To solve this problem, we’ll work through the information provided and calculate the total days required to complete the batch of pens.

Step-by-Step Solution:

1. Calculate the daily output of one machine:

   • The factory can complete the batch in 20 days using 12 machines.
   • So, total machine-days needed = $20 \times 12 = 240$ machine-days.

2. Calculate the output rate per day for each machine:

   • Each machine contributes to this output, so the work done per machine per day = $\frac{1}{240}$ of the batch.

3. Output from 8 machines for the first 9 days:

   • For the first 9 days, only 8 machines were operational.
   • The work done in 9 days by 8 machines = $8 \times 9 = 72$ machine-days.

4. Remaining work:

   • Total work needed is 240 machine-days.
   • Work completed in the first 9 days is 72 machine-days.
   • Remaining work = $240 - 72 = 168$ machine-days.

5. Calculate additional days with all 12 machines:

   • From day 10 onward, all 12 machines are used.
   • Days required to complete 168 machine-days of work with 12 machines = $\frac{168}{12} = 14$ days.

6. Total days taken:

   • The first phase took 9 days, and the second phase took 14 days.
   • Total days = $9 + 14 = 23$ days.

Answer:

The total number of days taken to make the batch of pens is 23 days.

Would you like a detailed explanation on any step, or have any further questions?

Here are five related questions to extend your understanding:

1. What would happen to the total time if only 6 machines were operational in the first 9 days?
2. How many days would it take if the factory had 15 machines instead of 12?
3. What is the effect on the total production time if the factory works an extra 2 hours per day?
4. How would the answer change if, instead of day 10, all 12 machines became available on day 8?
5. How much work does one machine complete in 5 days at this rate?

Tip: When solving similar problems, try breaking down the work into manageable chunks, as we did here by calculating machine-days. This approach helps simplify complex multi-step calculations.