If it takes 3 people 20 hours to dig a hole, how long does it take 5 people?

Answer 1

This is a simple inverse relationship:

(workers)(time) = k

Here 3(20) = 60

So 5(t) = 60

t = 12

Answer 2

Silver, the fundamental concept here is

Workproduct = Rate * Time

We know we have a hole to dig, we don’t care how big. All we know is it takes 3 people 20 hours each to do the job.

Often, we would have a Rate (what fraction of the hole can be dug per hour) and a Time ( how much time per digger).

We start by understanding the total workproduct. We can be arbitrary about that, by saying there’s 1 hole.

What is the Rate when 3 people do the job?

Since Workproduct = Rate * Time, we can see that

R = W / T

There are 3 workers, so the Rate per person = W / 3T

We know the Time = 20 hours, and W = 1, so

R₁ = 1/(3*20) = 1/60

Remember this is the fraction of the job that any one digger can do in one hour.

So, how much time does it take 5 diggers?

W = R T, so T = W/R

we know R₁= 1/60, and there 5 diggers so

R₅ = 5/60

T = 1/(5/60) = 60/5 = 12

Does this make sense? It takes 3 people 20, and we find it takes more people (5) less time (12). Yes, that makes sense.

Answer 3

What if we only had one person digging, instead of 3? Well, then it would take 3 times as long, or 60 hours. Now let's quintuple that 1 person into 5 people. The time gets divided by 5. The answer is 12 hours.

Notice that we could have simply multiplied 20 hours by 3/5, the inverse of the ratio of (new # of people) / (old # of people). But I find it's often best when starting out with this topic to always find the unit rate, in this case, one person digs one hole in 60 hours. That way, we can not only change the number of people easily, but also the number of holes.