# Difference Between Unit Rate and Rate

## Rates and Unit Rates

Welcome to this video on rates and, more specifically, **unit rates**. This math concept is practical and useful, so we hope that you come away with a solid understanding and confidence in interpreting the rates that are all around you. Let’s get started!

Rates are spoken about and used every day in many aspects of life. The concept is pretty straightforward, as rates are nothing more than ratios of values that represent different units of measure. The values that are being compared are called the **terms** of the rate.

Let’s consider the treasure trove of rates that may be in your grocery cart the next time you go shopping. In the examples listed here, the rate that you agree to pay is simply the **cost** related to the **quantity** of the product.

Let’s say we have a 12-ounce box of pasta, which costs $1.49. The rate is then $1.49 per 12 ounces. If we have a pound of deli meat and the cost is $9.99, our rate would then be $9.99 per pound. Lastly, let’s say we have an 8-pack of 20-ounce bottles of soda and the cost is $5.98. Our rate then is $5.98 per pack.

Item | Cost | Rate |
---|---|---|

12 oz. box of pasta | $1.49 | $1.49/12 oz. of pasta |

1 lb. of deli meat | $9.99 | $9.99/1 lb. of deli meat |

8-pack of 20 oz. soda | $5.98 | $5.98/eight 20 oz. bottles |

If we look at this last example a little closer, we’ll see that there is room to break down the rate even further. The example provides the price of an 8-pack, but what if I want to determine the cost of one 20-ounce bottle? The ratio $5.98 to 8 bottles \(\frac{$5.98}{8\text{ bottles}}\) provides that information. This quick calculation tells me that each 20-ounce bottle costs approximately 75 cents. $0.75.

This breakdown to determine the cost per bottle may be helpful to determine whether I buy the eight pack of one type of soda, or the individual 20-ounce bottles of another brand on sale for 50 cents $0.50 each.

The process of breaking down the cost to a smaller unit reveals the **unit rate** of the product. This is helpful to make informed decisions at the store, as the volumes of product in various packaging are often different. By comparing unit rates, savvy customers are able to make price comparisons based on common units of the product regardless of packaging and advertised “sale” prices.

Let’s break down the soda cost per bottle further to determine the cost **per ounce**. If one 20 ounce bottle costs roughly 75 cents $0.75, then dividing that cost by 20 ounces reveals the cost per ounce: So 75 cents per ounce divided by 20 ounces, gives you roughly 4 cents per ounce.

\(\frac{$0.75}{20\text{ oz.}}\) = gives you roughly 4 cents per ounce.

As you can see, breaking down costs to the smallest unit reveals the cost savings of the sale. Of course, saving a few cents per ounce of soda may not be the deciding factor of your purchase. Other factors come into play when consumers are shopping, like brand loyalty and personal preference. However, comparing unit costs provides an objective way of making consumer choices based on the price.

The important thing to remember when analyzing unit rates is that the units must be the same. Let’s consider another example to illustrate this point.

Suppose you are on a road trip in Wyoming and on the first day you covered 300 miles in 4 hours of mostly highway driving. You can quickly determine your average rate of speed as **miles per hour** with the following calculation:

\(\frac{300\text{ miles}}{4\text{ hours}}=75\) miles per hour

Coincidentally, your friend is traveling in Germany, where the standard unit of measure is in kilometers, and she reports that she covered approximately 513 kilometers in 4 hours on her first day of the road trip. Her average speed would be calculated as:

\(\frac{513\text{ kms}}{4\text{ hours}}=128.25\) kilometers per hour

Clearly, this is comparing “apples to oranges” in the sense that the underlying units are not the same. A conversion of either miles to kilometers or kilometers to miles must be made to make a fair comparison of average speed.

Keep in mind that a kilometer is a shorter unit of distance than a mile. One mile is equal to approximately 1.609 kilometers. To convert your average speed of 75 miles per hour to kilometers, simply multiply 75 by 1.609:

\(75 \times 1.609=120.6\) kilometers per hour

On the other hand, you could convert your friend’s reported kilometers per hour to miles per hour. 1 kilometer is equal to approximately .6215 of 1 mile. Multiply this conversion factor by your friend’s daily average speed to convert to miles per hour:

\(128.25 \times 0.6215=79.7\) miles per hour

The way that you convert does not matter as long as you compare the average speeds of the same unit. Both conversions show that your friend in Europe traveled at a faster rate on the first day of her trip.

I hope that you feel more equipped to interpret and solve the math “puzzles” that you may come across. Thanks for watching, and happy studying!