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Question Number 8230 by 314159 last updated on 03/Oct/16

$$\mathrm{If}\frac{x+y}{x+y+z}=\frac{y+z}{x+y+z}=\frac{x+z}{x+y+z}=p,\mathrm{then}\mathrm{which}\mathrm{of}$$$$\mathrm{the}\mathrm{following}\mathrm{can}\mathrm{be}\mathrm{the}\mathrm{value}\mathrm{of}p?$$

Answered by Rasheed Soomro last updated on 03/Oct/16

$$\frac{\mathrm{a}}{\mathrm{A}}=\frac{\mathrm{b}}{\mathrm{B}}=\frac{\mathrm{c}}{\mathrm{C}}=\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{A}+\mathrm{B}+\mathrm{C}}$$$$\mathrm{p}=\frac{x+y}{x+y+z}=\frac{y+z}{x+y+z}=\frac{x+z}{x+y+z}$$$$=\frac{(\mathrm{x}+\mathrm{y})+(\mathrm{y}+\mathrm{z})+(\mathrm{x}+\mathrm{z})}{(\mathrm{x}+\mathrm{y}+\mathrm{z})+(\mathrm{x}+\mathrm{y}+\mathrm{z})+(\mathrm{x}+\mathrm{y}+\mathrm{z})}$$$$=\frac{2\mathrm{x}+2\mathrm{y}+2\mathrm{z}}{3(\mathrm{x}+\mathrm{y}+\mathrm{z})}=\frac{2(\mathrm{x}+\mathrm{y}+\mathrm{z})}{3(\mathrm{x}+\mathrm{y}+\mathrm{z})}=\frac{2}{3}$$

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