Solving the Mystery: How Long to Complete the Job Alone?

In summary, machine A takes 8.4 hours to complete the job, whereas machine B can complete it in 12.4 hours.
  • #1
mindauggas
127
0

Homework Statement



Machine A can do a job, working alone, in 4 hours less than machine B. Working together, they can complete
the job in 5 hours. How long would it take each machine, working alone, to complete the job?

Homework Equations



Ans. Machine A: 8.4 hours; machine B: 12.4 hours, approximately

The Attempt at a Solution



Don't know how to reason this one through ...
 
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  • #2
Let the time that machine B takes to carry out the job be b hours.

Using the information we are given, we can write the time that machine A takes to carry out the job as the expression ... ?
 
  • #3
NascentOxygen said:
Let the time that machine B takes to carry out the job be b hours.

Using the information we are given, we can write the time that machine A takes to carry out the job as the expression ... ?

a=b-4

And then ... ?
 
  • #4
mindauggas said:
a=b-4

And then ... ?

In problems of this type you need to ASSUME something about how two machines work together. Clearly, the work times themselves don't add, so what does happen?

If we assume the RATES add (so that rate(A+B) = rate(A) + rate(B)) then we get two equations for the rates. If Ra = rate of A and Rb = rate of B (meaning the number of jobs per hour these machines can complete), time(A) = 1/Ra, time(B) = 1/Rb, and time(A+B) = 1/(Ra+Rb). You can take it from there.

RGV
 
  • #5
Ray Vickson said:
In problems of this type you need to ASSUME something about how two machines work together. Clearly, the work times themselves don't add, so what does happen?

If we assume the RATES add (so that rate(A+B) = rate(A) + rate(B)) then we get two equations for the rates. If Ra = rate of A and Rb = rate of B (meaning the number of jobs per hour these machines can complete), time(A) = 1/Ra, time(B) = 1/Rb, and time(A+B) = 1/(Ra+Rb). You can take it from there.

RGV

So in this particular case i have

time(A) = 1/Ra = B-4

time(B) = 1/Rb = B

time(A+B) = 1/(Ra+Rb) = [itex]\frac{1}{(\frac{1}{(B-4)})+(\frac{1}{B})}[/itex]

But is it really the case?
 
  • #6
mindauggas said:
So in this particular case i have

time(A) = 1/Ra = B-4

time(B) = 1/Rb = B

time(A+B) = 1/(Ra+Rb) = [itex]\frac{1}{(\frac{1}{(B-4)})+(\frac{1}{B})}[/itex]

But is it really the case?

If the addition-of-rates assumption holds, then YES, that is the case; if the addition-of-rates assumption does not hold, then NO, that is not the case. The question itself (at least as you stated it here) leaves undetermined the manner in which two machines work together; the additive-rates method applies in some real-world situations (at least approximately), but not in some others.

Instead of further agonizing about the problem, I suggest you just go ahead and solve it to see what you get.

RGV
 
  • #7
I get the quadratic B[itex]^{2}[/itex]-14B+20=0

B[itex]_{1}[/itex]=12,4 (so we get the correct answer)

B[itex]_{2}[/itex]=1,6 (i need to discard it, because the other assumption is A=B-4 and we can't have a negative work done (at least in this situation)).

Thanks RGV/
 

1. What is the purpose of solving the mystery of how long it takes to complete a job alone?

The purpose of solving this mystery is to determine the efficiency and productivity of an individual when working alone. This information can be useful in planning and organizing tasks, setting deadlines, and making decisions on whether to assign a job to a single individual or a team.

2. How can the time it takes to complete a job alone be calculated?

The time it takes to complete a job alone can be calculated by dividing the total amount of work by the work rate (or productivity) of the individual. This work rate can be determined by observing the time it takes for the individual to complete similar tasks or by conducting timed trials.

3. What factors can affect the time it takes to complete a job alone?

The time it takes to complete a job alone can be affected by various factors, such as the complexity and difficulty of the task, the skill level and experience of the individual, the availability of resources and tools, and any external distractions or interruptions.

4. Is there a specific formula for calculating the time it takes to complete a job alone?

There is no specific formula for calculating the time it takes to complete a job alone, as it can vary depending on the individual and the specific task at hand. However, the general concept is to divide the total amount of work by the work rate of the individual.

5. How can knowing how long it takes to complete a job alone benefit organizations and individuals?

Knowing how long it takes to complete a job alone can benefit organizations and individuals by allowing them to effectively plan and manage their time and resources. It can also help in setting realistic goals and expectations, improving productivity and efficiency, and making informed decisions on task delegation and workload distribution.

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