Question

All members of our painting team paint at the same rate. If $12$ members can paint a specific wall in $12$ minutes, then how long would it take $18$ members to paint the same wall, in minutes?

Answer 1

We start by determining the total amount of work done by the painting team. The work can be quantified in terms of member-minutes, where one member working for one minute contributes one unit of work.

Given that 12 members can complete the painting of a wall in 12 minutes, we can calculate the total work done in terms of member-minutes:

\[
\text{Total Work} = \text{Number of Members} \times \text{Time} = 12 \, \text{members} \times 12 \, \text{minutes} = 144 \, \text{member-minutes}
\]

Now, we want to find out how long it would take for 18 members to finish the same wall, which still requires a total of 144 member-minutes of work.

Let \( t \) be the time in minutes it takes for 18 members to paint the wall. The total work done by 18 members in \( t \) minutes can be expressed as:

\[
\text{Total Work} = 18 \, \text{members} \times t \, \text{minutes} = 18t \, \text{member-minutes}
\]

Since the total work required is the same (144 member-minutes), we set up the equation:

\[
18t = 144
\]

Next, we solve for \( t \):

\[
t = \frac{144}{18}
\]

Calculating the right-hand side:

\[
t = 8
\]

Thus, it would take 18 members 8 minutes to paint the wall. The final answer is:

\[
\boxed{8}
\]