josh80 wrote:Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?

2
3
4
6
8

Approach 1:
Since Work done = #_of_Machines * Rate * Time, we can say that the amount of work done by 6 machines over 12 days is 6 * r * 12 days = 72r.

If we have only 8 days to do the job, the equation becomes x * r * 8 = 72r where x is the number of machines. x=9 machines, so we need an extra 3 machines (on top of the original 6) to do the job in 8 days.

Approach 2:
If you notice that # of machines and time are inversely proportional (meaning that doubling the machines will cut the time in half), we can quickly solve.

the time goes from 12 days to 8 days, so time was multiplied by 2/3. As a result, # of machines will be multiplied by the inverse, 3/2. So the new # of machines will be 6 * 3/2 = 9 machines. We need an extra 3 machines.

josh80 wrote:Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?

2
3
4
6
8

Approach 1: Plug in a rate for EACH MACHINE

Let the rate for each machine = 1 unit per day.
Rate for 6 machines = 6 units per day.
In 12 days, the amount of work produced by 6 machines = r*t = 6*12 = 72 units.
To produce 72 units in 8 days, the required amount of work per day = w/t = 72/8 = 9 units per day.
To increase the rate from 6 units per day to 9 units per day, 3 more machines are needed.

The correct answer is B.

Approach 2: Inverse proportion method

The number of machines is INVERSELY PROPORTIONAL to the number of days:
(machines)(days) = (machines)(days).
As the number of machines INCREASES, the number of days must DECREASE, so that in each case the SAME AMOUNT OF WORK is produced.

Since 6 machines take 12 days, and the job is to be completed in 8 days, we get:
6 * 12 = m * 8
72 = 8m
m = 9.

Since 9 machines are required, the original number of machines -- 6 -- must increase by 3.

The correct answer is B.