SOLUTION: The sum of 2 numbers is 20, their product is 10. If the sum of their reciprocals is a, find a?

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Question 1071818: The sum of 2 numbers is 20, their product is 10. If the sum of their reciprocals is a, find a?
Found 4 solutions by josgarithmetic, stanbon, KMST, ikleyn:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Two numbers, x and y

system%28x%2By=20%2Cxy=10%29

x%2820-x%29=10
20x-x%5E2=10
-x%5E2%2B20x-10=0
x%5E2-20x%2B10=0
discrim, 400-4%2A10=400-40=360

x=%2820%2B-+sqrt%28360%29%29%2F2

x=%2820%2B-+sqrt%286%2A6%2A10%29%29%2F2

x=%2820%2B-+6%2Asqrt%2810%29%29%2F2

x=10%2B-+3sqrt%2810%29

The two numbers are system%2810-3sqrt%2810%29%2Cand%2C10%2B3sqrt%2810%29%29.


SUM OF THE RECIPROCALS
1%2F%2810-3sqrt%2810%29%29%2B1%2F%2810%2B3sqrt%2810%29%29

, to rationalize denominator and get common denominator

%2810%2B3sqrt%2810%29%2B10-3sqrt%2810%29%29%2F%28100-9%2A10%29, uses Difference Of Squares

20%2F%28100-90%29

20%2F10

highlight%282%29

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of 2 numbers is 20, their product is 10. If the sum of their reciprocals is a, find a?
----
x + y = 20
x*y = 10
-------
x(20-x) = 10
x^2 - 20x + 10 = 0
----
x = 19.487
y = 0.513
====
a = 1/x + 1/y = 2
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Cheers,
Stan H.
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Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Let x and y be the two numbers.
We are told that
x%2By=20 and x%2Ay=20 ,
and we are asked to find
a=1%2Fx%2B1%2Fy .
From x%2Ay=10 we get
x=10%2Fy and y=10%2Fx .
Substituting those expressions
for x and y
in x%2By=20 , we get
10%2Fy%2B10%2Fx=20
10%281%2Fy%2B1%2Fx%29=20
Does the expression in brackets look familiar?
It is a=1%2Fx%2B1%2Fy .
So, 10a=20 ,
a=20%2F10 , and
highlight%28a=2%29 .
You could solve for x and y ,
but it will be ugly.

Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
.
You are given 

x + y = 20,  xy = 10.


You are asked about 1%2Fx+%2B+1%2Fy.


1%2Fx+%2B+1%2Fy = y%2F%28xy%29+%2B+x%2F%28xy%29 = %28x+%2B+y%29%2Fxy%29.


Now replace the numerator by 20, and replace the denominator by 10. You will get


1%2Fx+%2B+1%2Fy = %28x+%2B+y%29%2Fxy%29 = 20%2F10 = 2.

Solved. You do not need solve any equations.


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What "josgarithmetic" is doing here is demonstration of his mathematical illiteracy.

He is doing here day after day despite my warnings.