9 3*3 3 So each time I found a number that divides both numbers evenly, I divide it out and keep looking. I never actually found the GCF, but eventually removed all the factors.
The important point here is that you don’t have to see everything at once; you can just do whatever you see. Each time you make the numbers smaller, you can see better and take another step.
Now, even what I said was "obvious" may not be obvious to you. It takes a lot of practice to recognize factors. One thing you can do to make it easier is just to "play" with fractions. If you get used to them, they'll become friends, and friends like to help you out when they can! What I mean is, if you take simple fractions and try to make them complicated by multiplying the numerator and denominator by something, you'll get used to what fractions that can be simplified look like. It may be very helpful that you're working with ratios, because a lot of ratio problems are really just fraction problems in disguise. If you see that the ratios 6/5 and 30/25 are the same, look at them and think, what does that tell me about simplifying 30/25?
This idea of playing with fractions by making up your own problems (by making small ones complicated and then trying to simplify them) is very much like the way kids learn by playing. Here, you’re playing hide and seek with a fraction, telling it to hide, and giving you a chance to practice finding it!
Here is a question from 1997 with several examples (only one of which turns out to be interesting):
Reducing Fractions I'm having trouble reducing fractions. I need to reduce 18/35, 35/48 and 6/20. Can you help?
Doctor Mike answered, with a focus on the need to understand, rather than just follow rules:
This is an interesting bunch of problems. Let's start with the last one. You will soon see why I want to do it that way. There are some tricks sometimes used for reducing fractions, but once you understand what is going on, you will not need them. Ten years old is a perfect age to gain this understanding. The numerator of 6/20 is six, which factors into 2*3. That is, 2 times 3 equals 6. The denominator of 6/20 is 20, which factors into 2*10. That is, 2 times 10 equals 20. Let's remember this.
So far, we’ve seen that 2 is a factor of both the numerator and the denominator.
Canceling is “un-multiplying”
I hope you have already seen the easy and very natural rule for multiplying fractions. You multiply the two numerators together to get the new numerator, and you multiply the two denominators together to get the new denominator. In symbols, it looks like this: A X A*X --- * --- = ----- B Y B*Y This shows how to multiply 2 fractions together. BUT ALSO, it shows how to "UN-multiply" if you want to do that. What I mean is that if you have a fraction like (A*X)/(B*Y), then this rule tells how you can change it to A/B times X/Y.
So we can pull a fraction apart into a product, which will be useful.
Okay, back to 6/20 and how we are going to reduce it. Remember: 6 2*3 ---- = ------ 20 2*10 I am going to use the rule for multiplying fractions to continue working with 6/20 like this: 6 2*3 2 3 3 3 ---- = ------ = --- * --- = 1 * --- = --- 20 2*10 2 10 10 10 What I did was to use the fraction multiplication rule, then change 2/2 to 1, then do the multiplication by 1. This shows EXACTLY WHY you can reduce 6/20 to 3/10.
So what’s happening when we remove a common factor? We’re pulling the fraction apart into a product of 1 with a simpler fraction.
It is important for you to know why, because if you really understand something you will never forget it. Some people say that about riding a bicycle, that if you learn how but then don't do it for 30 years you will still know how. Same thing about really understanding fractions and factoring numbers. If you really understand the ideas, you will still know it when your grandchildren are ten years old.
Really knowing means never forgetting.
Recognizing when it can’t be reduced
How about those other two examples? There’s a reason for putting them off:
Let's take a look at another one of those problems of yours, say 18/35. We want to factor the numerator and denominator to see if we can find the same number as a factor in both places. That is how we did the 6/20 problem; we found a 2 in both places: 18 2*3*3 ---- = ------- 35 5*7 I have factored the numerator and the denominator both as much as I possibly can. A number like 2 or 3 or 5 or 7 that cannot be factored any more is called a prime number. As you can clearly see, there is no prime number factor that is both on the top and on the bottom. That means that the fraction 18/35 is already in lowest terms. You should try the other one 35/48 and see if it can be reduced.
When you try and try and can’t simplify a fraction (that is, can’t find any common factors), it’s time to switch gears and aim for a new goal: to show that it is already in lowest terms – that is, it can’t be simplified. The best way to do that is to completely factor each number (down to a product of primes). If there were any common factors, you would find a common prime factor; if not, you know you are done.
Here's another practice problem that is a little harder. See if you can work with 130/143 to reduce it. Good luck.
Have you found the common factor yet?
GCF … or not
Here is a question from an adult in 1996:
Reducing Fractions I was placed in ESE from Pre-k through grade 12 and they never had me learn to reduce fractions. What is the easiest method of reducing fractions to lowest terms that you know? Greg
“ESE” appears to be Exceptional Student Education; presumably Greg wasn’t expected to need fractions.
Doctor Byron answered, first describing the use of the GCF:
In order to reduce a fraction, you need to find the biggest number that divides into both the numerator (the top number) and the denominator (the bottom number). After some practice, you may find that you can do this pretty quickly. At first, though, it is often helpful to list all the factors and simply look to see the biggest one shared by both numbers. Factors, by the way, are simply any number that divides into another number evenly. Let me give you an example: Let's look at the fraction - 48 ------ 32 The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 The factors of 32 are: 1, 2, 4, 8, 16, 32 If you look at both lists you'll see that 16 is the biggest number that is on both lists. So to reduce the fraction, we divide the top and bottom by sixteen. 48/16 3 ------- = ----- 32/16 2 If you find that there are no numbers (other than one) which are shared factors of both, then the fraction is as simple as it can get.
It can be difficult to find the GCF for larger numbers, but for reasonably small numbers this way (based directly on the definition) is sufficient. We’ll be looking at GCF soon in a little more depth.
But, as we’ve seen already, the GCF isn’t really needed, so it isn’t worth working too hard at it:
Also, you don't have to simply look for the biggest factor right away. If you notice a shared factor quickly, you can go ahead and divide by it right away. You will have to make sure that there aren't any factors left, though. Let's look at the same example again: You may have noticed that both 48 and 32 are even, so it's pretty obvious that they both share two as a factor. So you could start by dividing both by two: 48/2 24 ------ = ---- 32/2 16 Now you might notice that 24 and 16 are both divisible by four, so you can go ahead and divide again: 24/4 6 ------ = --- 16/4 4 Finally, we see that these are even and can be divided by 2 again: 6/2 3 ----- = --- 4/2 2
This is often the easiest way, as Greg requested.
What if the numbers are big?
In 2004, we got this question:
Reducing Fractions with Large Numbers If you reduced 32/48 to lowest term, what would the answer be? Those are big numbers, which makes it hard for me.
Hi, Sonya. Yes, they're big numbers; but you can reduce the fraction bit by bit rather than having to do it all in one step--it's a lot like dieting! That makes the problem a lot less scary. Let's take a worse example: 280/588
I chose an example where even I can’t see the GCF easily. But I can see something to do!
What we want to do is to divide the numerator and denominator by a common factor. Each time we do that, we are reducing the fraction; eventually we will reach a point where we can't do it any more, and then we're done. So we can start with any obvious common factors (such as 10, if they both ended with 0); otherwise, just start with the smallest prime number, 2, and see if it goes into both numbers. In this case, it does; they're both even. So we divide 280 by 2 and get 140; we divide 588 by 2 and get 294. Now we've reduced the fraction to 140/294.
What we have here is a double strategy: do whatever you readily see; and if you don’t see anything, just systematically try dividing by small primes.
Then we do the same thing again! Both numbers are still even, so we can divide them again by 2. (That's important: after you've divided by one prime number, you have to check the same prime again, because you may not be done with it.) So we get 70/147. Now only one number is even, so we move on to the next prime number, 3. Neither number is divisible by 3, so we move on to 5, which only divides 70; and then to 7, which does divide both numbers. Dividing by 7, we get 10/21. Now we don't need to try any more primes, because the only factors of 10 are 2 and 5, and we've already tried them. We know we're done when we can try all the prime factors of one number, and they aren't factors of the other. So we've reduced a really big, ugly fraction, 280/588, to 10/21, without ever having to divide by a big number. The weight came off in manageable bits.
This strategy will let us eventually simplify even the worst fractions. But there’s something more we can do to speed things up in a really extreme case …
But what if they’re REALLY big?
There are techniques we don’t teach to kids, that help when the numbers really get out of hand. Take a look at this question from 1998:
Reducing Fractions with Large Numbers Dear Dr. Math, A neighborhood child asked for help with a math problem involving reducing fractions. The numbers involved were relatively high (i.e. 1742/4395). Is there a "trick" that would allow me to go through the least amount of possibilities before declaring that 4,5,6,7,8,9, digit number fractions may or may not be reduced?
Doctor Rob answered:
Indeed, there is a trick, and it is never taught at the elementary school level. It is called "Euclid's Algorithm". The idea is to find the largest common factor of the numerator and denominator. It goes like this. Take the larger of the two numbers, and divide it by the smaller, getting a quotient and remainder. If the remainder is zero, the answer is the smaller of the two numbers. If the remainder is not zero, replace the larger number by this remainder, and repeat the above. Example: 52740 and 20904. 52740 = 2*20904 + 10932, replace 52740 by 10932. 20904 = 1*10932 + 9972, replace 20904 by 9972. 10932 = 1*9972 + 960, 9972 = 10*960 + 372, 960 = 2*372 + 216, 372 = 1*216 + 156, 216 = 1*156 + 60, 156 = 2*60 + 36, 60 = 1*36 + 24, 36 = 1*24 + 12, 24 = 2*12 + 0, so the answer is 12, and 52740/12 = 4395, 20904/12 = 1742, so, reduced to lowest terms, 20904/52740 = 1742/4395. This always works, and always gives the largest common factor. Furthermore, as you can see from the example, it doesn't take very many steps. In the worst case it takes five times the number of digits in the smaller number, but usually less than half that many.
In this case, we would have divided by 12 after three steps of my strategy above; but then it might take a while to be sure we couldn’t do any more. This method saves that work. We will have done 11 long divisions (or the equivalent on a calculator), then two more at the end to simplify the fraction.
For more on this method, see