1.3 Equivalent Rational Numbers

Learning Objectives

You will learn that every rational number can be written as the ratio of two integers.
You will recognize that pi is not rational but can be approximated.
You will understand that division of two integers will always result in a rational number.
You will recognize and generate the equivalent representations of positive and negative rational numbers including equivalent fractions.
You will locate positive and negative rational numbers on a number line and understand the concept of opposites by plotting pairs of points on a coordinate grid.
You can compare positive and negative rational numbers expressed in various forms using symbols <, >, =, ≤, and ≥.
You can solve problems in various contexts involving calculations with exponents as well as simple and compound interest

Introduction

Real Life Example

Comparing Distances

On the dive boat one morning, Antonio began talking with another boy named Pascal. Antonio and his family were from Colorado and Pascal was just two years older than Antonio. The boys struck up a great conversation about diving and fish and the things that they had seen on their dives.

After a little while, they spotted some dolphins swimming with the boat. This is something that often happens as dolphins love the rushing water generated by the motor on the back of the boat.

“Did you know that they can swim 0.83 miles in one minute?” Pascal asked Antonio.

“Really, no I didn’t know that. I do know that a swordfish can swim almost one-half mile in a minute. I think the exact number is $\frac{9}{20}$ of a mile.”

“Wow, which one can swim the farthest in one minute?” Pascal asked thinking carefully through the math.

By the time they reached the dive site, Antonio had figured out which one can swim the farthest in one minute.

Have you? The numbers that the boys are using are called rational numbers. When you understand rational numbers, you will also understand how to figure out which one can swim the farthest in one minute. Pay attention, and this lesson on rational numbers will teach you all that you need to know.

Guided Learning

Example A

Now that you know how to identify a rational number, you may need to compare or order them from time to time. For example, what if you have a loss of $\frac{1}{2}$ compared to a loss of 0.34. You would need to determine which loss is greater. Placing the numbers on a number line can help you do this. Let's review the inequality symbols which can help us compare and order rational numbers:

> means is greater than.

< means is less than.

= means is equal to.

Example B

Choose the inequality symbol that goes in the blank to make this statement true.

$-2.5 \ \underline{\;\;\;\;\;\;\;} \ -5$

First, Draw a number line from -5 to 5.

Place the numbers -2.5 and -5 on that number line. Since $0.5=\frac{1}{2}, \ -2.5$ will be halfway between -2 and -3 on the number line.

Since -2.5 is further to the right on the number line than -5 is, -2.5 is greater than -5

The symbol > goes in the blank because -2.5 > -5 .

Example C

Order these rational numbers from least to greatest.

$&\frac{4}{5} \qquad 0.6 \qquad 1 \qquad 0.\bar{6}$

It is often fairly easy to place decimals on a number line that is divided into tenths. So, we can draw a number line from 0 to 1 and divide it into tenths. Then we can place all four numbers on the number line.

First, we should change $\frac{4}{5}$ to a fraction with a denominator of 10:

$\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$

Since eight tenths is equivalent to $\frac{4}{5}$ , we can find eight tenths on the number line and place $\frac{4}{5}$ there.

0.6 means six tenths. So, we can find six tenths on the number line and place 0.6 there.

1 is shown on the number line, so we can add a dashed line to show that number also.

$0.\bar{6}$ means 0.666... So, $0.\bar{6}$ is a little greater than six tenths, but less than seven tenths. We can place $0.\bar{6}$ roughly where it belongs on the number line.

The number line will look like this when we are finished.

From the number line, we can see that $0.6< 0.\bar{6} < \frac{4}{5} < 1$ .

So, ordered from least to greatest, the numbers are $0.6, \ 0.\bar{6}, \ \frac{4}{5}, \ 1$ .

Compare the following rational numbers and compare your answers with a partner.

$-0.7 \ \underline{\;\;\;\;\;\;\;} \ -\frac{7}{10}$
$0.34 \ \underline{\;\;\;\;\;\;\;} \ \frac{1}{2}$
$67 \ \underline{\;\;\;\;\;\;\;} \ -10$

Check your answers with a peer.

Example D

Identify Equivalent Proper Fractions, Mixed Numbers, and Improper Fractions

This lesson is all about fractions. To understand fractions, you will need to think about whole numbers too. Without whole numbers, it is impossible to understand fractions because a fraction is a part of a whole.

Whole numbers are numbers like 1, 8, 56, and 278—numbers that don’t contain fractional parts. Not all numbers are whole.

A fraction describes a part of a whole number. You are certainly familiar with fractions in your everyday dealings with cooking. Consider a recipe that calls for $\frac{1}{2}$ cup of chocolate chips. You know that $\frac{1}{2}$ cup represents one-half of a whole cup.

A fraction has certain parts. What are those parts?

The number written below the bar in a fraction is the denominator, which tells how many parts the whole is divided into. The numerator is the number above the bar in a fraction, which tells how many parts of the whole you have. In the recipe that calls for $\frac{1}{2}$ cup, the denominator is 2, so we know that one whole cup is divided into 2 parts. The numerator is 1, so we know that we need 1 of the 2 parts of the whole cup.

A whole can be divided into an infinite number of parts. You can divide 1 cup of flour into thirds, sixths, tenths, and so on. Fractions which describe the same part of a whole are called equivalent fractions. Remember that the word equivalent means equal. For instance, if you measure out $\frac{2}{4}$ cup of flour, $\frac{3}{6}$ cup of flour, or $\frac{1}{2}$ cup of flour, you will have the same amount of flour.

Therefore $\frac{2}{4}, \frac{3}{6}$ , and $\frac{1}{2}$ are all equivalent fractions.

When we have a fraction, we can create a new fraction that is equivalent to that fraction. We call this making equal

fractions or making equivalent fractions.

Example E

Making Equivalent Fractions

The first way is to work on simplifying a fraction to make it smaller. To simplify a fraction, we can reduce the number in the numerator and denominator by dividing them by the same number. For example, $\frac{4}{8}$ can be rewritten as $\frac{1}{2}$ by dividing both the numerator and the denominator by 4. Note that not all fractions can be rewritten by dividing. If the only number that both the numerator and denominator are divisible by is 1, then the fraction is said to be in its simplest form.

Example F

Simplify $\frac{6}{18}$

To simplify this fraction, we look for a number that we can divide into both the numerator and the denominator. In this case, the number is 6. We call 6 the Greatest Common Factor (GCF) of the numerator and the denominator. To simplify, we divide the numerator and the denominator by 6.

$\frac{6 \div 6}{18 \div 6}= \frac{1}{3}$

The simplified answer is $\frac{1}{3}$ .

The second way to create an equivalent fraction is by multiplying. We can create an equivalent fraction by multiplying the numerator and denominator by the same number. It doesn’t matter which number you choose, as long as the numbers are the same numbers.

Create an equivalent fraction for $\frac{7}{8}$ .

To do this, we need to multiply the numerator and the denominator by the same number. Let’s choose 2.

$\frac{7 \times 2}{8 \times 2}= \frac{14}{16}$

The answer is $\frac{14}{16}$ .

Try this one:

Write four equivalent fractions for $\frac{8}{12}$ .

First, let’s see if we can reduce the numbers in the numerator and denominator. Are there any numbers that can be divided into both 8 and 12? 8 and 12 are both divisible by 2 and 4. So, the fraction $\frac{8}{12}$ is not in its simplest form.

$\frac{8}{12} &= \frac{8 \div 2}{12 \div 2}=\frac{4}{6}\\\frac{8}{12} &= \frac{8 \div 4}{12 \div 4}=\frac{2}{3}$

When we divide both the numerator and the denominator by 2, we get $\frac{4}{6}$ as an equivalent fraction. When we divide the numerator and the denominator by 4, we get $\frac{2}{3}$ as an equivalent fraction.

To find more equivalent fractions, we can multiply the numerator and denominator of $\frac{8}{12}$ by any number. Let’s multiply by 3. We get $\frac{24}{36}$ as an equivalent fraction to $\frac{8}{12}$ . If we multiply the numerator and denominator by 5, we get $\frac{40}{60}$ as an equivalent fraction to $\frac{8}{12}$ .

$\frac{8}{12} &= \frac{8 \times 3}{12 \times 3}=\frac{24}{36}\\\frac{8}{12} &= \frac{8 \times 5}{12 \times 5}=\frac{40}{60}$

The answers, $\frac{2}{3}, \frac{4}{6}, \frac{24}{36}, \frac{40}{60}$ , are all equivalent fractions of $\frac{8}{12}$ .

Notice that creating equivalent fractions in this example involved both simplifying and multiplying!

There are other types of fractions too.

Sometimes when working with fractions, you use numbers which consist of a whole number and a fraction. This is called a mixed number. For example, if a recipe calls for more than 1 cup of flour but less than 2 cups of flour, you need to use a mixed number to describe exactly how much flour you need. A mixed number is written as a whole number with a fraction to the right of it. Some common mixed numbers include: $1 \frac{1}{2}$ or $2 \frac{2}{3}$ .

When the numerator of a fraction is greater than or equal to the denominator, you have an improper fraction. Improper fractions are greater than or equal to 1.

$\frac{2}{2}, \frac{3}{3}$ , and $\frac{10}{10}$ are all improper fractions that equal 1.

Why is this? Well, to understand this, you have to think about what the numerator and the denominator mean. The denominator is how many parts the whole is divided into. The numerator is how many of those parts you have. If you have two out of two parts, then you have the whole thing.

$\frac{5}{2}, \frac{8}{3}$ , and $\frac{11}{4}$ are all fractions that are greater than 1. These are called improper fractions.

Mixed numbers and Improper Fractions can be equivalent or equal to each other.

Improper fractions can be written as mixed numbers by dividing the numerator by the denominator and keeping the remainder as the numerator (while keeping the same denominator). Mixed numbers can be rewritten as improper fractions by multiplying the whole number in the mixed number by the denominator and adding the product to the numerator.

Try this one:

$\frac{9}{2}=4 \frac{1}{2}$

These two quantities are equal. This improper fraction is equal to the mixed number.

Take a look at an improper fraction:

Write $3 \frac{2}{3}$ as an improper fraction.

Remember, to write a mixed number as an improper fraction, we first multiply the whole number (3) by the denominator in the fraction, $3 \times 3 = 9$ . Next, we add this number to the numerator of the fraction, 9 + 2 = 11 . We put this new number over the original denominator and we have our improper fraction.

Our answer is that $3 \frac{2}{3}$ can be written as the improper fraction $\frac{11}{3}$ . It may help you to think about it this way: $\frac{3}{3} = 1$ , so if we have a whole number of 3's, we also have $\frac{3}{3} + \frac{3}{3} + \frac{3}{3} + \frac{2}{3}$ or $\frac{11}{3}$

Try this one:

Write $\frac{7}{3}$ as a mixed number.

To write an improper fraction as a mixed number, we divide the numerator by the denominator. $7 \div 3 = 2R1$ . To finish, we write the remainder above the original denominator and write the whole number part of the quotient to the left of this new fraction.

Our answer is that $\frac{7}{3}$ can be written as the mixed number $2 \frac{1}{3}$ .

Practice working with equivalent fractions.

Simplify $\frac{10}{12}$
Create an equivalent fraction for $\frac{5}{6}$
Write $\frac{15}{2}$ as a mixed number

Check your answers with a peer.

Example G

Compare and Order Fractions and Mixed Numbers with and without Approximation

Now that you are able to look at a fraction you can compare and order them. In order to be exact when comparing and ordering fractions, you have to find a common denominator for all of the fractions. Then, compare or order the fractions by looking at the value of the numerator. This will give you an exact comparison.

Let’s look at another way to compare:

Use approximation to order $\frac{7}{8}$ . $\frac{2}{5}, 3 \frac{5}{8}, \frac{1}{29}$ , and $\frac{29}{30}$ from greatest to least.

We begin by getting an approximate sense of the value of each of the fractions in the group by comparing each fraction with the common benchmarks 0, $\frac{1}{2}$ , and 1.

Because the number 7, which is the numerator in the fraction $\frac{7}{8}$ is very close in value to the denominator (8), we say that $\frac{7}{8}$ is approximately 1.

In the fraction, $\frac{2}{5}$ , the numerator is approximately $\frac{1}{2}$ of the denominator. So, we say that $\frac{2}{5}$ is about $\frac{1}{2}$ .

The number $3 \frac{5}{8}$ is the only mixed number in the group, so we can see immediately that this number is larger than all of the other numbers in the group because it is greater than 1.

In the fraction, $\frac{1}{29}$ , the denominator is much greater than the numerator, so $\frac{1}{29}$ is closest to the benchmark 0.

The numerator of 29 in the fraction $\frac{29}{30}$ is close in value to the denominator, 30, so $\frac{29}{30}$ is approximately 1.

Now that we have the approximate values of each fraction in the group, we write the fractions in a preliminary greatest to least order with the benchmarks in parentheses: $3 \frac{5}{8} \left(3 \frac{1}{2} \right), \frac{7}{8} (1), \frac{29}{30} (1), \frac{2}{5} \left(\frac{1}{2}\right), \frac{1}{29} (0)$ .

This approximation technique helps with most of the fractions in the group, but there are two fractions which are close to 1. We know that both $\frac{7}{8}$ and $\frac{29}{30}$ are less than 1, but which of the two fractions is closest to 1? One helpful way to determine which fraction is closest to 1 is to draw two number lines between 0 and 1, arranged so that one number line is above the other. Divide the top number line into eight equal parts (eighths) and the bottom number line into thirty equal parts (thirtieths).

From this illustration, it is easy to see that $\frac{29}{30}$ is closer to 1 than $\frac{7}{8}$ and is therefore greater than $\frac{7}{8}$ .

The answer is $3 \frac{5}{8}, \frac{29}{30}, \frac{7}{8}, \frac{2}{5}, \frac{1}{29}$ .

Example H

Compare $\frac{2}{3}$ and $\frac{5}{7}$ . Write >, <, or =.

At first glance, it is hard to compare the two fractions because they have different denominators. Remember the second mentioned method in comparing fractions is to find the common denominator. Look at the two denominators. Sometimes when comparing fractions we see fractions that have denominators that are multiples of the smaller denominator, and thus could be simplified so that both fractions have common denominators, or that could be multiplied to match a larger denominator.

For example, consider $\frac{3}{9}$ and $\frac{2}{3}$ . We see that in the fraction $\frac{3}{9}$ the 9 is a multiple of 3 and could be simplified by dividing the top and the bottom by 3. When we do this we get $\frac{1}{3}$ . Now we can compare the two fractions: $\frac{1}{3}$ and $\frac{2}{3}$ , so $\frac{1}{3} < \frac{2}{3}$ .

Also, we could have changed $\frac{2}{3}$ to $\frac{6}{9}$ and compare the fractions in this way also.

$\frac{3}{9}$ and $\frac{6}{9}$ - our comparison is the same, $\frac{3}{9} < \frac{6}{9}$

In this problem, 7 is not a multiple of 3. The lowest common denominator in this instance can only be the product of the two denominators $(3 \times 7 = 21)$ . In order to find an equivalent fraction for $\frac{2}{3}$ with a denominator of 21, we multiply both the numerator and denominator of $\frac{2}{3}$ by 7. We get an equivalent fraction of $\frac{14}{21}$ . In order to find an equivalent fraction for $\frac{5}{7}$ with a denominator of 21, we multiply both the numerator and denominator of $\frac{5}{7}$ by 3. We get an equivalent fraction of $\frac{15}{21}$ .

$\frac{2}{3} \times \frac{7}{7}=\frac{14}{21}\\\frac{5}{7} \times \frac{3}{3}=\frac{15}{21}$

Now that we have a common denominator between the two fractions, we can simply compare the numerators.

The answer is that $\frac{2}{3} < \frac{5}{7}$ .

Compare using <, >, or =

$\frac{1}{3}$ and $\frac{5}{6}$
$\frac{2}{9}$ and $\frac{7}{11}$
$\frac{8}{9}$ and $\frac{3}{4}$

Check your answers with a peer.

Example I

Describe Real-World Portion or Measurement Situations by Comparing and Ordering Fractions with and without Approximation

With common measurement units, such as feet, cups, inches, and ounces, we often use fractions to describe more precise measurements in relation to other measurements.

Take a look at this one:

In the long jump contest, Peter jumped $5 \frac{3}{8}$ feet, Sharon jumped $6 \frac{3}{5}$ feet, and Juan jumped $6 \frac{2}{7}$ feet. Now order their jump distances from greatest to least.

The problem asks us to order the jump distances from greatest to least. We have three mixed numbers, so we should look first at the whole number parts of the mixed numbers to see if we can compare the jump distances.

Peter jumped more than 5 feet, but less than 6 feet. Sharon jumped more than 6 feet, but less than 7 feet. Juan also jumped more than 6 feet, but less than 7 feet.

Simply by comparing the whole numbers, we can see that Peter jumped the shortest distance because he jumped less than 6 feet. Because Sharon and Juan both jumped between 6 and 7 feet, we need to compare the fractional part of their jumps. Sharon jumped $\frac{3}{5}$ of a foot more than 6 feet and Juan jumped $\frac{2}{7}$ of a foot more than 6 feet. In order to compare these two fractions, we have to find a common denominator. The lowest common denominator for these two fractions is 35. We get an equivalent fraction of $\frac{21}{35}$ for $\frac{3}{5}$ when we multiply both the numerator and denominator by 7. We get an equivalent fraction of $\frac{10}{35}$ for $\frac{2}{7}$ when we multiply both the numerator and denominator by 5. Now we can order the distances.

The answer is Sharon $6 \frac{3}{5} \ ft$ ., Juan $6 \frac{2}{7} \ ft$ ., Peter $5 \frac{3}{8} \ ft$ .

Example J

Write Fractions and Mixed Numbers as Terminating Decimals

Now that you have mastered fractions and their corresponding operations, it’s time to discover how they relate to decimals. You know that fractions and decimals are related because they are both ways of describing numbers that are part of a whole. In essence, a fraction is simply another way of describing what a decimal describes. They both represent parts of a whole and both can show the same part using different formats. The fraction shows us the part using the fraction bar comparing part to whole and the decimal shows us the part using place value. To start off, we’ll see how fractions can be converted into decimals and how decimals can be converted into fractions.

How do we convert fractions to decimals and decimals to fractions?

First, remember that fractions and decimals are different ways of writing the same thing. Both show us how to represent a part of a whole. Think about how we talk about fractions and decimals because this will become useful as we convert them.

Say this value out loud 0.1. You can say “point one” or “one tenth.” Does the second version sound a little bit familiar? It sounds like the fraction $\frac{1}{10}$ . It turns out that $0.1 = \frac{1}{10}$ .

Yes, and a decimal is just another way of writing a fraction.

How do we convert fractions into decimals?

We can easily convert fractions into decimals. You’ve probably noticed by now that a fraction is really a short way of writing a division expression. Writing $\frac{3}{4}$ is really like writing $3 \div 4$ . The way that we find out how to write $\frac{3}{4}$ as a decimal is to go ahead and solve the division problem. Since 4 doesn’t go into 3, we have to expand the number over the decimal point.

${4 \overline{) {3.0 \;}}}$

How many times does 4 go into 3.0? Four goes into 3.0 0.7 times.

$& \overset{ \ \ \ \ .7}{4 \overline{ ) {3.0 \;}}}\\& \underline{-2.8 \ }\\& \quad \ .2$

Be sure when you are writing your quotient above the dividend to keep the original place of the decimal point. Since 4 does not divide evenly into 3.0 and we have a remainder of 0.2, we can go further to the other side of the decimal point by adding a 0 next to the remainder of 0.2.

$& \overset{ \ \ \ \ .75}{4 \overline{ ) {3.00 \;}}}\\& \ \underline{ \ -2.8 \ }\\& \quad \ \ .20\\& \ \underline{ \ \ -.20 \ }\\& \quad \ \ \ \ \ 0$

4 goes evenly into 0.20 0.05 times, so we have our final answer.

$\frac{3}{4} = 0.75$

Take a look at this one:

Convert $\frac{1}{4}$ to a decimal.

We start this by changing it into a division problem. We will be dividing 1 by 4. You already know that 1 can’t be divided by 4, so you will need to use a decimal point and add zeros as needed.

$& \overset{ \ \ \ \ .25}{4 \overline{ ) {1.00 \;}}}\\& \underline{-8 \quad \ }\\& \quad \ 20\\& \ \ \underline{-20 \ }\\& \qquad 0$

Our answer is 0.25.

How do we convert mixed numbers to decimals?

When you are working with mixed numbers like $1 \frac{3}{4}$ for example, it is easiest to simply set the whole number to the side and solve the division problem with the fraction. When you have completed the division problem with the fraction, make sure that you put the whole number back on the left side of the decimal point.

$1 \frac{3}{4} = 1.75$

Convert $3 \frac{1}{2}$ to a decimal.

First, set aside the 3. We will come back to that later.

Next, we divide 1 by 2. Use a decimal point and zeros as needed.

$& \overset{ \ \ \ \ .5}{2 \overline{ ) {1.0 \;}}}\\ & \underline{ \ -1.0}\\ & \quad \ 0$

Now we add in the 3.

Our final answer is 3.5.

Convert each fraction or mixed number to a decimal and check your answers with a partner

$\frac{1}{5}$
$\frac{3}{6}$
$4 \frac{4}{5}$

Example K

Write Fractions and Mixed Numbers as Repeating Decimals

By now you’ve gotten the hang of converting fractions to decimals. So far, we have been working with what are known as terminating decimals, or decimals that have an end like 0.75 or 0.5.

One reason that we sometimes use fractions instead of decimals is because some decimals are repeating decimals, or decimals that go on forever. If you try to find a decimal for $\frac{1}{3}$ by dividing, you can divide forever because $\frac{1}{3}$ written as a decimal = 0.3333333333 .... It goes on and on. That’s why we usually just simply write a line above the number that repeats. For $\frac{1}{3}$ , we write: $0.\overline{3}$ . Let’s check out some examples involving repeating decimals.

Write $\frac{5}{6}$ using decimals

First, we rewrite $\frac{5}{6}$ as the division problem $5 \div 6$ . We already know that we will have to go on the right side of the decimal point, so we are going to begin by dividing 6 into 5.0.

Six goes into 5.0 0.8 times, but we have the remainder of 0.2. Six goes into 0.2 0.03 times and we have a remainder of 0.02. Since 6 always goes into 20 three times, $(3 \cdot 6 = 18)$ and there will always be a remainder of 2, we can see that it will never evenly divide.

If you keep dividing, you will get 0.83333333333.... forever and ever.

Our final answer is $0.8\overline{3}$ .

What about mixed numbers?

Well, there are some mixed numbers where the fraction part is a repeating decimal. Let’s look at an example.

Write $2 \frac{2}{3}$ using decimals.

Just as we did with the terminating decimals, we are going to set the whole number, 2, to the side, and then come back to it when we are ready to add it to the final answer. So, we are simply solving for the decimal equivalent of $\frac{2}{3}$ . We write the division problem $2.0 \div 3$ . How many times does 3 go into 2.0? It goes into 2.0 0.6 times.

We have 0.20 as the remainder. How many times does 3 go into 0.20? The answer is 0.06 times.

Are you noticing a pattern here? It is obvious that there will always be a remainder whether we divide 3 into 2.0, 0.2, 0.02, 0.002, or 0.0002 and on and on. Clearly $\frac{2}{3}$ is a repeating decimal.

For our final answer we write $2.\overline{6}$ .

Convert each to a repeating decimal.

$\frac{1}{6}$
$4 \frac{4}{6}$

Example L

Write Decimals as Fractions

Now that we have mastered writing fractions as decimals, it will be good to know how to write decimals as fractions.

Consider again the decimal 0.1. We already know that we can say that this number is “one-tenth.” It’s very easy to rewrite decimals as fractions because decimals are already expressed as fractions with a denominator that is a factor of ten.

$0.1 = \frac{1}{10}, 0.01 = \frac{1}{100}$

We can also say that $0.86 = \frac{86}{100}$ .

To convert decimals to fractions, we write the number to the right of the decimal place over a denominator equivalent to the last place value of the decimal number. So, if we have 0.877, we would write $\frac{877}{1000}$ .

If we have simply 0.6, we can write $\frac{6}{10}$ , or in simplest terms, $\frac{3}{5}$ . Always make sure to write your fraction in simplest terms.

Let’s look at an example.

Convert 0.35 to a fraction.

Start by saying the decimal to yourself out loud. To say 0.35 out loud, we can say “35 hundredths,” so we can go ahead and write the fraction down.

$\frac{35}{100}$

That’s a big fraction. We want to make our lives a little bit easier, so we will reduce the fraction to simplest terms. This fraction expressed in simplest terms is $\frac{7}{20}$ .

Our final answer is $\frac{7}{20}$ .

Try this one:

Convert 2.4 to a mixed number.

Just as we set aside the whole number when converting mixed numbers to decimals, we will set aside the numbers to the left of the decimal point when converting decimals to fractions. So, in this case, we just have to find out what 0.4 is when expressed as a fraction.

We write it directly as the fraction “four tenths” or $\frac{4}{10}$ . Can we simplify it? You bet. $\frac{4}{10}=\frac{2}{5}$ .

2.4 expressed as a mixed number $= 2 \frac{2}{5}$ .

Convert each decimal to a fraction in simplest form. (Don't forget to return set-aside whole numbers to your final answer.)

0.5
0.67
3.21

Example M

Compare and Order Decimals and Fractions

Eventually, you will be able to convert common fractions to decimals and common decimals to fractions in your head. You already know some of the more common ones like $0.5 = \frac{1}{2}$ . Knowing this off the top of your head will make it easy for you to compare and order between fractions and decimals. For now, we will practice our newly acquired skills at converting to compare and order. It’s always helpful to check.

Look at another comparison:

Compare $\frac{1}{4}$ and 0.25 using <, > or =

To compare a fraction to a decimal or a decimal to a fraction, we will need to convert one of them, so that we can compare a fraction to a fraction or a decimal to a decimal. For this one, we will convert $\frac{1}{4}$ to a decimal. we divide 1.0 by 4. 4 goes into 1.0 .2 times. There is also a reminder of 0.20 and 4 goes into 0.20 0.05 times. Now we know that $\frac{1}{4}$ written as a decimal is 0.25.

We compare it as $\frac{1}{4} = 0.25$ .

Take a look at another problem:

Compare $1 \frac{2}{20}$ and 1.30

Our work in estimating the value of fractions and rounding decimals can be helpful when comparing fractions and decimals because you can look at a fraction or a decimal and quickly have a sense of what the approximate value is. Take a look at the mixed number $1 \frac{2}{20}$ . Don’t be intimidated by the large denominator, it looks like we can simplify it. Simplify it to $1 \frac{1}{10}$ .

Now we can take 1.30 and make it a mixed number. 1.30 becomes $1 \frac{3}{10}$ .

Our final answer is that $1 \frac{2}{20} < 1.30$ .

When ordering fractions and decimals be sure to use the strategies we just practiced. Make sure that they are in the same form and then order them from least to greatest or from greatest to least.

Identify and Write Different Forms of Equivalent Ratios among Rational Numbers

In math and in real-life, we compare things all the time. We look at what we have and what someone else has or we look at the difference between values and we compare them. Comparing comes very naturally. Using ratios comes naturally too, because ratios are a way that we can compare numbers and values.

What is a ratio?

A ratio compares two numbers or quantities called terms. For example, suppose there are 3 green (G) apples and 4 red (R) apples in this basket.

We can express the ratio of green apples to red apples in the basket as a fraction.

We can also express this ratio in words, 3 to 4, or using a colon, 3:4.

The ratio above compares one part of the apples in the basket to another part. The ratio above compares the apples that are green to the apples that are red.

A ratio may also express a part to a whole. For example, we can express the ratio of green apples to the total number of apples in the basket.

There are a total of 7 apples in the basket, so the ratio of green apples to total apples is 3 to 7 or 3:7.

Here are the three ways that we can write a ratio:

In fraction form using a fraction bar
Using the word “to”
Using a colon:

Look at the drawing and write each the ratio three different ways.

Compare stars to suns

Now that you know how to write a ratio, let's look at equivalent ratios.

A ratio shows the relationship between two quantities. Equivalent ratios can be used to show the same relationship between two quantities. Remember that the word “equivalent” means equal.

Because we can write ratios in fraction form, we can use what we know about finding equivalent fractions to help us identify equivalent ratios. Here is where simplifying fractions is going to help us. We can simplify ratios to discover equivalence just as we can simplify fractions.

Take a look at another sample:

Determine if these two ratios are equivalent and .

We can start by simplifying the larger of the two fractions. If it simplifies to the same number as the smaller fraction, then we know that the two ratios are equivalent.

These are equal ratios.

Simplifying is one way to check for equivalence. We can also create equivalent ratios by multiplying just as we would make equivalent fractions.

Change to a ratio with 15 as the second term (the denominator).

Since , multiply both terms of the ratio by 5.

This shows that the ratio is equivalent to the ratio .

So, the two ratios listed above are equivalent.

Here is one more to view:

Determine if these two ratios are equivalent 7:6 and 13:12.

Rewrite the ratios as fractions and .

Change to a ratio with 12 as the second term.

Since , multiply both terms of the ratio by 2.

When the second term (the denominator) is 12, the equivalent ratio for the numerator is 14, not 13 .

So, 7:6 and 13:12 are not equivalent ratios.

Another way to determine if two ratios are equivalent is to cross multiply the terms in the ratio. If the cross products are equal, then the two ratios are equivalent. If the cross products are not equal, then the two ratios are not equivalent.

Compare these ratios:

Determine if these two ratios are equivalent and .

We use the symbol to show that the two ratios below may or may not be equal.

Now, we cross multiply. To cross multiply, multiply the circled pairs of numbers shown below. The product we get when we multiply each circled pair of numbers is called a cross product.

Since , the cross products are equal. This means that , and those two ratios are equivalent.

Determine whether or not each pair of ratios is equivalent. Write yes or no. and and and

Example N

Write Ratios in Simplest Form

In the last section, we touched briefly on how to simplify ratios that are in fraction form to see if they are equivalent or not. We will explore this further in this section.

What does it mean to simplify?

To simplify means to make smaller. When we simplify, we make a fraction smaller. It is still equivalent to the larger form of the fraction, but it is simpler.

How do we simplify a ratio in fraction form?

To write a ratio in simplest form, find the greatest common factor of both terms in the ratio. The greatest common factor of two numbers is the greatest number that divides both numbers evenly. Then, divide both terms of the ratio by the greatest common factor.

This is basically the same procedure you use to rewrite a fraction in simplest form. Let’s look at an example.

Take a look at this ratio:

Write this ratio in simplest form .

First, find the greatest common factor of the terms 20 and 24.

The factors of 20 are 1, 2, 4, 5, 10, and 20.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

The factors that both 20 and 24 have in common are 1, 2 and 4.

The greatest of those common factors is 4, so divide both terms by 4 to write the ratio in simplest form.

So, the simplest form of the ratio is .

How is this useful when working with ratios?

It is useful because we can simplify ratios in fraction form to see if they are equivalent. We can also use a simplified ratio to find an equivalent ratio. To do this, you can multiply both terms of the original ratio by the same number to find an equivalent ratio.

Is there way you can compare three equivalent ratios?

Write three equivalent ratios for this ratio 1:9.

The ratio 1:9 or is already in simplest form. Notice that we wrote the ratio into fraction form so that it is easier to work with.

Now, we will write three equivalent ratios by multiplying both terms by the same number.

It does not matter by which numbers we choose to multiply the terms. Let's multiply by 2, first.

Let's multiply by 5, next.

Let's multiply by 100, next.

So, three ratios that are equivalent to 1:9 are 2:18, 5:45, and 100:900.

Simplify this ratio 4:16
Simplify this ratio 3 to 18
Write an equivalent ratio to 4:5

Example O

Write and Compare Ratios in Decimal Form.

Just as fractions can be written in decimal form, ratios can be written in fraction form, so they can also be written as decimals. Now we will look at how to write a ratio as a decimal.

How do we write a ratio as a decimal?

To convert a ratio to decimal form, write the ratio as a fraction. Then divide the term above the fraction bar by the term below the fraction bar.

Let’s look at how to do this.

Example P

Rewrite the ratio 1 to 4 in decimal form.

The ratio 1 to 4 can be expressed as the fraction . This is our first step.

Next, divide the term above the fraction bar, 1, by the term below the fraction bar, 4.

Since 1 cannot be evenly divided by 4, rewrite 1 as a decimal with a zero after the decimal point You can do this because . Before you divide, write a decimal point in the quotient directly above the decimal point in the dividend. Then divide.

Continue adding zeroes after the decimal point and diving until the quotient has no remainder. The decimal form of the ratio is 0.25.

Take a look at how to rewrite the ratio in decimal form below:

Rewrite the ratio 9:5 in decimal form.

The ratio 9:5 can be expressed as the fraction .

Next, divide the term above the fraction bar, 9, by the term below the fraction bar, 5. There is a remainder. So, add zeroes after the decimal point in 9 to continue dividing. The decimal form of 9:5 is 1.8.

What about comparing? Can we use decimals to compare ratios?

Sometimes, you may want to compare two ratios and determine if they are equivalent or not. Rewriting both ratios in decimal form is one way to do this.

Here is another way to make a comparison:

Compare these two ratios and determine if they are equivalent and .

Rewrite in decimal form. Rewrite in decimal form. To compare the ratios in decimal form, give each decimal the same number of decimal places. In other words, give 0.5 two decimal places: 0.5 = 0.50.

Now compare. Since both decimals have a 0 in the ones place and a 5 in the tenths place, compare the digits in the hundredths place.

Since . So, the ratios, and , are not equivalent.

In fact, .

Write each ratio as a decimal.

5 to 10
4 to 10
Compare 6 to 10 and 1 to 4

Example Q

Solve Real-World Problems Involving Ratios Among Rational Numbers

Being able to rewrite ratios in different forms can help us solve real-world problems. Let’s look at using ratios to solve some real-world problems.

Elena and Jake have a box with only two colors of marbles in it. There are 28 blue marbles and 16 gray marbles in the box. Elena says that the ratio of gray marbles to blue marbles is . Jake says that the ratio of gray marbles to blue marbles is . Who is correct, or are they both correct? They also want to find the ratio of gray marbles to the total number of marbles in the box.

Consider the first question. It is a question about comparing and determining equivalence.

Write the ratio of gray marbles to blue marbles as a fraction. Be careful that the ratio you write compares gray marbles to blue marbles, not blue marbles to gray marbles.

Since there are 16 gray marbles and 28 blue marbles, the ratio is:

So, Elena is correct that the ratio of gray marbles to blue marbles is .

If that ratio is equivalent to , then Jake is correct too. One way to determine if those two ratios are equivalent is to cross multiply.

Since the cross products are equal, the two ratios are equivalent.

So, the answer to the first question is that Elena and Jake are both correct.

Now let’s think about the second part of the question. To figure this out, we need to figure out the ratio of gray marbles to total marbles in the box.

Write that ratio in simplest form.

The factors of 16 are: 1, 2, 4, 8, and 16.

The factors of 44 are: 1, 2, 4, 11, 22, and 44.

The greatest common factor of 16 and 44 is 4. So, divide both terms by 4.

The ratio of gray marbles to the total number of marbles, written in simplest form, is 4:11.

Real Life Example Completed

Comparing Distances

Here is the original problem once again. Reread it and underline any important information.

After a little while, they spotted some dolphins swimming with the boat. This is something that often happens as dolphins love the rushing water generated by the motor on the back of the boat.

“Did you know that they can swim .83 miles in one minute?” Pascal asked Antonio.

“Really, no I didn’t know that. I do know that a swordfish can swim almost one-half mile in a minute. I think the exact number is $\frac{9}{20}$ of a mile.”

“Wow, which one can swim the farthest in one minute?” Pascal asked, thinking carefully through the math.

By the time they reached the dive site, Antonio had figured out which one can swim farthest in one minute.

To figure out which one can swim the farthest in one minute, we will need to compare these two rational numbers.

A dolphin = 0.83 of a mile in one minute

A swordfish $= \frac{9}{20}$ of a mile in one minute

To figure this out, we first need to change the fraction into a decimal so that both numbers are in the same form.

$\frac{9}{20}=\frac{45}{100}=0.45$

Next, we compare 0.83 to 0.45.

0.83 > 0.45

A dolphin can swim farther than a swordfish in one minute.

Review

When comparing fractions, be sure to find common denominators to compare fractions on the same terms
When comparing a fraction and a decimal change both numbers to decimals using division or change both numbers to fractions
When finding equivalent ratios, you may cross multiply to see if the integers are the same. You may also reduce one of the fractions using division or you may increas one of the fractions using multiplication to find equivalent ratios.

Whole numbers

Whole numbers don’t contain fractional parts.

Fraction

A fraction describes a part of a whole number.

Denominator

The number written below the bar in a fraction is the denominator, which tells how many parts the

whole is divided into.

Numerator

The number above the bar in a fraction is the numerator, which tells how many parts of the whole

you have.

Equivalent Fractions

Fractions which describe the same part of a whole are called equivalent fractions.

Equivalent

Equivalent means equal.

Equivalent ratios

These can be used to show the same relationship between two quantities.

Simplifying

To simplify a fraction, we can reduce the number in the numerator and denominator by dividing

them by the same number.

Greatest Common Factor

The Greatest Common Factor (GCF) is the greatest factor that divides two numbers.

Mixed Number

A numbers which consists of a whole number and a fraction is called a mixed number.

Improper Fraction

When the numerator of a fraction is greater than or equal to the denominator, you have an improper fraction. Improper fractions are greater than or equal to 1.

Video Resources

Khan Academy, Comparing Rational Numbers

Khan Academy Mixed Numbers and Improper Fractions

James Sousa, Comparing Fractions with Different Denominators Using Inequality Symbols

James Sousa, Example of Ordering Fractions with Different Denominators from Least to Greatest

Khan Academy Converting Fractions to Decimals

James Sousa, Example of Converting Fractions to a Terminating Decimal

James Sousa, Example of Converting Fractions to a Repeating Decimal

Learning Objectives

You will learn that every rational number can be written as the ratio of two integers.

You will understand that division of two integers will always result in a rational number.

You will recognize and generate the equivalent representations of positive and negative rational numbers including equivalent fractions.

Introduction

Real Life Example

Guided Learning

Example A

Example B

Example C

Example D

Example E

Making Equivalent Fractions

Example F

Example G

Compare and Order Fractions and Mixed Numbers with and without Approximation

Example H

Example I

Describe Real-World Portion or Measurement Situations by Comparing and Ordering Fractions with and without Approximation

Example J

Write Fractions and Mixed Numbers as Terminating Decimals

Example K

Write Fractions and Mixed Numbers as Repeating Decimals

Example L

Write Decimals as Fractions

Example M

Compare and Order Decimals and Fractions

Identify and Write Different Forms of Equivalent Ratios among Rational Numbers

What is a ratio?

A ratio compares two numbers or quantities called terms. For example, suppose there are 3 green (G) apples and 4 red (R) apples in this basket.

We can express the ratio of green apples to red apples in the basket as a fraction.

We can also express this ratio in words, 3 to 4, or using a colon, 3:4.

The ratio above compares one part of the apples in the basket to another part. The ratio above compares the apples that are green to the apples that are red.

A ratio may also express a part to a whole. For example, we can express the ratio of green apples to the total number of apples in the basket.

There are a total of 7 apples in the basket, so the ratio of green apples to total apples is 3 to 7 or 3:7.

Here are the three ways that we can write a ratio:

In fraction form using a fraction bar

Using the word “to”

Using a colon:

Look at the drawing and write each the ratio three different ways.

Compare stars to suns

Now that you know how to write a ratio, let's look at equivalent ratios.

A ratio shows the relationship between two quantities. Equivalent ratios can be used to show the same relationship between two quantities. Remember that the word “equivalent” means equal.

Because we can write ratios in fraction form, we can use what we know about finding equivalent fractions to help us identify equivalent ratios. Here is where simplifying fractions is going to help us. We can simplify ratios to discover equivalence just as we can simplify fractions.

Take a look at another sample:

Determine if these two ratios are equivalent and .

We can start by simplifying the larger of the two fractions. If it simplifies to the same number as the smaller fraction, then we know that the two ratios are equivalent.

These are equal ratios.

Simplifying is one way to check for equivalence. We can also create equivalent ratios by multiplying just as we would make equivalent fractions.

Change to a ratio with 15 as the second term (the denominator).

Since , multiply both terms of the ratio by 5.

This shows that the ratio is equivalent to the ratio .

So, the two ratios listed above are equivalent.

Here is one more to view:

Determine if these two ratios are equivalent 7:6 and 13:12.

Rewrite the ratios as fractions and .

Change to a ratio with 12 as the second term.

Since , multiply both terms of the ratio by 2.

When the second term (the denominator) is 12, the equivalent ratio for the numerator is 14, not 13 .

So, 7:6 and 13:12 are not equivalent ratios.

Another way to determine if two ratios are equivalent is to cross multiply the terms in the ratio. If the cross products are equal, then the two ratios are equivalent. If the cross products are not equal, then the two ratios are not equivalent.

Compare these ratios:

Determine if these two ratios are equivalent and .

We use the symbol to show that the two ratios below may or may not be equal.

Now, we cross multiply. To cross multiply, multiply the circled pairs of numbers shown below. The product we get when we multiply each circled pair of numbers is called a cross product.

Since , the cross products are equal. This means that , and those two ratios are equivalent.

Determine whether or not each pair of ratios is equivalent. Write yes or no. and and and

Example N

Write Ratios in Simplest Form

In the last section, we touched briefly on how to simplify ratios that are in fraction form to see if they are equivalent or not. We will explore this further in this section.

What does it mean to simplify?

To simplify means to make smaller. When we simplify, we make a fraction smaller. It is still equivalent to the larger form of the fraction, but it is simpler.

How do we simplify a ratio in fraction form?

To write a ratio in simplest form, find the greatest common factor of both terms in the ratio. The greatest common factor of two numbers is the greatest number that divides both numbers evenly. Then, divide both terms of the ratio by the greatest common factor.

This is basically the same procedure you use to rewrite a fraction in simplest form. Let’s look at an example.

Take a look at this ratio:

Write this ratio in simplest form .

First, find the greatest common factor of the terms 20 and 24.

The factors of 20 are 1, 2, 4, 5, 10, and 20.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.