1. **State the problem:** We have two types of printers: old and new. - 5 old printers take 30 minutes to produce a batch. - 4 new printers take 18 minutes to produce the same batch. We want to find: a) Which is faster: 6 old printers or 2 new printers? b) How much less time does the faster option take? 2. **Find the rate of one old printer:** The total work is 1 batch. 5 old printers take 30 minutes, so their combined rate is $$\frac{1}{30}$$ batches per minute. Therefore, one old printer's rate is $$\frac{1}{30} \div 5 = \frac{1}{150}$$ batches per minute. 3. **Find the rate of one new printer:** 4 new printers take 18 minutes, so their combined rate is $$\frac{1}{18}$$ batches per minute. One new printer's rate is $$\frac{1}{18} \div 4 = \frac{1}{72}$$ batches per minute. 4. **Calculate time for 6 old printers:** Combined rate of 6 old printers is $$6 \times \frac{1}{150} = \frac{6}{150} = \frac{1}{25}$$ batches per minute. Time taken is the reciprocal: $$\frac{1}{\frac{1}{25}} = 25$$ minutes. 5. **Calculate time for 2 new printers:** Combined rate of 2 new printers is $$2 \times \frac{1}{72} = \frac{2}{72} = \frac{1}{36}$$ batches per minute. Time taken is the reciprocal: $$\frac{1}{\frac{1}{36}} = 36$$ minutes. 6. **Compare times:** 6 old printers take 25 minutes. 2 new printers take 36 minutes. 7. **Answer:** a) 6 old printers produce the batch faster. b) The time difference is $$36 - 25 = 11$$ minutes less for 6 old printers.