SOLUTION: The sum of the reciprocals of two consecutive even integers is 23/264 . Find the two integers.

Algebra ->  Problems-with-consecutive-odd-even-integers -> SOLUTION: The sum of the reciprocals of two consecutive even integers is 23/264 . Find the two integers.      Log On


   

Question 1204547: The sum of the reciprocals of two consecutive even integers is 23/264
. Find the two integers.
Found 3 solutions by math_tutor2020, Alan3354, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

x = first even integer
x+2 = second even integer just after x.

Their reciprocals add to 23/264.
1%2Fx+%2B+1%2F%28x%2B2%29+=+23%2F264

%28x%2B2%29%2F%28x%28x%2B2%29%29+%2B+x%2F%28x%28x%2B2%29%29+=+23%2F264

%28x%2B2%2Bx%29%2F%28x%28x%2B2%29%29+=+23%2F264

%282x%2B2%29%2F%28x%28x%2B2%29%29+=+23%2F264

%282x%2B2%29%2A264+=+x%28x%2B2%29%2A23

264%282x%2B2%29+=+23x%28x%2B2%29

528x%2B528+=+23x%5E2%2B46x

0+=+23x%5E2%2B46x-528x-528

23x%5E2-482x-528+=+0

Use the quadratic formula from here.
a = 23, b = -482, c = -528
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-482%29%2B-sqrt%28%28-482%29%5E2-4%2823%29%28-528%29%29%29%2F%282%2823%29%29

x+=+%28482%2B-sqrt%28232324+%2B+-48576%29%29%2F%2846%29

x+=+%28482%2B-sqrt%28280900%29%29%2F%2846%29

x+=+%28482%2B-++530%29%2F%2846%29

x+=+%28482%2B530%29%2F%2846%29 or x+=+%28482-530%29%2F%2846%29

x+=+%281012%29%2F%2846%29 or x+=+%28-48%29%2F%2846%29

x+=+22 or x+=+-24%2F23

x+=+22 or matrix%281%2C2%2Cx+=+-1.043478%2C%22approximately%22%29
Ignore the non-integer result.

Therefore, 1%2F22+%2B+1%2F24+=+23%2F264


Answer: The two consecutive even integers are 22 and 24.

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 23/264
. Find the two integers.
-------------
23/264 = 46/528
sqrt%28528%29+=+23 --- close to 23
---> 22 & 24

Answer by ikleyn(52898) About Me  (Show Source):
You can put this solution on YOUR website!
.

I will use an idea, which is close to Alan' solution, but is slightly different.


If the sum of the reciprocals of two consecutive even integers is 23/264,

then each reciprocal is close to  %281%2F2%29%2A%2823%2F264%29 = 23%2F528.


Hence, each of the two consecutive even integer is close to  528%2F23 = 22.96.


From it, I conclude that the two consecutive even integer numbers are 22 and 24.


Now it is a matter of arithmetic to check that it is true.

Solved.