The amount of time p people to paint d doors varies directly with the number of doors and inversely with the number of people. Four people can paint 10 doors in 2 hours How many people will take to paint 25 doors in 5 hours?

Question

4054 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-04-18T06:17:46

$4$ $4$

The first sentence tells us that the time $t$ $t$ taken for $p$ $p$ people to paint $d$ $d$ doors can be described by a formula of the form:

$t = \frac{k d}{p}$ $t = (kd)/p" "$ ... (i)

for some constant $k$ $k$ .

Multiplying both sides of this formula by $\frac{p}{d}$ $p/d$ we find:

$\frac{t p}{d} = k$ $(tp)/d = k$

In the second sentence, we are told that one set of values satisfying this formula has $t = 2$ $t=2$ , $p = 4$ $p=4$ and $d = 10$ $d=10$ .

So:

$k = \frac{t p}{d} = \frac{2 \cdot 4}{10} = \frac{8}{10} = \frac{4}{5}$ $k = (tp)/d = (2*4)/10 = 8/10 = 4/5$

Taking our formula (i) and multiplying both sides by $\frac{p}{t}$ $p/t$ , we find:

$p = \frac{k d}{t}$ $p = (kd)/t$

So substituting $k = \frac{4}{5}$ $k=4/5$ , $d = 25$ $d=25$ and $t = 5$ $t=5$ , we find that the number of people required is:

$p = \frac{(\frac{4}{5}) \cdot 25}{5} = \frac{20}{5} = 4$ $p = ((4/5)*25)/5 = 20/5 = 4$