Three machines, K, M, and P, working simultaneously and

Three machines, K, M, and P, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes. How long does it take Machine K, working alone at its constant rate, to complete the task?

Say k, m, and p are the numbers of minutes machines K, M, and P take, respectively, to complete the task. Then we have that \(\frac{1}{k}+\frac{1}{m}+\frac{1}{p}=\frac{1}{24}\).

(1) Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes --> \(\frac{1}{m}+\frac{1}{p}=\frac{1}{36}\), thus \(\frac{1}{k}+\frac{1}{36}=\frac{1}{24}\) --> we can find the value of \(k\). Sufficient.

(2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes --> \(\frac{1}{k}+\frac{1}{p}=\frac{1}{48}\). The value of k cannot be determined from the data we have. Not sufficient.

Answer: A.
_________________

Three machines, K, M, and P, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes. How long does it take Machine K, working alone at its constant rate, to complete the task?

(1) Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes.
RATE K + M + P) - RATE : (M+P) = RATE : K. SUFFICIENT
(2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.
RATE K + M + P) - RATE : (K+P) = RATE : M. NOT SUFFICIENT
_________________

I approached this pbm a little differently. Pls. Explain where I am going wrong...is it OK to reason this way?

Let Rk, Rm and Rp be the rates for the machines K,M and P respectively.
Then 1/Rk +1/Rm+1/Rp = 24

St 1 gives ---> 1/Rm + 1/Rp = 36

So, we get 1/Rk + 36 = 24.

Solving, 1/Rk = 24-36=-12

Why am I getting a negative value?

1/Rk, 1/Rm, and 1/Rp are the numbers of minutes machines K, M, and P take to complete the task alone. Each must be greater than the time needed for three machines to complete a certain task together (24 minutes), thus 1/Rk +1/Rm+1/Rp = 24 is not right. The same for 1/Rm + 1/Rp = 36.

Hope it's clear.
_________________

Got it. So would it be correct to say that 1/ (Rk+Rm+Rp) = 24? (since the denominator has combined rate now)?

We are given that machines K, M, and P, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes. If we consider the entire task to be equal to 1, and the time in minutes for machines K, M, and P to complete the task to be k, m, and p, respectively, then the rates of machines K, M, and P are:

1/k = rate of machine K to complete the task

1/m = rate of machine M to complete the task

1/p = rate of machine P to complete the task

Since it takes machines K, M, and P, working simultaneously and independently, 24 minutes, the combined rate of machines K, M, and P is 1 task per 24 minutes. That is:

1/k + 1/m + 1/p = 1/24

We need to determine how long it takes machine K to complete the task, or in other words, the value of k. Since 1/k + 1/m + 1/p = 1/24, the rate of machine K is:

1/k = 1/24 - 1/m - 1/p

1/k = 1/24 - (1/m + 1/p)

Thus, if we can determine the value of (1/m + 1/p), we can determine the value of 1/k and hence the value of k.

Statement One Alone:

Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes.

From statement one we know:

1/m + 1/p = 1/36

Thus, the rate for machine K to complete the task is 1/24 - 1/36 = 3/72 - 2/72 = 1/72, and therefore, the time for machine K to complete the task is 72 minutes.

Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.

From statement two we know:

1/k + 1/p = 1/48

Since we don’t know the value of p, this is not enough information to determine the value of k.

Statement two alone is not sufficient to answer the question.

Answer: A
_________________

At first glance it was (D) for me as both statement looks identical. Bunuel, why we are not able to answer the question with Statement 2.
_________________

Good question.

We are given \(\frac{1}{k}+\frac{1}{m}+\frac{1}{p}=\frac{1}{24}\). and want to find the value of k.

(2) says that \(\frac{1}{k}+\frac{1}{p}=\frac{1}{48}\). If we substitute this above, we'll get: \(\frac{1}{m}+\frac{1}{48}=\frac{1}{24}\) (linear equation with one unknown m). From this we can find that m = 48 but still no way of finding k.

In (1) on the other hand we are also getting a linear equation with one unknown, but that unknown there is k itself: \(\frac{1}{k}+\frac{1}{36}=\frac{1}{24}\).

Hope it helps.
_________________

Given: Three machines, K, M, and P, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes.

Asked: How long does it take Machine K, working alone at its constant rate, to complete the task?

1/K + 1/M + 1/P = 1/24

(1) Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes.
1/M + 1/P = 1/36
1/K = 1/24 -1/36 = 1/72
K = 72 mins
SUFFICIENT

(2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.
1/K + 1/P = 1/48
Value of K can not be found out
NOT SUFFICIENT

IMO A

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________