Federico's able to finish a job in $15$ days, and Relen's able to finish a job in $30$ days. In how many days can they finish this job together?

Question

Federico's able to finish a job in $15$ days, and Relen's able to finish a job in $30$ days. In how many days can they finish a job if they work together?

Let me show what I tried:

$$F \implies 15 $$ $$R \implies 30$$

Which means that

$$\frac{1}{15} + \frac{1}{30} = 1$$

However, I'm wondering if there's a method that I can use for all work problems.

Answer 1

$$\frac{1}{15} + \frac{1}{30}$$

Is how much work the two of them together can achieve in one day. Thus, the number of days it takes them to finish the job is

$$\frac{1}{\frac{1}{15} + \frac{1}{30}}$$

In general, then, if you have $n$ people, and if it takes them individually $m_1$ through $m_n$ units of time respectively to finish the job, then working together it will take them

$$\frac{1}{\frac{1}{m_1} + ... + \frac{1}{m_n}}$$

units of time to complete the job.

Of course, this is all assuming that working together does not increase or decrease their working speed.

Answer 2

Notice that $$\frac{1}{15} + \frac{1}{30} = 1$$

is an absolute malarkey.

It's

$$\frac{1}{15} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10}$$

Hence together it takes...