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Rate is a very important type of ratio, used in many everyday problems, such as grocery shopping, traveling, medicine--in fact, almost every activity involves some type of rate. Miles per hour or feet per second are both rates of speed. Number of heartbeats per minute is called "heart rate." If you ask a babysitter, "What is your rate?", you are asking how many dollars per hour you will be charged. The little word "per" is always a clue that you are dealing with a rate. Unit price is a particular rate that compares a price to some unit of measure. For example, suppose eggs are on sale for $.72 per dozen. The unit price is $.72 divided by 12, or 6 cents per egg.

The word "per" can be replaced by the "/" in problems, so 6 cents per egg can also be written 6 cents/egg.

Smart shoppers know how to estimate unit prices when deciding whether it's better to buy a larger size of an item. Many everyday problems involve rates of speed, using distance and time. We can solve these problems using proportions and cross products. However, it's easier to use a handy formula: rate equals distance divided by time: r = d/t. Actually, this formula comes directly from the proportion calculation -- it's just that one multiplication step has already been done for you, so it's a shortcut to learn the formula and use it. You can write this formula in two other ways, to solve for distance (d = rt) or time (t = d/r).

Let's say you rode your bike 2 hours and traveled 24 miles. What is your rate of speed? Use the formula r = d/t. Your rate is 24 miles divided by 2 hours, so:

r = 24 miles ÷ 2 hours = 12 miles per hour.

Now let's say you rode your bike at a rate of 10 miles per hour for 4 hours. How many miles did you travel? This time, use the distance formula d = rt:

d = 10 miles per hour × 4 hours = 40 miles.

Next, you ride 18 miles and travel at a rate of 12 miles per hour. How long did this take you? Use the time formula t = d/r:

t = 18 miles ÷ 12 miles per hour = 1.5 hours, or 1 ½ hours.

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    and time

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First Glance   In Depth   Examples   Workout

Distance, rate and time