Problem Breakdown and Solution:

The problem compares the productivity of two groups of printers: old and new. Here's how we'll solve the problem:

Step 1: Calculate the productivity of each printer type

• Old printers:
  5 old printers take 30 minutes to produce 1 batch.
  Therefore, the work rate for 1 old printer is:
  $\frac{1}{5 \times 30} = \frac{1}{150} \, \text{batches per minute.}$

• New printers:
  4 new printers take 18 minutes to produce 1 batch.
  Therefore, the work rate for 1 new printer is:
  $\frac{1}{4 \times 18} = \frac{1}{72} \, \text{batches per minute.}$

Step 2: Calculate the total work rate for each group

• 6 old printers:
  Total work rate is:
  $6 \times \frac{1}{150} = \frac{6}{150} = \frac{1}{25} \, \text{batches per minute.}$

• 2 new printers:
  Total work rate is:
  $2 \times \frac{1}{72} = \frac{2}{72} = \frac{1}{36} \, \text{batches per minute.}$

Step 3: Calculate the time to produce 1 batch for each group

• 6 old printers:
  Time taken = $\frac{1}{\text{work rate}} = \frac{1}{1/25} = 25 \, \text{minutes.}$

• 2 new printers:
  Time taken = $\frac{1}{\text{work rate}} = \frac{1}{1/36} = 36 \, \text{minutes.}$

Step 4: Compare the results

• Which is faster?
  6 old printers are faster, taking 25 minutes compared to 36 minutes for 2 new printers.

• How much less time?
  The difference in time is:
  $36 - 25 = 11 \, \text{minutes.}$

Final Answer

a) 6 old printers would produce the batch in less time.
b) They would take 11 minutes less.

Would you like additional details or clarifications? Here are some related questions for further exploration:

1. How does the efficiency of a single new printer compare to an old printer?
2. What is the time taken if 5 new printers are used instead of 4?
3. If the batch size doubles, how would this affect the time taken by each group?
4. How many old printers are needed to match the efficiency of 4 new printers?
5. What is the minimum number of new printers required to beat 6 old printers' time?

Tip: To solve problems involving work rates, always break tasks into individual contributions and add rates when workers work together.