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Working together at their respective constant rates, robot A

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Working together at their respective constant rates, robot A [#permalink] New post 06 Jul 2018, 16:20
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Working together at their respective constant rates, robot A and robot B polish 88 pounds of gemstones in 6 minutes. If robot A’s rate of polishing is \(\frac{3}{5}\)that of robot B, how many minutes would it take robot A alone to polish 165 pounds of gemstones?

(A) 15.75
(B) 18
(C) 18.75
(D) 27.5
(E) 30
[Reveal] Spoiler: OA

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Re: Working together at their respective constant rates, robot A [#permalink] New post 07 Jul 2018, 19:10
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Let us assume,

Machine B can do x work in 1 min
or, Machine B can do \(6x\) work in \(6 min\)

It follows that Machine A is only 3/5 efficient as machine A hence,
Machine A can do \(\frac{18}{5} x work in 6 min.\)

From question,

\(\frac{18}{5} x + 6x = 88\)
because working together the two machines can polish \(88 pounds\) of gemstone in \(6 min.\)

solving for x we get \(x = \frac{440}{48}\)
This is the pound of gemstone machine B can polish in 1 min.
The amount of gemstone that machine A can polish in 1 min is \(\frac{440}{48} * \frac{3}{5}\)
Hence, machine A can polish \(\frac{1320}{240}\) pound of gemstone in 1 min.
or, Machine A can polish 1 pound of gemstone in \(\frac{240}{1320}\) min
or, Machine A can polish 165 pound of gemstone in \(\frac{240}{1320} * 165\) min = 30 min
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Re: Working together at their respective constant rates, robot A [#permalink] New post 10 Jul 2018, 06:12
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Explanation

When rate problems involve multiple situations, it can help to set up an initial “skeleton” W= RT chart for the solution. That way, you can determine what data is needed and fill in that data as you find it. Since the question asks how long robot A will take alone, the chart will look like this:

Attachment:
image1.jpg
image1.jpg [ 26.04 KiB | Viewed 3215 times ]


Work is known and the question asks for time, so robot A’s rate is needed. Call the rates a and b. Now set up another chart representing what you know about the two robots working together.

Attachment:
image1.jpg
image1.jpg [ 50.99 KiB | Viewed 3215 times ]


Now, 6(a + b) = 88 and, from the question stem, robot A’s rate is \(\frac{3}{5}\) of B’s rate. This can be written as \(a = \frac{3}{5}b\). To solve for a, substitute for b:

\(a = \frac{3}{5} b\)
\(\frac{5}{3}a = b\)

\(6(a+ \frac{5}{3}a) = 88\)

\(6 \frac{8}{3} a = 88\)
\(a = \frac{11}{2}\)

So A’s rate is pounds per minute. Now just plug into the original chart:

Attachment:
image1.jpg
image1.jpg [ 28.05 KiB | Viewed 3216 times ]


The time robot A takes to polish 165 pounds of gems is \(165/\frac{11}{2} =\frac{330}{165} = 30\) minutes.
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Re: Working together at their respective constant rates, robot A [#permalink] New post 10 Jul 2019, 03:47
amorphous wrote:
Let us assume,

Machine B can do x work in 1 min
or, Machine B can do \(6x\) work in \(6 min\)

It follows that Machine A is only 3/5 efficient as machine A hence,
Machine A can do \(\frac{18}{5} x work in 6 min.\)

From question,

\(\frac{18}{5} x + 6x = 88\)
because working together the two machines can polish \(88 pounds\) of gemstone in \(6 min.\)

solving for x we get \(x = \frac{440}{48}\)
This is the pound of gemstone machine B can polish in 1 min.
The amount of gemstone that machine A can polish in 1 min is \(\frac{440}{48} * \frac{3}{5}\)
Hence, machine A can polish \(\frac{1320}{240}\) pound of gemstone in 1 min.
or, Machine A can polish 1 pound of gemstone in \(\frac{240}{1320}\) min
or, Machine A can polish 165 pound of gemstone in \(\frac{240}{1320} * 165\) min = 30 min



What's the reason the changing the from here, Hence, machine A can polish \(\frac{1320}{240}\) pound of gemstone in 1 min. to here or, Machine A can polish 1 pound of gemstone in \(\frac{240}{1320}\) min ??????
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Re: Working together at their respective constant rates, robot A [#permalink] New post 10 Jul 2019, 05:55
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huda wrote:
amorphous wrote:
Let us assume,

Machine B can do x work in 1 min
or, Machine B can do \(6x\) work in \(6 min\)

It follows that Machine A is only 3/5 efficient as machine A hence,
Machine A can do \(\frac{18}{5} x work in 6 min.\)

From question,

\(\frac{18}{5} x + 6x = 88\)
because working together the two machines can polish \(88 pounds\) of gemstone in \(6 min.\)

solving for x we get \(x = \frac{440}{48}\)
This is the pound of gemstone machine B can polish in 1 min.
The amount of gemstone that machine A can polish in 1 min is \(\frac{440}{48} * \frac{3}{5}\)
Hence, machine A can polish \(\frac{1320}{240}\) pound of gemstone in 1 min.
or, Machine A can polish 1 pound of gemstone in \(\frac{240}{1320}\) min
or, Machine A can polish 165 pound of gemstone in \(\frac{240}{1320} * 165\) min = 30 min



What's the reason the changing the from here, Hence, machine A can polish \(\frac{1320}{240}\) pound of gemstone in 1 min. to here or, Machine A can polish 1 pound of gemstone in \(\frac{240}{1320}\) min ??????


We want to find the time it takes to polish a certain amount of gemstone not the other way round i.e. we are not interested in finding how many gemstones can be polished in a given time.
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Re: Working together at their respective constant rates, robot A [#permalink] New post 13 Oct 2019, 01:22
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sandy wrote:
Working together at their respective constant rates, robot A and robot B polish 88 pounds of gemstones in 6 minutes. If robot A’s rate of polishing is \(\frac{3}{5}\)that of robot B, how many minutes would it take robot A alone to polish 165 pounds of gemstones?

(A) 15.75
(B) 18
(C) 18.75
(D) 27.5
(E) 30


Let, Efficiency of Robot B = 5e
So, Efficiency of Robot A = 3e

Combined efficiency of A and B is 8e = \(\frac{88}{6}\) pounds/min

Or, e = \(\frac{11}{6}\) pounds/min

So, Efficiency of A = \(\frac{33}{6}\) pound/min

Thus, Time taken for robot A to polish 165 gemstones is \(\frac{165*6}{33}\) = 30 minutes, Answer must be (E)

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Re: Working together at their respective constant rates, robot A [#permalink] New post 30 Apr 2020, 04:14
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Can someone explain why the combined work formula doesn't work here? I understand your methodology, but keep getting 18 using that formula. Thanks!
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Re: Working together at their respective constant rates, robot A [#permalink] New post 30 Apr 2020, 05:33
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dsmaier wrote:
Can someone explain why the combined work formula doesn't work here? I understand your methodology, but keep getting 18 using that formula. Thanks!


Question: Working together at their respective constant rates, robot A and robot B polish 88 pounds of gemstones in 6 minutes. If robot A’s rate of polishing is 3/5 that of robot B, how many minutes would it take robot A alone to polish 165 pounds of gemstones?

You wish to use combined work formula: Let us do it:

Time taken by A and B to polish 88 pounds of gems = 6 minutes
Let time by B to polish 88 pounds of gems = x min
So time by A to polish 88 pounds of gems = 5x/3 min
=> time taken by A and B to polish 88 pounds of gems = (x)(5x/3)/(x + 5x/3) = 5x/8 minutes = 6 => x = 48/5 minutes
=> time by A to polish 88 pounds of gems = 5x/3 = 5/3 * 48/5 = 16 minutes
=> time by A to polish 1 pound of gems = 16/88 minutes
=> time by A to polish 165 pound of gems = 16/88 * 165 = 30 minutes

Obviously, it doesn't make sense to do it like this, since it is lengthy

So, let us improvise:

Question: Working together at their respective constant rates, robot A and robot B polish 88 pounds of gemstones in 6 minutes. If robot A’s rate of polishing is 3/5 that of robot B, how many minutes would it take robot A alone to polish 165 pounds of gemstones?

Rate of A = 3/5 of rate of B

thus: If B polishes 5x gems per minute => A will polish 3x gems per minute => Together they polish 8x gems per minute
thus, in 6 minutes they will polish 8x * 6 = 48x gems
thus, this 48x is actually 88
thus: 88 gems is 48x
=> 165 gems is 48x/88 * 165 = 90x gems
A was polishing 3x gems per minute
So, time = 90x/3x = 30 minutes
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Re: Working together at their respective constant rates, robot A [#permalink] New post 30 Apr 2020, 12:23
Thanks a ton - great seeing the comparison, much appreciated.
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Re: Working together at their respective constant rates, robot A [#permalink] New post 25 Aug 2020, 09:43
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sandy wrote:
Working together at their respective constant rates, robot A and robot B polish 88 pounds of gemstones in 6 minutes. If robot A’s rate of polishing is \(\frac{3}{5}\)that of robot B, how many minutes would it take robot A alone to polish 165 pounds of gemstones?

(A) 15.75
(B) 18
(C) 18.75
(D) 27.5
(E) 30


Working together at their respective constant rates, robot A and robot B polish 88 pounds of gemstones in 6 minutes.

rate = output/time
So, if 88 pounds of gemstones are polished in 6 minutes, their combined RATE = 88/6 = 44/3 gemstones per minute

Let A = robot A's RATE in gemstones per minute
Let B = robot B's RATE in gemstones per minute

We can now write: A + B = 44/3 gemstones per minute


Robot A’s rate of polishing is 3/5 that of robot B
So, we can write: A = (3/5)B
If we want to solve this equation for B, we can multiply both sides by 5/3 to get: (5/3)A = B
Or we can express this as: B = 5A/3

We can now take our original equation: A + B = 44/3
And replace B with 5A/3 to get: A + 5A/3 = 44/3
Let's eliminate the fractions by multiplying both sides of the equation by 3 to get: 3A + 5A = 44
Simplify: 8A = 44
Solve: A = 44/8 = 11/2

In other words, robot A's RATE = 11/2 gemstones per minute

How many minutes would it take robot A alone to polish 165 pounds of gemstones?

time = output/rate
So, time = 165/(11/2)
= (165)(2/11)
= 330/11
= 30 minutes

Answer: E

Cheers,
Brent
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Re: Working together at their respective constant rates, robot A [#permalink] New post 25 Aug 2020, 10:26
Robot A and B can polish 88 pounds of gemstones in 6 minutes.
So their combines rate is\(\frac{88}{6}\)
We can write this as:

\(A+B = \frac{88}{6}\) where A and B are the rates of Robot A and Robot B respectively.

Now, we are told that obot A’s rate of polishing is \(\frac{3}{5}\) that of robot B.
This can be written as:

\(B = \frac{5A}{3}\)

Substituting in our rate equation, we get:
\(A+\frac{5A}{3}\) = \(\frac{88}{6}\)

Solving this, we get \(A = \frac{33}{6}\)

We know, \(Work = Rate * Time\) which implies \(Time = \frac{Work}{Rate}\)

plugging the values into this formula:

\(Time = \frac{165*6}{33}\) which is 30.

Therefore the answer is (E)
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Re: Working together at their respective constant rates, robot A [#permalink] New post 25 Aug 2020, 19:18
Since we were given that the rate of a is 3/5 of B
This can be re written as ratio inform of A:B = 3:5
From the information above we can deduce that A polish 33 gems in 6mins while B Polish 55 gems

Then we can say 33gems = 6mins
Then 165gems = xmins

Cross multiply this we have 165*6/33
Which equal to 30

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Re: Working together at their respective constant rates, robot A   [#permalink] 25 Aug 2020, 19:18
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