Question

Working alone at its constant rate, machine A produces k liters of a chemical in 10 minutes. Working alone at its constant rate, machine B produces k liters of the chemical in 15 minutes. How many minutes does it take machines A and B, working simultaneously at their respective constant rates, to produce k liters of the chemical?

Hint:

For work-rate problems, it's often easiest to think in terms of "job per unit of time". If Machine A does the job in 10 mins, its rate is 1/10 of the job per minute. Machine B's rate is 1/15 of the job per minute. Their combined rate is \(1/10 + 1/15 = 1/6\) of the job per minute. The time to do 1 job is the reciprocal of the rate, which is 6 minutes.

Answer 1

Step 1: Understanding the Concept:
This is a classic "combined work rate" problem. To find the time it takes for two machines to complete a job together, we first need to determine their individual rates of work, then add those rates to get a combined rate.
Step 2: Key Formula or Approach:
- Rate of work = \(\frac{\text{Work Done}}{\text{Time Taken}}\)
- Combined Rate = Rate of Machine A + Rate of Machine B
- Time Taken = \(\frac{\text{Work Done}}{\text{Combined Rate}}\)
Step 3: Detailed Explanation:
Let the amount of work be producing k liters of the chemical.
Rate of Machine A:
Machine A produces k liters in 10 minutes.
\[ \text{Rate}_A = \frac{k \text{ liters}}{10 \text{ minutes}} \] Rate of Machine B:
Machine B produces k liters in 15 minutes.
\[ \text{Rate}_B = \frac{k \text{ liters}}{15 \text{ minutes}} \] Combined Rate:
When working together, their rates add up.
\[ \text{Combined Rate} = \text{Rate}_A + \text{Rate}_B = \frac{k}{10} + \frac{k}{15} \] To add these fractions, find a common denominator, which is 30.
\[ \text{Combined Rate} = \frac{3k}{30} + \frac{2k}{30} = \frac{5k}{30} = \frac{k}{6} \text{ liters per minute} \] Time to Produce k Liters Together:
The work to be done is to produce k liters. We use the combined rate to find the time.
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{k \text{ liters}}{\frac{k}{6} \text{ liters/minute}} \] \[ \text{Time} = k \times \frac{6}{k} = 6 \text{ minutes} \] Step 4: Final Answer:
It takes the two machines 6 minutes to produce k liters of the chemical when working together.