Problem Solving

At Supersonic Corporation, the time required for a machine to complete a job is determined by the formula: $$t=\sqrt{w}+\sqrt{\left(w-1\right)}$$ where w = the weight of the machine in pounds and t = the hours required to complete the job. If machine A weighs 8 pounds, and machine B weighs 7 pounds, how many hours will it take the two machines to finish one job if they work together?

$$A.\ \frac{6}{7-\sqrt{3}}$$
$$B.\ \frac{1}{2}\left(\sqrt{8}+\sqrt{6}\right)$$
$$C.\ \frac{1}{3}\left(6-\sqrt{3}\right)$$
$$D.\ 3\left(\sqrt{3}+\sqrt{2}\right)$$
$$E.\ \sqrt{8}+2\sqrt{7}+\sqrt{6}$$

The OA is B.

In this case, I just need to do the following,

Can I say that the total time will be, Time of machine A + time of machine B, right?

Then,
$$T_A=\sqrt{8}+\sqrt{8-1}=\sqrt{8}+\sqrt{7}$$
And
$$T_B=\sqrt{7}+\sqrt{7-1}=\sqrt{7}+\sqrt{6}$$
Finally,
$$T_{total}=\left(\sqrt{8}+\sqrt{7}\right)+\left(\sqrt{7}+\sqrt{6}\right)=\sqrt{8}+2\sqrt{7}+\sqrt{6}$$
But, how can I get the combined time?

I'm really confused by this PS question. Experts, any suggestion? Thanks in advance.

At Supersonic Corporation, the time required for a machine to complete a job is determined by the formula: $$t=\sqrt{w}+\sqrt{\left(w-1\right)}$$ where w = the weight of the machine in pounds and t = the hours required to complete the job. If machine A weighs 8 pounds, and machine B weighs 7 pounds, how many hours will it take the two machines to finish one job if they work together?

$$A.\ \frac{6}{7-\sqrt{3}}$$
$$B.\ \frac{1}{2}\left(\sqrt{8}+\sqrt{6}\right)$$
$$C.\ \frac{1}{3}\left(6-\sqrt{3}\right)$$
$$D.\ 3\left(\sqrt{3}+\sqrt{2}\right)$$
$$E.\ \sqrt{8}+2\sqrt{7}+\sqrt{6}$$

The OA is B.

In this case, I just need to do the following,

Can I say that the total time will be, Time of machine A + time of machine B, right?

Then,
$$T_A=\sqrt{8}+\sqrt{8-1}=\sqrt{8}+\sqrt{7}$$
And
$$T_B=\sqrt{7}+\sqrt{7-1}=\sqrt{7}+\sqrt{6}$$
Finally,
$$T_{total}=\left(\sqrt{8}+\sqrt{7}\right)+\left(\sqrt{7}+\sqrt{6}\right)=\sqrt{8}+2\sqrt{7}+\sqrt{6}$$
But, how can I get the combined time?

I'm really confused by this PS question. Experts, any suggestion? Thanks in advance.

Hi LUANDATO,
Let's take a look at your question.

The time required to complete a job is:
$$=\sqrt{w}+\sqrt{\left(w-1\right)}$$

The rate is inverse of time, therefore we can find the rate as:
$$Rate\ =\frac{1}{\sqrt{w}+\sqrt{\left(w-1\right)}}$$
Rationalizing the denominator:
$$Rate\ =\frac{1}{\sqrt{w}+\sqrt{\left(w-1\right)}}\times\frac{\sqrt{w}-\sqrt{\left(w-1\right)}}{\sqrt{w}-\sqrt{\left(w-1\right)}}$$
$$Rate\ =\frac{\sqrt{w}-\sqrt{\left(w-1\right)}}{\left(\sqrt{w}+\sqrt{\left(w-1\right)}\right)\left(\sqrt{w}-\sqrt{\left(w-1\right)}\right)}$$
$$Rate\ =\frac{\sqrt{w}-\sqrt{\left(w-1\right)}}{w-\left(w-1\right)}$$
$$Rate\ =\frac{\sqrt{w}-\sqrt{\left(w-1\right)}}{w-w+1}$$
$$Rate\ =\frac{\sqrt{w}-\sqrt{\left(w-1\right)}}{1}$$
$$Rate\ =\sqrt{w}-\sqrt{\left(w-1\right)}$$

Since, Machine A weighs 8 pounds, therefore, rate of machine A is:
$$Rate\ =\sqrt{8}-\sqrt{\left(8-1\right)}$$
$$Rate\ =\sqrt{8}-\sqrt{\left(7\right)}$$

Machine B weighs 7 pounds, therefore, rate of machine B is:
$$Rate\ =\sqrt{7}-\sqrt{\left(7-1\right)}$$
$$Rate\ =\sqrt{7}-\sqrt{\left(6\right)}$$

Combined rate of Machine A and B is:
$$Combined\ Rate\ =\left(\sqrt{8}-\sqrt{7}\right)+\left(\sqrt{7}-\sqrt{6}\right)$$
$$Combined\ Rate\ =\sqrt{8}-\sqrt{7}+\sqrt{7}-\sqrt{6}$$
$$Combined\ Rate\ =\sqrt{8}-\sqrt{6}$$

Now, we can find the time time taken by both the machines to finish the job by finding the reciprocal of the combined rate.
$$Time=\frac{1}{\sqrt{8}-\sqrt{6}}$$
Rationalizing the denominator to simplify:
$$Time=\frac{1}{\sqrt{8}-\sqrt{6}}\times\frac{\sqrt{8}+\sqrt{6}}{\sqrt{8}+\sqrt{6}}$$
$$Time=\frac{\sqrt{8}+\sqrt{6}}{8-6}$$
$$Time=\frac{\sqrt{8}+\sqrt{6}}{2}$$
Therefore, Option B is correct.

I am available if you'd like any follow up.