Mowing the lawn alone vs. together

Question

The question is: Nils mows the lawn in 3 hrs alone. Nils and Jonas mow the lawn together in 2 hrs. How quickly would Jonas mow the lawn alone?

My thinking: If Nils mows the lawn alone in 3 hrs, he mows 1/3 the lawn in 1 hr. If Nils and Jonas mow the lawn together in 2 hrs, they mow 1/2 the lawn in one hr., which means they each mow 1/4 of the lawn in one hr.

So, I would think that Jonas would take 4 hrs to mow the lawn alone, but the answer provided is 6 hrs.

What am I doing wrong?

Answer 1

This kind of problem is always confusing, and this is why the use of variables is really useful, even in everyday life.

Let's identify things that won't change with the number of people mowing the lawn simultaneously. The most obvious pick is what you identified : the fraction of the lawn that each one of them mow in 1 hour.

Let's call $x$ this value for Nils and $y$ for Jonas.

"Nils mows the lawn in 3 hrs alone" translates in $3x=1$. So $x = \frac13$ as you said. "Nils and Jonas mow the lawn together in 2 hrs" translates in $2(x+y) = 1$. So $2x + 2y = 1$, and $y = \frac{1-2x}{2}$. You know that $x = \frac13$ so $y = \frac{1-2x}{2} = \frac{1-2*\frac13}{2} = \frac16$.

We then know that Jonas mows $\frac16$ of the lawn in 1 hour, so he takes 6 hours to do the whole job alone.

Your mistake was in this sentence : "they mow 1/2 the lawn in one hr., which means they each mow 1/4 of the lawn in one hr", because you assume that Nils and Jonas take the same time to mow.

Addendum : You could also solve the problem with a less formal way. You know that in $2$ hours, Nils hae mowed $\frac23$ of the lawn, and that Nils and Jonas have finished the job at this time. This means that Jonas has mowed $\frac13$ of the lawn in this time. Thus, he is $2$ times slower than Nils. Hence the result.

Answer 2

You say:

...they mow 1/2 the lawn in one hr., which means they each mow 1/4 of the lawn in one hr.

but here, you're assuming that Nils and Jonas mow at the same speed. This is not necessarily true. Nils and Jonas do mow half the lawn in one hour, but remember that Nils has mowed a third of it and Jonas has done the rest. So Jonas has mowed $\frac{1}{2} - \frac{1}{3} = \frac{1}{6}$ of the lawn in one hour, implying that he will mow the entire lawn alone in $6$ hours.

Answer 3

If the lawn were 6 acres, apparently Nils mows 2 acres per hour, while Nils and Jonas together mow 3 acres per hour. Hence Jonas mows one acre per hour. So Jonas would require 6 hours to mow the lawn by himself.