Can someone explain this result? (Rule of three)

Question

I have helped my cousin with homework.

The homework goes: $6$ machines do the work in $210$ hours, how long will it take $10$ machines. Assuming the work is linear.

I have solved that it is $0.6 \times 210 = 126$, as they will be working $40$ percent faster, therefore making the same in $60\%$ of the time.

But, I have also calculated $210/6$, which gives $35$ and tells how many hours it takes a machine to do. But, I don't understand that, as that would mean that one machine gets the job done in less time than $6$.

Can anyone tell what does that really mean?

Answer 1

It takes $1$ machine working alone $210 \times 6= 1260$ hours, not the $210/6 = 35$ hours in your second attempt

So it takes $10$ machines working together $1260/10=126$ hours, as in your first attempt

Answer 2

$\frac {210}6$ does not have any meaning in this context. The important thing is that the work takes $6 \cdot 210=1260$ machine-hours. You can divide that by the number of machines to get the number of hours needed, which gives $210$ for $6$ machines and $126$ for $10$ machines. It is often helpful to write out the units. What would hours/machine (which is what $\frac {210}6$ gives) mean? If the machines worked one after another on the same item it would be meaningful as the amount of time each machine takes.