Inverse Variation-People And Days

[lingping7]

I cannot explain the problem so i will write the problem itself down.
A&B can do a piece of work in 8 days, B&C in 6 days and C&A in 12 days. How many days will it take if all of them do the work together?
Please tell me how to solve questions like these

Thanks in advance ;-)

 

[pappus]

I cannot explain the problem so i will write the problem itself down.
A&B can do a piece of work in 8 days, B&C in 6 days and C&A in 12 days. How many days will it take if all of them do the work together?
Please tell me how to solve questions like these

Thanks in advance ;-)

1. Let
a = work done by A in 1 day
b = work done by B in 1 day
c = work done by C in 1 day

w = the complete work.

2. Then you know:

\(\displaystyle \displaystyle{\left| \begin{array}{lcr}8a+8b&=&w \\ 6b+6c& =& w \\12c + 12a &=& w \end{array} \right.}\)

Solve for (a, b, c) with respect to w.

3. You should come out with \(\displaystyle \displaystyle{(a,b,c) = \left(\frac w{48} , \frac{5w}{48} , \frac w{16} \right)}\)

You know now how much work is finished by a or b or c in one day.

4. Working together means

\(\displaystyle \displaystyle{\left(\frac w{48} + \frac{5w}{48} + \frac w{16} \right)}\)

is the work done in one day when all three work together.

So how much time does it take to complete the work?

 

[HallsofIvy]

Yet another way to do this: if people work together (or machines, or pipes filling a reservoir, etc.), it is their rates that add.

Let A, B, and C represent the number of days it would take "A", "B", and "C", respectively, to do the job. The rates are 1/A, 1/B, and 1/C "jobs per day" respectively. When A and B work together, their rate is 1/A+ 1/B= (A+ B)/AB. Since it takes them, together, 8 days to do the job so their rate is "1 job per 8 days" or 1/8 job/day, (1/A+ 1/B))= (A+ B)/AB= 1/ 8 job/day. When B and C work together, their rate is 1/B+ 1/C= (B+ C)/BC. Since it takes them, together, 6 days to do the job, their rate is "1 job per 6 days" or 1/6 job/day. When A and C work together, their rate is 1/A+ 1/C= (A+ C)/AC. Since it takes them, together, 12 days to do the job, their rate is "1 job per 12 days" or 1/12 job/day.

All there working together do work at 1/A+ 1/B+ 1/C= (BC+ AC+ AB)/(ABC) job/day.

(A+ B)/AB+ (B+ C)/BC+ (A+ C)/AC= (A+ B)C/ABC+ (B+ C)A/ABC+ (A+ C)B/ABC= (AC+ BC+ AB+ AC+ AB+ BC)/ABC= 2(AB+AC+ BC)/ABC so that we have
1/8+ 1/6+ 1/12= 3/24+ 4/24+ 2/24= 9/24= 3/8. That is, the three together work at a rate of 3/8 "job per day" and so it will take them 8/2 or 2 and 2/3 days to complete the job.