SOLUTION: 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 alon

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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

Found 2 solutions by Edwin McCravy, oberobic:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
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.
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%282%2FM+%2B+5%2FB+=+1%2F4%2C%0D%0A4%2FM+%2B+4%2FB+=+1%2F3%29

Do not clear of fractions.  Use elimination

Multiply the first equation through by -2

system%28-4%2FM+-+10%2FB+=+-1%2F2%2C%0D%0A4%2FM+%2B+4%2FB+=+1%2F3%29

Adding them term by term:

-6%2FB+=+-1%2F2+%2B+1%2F3
-6%2FB+=+-3%2F6+%2B+2%2F6
-6%2FB+=+-1%2F6
6%2FB+=+1%2F6
B+=+36

It would take a boy 36 days.

Substitute in

2%2FM+%2B+5%2FB+=+1%2F4
2%2FM+%2B+5%2F36+=+1%2F4
2%2FM+=+1%2F4+-+5%2F36
2%2FM+=+9%2F36+-+5%2F36
2%2FM+=+4%2F36
2%2FM+=+1%2F9
M+=+18

It would take one man 18 days.

Edwin

Answer by oberobic(2304) About Me  (Show Source):
You can put this solution on YOUR website!
With work problems you always determine the amount of work each type of resource can do per day.
Then add up the fractions total 1 whole job.
.
m = work done by a man per day
b = work done by a boy per day
.
(2m + 5b) * 4 = 1 whole job
(4m + 4b) * 3 = 1 whole job
.
8m + 20b = 1
12m + 12b = 1
.
adjust the equations to solve using elimination
.
3(8m + 20b) = 3
5(12m + 12b) = 5
.
24m + 60b = 3
60m + 60b = 5
-------------
-36m = -2
m = 1/18
.
So each man does 1/18 of the whole job per day.
Therefore, one man would take 18 days to do the job.
.
Either repeat this process or use substitution to find the boys' rate of work.
.
(2m + 5b) * 4 = 1
(2/18 + 5b) = 1/4
5b = 1/4 - 2/18
5b = 1/4 - 1/9
5b = 9/36 - 4/36
5b = 5/36
b = 1/36
.
So each boy does 1/36 of the whole job per day.
Therefore, one boy would take 36 days to do the job.
.
Always check your work.
Substitute these values into one of the equations to check.
.
(2m + 5b) * 4 = 1 whole job
(2/18 + 5/36) * 4 = ??
(4/36 + 5/36) * 4 = ??
9/36 * 4 = 36/36 = 1
Correct.
.
Done.