Solving a 'Stanford' University entrance exam

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[00:00:00]Welcome to Smart Math. Today, I'm solving this math Olympic problem in a very crazy way.

[00:00:06]Where our equation is 1 over a plus 1 over b plus 1 over ab is equals to 1 over 5 1 over 4.

[00:00:15]Where a and b have positive integer.

[00:00:18]So, let's find out the solution.

[00:00:21]So, first of all, we all uh we take LCM.

[00:00:26]Take an LCM.

[00:00:28]ab means ab is 1.

[00:00:32]So, we have no a means multiply by 1 into a.

[00:00:37]a have no b means multiply by b.

[00:00:40]into 1 that is equals to 1 by 4.

[00:00:44]Now, I'm taking cross multiplication.

[00:00:47]Cross multiplication mean this equation multiply by 4.

[00:00:52]4 into b plus a plus 1 and 1 into ab.

[00:00:59]So, equation will be 4b plus 4a plus 4 that is equals to ab.

[00:01:10]Now, as we saw or now as we wrote or ab is equals to 4b plus 4a plus 4.

[00:01:25]So, these terms will be all as positive goes opposite sign negative negative.

[00:01:30]This means ab minus 4b minus 4a that is equals to 4.

[00:01:39]So, add both side by 16.

[00:01:43]I'm adding both side by 16.

[00:01:46]Here and here.

[00:01:48]So, equation will be ab minus 4b minus 4a plus 16 is to 20 So, if I'm taking common B here.

[00:02:07]Taking common B that's mean A - 4 and I'm taking -4 here A - 4 that is equals to 20 So this means that that if I'm taking common A - 4 and resultant is B - 4 that is equals to 20 So now as well all possible possible factor of 20 that is 1 into 20 20 into 1 2 into 10 10 into 2 5 into 4 or 4 into 5 So we have six cases.

[00:03:09]So let's find out case number one.

[00:03:13]So recall negative value will be rejected.

[00:03:25]So first we solve case number 1 2 3 4 5 6 six case we have.

[00:03:34]Case number one A - A - 4 into B - 4 that is equals to 1 into 20 so A - 4 is equals to 1 and B - 4 is equals to 20 A - 4 that is equals to 1 So this term positive goes opposite mean negative. A is equals to 1 + 4 where A is equals to 5 and now B - 4 that is equals to 20.

[00:04:07]So, minus goes opposite side, mean positive.

[00:04:10]20 + 4 B is equals to 24.

[00:04:17]So, now case number two.

[00:04:20]Now, case number two. A - 4 is equals to 20 and B - 4 is equals to 1.

[00:04:25]So, A - 4 is equals to 20. Minus here goes opposite side.

[00:04:31]A is equals to 20 + 4 that is equals to A is A 24.

[00:04:38]So, now B - 4 is equals to 1. Minus here goes positive.

[00:04:45]1 + 4 and B is equals to 5.

[00:04:50]Now, case number three.

[00:04:53]So, now A is A - 4 is 2 and B - 4 is 10.

[00:04:58]A - 4 is equals to 2. So, this term is minus goes opposite side, mean positive.

[00:05:05]A is equals to 2 + 4 and A is equals to 6.

[00:05:12]So, now B - 4 that is equals to 10. Minus here goes opposite side, mean positive. B is equals to 10 + 4.

[00:05:22]B is equals to 20 10 + 4 is 14.

[00:05:29]Now, case number four.

[00:05:32]So, case number four. A - 4 is equals to 10 and B - 4 is equals to 2.

[00:05:38]A - 4 is equals to 10. Minus here goes positive.

[00:05:43]10 + 4 and A is equals to 14.

[00:05:49]And now B minus four that is equals to two.

[00:05:53]Minus here goes opposite side mean positive. B is equals to two plus four and B is equals to six.

[00:06:02]B is equals to six. Now case number five.

[00:06:06]Case number five, A minus four is five and B minus four is four.

[00:06:11]A minus four is equals to five. So this time minus goes opposite side mean positive. A is five plus four.

[00:06:20]A will be nine.

[00:06:23]And now B minus four is equals to four.

[00:06:27]So B here minus goes opposite side mean positive. Four plus four and B is equals to eight.

[00:06:37]So now case number six.

[00:06:40]Now case number six, A minus four that is equals to four. Minus here goes positive.

[00:06:49]Positive four. A is equals to eight.

[00:06:53]And now B minus four that is equals to five.

[00:06:57]Minus here goes on positive. Five plus four that is equals to nine.

[00:07:06]So I can conclude that the value of our first two cases is five and A is five and B is 24. So this is symmetric.

[00:07:22]Case number one and two. One and two case.

[00:07:28]So three and four cases where the the value of A is six and B is equals to 14.

[00:07:37]And we will also call the symmetric.

[00:07:42]Five and 6 is value of A that is 8 and value of B that is 9.

[00:07:50]is called also be called for symmetric value.

[00:07:55]Thanks for watching.

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