Let's break down and solve the problem step by step:

Problem Statement:

1. With "p" machines, a job is completed in 30 days.
2. With "p + 4" machines, a job that is twice as difficult is completed in 40 days.
3. We need to determine the time it will take for "p + 2" machines to complete a job of equal difficulty to the initial one.

Step-by-Step Solution:

Step 1: Determine the work rate of the machines.

• Let the total amount of work for the initial job be $W$.
• The work rate for "p" machines is $\frac{W}{30}$ (work per day).

Thus, the work done by one machine per day is: $\text{Rate per machine} = \frac{W}{30p}$

Step 2: Establish the work rate for the more difficult job.

• The more difficult job has double the difficulty, so the total work is $2W$.
• The rate with "p + 4" machines is $\frac{2W}{40}$.

So, the rate per machine in this scenario is: $\text{Rate per machine} = \frac{2W}{40(p+4)} = \frac{W}{20(p+4)}$

Step 3: Find the relationship between $p$ and $W$.

Since the rate per machine must be the same in both cases, we equate the rates: $\frac{W}{30p} = \frac{W}{20(p+4)}$ Canceling $W$ from both sides: $\frac{1}{30p} = \frac{1}{20(p+4)}$ Cross-multiply: $20(p+4) = 30p$ Expanding and solving for $p$: $20p + 80 = 30p$ $10p = 80$ $p = 8$

Step 4: Calculate the time for "p + 2" machines.

• $p + 2 = 8 + 2 = 10$ machines.
• We know the work rate for one machine is $\frac{W}{240}$.

The total time required for 10 machines: $\text{Time} = \frac{W}{10 \times \frac{W}{240}} = \frac{240}{10} = 24 \text{ days}$

Thus, the correct answer is $\boxed{24}$ days.

Final Answer: Option C (24).

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Would you like further details on any step, or do you have any other questions?

Here are five related questions you might find interesting:

1. How would the problem change if the job difficulty was tripled instead of doubled?
2. How can the work rate of machines be calculated if the job was done in parts by different teams?
3. What would happen if the number of machines was variable over time?
4. How can we generalize this problem to any number of machines and job difficulties?
5. How does this problem relate to concepts of efficiency and optimization?

Tip: In problems like these, identifying and equating the rates of work is a crucial step that simplifies the calculations significantly.