Working simultaneously and independently at an identical constant rate

1) First let's set up an equation for the 4 machines and find a rate of each machine: \(4r*6=x; 4r=\frac{x}{6}, r=\frac{x}{6}*\frac{1}{4}, r=\frac{x}{24}\)
2) Now let's set up an equation for 3x units: let N be the number of machines, then \(N*\frac{x}{24}*4=3x; N*\frac{x}{6}=3x, N=\frac{3x}{x/6}=18\)

The correct answer is B.

This is a wonderful problem to make an educated guess on. Rates & work problems are great for guessing in general.

Four machines make a certain amount of stuff in 6 days. We want to make three times as much stuff. That means we need at least three times as many machines, unless we also get extra time! (We don't.) So, eliminate 8.

We also have fewer days to spend making the stuff than we did originally. So, we can't get away with just having three times as many machines. We need more than three times as many. Eliminate 12.

Do we need 24 machines, though? That's 6 times as many as we had before. If you have six times as many machines, and you're only trying to make three times as much stuff, you should need half as much time. But we needed 4 days, not 3. Eliminate 24.

Guess 16 or 18 and move on!
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Let’s say x=600 units of P. Then, there are 100 units produced per day (600/6).

And since there are 4 machines, each machine produces 25 units per day (100/4).

We want to know how many machines it takes to produce 3x. Since we assumed x is 600, 3x is 1800 units of P. There must be 450 units produced per day (1800/4). And because one machine produces 25 units per day we will need 18 machines (450/25).


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Hi math expert Bunuel,
Can you please let me know about any practice problems similar to this?
Would really appreciate your help!
Thanks

17. Work/Rate Problems

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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

Answer: Option B

Video solution by GMATinsight



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4 machines - 6 days - x units
4 machines -18 days - 3x units
MM machines - 4 days - 3x units
then (4 machines)*(18 days)=(M machines)*(4 days) --> M = 18

Video solution from Quant Reasoning:
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4M*6 = X

Therefore: 3X = 4M*6*3

Now 3X work is 4M*6*3, therefore 4M*6*3/m*4 = 18 (in denominator 4 stands for days)
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Thanks GMATinsight for the brilliant work rate machine formula here that make it easy peasy like a lemon squeezy

I solved it in the unitary method.
6 days are required for 4 machines (to produce x units)
1 day is required for 4*6= 24 machines (to produce x units)
4 days are required for 24/4= 6 machines (to produce x units)

Now for 3x units, 3*6 =18 machines are required.

Each machine has a rate of x/6 divided by 4 = x/24

To complete 3x work in 4 days you need a rate of

R T W
(3x/4 days) 4days 3X

Therefore number of machines needed is

Rate to complete in 4 days / individual machine rate

3x/4 / x/24 = 18 days

Let's say one machine can produce r units of product P per day. Then, in 6 days, four machines can produce a total of 4r x 6 = 24r units of product P. We are given that this equals x, so:

24r = x

We want to know how many machines are needed to produce 3x units of product P in 4 days. Let's call the number of machines we need n. Then, in 4 days, n machines can produce:

nr x 4

units of product P. We want this to be equal to 3x, so:

nr x 4 = 3x

We can substitute x = 24r into this equation to get:

nr x 4 = 3 x 24r

Simplifying:

4nr = 72r

Dividing both sides by r (since r is not equal to 0):

4n = 72

So:

n = 18

Therefore, we need 18 machines to produce a total of 3x units of product P in 4 days. The answer is (B) 18.

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?

4 machine ---- x units ---- 6 days

4 machine ---- 3x units --- 18 days

1 machine ---- 3x units ---- 72 days

N machine ---- 3x units ---- 4 days

=> 74/4 = 18 machines

Hope it helps!

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

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