Question

It takes 23 minutes for 6 people to paint 6 walls. How many minutes does it take 9 people to paint 7 walls?

Answer 1

For this problem you need to first find a unit rate (rate = 1/time), how fast per minute can 1 person paint a wall?

Once you have this rate you can find how many minutes it takes to do any number of walls with any number of people.

Lets call this unit rate 'r'. Number of people = n, Number of walls = w.
By multiplying the unit rate by 'n', you get the speed n people can paint a wall. Example, if it takes 4 min to paint 1 wall the unit rate = 1/4. The speed for 8 people is 8*(1/4) = 2 walls per minute. Time = 1/rate. So 2 walls per minute = 1/2 min per wall.  Now multiply time by w. If there are 10 walls, (1/2)*10 = 5 , It takes 8 people 5 min to paint 10 walls.

The general formula is:
$t = \frac{w}{n*r} \\ \\ r = \frac{w}{n*t}$

Now apply to given problem:
Time = 23 min, n = 6, w = 6
Solve for unit rate 'r'
$r = \frac{6}{6*23} = \frac{1}{23}$
Find time, given
n = 9, w = 7
$t = \frac{7}{9*(1/23)} = \frac{7*23}{9} = 17.89$
Answer:
it takes 17.89 min or 17 min 53 sec