Each of x, y and z is a positive real number.

Determine the ratio x:y:z in each of the following equations.

(I) (x+z)/y = z/x = x/(z-y)

(II) 21(x² + y² + z²) = (x + 2y + 4z)²

One way of doing part 1 is to assume, say, z is 1, and see what the other variables come out to:

(x+1)/y = 1/x

x^2 + x = y

we're also given 1/x = x/(1-y)

x^2 = 1-y

then substituting,

1 - y + x = y

x = 2y - 1

Then, substituting again,

2y/y = 1/x

1/x = 2

x = 1/2

then

y = 1/4 + 1/2 = 3/4

Summarizing:

x = 2/4; y = 3/4; z = 4/4

The ratio is 2:3:4.

Posted by Charlie on 2011-06-15 14:26:17