Let's break this down step by step.

Step 1: Taskmaker's rate of work

The taskmaker can perform 1 task in 40 minutes. This means the taskmaker’s rate of work is:

$\text{Taskmaker's rate} = \frac{1 \, \text{task}}{40 \, \text{minutes}} = \frac{1}{40} \, \text{tasks per minute}$

Step 2: Apprentice's rate of work

The apprentice can perform 1 task in 1 hour (which is 60 minutes). Therefore, the apprentice's rate of work is:

$\text{Apprentice's rate} = \frac{1 \, \text{task}}{60 \, \text{minutes}} = \frac{1}{60} \, \text{tasks per minute}$

Step 3: Work done by the taskmaker in the first hour

The taskmaker works alone for 1 hour (which is 60 minutes). During this time, the amount of work completed by the taskmaker is:

$\text{Work by taskmaker in 1 hour} = \text{Rate of taskmaker} \times 60 \, \text{minutes} = \frac{1}{40} \times 60 = 1.5 \, \text{tasks}$

So after the first hour, the taskmaker completes 1.5 tasks, leaving $5 - 1.5 = 3.5$ tasks still to be completed.

Step 4: Combined work rate of taskmaker and apprentice

Once the apprentice joins, both the taskmaker and the apprentice work together. Their combined rate of work is:

$\text{Combined rate} = \frac{1}{40} + \frac{1}{60} = \frac{3}{120} + \frac{2}{120} = \frac{5}{120} = \frac{1}{24} \, \text{tasks per minute}$

This means they can complete $\frac{1}{24}$ tasks per minute working together.

Step 5: Time taken to complete the remaining 3.5 tasks

The remaining work is 3.5 tasks, and their combined rate is $\frac{1}{24}$ tasks per minute. Therefore, the time it takes for them to complete the remaining tasks is:

$\text{Time} = \frac{3.5 \, \text{tasks}}{\frac{1}{24} \, \text{tasks per minute}} = 3.5 \times 24 = 84 \, \text{minutes}$

Step 6: Total time

The total time is the 1 hour (60 minutes) that the taskmaker worked alone, plus the 84 minutes they worked together:

$\text{Total time} = 60 \, \text{minutes} + 84 \, \text{minutes} = 144 \, \text{minutes}$

Converting this to hours and minutes:

$144 \, \text{minutes} = 2 \, \text{hours and} \, 24 \, \text{minutes}$

Final Answer:

It will take 2 hours and 24 minutes for both to perform 5 tasks, with the taskmaker working alone for the first hour.

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Would you like more details on this solution? Here are some related questions:

1. What if the taskmaker worked alone for 30 minutes instead of 1 hour?
2. How would the solution change if the apprentice could perform 1 task in 50 minutes?
3. Can we calculate the time if both the taskmaker and apprentice worked the entire time together?
4. What is the total amount of work completed after both work together for 2 hours?
5. How would the total time change if they needed to complete 10 tasks instead?

Tip: To find combined rates, always sum individual rates and then divide the remaining work by this combined rate to find the time needed.