Short Answer 
The two machines take approximately 13.64 minutes to produce one bale of cotton when working together. The first machine has a rate of 1 bale every 25 minutes, while the second produces 1 bale every 30 minutes, resulting in a combined rate of 11/150 bales per minute.
Step 1: Determine the Rates of the Machines
Firstly, you need to find the working rates of both machines. The first machine can produce 1 bale every 25 minutes, which translates to a rate of x = 1/25 bales/min. The second machine, on the other hand, produces 1 bale every 30 minutes, giving it a rate of y = 1/30 bales/min.
Step 2: Calculate the Combined Rate
Next, you will combine the rates of both machines to figure out how quickly they can work together. This is done by adding their individual rates:
- Combined Rate = x + y = 1/25 + 1/30
- This can be simplified to = (30 + 25)/(25 * 30) = 55/750
- Thus, the combined rate is 11/150 bales/min.
Step 3: Find the Time Taken to Pick One Bale
The final step involves calculating the time required for both machines to produce 1 bale of cotton using their combined rate. This is done by dividing the desired quantity by the combined rate:
- Time = (1 bale)/(11/150 bales/min)
- This simplifies to = 150/11 minutes.
- So, the time taken is approximately 13.64 minutes.