Question 1091579: The difference between two positive numbers is 18 and 4 times their GM is equal to 5 times their HM find the numbers
Found 2 solutions by rothauserc, ikleyn:
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! 1) x - y = 18, with x, y > 0
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GM is geometric mean, GM for x, y is defined as square root (x * y)
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HM is harmonic mean, HM for x, y is defined as 2 / (1/x + 1/y)
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2) 4 * square root(x * y) = 5 * (2 / (1/x + 1/y))
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solve equation 1) for x
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x = y + 18
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substitute for x in equation 2
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4 * square root((y+18) * y) = 5 * (2 / (1/(y+18) + 1/y))
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4 * square root((y+18) * y) = 10 / (1/(y+18) + 1/y)
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cross multiply the fractions
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4 * square root((y+18) * y) * (1/(y+18) + 1/y) = 10
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divide both sides of = by 4
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square root((y+18) * y) * (1/(y+18) + 1/y) = 5/2
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divide both sides of = by (1/(y+18) + 1/y)
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square root((y+18) * y) = (5y * (y+18)) / (4 * (y+9))
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square both sides of the =
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(y+18) * y = (25y^2 * (y+18)^2) / (16 * (y+9)^2
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cross multiply the fractions
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16y * (y+18) * (y+9)^2 = 25y^2 * (y+18)^2
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write the left side of the = in standard form
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16y^4 +576y^3 +6480y^2 +23328y = 25y^2 * (y+18)^2
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write the right side of the = in standard form
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16y^4 +576y^3 +6480y^2 +23328y = 25y^4 +900y^3 +8100y^2
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consolidate the terms to the left side of =
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-9y^4 -324y^3 -1620y^2 +23328y = 0
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this factors into
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-9y * (y-6) * (y+18) * (y+24) = 0
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divide both sides of = by -9
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our solutions for y are 0, 6, -18, -24
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we reject solutions 0, -18, -24 since y > 0
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x = 6 + 18 = 24
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x = 24 and y = 6
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Answer by ikleyn(52788)  (Show Source):
You can put this solution on YOUR website! .
1) x - y = 18, with x, y > 0. (1)
GM is geometric mean, GM for x, y is defined as  .
HM is harmonic mean, HM for x, y is defined as  .
2)  =  ====>
 =  ====>
=  ====> square both sides ====>
 =  ====> cancel xy in both sides ====>
16 =  ====>  = 100xy ====>  = 25xy (2)
Now from (1) express x = y + 18 and substitute it into (2). You will get
 = 25*(y+18)*y, or
 = 25*y*(y+18) ====>  =  ====>  = 0 ====> cancel 9 in both sides ====>
 = 0
(y+24)*(y-6) = 0.
Since y must be positive, we have only ONE solution for y: y = 6.
Then x = y + 18 = 24.
Answer. x = 24, y = 6 is the only solution.
The lesson to learn from this solution:
If you choose the correct way, you reduce the problem to the quadratic equation, and there is no need to solve the equation of degree 4.
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