Working together

Let  the part of the job that A can do in one hour  = A

Let the part of the job that B can do in one hour  = B

And let the part of the job that C can do in one hour  = C

And using....... rate per hour * hrs. worked = part of the job done......

we  have this system of equations

[ Note....1.5 hrs = 3/2 hrs and 1 + 2/3 hrs  = 5/3 hrs  ]

(A) (3/2) + (B)(3/2)  = 1  →  3A + 3B  = 2   → B =  [2 - 3A] / 3    ( 1)

(A)(5/3) + (C)((5/3)  = 1  →  5A + 5C  = 3 → C =  [3 - 5A ] / 5     (2)

A + B + C  =  1               → A + B + C  = 1    (3)

Subbing (1) and (2)  into (3)...we have that

A + [2 - 3A]/3 + [3 - 5A]/5  = 1      multiply through by 15

15A  + 5[ 2 - 3A] + 3[ 3 - 5A] = 15    simplify

15A + 10 - 15A + 9 - 15A  = 15

-15A  + 19 = 15

-15A = -4    divide both sides by -15

A  = 4/15   ...so A can complete 4/15 of the job in one hour

B  completes  .... [2 - 3(4/15)] / 3  =  [ 30 -12] / 45  = 18/45  = 2/5  of the job in one hour

C  completes ...  [ 3 - 5(4/15)] / 5  = [ 45 - 20] / 75 = 25/75  =1/3 of the job in one hoiur

So....B and C working together complete    2/5 + 1/3  = [6 + 5] / 15  =

11/15  of the job in one hour

Flip this fraction over to find the total time it takes B + C working together to finish the whole job  = 

15/11 hrs ≈  81.81 minutes  ≈ 1 hr, 21minutes,48.6 seconds

If A and B working together can finish a task in 1 1/2 hours,
and if A and C working together can finish the same task in 1 2/3 hours,
and if A, B, and C working together can finish the same task in 1 hour,
how long would it take for B and C working together to finish the same task?

Let W = Work

\(\begin{array}{|rcll|} \hline \frac{W}{A} &+& \frac{W}{B} && &=& \frac{W}{1\frac12\ h} \quad & | \quad : W \\ \frac{W}{A} & & &+& \frac{W}{C} &=& \frac{W}{1\frac23\ h} \quad & | \quad : W \\ \frac{W}{A} &+& \frac{W}{B} &+& \frac{W}{C} &=& \frac{W}{1\ h} \quad & | \quad : W \\ & & \frac{W}{B} &+& \frac{W}{C} &=& \frac{W}{x} \quad & | \quad : W \\\\ \frac{1}{A} &+& \frac{1}{B} && &=& \frac{1}{1+\frac12} & (1) \\ \frac{1}{A} & & &+& \frac{1}{C} &=& \frac{1}{1+\frac23} & (2) \\ \frac{1}{A} &+& \frac{1}{B} &+& \frac{1}{C} &=& \frac{1}{1} & (3) \\ & & \frac{1}{B} &+& \frac{1}{C} &=& \frac{1}{x} & (4) \\ \hline \end{array} \)

\(\frac{1}{A} =\ ?\)

\(\small{ \begin{array}{|lrcll|} \hline (1)+(2)-(3): \\\\ & (\frac{1}{A}+\frac{1}{B}) + (\frac{1}{A}+\frac{1}{C})-(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}) &=& \frac{1}{1+\frac12} + \frac{1}{1+\frac23} - \frac{1}{1} \\ & \frac{1}{A}+\frac{1}{B} + \frac{1}{A}+\frac{1}{C}-\frac{1}{A}-\frac{1}{B}-\frac{1}{C} &=& \frac{1}{1+\frac12} + \frac{1}{1+\frac23} - \frac{1}{1} \\ & \frac{1}{A} &=& \frac{1}{1+\frac12} + \frac{1}{1+\frac23} - \frac{1}{1} \\ & \frac{1}{A} &=& \frac{1}{\frac32} + \frac{1}{\frac53} - 1 \\ & \frac{1}{A} &=& \frac23 + \frac{3}{5} - 1 \\ & \frac{1}{A} &=& \frac{10+9}{15} - 1 \\ & \frac{1}{A} &=& \frac{19}{15} - \frac{15}{15} \\ & \mathbf{\frac{1}{A}} & \mathbf{=} & \mathbf{\frac{4}{15}} \\ \hline \end{array} }\)

\(x =\ ?\)

\(\begin{array}{|lrcll|} \hline (3) & \frac{1}{A} + \frac{1}{B} + \frac{1}{C} &=& \frac{1}{1} \\ & \frac{1}{B} + \frac{1}{C} &=& \frac{1}{1} - \frac{1}{A} \quad & | \quad \frac{1}{B} + \frac{1}{C} = \frac{1}{x} \\ & \frac{1}{x} &=& \frac{1}{1} - \frac{1}{A} \\ & \frac{1}{x} &=& \frac{1}{1} - \mathbf{\frac{4}{15}} \\ & \frac{1}{x} &=& 1 - \mathbf{\frac{4}{15}} \\ & \frac{1}{x} &=& \frac{15}{15} - \mathbf{\frac{4}{15}} \\ & \frac{1}{x} &=& \frac{15-4}{15} \\ & \frac{1}{x} &=& \frac{11}{15} \\ & x &=& \frac{15}{11} \\ & \mathbf{x} & \mathbf{=}& \mathbf{1\frac{4}{11}\ h} \\ \hline \end{array} \)

It would take for B and C working together to finish the same task \(\mathbf{1\frac{4}{11}\ h}\)

1/a + 1/b = 2/3.......................(1)
1/a + 1/c = 3/5....................... (2) 
1/a + 1/b + 1/c = 1..................(3) subbing 1 into 3 we get:
2/3 + 1/c = 1, and c =3 and subbing 2 into 3 we get:
3/5 + 1/b = 1, and b =5/2. adding the reciprocals of b, and c we have:
1/3 + 2/5 = 11/15, and the reciprocal of this is:
15/11 = 1 4/11 hours for B and C to finish the job.