Japanese | Can you solve this ? | Math Olympiad a + b + c = ?

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[00:00:01]In this video, let us solve for a plus b plus c given ab is 40, bc is 50, and ac is 60.

[00:00:24]Let us call this equation one.

[00:00:28]This equation two.

[00:00:31]This equation three.

[00:00:35]Then I'm going to multiply equation one by equation two by equation three to give us ab times bc times ac equal to This will be 40 * 50 * 60.

[00:00:59]So, I have 40 * 50 * 60.

[00:01:09]On the left, we have a * a here a squared.

[00:01:14]b * b here b squared.

[00:01:18]c * c here c squared.

[00:01:22]Then 4 * 5 * 6. 4 * 5 is 20.

[00:01:27]20 * 6 is 120.

[00:01:30]So, that'll be 120 * Since we have three zeros, 1,000.

[00:01:38]Then we have by law of indices, this can be written as abc raised to power two is equal to 120,000.

[00:01:52]Then I'm going to take the square root of both sides.

[00:01:57]Here we have square root of abc raised to power two equal to the square root, positive square root of 120 thousand.

[00:02:13]This here takes care of this. So, we have abc is equal to We can break this down into square root of 12 times 10,000.

[00:02:37]So, a * b * c is equal to square root of 12 * square root of 10,000.

[00:02:53]So, we have abc is equal to We can still break this into square root of 4 * 3.

[00:03:03]Then here, this will just give us 100.

[00:03:09]So, abc is equal to square root of 4 * square root of 3 * 100.

[00:03:23]So, abc is equal to square root of 4, positive square root of 4 is 2.

[00:03:30]So, we have 2 here * root 3 * 100.

[00:03:38]Then putting it all together, abc will give us 2 * 100 here will have 200 root 3.

[00:03:50]From equation one, we have ab is equal to 40.

[00:03:58]From here, we have ab * c.

[00:04:01]So, this is going to be the same thing as saying ab * c is equal to 200 root 3.

[00:04:15]And we already have a value for ab from equation one.

[00:04:19]So, this will imply 40 * c is equal to 200 root 3.

[00:04:30]I'll divide both sides by 40.

[00:04:37]So, that this takes care of this.

[00:04:40]4 here is 5.

[00:04:43]So, the value for c is 5 root 3.

[00:04:51]I'm going to repeat the same thing for equation two. Equation two, we have bc is equal to 50.

[00:05:00]And we're given Well, we just arrived at abc is equal to 200 root 3.

[00:05:09]So, this is the same thing as saying a into bc is equal to 200 root 3.

[00:05:19]Now, bc is 50.

[00:05:22]bc here will be replaced with 50.

[00:05:28]is equal to 200 root 3.

[00:05:34]So, we divide both sides by 50.

[00:05:41]This takes care of this.

[00:05:43]50 here 50 here is 1.

[00:05:47]50 here is 4.

[00:05:49]So, we have a is equal to 4 root 3.

[00:05:56]So far, we've gotten value for a.

[00:05:59]I've gotten value for b, for c.

[00:06:01]Now, let us do for the third equation, ac is equal to 60 from equation three.

[00:06:12]So, abc is equal to 200 root 3.

[00:06:21]This is ac. This is ac. So, that's saying ac * b is equal to 200 root 3. Now, ac is 60.

[00:06:35]So, we have 60 here * b is equal to 200 root 3.

[00:06:45]To solve for b, we divide both sides by 60.

[00:06:52]60 cancels 60.

[00:06:55]This takes care of this.

[00:06:57]And two here is three. Two here is 10.

[00:07:01]So, we get b is equal to 10 root 3 divided by 3.

[00:07:11]So, now we've gotten value for a, b, and c. The value we got for a just now, before this, was 4 root 3.

[00:07:24]The value we got for b, which is this, is 10 root 3 divided by 3.

[00:07:36]And the very first value we got, which was for c was 5 root 3.

[00:07:44]Therefore our problem, which is to get the sum of a plus b plus c is then going to be 4 root 3 plus b which is 10 root 3 over 3.

[00:08:13]Then plus c.

[00:08:16]c is 5 root 3.

[00:08:24]So, we just added this up. And whatever we get would be the answer for the sum of a plus b plus c.

[00:08:33]This is divided by 3. So, let's find the lcm of all of this. This is divided by 1 divided by 1.

[00:08:43]So, lcm is going to be 3.

[00:08:46]1 in 3 is 3. 3 * 4 is 12. So, here we have 12 root 3.

[00:08:54]3 in 3 is 1. 1 * 10 is 10. So, we have 10 root 3 here.

[00:09:03]1 in 3 is 3. 3 * 5 is 15.

[00:09:07]So, we have 15 root 3 here.

[00:09:13]Then a plus b plus c is equal to root 3 is common to all of this. Let's factor that out.

[00:09:25]Then we have 12 plus 10 plus 15 then divided by 3.

[00:09:41]We have a plus b plus c is equal to root 3 into 12 here plus 10 is 22.

[00:09:56]22 plus 15 That is going to give us 37.

[00:10:03]Then divided by three.

[00:10:06]Which we can write as a plus b plus c is equal to 37 root three divided by three giving us the final answer to this problem.

[00:10:25]Thanks for watching. Please like and share. And also remember to subscribe to my channel. And I'll see you in my next video.

[00:10:33]Bye-bye.