BigGlenntheHeavy said:
To complete a certain job, A alone would take m times as many days as B and C working together; B alone would take n times as many days as A and C together; C alone would take p times as many days as A and B together.
Prove that:
\(\displaystyle \frac{m}{m+1}+\frac{n}{n+1}+\frac{p}{p+1} \ = \ 2\)
Let
time taken by A alone = t[sub:8me42j1x]A[/sub:8me42j1x]
time taken by B alone = t[sub:8me42j1x]B[/sub:8me42j1x]
time taken by C alone = t[sub:8me42j1x]C[/sub:8me42j1x]
time taken by A&B together = t[sub:8me42j1x]AB[/sub:8me42j1x]
time taken by B&C together = t[sub:8me42j1x]BC[/sub:8me42j1x]
time taken by C&A together = t[sub:8me42j1x]CA[/sub:8me42j1x]
then
1/t[sub:8me42j1x]B[/sub:8me42j1x] + 1/t[sub:8me42j1x]C[/sub:8me42j1x] = 1/t[sub:8me42j1x]BC[/sub:8me42j1x]
t[sub:8me42j1x]BC[/sub:8me42j1x] = t[sub:8me42j1x]B[/sub:8me42j1x] * t[sub:8me42j1x]C[/sub:8me42j1x]/(t[sub:8me42j1x]B[/sub:8me42j1x]+t[sub:8me42j1x]C[/sub:8me42j1x]) = t[sub:8me42j1x]A[/sub:8me42j1x]/m
m = t[sub:8me42j1x]A[/sub:8me42j1x](t[sub:8me42j1x]B[/sub:8me42j1x] + t[sub:8me42j1x]C[/sub:8me42j1x])/(t[sub:8me42j1x]B[/sub:8me42j1x] * t[sub:8me42j1x]C[/sub:8me42j1x])
m/(m+1) = (t[sub:8me42j1x]A[/sub:8me42j1x] * t[sub:8me42j1x]B[/sub:8me42j1x] + t[sub:8me42j1x]C[/sub:8me42j1x] * t[sub:8me42j1x]A[/sub:8me42j1x])/(t[sub:8me42j1x]A[/sub:8me42j1x]*t[sub:8me42j1x]B[/sub:8me42j1x] + t[sub:8me42j1x]B[/sub:8me42j1x]*t[sub:8me42j1x]C[/sub:8me42j1x] + t[sub:8me42j1x]C[/sub:8me42j1x]*t[sub:8me42j1x]A[/sub:8me42j1x])
Rest is obvious...
