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Working simultaneously at their respective constant rates, M [#permalink]

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06 Dec 2012, 09:58

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Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

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06 Dec 2012, 12:28

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Walkabout wrote:

Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

Say x=1 hour and y=2 hours (notice that y must be greater than x, since the time for machine A to do the job, which is y hours, must be more than the time for machines A and B working together to do the same job, which is x hours).

In this case, the time needed for machine B to do the job must also be 2 hours: 1/2+1/2=1.

Now, plug x=1 and y=2 in the options to see which one yields 2. Only option E fits.

Q: I get the algebra but I got confused with this question because I thaught adding and deviding rates was a NoNo? Why is it diffrent in this case?

No, we CAN easily sum the rates. For example:

If we are told that A can complete a job in 2 hours and B can complete the same job in 3 hours, then A's rate is 1/2 job/hour and B's rate is 1/3 job/hour. The combined rate of A and B working simultaneously would be 1/2+1/3=5/6 job/hours, which means that the will complete 5/6 job in hour working together.

THEORY There are several important things you should know to solve work problems:

1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems.

\(time*speed=distance\) <--> \(time*rate=job \ done\). For example when we are told that a man can do a certain job in 3 hours we can write: \(3*rate=1\) --> \(rate=\frac{1}{3}\) job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then \(5*(2*rate)=1\) --> so rate of 1 printer is \(rate=\frac{1}{10}\) job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then \(3*(2*rate)=12\) --> so rate of 1 printer is \(rate=2\) pages per hour;

So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate).

2. We can sum the rates.

If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is \(rate_a=\frac{job}{time}=\frac{1}{2}\) job/hour and B's rate is \(rate_b=\frac{job}{time}=\frac{1}{3}\) job/hour. Combined rate of A and B working simultaneously would be \(rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\) job/hour, which means that they will complete \(\frac{5}{6}\) job in one hour working together.

3. For multiple entities: \(\frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T}\), where \(T\) is time needed for these entities to complete a given job working simultaneously.

For example if: Time needed for A to complete the job is A hours; Time needed for B to complete the job is B hours; Time needed for C to complete the job is C hours; ... Time needed for N to complete the job is N hours;

Then: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T}\), where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously.

For two and three entities (workers, pumps, ...):

General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:

Given that \(t_1\) and \(t_2\) are the respective individual times needed for \(A\) and \(B\) workers (pumps, ...) to complete the job, then time needed for \(A\) and \(B\) working simultaneously to complete the job equals to \(T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2}\) hours, which is reciprocal of the sum of their respective rates (\(\frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T}\)).

General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:

Re: Working simultaneously at their respective constant rates, M [#permalink]

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17 Nov 2013, 02:11

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Simplest solution Here-

RA + RB = 1/X

RA = 1/Y

RB = (1/X - 1/Y) = (Y-X)/XY

Time = 1/RB = XY/(X+Y)

I have treated 800 as equivalent to unity(= 1), as it's presence in final answer was trivial, as it will eventually cancel out, taking it unity has make the solution quite Un Complex..
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Re: Working simultaneously at their respective constant rates, M [#permalink]

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04 Oct 2013, 08:31

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Jp27 wrote:

Walkabout wrote:

Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

The above sol is awesome.... but i did it the longer way, algebraically...

rate of A be a and B be b

a + b = \(800/x\) ..... 1 a = \(800/y\) ..... 2

Use 2 in 1... we get

b = \(800 (y-x) / xy\)

Finally

Rate of B * time = Work done by B (we want time)

\(800 (y-x) / xy * t = 800\)

t = \(xy / (y-x)\)

Yeah, R*T=W is a lengthy way to solve these problems but, I have seen that it is almost a sure shot way to solve most of the problems on this concept. Picking up the smart numbers may be a neat way to solve these questions but it highly depends on the mental state when you are taking the exam.
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Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

Re: Working simultaneously at their respective constant rates, M [#permalink]

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29 Mar 2014, 03:50

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Another approach:

Rate(A) = 800/y Rate(A+B) = 800/x

Rate A + Rate B = Rate(A+B)

=> Rate(B) = Rate(A+B) - Rate(A) = 800(y-x)/xy

Then the GODFATHER equation Rate * Time = Work

800(y-x)/xy * Time = 800

Time = xy/(y-x)

Rgds, TGC!
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HI Brunel I get it sir, just got confused with this "If an object moves the same distance twice, but at different rates, then the average rate will NEVER be the average of the two rates given for the two legs of the journey.(MGMAT)"

This is about completely different matter: it says that if an object covers 100 miles at 10 miles per hour and another 100 miles at 20 miles per hour, then the average speed for 200 miles won't be (10+20)/2=15 miles per hour.

(average speed) = (total distance)/(total time):

(total distance) = 100 + 100 = 200 miles.

(total time) = 100/10 + 100/20 = 15 hours.

(average speed) = (total distance)/(total time) = 200/15 miles per hour.

Notice here though that we can add or subtract rates (speeds) to get relative rate.

For example if two cars are moving toward each other from A to B (AB=100 miles) with 10mph and 15mph respectively, then their relative (combined) rate is 10+15=25mph, and they'll meet in (time)=(distance)/(rate)=100/25=4 hours;

Similarly if car x is 100 miles ahead of car y and they are moving in the same direction with 10mph and 15mph respectively then their relative rate is 15-10=5mph, and y will catch up x in 100/5=20 hours.

Re: Working simultaneously at their respective constant rates, M [#permalink]

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02 Nov 2013, 08:20

Jp27 wrote:

Walkabout wrote:

Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

Re: Working simultaneously at their respective constant rates, M [#permalink]

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06 Apr 2014, 16:49

I managed to solve this via plugging in numbers but I went over 3 minutes! With these plug-in-numbers type of problems, I strive to pick simple numbers but there always seems to be one, usually the one i'm solving for, that ends up being a rather complicated number. My question is:

1) I plugged in Rate A and Rate A+B and then solved for time. Should I have plugged in numbers for time directly. Is there a general rule as to what number I should be plugging in? 2) In this case, to keep things super simple, I could have plugged the total Rate to be 8 and Ra and Rb to both be 4. Is it bad practice to choose the same numbers for the individual rates/times?

Re: Working simultaneously at their respective constant rates, M [#permalink]

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07 Apr 2014, 01:04

russ9 wrote:

I managed to solve this via plugging in numbers but I went over 3 minutes! With these plug-in-numbers type of problems, I strive to pick simple numbers but there always seems to be one, usually the one i'm solving for, that ends up being a rather complicated number. My question is:

1) I plugged in Rate A and Rate A+B and then solved for time. Should I have plugged in numbers for time directly. Is there a general rule as to what number I should be plugging in? 2) In this case, to keep things super simple, I could have plugged the total Rate to be 8 and Ra and Rb to both be 4. Is it bad practice to choose the same numbers for the individual rates/times?

Just refer to method of Bunuel; he did using plug-ins.

I used same variables available & got correct answer (Had taken 800 = 1 as done by honchos)
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Re: Working simultaneously at their respective constant rates, M [#permalink]

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22 Apr 2014, 04:24

HI Brunel I get it sir, just got confused with this "If an object moves the same distance twice, but at different rates, then the average rate will NEVER be the average of the two rates given for the two legs of the journey.(MGMAT)"

Re: Working simultaneously at their respective constant rates, M [#permalink]

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09 May 2014, 14:24

Bunuel wrote:

Walkabout wrote:

Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

Say x=1 hour and y=2 hours (notice that y must be greater than x, since the time for machine A to do the job, which is y hours, must be more than the time for machines A and B working together to do the same job, which is x hours).

In this case, the time needed for machine B to do the job must also be 2 hours: 1/2+1/2=1.

Now, plug x=1 and y=2 in the options to see which one yields 2. Only option E fits.

Answer: E.

Hi Bunuel,

I managed to solve this via algebra but it took 2+ minutes. When it comes to plugging in "smart numbers", I always get confused as to which variables I should use to plug in smart numbers vs. which numbers I should solve for.

In this example, I get completely throw off if I should be plugging in numbers for time(numbers that factor in 800) or if I should plug in numbers for Rate. I get paralysis by analysis when I think about whether the numbers I pick will go flawlessly and thereby Ra and Rb will add up to R a + b or will the factors yield decimals?

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