For

**"work" word problems**, it's often best to start by converting the times to rates. For instance, if you can complete some task in three hours (the time), then you can do 1/3 of it each hour (the rate). If I take four hours to do the same thing, then I do only 1/4 of it each hour. If we worked together, we'd do 1/3 + 1/4 = 7/12 of it each hour.

In your case:

. . . . .A's rate: 1/a

. . . . .B's rate: 1/b

. . . . .C's rate: 1/c

You are given that the time for A & B together is 12/7 hours, the time for B & C together is 12/5 hours, and the time for all three together is

4/3 hours. Then:

. . . . .A & B together: 1/a + 1/b = 7/12

. . . . .B & C together: 1/b + 1/c = 5/12

. . . . .all three together: 1/a + 1/b + 1/c =

3/4
This gives you three

*rational* equations in three unknowns (ouch!). I'd start by subtracting the first line above from the third, giving us:

. . . . .$\displaystyle \frac{1}{c}\, =\, \frac{2}{12}\, =\, \frac{1}{6}$

Solve this for the value of "c". Then subtract the second line from the third line, and solve the result for the value of "a". Plug this value into the first equation, and back-solve for the value of "b".

If you get stuck, please reply showing your steps and reasoning so far. Thank you!