# 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 \(\frac{$5.98}{8\text{ bottles}}\) provides that information. This quick calculation tells me that each 20-ounce bottle costs approximately $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 $0.50 each.

## How to find Unit Rate

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 $0.75, then dividing that cost by 20 ounces reveals the cost per ounce.

\(\frac{\$0.75}{20\text{ oz}}\)\(\approx \$0.04\)

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:

\(\frac{300\text{ mi}}{4\text{ hrs}}\) \(=75\text{ mph}\)

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:

\(\frac{513\text{ km}}{4\text{ hrs}}\) \(=128.25\text{ kph}\)

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. \(1\text{ mi}\approx 1.609\text{ km}\). To convert your average speed of 75 miles per hour to kilometers, simply multiply \(75\times 1.609\):

\(75\text{ mph} \times 1.609\text{ km/mi}=120.6\text{ kph}\)

On the other hand, you could convert your friend’s reported kilometers per hour to miles per hour. >span style=”font-style:normal; font-size:90%”>\(1\text{ km}\approx 0.6215\text{ mi}\). Multiply this conversion factor by your friend’s daily average speed to convert to miles per hour:

\(128.25\text{ kph} \times 0.6215\text{ mi/km}=79.7\text{ mph}\)

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!

## Frequently Asked Questions

#### Q

### How do you find the rate and unit rate?

#### A

Since rate is a ratio that compares two different units, find the rate of something by comparing one unit to another. For example, a bag of 8 apples for $3.60, can be described as $3.60 per bag of apples. You are comparing price to a bag of apples. A unit rate is a rate expressed for the quantity of one. In this example, that would be the price in dollars for 1 single apple. To find the unit rate, we will divide $3.60 by the number of apples in the bag, which is 8, to get the unit rate $0.45/apple.

#### Q

### What are examples of unit rates?

#### A

A very common example of a unit rate, which we encounter on a daily basis, is miles per hour. Since this rate is describing the number of miles in one hour, it is a unit rate. Another common unit rate is the price of gas per gallon. You can see this unit rate when passing any gas station sign. When buying meat, the price is displayed in dollars per one pound, which is also a unit rate. The price of most snack foods in the grocery store are also unit rates, such as the price of a bag of chips or a bag of pretzels.

#### Q

### What does unit rate mean?

#### A

A unit rate describes the ratio of two different units for the quantity of one. Some of the unit rates we use every day are miles per hour someone travels, price per pound of meat, and price per day for a rental car. We can find the unit rate when given a rate by dividing the unit in the numerator by the quantity in the denominator. For example, if we are given the rate $4.50/5 gallons of milk, we can divide 4.50 by 5 to find the unit price, which is dollars per one gallon of milk.

#### Q

### How do I calculate rates?

#### A

We do not actually calculate rate; rate is a ratio between two things that have different units of measure. For example, if we know that a pack of 3 Romaine Hearts costs $2.99, then the rate is $2.99/pack of Romaine Hearts.

The distance between Chicago and New York is approximately 790 miles. On average a flight from Chicago to New York is 3 hours. We can show the rate of travel for a plane as 790 miles/3 hours.

#### Q

### What is the difference between a rate and a unit rate?

#### A

The difference between a rate and a unit rate is that a rate is the ratio between two different units of measure, while a unit rate is the ratio of between two different units of measure for a single thing.

#### Q

### Can a unit rate be used to compare two rates?

#### A

A unit rate can be used to compare two rates as long as the units of measure are the same. For example, in the US we use $/gallon when describing the price of gas. In Europe they use Euro/liter. When comparing the price of gas in the US and Europe, it would be misleading if we did not convert the units to match because $1 = 0.82 Euro and 1 gallon = 3.785 liters.

## Fact Sheet

## Practice Questions

**Question #1:**

Calculate the following rate as a unit rate: $36 for 4 movie tickets

$9 for one movie ticket

$9 for two movie tickets

$18 for 2 movie tickets

$18 for one movie ticket

**Answer:**

The correct answer is $9 for one movie ticket.

In order to express $36 for one movie ticket as a unit rate, we need to determine the cost for one movie ticket. This requires us to divide $36 by 4, which equals $9.

**Question #2:**

Olivia can bake 95 cookies in 5 hours. At this rate, how many cookies can she bake in 11 hours?

100 cookies

90 cookies

475 cookies

209 cookies

**Answer:**

The correct answer is 209 cookies.

Olivia bakes 95 cookies in 5 hours, which can be expressed as the following rate: \(\frac{95\text{ cookies}}{5\text{ hours}}\). From here we can divide \(\frac{95}{5}\) in order to determine how many cookies she bakes per one hour. \(\frac{95}{5}=19\), so 19 cookies per hour. If Olivia bakes 19 cookies per hour, and we need to know how many cookies she can bake in 11 hours, we can simply multiply \(19×11\), which equals 209 cookies.

**Question #3:**

Franklin can jog approximately 2,000 meters in 4 minutes. Express this rate as a unit rate.

800 meters per minute

500 meters per minute

4,000 meters per 8 minutes

300 meters per 5 minutes

**Answer:**

The correct answer is 500 meters per minute.

A unit rate describes a ratio that represents “per one”. In order to express 2,000 meters in 4 minutes as a unit rate we need to divide 2,000 by 4. \(\frac{2,000}{4}=500\), which is the unit rate 500 meters per one minute.

**Question #4:**

Jamie wants to buy a 6 pack of pineapple juice boxes. The price for a 6 pack is $2.64. Jamie notices that one juice box is leaking, so he only wants to buy five out of the six. How much would five juice boxes cost?

$2.35

$2.80

$2.20

$2.99

**Answer:**

The correct answer is $2.20.

In order to calculate the price for 5 juice boxes, we need to determine the price for one juice box. This can be calculated by dividing \(\frac{$2.64}{6}\), which equals $0.44, or 44 cents. Each juice box costs 44 cents. If Jamie wants to purchase 5 juice boxes at 44 cents each, we simply multiply \(5×0.44\), which is 2.2, or $2.20.

**Question #5:**

Your local grocery stores are having sales on bags of candy. Store A is selling 40 oz. bags of candy for $8.75. Store B is selling 30 oz. bags of candy for $7.49. Which store has the better deal?

Store A

Store B

**Answer:**

The correct answer is Store A.

In order to identify the better deal, we need to calculate the unit cost (unit rate) for each store. Store A sells candy for \(\frac{$8.75}{40\text{ oz}}\), so let’s divide $8.75 by 40 in order to determine how much one ounce of candy will cost. \(\frac{$8.75}{40\text{ oz}}\) is approximately 22 cents per one ounce. Store B sells candy for \(\frac{$7.49}{30\text{ oz}}\) so let’s divide $7.49 by 30 to determine its unit cost. \(\frac{$7.49}{30\text{ oz}}\) is approximately 25 cents per one ounce. Now that we have calculated the unit cost for each store, we can clearly see that Store A has the better deal on bags of candy! Better hurry before they sell out!