Simplifying Ratios and Rates

 

Learning Objective(s)

·         Write ratios and rates as fractions in simplest form.

·         Find unit rates.

·         Find unit prices.

 

Ratios are used to compare amounts or quantities or describe a relationship between two amounts or quantities. For example, a ratio might be used to describe the cost of a month’s rent as compared to the income earned in one month. You may also use a ratio to compare the number of elephants to the total number of animals in a zoo, or the amount of calories per serving in two different brands of ice cream.

 

Rates are a special type of ratio used to describe a relationship between different units of measure, such as speed, wages, or prices. A car can be described as traveling 60 miles per hour; a landscaper might earn $35 per lawn mowed; gas may be sold at $3 per gallon.

 

Ratios compare quantities using division. This means that you can set up a ratio between two quantities as a division expression between those same two quantities.

 

Here is an example. If you have a platter containing 10 sugar cookies and 20 chocolate chip cookies, you can compare the cookies using a ratio.

 

The ratio of sugar cookies to chocolate chip cookies is:

 

 

The ratio of chocolate chip cookies to sugar cookies is:

 

 

 

 

You can write the ratio using words, a fraction, and also using a colon as shown below.

 

 

Some people think about this ratio as: “For every 10 sugar cookies I have, I have 20 chocolate chip cookies.”

 

You can also simplify the ratio just as you simplify a fraction.

 

 

So we can also say that:

 

 

 

How to Write a Ratio   A ratio can be written in three different ways:   ·         with the word “to”: 3 to 4. ·         as a fraction: . ·         with a colon: 3 : 4.   A ratio is simplified if it is equivalent to a fraction that has been simplified.

 

Below are two more examples that illustrate how to compare quantities using a ratio, and how to express the ratio in simplified form.

 

 

Often, one quantity in the ratio is greater than the second quantity. You do not have to write the ratio so that the lesser quantity comes first; the important thing is to keep the relationship consistent.

 

 

Ratios can compare a part to a part or a part to a whole. Consider the example below that describes guests at a party.

 

 

 

A poll at Forrester University found that 4,000 out of 6,000 students are unmarried. Find the ratio of unmarried to married students. Express as a simplified ratio.   A) 3 to 2   B) 1 to 3   C) 2 to 1   D) 2 to 3   Show/Hide Answer A) 3 to 2 Incorrect. The ratio 3 to 2 compares the total number of students to the number of unmarried students. The correct answer is 2 to 1.   B) 1 to 3 Incorrect. The ratio 1 to 3 compares the number of married students to the total number of students. The correct answer is 2 to 1.   C) 2 to 1 Correct. If 4,000 students out of 6,000 are unmarried, then 2,000 must be married. The ratio of unmarried to married students can be represented as 4,000 to 2,000, or simply 2 to 1.   D) 2 to 3 Incorrect. The ratio 2 to 3 compares the number of unmarried students to the total number of students. The correct answer is 2 to 1.

 

 

A rate is a ratio that compares two different quantities that have different units of measure. A rate is a comparison that provides information such as dollars per hour, feet per second, miles per hour, and dollars per quart, for example. The word “per” usually indicates you are dealing with a rate.  Rates can be written using words, using a colon, or as a fraction. It is important that you know which quantities are being compared.

 

For example, an employer wants to rent 6 buses to transport a group of 300 people on a company outing. The rate to describe the relationship can be written using words, using a colon, or as a fraction; and you must include the units.

 

six buses per 300 people

 

 6 buses : 300 people

 

 

As with ratios, this rate can be expressed in simplest form by simplifying the fraction.

 

 

This fraction means that the rate of buses to people is 6 to 300 or, simplified, 1 bus for every 50 people.

 

 

 

 

 

Anyla rides her bike 18 blocks in 20 minutes. Express her rate as a simplified fraction.   A) 18:20   B)   C)   D)   Show/Hide Answer A) 18:20 Incorrect. Anyla’s trip compares quantities with different units, so it can be described as a rate. Since rates compare two quantities measured in different units of measurement they must include their units. The correct answer is .   B) Correct. Anyla’s trip compares quantities with different units (blocks and minutes) so it is a rate and can be written . This fraction can be simplified by dividing both the numerator and the denominator by 2.   C) Incorrect. 18 blocks in 20 minutes is not equivalent to 10 blocks in 9 minutes. Check the units again in your answer. The correct answer is .   D) Incorrect. Anyla’s trip compares quantities with different units, so it can be described as a rate. This is a correct representation and includes the units, but the fraction can be simplified. The correct answer is.

 

 

A unit rate compares a quantity to one unit of measure. You often see the speed at which an object is traveling in terms of its unit rate.

 

For example, if you wanted to describe the speed of a boy riding his bike—and you had the measurement of the distance he traveled in miles in 2 hours—you would most likely express the speed by describing the distance traveled in one hour. This is a unit rate; it gives the distance traveled per one hour. The denominator of a unit rate will always be one.

 

Consider the example of a car that travels 300 miles in 5 hours. To find the unit rate, you find the number of miles traveled in one hour.

 

 

A common way to write this unit rate is 60 miles per hour.

 

 

 

A unit price is a unit rate that expresses the price of something. The unit price always describes the price of one unit, so that you can easily compare prices.

 

You may have noticed that grocery shelves are marked with the unit price (as well as the total price) of each product. This unit price makes it easy for shoppers to compare the prices of competing brands and different package sizes.

 

Consider the two containers of blueberries shown below. It might be difficult to decide which is the better buy just by looking at the prices; the container on the left is cheaper, but you also get fewer blueberries. A better indicator of value is the price per single ounce of blueberries for each container.

 

Unit Price 25¢ per oz.   Total Price $2.00 8 oz.	Unit Prices 20¢ per oz.   Total Price $2.40 12 oz.

 

Look at the unit prices—the container on the right is actually a better deal, since the price per ounce is lower than the unit price of the container on the left. You pay more money for the larger container of blueberries, but you also get more blueberries than you would with the smaller container. Put simply, the container on the right is a better value than the container on the left.

 

So, how do you find the unit price?

 

Imagine a shopper wanted to use unit prices to compare a 3-pack of tissue for $4.98 to a single box of tissue priced at $1.60. Which is the better deal?

Find the unit price of the 3-pack:

 

Since the price given is for 3 boxes, divide both the numerator and the denominator by 3 to get the price of 1 box, the unit price. The unit price is $1.66 per box.

 

The unit price of the 3-pack is $1.66 per box; compare this to the price of a single box at $1.60. Surprisingly, the 3-pack has a higher unit price! Purchasing the single box is the better value.

 

Like rates, unit prices are often described with the word “per.” Sometimes, a slanted line / is used to mean “per.” The price of the tissue might be written $1.60/box, which is read “$1.60 per box.”

 

 

The following example shows how to use unit price to compare two products and determine which has the lower price.

 

 

 

A shopper is comparing two packages of rice at the grocery store. A 10-pound package costs $9.89 and a 2-pound package costs $1.90. Which package has the lower unit price to the nearest cent? What is its unit price?   A) The 2-pound bag has a lower unit price of $.95/pound.   B) The 10-pound bag has a lower unit price of $0.99/pound.   C) The 10-pound bag has a lower unit price of $.95/pound.   D) The 2-pound bag has a lower price of $1.89/2 pounds.   Show/Hide Answer A) The 2-pound bag has a lower unit price of $.95/pound. Correct. The unit price per pound for the 2-pound bag is $1.90 ÷ 2 = $0.95. The unit price per pound for the 10-pound bag is $9.89 ÷ 10 = $0.989 which rounds to $0.99.   B) The 10-pound bag has a lower unit price of $0.99/pound. Incorrect. $9.89 ÷ 10 = $0.989 which rounds to $0.99. $1.90 ÷ 2 = $0.95. The 2-pound bag has a lower unit price. The correct answer is A.   C) The 10-pound bag has a lower unit price of $.95/pound. Incorrect. $9.89 ÷ 10 = $0.989 which rounds to $0.99. The correct answer is A.   D) The 2-pound bag has a lower price of $1.89/2 pounds. Incorrect. A unit price is the price for one unit; in this case you need to find the cost of one pound, not two pounds. The correct answer is A.

 

Ratios and rates are used to compare quantities and express relationships between quantities measured in the same units of measure and in different units of measure. They both can be written as a fraction, using a colon, or using the words “to” or “per”. Since rates compare two quantities measured in different units of measurement, such as dollars per hour or sick days per year, they must include their units. A unit rate or unit price is a rate that describes the rate or price for one unit of measure.