Type 1 workers are three times as productive as Type 2 workers. Twelve Type 1 workers can complete a certain task in 10 days. If instead a group of 4 Type 1 workers and 8 Type 2 workers work together, in how many days will they complete the same task?

Introduction / Context:
This time and work question compares two types of workers with different productivities. Type 1 workers are more efficient than Type 2 workers. We know how long a group of Type 1 workers takes to complete a task and must find the time required by a mixed group containing both worker types. This tests proportional reasoning and the concept of equivalent work rates.

Given Data / Assumptions:

• Type 1 workers can each do three times as much work as a Type 2 worker in the same time.
• Twelve Type 1 workers complete the task in 10 days.
• We need the time taken when 4 Type 1 workers and 8 Type 2 workers work together.
• All workers of the same type are equally efficient, and work rates are constant.
• Total work is assumed to be 1 unit.

Concept / Approach:
First, we determine the work rate of a single Type 1 worker by using the total work completed by 12 such workers in 10 days. Knowing that Type 1 workers are three times as efficient as Type 2 workers allows us to find the rate of a Type 2 worker. We then compute the combined daily work rate of the new group (4 Type 1 and 8 Type 2 workers) and invert that rate to find the total time required for the job.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: Twelve Type 1 workers complete the work in 10 days, so total work = 12 * (rate of Type 1) * 10 = 1. Step 3: Let the rate of one Type 1 worker be r units per day. Then 12 * r * 10 = 1, so 120r = 1, giving r = 1/120 units per day. Step 4: A Type 2 worker is one third as efficient as a Type 1 worker, so rate of one Type 2 worker = r / 3 = (1/120) / 3 = 1/360 units per day. Step 5: In the new group, we have 4 Type 1 workers and 8 Type 2 workers. Step 6: Combined rate of 4 Type 1 workers = 4 * (1/120) = 4/120 = 1/30 units per day. Step 7: Combined rate of 8 Type 2 workers = 8 * (1/360) = 8/360 = 1/45 units per day. Step 8: Total rate of the mixed group = 1/30 + 1/45. Use LCM of 90: 1/30 = 3/90 and 1/45 = 2/90, so total rate = 5/90 = 1/18 units per day. Step 9: Time taken to complete the work = 1 / (1/18) = 18 days.

Verification / Alternative check:
We can check the reasonableness by comparing rates. Twelve Type 1 workers alone take 10 days, so their rate is 1/10 of the job per day. The new group has fewer Type 1 workers and some slower Type 2 workers. Hence, its rate must be smaller than 1/10, meaning the time must be more than 10 days. Our answer 18 days is indeed greater than 10, which makes logical sense.

Why Other Options Are Wrong:

• 16 days: This suggests a higher rate than 1/18 and does not match the calculated combined rate of the group.
• 17 days: Not supported by any correct combination of rates and is slightly faster than the correct answer.
• 20 days: This would imply a slower rate of 1/20 units per day, which does not agree with the derived total rate of 1/18 units per day.

Common Pitfalls:
Students may misinterpret the statement "Type 1 workers can do three times the work of Type 2 workers" and invert the ratio. Another error is to average the days directly instead of using rates. Properly calculating individual and combined work rates and keeping track of fractions is essential to solving such problems accurately.

Final Answer:
The group of 4 Type 1 and 8 Type 2 workers will complete the task in 18 days.