Question 616507
f 12 men and 16 boys can do a piece of work in 5 days and 
13men and 24 boys can do the same piece of work in 4 days.
 how long will 7 men and 10 boys take to complete the same work?
:
Let b = days required for one boy to do the job
Let m = days required for one man to do the job
Let the completed job = 1
:
"If 12 men and 16 boys can do a piece of work in 5 days"
{{{(12(5))/m}}} + {{{(16(5))/b}}} = 1
{{{60/m}}} + {{{80/b}}} = 1
:
"13men and 24 boys can do the same piece of work in 4 days."
{{{(13(4))/m}}} + {{{(24(4))/b}}} = 1
{{{52/m}}} + {{{96/b}}} = 1
:
We can combine the two equations
{{{60/m}}} + {{{80/b}}} = {{{52/m}}} + {{{96/b}}}
combine like terms
{{{60/m}}} - {{{52/m}}} = {{{96/b}}} - {{{80/b}}}
{{{8/m}}} = {{{16/b}}}
cross multiply
8b = 16m
divide by 8
b = 2m 
:
Use the 1st scenario equation to find m, replace b with 2m
{{{60/m}}} + {{{80/(2m)}}} = 1
reduce the fraction
{{{60/m}}} + {{{40/m}}} = 1
Multiply by m
60+40 = m
m = 100 days for one man to do the job
then
b = 2(100)
b = 200 days for one boy to do the job
:
" how long will 7men and 10 boys take to complete the same work?
let t = time required for this to happen
{{{(7t)/100}}} + {{{(10t)/200}}} = 1
reduce the fraction, to give us a common denominator
{{{(7t)/100}}} + {{{5t/100}}} = 1
mult by 100
7t + 5t = 100
12t = 100
t = 100/12
t = 8{{{1/3}}} days for 7 men and 10 boys to do the job