1/a + 1/b = 1/t
This formula will give you the combined RATE of the two entities. So it if takes Bill 3hr to do a job, and Sam 4 hr, then that formula will give you the RATE when they work together (7/12 job/hr).
If we are looking for the time it takes them to do 1 job, then you just mupltiply 7/12 by its reciprocal (12/7), in order to get to 1 job. So it would take them 12/7 hour to complete the job together.
T/H1 + T/H2 = 1
T is the time H1 and H2 worked together (note that you defined T, but not H1 and H2, which makes it difficult to know what you are expressing in this equation. However, I will give it a shot).
Let's say that H1 = Time it takes H1 to do a job alone
H2 = time it takes H2 to do a job alone
T/H1 = the fraction of the job completed by H1
T/H2 = the fraction of the job completed by H2
This equation expresses the relative contribution of each individual to the total job. It is different from equation 1 in that it is expressing a fundamental truth about all work problems, which is that the total amount of work for a single job always equals 1. This is what makes these problems confusing - because all RATES are in fractions!
So T/H1 = the amount of the job contributed by H1.
T/H2 = the amount of the job contributed by H2.
So I would say that they aren't the same equation, but you can derive one from the other (I think).
