Let J and K be the times of Jeff and Kirk respectively
To know the time it takes two people to perform a task together use the equation:
[tex] \frac{1}{J}+ \frac{1}{K}= \frac{1}{10} [/tex]
We also know from the problem that J=K-1, so
[tex]\frac{1}{K-1}+ \frac{1}{K}= \frac{1}{10}
[/tex]
Now get common denominators
[tex]\frac{K}{K(K-1)}+ \frac{K-1}{K(K-1)}= \frac{1}{10}[/tex]
[tex]\frac{K+K-1}{K(K-1)}= \frac{1}{10}[/tex]
Take reciprocals
[tex]\frac{K(K-1)}{2K-1}=10[/tex]
[tex]K(K-1)=10(2K-1)[/tex]
[tex]K^{2} -K=20K-10[/tex]
[tex]K^{2} -21K+10=0[/tex]
By the quadratic equation:
K=20.512
J=K-1=19.512
Putting these into the original formula for two people working together on a task
[tex] \frac{1}{20.512}+ \frac{1}{19.512}= \frac{1}{10} [/tex]
So it checks out.
