How to Decompose Any Fraction In Seconds! (LCM Math Trick)

[00:00:00]Anyone who can solve this problem is definitely a top student. Let's look at the problem. We need to decompose 17/30 into a sum of four fractions where all numerators are one and all denominators are different. For problems like this, when we usually solve them, we first write these four fractions with a common denominator of 30.

[00:00:25]When we do it this way, it becomes very straightforward because all I really need to ensure is that these four numbers add up to 17. But how can I make sure that all of their numerators are just one? Well, I just need to simplify the denominator with the numerator. If I want to simplify, the numerator has to be a factor of the denominator. That's the only way we can actually simplify it. So, what exactly are the factors of the number 30?

[00:00:47]Let's go ahead and write this down. So, 30 can be written as 1 * 30. Are there any other ways? Yes, there's also 2 * 15. And then we have 3 * 10. Are there any more combinations? Oh, and there's 5 * 6. I believe that's all of them, isn't it?

[00:01:02]So, when we were picking, I realized I just needed to find four numbers that add up to 17. But after trying for a long time, I couldn't find them. So, are we just going to give up on this problem? Wait, we have other methods.

[00:01:13]For example, I can expand this denominator, making it 60. Okay, if the denominator is multiplied by two to become 60, the numerator also needs to be multiplied by two to become 34.

[00:01:24]Although they were both scaled up at the same time, the value of the fraction remained unchanged. What's the benefit of doing that? For these four fractions, what I need to find are the ones that have a denominator of 60. Isn't that correct? Let's write it out again. It will then become 34/60.

[00:01:42]I'm looking for four fractions that all have a denominator of 60.

[00:01:50]So, all I need to do is find four numbers that add up to 34. And what do I need to ensure? I need to ensure the numerators are all one. So, they must also be able to simplify with 60. In other words, they are all factors of 60.

[00:02:06]So, for the next step, I'm going to start looking for the factors of the number 60. 60 can be written as 1 * 60, and then we also have 2 * 30, and then 3 * 20. Moving on from there, we find 4 * 15, and then 5 * 12. Are there any more pairs that we've missed? Yes, there's also 6 * 10.

[00:02:27]So, what I'm going to do now is search through this collection of numbers and find four of them that will add up to 34.

[00:02:34]Everyone, pay attention. When we're looking for numbers, we usually start by finding the relatively larger ones. For instance, who could I look for? I could pick this 20, right? And who else could I find? I could also find 10, because these two numbers together immediately add up to 30. Let's write that down. 10, 20. So, after that, I just need to find four more. Which two numbers add up to four?

[00:02:56]I've noticed that one of them is a one, and the other one is a three. All right, so we have one and three. With this in mind, I can now confidently proceed with the process of simplifying fractions.

[00:03:03]For example, if we take the fraction 10/60, I can divide both its numerator and its denominator by 10 at the same time. Won't the entire value of this fraction then simply become 1/6? You see, when 60 is divided by 10, it results in six, and 10 divided by 10 is just one. Isn't that right?

[00:03:16]And when you're working on this, there's actually a pretty clever method you can use. Now, everyone, think about this. If I want to divide 60 by 10, don't I just need to look right here? Well, 60 divided by 10 simply leaves us with six, doesn't it? All right, so what if it's this next? The denominator and numerator both need to be divided by 20. Both divided by 20. I'll just look here. 60 divided by 20, doesn't that just leave us with three? So, when 60 is divided by 20, the denominator becomes three, and the numerator is one.

[00:03:39]More. No further simplification is needed. It's directly 1/6, then divide both top and bottom by three to get 1/2.

[00:03:46]This problem is done. I suggest you all watch this to the end.