SOLUTION: The difference between two positive numbers is 18 and 4 times their GM is equal to 5 times their HM find the numbers

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
:
GM is geometric mean, GM for x, y is defined as square root (x * y)
:
HM is harmonic mean, HM for x, y is defined as 2 / (1/x + 1/y)
:
2) 4 * square root(x * y) = 5 * (2 / (1/x + 1/y))
:
solve equation 1) for x
:
x = y + 18
:
substitute for x in equation 2
:
4 * square root((y+18) * y) = 5 * (2 / (1/(y+18) + 1/y))
:
4 * square root((y+18) * y) = 10 / (1/(y+18) + 1/y)
:
cross multiply the fractions
:
4 * square root((y+18) * y) * (1/(y+18) + 1/y) = 10
:
divide both sides of = by 4
:
square root((y+18) * y) * (1/(y+18) + 1/y) = 5/2
:
divide both sides of = by (1/(y+18) + 1/y)
:
square root((y+18) * y) = (5y * (y+18)) / (4 * (y+9))
:
square both sides of the =
:
(y+18) * y = (25y^2 * (y+18)^2) / (16 * (y+9)^2
:
cross multiply the fractions
:
16y * (y+18) * (y+9)^2 = 25y^2 * (y+18)^2
:
write the left side of the = in standard form
:
16y^4 +576y^3 +6480y^2 +23328y = 25y^2 * (y+18)^2
:
write the right side of the = in standard form
:
16y^4 +576y^3 +6480y^2 +23328y = 25y^4 +900y^3 +8100y^2
:
consolidate the terms to the left side of =
:
-9y^4 -324y^3 -1620y^2 +23328y = 0
:
this factors into
:
-9y * (y-6) * (y+18) * (y+24) = 0
:
divide both sides of = by -9
:
our solutions for y are 0, 6, -18, -24
:
we reject solutions 0, -18, -24 since y > 0
:
x = 6 + 18 = 24
:
*******************************************
x = 24 and y = 6
*******************************************
:

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.

RELATED QUESTIONS
The difference between two numbers is 8 and twice the sum of their reciprocols is equal... (answered by ankor@dixie-net.com)
The difference between two numbers is 8 and twice the sum of their reciprocals are equal... (answered by amarjeeth123)
if AM between 2 no's is 5 and their Gm is 4 then the HM will... (answered by richwmiller)
if AM between 2 no's is 5 and their Gm is 4 then the HM will... (answered by richwmiller)
the difference between two numbers is 3. IF the square of the smaller number is equal to... (answered by anantha)
I need help with this question find two consecutive numbers such that their sum is equal... (answered by josgarithmetic,Boreal,MathTherapy)
The difference between two positive numbers is 5.Their product is 104.Find the numbers.... (answered by edjones)
The difference between two number is 10 and their sum is equal to 4 times the smaller... (answered by rfadrogane)
The difference between two numbers is 10 and their sum is four times the smaller number.... (answered by Alan3354)