SOLUTION: The sum of the reciprocals of two even numbers equals 11/60. Write an equation and find the two numbers. This really stumped me....please help!!!!!

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Question 184875: The sum of the reciprocals of two even numbers equals 11/60.
Write an equation and find the two numbers.
This really stumped me....please help!!!!!
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
The sum of the reciprocals of two even numbers equals 11/60.
Write an equation and find the two numbers.
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Let the two even numbers be (2x) and (2y)
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Equation:
1/(2x) + 1/(2y) = 11/60
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You do not have enough information to solve this equation.
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Cheers,
Stan H.

Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

I don't blame you a bit for being stumped on this one. I had to chew on it all afternoon. But I did come up with two solutions.

Now if you add the two fractions containing the variables:

From which you could derive two equations by equating the numerators and equating the denominators. But there is a problem. Setting the numerators equal to each other gives you:

And there are no two even numbers that will satisfy this equation. This is where I was stumped for a while until it occured to me to make a little change to the equation, thus:

Now we can say:

And

Rearranging the first equation:

And substituting in the second equation:

And this quadratic factors rather tidily:

Therefore:

or

And going back to:

We can see that y is 12 when x is 10, or vice versa, so the two numbers are 10 and 12.

Check:

Done, right? Not so fast, Sparky. Like the guy on TV selling Ginzu knives says, "But wait...there's more!"

After obtaining that result, it occurred to me that if multiplying by resulted in a solution, perhaps there was some integer k > 1 such that multiplying by would result in a solution to this problem.

So I tried but that yielded:

which has irrational roots, and is therefore not a solution (verification of this left as an exercise for the student)

Then I tried So I tried yielding:

Which factors to:

Giving us 6 and 60 as the two numbers.

Check:

I actually tried about a hundred different values of k but only found the two I showed you that worked, so I'm pretty certain that these are the only two solutions to the problem as stated.

Thanks. I had fun with this one.

John