rates

fruti_yum wrote:Working simultaneously and independently at an identical constant rate, 4 machines of a certain type can produce a total of x units of product P in 6 days. How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 3x units of product P in 4 days?

A. 24
B. 18
C. 16
D. 12
E. 8

Let's assign a nice value to x (a value that will work well with all of the numbers 3, 4 and 6.

Let's say x = 24

GIVEN: 4 machines make x units in 6 days
This means 4 machines make 24 units in 6 days
So, 4 machines make 4 units in 1 day [if you divide the work time by 6, the output is also divided by 6]
So, 1 machine makes 1 unit in 1 day [if you divide the number of machines by 4, the output is also divided by 4]

From here, we can answer the question How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 3x units of product P in 4 days?
If x = 24, 3x = 72
Our goal is to make 72 units in 4 days.

So, 1 machine makes 4 units in 4 days [if you multiply the work time by 4, the output is also multiplied by 4]
So, 18 machines make 72 units in 4 days [if you multiply the number of machines by 18, the output is also multiplied by 18]

Answer: B

Cheers,
Brent

fruti_yum wrote:Working simultaneously and independently at an identical constant rate, 4 machines of a certain type can produce a total of x units of product P in 6 days. How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 3x units of product P in 4 days?

A. 24
B. 18
C. 16
D. 12
E. 8

We are given that 4 machines can complete x units in 6 days. Thus, the rate of the 4 machines is x/6.
Now, we need to determine the number of machines needed to produce a rate of 3x/4. To calculate that number of machines, we can use the following proportion in which the value in each numerator is the number of machines and the value in each denominator is the corresponding rate of those machines. We can let n = the number of machines needed:

4/(x/6) = n/(3x/4)

24/x = 4n/3x

72x = 4nx

18 = n

Alternate Solution:

If 4 machines can produce a total of x units of product P in 6 days, then 12 machines can produce 3x units of product P in 6 days. To find the number of machines needed to produce 3x units in 4 days, let’s set up an inverse proportion, denoting the number of machines needed by n.

12 x 6 = 4 x n

n = 3 x 6 = 18

Answer: B