Question 313753
A building contractor employs a master mason, a journeyman mason and an apprentice mason.
The master mason and the journeyman working together can finish a brick wall in 10 hours. 
It takes the master and apprentice 12 hours to do the same job,
 while the journeyman and apprentice require 15 hours to complete the task.
:
Find how many hrs each require working alone
:
Let the individual times of the three masons be: m, j, a
:
Let the completed job = 1
:
Write an equation for each statement:
:
"The master mason and the journeyman working together can finish a brick wall in 10 hours"
Eq1:{{{10/m}}} + {{{10/j}}} = 1
:
"It takes the master and apprentice 12 hours to do the same job,"
Eq2:{{{12/m}}} + {{{12/a}}} = 1
:
"the journeyman and apprentice require 15 hours to complete the task."
Eq3:{{{15/j}}} + {{{15/a}}} = 1
:
Multiply Eq1 by 6 and Eq2 by 5:
{{{60/m}}} + {{{60/j}}} = 6
{{{60/m}}} + {{{60/a}}} = 5
-------------------------------Subtraction eliminates m, so we have
{{{60/j}}} - {{{60/a}}} = 1
:
Multiply eq3 by 4 and add to the above equation
{{{60/j}}} - {{{60/a}}} = 1
{{{60/j}}} + {{{60/a}}} = 4
----------------------------- adding eliminates a, find j
{{{120/j}}} = 5
5j = 120
j = {{{120/5}}}
j = 24 hrs to do the job alone
:
Substitute 24 for j in eq1, find m
{{{10/m}}} + {{{10/24}}} = 1
Multiply by 24m
10(24) + 10m = 24m
240 = 24m - 10m
m = {{{240/14}}}
m = 17.14 hrs to do the job alone
:
Using eq3, find a
{{{15/24}}} + {{{15/a}}} = 1
Multiply by 24a
15a + 15(24) = 24a
360 = 24a - 15a
a = {{{360/9}}}
a = 40 hrs to do the job alone
:
Summarize: a = 40; j = 24; m = 17.14 hrs
:
"If all three work together, how long does it take to build the wall?"
Let t = time required when all three work together
{{{t/17.14}}} + {{{t/24}}} + {{{t/40}}} = 1
multiply equation by 17.14*24*40 = 16454.4, results:
960t + 685.6t + 411.36t = 16454.4
2056.96t = 16454.4
t = {{{16454.4/2056.96}}}
t ~ 8 hrs all three working
;
:
Check solution using a calc:
8/17.14 + 8/24 + 8/40 =
.467 + .333 + .2 = 1