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This is the problem: "A mountain climber is at an elevation of $10,000$ feet. If she descends $2,000$ feet a day, which equation would be used to show how many days it will take to reach sea level ($0$ feet)?"

My online school says the problem looks like this; $10,000 ÷ 2,000$ but I thought it should be $-2,000$ since climber is descending.

The online teacher said this, "It would not be a negative number since we are given $10,000$ feet and $2,000$ each day. This answer is correct."

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    $\begingroup$ How would you interpret an answer of $-5$ days? $\endgroup$ – JB King Apr 13 '15 at 17:53
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If you define "going down" as negative, then the mountain climber is trying to travel $10 \ 000$ feet down, which is $-10 \ 000$. If she's going down $-2 \ 000$ a day, then you have:

$$\text{days} = \dfrac{-10 \ 000}{- 2 \ 000}$$

On the other hand, if you define "going down" as positive, then the mountain climber is still traveling $10 \ 000$ feet down, but it's defined as positive. If going down is positive, so is going down at a rate of $2 \ 000$ a day:

$$\text{days} = \dfrac{10 \ 000}{ 2 \ 000}$$

Both your answer and the teacher's answer are correct, but yours feels more natural to me.

I just noticed: her velocity is not given as $2 \ 000$, that's her speed. Her velocity is a vector quantity and so is given by ${2 \ 000}\dfrac {\text{ft}}{\text{day}}\left[{\text{towards the ground}}\right]$, and you have to write that whichever way is useful in solving the problem. In these word problems, it's often a matter of preference which way is positive and which way is negative.

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    $\begingroup$ $-2000$ can be seen as her velocity as well because in one dimension we can deal with vectors as scalars. The question has cleared up already which way is negative and which is positive. It said, "...at an elevation of $10000$...". That means upward is taken positive. Now of course her velocity with which she descends is $-2000\frac{ft}{d}$. $\endgroup$ – Sufyan Naeem Feb 9 '16 at 19:40
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$\frac{10,000}{2,000}$ is defined to be the solution to $10,000-2,000x=0$.

Now it's left to prove that the solution to $10,000-2,000x=0$ should be the answer.

Well, after $1$ day we have $10,000-2,000=8,000$ feet left, which is intuitive -- if she descends $2,000$ a day, then after one day she will have $8,000$ left.

So intuitively $x$, i.e. the solution to $10,000-2,000x=0$, should indeed be the answer, but, as I said, it is by the definition of division equal to $\frac{10,000}{2,000}$.

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