# John can finish a job in 8 hours whereas Sally only needs 5 finish the job. How quickly can they finish the job if they are working together?

Dec 7, 2016

$3$ hours, $4$ minutes and $37$ seconds

#### Explanation:

In $1$ hour, John will complete $\frac{1}{8}$ of a job and Sally $\frac{1}{5}$.

So if they work together, in one hour they can complete:

$\frac{1}{8} + \frac{1}{5} = \frac{5}{40} + \frac{8}{40} = \frac{13}{40}$

To complete a whole job will therefore take:

$\frac{40}{13} = \frac{39}{13} + \frac{1}{13} = 3 \frac{1}{13}$ hours

In minutes, $\frac{1}{13}$ hour is:

$\frac{60}{13} = \frac{52}{13} + \frac{8}{13} = 4 \frac{8}{13}$ minutes

In seconds, $\frac{8}{13}$ minutes is:

$\frac{8 \cdot 60}{13} = \frac{480}{13} = 36. \overline{923076} \approx 37$

So the total time for the job is:

$3$ hours, $4$ minutes and $37$ seconds

Dec 7, 2016

$3 \frac{1}{13}$ hours (Assuming they can work at the same rate together as individually)

#### Explanation:

First we need to make an assumption that John and Sally can work at the same rate together as they do individually. This is quite a large assumption and quite possibly untrue in real life. However, since we haven't been given any information to the contrary, we'll go with that.

Let the total amount of work in the job be $x$ units.

Let ${R}_{j}$ be John's work rate per hour
Let ${R}_{s}$ be Sally's work rate per hour

From the question we know:

${R}_{j} = \frac{x}{8}$ Units of work per hour

${R}_{s} = \frac{x}{5}$ Units of work per hour

Now let $t$ be the time in hours they need to complete the job working together (With our assumption above)

Then their combined work rates will be ${R}_{j} + {R}_{s}$ to complete $x$ units of work in $t$ hours.

$\frac{x}{8} + \frac{x}{5} = \frac{x}{t}$

$\frac{1}{8} + \frac{1}{5} = \frac{1}{t}$

$\frac{13}{40} = \frac{1}{t}$

$t = \frac{40}{13} = 3 \frac{1}{13}$ hours