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The equation that relates distance, rate, and time is D = RT, where D is the distance, R is the rate, and T is the time.

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From your own personal experience, this should make perfect sense. If you are driving somewhere at 60 miles per hour for 5 hours, you’d drive a total of D = 60 * 5 = 300 miles. If you were driving half as fast and only going 30 miles per hour for 5 hours, you’d drive a total of D = 30 * 5 = 150 miles.

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So, when you go half as fast, you also go half the distance in the same amount of time! This is a very straightforward description for how we often think about distance, rate, and time.

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We also need to rearrange the equation to find some other useful pieces of information.

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The first rearrangement is dividing both sides by T to find the equation D/T=R.

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The second rearrangement is dividing both sides by R to find the equation D/R=T.

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Therefore, we have 3 total equations you should be familiar with!

- D=RT
- D/T=R
- D/R=T

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This seems a bit like a pointless exercise but considering the units that each piece is measured in is particularly helpful to seeing why we do this.

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Distance is measured in meters, miles, yards, kilometers, and many others.

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Time is measured in seconds, minutes, hours, days, and a few others (but those aren’t as useful as these basic units).

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Rate is measured in some unit of distance divided by some unit of time. For example, we can have miles per hour or meters per second. This is definitely the trickiest unit to deal with since it includes TWO parts.

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Most often, you will see word problems that test your ability to calculate distance, rate, or time.

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Consider the following problem.

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Someone is flying on an airplane between New York and Los Angeles, the distance traveled is approximately 2,500 miles. The total flight takes 4.5 hours. How fast was the airplane traveling, in miles per hour?

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The first question to ask is which equation will we be using? Since we are solving for rate, we will use equation number 2 from earlier.

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This tells us that R=D/T.

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Simply plugging in the values for distance and time tells us that R=2,500 miles / 4.5 hours.

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Dividing these numbers gives us that the plane was traveling at around 555.56 miles per hour. Notice how the units in the fraction actually guaranteed that we had the correct numbers on the numerator and denominator!

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I cannot emphasize enough how important it is to include units, no matter how smart you are. This will prevent a simple mistake such as saying the plane traveled 0.0018 miles per hour.

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Finally, let’s consider an example that doesn’t initially seem like a distance, rate, and time problem.

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Johnny can mow 2 lawns every 3 hours, while Sally can mow 1 lawns every 3 hours. How long, in minutes, will it take Sally and Johnny to mow 1 lawn?

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To solve this problem, we’ll actually use all of the tools we’ve built up so far. The first thing to determine is how fast Johnny and Sally work together. We actually have a rate for how fast each of them mows individually, Johnny’s rate is 2 lawns / 3 hours and Sally’s rate is 1 lawns / 3 hours. Notice how the distance in this example is actually the number of lawns. This is because this is how “far” they are going in a given amount of time, so to speak.

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Simply add these two fractions together to find 2 lawns / 3 hours + 1 lawns / 3 hours = (3 lawns) / 3 hours = 1 lawn per hour.

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Since their combined rate is mowing 1 lawn per hour, we know that this is exactly 60 minutes to mow one lawn when they work together!

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This last example is definitely trickier but uses the exact same skills from the foundations of distance, rate, and time. Always write out your units as you do work to ensure that you keep everything in the correct order and you’ll find that the basics of distance, rate, and time are easy to apply to many of the problems you will encounter!

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