Finding Constant and Average Rates

Finding Constant and Average Rates

Watch this video lesson to learn the differences between constant and average rates. You will learn the steps you need to take to find them and get two great visuals you can keep in mind to help you solve these rate problems.

What Is a Constant Rate?

So, what exactly is a constant rate? Well, a constant rate is something that changes steadily over time. Picture the burning of a candle. That is a constant rate because a candle burns down steadily over time. We can be confident that if a particular candle takes so many hours to burn down, and we make the candle twice as big, we will have a candle that will burn twice as long.

What Is an Average Rate?

Now, what about the average rate? An average rate is different from a constant rate in that an average rate can change over time. An average rate is actually the average or overall rate of an object that goes at different speeds or rates over a period of time. For this one, picture the flight of a bumblebee.

The bumblebee sees a flower and rushes over to the flower at a quick speed. Once it's there, the bumblebee slows down and slowly buzzes around the flower to inspect it. Oh, but look!

There's another even bigger flower in the distance. The bumblebee sees it and buzzes off at an even quicker speed.

The average rate of the bumblebee would take all these different speeds and find the average speed the bumblebee had over his whole trip.

Now that we've covered these definitions, let's see how we can find each of these rates. We will stick to our original visuals to help us out with our problems.

Finding a Constant Rate

Let's go back to our burning candle to help us with finding a constant rate.

We are taking a test, and we see this problem in front of us: Object A takes 1 hour to burn 1 inch, and Object B takes 2 hours to burn down 1 inch. What is the constant rate of each object?

Okay, another dry math problem. But, that's okay! We have our useful visual we can refer to, so we won't be bored. We can picture two differently sized candles. We can get creative here and make the problem more fun. We can have one rainbow colored candle and another candle with a lightning bolt drawing on the side. Picture anything that keeps your interest.

Now that we have our candles in front of us, we can get to the meat of the problem and find the constant rates of each candle.

We will call the rainbow colored candle Object A, and we will call the lightning bolt candle Object B. I'm picturing my rainbow colored candle burning. I see that for each hour that passes, my rainbow colored candle goes down by an inch. To calculate my constant rate for this candle, this object, I recall the formula for rate, which is rate = distance / time.

Okay. So, I don't have a distance, per se. But I do have the amount of candle burned, which is 1 inch. I also have a time, which is 1 hour. I plug these numbers into my formula to get my constant rate. I get 1 / 1, which gives me 1. So, my constant rate for my rainbow colored candle, or Object A, is 1 inch per hour. That is part of my answer.

The other part of my answer is the rate for the lightning bolt candle, or Object B. I will do the same and plug in 1 for the amount of candle burned, or the distance, and 2 hours for my time. My constant rate here, then, is 1 / 2. or 0.5 inches per hour.

Now I'm done with this problem. Object A takes 1 hour to burn 1 inch, and Object B takes 2 hours to burn down 1 inch. What is the constant rate of each object? I have found that my answer for this problem is 1 inch per hour for Object A and 0.5 inches per hour for Object B.

Finding an Average Rate

To help us out with an average rate problem, let's go back to our bumblebee visual.

Our math problem might say something like this: An object has three parts to its travel from point A to point B. In part 1, the object travels at a speed of 10 km per hour for half an hour. In part 2, it travels at 2 km per hour for 15 minutes. And in part 3, the object travels at 15 km per hour for another half hour. What is the average rate for this object?

This one's a little more complicated. We can picture our bumblebee for the object, and we can relate each part of the travel to each part of the bumblebee's journey. The first part can be our bumblebee going fast to its first flower; the second part is then our bumblebee checking out the flower; and the third part is our bumblebee flying fast to the second, larger flower.

To find our average, I will use the same rate formula I did for finding the constant rate. The only additional information I need is the total distance traveled and the total time. But, I can find this information from what was provided in the problem. Re-reading the problem, I can see that the bumblebee traveled 5 km in half an hour for the first part of the trip, because if the bumblebee is flying at 10 km per hour, then at the half-time point, the bumblebee will have gone 5 km. For the second part of the trip, because the bumblebee is traveling at 2 km per hour, after 15 minutes, or a quarter of an hour, the bumblebee will have gone 0.5 km, which is a quarter of 2 km. Then, in the last part of the trip, because the bumblebee is traveling at 15 km per hour in half an hour, the bumblebee will have gone 7.5 km, or half of 15 km.

Now that I have my distances and times for each part of the trip, I will add them up to get the total distance and total time. My total distance traveled is 5 + 0.5 + 7.5, which equals 13 km. My total time is 0.5 + 0.25 + 0.5, which equals an hour and a quarter, or 1.25 hours.

Plugging these two numbers into my formula, I get 13 km / 1.25 hours, which equals 10.4 km per hour. That is my answer, and I am done.

Lesson Summary

In review, recall that a constant rate is something that changes steadily over time and that an average rateis the overall speed or rate of something that travels at different speeds or rates over time.

To find both rates, we use the rate formula, which is rate = distance/time. The distance in the formula is the amount of change in the problem over a period of time. To find the constant rate, you would plug in the change and divide by time. Because it is constant, this number won't change.

For the average rate, though, you would first have to find the total distance or change and the total time involved. You would then divide the total change or distance by the total time to find your average rate.