## A ratio compares values .

A ratio says how much of one thing there is compared to another thing.

Ratios can be shown in different ways:

A ratio can be scaled up:

## Try it Yourself

Using ratios.

The trick with ratios is to always multiply or divide the numbers by the same value .

## Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk.

So the ratio of flour to milk is 3 : 2

To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4:

3 ×4 : 2 ×4 = 12 : 8

In other words, 12 cups of flour and 8 cups of milk .

The ratio is still the same, so the pancakes should be just as yummy.

## "Part-to-Part" and "Part-to-Whole" Ratios

The examples so far have been "part-to-part" (comparing one part to another part).

But a ratio can also show a part compared to the whole lot .

## Example: There are 5 pups, 2 are boys, and 3 are girls

Part-to-Part:

The ratio of boys to girls is 2:3 or 2 / 3

The ratio of girls to boys is 3:2 or 3 / 2

Part-to-Whole:

The ratio of boys to all pups is 2:5 or 2 / 5

The ratio of girls to all pups is 3:5 or 3 / 5

## Try It Yourself

We can use ratios to scale drawings up or down (by multiplying or dividing).

## Example: To draw a horse at 1/10th normal size, multiply all sizes by 1/10th

This horse in real life is 1500 mm high and 2000 mm long, so the ratio of its height to length is

1500 : 2000

What is that ratio when we draw it at 1/10th normal size?

We can make any reduction/enlargement we want that way.

"I must have big feet, my foot is nearly as long as my Mom's!"

But then she thought to measure heights, and found she is 133cm tall, and her Mom is 152cm tall.

In a table this is:

The "foot-to-height" ratio in fraction style is:

We can simplify those fractions like this:

And we get this (please check that the calcs are correct):

"Oh!" she said, "the Ratios are the same".

"So my foot is only as big as it should be for my height, and is not really too big."

You can practice your ratio skills by Making Some Chocolate Crispies

## 5.4 Ratios and Proportions

Learning objectives.

After completing this section, you should be able to:

- Construct ratios to express comparison of two quantities.
- Use and apply proportional relationships to solve problems.
- Determine and apply a constant of proportionality.
- Use proportions to solve scaling problems.

Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15 , we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 million. These types of comparisons are ratios.

## Constructing Ratios to Express Comparison of Two Quantities

Note there are three different ways to write a ratio , which is a comparison of two numbers that can be written as: a a to b b OR a : b a : b OR the fraction a / b a / b . Which method you use often depends upon the situation. For the most part, we will want to write our ratios using the fraction notation. Note that, while all ratios are fractions, not all fractions are ratios. Ratios make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only.

## Example 5.28

Expressing the relationship between two currencies as a ratio.

The Euro (€) is the most common currency used in Europe. Twenty-two nations, including Italy, France, Germany, Spain, Portugal, and the Netherlands use it. On June 9, 2021, 1 U.S. dollar was worth 0.82 Euros. Write this comparison as a ratio.

Using the definition of ratio, let a = 1 a = 1 U.S. dollar and let b = 0.82 b = 0.82 Euros. Then the ratio can be written as either 1 to 0.82; or 1:0.82; or 1 0.82 . 1 0.82 .

## Your Turn 5.28

Example 5.29, expressing the relationship between two weights as a ratio.

The gravitational pull on various planetary bodies in our solar system varies. Because weight is the force of gravity acting upon a mass, the weights of objects is different on various planetary bodies than they are on Earth. For example, a person who weighs 200 pounds on Earth would weigh only 33 pounds on the moon! Write this comparison as a ratio.

Using the definition of ratio, let a = 200 a = 200 pounds on Earth and let b = 33 b = 33 pounds on the moon. Then the ratio can be written as either 200 to 33; or 200:33; or 200 33 . 200 33 .

## Your Turn 5.29

Using and applying proportional relationships to solve problems.

Using proportions to solve problems is a very useful method. It is usually used when you know three parts of the proportion, and one part is unknown. Proportions are often solved by setting up like ratios. If a b a b and c d c d are two ratios such that a b = c d , a b = c d , then the fractions are said to be proportional . Also, two fractions a b a b and c d c d are proportional ( a b = c d ) ( a b = c d ) if and only if a × d = b × c a × d = b × c .

## Example 5.30

Solving a proportion involving two currencies.

You are going to take a trip to France. You have $520 U.S. dollars that you wish to convert to Euros. You know that 1 U.S. dollar is worth 0.82 Euros. How much money in Euros can you get in exchange for $520?

Step 1: Set up the two ratios into a proportion; let x x be the variable that represents the unknown. Notice that U.S. dollar amounts are in both numerators and Euro amounts are in both denominators.

Step 2: Cross multiply, since the ratios a b a b and c d c d are proportional, then a × d = b × c a × d = b × c .

You should receive 426.4 426.4 Euros ( 426.4 € ) ( 426.4 € ) .

## Your Turn 5.30

Example 5.31, solving a proportion involving weights on different planets.

A person who weighs 170 pounds on Earth would weigh 64 pounds on Mars. How much would a typical racehorse (1,000 pounds) weigh on Mars? Round your answer to the nearest tenth.

Step 1: Set up the two ratios into a proportion. Notice the Earth weights are both in the numerator and the Mars weights are both in the denominator.

Step 2: Cross multiply, and then divide to solve.

So the 1,000-pound horse would weigh about 376.5 pounds on Mars.

## Your Turn 5.31

Example 5.32, solving a proportion involving baking.

A cookie recipe needs 2 1 4 2 1 4 cups of flour to make 60 cookies. Jackie is baking cookies for a large fundraiser; she is told she needs to bake 1,020 cookies! How many cups of flour will she need?

Step 1: Set up the two ratios into a proportion. Notice that the cups of flour are both in the numerator and the amounts of cookies are both in the denominator. To make the calculations more efficient, the cups of flour ( 2 1 4 ) ( 2 1 4 ) is converted to a decimal number (2.25).

Step 2: Cross multiply, and then simplify to solve.

Jackie will need 38.25, or 38 1 4 38 1 4 , cups of flour to bake 1,020 cookies.

## Your Turn 5.32

Part of the definition of proportion states that two fractions a b a b and c d c d are proportional if a × d = b × c a × d = b × c . This is the "cross multiplication" rule that students often use (and unfortunately, often use incorrectly). The only time cross multiplication can be used is if you have two ratios (and only two ratios) set up in a proportion. For example, you cannot use cross multiplication to solve for x x in an equation such as 2 5 = x 8 + 3 x 2 5 = x 8 + 3 x because you do not have just the two ratios. Of course, you could use the rules of algebra to change it to be just two ratios and then you could use cross multiplication, but in its present form, cross multiplication cannot be used.

## People in Mathematics

Eudoxus was born around 408 BCE in Cnidus (now known as Knidos) in modern-day Turkey. As a young man, he traveled to Italy to study under Archytas, one of the followers of Pythagoras. He also traveled to Athens to hear lectures by Plato and to Egypt to study astronomy. He eventually founded a school and had many students.

Eudoxus made many contributions to the field of mathematics. In mathematics, he is probably best known for his work with the idea of proportions. He created a definition of proportions that allowed for the comparison of any numbers, even irrational ones. His definition concerning the equality of ratios was similar to the idea of cross multiplying that is used today. From his work on proportions, he devised what could be described as a method of integration, roughly 2000 years before calculus (which includes integration) would be fully developed by Isaac Newton and Gottfried Leibniz. Through this technique, Eudoxus became the first person to rigorously prove various theorems involving the volumes of certain objects. He also developed a planetary theory, made a sundial still usable today, and wrote a seven volume book on geography called Tour of the Earth , in which he wrote about all the civilizations on the Earth, and their political systems, that were known at the time. While this book has been lost to history, over 100 references to it by different ancient writers attest to its usefulness and popularity.

## Determining and Applying a Constant of Proportionality

In the last example, we were given that 2 1 4 2 1 4 cups of flour could make 60 cookies; we then calculated that 38 1 4 38 1 4 cups of flour would make 1,020 cookies, and 720 cookies could be made from 27 cups of flour. Each of those three ratios is written as a fraction below (with the fractions converted to decimals). What happens if you divide the numerator by the denominator in each?

The quotients in each are exactly the same! This number, determined from the ratio of cups of flour to cookies, is called the constant of proportionality . If the values a a and b b are related by the equality a b = k , a b = k , then k k is the constant of proportionality between a a and b b . Note since a b = k , a b = k , then b = a k . b = a k . and b = a k . b = a k .

One piece of information that we can derive from the constant of proportionality is a unit rate. In our example (cups of flour divided by cookies), the constant of proportionality is telling us that it takes 0.0375 cups of flour to make one cookie. What if we had performed the calculation the other way (cookies divided by cups of flour)?

In this case, the constant of proportionality ( 26.66666 … = 26 2 3 ) ( 26.66666 … = 26 2 3 ) is telling us that 26 2 3 26 2 3 cookies can be made with one cup of flour. Notice in both cases, the "one" unit is associated with the denominator. The constant of proportionality is also useful in calculations if you only know one part of the ratio and wish to find the other.

## Example 5.33

Finding a constant of proportionality.

Isabelle has a part-time job. She kept track of her pay and the number of hours she worked on four different days, and recorded it in the table below. What is the constant of proportionality, or pay divided by hours? What does the constant of proportionality tell you in this situation?

To find the constant of proportionality, divide the pay by hours using the information from any of the four columns. For example, 87.5 7 = 12.5 87.5 7 = 12.5 . The constant of proportionality is 12.5, or $12.50. This tells you Isabelle's hourly pay: For every hour she works, she gets paid $12.50.

## Your Turn 5.33

Example 5.34, applying a constant of proportionality: running mph.

Zac runs at a constant speed: 4 miles per hour (mph). One day, Zac left his house at exactly noon (12:00 PM) to begin running; when he returned, his clock said 4:30 PM. How many miles did he run?

The constant of proportionality in this problem is 4 miles per hour (or 4 miles in 1 hour). Since a b = k , a b = k , where k k is the constant of proportionality, we have

a miles b hours = k a miles b hours = k

a 4 .5 = 4 a 4 .5 = 4 (30 minutes is ½ ½ , or 0.5 0.5 , hours)

a = 4 ( 4.5 ) a = 4 ( 4.5 ) , since from the definition we know a = k b a = k b

a = 18 a = 18

Zac ran 18 miles.

## Your Turn 5.34

Example 5.35, applying a constant of proportionality: filling buckets.

Joe had a job where every time he filled a bucket with dirt, he was paid $2.50. One day Joe was paid $337.50. How many buckets did he fill that day?

The constant of proportionality in this situation is $2.50 per bucket (or $2.50 for 1 bucket). Since a b = k , a b = k , where k k is the constant of proportionality, we have

a dollars b buckets = k 337.50 b = 2.50 a dollars b buckets = k 337.50 b = 2.50

Since we are solving for b b , and we know from the definition that b = a k : b = a k :

b = 337.50 2.50 b = 135 b = 337.50 2.50 b = 135

Joe filled 135 buckets.

## Your Turn 5.35

Example 5.36, applying a constant of proportionality: miles vs. kilometers.

While driving in Canada, Mabel quickly noticed the distances on the road signs were in kilometers, not miles. She knew the constant of proportionality for converting kilometers to miles was about 0.62—that is, there are about 0.62 miles in 1 kilometer. If the last road sign she saw stated that Montreal is 104 kilometers away, about how many more miles does Mabel have to drive? Round your answer to the nearest tenth.

The constant of proportionality in this situation is 0.62 miles per 1 kilometer. Since a b = k , a b = k , where k k is the constant of proportionality, we have

a miles b kilometers = k a 104 = 0.62 a = 0.62 ( 104 ) a = 64.48 a miles b kilometers = k a 104 = 0.62 a = 0.62 ( 104 ) a = 64.48

Rounding the answer to the nearest tenth, Mabel has to drive 64.5 miles.

## Your Turn 5.36

Using proportions to solve scaling problems.

Ratio and proportions are used to solve problems involving scale . One common place you see a scale is on a map (as represented in Figure 5.16 ). In this image, 1 inch is equal to 200 miles. This is the scale. This means that 1 inch on the map corresponds to 200 miles on the surface of Earth. Another place where scales are used is with models: model cars, trucks, airplanes, trains, and so on. A common ratio given for model cars is 1:24—that means that 1 inch in length on the model car is equal to 24 inches (2 feet) on an actual automobile. Although these are two common places that scale is used, it is used in a variety of other ways as well.

## Example 5.37

Solving a scaling problem involving maps.

Figure 5.17 is an outline map of the state of Colorado and its counties. If the distance of the southern border is 380 miles, determine the scale (i.e., 1 inch = how many miles). Then use that scale to determine the approximate lengths of the other borders of the state of Colorado.

When the southern border is measured with a ruler, the length is 4 inches. Since the length of the border in real life is 380 miles, our scale is 1 inch = 95 = 95 miles.

The eastern and western borders both measure 3 inches, so their lengths are about 285 miles. The northern border measures the same as the southern border, so it has a length of 380 miles.

## Your Turn 5.37

Example 5.38, solving a scaling problem involving model cars.

Die-cast NASCAR model cars are said to be built on a scale of 1:24 when compared to the actual car. If a model car is 9 inches long, how long is a real NASCAR automobile? Write your answer in feet.

The scale tells us that 1 inch of the model car is equal to 24 inches (2 feet) on the real automobile. So set up the two ratios into a proportion. Notice that the model lengths are both in the numerator and the NASCAR automobile lengths are both in the denominator.

This amount (216) is in inches. To convert to feet, divide by 12, because there are 12 inches in a foot (this conversion from inches to feet is really another proportion!). The final answer is:

The NASCAR automobile is 18 feet long.

## Your Turn 5.38

Check your understanding, section 5.4 exercises.

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Home / United States / Math Classes / 7th Grade Math / Writing and Interpreting Ratios

## Writing and Interpreting Ratios

We use ratios to compare two quantities having the same unit. Learn how to write ratios, interpret comparisons from real -life problems, the concept of equivalent ratios, and how ratios can be expressed using a ratio table. ...Read More Read Less

## Table Of Contents

- What are Ratios?
- How do we express Ratios?

## Application of Ratios

Equivalent ratios and ratio table, solved examples.

- Frequently Asked Questions

## What are Ratios ?

A ratio is a comparison of quantities having the same unit. A ratio helps us indicate how big or how small a quantity is when compared to another quantity. Two known quantities can be compared by dividing them. We can use division to compare the quantities \(a\) and \(b\): \(\frac{a}{b}\) . Here, \(a\) is the dividend and \(b\) is the divisor. Ratio is an efficient method of representing this division operation in a different manner.

## How do we express Ratios ?

A ratio can be expressed in three ways:

- A ratio is denoted using the ‘:’ symbol. A ratio is expressed by writing the ‘:’ symbol in the middle of the two quantities that are being compared. In the ratio \(a : b\) , \(a\) and \(b\) are together known as the terms of the ratio. Also, \(a\) is known as the antecedent or the first term, and \(b\) is known as the consequent or the second term.
- But ratios needn’t always be represented using the ‘:’ symbol. As ratios are basically fractions, a fraction like \(\frac{a}{b}\) can also be interpreted as a ratio.
- A ratio can also be expressed in words as \(a\) to \(b\) .

∴ \(a : b\) = \(\frac{a}{b}\) = \(a\) to \(b\)

As discussed earlier, ratios can be used to compare quantities in our day-to-day lives. You can easily compare the number of boys and the number of girls in your classroom by taking the ratio.

Ratios can help make decision-making easier while performing activities like grocery shopping and cooking easier.

Suppose you need 3.5 ounces of ice cream and 10 ounces of milk while making a milkshake for one person. Ratios make it easy to scale up this recipe if we need to make this milkshake for multiple people. Let’s see how ratios can be scaled up by performing operations on them.

Two ratios that describe the same relationship are known as equivalent ratios. Equivalent ratios can be represented using a ratio table by adding or subtracting quantities in equivalent ratios, or by multiplying and dividing with the same values.

In the previous example of the recipe of milkshakes, a ratio table can be used to scale up the number of servings.

Ingredients for 1 milkshake: 3.5 ounces of ice cream for 10 ounces of milk.

With this information in hand, it is possible to scale up the recipe to the required amount.

Missing values in a ratio table can be calculated by performing addition, subtraction, multiplication, or division.

Example 1: If there are 25 students in a class and 15 of them are girls, find the simplest ratio of the number of boys to the number of girls.

Total number of students = 25

Number of girls = 15

Number of boys = 25 – 15 = 10

Ratio of number of boys to the number of girls: 15 : 10

Divide both sides by 5.

\(\frac{15 \div 5}{10 \div 5} = \frac{3}{2}\)

The simplest form of the ratio 15 : 10 is 3 : 2.

Example 2: You have to mix \(\frac{1}{3}\) cup of yellow paint for every \(\frac{1}{2}\) cup of red paint to make 10 cups of orange paint. Find the required number of cups of yellow paint.

We can use a ratio table to find the number of cups of yellow paint required to get 10 cups of orange paint,

By looking at the ratio table, we can conclude that we need 4 cups of yellow paint to get a cup of orange paint.

Example 3: Find the missing values from the ratio table.

Solution: In this case, we can perform math operations to find the missing values of the ratio table.

Note that during multiplication or division, the same operation was done on both sides. But during addition or subtraction, the values used for the operation are different on both sides: they are equivalent ratios.

## Should the quantities in a ratio have the same unit?

The quantities or terms in a ratio should be of the same unit. If we want to compare two quantities of different units, we use rate instead of ratios.

## Are ratios and fractions the same?

Ratios and fractions are mathematically the same: 1 : 2 is the same as \(\frac{1}{2}\). But in most cases, they carry a different meaning. For example, \(\frac{1}{2}\) of a sandwich means half of a sandwich. But a ratio is a comparison of two different quantities, i.e. 1 : 2 represents the ratio of non-vegetarian sandwiches to vegetarian sandwiches, which means there are two vegetarian sandwiches for every non-vegetarian sandwich.

## Can we multiply or divide any values in the ratio table?

The values in the ratio table can be multiplied or divided by any values. We just need to make sure that we use the same value while multiplying or dividing both sides.

## Can we add or subtract any values in the ratio table?

While multiplication and division operations can be done with any values, this is not possible with addition and subtraction. In the case of addition and subtraction, the operation is done with equivalent ratios

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## Writing and Using Ratios

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A ratio is a comparison of quantities. For example, for most mammals, the ratio of legs to noses is \(4:1,\) but for humans, the ratio of legs to noses is \(2:1.\)

## Writing Ratios

Using ratios to problem solve.

Ratios can be written using the word "to," a colon, or a fraction.

For example, in a group of 3 girls and 5 boys, the ratio of girls to boys can be written as \(3 \text{ to } 5, \) \(3:5,\) or \(\dfrac{3}{5}.\)

A bowl contains 3 apples, 5 oranges, and 9 kiwi. What is the ratio of kiwi to apples? ANSWER There are 9 kiwi and 3 apples, so the ratio of kiwi to apples is \(9:3,\) or in simplified form, \(3:1.\)

A bowl contains 3 apples, 5 oranges, and 9 kiwi. What is the ratio of oranges to total pieces of fruit? ANSWER There are 5 oranges and \(3+5+9=17\) total pieces of fruit, so the ratio of oranges to total pieces of fruit is \(5:17.\)

Marshall has 18 bananas. Joey comes up to Marshall and says that he has 30 apples. What is the ratio of apples to bananas?

In a parking lot, the ratio of white cars to blue cars is 2:3, and the ratio of blue cars to silver cars is 2:5. What is the least number of cars in the parking lot?

Assume that there is at least 1 car in the parking lot.

Ratios are often easiest to use if written in fraction form. To find an unknown value in a ratio, we can use two equivalent ratios.

For example, if the ratio \(3:4\) is equivalent to the ratio \(18:x,\) then \(\frac{3}{4}=\frac{18}{x}\) and \(x=24\) because the values in the fraction on the right are six times greater than the corresponding values in the fraction on the left.

If \( 6: 15 = 10 : x, \) what is \(x\,?\) ANSWER Expressing the ratios as fractions, we get \( \frac{6}{15} = \frac{10}{x} .\) The fraction \(\frac{6}{15}\) simplifies to \(\frac{2}{5},\) so now we have \( \frac{2}{5} = \frac{10}{x} .\) The values in the ratio on the right are five times greater than the values in the ratio on the left, so \(x=25.\) We can also rewrite the ratios as fractions and cross multiply to solve. \[\begin{align} \frac{6}{15} &= \frac{10}{ x} \\ 6x &= (10)(15) \\ 6x &= 150 \\ x &= 25.\end{align}\]

If Calvin paid $5 for 7 pencils, how much would he pay for 56 pencils? ANSWER Let \(x\) be the price of \(56\) pencils. Since the price of a single pencil does not change, we have \[\begin{align} \frac{\$5}{7} &= \frac{x}{56} \\ 7x &= \$5 \times 56 \\ x &= \$40. \end{align}\] Hence, Calvin would pay \(\$40\) for \(56\) pencils.

The ratio of Alice's pay to Bob's pay is \( \frac{5}{4} \). The ratio of Bob's pay to Charlie's pay is \( 10:9 \). If Alice is paid $75, how much is Charlie paid? ANSWER Since the ratio of Alice's pay to Bob's pay is \( 5:4 \), Bob's pay must be \( b \), where \( \frac{5}{4}=\frac{75}{b} \). Cross-multiplying by the denominators, we get \( 5b = 4(75) \), so \( b = 60 \). Continuing in the same way, we compare Bob to Charlie: \[ \frac{10}{9}=\frac{60}{c} \implies 10c = 9(60) \implies c = 54. \] Thus, Charlie is paid $54. \( _\square \)

The ratio of boys to girls in Mr. John's math class is 2 : 3 . If there are 4030 students in the class, how many more girls than boys are in the class?

Ten years ago, the ratio of the ages of two brothers was 1:4. Fourteen years from now, the ratio will be 4:7. What is the ratio of their present ages?

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- How to Read and Write Ratios

Are you ready to embark on a mathematical journey? If so, let's explore together the world of ratios. This concept, while seemingly simple, forms the basis of many mathematical calculations and real-world applications.

- What is a Ratio?

Let's start at the very beginning. In the simplest terms, a ratio is a way of comparing quantities. It tells us how much of one thing there is compared to another thing. Ratios can be written in different ways, but they all serve the same purpose of comparison.

## How to Write Ratios?

There are three main ways to write ratios. You can write them using the word "to", using a colon (:), or as a fraction. For example, if there are four apples and three oranges, we can express the ratio of apples to oranges as "4 to 3", "4:3", or "4/3". All three methods represent the same ratio and can be used interchangeably.

## How to Read Ratios?

When reading ratios, the order is very important. The number before the "to" or the colon is always compared to the number after. So, in the ratio 4:3, we say "four is to three". This means there are four parts of the first quantity for every three parts of the second quantity.

## The Concept of Equivalent Ratios

Equivalent ratios are ratios that express the same relationship between numbers. They are essentially the same fraction but multiplied or divided by the same number. For example, the ratios 2:1, 4:2, 6:3 are all equivalent because they all represent the same relationship "two is to one".

## Famous Mathematicians and Ratios

Many famous mathematicians have made significant contributions to the understanding and development of ratios. For instance, Pythagoras, a Greek mathematician, discovered a profound relationship between ratios and musical harmony. His findings laid the groundwork for our understanding of music theory today.

Similarly, Euclid, another ancient Greek mathematician, wrote extensively about ratios in his work "Elements", one of the most influential works in the history of mathematics. He used ratios to develop the concept of proportion, which has wide-ranging applications in fields such as geometry, physics, and engineering.

## Important Considerations When Working with Ratios

When working with ratios, it's important to remember that they are a form of comparison, not an absolute measure. A ratio tells us about the relationship between quantities, not their absolute values. For example, a ratio of 2:1 tells us that there are twice as many of one quantity as there are of another, but it doesn't tell us the exact amounts.

Another important point is that ratios must always be simplified. Similar to fractions, ratios should be expressed in their simplest form for clarity and ease of understanding. For example, the ratio 8:4 should be simplified to 2:1.

## Practical Applications of Ratios

Understanding ratios can be incredibly useful in a variety of practical situations. From cooking recipes to financial analysis, from art and design to health and fitness, ratios help us quantify relationships and make informed decisions.

Hopefully, this tutorial has shed some light on how to read and write ratios. This fundamental mathematical concept, introduced by pioneering mathematicians like Pythagoras and Euclid, plays a crucial role in many areas of study and everyday life.

Reading and writing ratios correctly is a skill that can be easily mastered with practice. Remember, ratios are simply a way of comparing quantities and should be expressed in their simplest form.

As we move forward in our journey of understanding ratios, always remember their origin and importance. The concept of ratios has been at the heart of many great mathematical discoveries and continues to be a powerful tool for understanding the world around us.

## Introduction to Ratios Tutorials

If you found this ratio information useful then you will likely enjoy the other ratio lessons and tutorials in this section:

- Examples of Ratios in Everyday Life
- Identify the Ratios: Exercise

Let's use this illustration of shapes to learn more about ratios. How can we write the ratio of squares to circles, or 3 to 6? The most common way to write a ratio is as a fraction, 3/6. We could also write it using the word "to," as "3 to 6." Finally, we could write this ratio using a colon between the two numbers, 3:6. Be sure you understand that these are all ways to write the same number.

Which way you choose will depend on the problem or the situation.

- ratio of squares to circles is 3/6
- ratio of squares to circles is 3 to 6
- ratio of squares to circles is 3:6

There are still other ways to make the same comparison, by using equal ratios. To find an equal ratio, you can either multiply or divide each term in the ratio by the same number (but not zero). For example, if we divide both terms in the ratio 3:6 by the number three, then we get the equal ratio, 1:2. Do you see that these ratios both represent the same comparison? Some other equal ratios are listed below. To find out if two ratios are equal, you can divide the first number by the second for each ratio. If the quotients are equal, then the ratios are equal. Is the ratio 3:12 equal to the ratio 36:72? Divide both, and you discover that the quotients are not equal. Therefore, these two ratios are not equal.

Some other equal ratios: 3:6 = 12:24 = 6:12 = 15:30

Are 3:12 and 36:72 equal ratios?

Find 3÷12 = 0.25 and 36÷72 = 0.5

The quotients are not equal —> the ratios are not equal.

You can also use decimals and percents to compare two quantities. In our example of squares to circles, we could say that the number of squares is "five-tenths" of the number of circles, or 50%.

Here is a chart showing the number of goals made by five basketball players from the free-throw line, out of 100 shots taken. Each comparison of goals made to shots taken is expressed as a ratio, a decimal, and a percent. They are all equivalent, which means they are all different ways of saying the same thing. Which do you prefer to use?

## What Is a Ratio? Definition and Examples

How to use ratios in math

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- Math Tutorials
- Pre Algebra & Algebra
- Exponential Decay
- Worksheets By Grade
- B.B.A., Finance and Economics, University of Oklahoma

Ratios are a helpful tool for comparing things to each other in mathematics and real life, so it is important to know what they mean and how to use them. These descriptions and examples will not only help you to understand ratios and how they function but will also make calculating them manageable no matter what the application.

## What Is a Ratio?

In mathematics, a ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent .

Example: you have polled a group of 20 people and found that 13 of them prefer cake to ice cream and 7 of them prefer ice cream to cake. The ratio to represent this data set would be 13:7, with 13 being the antecedent and 7 the consequent.

A ratio might be formatted as a Part to Part or Part to Whole comparison. A Part to Part comparison looks at two individual quantities within a ratio of greater than two numbers, such as the number of dogs to the number of cats in a poll of pet type in an animal clinic. A Part to Whole comparison measures the number of one quantity against the total, such as the number of dogs to the total number of pets in the clinic. Ratios like these are much more common than you might think.

## Ratios in Daily Life

Ratios occur frequently in daily life and help to simplify many of our interactions by putting numbers into perspective. Ratios allow us to measure and express quantities by making them easier to understand.

Examples of ratios in life:

- The car was traveling 60 miles per hour, or 60 miles in 1 hour.
- You have a 1 in 28,000,000 chance of winning the lottery. Out of every possible scenario, only 1 out of 28,000,000 of them has you winning the lottery.
- There were enough cookies for every student to have two, or 2 cookies per 78 students.
- The children outnumbered the adults 3:1, or there were three times as many children as there were adults.

## How to Write a Ratio

There are several different ways to express a ratio. One of the most common is to write a ratio using a colon as a this-to-that comparison such as the children-to-adults example above. Because ratios are simple division problems, they can also be written as a fraction . Some people prefer to express ratios using only words, as in the cookies example.

In the context of mathematics, the colon and fraction format are preferred. When comparing more than two quantities, opt for the colon format. For example, if you are preparing a mixture that calls for 1 part oil, 1 part vinegar, and 10 parts water, you could express the ratio of oil to vinegar to water as 1:1:10. Consider the context of the comparison when deciding how best to write your ratio.

## Simplifying Ratios

No matter how a ratio is written, it is important that it be simplified down to the smallest whole numbers possible, just as with any fraction. This can be done by finding the greatest common factor between the numbers and dividing them accordingly. With a ratio comparing 12 to 16, for example, you see that both 12 and 16 can be divided by 4. This simplifies your ratio into 3 to 4, or the quotients you get when you divide 12 and 16 by 4. Your ratio can now be written as:

- 0.75 (a decimal is sometimes permissible, though less commonly used)

## Practice Calculating Ratios With Two Quantities

Practice identifying real-life opportunities for expressing ratios by finding quantities you want to compare. You can then try calculating these ratios and simplifying them into their smallest whole numbers. Below are a few examples of authentic ratios to practice calculating.

- What is the ratio of apples to the total amount of fruit? (answer: 6:8, simplified to 3:4)
- If the two pieces of fruit that are not apples are oranges, what is the ratio of apples to oranges? (answer: 6:2, simplified to 3:1)
- What is the ratio of cows to horses that she treated? (answer: 12:16, simplified to 3:4. For every 3 cows treated, 4 horses were treated)
- What is the ratio of cows to the total number of animals that she treated? (answer: 12 + 16 = 28, the total number of animals treated. The ratio for cows to total is 12:28, simplified to 3:7. For every 7 animals treated, 3 of them were cows)

## Practice Calculating Ratios With Greater Than Two Quantities

Use the following demographic information about a marching band to complete the following exercises using ratios comparing two or more quantities.

Instrument type

- 160 woodwinds
- 84 percussion
- 127 freshmen
- 63 sophomores

1. What is the ratio of boys to girls? (answer: 2:3)

2. What is the ratio of freshmen to the total number of band members? (answer: 127:300)

3. What is the ratio of percussion to woodwinds to brass? (answer: 84:160:56, simplified to 21:40:14)

4. What is the ratio of freshmen to seniors to sophomores? (answer: 127:55:63. Note: 127 is a prime number and cannot be reduced in this ratio)

5. If 25 students left the woodwind section to join the percussion section, what would be the ratio for the number of woodwind players to percussion? (answer: 160 woodwinds – 25 woodwinds = 135 woodwinds; 84 percussionists + 25 percussionists = 109 percussionists. The ratio of the number of players in woodwinds to percussion is 109:135)

- Sixth-Grade Lesson Plan: Ratios
- How to Calculate a Male to Female Ratio and Other Quantities
- Empirical Formula: Definition and Examples
- Mole Ratio: Definition and Examples
- Math Glossary: Mathematics Terms and Definitions
- Learn About Molecular and Empirical Formulas
- Definition and Examples of Colons
- Learn to Calculate Percent Change
- How Are Odds Related to Probability?
- Probabilities for Rolling Three Dice
- Example of an ANOVA Calculation
- Exponents and Bases
- Computations With Fractions
- What Is the Median?
- Histogram Classes
- Calculate Simplest Formula From Percent Composition

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Unit 1: Ratio, Rate, & Proportion

## Topic A: Writing Ratios

Introduction to ratios.

Ratio is a comparison of one number or quantity with another number or quantity. Ratio shows the relationship between the quantities.

Ratio is pronounced “rā’shō” or it can be pronounced “rā’shēō.” Check out this YouTube video to listen to someone pronouce the word: How to Pronounce Ratio .

You often use ratios, look at these examples:

- Scores in games are ratios. For example, the Penguins won 4 to 3 or the Canucks lost 1 to 5.
- Directions for mixing can be ratios. For example, use 1 egg to each cup of milk or mix 25 parts gas to 1 part oil for the motorcycle.
- Betting odds are given as ratios. For example, Black Jade is a 3 to 1 favourite or the heavyweight contender is only given a 2 to 5 chance to win.
- The scale at the bottom of maps is a ratio. For example, 1 centimetre represents 10 kilometres.
- Prices are often given as ratios. For example, 100 grams for $0.79 or 2 cans for $1.85.

For ratios to have meaning you must know what is being compared and the units that are being used. Read these examples of ratios and the units that are used. A general ratio may say “parts” for the units.

- It rained four days and was sunny for three days last week. The ratio of rainy days to sunny days was [latex]4:3[/latex]. ([latex]4:3[/latex] is properly read “4 is compared to 3” but is often read “4 to 3”).
- The class has 12 men and 15 women registered. The ratio of men to women in the class is [latex]12:15[/latex].
- At the barbeque, 36 hot dogs and 18 hamburgers were eaten. The ratio of hot dogs eaten to hamburgers eaten is [latex]36:18[/latex].
- The class spends 3 hours on English and 2 hours on math each day. The ratio of time spent on English compared to math is [latex]3:2[/latex].

Write the ratios asked for in these questions using the [latex]:[/latex] symbol (for example, [latex]4:1[/latex]). Write the units and what is being compared beside the ratio.

- Powdered milk is mixed using 1 part of milk powder to 3 parts of water. Write a ratio to compare the milk powder to the water. Answer: [latex]1:3[/latex] — 1 part of milk powder to 3 parts of water
- One kilogram of ground beef will make enough hamburger for 5 people. Write a ratio to express the amount of ground beef for hamburgers to the number of people.
- Seventy-five vehicles were checked by the police. Fifteen vehicles did not meet the safety standards, but 60 of them did. Write a ratio comparing the unsafe vehicles to the safe vehicles.
- The recipe says to roast a turkey according to its weight. For every kilogram, allow 40 minutes of cooking. Write a ratio comparing time to weight.
- The 4-litre pail of semi-transparent oil stain should cover 24 square metres of the house siding if the wood is smooth. Write the ratio comparing quantity of stain to the smooth wood surface area.
- The same 4 L of stain will only cover 16 square metres of the house siding if the wood is rough. Write that ratio.

Answers to Exercise 1

- [latex]1:5[/latex] 1 kg of beef to 5 people
- [latex]15:60[/latex] 15 unsafe vehicles to 60 safe vehicles
- [latex]40:1[/latex] 40 minutes to 1 kg of turkey
- [latex]4:24[/latex] 4 L of stain to 24 m 2 of smooth wood
- [latex]4:16[/latex] 4 L of stain to 16 m 2 of rough wood

The numbers that you have been using to write the ratios are called the terms of the ratio.

The order that you use to write the terms is very important. Read a ratio from left to right and the order must match what the numbers mean. For example, 3 scoops of coffee to 12 cups of water must be written [latex]3:12[/latex] as a ratio because you are comparing the quantity of coffee to the amount of water.

If you wish to talk about the amount of water compared to the coffee you have, you would say, “Use 12 cups of water for every 3 scoops of coffee” and the ratio would be written [latex]12:3[/latex].

Ratios can be written 3 different ways:

- Using the [latex]:[/latex] symbol — [latex]2:5[/latex]
- The first number in the ratio is the numerator; the second number is the denominator.
- Ratios written as a common fraction are read as a ratio, not as a fraction. Say “2 to 5,” not “two-fifths.”
- Using the word “to” — 2 to 5

Use the ratios you wrote in Exercise 1 to complete the chart.

Answers to Exercise 2

- [latex]1:5[/latex] or [latex]\frac{1}{5}[/latex] or 1 to 5
- [latex]15:60[/latex] or [latex]\frac{15}{60}[/latex] or 15 to 60
- [latex]40:1[/latex] or [latex]\frac{40}{1}[/latex] or 40 to 1
- [latex]4:24[/latex] or [latex]\frac{4}{24}[/latex] or 4 to 24
- [latex]4:16[/latex] or [latex]\frac{4}{16}[/latex] or 4 to 16

Answers to Exercise 3

- [latex]9:6[/latex] or [latex]\frac{9}{6}[/latex] or 9 to 6
- [latex]7:9[/latex] or [latex]\frac{7}{9}[/latex] or 7 to 9
- [latex]6:9[/latex] or [latex]\frac{6}{9}[/latex] or 6 to 9

## Equivalent Ratios

Like equivalent fractions, equivalent ratios are equal in value to each other.

[latex]10:100 = 1:10[/latex]

Ratios can be written as common fractions. It is convenient to work with ratios in the common fraction form.

You can then easily:

- Find equivalent ratios in higher terms
- Find equivalent ratios in lower terms
- Find a missing term

Express [latex]4:5[/latex] in higher terms.

[latex]4:5=\dfrac{4}{5}\longrightarrow \dfrac{4}{5} \times \left(\dfrac{2}{2}\right) \longrightarrow\left(\dfrac{4\times 2}{5\times 2}\right)\longrightarrow\dfrac{8}{10}[/latex]

[latex]4:5[/latex] is equivalent to [latex]8:10[/latex]

Express [latex]3:6[/latex] in lower terms.

[latex]3:6=\dfrac{3}{6}\longrightarrow \dfrac{3}{6} \div \left(\dfrac{3}{3}\right) \longrightarrow \left(\dfrac{3\div 3}{6\div 3}\right)\longrightarrow\dfrac{1}{2}[/latex]

[latex]3:6[/latex] is equivalent to [latex]1:2[/latex]

To find equivalent ratios in higher terms, multiply each term of the ratio by the same number. To find equivalent ratios in lower terms, divide each term of the ratio by the same number.

Write equivalent ratios in any higher term. You may want to write the ratio as a common fraction first. Ask your instructor to mark this exercise.

- [latex]5:6= \dfrac{5}{6} \times \left(\dfrac{3}{3}\right)=\left(\dfrac{5\times 3}{6\times 3}\right)=\dfrac{15}{18}=15:18[/latex]
- [latex]4:3[/latex]
- [latex]10:2[/latex]
- [latex]50:1[/latex]
- [latex]9:4[/latex]
- [latex]3:5[/latex]

Answers to Exercise 4 See your instructor.

Write these ratios in lowest terms—that is, simplify the ratios.

- [latex]4:12=\dfrac{4}{12}\div\left(\dfrac{4}{4}\right)=\left(\dfrac{4\div4}{12\div4}\right)=\dfrac{1}{3}=1:3[/latex]
- [latex]10:5[/latex]
- [latex]7:21[/latex]
- [latex]20:5[/latex]
- [latex]6:14[/latex]
- [latex]2:4[/latex]
- [latex]6:3[/latex]
- [latex]16:8[/latex]

Answers to Exercise 5 Ratios written as a common fraction or using the word “to” will also be correct in this exercise. The terms must be the same.

- [latex]1:3[/latex]
- [latex]2:1[/latex]
- [latex]4:1[/latex]
- [latex]3:7[/latex]
- [latex]1:2[/latex]

Using a colon, write a ratio in lowest terms for the information given.

- In the class of 25 students, only 5 are smokers. Write the ratio of smokers to non-smokers in the class. ( Note —you must first calculate the number of non-smokers.)
- The police issued 12 roadside suspensions to drivers out of the 144 who were checked in the road block last Friday. Write the ratio of suspended drivers to the number checked.
- Twenty-seven students registered for the course and 24 completed it. Write a ratio showing number of completions compared to number enrolled.
- During an hour (60 minutes) of television viewing last night there were 14 minutes of commercials, so there were only 46 minutes of the actual program! Write the ratio of commercial time to program time.
- A nickel to a dime [latex]5:10=1:2[/latex]
- A nickel to a quarter
- A nickel to a dollar
- A dime to a nickel
- A dime to a quarter
- A dime to a dollar
- A dollar to a dime

Answers to Exercise 6

- [latex]1:4[/latex]
- [latex]1:12[/latex]
- [latex]8:9[/latex]
- [latex]1:5[/latex]
- [latex]1:20[/latex]
- [latex]2:5[/latex]
- [latex]1:10[/latex]
- [latex]10:1[/latex]

## Topic A: Self-Test

Mark /12 Aim 10/12

- Terms of the ratio
- Equivalent ratios
- The campground had three vacant campsites and 47 occupied sites. Write the ratio of occupied sites to vacant sites. Ratio: Read:
- For every ten dogs in the city, only 2 have current dog licences. Write the ratio of licensed dogs to unlicensed dogs. (Find the number of unlicensed dogs first). Ratio: Read:
- [latex]9:12[/latex]
- [latex]6:4[/latex]
- [latex]500:1000[/latex]
- [latex]2:9[/latex]
- [latex]35:15[/latex]

## Answers to Topic A Self-Test

- A ratio is a comparison of one number or quantity with another number or quantity. Ratios show the relationship between the quantities or amounts.
- Terms of a ratio are the numbers used in the ratio, the parts of the ratio.
- Equivalent ratios are ratios of equal value to each other.
- [latex]47:3[/latex] Read: ““47 occupied sites to 3 vacant sites.”
- [latex]1:4[/latex] Read: “1 licensed dog to 4 unlicensed dogs.”
- [latex]3:4[/latex]
- [latex]3:2[/latex]
- [latex]7:3[/latex]

Adult Literacy Fundamental Mathematics: Book 6 - 2nd Edition Copyright © 2022 by Liz Girard and Wendy Tagami is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

## Share This Book

Simplifying Ratios and Rates

Learning Objective(s)

· Write ratios and rates as fractions in simplest form.

· Find unit rates.

· Find unit prices.

Introduction

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:

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 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.

Finding Unit Rates

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.

Finding Unit Prices

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.

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?

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.

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.

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## 4.1.1: Simplifying Ratios and Rates

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## Learning Objectives

- Write ratios and rates as fractions in simplest form.
- Find unit rates.
- Find unit prices.

## Introduction

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:

\(\ \frac{\text { sugar cookies }}{\text { chocolate chip cookies }}=\frac{10}{20}\)

The ratio of chocolate chip cookies to sugar cookies is:

\(\ \frac{\text { chocolate chip cookies }}{\text { sugar cookies }}=\frac{20}{10}\)

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

\(\ \begin{array}{c} \text { ratio of } {\color{red}\text{sugar cookies to}}\\ \color{blue}\text{chocolate chip cookies}\\ {\color{red} 10} \text{ to } \color{blue}20\\ \frac{\color{red}10}{\color{blue}20}\\ {\color{red}10}:\color{blue}20 \end{array}\)

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.

\(\ \frac{10}{20}=\frac{10 \div 10}{20 \div 10}=\frac{1}{2}\)

So we can also say that:

\(\ \begin{array}{c} \text { ratio of } {\color{red}\text{sugar cookies to}}\\ \color{blue}\text{chocolate chip cookies}\\ {\color{red} 1} \text{ to } \color{blue}2\\ \frac{\color{red}1}{\color{blue}2}\\ {\color{red}1}:\color{blue}2 \end{array}\)

## How to Write a Ratio

A ratio can be written in three different ways:

- with the word “to”: 3 to 4
- as a fraction: \(\ \frac{3}{4}\)
- 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.

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

The ratio of shots made to shots taken is \(\ \frac{14}{25}\), 14:15, or 14 to 15.

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.

Paul is comparing the amount of calories in a large order of French fries from his two favorite fast food restaurants. Fast Foodz advertises that an order of fries has 450 calories, and Beef Stop states that its fries have 300 calories. Write a ratio that represents the amount of calories in the Fast Foodz fries compared to the calories in Beef Stop fries.

The ratio of calories in Fast Foodz fries to Beef Stop fries is \frac{3}{2}, 3:2, or 3 to 2.

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

Luisa invites a group of friends to a party. Including Luisa, there are a total of 22 people, 10 of whom are women.

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

The ratio of women to men at the party, \(\ \frac{5}{6}\), is greater than the ratio of women to the total number of people, \(\ \frac{5}{11}\).

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.

- 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.
- 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.
- 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.
- 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.

\(\ \begin{array}{c} \text{six buses per }300 \text{ people}\\ 6 \text{ buses : }300 \text{ people}\\ \frac{6 \text{ buses}}{300 \text{ people}} \end{array}\)

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

\(\ \frac{6 \text { buses}{\div6 }}{300 \text { people}{\div6 }}=\frac{1 \text { bus }}{50 \text { people }}\)

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

Write the rate as a simplified fraction: 8 phone lines for 36 employees.

The rate of phone lines for employees can be expressed as \(\ \frac{2 \text { phone lines }}{9 \text { employees }}\).

Write the rate as a simplified fraction: 6 flight attendants for 200 passengers.

The rate of flight attendants to passengers is \(\ \frac{3 \text { flight attendants }}{100 \text { passengers }}\).

Anyla rides her bike 18 blocks in 20 minutes. Express her rate as a simplified fraction.

- \(\ 18: 20\)
- \(\ \begin{array}{cc} 9 & \text { blocks } \\ \hline 10 & \text { minutes } \end{array}\)
- \(\ \frac{9 \text { minutes }}{10 \text { blocks }}\)
- \(\ \frac{18 \text { blocks }}{20 \text { minutes }}\)

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 \(\ \begin{aligned} 9 & \text { blocks } \\ \hline 10 & \text { minutes } \end{aligned}\).

Correct. Anyla’s trip compares quantities with different units (blocks and minutes), so it is a rate and can be written \(\ \begin{array}{cc} 18 & \text { blocks } \\ \hline 20 & \text { minutes } \end{array}\). This fraction can be simplified by dividing both the numerator and the denominator by 2.

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 \(\ \begin{aligned} 9 & \text { blocks } \\ \hline 10 & \text { minutes } \end{aligned}\).

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 \(\ \begin{aligned} 9 & \text { blocks } \\ \hline 10 & \text { minutes } \end{aligned}\).

## Finding Unit Rates

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.

\(\ \frac{300 \text { miles}{\div5 }}{5 \text { hours}{\div5 }}=\frac{60 \text { miles }}{1 \text { hour }}\)

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

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

The unit rate of the subway car is 75 riders per subway car.

## Finding Unit Prices

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.

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: \(\ \frac{\$ 4.98}{3 \text { boxes }}\)

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."

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

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

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

Sami is trying to decide between two brands of crackers. Which brand has the lower unit price?

Brand A: $1.12 for 8 ounces

Brand B: $1.56 for 12 ounces

The unit price of Brand A crackers is 14 cents/ounce and the unit price of Brand B is 13 cents/ounce. Brand B has a lower unit price and represents the better value.

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?

- The 2-pound bag has a lower unit price of $.95/pound.
- The 10-pound bag has a lower unit price of $0.99/pound.
- The 10-pound bag has a lower unit price of $.95/pound.
- The 2-pound bag has a lower price of $1.89/2pound.
- Correct. The unit price per pound for the 2-pound bag is \(\ \$ 1.90 \div 2=\$ 0.95\). The unit price per pound for the 10-pound bag is \(\ \$ 9.89 \div 10=\$ 0.989\), which rounds to $0.99.
- Incorrect. \(\ \$ 9.89 \div 10=\$ 0.989\), which rounds to $0.99. \(\ \$ 1.90 \div 2=\$ 0.95\). The 2-pound bag has a lower unit price. The correct answer is A.
- Incorrect. \(\ \$ 9.89 \div 10=\$ 0.989\), which rounds to $0.99. The correct answer is A.
- 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.

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Example 1: writing a ratio about an everyday life situation. Ms. Holly is looking after 7 children. She has 5 toys in her nursery. Write the ratio of children to toys. Identify the different quantities being compared and their order. There are 7 children and 5 toys. The order of the ratio is children to toys. 2 Write the ratio using a colon ...

Ratios can be shown in different ways: Use the ":" to separate the values: 3 : 1 : Or we can use the word "to": 3 to 1 : Or write it like a fraction: 31: A ratio can be scaled up: Here the ratio is also 3 blue squares to 1 yellow square, even though there are more squares.

Constructing Ratios to Express Comparison of Two Quantities. Note there are three different ways to write a ratio, which is a comparison of two numbers that can be written as: a a to b b OR a: b a: b OR the fraction a / b a / b. Which method you use often depends upon the situation. For the most part, we will want to write our ratios using the ...

Ratio review. Google Classroom. Learn how to find the ratio between two things given a diagram. A ratio compares two different quantities. For example, those two quantities could be monkeys and bananas: Notice that there are 4 monkeys and 5 bananas. Here are a few different ways we can describe the ratio of monkeys to bananas:

Consider the ratio \(6 \colon 11\). Write \(6 \colon 11\) in two other ways. Write three equivalent ratios. Is the ratio \(6\colon 11\) in lowest terms? Explain why or why not in a complete sentence. In a given pond, there are 22 rock fish for every 121 minnows. Express the ratio of minnows to rock fish in lowest terms.

Ratios can be written 3 different ways: Using the :: symbol — 2: 5 2: 5. As a common fraction — 25 2 5. The first number in the ratio is the numerator; the second number is the denominator. Ratios written as a common fraction are read as a ratio, not as a fraction. Say "2 to 5," not "two-fifths.".

A ratio can be expressed in three ways: A ratio is denoted using the ':' symbol. A ratio is expressed by writing the ':' symbol in the middle of the two quantities that are being compared. In the ratio \ (a : b\), \ (a\) and \ (b\) are together known as the terms of the ratio. Also, \ (a\) is known as the antecedent or the first term ...

Constructing Ratios to Express Comparison of Two Quantities. Note there are three different ways to write a ratio, which is a comparison of two numbers that can be written as: /**/(\$ 1 =0.82\,{€})/**/, how many dollars should you receive? Round to the nearest cent if necessary.

Writing Ratios. Ratios can be written using the word "to," a colon, or a fraction. For example, in a group of 3 girls and 5 boys, the ratio of girls to boys can be written as 3 \text { to } 5, 3 to 5, 3:5, 3: 5, or \dfrac {3} {5}. 53. A bowl contains 3 apples, 5 oranges, and 9 kiwi. What is the ratio of kiwi to apples?

A ratio is a way to show a relationship two numbers.Ratios can be used to compare things of the same type. For example, we may use a ratio to compare the nu...

Let's start at the very beginning. In the simplest terms, a ratio is a way of comparing quantities. It tells us how much of one thing there is compared to another thing. Ratios can be written in different ways, but they all serve the same purpose of comparison. How to Write Ratios? There are three main ways to write ratios.

It compares the amount of one ingredient to the sum of all ingredients. part: whole = part: sum of all parts. To write a ratio: Determine whether the ratio is part to part or part to whole. Calculate the parts and the whole if needed. Plug values into the ratio. Simplify the ratio if needed.

Ratios can be written in several different ways: as a fraction, using the word "to", or with a colon. ... Finally, we could write this ratio using a colon between the two numbers, 3:6. Be sure you understand that these are all ways to write the same number. Which way you choose will depend on the problem or the situation. ratio of squares to ...

As you can see, the parts consist of the number of girls and boys which sum up to the whole or the total number of students. [latex]3 [/latex] girls + [latex]1 [/latex] boy = [latex]4 [/latex] students. With this setup, it is now easy to come up with various kinds of ratios. Examples: Find the required ratios in three different formats.

Intro to ratios. Google Classroom. About. Transcript. The video explains ratios, which show the relationship between two quantities. Using apples and oranges as an example, it demonstrates how to calculate and reduce ratios (6:9 to 2:3) and how to reverse the ratio (9:6 to 3:2). Created by Sal Khan. Questions.

There are several different ways to express a ratio. One of the most common is to write a ratio using a colon as a this-to-that comparison such as the children-to-adults example above. Because ratios are simple division problems, they can also be written as a fraction. Some people prefer to express ratios using only words, as in the cookies ...

Ratios can be written 3 different ways: Using the :: symbol — 2: 5 2: 5. As a common fraction — 2 5 2 5. The first number in the ratio is the numerator; the second number is the denominator. Ratios written as a common fraction are read as a ratio, not as a fraction. Say "2 to 5," not "two-fifths.".

Write a ratio comparing the number of circles to triangles using a symbol. Step 1: Determine the number of both items asked in the problem. The problem asks for circles and triangles. There are 2 ...

Which is greater: the ratio of women to men at the party, or the ratio of women to the total number of people present? Identify the first relationship. Write a ratio comparing women to men. Since there are 22 people and 10 are women, 12 must be men. Simplify the ratio. 10 and 12 have a common factor of 2; the ratio of women to men at the party is .

A ratio is an association between two or more quantities. There are many ways to describe a situation in terms of ratios. For example, look at this collection: Figure 9.1.3 9.1. 3. Here are some correct ways to describe the collection: The ratio of squares to circles is 6: 3 6: 3.

The ratio of cookies to cupcakes in the given image can be expressed in three different ways: 6 to 7; 6 : 7 $\frac{6}{7}$ Ratio is the quantitative relationship between two quantities or numbers. In the ratio a : b, the first quantity is called an antecedent and the second quantity is called consequent. ... Write the given ratios and their ...

Consider the two other ways to write a ratio. You'll want to express your answer in a particular format if required. The ratio of shots made to shots taken is \(\ \frac{14}{25}\), 14:15, or 14 to 15. ... A rate is a ratio that compares two different quantities that have different units of measure. A rate is a comparison that provides ...