Question 514070
We'll assume that Barry and Samuel complete their jobs at a constant pace.  If Barry can do this job in 10 hours, then he can do {{{1/10}}} of a job in 1 hour.  Likewise, if Samuel can takes 12 hours to do the same job, he can do {{{1/12}}} of a job in 1 hour.  In both cases, we took the reciprocal to go from hours per job to job per hour.  So together,

{{{ 1/10 + 1/12 }} of a job in 1 hour.  Let's simplify this sum:

{{{ 1/10 + 1/12 }}}
{{{ 6/60 + 5/60 }}}
{{{ 11/60 }}} (putting both fractions over the least common denominator of 10 and 12, which is 60, and adding)

Thus together, Barry and Samuel complete 11/60 of a job in one hour.  The number of hours it takes to complete the job will be the reciprocal of this fraction: it takes 60/11 of a hour, or 5 5/11 hours, to Barry and Samuel to complete the job together.  We're taking the reciprocal because we are going from jobs per hour to hours per job.