Calculating Unit Rates and Unit Prices

Now we're going to learn about unit rates and unit prices. A unit rate is a rate where the second value, or denominator if it's written as a fraction, is the number one.

So for our first example, we have… after 1 hour of driving, the car has traveled 60 miles. We can write this as 60 miles per 1 hour, or as a fraction, 60 miles over 1 hour. And we can recognize that it's a unit rate because the second value, or the denominator, is the number one. However, it's more common for us to not even write the number one.

So if you ever see a rate written this way with no number with the second unit of measurement, then it's implied that it's just a one. We can also write it a third way using the front slash symbol instead of the word “per.” So we would read this as 60 miles per hour, but we use the front slash instead of the word “per.”

Let's look at our first example. Ian can type 170 words in five minutes. Find his typing speed as a unit rate. So another way we can read this problem is… find how many words Ian can type per minute. Now let's look at the information that we have. We know that he can type 170 words in five minutes. We can also represent this ratio as a fraction as 170 words over five minutes.

And remember that whenever one of your values represents time, you want to write that as the second value, or if it's a fraction, write that value in the denominator. So that's why we put five minutes in the denominator of our fraction. But keep in mind here, this is not a unit rate because we have five minutes as our second value instead of one. So we're going to convert this to a unit rate.

So we'll start with our fraction. And remember that fractions represent division. So by simplifying this fraction with division, we're actually going to be able to convert this to a unit rate. So when we're dividing, we'll take the numerator and divide by the denominator.

So it becomes 170 divided by five, which gives us 34. And that's how many words Ian can type in 1 minute. So we can write this as 34 words per minute, or if we want to use the forward slash instead of the word “per,” we can write it this way, but we would still read it as 34 words per minute.

Next, we have unit price. So unit price is a type of unit rate where the first value, or the numerator, is a measurement of money. But since it is still a unit rate, the second value will be a one. So for example, a box of cookies costs $3. Let's write this as a unit price.

We could say that it costs $3 per one box, or as a fraction, $3 over one box. And most commonly, we don't even bother writing the number one for the second value. And then the third way is to use the front slash instead of the word “per.” So any of these three variations represent the unit price for this box of cookies.

Shayla bought three boxes of cookies for $12.75. Find the unit price for a box of cookies. We can also think of this problem as asking us to find the cost of one box of cookies, because that's the same thing as a unit price for a box of cookies.

So let's set this problem up based on the information they give us. We have $12.75 for three boxes, or as a fraction, $12.75 over three boxes. And notice these are not unit prices, because even though our first value does represent a dollar amount, our second value is not the number one.

So we'll need to convert that fraction into a value that has a one as the second value. And we'll use division - numerator divided by denominator, which gives us $12.75 divided by three, and that's $4.25. So when we write it out with our units, we have $4.25 per box. Or we can use the front slash symbol and still read it as $4.25 per box.

So keep in mind, when you are setting up a problem that asks you to find the unit price or the unit rate, we need to first set it up based on the information they give us, ideally as a fraction. And if it does involve some type of money, we place that as the numerator of the fraction. And then we use division to figure out the value of the unit price or unit rate. And remember to keep the units with the values.