Question 503738
2 men and 5 boys can do piece of work in 4 days , while 4 men and 4 boys can do in 3 day. how long would it take one man alone to do it and how many days would it take one boy alone to do it.
<pre>
Make this chart:

                   Number of          Time             
                   jobs done        required        Rate
                   or fraction         in            in
                     thereof          days        jobs/day
1 Man                                
1 Boy                                 
2 Men                               
5 Boys                                 
4 Men                                
4 Boys                              
2 Men and 5 Boys                       
4 Men and 4 Boys                     

Fill in 1 for the number of jobs done in each case:

                   Number of          Time
                   jobs done        required        Rate
                   or fraction         in            in
                     thereof          days        jobs/day
1 Man                    1            
1 Boy                    1            
2 Men                    1            
5 Boys                   1             
4 Men                    1            
4 Boys                   1          
2 Men and 5 Boys         1            
4 Men and 4 Boys         1            

Let M be the time required for 1 man to complete the job
Let B be the time required for 1 boy to complete the job
Therefore the number of days required for 2 Men will by
only half of M that or M/2
And the number of days required for 5 boys will be only
one-fifth of B or B/5
We are given the times in the last two cases.
Fill all those in:
               

                   Number of          Time
                   jobs done        required        Rate
                   or fraction         in            in
                     thereof          days        jobs/day
1 Man                    1              M           
1 Boy                    1              B           
2 Men                    1             M/2          
5 Boys                   1             B/5            
4 Men                    1             M/4          
4 Boys                   1             B/4          
2 Men and 5 Boys         1              4           
4 Men and 4 Boys         1              3           

Next we fill in the Rates in jobs/day by dividing
number of jobs by the number of days. Notice that
1 over a fraction is the reciprocal of that fraction:



                   jobs done        required        Rate
                   or fraction         in            in
                     thereof          days        jobs/day
1 Man                    1              M            1/M
1 Boy                    1              B            1/B 
2 Men                    1             M/2           2/M 
5 Boys                   1             B/5           5/B  
4 Men                    1             M/4           4/M 
4 Boys                   1             B/4           4/B
2 Men and 5 Boys         1              4            1/4
4 Men and 4 Boys         1              3            1/3


The sum of the rates for 2 Men and 5 Boys must equal to their 
combined rate, so

2/M + 5/B = 1/4

The sum of the rates for 4 Men and 4 Boys must equal to their 
combined rate, so

4/M + 4/B = 1/3

So we have this system of equations:

{{{system(2/M + 5/B = 1/4,
4/M + 4/B = 1/3)}}}

Do not clear of fractions.  Use elimination

Multiply the first equation through by -2

{{{system(-4/M - 10/B = -1/2,
4/M + 4/B = 1/3)}}}

Adding them term by term:

{{{-6/B = -1/2 + 1/3}}}
{{{-6/B = -3/6 + 2/6}}}
{{{-6/B = -1/6}}}
{{{6/B = 1/6}}}
{{{B = 36}}}

It would take a boy 36 days.

Substitute in

{{{2/M + 5/B = 1/4}}}
{{{2/M + 5/36 = 1/4}}}
{{{2/M = 1/4 - 5/36}}}
{{{2/M = 9/36 - 5/36}}}
{{{2/M = 4/36}}}
{{{2/M = 1/9}}}
{{{M = 18}}}

It would take one man 18 days.

Edwin</pre>