Word problem : Finding the time in which workers can do a job .

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Homework Help Overview

The problem involves determining the time it takes for five workers to complete a job based on their individual and combined work rates. The workers' combined efforts are described through several equations representing different groupings and their respective completion times.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss converting the conditions into equations based on work rates. There is a focus on the relationship between individual rates and combined rates, with some questioning how to handle the equations with more unknowns than equations. Others explore the implications of the provided information and its potential to simplify the problem.

Discussion Status

The discussion is active, with participants offering insights into the nature of the equations and the relationships between the workers' rates. Some have identified a special characteristic of the data that may simplify the solution process. There is an ongoing exploration of how to approach the final calculation without needing to solve for each worker's individual rate.

Contextual Notes

Participants note that there are four equations available for five unknowns, leading to discussions about the implications of this setup. The problem does not explicitly require finding each worker's rate, which is a point of clarification among participants.

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Homework Statement

Each one of the five workers W1, W2, W3, W4 and W5 can do a certain job.
W1, W2, W3 together can do it in 7.5 hours.
W1, W3, W5 together can do it in 5 hours.
W1, W3, W4 together can do it in 6 hours.
W2, W4, W5 together can do it in 5 hours.
Find the time in which all five together can complete the job.

Homework Equations



-

The Attempt at a Solution



Can I convert these conditions into equations like this - 1/w1 + 1/w2 + 1/w3 = 1/7.5 ??
Please help !
 
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Yes, that's exactly right. When people work together, their rates add. If W1 can do a job in w1 hours, he/she works at a rate of 1/w1 "job per hour".
So
"Each one of the five workers W1, W2, W3, W4 and W5 can do a certain job.
W1, W2, W3 together can do it in 7.5 hours." 1/w1+ 1/w2+ 1/w3= 1/7.5
"W1, W3, W5 together can do it in 5 hours." 1/w1+ 1/w3+ 1/w5= 1/5
"W1, W3, W4 together can do it in 6 hours." 1/w1+ 1/w3+ 1/w4= 1/6
"W2, W4, W5 together can do it in 5 hours." 1/w1+ 1/w4+ 1/w5= 1/5

Since you have only four equations in five unknowns, you cannot solve for the five values separately. Fortunately, the problem does not ask you to. It asks for 1/(w1+w2+ w3+ w4+ w5).
 
HallsofIvy said:
Yes, that's exactly right. When people work together, their rates add. If W1 can do a job in w1 hours, he/she works at a rate of 1/w1 "job per hour".
So
"Each one of the five workers W1, W2, W3, W4 and W5 can do a certain job.
W1, W2, W3 together can do it in 7.5 hours." 1/w1+ 1/w2+ 1/w3= 1/7.5
"W1, W3, W5 together can do it in 5 hours." 1/w1+ 1/w3+ 1/w5= 1/5
"W1, W3, W4 together can do it in 6 hours." 1/w1+ 1/w3+ 1/w4= 1/6
"W2, W4, W5 together can do it in 5 hours." 1/w2+ 1/w4+ 1/w5= 1/5

Since you have only four equations in five unknowns, you cannot solve for the five values separately. Fortunately, the problem does not ask you to. It asks for 1/(w1+w2+ w3+ w4+ w5).
Does it ask for 1/(w1+w2+ w3+ w4+ w5) or 1/w1+1/w2+1/w3+1/w4+1/w5 ?
 
Last edited:
Guys please help ! I'm not able to solve it.
 
agoogler said:
Does it ask for 1/(w1+w2+ w3+ w4+ w5) or 1/w1+1/w2+1/w3+1/w4+1/w5 ?
You're right, almost. Halls confused himself at the final step. You need 1/(1/w1+1/w2+1/w3+1/w4+1/w5).
It would be easier to think about working in terms of rates, r1..r5, rather than these inverted rates. But the equations are essentially the same: r1+r2+r3 = 1/7.5 etc. All working together, the total rate is r1+r2+r3+r4+r5. How do you turn that into the time they'll take?
Btw, there's something a bit special about the provided information which allows you to get the answer without finding all of r1 to r5. Can you see what it is?
 
haruspex said:
You're right, almost. Halls confused himself at the final step. You need 1/(1/w1+1/w2+1/w3+1/w4+1/w5).
It would be easier to think about working in terms of rates, r1..r5, rather than these inverted rates. But the equations are essentially the same: r1+r2+r3 = 1/7.5 etc. All working together, the total rate is r1+r2+r3+r4+r5. How do you turn that into the time they'll take?
Btw, there's something a bit special about the provided information which allows you to get the answer without finding all of r1 to r5. Can you see what it is?
The reciprocal of the total rate should be the time required . Um , since the first condition says r1+r2+r3=1/7.5 , I think I only need to find r4 and r5 then add it to that. Right ?
 
haruspex said:
You're right, almost. Halls confused himself at the final step.
I do that a lot!

You need 1/(1/w1+1/w2+1/w3+1/w4+1/w5).
It would be easier to think about working in terms of rates, r1..r5, rather than these inverted rates. But the equations are essentially the same: r1+r2+r3 = 1/7.5 etc. All working together, the total rate is r1+r2+r3+r4+r5. How do you turn that into the time they'll take?
Btw, there's something a bit special about the provided information which allows you to get the answer without finding all of r1 to r5. Can you see what it is?
 
agoogler said:
The reciprocal of the total rate should be the time required . Um , since the first condition says r1+r2+r3=1/7.5 , I think I only need to find r4 and r5 then add it to that. Right ?
Yes. (The special fact about the given data is that W1 and W3 always occur together, so as far as the sum of all five is concerned they constitute only one unknown.)
 
haruspex said:
Yes. (The special fact about the given data is that W1 and W3 always occur together, so as far as the sum of all five is concerned they constitute only one unknown.)
So I just tried to solve it and got the answer 10/3 , Am I right?
 
  • #10
agoogler said:
So I just tried to solve it and got the answer 10/3 , Am I right?

Yes. Fwiw, the easiest way is to take all the equations like r1+r2+r3=1/7.5, add them up, and add the last one (r2, r4, r5) in again. On the LHS you then have 3(r1+r2+r3+r4+r5).
 
  • #11
haruspex said:
Yes. Fwiw, the easiest way is to take all the equations like r1+r2+r3=1/7.5, add them up, and add the last one (r2, r4, r5) in again. On the LHS you then have 3(r1+r2+r3+r4+r5).
LOL , I didn't notice that. Thanks !
 

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