Graphing Proportional Relationships

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Two values are proportional if they maintain the same ratio and their graph will always be a straight line that goes through the origin (0,0). In this lesson we'll explore how to draw proportional graphs and use them to solve desperate situations!

Save the Merchant Ship!

It was a race against time. The Serendipity, a large, slow merchant ship, sailed toward South America. The passengers were all enjoying the voyage, but what they didn't know was that just off the tip of Florida, three small, fast, heavily-armed pirate ships were eagerly awaiting the merchant's approach. The happy vessel was now only 70 miles from disaster.

At that point, the Avenger, a fast Yankee warship, received a telegram about the Serendipity's danger, and sailed from St. Augustine, Florida. Bristling with cannons and marines, the warship could handle the pirates but was 250 miles away from the ambush point. Is there enough time to catch up? Can the Avenger sail 250 miles before the Serendipity covers 70? It's a question of proportions.

Many situations in life involve proportional relationships, and sometimes the best way to understand those relationships is to draw a graph. Two values have a proportional relationship if their ratio (the fraction formed when one is divided by the other) is always the same. If one doubles, so does the other.

A graph is a picture of a mathematical relationship that helps us visualize how the numbers work together. It can help us look up the information we need when we have a problem involving proportions.

In our example, the distance each ship has moved is proportional to how much long it has been sailing. Figure 1 will tell us if the Avenger can travel 250 miles before the Serendipity travels 70. Let's take a look.

Figure 1. Graph showing distance versus time of the two ships
Proportional Graph

If you follow the blue line, which is the voyage of the Serendipity, you'll see when the ship goes 70 miles it encountered the pirates after 22 hours. What about the Avenger? Well, follow the red line to the same 22-hour point. Did the Avenger travel 250 miles by that time? No, at 22 hours the Avenger had only covered about 236 miles. It was 14 miles away when the merchant was attacked.

So what happened? The Avenger was too late to save the Serendipity, arriving at the scene just over an hour after the merchant vessel arrived. The Serendipity had been sunk, her cargo plundered and all passengers lost. The pirates were gone.

Drawing a Proportional Graph

When you're drawing a graph for a proportional relationship, keep two things in mind:

  1. Proportional graphs are always a straight line.
  2. Proportional graphs always start at the origin, or (0, 0) point.

So now let's see if we can save the Serendipity. The Avenger averaged just over 10 miles per hour, and the Serendipity averaged about 3. What if the Americans sent a newer, faster warship to catch the Serendipity?

The Aggressor, a fast clipper ship, was also anchored at St. Augustine. Lighter and slimmer than the bulky Avenger, the Aggressor averaged 13.5 miles per hour. Would it be able to catch the Serendipity before the slow merchant reached the Bahamas? Was it able to cover 250 miles from St. Augustine to the Bahamas in less than 22 hours? Let's draw the graph and find out.

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