Working together, a job takes Tom and Joe 8 days, Tom and Dave 13 1/3 days, and Joe and Dave 6 2/3 days. How long does it take each person to do the job alone?
Let x = the number of days Tom requires working alone.
Let y = the number of days Joe requires working alone.
Let z = the number of days Dave requires working alone.
Make this chart and fill in the unknown times x, y, and z
Number of Number of Rate in
jobs done days required jobs/day
------------------------------------------------------------------------
Tom & Joe working together | | | |
Tom & Dave working together | | | |
Joe & Dave working together | | | |
Tom working alone | | x | |
Joe working alone | | y | |
Dave working alone | | z | |
In all 6 cases exactly 1 whole job is done. So we fill in 1 for all the number of
jobs done:
Number of Number of Rate in
jobs done days required jobs/day
------------------------------------------------------------------------
Tom & Joe working together | 1 | | |
Tom & Dave working together | 1 | | |
Joe & Dave working together | 1 | | |
Tom working alone | 1 | x | |
Joe working alone | 1 | y | |
Dave working alone | 1 | z | |
Two of the given times are in mixed fractions, so let's make them into improper fractions:
Number of days required by Tom and Dave working together = =
Number of days required by Joe and Dave working together = =
So we fill in the given number of days required in the first three cases:
Number of Number of Rate in
jobs done days required jobs/day
------------------------------------------------------------------------
Tom & Joe working together | 1 | 8 | |
Tom & Dave working together | 1 | 40/3 | |
Joe & Dave working together | 1 | 20/3 | |
Tom working alone | 1 | x | |
Joe working alone | 1 | y | |
Dave working alone | 1 | z | |
Next we fill in the rates in jobs/day by dividing jobs by days:
1÷8 = , 1÷ = 1× = , 1÷ = 1× =
Number of Number of Rate in
jobs done days required jobs/day
------------------------------------------------------------------------
Tom & Joe working together | 1 | 8 | 1/8 |
Tom & Dave working together | 1 | 40/3 | 3/40 |
Joe & Dave working together | 1 | 20/3 | 3/20 |
Tom working alone | 1 | x | 1/x |
Joe working alone | 1 | y | 1/y |
Dave working alone | 1 | z | 1/z |
The three equations come from adding the rates to get the combined rates:
+ =
+ =
+ =
So we have this system of three equations in three unknowns:
Important: DO NOT clear of fractions! That's because doing so would introduce
terms in two variables multiplied together, that is, terms in xy, xz, and yz,
which would complicate matters further.
Instead let u = , v = , and w =
Then we have
ì u + v =
í u + w =
î v + w =
NOW we MAY clear of fractions because we won't get multiplied variables.
ì 8u + 8v = 1
í40u + 40w = 3
î 20v + 20w = 3
Solve that system by elimination (I'm sure you know how).
You get u = , v = , and w =
Then since u = , then = , and x = 40, so Tom
working alone takes 40 days to do the job.
Since v = , then = , and y = 10, so Joe
working alone takes 10 days to do the job.
Since w = , then = , and z = 20, so Joe
working alone takes 20 days to do the job.
Edwin