SOLUTION: The sum of the reciprocals of two consecutive even integers is 5/12. Find the integers.

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Question 1114940: The sum of the reciprocals of two consecutive even integers is 5/12.
Find the integers.
Found 3 solutions by josgarithmetic, Alan3354, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
1%2Fn%2B1%2F%28n%2B2%29=5%2F12
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%28n%2B2%29%2Bn=%285%2F12%29%28n%28n%2B2%29%29
2n%2B2=%285%2F12%29%28n%5E2%2B2n%29
24n%2B24=5%28n%5E2%2B2n%29
24n%2B24=5n%5E2%2B10n
5n%5E2-14n-24=0
discriminant, 196%2B4%2A5%2A24=676=26%5E2
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n=%2814%2B-+26%29%2F10

system%28n=cross%28-1.2%29%2C+or%2C+n=4%29

The solution of the equation which is integer is n=4.
The integers for the question description are 4 and 6.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the reciprocals of two consecutive even integers is 5/12.
Find the integers.
1%2Fx+%2B+1%2F%28x%2B2%29+=+5%2F12
%282x%2B2%29%2F%28x%5E2%2B2x%29+=+5%2F12
24x+%2B+24+=+5x%5E2+%2B+10x
Can you do the rest?

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Knowing how to solve problems like this using formal algebra is of course valuable....

But logical analysis and a tiny bit of mental arithmetic finds the answer much more quickly.

The reciprocals of integers get very small very quickly. If the sum of the reciprocals of two integers is 5/12, the integers have to be very small.

Given that the two are consecutive even integers, the solution is found with little effort:
2 and 4? 1%2F2%2B1%2F4+=+3%2F4 no...
4 and 6? 1%2F4%2B1%2F6+=+3%2F12%2B2%2F12+=+5%2F12 yes!