If three people can paint a house in 12 days, how many days would it take for 6 people to paint the same house?

Step 1: Understand the problem. The problem is about the relationship between the number of people painting a house and the time it takes to finish the job.

Step 2: Identify the knowns. We know that 3 people can paint a house in 12 days.

Step 3: Identify the unknown. We want to find out how many days it would take for 6 people to paint the same house.

Step 4: Set up the relationship. The time it takes to paint a house is inversely proportional to the number of people painting. This means that if you double the number of people, it should take half the time.

Step 5: Use the relationship to solve for the unknown. If 3 people can paint a house in 12 days, then doubling the number of people to 6 should halve the time it takes to 12/2 = 6 days.

So, it would take 6 people 6 days to paint the same house.

Sure, let's solve this step by step.

1. Determine the total amount of work needed to paint the house:

   • If 3 people can paint the house in 12 days, we can calculate the total amount of work in "person-days."
   • Total work = Number of people × Number of days
   • Total work = 3 people × 12 days = 36 person-days

2. Calculate the number of days it would take for 6 people to complete the same amount of work:

   • We know the total work required is 36 person-days.
   • If 6 people are working, we need to find out how many days it will take for them to complete 36 person-days of work.
   • Let $D$ be the number of days it takes for 6 people to paint the house.
   • Total work = Number of people × Number of days
   • 36 person-days = 6 people × $D$ days

3. Solve for $D$:

   • $D = \frac{36 \text{ person-days}}{6 \text{ people}}$
   • $D = 6 \text{ days}$

So, it would take 6 people 6 days to paint the house.