Question 167380
A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together?


A and B can do a piece of work in 42 days,
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So A's and B's combined rate is 1 job per 42 days or {{{(1_job)/(42_days)}}} or {{{1/42}}}{{{jobs/day}}}
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B and C in 31 days
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So B's and C's combined rate is 1 job per 31 days or {{{(1_job)/(31_days)}}} or {{{1/31}}}{{{jobs/day}}}
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C and A in 20 days
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So C's and A's combined rate is 1 job per 31 days or {{{(1_job)/(31_days)}}} or {{{1/31}}}{{{jobs/day}}}

Suppose A's rate working alone is is 1 job per x days or {{{(1_job)/(x_days)}}} or {{{1/x}}}{{{jobs/day}}}.

Suppose B's rate working alone is is 1 job per y days or {{{(1_job)/(y_days)}}} or {{{1/y}}}{{{jobs/day}}}.

Suppose C's rate working alone is is 1 job per z days or {{{(1_job)/(z_days)}}} or {{{1/z}}}{{{jobs/day}}}.

Suppose their combined rate is 1 job per d days or {{{(1_job)/(d_days)}}} or {{{1/d}}}{{{jobs/day}}}.


The four equations come from:

      {{{(matrix(4,1,

"A's", rate, in, "jobs/day"))}}} + {{{(matrix(4,1,

"B's", rate, in, "jobs/day"))}}} = {{{(matrix(5,1,

Their, combined, rate, in, "jobs/day"))}}}

            {{{1/x}}} + {{{1/y}}} = {{{1/42}}}

      {{{(matrix(4,1,

"B's", rate, in, "jobs/day"))}}} + {{{(matrix(4,1,

"C's", rate, in, "jobs/day"))}}} = {{{(matrix(5,1,

Their, combined, rate, in, "jobs/day"))}}}

            {{{1/y}}} + {{{1/z}}} = {{{1/31}}}

      {{{(matrix(4,1,

"C's", rate, in, "jobs/day"))}}} + {{{(matrix(4,1,

"A's", rate, in, "jobs/day"))}}} = {{{(matrix(5,1,

Their, combined, rate, in, "jobs/day"))}}}

            {{{1/z}}} + {{{1/x}}} = {{{1/20}}}



      {{{(matrix(4,1,

"A's", rate, in, "jobs/day"))}}} + {{{(matrix(4,1,

"B's", rate, in, "jobs/day"))}}} + {{{(matrix(4,1,

"C's", rate, in, "jobs/day"))}}} = {{{(matrix(5,1,

Their, combined, rate, in, "jobs/day"))}}}

            {{{1/x}}} + {{{1/y}}} + {{{1/z}}} = {{{1/d}}}

{{{1/x}}} + {{{1/y}}} = {{{1/42}}}
{{{1/y}}} + {{{1/z}}} = {{{1/31}}}
{{{1/z}}} + {{{1/x}}} = {{{1/20}}}
{{{1/x}}} + {{{1/y}}} + {{{1/z}}} = {{{1/d}}}

Now we must find their combined rate which is

So we line up the first three equations like this and add them all:

          {{{1/x}}} + {{{1/y}}}         = {{{1/42}}}
              {{{1/y}}} + {{{1/z}}}     = {{{1/31}}}
          {{{1/x}}}     + {{{1/z}}}     = {{{1/20}}}
         ---------------------
          {{{2/x}}} + {{{2/y}}} + {{{2/z}}}     = {{{1/42+1/31+1/20}}}

          {{{2/x}}} + {{{2/y}}} + {{{2/z}}}     = {{{1381/13020}}}

Dividing both side by 2

          {{{1/x}}} + {{{1/y}}} + {{{1/z}}}     = {{{1381/26040}}}

And since the fourth equation is

          {{{1/x}}} + {{{1/y}}} + {{{1/z}}} = {{{1/d}}}

Since things equal to the same thing are equal to each other,

           {{{1/d}}} = {{{1381/26040}}}

Cross-multiplying:

       1381d = 26040
           d = {{{26040/1381}}}
           d = 18.85590152 days

Edwin</pre>