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:

$$\boxed{{\displaystyle 8}}$$