translate the given information into a system of equations. the question is below

Question

working together, the father, the mother, and the son finish a job in 10 hrs. the father and the mother, working together, finish the job in 12 hrs; the mother and the son,working together,finish it in 20 hrs.working alone,how long does it take each of them to finish the job ?

Answer 1

f=# of hours  it takes the father

m=# of hours it takes the mother

s=#of hours it takes the son

1/f=that part of the job the father does in 1 hour

1/m=that part of the job the mother does in 1 hour

1/s=that part of the job the son does in 1 hour

10(1/f)+10(1/m)+10(1/s)=1

12(1/f)+12(1/m)=1

20(1/m)+20(1/s)=1

multiply first equation by 2

20(1/f)+20(1/m)+20(1/s)=2

20(1/f)+1=2

20(1/f)=1

f=20 hours (answer)

12/20+12/m=1

12/m=1-12/20

12/m=8/20=2/5

cross multiply

2m=60

m=30 hours (answer)

use the third equation

20(1/30)+(20/s)=1

20/s=1-20/30

20/s=10/30

20/s=1/3

cross multiply

s=20*3

s=60 hours (answer)

Answer 2

The work rates for mother, father, and son is 1/M, 1/F, and 1/S respectively.

So the father finished the job in F hours, the mother finished the job in M hours

and the son will finish the job in S hours if they each work alone.

 

Working together, they finish the job in 10 hours.

 

 10 ( 1/M + 1/F + 1/S) = 1

 

1/M  + 1/F + 1/S = 1/10  <--- this is equation #1

 

 

LIkewise,   1/M + 1/F  = 1/12  <---- this is equation #2: mother and father finish the job is 12 hours

 

                  1/M  + 1/S = 1/20  <---- this is equation #3: mother and son finish the job in 20 hours

 

 

Equation #1 minus Equation #2:

 1/S = 1/10 - 1/12 <---- 1/M and 1/F cancel out

 

        = 6/60 - 5/60 <--- common denominator

        = 1/60

 

The son finished the job by himself in 60 hours

 

  Per equation #3:

 

1/M + 1/60 = 1/20

 

1/M = 1/20 - 1/60 = 3/60 - 1/60 = 2/60 = 1/30

Mom finishes the job by herself in 30 hours

 

 

Per first equation

 

1/F + 1/30 + 1/60 = 1/10

 

1/F + 2/60 + 1/60 = 1/10

 

1/F + 3/60 = 1/10

 

1/F + 1/20 = 1/10

 

1/F = 1/10 - 1/20

 

1/F = 2/20 - 1/20

 

1/F = 1/20

 

Father finished the job in 20 hours