SOLUTION: 1) 2 men and 7 boys can do a piece of work in 4 days. The same work is done by 4 men and 4 boys in 3 days. How long would it take 1 man and 1 boy to do the same?

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: 1) 2 men and 7 boys can do a piece of work in 4 days. The same work is done by 4 men and 4 boys in 3 days. How long would it take 1 man and 1 boy to do the same?      Log On


   

Question 530267: 1) 2 men and 7 boys can do a piece of work in 4 days. The same work is done by 4 men and 4 boys in 3 days. How long would it take 1 man and 1 boy to do the same?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
2 men and 7 boys can do a piece of work in 4 days. The same work is done by 4 men and 4 boys in 3 days. How long would it take 1 man and 1 boy to do the same?
Let x = the number of days it would take a man to do 1 job working alone
Let y = the number of days it would take a boy to do 1 job working alone
 

Make this chart:

                jobs   days    jobs/day
1 man                         
1 boy            
2 men            
7 boys           
2 men & 7 boys    
4 men             
4 boys             
4 men & 4 boys    

Since x = the number of days it would take 1 man to do 1 job working alone
and y = the number of days it would take 1 boy to do 1 job working alone
we fill this much in:

                jobs   days    rate in jobs/day
1 man             1     x                         
1 boy             1     y
2 men            
7 boys           
2 men & 7 boys    
4 men             
4 boys             
4 men & 4 boys    

Now we fill in their rates in jobs/day by dividing
jobs by days:

                jobs   days    rate in jobs/day
1 man             1     x            1/x                         
1 boy             1     y            1/y 
2 men            
7 boys           
2 men & 7 boys    
4 men             
4 boys             
4 men & 4 boys    

The rate for 2 men is 2 times the rate for 1 man
The rate for 4 men is 4 times the rate for 1 man
The rate for 7 boys is 7 times the rate for 1 boy
The rate for 4 boys is 4 times the rate for 1 boy

So fill those rates in:

                jobs   days    rate in jobs/day
1 man             1     x            1/x                         
1 boy             1     y            1/y 
2 men                                2/x            
7 boys                               7/y
2 men & 7 boys    
4 men                                4/x 
4 boys                               4/y
4 men & 4 boys    


Since in every case above we are talking about completing 1 job,
we put 1's all the way down the "number of jobs" list.

                jobs   days    rate in jobs/day
1 man             1     x            1/x                         
1 boy             1     y            1/y 
2 men             1                  2/x            
7 boys            1                  7/y
2 men & 7 boys    1
4 men             1                  4/x 
4 boys            1                  4/y
4 men & 4 boys    1

>>...2 men and 7 boys can do a piece of work in 4 days...<<
>>...The same work is done by 4 men and 4 boys in 3 days...<<

So we fill in 4 days and 3 days for those cases:

                jobs   days    rate in jobs/day
1 man             1     x            1/x                         
1 boy             1     y            1/y 
2 men             1                  2/x            
7 boys            1                  7/y
2 men & 7 boys    1     4
4 men             1                  4/x 
4 boys            1                  4/y
4 men & 4 boys    1     3 

Now we get their rates by dividing jobs by days:


                jobs   days    rate in jobs/day
1 man             1     x            1/x                         
1 boy             1     y            1/y 
2 men             1                  2/x            
7 boys            1                  7/y
2 men & 7 boys    1     4            1/4
4 men             1                  4/x 
4 boys            1                  4/y
4 men & 4 boys    1     3            1/3

Now we make our equations from:

%28matrix%284%2C1%2Crate%2Cfor%2C2%2Cmen%29%29 + %28matrix%284%2C1%2Crate%2Cfor%2C7%2Cboys%29%29 = %28matrix%287%2C1%2Crate%2Cfor%2C4%2Cmen%2Cand%2C7%2Cboys%29%29

and

%28matrix%284%2C1%2Crate%2Cfor%2C4%2Cmen%29%29 + %28matrix%284%2C1%2Crate%2Cfor%2C4%2Cboys%29%29 = %28matrix%287%2C1%2Crate%2Cfor%2C4%2Cmen%2Cand%2C4%2Cboys%29%29

2/x + 7/y = 1/4
4/x + 4/y = 1/3

DO NOT CLEAR OF FRACTIONS! Do by elimination:

Multiply the first equation by -2 to eliminate the terms in x:

-4/x - 14/y = -1/2
 4/x +  4/y =  1/3
------------------
      -10/y = -1/6
         -y = -60
          y = 60

2/x +  7/y = 1/4
2/x + 7/60 = 1/4
   
Multiply through by LCD of 60x

120 + 7x = 15x
     -8x = -120 
       x = 15

So it takes 1 man 15 days to do one job,
and it takes 1 boy 60 days to do one job. 

Now we have a new problem:

Let z = the number of hours required for 1 man and 1 boy

Make this new chart

                jobs   days    rate in jobs/day
1 man             1     15            1/15                         
1 boy             1     60            1/60 
1 man & 1 boy     1      z            1/z

We get our equation from:

%28matrix%284%2C1%2Crate%2Cfor%2C1%2Cman%29%29 + %28matrix%284%2C1%2Crate%2Cfor%2C1%2Cboy%29%29 = %28matrix%287%2C1%2Crate%2Cfor%2C1%2Cman%2Cand%2C1%2Cboy%29%29

1/15 + 1/60 = 1/z

Multiply through by 60z

4z + z = 60
    5z = 60
     z = 12

 So it would take 1 man and 1 boy 12 hours to do the job.

Edwin