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# Proportions Using Cross Products

## Cross-multiply to solve proportions with one variable

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Jellyfish Journeys

Credit: Dana Le
Source: http://www.flickr.com/photos/15493310@N07/6538725999

Have you ever seen a jellyfish? If you've ever been swimming in the ocean or to an aquarium, you may have seen a translucent umbrella with long tentacles, drifting in the water. You probably were mesmerized by the billowing sides the animal—but learned to be cautious of those long tentacles. These amazing creatures seem to float like mythical spaceships through the water. They are beautiful to behold but can be dangerous to encounter—the sting of some species can be extremely painful, or even deadly.

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There are many different types of jellyfish in many different sizes. Some species, like the Nomura's jellyfish pictured below, can grow to be more than six feet across, while more common jellyfish are usually around the size of a baseball glove. These are the ones that you can see washed up on sandy beaches in the summertime. The larger ones are often found deep below the surface of the ocean, and the tiniest ones are so small that you can swim through a school of them while scuba diving and not even notice that you have been stung until afterwards.

Credit: Janne Hellsten
Source: http://www.flickr.com/photos/nurpax/3768366525/

One tiny species of jellyfish that lives in the Mediterranean Sea is a powerful swimmer. It only measures 1.5 inches in diameter, but it can travel up to 3,600 feet per day! This jellyfish swims up and down a vertical column of water in search of plankton to eat. Despite its small size, its body can travel amazing distances. Its feat is equivalent to a 6-foot-tall person swimming 33 miles in one day! For the math behind these numbers, check out the "Explore More" section below.

Watch some exotic jellyfish in action: http://www.youtube.com/watch?v=aJUuotjE3u8

#### Explore More

We stated above that a 1.5-inch jellyfish traveling 3,600 feet per day is the same as a person swimming 33 miles in a day. In order to mathematically show this statement is true, we'll need to solve a proportion using cross-multiplication.

First, let's write a ratio that compares the size of the jellyfish to the distance it travels in a day.

1.5 inches3,600 feet\begin{align*}\frac{1.5 \text{ inches}}{3,600 \text{ feet}}\end{align*}

1.5 inches is the same as 0.125 feet. We can now rewrite the jellyfish ratio as:

0.125 feet3,600 feet\begin{align*}\frac{0.125 \text{ feet}}{3,600 \text{ feet}}\end{align*}

Now, consider a 6-foot-tall human being. How far could a human travel if he traveled at the same rate as the jellyfish? We can write a proportion to represent this scenario.

0.125 feet3,600 feet=6 feetx\begin{align*}\frac{0.125 \text{ feet}}{3,600 \text{ feet}}=\frac{6 \text{ feet}}{x}\end{align*}

Next, we cross-multiply and divide.

0.125xx=21,600 feet=172,800 feet\begin{align*}0.125x &= 21,600 \text{ feet}\\ x & = 172,800 \text{ feet}\end{align*}

Finally, convert feet to miles. There are 5,280 feet in a mile.

172,800 feet×1 mile5,280 feet33 miles\begin{align*}172,800 \text{ feet}\times\frac{1 \text{ mile}}{5,280 \text{ feet}}\approx33 \text{ miles}\end{align*}

Thus, a 6-foot-tall human would have to swim 33 miles in a day to swim a distance proportional to the distance that the jellyfish swims in a day.

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