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

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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
Bunuel wrote:
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.

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?
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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
audiogal101 wrote:
Bunuel wrote:
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.

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.
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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
Bunuel wrote:
audiogal101 wrote:
Bunuel wrote:
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.

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)?
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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
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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.
(2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.

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.

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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
Bunuel wrote:
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.

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.
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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
QZ wrote:
Bunuel wrote:
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.

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.
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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
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.
(2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.

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
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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
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Re: Three machines, K, M, and P, working simultaneously and [#permalink]
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