A manufacturer builds a machine which will address 500 envelopes in 8 minutes. He wishes to build another machine so that when both are operating together they will address 500 envelopes in 2 minutes. The equation used to find how many minutes $x$ it would require the second machine to address 500 envelopes alone, is

### Concept & Work Rate Equation When two machines or people work together, their combined rate of work is the sum of their individual rates. If a machine completes a full task in $T$ minutes, its work rate is $\frac{1}{T}$ of the task per minute. $$ \text{Combined Rate} = \text{Rate}_1 + \text{Rate}_2 $$ ### Step-by-Step Solution * **Define the standard unit of work:** Let addressing a batch of 500 envelopes equal $1$ complete job. * **Determine Machine 1's rate:** It takes $8$ minutes to do $1$ job. Therefore, its rate is $\frac{1}{8}$ of the job per minute. * **Determine Machine 2's rate:** It takes $x$ minutes to do $1$ job. Therefore, its rate is $\frac{1}{x}$ of the job per minute. * **Determine Combined rate:** Together, they take $2$ minutes to do the same $1$ job. Their combined rate is $\frac{1}{2}$ of the job per minute. * **Formulate Equation:** Summing the individual rates equals the combined rate: $\frac{1}{8} + \frac{1}{x} = \frac{1}{2}$ ### Exam Strategy & Shortcut Always translate "time taken for a fixed task" directly into fractions representing the "portion of the task done per unit time". Since the quantity of envelopes (500) remains constant across all conditions, it acts as a distracting variable and simplifies to $1$ job unit, making the fractional rate equation obvious. ### Common Pitfall A common mistake is trying to add the total times directly (e.g., $8 + x = 2$) or multiplying the times, rather than converting them into reciprocal rates first. ### Final Answer Therefore, the correct answer is **$\frac{1}{8} + \frac{1}{x} = \frac{1}{2}$**.