To use the rate formula, we can go through a process like this:
We know that our rate formula is 1/A + 1/B = 1/A and B, where A is the amount of time worker A takes to finish a job alone, B is the amount of time worker B takes to finish a job alone, and A and B is the amount of time workers A and B take to finish a job together. We can use this formula for as many workers as we want. Here, we have six workers, so we use the equation 1/A + 1/B + 1/C + 1/D + 1/E + 1/F = 1/A, B, C, D, E, and F.
We know that the six workers build 4 cars in 2 days. This also means that they build 1 car in 1/2 days. So we'll fill in 1/2 for A, B, C, D, E, and F:
1/A + 1/B + 1/C + 1/D + 1/E + 1/F = 1/(1/2)
We assume that all individual workers work at the same pace, so we'll call the number of days it takes for a single worker to build 1 car x:
1/x + 1/x + 1/x + 1/x + 1/x + 1/x = 2
6/x = 2
Now we solve for x:
x = 2
So we know it takes a single worker 3 days to build 1 car. Let's set up an equation for how long it would take 8 workers to build one car:
8(1/3) = 1/A, B, C, D, E, F, G, and H
Now we solve for the number of days is would take 8 workers to build one car:
8/3 = 1/A, B, C, D, E, F, G, and H
1/A, B, C, D, E, F, G, and H = 3/8
So it would take 8 workers 3/8 of a day to build 1 car. To find the number of days it would take to build 6 cars, we multiply by 6:
3/8 * 6 = 18/8 = 9/4
So it would take 8 workers 9/4 days to build 6 cars.
We can also go through this logically by understanding that time will be proportional to number of cars and inversely proportional to number of workers:
6 workers 4 cars = 2 days
1 worker 4 cars = 2 * 6 = 12 days
1 worker 1 car = 12 / 4 = 3 days
1 worker 6 cars = 3 * 6 = 18 days
8 workers 6 cars = 18 / 8 = 9/4 days
