SOLUTION: The sum of two numbers is 25. The sum of their reciprocals is 1/4. Determine the two numbers.

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Question 1189179: The sum of two numbers is 25. The sum of their reciprocals is 1/4. Determine the two numbers.
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of two numbers is 25. The sum of their reciprocals is 1/4. Determine the two numbers.
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x+y = 25 --> y = 25-x
1%2Fx+%2B+1%2Fy+=+1%2F4
%28x+%2B+y%29%2Fxy+=+1%2F4
4x+%2B+4y+=+xy
Sub for y
4x+%2B+4%2825-x%29+=+x%2A%2825-x%29+=+25x+-+x%5E2
100+=+25x+-+x%5E2
x%5E2+-+25x+%2B+100+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-25x%2B100+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-25%29%5E2-4%2A1%2A100=225.

Discriminant d=225 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--25%2B-sqrt%28+225+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-25%29%2Bsqrt%28+225+%29%29%2F2%5C1+=+20
x%5B2%5D+=+%28-%28-25%29-sqrt%28+225+%29%29%2F2%5C1+=+5

Quadratic expression 1x%5E2%2B-25x%2B100 can be factored:
1x%5E2%2B-25x%2B100+=+%28x-20%29%2A%28x-5%29
Again, the answer is: 20, 5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-25%2Ax%2B100+%29

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5 and 20




Answer by ikleyn(53765) About Me  (Show Source):
You can put this solution on YOUR website!
.
The sum of two numbers is 25. The sum of their reciprocals is 1/4. Determine the two numbers.
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Your starting equations are

    x + y = 25           (1)

    1%2Fx + 1%2Fy = 1%2F4        (2)


From equation (2)

    %28x+%2B+y%29%2Fxy ] 1%2F4

    4*(x+y) = xy


Replace here x+y by 25, based on equation (1).  You will get

    xy = 4*25 = 100.


So, you have two equations

    x + y = 25       (3)

     xy   = 100      (4)


From here, the  ANSWER  is OBVIOUS:  (x,y) = (20,5)  or  (5,20).      (*)


If you want to get formal solution, express  x = 25-y from (3) and substitute it to (4).  You will get

    (25-y)*y = 100

    25y - y^2 = 100

    y^2 - 25y + 100 = o

    (y-20)*(y-5) = 0

giving two possibilities  y= 20  or  y= 5.


They lead to answer (*): the numbers are 5 and 20, in any order.

Solved.