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

1. Jun 25, 2013

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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3. The attempt at a solution

Can I convert these conditions into equations like this - 1/w1 + 1/w2 + 1/w3 = 1/7.5 ??

2. Jun 25, 2013

HallsofIvy

Staff Emeritus
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).

3. Jun 25, 2013

Does it ask for 1/(w1+w2+ w3+ w4+ w5) or 1/w1+1/w2+1/w3+1/w4+1/w5 ?

Last edited: Jun 25, 2013
4. Jun 25, 2013

5. Jun 25, 2013

haruspex

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?

6. Jun 26, 2013

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 ?

7. Jun 26, 2013

HallsofIvy

Staff Emeritus
I do that a lot!

8. Jun 26, 2013

haruspex

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.)

9. Jun 27, 2013

So I just tried to solve it and got the answer 10/3 , Am I right?

10. Jun 27, 2013

haruspex

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. Jun 28, 2013