What is \frac{\frac{x}{x-y} + \frac{y}{y-z} + \frac{z}{z-x}}{\frac{x+y}{x-y} + \frac{y+z}{y-z} + \frac{z+x}{z-x} + 3} equal to?

Explanation

Let the numerator be p = \frac{x}{x-y} + \frac{y}{y-z} + \frac{z}{z-x}. By adding the 3 into each term of the denominator as 1 + 1 + 1, it becomes \left(\frac{x+y}{x-y} + 1\right) + \left(\frac{y+z}{y-z} + 1\right) + \left(\frac{z+x}{z-x} + 1\right). This simplifies to \frac{2x}{x-y} + \frac{2y}{y-z} + \frac{2z}{z-x}, which is exactly 2p. Therefore, the ratio is \frac{p}{2p} = \frac{1}{2}.

Related questions on Algebra

• If p + q + r = 0, then what is z^{\frac{p^2}{qr}} \times z^{\frac{q^2}{rp}} \times z^{\frac{r^2}{pq}} equal to ?
• What is the value of k for which (k^2 - 5k + 4)x^2 + (k^2 - 3k - 4)x + (k^2 - 4k) = 0 is an identity ?
• If \frac{a^2}{b^2 + c^2} = \frac{b^2}{c^2 + a^2} = \frac{c^2}{a^2 + b^2}, then what is the value of a^4 + b^4 + c^4 equal to ?
• If \frac{1}{x} = \frac{1}{p} + \frac{1}{q}, then what is \frac{pq}{p^2 - q^2}\left(\frac{x + p}{x - p} - \frac{x + q}{x - q}\right) equa...
• If (x - 5) is the HCF of x^2 - x - p and x^2 - qx - 10, then what is the value of (p + q) ?