# It takes John 20 hours to paint a building. It takes Sam 15 hours to paint the same building. How long will it take for them to paint the building if they work together, with Sam starting one hour later than John?

Oct 31, 2017

$t = \frac{60}{7} \text{ hours exactly}$

$t \approx 8 \text{ hours "34.29" minutes}$

#### Explanation:

Let the total amount of work to paint 1 building be ${W}_{b}$

Let the work rate per hour for John be ${W}_{j}$

Let the work rate per hour for Sam be ${W}_{s}$

Known: John takes 20 hours on his own $\implies {W}_{j} = {W}_{b} / 20$

Known: Sam takes 15 hours on his own $\implies {W}_{s} = {W}_{b} / 15$

Let the time in hours be $t$

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$t {W}_{j} + t {W}_{s} = {W}_{b}$

$t \left({W}_{j} + {W}_{s}\right) = {W}_{b}$

but ${W}_{j} = {W}_{b} / 20 \mathmr{and} {W}_{s} = {W}_{b} / 15$

$t \left({W}_{b} / 20 + {W}_{b} / 15\right) = {W}_{b}$

$t {W}_{b} \left(\frac{1}{20} + \frac{1}{15}\right) = {W}_{b}$

Divide both sides by ${W}_{b}$

$t \left(\frac{1}{20} + \frac{1}{15}\right) = 1$

$t \left(\frac{3 + 4}{60}\right) = 1$

$t = \frac{60}{7} \text{ hours}$

$t \approx 8 \text{ hours "34.29" minutes}$