Suppose that there are two machines A and B that complete a task within $4$ and $8$ hours respectively. The speed of A is reduced to $1/3$ of its current speed, while the speed of B is increased by $3$ times. Compute how long it takes to complete the task when the machines A and B work together.
My approach:
Machine A
$$\dfrac{1\space \text{task}}{4 \space \text{hours}}\times \dfrac{8}{3}\space \text{hours} = \dfrac{2}{3}\space \text{tasks}$$ Machine B
$$\dfrac{1\space \text{task}}{8 \space \text{hours}}\times 24\space \text{hours} = 3\space \text{tasks}$$
However, I'm not sure how that info is useful.
It would take the new A 12 hours to finish the job, so it does 1/12 of the job each hour. Also, it takes the new B 8/3 hours to do the job, so it does 3/8 of the job each hour.
Therefore, working together they do $1/12+3/8=11/24$ of the job each hour. So it takes them 24/11 hours to complete the job working together.