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This problem can be solved using the concept of *work rate* or inverse proportionality between the number of workers and the time they take to complete a task.

### Given:
- 6 builders can complete the work in 16 days.
- Sophia wants to complete the same work in 12 days.

Let's call the number of builders required to finish the work in 12 days as \( x \).

The relationship between the number of builders and the number of days required to complete the work is inversely proportional, meaning:

\[
\text{Number of builders} \times \text{Number of days} = \text{Constant work load}
\]

So, we can set up the following equation:

\[
6 \times 16 = x \times 12
\]

### Solve for \( x \):

\[
96 = 12x
\]

\[
x = \frac{96}{12}
\]

\[
x = 8
\]

### Conclusion:
Sophia would need to hire **8 builders** to complete the extension in 12 days.

Would you like further details on the method or have any other questions?

### Related Questions:
1. What would happen if Sophia wanted the work done in 8 days instead?
2. How long would it take 4 builders to complete the same work?
3. If Sophia could only hire 10 builders, how many days would it take to complete the extension?
4. What if the builders work at different rates? How would you approach the problem then?
5. Can this concept of inverse proportionality be applied to other real-life situations, like machine efficiency?

### Tip:
For work-rate problems, always remember that more workers usually means less time, assuming they all work at the same pace. This inverse relationship is key to solving such questions quickly.

Sophia wants to have an extension built on her house. She is told it will take 6 builders 16 days to build the extension. How many builders would Sophia need to hire if she wanted her extension built in 12 days?

Solve this work-rate problem: If 6 builders can complete a house extension in 16 days, how many builders are needed to finish it in 12 days? This problem explores inverse proportionality and work rates, perfect for students in Grades 6-8.

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