I'm having trouble determining when you divide and when you multiply to solve a scientific notation problem.
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??????
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.
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?
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.
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.
In scientific notation problems, whether to multiply or divide depends on what you're trying to find or calculate.
In the first problem about the New York City marathon, you want to determine how many times around the globe the marathon runners would have gone. We know that each runner runs a distance of 26 miles, and there are 20,000 runners. To find the total distance covered by all the runners, you need to multiply the number of runners (20,000) by the distance each runner runs (26 miles). This gives you a product of 520,000 miles, which represents the total distance covered by all the runners.
Next, you want to determine how many times this total distance (520,000 miles) is equivalent to the circumference of the Earth (2.5 x 10^4 miles). To do this, you need to divide the total distance covered by the circumference of the Earth. So, you divide 520,000 miles by 2.5 x 10^4 miles (or, equivalently, you can multiply 520,000 miles by the reciprocal of 2.5 x 10^4). This calculation gives you the answer that the marathon runners would have gone around the globe approximately 20.8 times.
In the second problem about the distance to Proxima Centauri, you're given that Proxima Centauri is 4.2 light-years away, and you want to determine the distance in miles. The given value, 4.2 light-years, represents a distance. You're not trying to compare it to any other distance or find a ratio. So, to calculate the distance in miles, you need to multiply the given value (4.2 light-years) by the number of miles in one light-year, which is 5.99 x 10^12 miles. This multiplication gives you the answer of 2.5158 x 10^13 miles.
In summary, when solving scientific notation problems, whether to multiply or divide depends on the relationship you're trying to find or calculate between the given values. In the first problem, you multiplied to find the total distance and then divided to find the comparison between the total distance and the Earth's circumference. In the second problem, you multiplied to convert the given distance in light-years to miles.