How to solve complex fraction sums instantly!🧠#maths #education #learning #mathstricks #solution #yt

Ajouté : 2026-05-28

[00:00:01]Here's a tricky final exam question from Shaoji, and it really stumped quite a few top students. Let's take a look at the problem. The question asks us to calculate 1 * 3/5 + 5 * 3/9 + 9 * 3/13 + 13 * 3/17 + 17 * 3/21.

[00:00:19]Just reading the question takes a while.

[00:00:21]A lot of students, upon seeing these fractions with different denominators being added, might think, "Shouldn't we just find a common denominator?" But first of all, finding a common denominator here would be really painful. But for problems like this, there's actually a clever way to solve them. Let's take a closer look.

[00:00:38]We notice that the denominators of these fractions are all written as the product of two numbers, right? And secondly, the numerators are all three.

[00:00:48]Next, what is the difference between the two numbers being multiplied in the denominators? It's always four. There seem to be some common features here, right? So, how can we make use of these observations? Actually, everyone needs to be familiar with a certain type of fraction. Once you're familiar with this type, you'll be able to solve this problem. What kind of fraction is it? We call it denominator product numerator difference.

[00:01:15]All right, let me explain what denominator product numerator difference means. For example, let's look at a fraction like this.

[00:01:23]5 * 4 over 9. So, what does this mean?

[00:01:28]You see the denominator is in the form of two numbers multiplied together, and the numerator, four, is actually 9 - 5.

[00:01:35]Isn't that just right? 9 - 5 = 4.

[00:01:39]So, it can be written as 5 * 9 over 9 - 5.

[00:01:44]Next, let's break it down a bit further into the form of two fractions being subtracted from each other.

[00:01:49]Look, that would be 5 * 9 over 9 - 5 * 5 over 9, right?

[00:01:54]This way, you can more clearly see how the original expression is split into two separate fractional terms. All right, next let's look at the numerator and denominator. If we divide both by nine, we can cancel out the nine, and what's left is 1/5, right? Now, if we divide both the numerator and denominator by five, we can cancel out the five, right? This becomes one, this becomes one. So, isn't that just 1/9?

[00:02:19]So, you see, if a fraction satisfies the condition where the denominator minus the numerator equals the numerator, it can be broken down into the form of two unit fractions being subtracted.

[00:02:30]Let's take a look at this part.

[00:02:33]For each of these fractions, even though the denominator is written as the product of two numbers, it's still the denominator, right? It's a product. But, does the numerator match the difference?

[00:02:43]Our difference here should be 5 minus 1, 9 minus 5, and 4, right? Since it doesn't satisfy the condition, I'll first write it in a way that does. Let's write it out. 1 * 4 over 5 + 5 * 4 over 9. Take a look, isn't it true that all the ones written by the teacher satisfy the condition where the denominator is the product and the numerator is the difference? Next, 9 * 4 over 13 + 13 * 4 over 17 + 17 * 4 over 21. Why do we need to add parentheses?

[00:03:17]Because are the numbers inside here the same as the ones above?

[00:03:22]They're not the same, so you need to keep it consistent with the ones above.

[00:03:26]So, in order to change the numerator from four to three, don't I need to multiply by three and then divide by four? Do you understand that? Multiply by three to put it back and divide by four to remove it. And what does this multiplying by three and dividing by four amount to? Isn't it basically the same as multiplying by 3/4? Can you understand that? Try to grasp how multiplication of fractions works. All right, next let's break up the parentheses and split each fraction into two terms in the form of a difference of fractions. Can you do the first one?

[00:04:00]How can you split it? Split it into 1 minus 1/5, right? Okay, now for the second fraction, how should it be split?

[00:04:09]Plus 1/5 minus 1/9, right? If you forgot, take a look here. All right, let's keep going.

[00:04:16]Plus 1/9 minus 1/13 plus 1/13 minus 1/17.

[00:04:20]Isn't each term corresponding to the ones above? All right, moving on. Plus 1/17 then minus 1/21. And after that, it's multiplied by 3/4. Can you understand that? Okay, now everyone pay close attention to the whole problem.

[00:04:36]Minus 1/5 plus 1/5 minus plus minus plus. What I just crossed out are all terms where you subtract a number and then add it back, so doesn't that just equal zero? Now look at what's left inside the parentheses. What's left is just 1 minus 1/21. What's inside the parentheses? Isn't it 20 over 21? And don't forget, what's still left at the end? There's still a multiplication by 1/4. Now let's simplify by canceling out. Divide both by three, this becomes one, this becomes seven. Now divide both by four, this becomes five, this becomes one. So the final result is 5/7, right?

[00:05:16]All right, that's how we finish this problem. I suggest everyone give it a like and save it to watch a few more times. Otherwise, if you want to use such a handy method again, you might not be able to find it.