Understanding and Writing Rates

In our last lesson, we learned about ratios. Now we're going to learn about a special type of ratios called rates. Rate is a type of total to total ratio. It describes how one value changes as another value changes. And the two values will be represented with different units of measurement.

So as we go through different problems and we work on writing rates to represent them, we will need to pay attention to the units. And we may see units that represent how much time has passed or how far a car has traveled or even how much money something costs.

So let's say we're driving along in this car and we want to keep track of how far we've traveled as time has passed on, after 1 hour of driving, the car has traveled 60 miles. After 2 hours of driving, the car has traveled 120 miles. And after 3 hours of driving, the car has traveled 180 miles. So as the time increases, the distance traveled increases.

So we have two different values that we're comparing. We're showing that as the time changes, the distance also changes. So we can use a rate to represent this relationship between time and distance.

So let's look at the different ways that we can represent rates for that scenario of the car traveling. We can represent it with words. And the words that we typically use to represent rates are “per,” “for,” or “in.” So we could say the cars traveled 60 miles in 1 hour, 120 miles in 2 hours, or 180 miles in 3 hours.

And just like we could represent ratios as fractions, we can also represent rates as fractions, which is how we'll normally see them in math. So we could write it as 60 miles over 1 hour, 120 miles over 2 hours, and 180 miles over 3 hours. And notice that when we write it with words or as a fraction, we keep the units with the numbers, and that's very important with rates. We always have to keep the unit of measurement with the numbers.

So does it matter what order we write those numbers? When we set up our rates, we talked about writing ratios based on how the word problem is presented to us. And sometimes we can refer to the problem to figure out what order to write the numbers in rates. But with some types of measurement, we do write them a certain way.

So when we have a rate that involves measurements of time, we place time as the second value. Or if it's a fraction, we place time as the denominator. So we could write it as 120 miles in 2 hours with 2 hours written as the second value, or 120 miles over 2 hours with the 2 hours written as the denominator. So notice where we have our number of hours placed as the second value and as the denominator.

It also matters when we're using measurements of money. So when we're writing rates that have money involved, we place that money value as the first value or as the numerator if we have a fraction. So, for example, we could have $5 for eight candies or $5 over eight candies. And either way that we write it, we have to put the dollar amount, the money, as the first number or as the numerator when we have it, as a fraction.

So when you're working with rates, pay extra careful attention to the units that you're using and how you're placing measurements of time and money.