The problem in the image can be broken down as follows:

• The factory can produce a batch of pens in 20 days using 12 machines.
• Due to a fault, only 3 machines were operational for the first 8 days.
• From day 9 onwards, all 12 machines were operational.

We are tasked with calculating the total number of days it takes to complete the batch of pens.

Step 1: Calculate the total work required to complete the batch.

If 12 machines can finish the batch in 20 days, we can consider the total work required to make the batch as:

$\text{Total work} = 12 \times 20 = 240 \text{ machine-days of work}$

Step 2: Work done by 3 machines in the first 8 days.

For the first 8 days, only 3 machines were working, so the work done in this period is:

$\text{Work done} = 3 \times 8 = 24 \text{ machine-days of work}$

Step 3: Remaining work after the first 8 days.

After the first 8 days, the remaining work is:

$\text{Remaining work} = 240 - 24 = 216 \text{ machine-days of work}$

Step 4: Calculate the days required for 12 machines to finish the remaining work.

From day 9 onwards, all 12 machines are working. The number of days required for these 12 machines to complete the remaining work is:

$\text{Days required} = \frac{216}{12} = 18 \text{ days}$

Step 5: Calculate the total number of days.

The total number of days taken to complete the batch is the sum of the first 8 days and the 18 days afterward:

$\text{Total days} = 8 + 18 = 26 \text{ days}$

Thus, the total number of days to complete the batch of pens is 26 days.

Do you want further clarification or additional details on this?

Here are 5 related questions you might find interesting:

1. What would happen if only 6 machines were working for the first 8 days instead of 3?
2. How would the calculation change if all machines were operational from the beginning?
3. What if the production rate per machine varied daily?
4. How can we express this type of problem as a system of equations?
5. How can we generalize this for any number of machines and days?

Tip: Always define the total amount of work (in terms of machine-days) first, as it helps in understanding the distribution of work over time.