Ratios are provided to compare quantities or quantities or explain a relationship between two quantities or quantities. Because that example, a ratio could be offered to define the expense of a month’s rent as compared to the revenue earned in one month. Girlfriend may also use a ratio to compare the number of elephants come the total number of animals in a zoo, or the quantity of calories per serving in two different brands of ice cream.

You are watching: What is a ratio that compares two quantities with different units

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


Ratios


Ratios compare amounts using division. This means that friend can collection up a ratio in between two amounts as a department expression in between those very same two quantities.

Here is an example. If you have actually a platter containing 10 street cookies and also 20 cacao chip cookies, you have the right to compare the cookies making use of a ratio.

The proportion of sugar cookie to cacao chip cookie is:

*

The ratio of coco chip cookies to sugar cookies is:

*

You deserve to write the ratio using words, a fraction, and likewise using a colon as displayed below.


ratio that sugar cookie to

chocolate chip cookies

10 come 20

*

10 : 20


Some civilization think around this proportion as: “For every 10 sugar cookies I have, I have 20 cacao chip cookies.”

You can likewise simplify the ratio simply as you simplify a fraction.

*

So us can likewise say that:


ratio of sugar cookies to

chocolate chip cookies

1 to 2

*

1 : 2


How to write a Ratio

A ratio can be written in three different ways:

· with words “to”: 3 to 4.

· together a fraction:

*
.

· v a colon: 3 : 4.

A proportion is streamlined if that is identical to a portion that has actually been simplified.

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


Example

Problem

A basketball player takes 50 jump shots during a practice. She makes 28 that them. What is the ratio of shots made to shots taken? simplify the ratio.

*

Identify the relationship.

*

Express the two quantities in portion form.

*

Simplify the portion to express the proportion in easiest form.

14 : 25

14 come 25

Consider the two other methods to write a ratio. You’ll desire to express her answer in a details format if required.

Answer

The ratio of shots made come shots take away is

*
,

14 : 25, or 14 to 25.


Often, one quantity in the proportion is greater than the second quantity. You carry out not have to write the proportion so the the lesser quantity comes first; the crucial thing is to store the connection consistent.


Example

Problem

Paul is comparing the amount of calorie in a big order the French fries indigenous his two favorite rapid food restaurants. Quick Foodz advertises the an bespeak of fries has actually 450 calories, and Beef avoid states the its fries have actually 300 calories. Write a proportion that to represent the amount of calorie in the rapid Foodz fries contrasted to the calories in Beef stop fries.

*

Identify the relationship.

*

Write a proportion comparing calories.

*

Simplify the ratio.

450 and 300 have actually a usual factor that 150.

Answer

The ratio of calorie in quick Foodz fries to Beef protect against fries is

*
, 3 : 2, or 3 come 2.


Ratios have the right to compare a part to a part or a component to a whole. Think about the example listed below that defines guests at a party.


Example

Problem

Luisa invites a team of friends to a party. Consisting of Luisa, there are a full of 22 people, 10 of whom are women.

Which is greater: the proportion of ladies to guys at the party, or the ratio of women to the total number of people present?

*

Identify the an initial relationship.

*

Write a proportion comparing women to men. Since there are 22 people and also 10 are women, 12 have to be men.

*

Simplify the ratio. 10 and 12 have actually a common factor the 2; the proportion of ladies to guys at the party is .

*

Identify the following relationship.

*

Write a proportion comparing the number of women come the total variety of people at the party.

*

Simplify the ratio.10 and also 22 have actually a usual factor the 2.

*

*

Rewrite each portion with a common denominator, 66.

*

*

Since

*
.

Answer

The proportion of women to men at the party, , is higher than the proportion of females to the total number of people,

*
.


A poll in ~ Forrester University uncovered that 4,000 out of 6,000 students room unmarried. Find the ratio of unmarried come married students. Express as a simplified ratio.

A) 3 come 2

B) 1 to 3

C) 2 to 1

D) 2 to 3


Show/Hide Answer

A) 3 come 2

Incorrect. The proportion 3 come 2 compares the total variety of students come the number of unmarried students. The exactly answer is 2 come 1.

B) 1 to 3

Incorrect. The proportion 1 to 3 compare the number of married students to the total number of students. The exactly answer is 2 come 1.

C) 2 come 1

Correct. If 4,000 students out of 6,000 are unmarried, climate 2,000 should be married. The proportion of unmarried to married students deserve to be stood for as 4,000 to 2,000, or merely 2 to 1.

D) 2 to 3

Incorrect. The proportion 2 come 3 to compare the variety of unmarried students come the total variety of students. The correct answer is 2 come 1.

Rates


A rate is a proportion that compares two various quantities the have different units the measure. A rate is a compare that offers information such as dollars per hour, feet every second, miles every hour, and also dollars every quart, because that example. Words “per” usually indicates you are managing a rate. Rates have the right to be written making use of words, making use of a colon, or as a fraction. It is vital that you know which amounts are gift compared.

For example, an employer desires to rent 6 buses to transfer a group of 300 world on a firm outing. The rate to define the relationship deserve to be written utilizing words, utilizing a colon, or as a fraction; and you must include the units.

six buses per 300 people

 6 buses : 300 people

*

As v ratios, this rate deserve to be expressed in simplest kind by simplifying the fraction.

*

This portion means the the price of buses to people is 6 come 300 or, simplified, 1 bus for every 50 people.


Example

Problem

Write the price as a streamlined fraction: 8 phone lines because that 36 employees.

*

Write together a fraction.

*

Simplify the fraction by making use of the typical factor that 4.

Answer

The price of phone call lines for employees deserve to be expressed together

*
.


Example

Problem

Write the price as a simplified fraction: 6 flight attendants because that 200 passengers.

*

Write as a fraction.

*

Simplify the portion by utilizing the typical factor that 2.

Answer

The rate of flight attendants and passengers is

*
.


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 pilgrimage compares amounts with various units, therefore it can be explained as a rate. Because rates compare two quantities measured in different units the measurement lock must encompass their units. The correct answer is .

B)

Correct. Anyla’s expedition compares amounts with various units (blocks and minutes) so that is a rate and also can be written . This portion can be streamlined by separating both the numerator and the denominator by 2.

C)

Incorrect. 18 block in 20 minute is not tantamount to 10 block in 9 minutes. Inspect the devices again in your answer. The exactly answer is .

D)

Incorrect. Anyla’s expedition compares quantities with different units, therefore it can be explained as a rate. This is a exactly representation and also includes the units, however the fraction can it is in simplified. The correct answer is.

Finding Unit Rates


A unit rate compares a amount to one unit the measure. You regularly see the speed at which an item is traveling in regards to its unit rate.

For example, if you wanted to describe the rate of a young riding his bike—and you had actually the measure of the street he traveled in mile in 2 hours—you would certainly most likely express the rate by explicate the street traveled in one hour. This is a unit rate; it gives the street traveled every one hour. The denominator that a unit price will always be one.

Consider the example of a vehicle that travel 300 miles in 5 hours. To discover the unit rate, you uncover the number of miles traveled in one hour.

*

A common way to compose this unit rate is 60 miles every hour.


Example

Problem

A crowded subway train has actually 375 passengers distributed evenly amongst 5 cars. What is the unit price of passengers per subway car?

*

Identify the relationship.

*

Write the rate as a fraction.

*

Express the portion with 1 in the denominator to uncover the number of passengers in one subway car.

Answer

The unit rate of the subway vehicle is 75 riders every subway car.


Finding Unit Prices


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

You may have noticed the grocery shelves are significant with the unit price (as well as the complete price) of each product. This unit price renders it basic for shoppers to to compare the prices of completing brands and different package sizes.

Consider the two containers that blueberries presented below. It might be challenging to decide which is the better buy just by looking in ~ the prices; the container ~ above the left is cheaper, but you likewise get fewer blueberries. A far better indicator of worth is the price per single ounce of blueberries because that each container.

*

*

Unit Price

25¢ every 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 top top the appropriate is actually a far better deal, since the price per ounce is reduced than the unit price that the container top top the left. You pay much more money for the larger container the blueberries, however you also get much more blueberries 보다 you would v the smaller container. Placed simply, the container ~ above the appropriate is a better value 보다 the container top top the left.

So, how do you discover the unit price?

Imagine a shopper want to usage unit prices to compare a 3-pack of organization for $4.98 come a solitary box of organization priced at $1.60. Which is the far better deal?

Find the unit price that the 3-pack:

*

Since the price provided 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 that 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 greater unit price! purchasing the single box is the better value.

Like rates, unit prices are often described with words “per.” Sometimes, a slanted heat / is provided to median “per.” The price the the tissue could be composed $1.60/box, i beg your pardon is review “$1.60 per box.”


Example

Problem

3 pounds that sirloin tips cost $21. What is the unit price per pound?

*

Write a rate to represent the expense per number of pounds.

*

Express the portion with 1 in the denominator by splitting both the numerator and the denominator through 3.

Answer

The unit price the the sirloin tips is $7.00/pound.


The following example shows how to usage unit price to compare 2 products and also determine which has the lower price.


Example

Problem

Sami is trying come decide between two brand of crackers. Which brand has actually the reduced unit price?

Brand A: $1.12 because that 8 ounces

Brand B: $1.56 for 12 ounces

Brand A

*

Write a price to stand for the price per oz for Brand A.

*

Find the unit price the Brand A by separating both the numerator and also the denominator by 8.

Brand B

*

Write a rate to represent the expense per ounce for Brand B.

*

Find the unit price that Brand B by separating both the numerator and also the denominator by 12.

*

Compare unit prices.

Answer

The unit price of Brand A crackers is 14 cents/ounce and also the unit price that Brand B is 13 cents/ounce. Brand B has actually a reduced unit price and also represents the far better value.


A shopper is comparing 2 packages that rice in ~ the grocery store. A 10-pound package expenses $9.89 and also a 2-pound package costs $1.90. I beg your pardon package has actually the reduced unit price come the nearest cent? What is its unit price?

A) The 2-pound bag has actually a lower unit price the $.95/pound.

B) The 10-pound bag has actually a reduced unit price that $0.99/pound.

C) The 10-pound bag has actually a reduced unit price the $.95/pound.

D) The 2-pound bag has actually a reduced price that $1.89/2 pounds.


A) The 2-pound bag has actually a lower unit price that $.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 actually a reduced unit price the $0.99/pound.

Incorrect. $9.89 ÷ 10 = $0.989 which rounds to $0.99. $1.90 ÷ 2 = $0.95. The 2-pound bag has actually a reduced unit price. The exactly answer is A.

C) The 10-pound bag has a reduced unit price the $.95/pound.

Incorrect. $9.89 ÷ 10 = $0.989 which ring to $0.99. The exactly answer is A.

D) The 2-pound bag has actually a lower price that $1.89/2 pounds.

Incorrect. A unit price is the price for one unit; in this situation you need to find the price of one pound, not 2 pounds. The correct answer is A.

See more: What Is The Difference Between Thermal Energy And Heat? ? Thermodynamics


Ratios and rates are offered to to compare quantities and also express relationships between quantities measure up in the exact same units of measure and in different units that measure. They both can be written as a fraction, making use of a colon, or using the words “to” or “per”. Due to the fact that rates to compare two quantities measured in different units of measurement, such as dollars every hour or ailing days per year, they must include their units. A unit rate or unit price is a price that explains the price or price because that one unit the measure.