A manufacturer builds a machine which will address 500 envelopes in 8 minutes.…

Concept

When two or more agents work on the same job together, their individual work rates — the fraction of the job each completes per unit time, equal to 1 divided by the time each would take alone — add up to give the combined work rate. If one agent alone takes t1 units of time and another alone takes t2 units of time for the same job, and together they take T units of time, then 1/t1 + 1/t2 = 1/T.

Applying it to this problem

1. The first machine addresses 500 envelopes alone in 8 minutes, so its rate is 1/8 of the job per minute.

2. Let the second machine take x minutes to address 500 envelopes alone, so its rate is 1/x of the job per minute.

3. Working together they address 500 envelopes in 2 minutes, so the combined rate is 1/2 of the job per minute.

4. Adding the two individual rates gives the combined rate: 1/8 + 1/x = 1/2 — this is the equation that models the situation.

Cross-check

Solving confirms the setup is consistent: 1/x = 1/2 − 1/8 = 3/8, so x = 8/3 ≈ 2.6667 minutes. Since the two machines together finish in 2 minutes — faster than the first machine's 8 minutes alone — the second machine must be faster than the first, and 2.6667 minutes is indeed less than 8 minutes, as expected. Adding the rates back also checks out: 1/8 + 3/8 = 4/8 = 1/2, which matches the combined rate.