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

[agoogler]

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

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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 !

 

[HallsofIvy]

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).

 

[agoogler]

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 ?

 

[agoogler]

Guys please help ! I'm not able to solve it.

 

[haruspex]

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?

 

[agoogler]

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 ?

 

[HallsofIvy]

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?

 

[haruspex]

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.)

 

[agoogler]

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?

 

[haruspex]

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).

 

[agoogler]

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 !