In this problem, we are asked to determine the ratio of the contribution of team member M to the contribution of team member E in a project. The overall contribution of an individual to a project depends on three main factors: their efficiency, the number of hours they work per day, and the total number of days they work. We need to analyze the given information for M and E and then calculate their individual contributions to find the required ratio.

Project Team Parameters: Efficiency and Shifts

Let's first define the efficiency levels and daily shift durations for the team members involved:

• Efficiency:
  • The problem states that M works with twice the efficiency of others. Let's assume a base efficiency unit, say \(\text{Eff}\), for team members S, E, and F.
  • Therefore, the efficiency of E (\(\text{Eff}_{\text{E}}\)) = \(\text{Eff}\).
  • And the efficiency of M (\(\text{Eff}_{\text{M}}\)) = \(2 \times \text{Eff}\).
• Shift Duration (Hours per Day):
  • S and M have 6-hour shifts in a day. So, the hours worked per day by M (\(\text{H}_{\text{M}}\)) = 6 hours.
  • E and F have 12-hour shifts. So, the hours worked per day by E (\(\text{H}_{\text{E}}\)) = 12 hours.

Determining Workdays for M and E

The problem provides a direct relationship regarding the number of days M and E worked on the project:

• Let the number of days E worked on the project be \(\text{D}_{\text{E}}\).
• The problem states that M worked for half as many days as E. Therefore, the number of days M worked (\(\text{D}_{\text{M}}\)) = \(\frac{1}{2} \times \text{D}_{\text{E}}\).

Calculating Individual Contributions to the Project

The total contribution of any team member to a project can be calculated using the following formula:

Contribution = Efficiency \(\times\) Hours per Day \(\times\) Number of Days Worked

Contribution of M

Let's apply the formula for team member M:

• Efficiency of M (\(\text{Eff}_{\text{M}}\)) = \(2 \times \text{Eff}\)
• Hours per day for M (\(\text{H}_{\text{M}}\)) = 6 hours
• Number of days M worked (\(\text{D}_{\text{M}}\)) = \(\frac{1}{2} \times \text{D}_{\text{E}}\)

Substituting these values into the contribution formula:

\(\text{Contribution}_{\text{M}} = \text{Eff}_{\text{M}} \times \text{H}_{\text{M}} \times \text{D}_{\text{M}}\)

\(\text{Contribution}_{\text{M}} = (2 \times \text{Eff}) \times 6 \text{ hours/day} \times \left(\frac{1}{2} \times \text{D}_{\text{E}}\right) \text{ days}\)

\(\text{Contribution}_{\text{M}} = 2 \times 6 \times \frac{1}{2} \times \text{Eff} \times \text{D}_{\text{E}}\)

\(\text{Contribution}_{\text{M}} = 6 \times \text{Eff} \times \text{D}_{\text{E}}\)

Contribution of E

Now, let's apply the formula for team member E:

• Efficiency of E (\(\text{Eff}_{\text{E}}\)) = \(\text{Eff}\)
• Hours per day for E (\(\text{H}_{\text{E}}\)) = 12 hours
• Number of days E worked (\(\text{D}_{\text{E}}\)) = \(\text{D}_{\text{E}}\)

Substituting these values into the contribution formula:

\(\text{Contribution}_{\text{E}} = \text{Eff}_{\text{E}} \times \text{H}_{\text{E}} \times \text{D}_{\text{E}}\)

\(\text{Contribution}_{\text{E}} = \text{Eff} \times 12 \text{ hours/day} \times \text{D}_{\text{E}} \text{ days}\)

\(\text{Contribution}_{\text{E}} = 12 \times \text{Eff} \times \text{D}_{\text{E}}\)

Finding the Contribution Ratio of M to E

To find the ratio of M's contribution to E's contribution, we divide the contribution of M by the contribution of E:

\(\text{Ratio} = \frac{\text{Contribution}_{\text{M}}}{\text{Contribution}_{\text{E}}}\)

\(\text{Ratio} = \frac{6 \times \text{Eff} \times \text{D}_{\text{E}}}{12 \times \text{Eff} \times \text{D}_{\text{E}}}\)

Notice that \(\text{Eff}\) and \(\text{D}_{\text{E}}\) are common terms in both the numerator and the denominator, so they cancel each other out:

\(\text{Ratio} = \frac{6}{12}\)

Simplifying the fraction:

\(\text{Ratio} = \frac{1}{2}\)

Thus, the ratio of the contribution of M to the contribution of E in the project is 1:2.

The final ratio of contributions is M : E = 1 : 2.