I'm having trouble determining when you divide and when you multiply to solve a scientific notation problem.

Question

For example:
Assume that there are 20,000 runners in the New York City marathon. Each runner runs a distance of 26 miles. If you add together the total number of miles for all the runners, how many times around the globe would the marathon runners have gone? Consider the circumference of the earth to be 2.5 x 10^4 miles. Round your answer to the nearest tenth.

I know to solve you multiply 26 x 20,000 to get 520,000 miles. Then you divide 520,000 by 2.5 x 10^4 to get 20.8 times around the globe.

Then, the number of miles that light travels in one year is called a light-year, which is equal to 5.99 x 10^12 miles. The nearest star (other than the sun) to the earth is Proxima Centauri, at 4.2 light-years away. How far is that in miles?

To solve you multiply 4.2 x 5.99 x 10^12 to get 2.5158 x 10^13 miles

I don't understand my one you multiply and one you divide??????

Answer 1

First, online, “*” is used to indicate multiplication to avoid confusion with “x” as an unknown.

In the first, you are converting to a larger unit, while the second you are converting to a smaller unit.

Answer 2

How do I know that I'm converting to a larger unit or a smaller unit?

In the first I'm trying to figure out how many times around the globe and in the second, I'm trying to figure out how far away something is in miles - how does that mean larger or smaller?

Answer 3

1. 5.2*10^5mi/2.5*10^4mi/rev = 2.08*10^1 = 20.8 Rev. or times around.
Your answer is 20.8 revolutions; however, if you had multiplied, your answer would have
been (mi)^2/rev. which does not make sense for your problem.

Answer 4

5.99*10^12mi/Light-yr. * 4.2Light-yrs. = 2.516*10^13 Miles.
If you had divided, your units would not have been miles.

I hope this helps.