SOLUTION: Suppose that it takes Jack and Bob 2 hours to do a job, it takes Bob and Danny 3 hours to do the same job and it takes Jack and Danny 4 hours to do the same job. How many hours wo

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Question 1086523: Suppose that it takes Jack and Bob 2 hours to do a job, it takes Bob and Danny 3 hours to
do the same job and it takes Jack and Danny 4 hours to do the same job. How many hours would it take Jack, Bob and Danny to do the same job?
Found 3 solutions by josgarithmetic, ikleyn, josmiceli:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
RATES, J, B, D, in JOBS/HOUR
system%28J%2BB=1%2F2%2CB%2BD=1%2F3%2CJ%2BD=1%2F4%29
J%2BB%2BB%2BD%2BJ%2BD=1%2F2%2B1%2F3%2B1%2F4
2J%2B2B%2B2D=6%2F12%2B4%2F12%2B3%2F12
2%28J%2BB%2BD%29=13%2F12
J%2BB%2BD=13%2F24---------combined rate, 13%2F24 jobs%2Fhour

Combined TIME for the three together to do 1 job:
24%2F13 hours;
--
1.846 hours
OR
1 hour 51 minutes

Answer by ikleyn(52810) About Me  (Show Source):
You can put this solution on YOUR website!
.
Suppose that it takes Jack and Bob 2 hours to do a job, it takes Bob and Danny 3 hours to
do the same job and it takes Jack and Danny 4 hours to do the same job.
How many hours would it take Jack, Bob and Danny to do the same job highlight%28working_together%29 ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let "j" be the Jack's rate of work;
    "b" be the Bob's  rate of work;  and
    "d" be the Danny' rate of work.


Then from condition you have this system of equations

j + b     = 1%2F2    (1)
    b + d = 1%2F3    (2)
j +     d = 1%2F4    (3)


Add all three equations (1), (2) and (3) (both sides).  You will get

2j + 2b + 2d = 1%2F2+%2B+1%2F3+%2B+1%2F4 = 6%2F12+%2B+4%2F12+%2B+3%2F12 = 13%2F12.    (4)


Divide both sides of (4) by 2. You will get

j + b + d = 13%2F24.


Thus the combined rate of work of three workers is 13%2F24.

It means that they will complete the job in 24%2F13 of an hour.

Answer. It will take 24%2F13 of an hour for three workers to complete the job working together.


See the lessons on  joint work  problems
    - Using Fractions to solve word problems on joint work
    - Solving more complicated word problems on joint work
    - Using quadratic equations to solve word problems on joint work
    - Solving rate of work problem by reducing to a system of linear equations
    - Selected joint-work word problems from the archive
    - Joint-work problems for 3 participants (*)
    - Had there were more workers, the job would be completed sooner
    - One unusual joint work problem
    - OVERVIEW of lessons on rate-of-work problems
in this site, especially the lesson marked (*).


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Rate of work and joint work problems" of the section "Word problems".



Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
You can add their rates of working to
get their rate working together
Let +R%5BJ%5D+ = Jack's rate working alone
Let +R%5BB%5D+ = Bob's rate working alone
Let +R%5BD%5D+ = Danny's rate working alone
---------------------------------------
[ Jack's rate + Bob's rate ] = [ 1 job / 2 hrs ]
(1) +R%5BJ%5D+%2B+R%5BB%5D+=+1%2F2+
[ Bob's rate + Danny's rate ] = [ 1 job / 3 hrs ]
(2) +R%5BB%5D+%2B+R%5BD%5D+=+1%2F3+
[ Jack's rate + Danny's rate ] = [ 1 job / 4 hrs ]
(3) +R%5BJ%5D+%2B+R%5BD%5D++=+1%2F4+
---------------------------------------------
Subtract (3) from (1)
(1) +R%5BJ%5D+%2B+R%5BB%5D+=+1%2F2+
(3) +-R%5BJ%5D+-+R%5BD%5D++=+-1%2F4+
----------------------------
+R%5BB%5D+-+R%5BD%5D+=+1%2F4+
Add this result to (2)
(3) +R%5BB%5D+-+R%5BD%5D++=+1%2F4+
(2) +R%5BB%5D+%2B+R%5BD%5D+=+1%2F3+
----------------------------
+2R%5BB%5D+=+3%2F12+%2B+4%2F12+
+2R%5BB%5D+=+7%2F12+
+R%5BB%5D+=+7%2F24+
----------------------------
(1) +R%5BJ%5D+%2B+R%5BB%5D+=+1%2F2+
(1) +R%5BJ%5D+%2B+7%2F24+=+12%2F24+
(1) +R%5BJ%5D+=+5%2F24+
----------------------------
(3) +R%5BJ%5D+%2B+R%5BD%5D++=+1%2F4+
(3) +5%2F24+%2B+R%5BD%5D++=+6%2F24+
(3) +R%5BD%5D+=+1%2F24+
----------------------------
Let +t+ = time in hrs for Jack, Bob and Danny
to do the same job
+R%5BJ%5D+%2B+R%5BB%5D+%2B+R%5BD%5D+=+1%2Ft+
+5%2F24+%2B+7%2F24+%2B+1%2F24+=+1%2Ft+
+13%2F24+=+1%2Ft+
+t+=+24%2F13+ hrs
+t+=+1+%2B+11%2F13+
+%28+11%2F13+%29%2A60+=+50.769+
+.769%2A60+=+46+
They would have to work 1 hr 50 min 46 sec
------------------------------------------
check answer:
(1) +R%5BJ%5D+%2B+R%5BB%5D+=+1%2F2+
(1) +5%2F24+%2B+7%2F24+=+1%2F2+
(1) +12%2F24+=+1%2F2+
(1) +1%2F2+=+1%2F2+
OK
(2) +R%5BB%5D+%2B+R%5BD%5D+=+1%2F3+
(2) +7%2F24+%2B+1%2F24+=+1%2F3+
(2) +8%2F24+=+1%2F3+
(2) +1%2F3+=+1%2F3+
OK
(3) +R%5BJ%5D+%2B+R%5BD%5D++=+1%2F4+
(3) +5%2F24+%2B+1%2F24++=+1%2F4+
(3) +6%2F24+=+1%2F4+
(3) +1%2F4+=+1%2F4+
OK