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Pre-Algebra View details

4. Compare and Order Fractions, Decimals, and Percents

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eCourses /

Pre-Algebra /

Chapter 2

Compare and Order Fractions, Decimals, and Percents

This lesson delves into the fundamental skill of comparing and ordering numbers, specifically focusing on fractions, decimals, and percents. This skill is not just for academic purposes; it has practical applications in everyday life. For example, when shopping for groceries, you might need to compare prices that are listed as fractions, decimals, or percentages to get the best deal. Similarly, in finance, understanding how to order these types of numbers can help you make smarter investment choices. In the realm of data analysis, being able to compare numbers in various formats is crucial for drawing accurate conclusions. Overall, mastering this skill can significantly improve your problem-solving abilities in a wide range of situations.

Lesson Settings & Tools

	12 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Slide of 12

Numbers appear in different forms from whole numbers to fractions and percents. What is more, numbers are often connected and need to be analyzed in comparison to each other. This is why it is useful to know how to compare and order different forms of numbers. This topic will be covered and practiced in this lesson.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Background to Help Understand Numbers

Background to Help Understand Multiples

Multiple
Least Common Multiple
Finding the LCM

Challenge

Comparing Math Homework Progress

Tiffaniqua, LaShay, and Kevin are really good friends. They spend most of their free time together. On Wednesday evening, LaShay called Tiffaniqua and asked her how much of her math homework she completed.

Tiffaniqua's, LaShay's, and Kevin's room

Tiffaniqua said that she did about

60 %

of the homework. LaShay replied that she had completed

0.55

of the homework. Together they called Kevin, who told that he had finished

\frac{3 1}{5 0}

of all the exercises. Who has completed the greatest part of the homework?

Discussion

Can Any Two Numbers Be Compared?

Imagine that two numbers need to be compared. Consider this pair of numbers.

15 % and 0.32

Which one is greater? Which is less? The first thing to check is whether the numbers have the same form. In this case, the first number is a percent, while the second is a decimal number.

Percent ↓ 15 % Decimal ↓ 0.32

They do not have the same form, so they cannot be directly compared. It is like comparing strawberries and dogs — they are just not comparable because they belong to totally different categories and have drastically different features.

It is the same way with numbers. Numbers can only be compared if they are written in the same format.

Percents : Decimals : 15 % 0.32 vs . vs . 54 % 2.19

Therefore, to compare two numbers, always begin by making sure that they have the same form. If they do, go ahead and compare them! If they do not, first convert one or both numbers such that they are written in the same format.

Discussion

Comparing Two Numbers in the Same Form

Consider two numbers in the same form. The good news is that they can totally be compared! But how? First, take a look at two decimal numbers.

0.285 and 0.281

The numbers look pretty similar. To determine which is greater, compare their digits one by one moving from left to right until a greater one is found. Remember, it is important to compare the corresponding digits — tenths versus tenths, hundredths versus hundredths, and so on.

An applet that compares 0.285 and 0.281 and concludes that 0.285 is greater

Since the last digit of

0.285

is greater, the first number is greater than the second one.

0.285 > 0.281

What about a pair of percents?

65 % and 67.3 %

The same method can be used to determine which one is greater. Compare the numbers digit by digit, ignoring the

%

sign. The second digit

7

is greater than

5,

so the second percent is greater.

65 % < 67.3 %

Pop Quiz

Comparing Two Numbers in Decimal and Percent Forms

Consider a pair of numbers. The numbers might be a pair of decimals, a pair of percents, or one decimal number and one percent. Which number is greater? Compare the numbers and pick a correct sign.

A random generator that generates two numbers and asks to compare them

Example

Ordering Decimals and Percents From Least to Greatest

Tiffaniqua, LaShay, and Kevin, along with a couple of their friends, are mastering the skill of blind typing. After finishing their homework, they held a little competition to figure out who types the fastest at the moment.

A laptop with the website type fast dot org is open and the text to type is shown

Tiffaniqua said that her typing speed is

254.6

characters per minute. LaShay said that her result is

279.5

characters per minute. Kevin's speed is

252

characters per minute.

a Order their speeds from least to greatest.

b Their friends told them that their results were given as percents or decimals that represent their speeds compared to the average speed of typing

250

characters.

125 %, 1.03, 98 %, 1.17

Order their results from least to greatest, too.

Hint

a Compare the decimals digit by digit, or compare their locations on a number line.

b Start by rewriting the numbers so that they are in the same form.

Solution

a Start by considering the typing speeds of Tiffaniqua, LaShay, and Kevin.

Tiffaniqua : LaShay : Kevin : 254.6 279.5 252

All these speeds are given as decimal numbers. One way to order these numbers from least to greatest is to compare them digit by digit. Another method is to plot them as points on a number line. On a number line, the farther to the right the number is, the greater it is.

Notice that point

K,

which represents Kevin's speed, is the least because it is farthest to the left. Next comes the point

T,

Tiffaniqua's speed. Finally, point

L

is the farthest to the right, meaning that LaShay's speed is the greatest.

K ↓ 252 < T ↓ 254.6 < L ↓ 279.5

Therefore, Kevin is the slowest typist at the moment, Tiffaniqua is in the middle, and LaShay is the fastest. Not to worry — Kevin still has time to practice and beat Tiffaniqua and LaShay.

b Start by considering the given typing speeds. Some of them are percents and some are decimal numbers.

125 %, 1.03, 98 %, 1.17

To be able to compare the numbers, they should be written in the same form — either all as percents or all as decimal numbers. Rewrite the decimals as percents by multiplying them by

100

and adding the percent sign.

1.03 \cdot 100 = 103 % 1.17 \cdot 100 = 117 %

Now all the numbers are written as percents!

125 %, 103 %, 98 %, 117 %

Compare the numbers by plotting them on a number line.

Finally, the percents can be ordered from least to greatest.

98 %, ↓ 98 %, 103 %, ↓ 1.03, 117 %, ↓ 1.17, 125 % ↓ 125 %

Discussion

Comparing Fractions

Consider a pair of fractions. How can the fractions be compared when they have different numerators and denominators?

\frac{4}{7} and \frac{5}{8}

Both fractions represent a part of a whole, but it is difficult to say which one is greater straight away. The best way to compare these fractions would be to convert them to equivalent fractions that have the same denominator.

\frac{4}{7} \to \frac{?}{New Denom .} and \frac{?}{New Denom .} \leftarrow \frac{5}{8}

When two fractions have the same denominator, they are said to have a common denominator.

Concept

Common Denominator

A common denominator is a denominator that is shared between two or more fractions. Consider a few examples.

Pair of Fractions	Common Denominator
$\frac{2}{3}$ and $\frac{5}{3}$	$3$
$\frac{8}{1 0}$ and $\frac{5}{1 0}$	$10$
$\frac{1 1}{1 7}$ and $\frac{6}{1 7}$	$17$

Fractions can always be rewritten to all have a common denominator. This process requires writing equivalent fractions by expanding or simplifying the fractions. As an example, take a look at a pair of fractions with different denominators.

\frac{1}{3} and \frac{1}{2}

These fractions are in their simplest form, which means that they can only be expanded. Write the multiples of their denominators,

3

and

2,

to find the factor of expansion.

Multiples of 3 : Multiples of 2 : 3, 6, 9, 12, 15, 18, \dots 2, 4, 6, 8, 10, 12, 14, \dots

There are two potential common denominators in these lists. Expand the first fraction by

\frac{1 2}{3} = 4

and the second fraction by

\frac{1 2}{2} = 6

to make them have a common denominator of

12 .

\frac{1 \times 4}{3 \times 4} \frac{1 \times 6}{2 \times 6} = \frac{4}{1 2} = \frac{6}{1 2}

Now the fractions share a common denominator.

\frac{4}{1 2} and \frac{6}{1 2}

Discussion

The Most Convenient Common Denominator

There are a lot of possible common denominators that fractions can have. However, it is almost always easier to deal with smaller numbers. This is when the least common denominator comes in handy!

Concept

Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of the denominators of the fractions. In other words, the least common denominator is the smallest of all the common denominators. Some examples are provided in the table below.

Fractions	Denominators	Multiples of Denominators	Common Denominators	LCM of Denominators (LCD)
$\frac{2}{3}$ and $\frac{1}{2}$	$3$ and $2$	$Multiples of 3 : Multiples of 2 : 3, 6, 9, 12, 15, \dots 2, 4, 6, 8, 10, 12, \dots$	$6,$ $12$	$6$
$\frac{5}{6}$ and $\frac{1}{4}$	$6$ and $4$	$Multiples of 6 : Multiples of 4 : 6, 12, 18, 24, 30, \dots 4, 8, 12, 16, 20, 24, \dots$	$12,$ $24$	$12$
$\frac{1}{4}$ and $\frac{5}{2}$	$4$ and $2$	$Multiples of 4 : Multiples of 2 : 4, 8, 12, \dots 2, 4, 6, 8, 10, 12, \dots$	$4,$ $8,$ $12$	$4$

The least common denominator is used when adding or subtracting fractions with different denominators.

\frac{2}{1 0} - \frac{1}{1 1} = \frac{2 2}{LCD 1 1 0} - \frac{1 0}{LCD 1 1 0}

To sum up, finding the least common denominator is the same as finding the least common multiple of the denominators. When fractions have a common denominator, which is most often the LCD, they are ready to be compared by comparing their numerators.

Fractions	$\frac{4}{1 1}$ and $\frac{9}{1 1}$
Numerators	$4 < 9$
Conclusion	$\frac{4}{1 1} < \frac{9}{1 1}$

Extra

Comparing Fractions With Common Numerators

There are cases when two fractions share the same numerator. In this case, the fractions can be compared based on their denominators. Consider a pair of fractions.

\frac{5}{7} and \frac{5}{1 6}

When fractions have the same denominator, the fraction with the greater numerator is greater. However, when fractions have the same numerator, it is the opposite — the fraction with the smaller denominator is greater.

Same Denominator	Same Numerator
The greater numerator, the greater the fraction.	The smaller denominator, the greater the fraction.

Since

7

is less than

16,

the first fraction must be greater.

\frac{5}{7} > \frac{5}{1 6}

To understand why this is true, think of a whole represented by

1 .

The denominators indicate how many pieces this whole is split into. Here, it is split into

7

pieces for the first fraction and

16

pieces for the second fraction.

$Two circles, each representing a whole, show how the denominator in a fraction determines the number of pieces into which the whole is split. The left circle is divided into 7 parts, while the right is divided into 16.$

Notice that the $7$ pieces of the first fraction are much larger than the $16$ pieces of the second. The numerator of each fraction indicates how many pieces get picked. Since the numerators are the same, $5$ pieces are selected from each whole. Which fraction has a larger selected section?

The total selected area is greater in the first fraction than in the second because each of the pieces is bigger. This is why the fraction with the smaller denominator is greater if the numerators are the same.

Pop Quiz

Comparing Two Fractions

Consider the given pair of fractions. Which one is greater? Choose the correct sign to complete the expression. If necessary, rewrite the fractions such that they have a common denominator.

$A random generator that generates two fractions and asks to compare them$

Example

It All Started With a Ball

The next day, Kevin, LaShay, and Tiffaniqua excitedly discussed the details of a local story published in the morning newspaper.

An article in the newspaper shown on a tablet

Four fractions were mentioned in the article.

\frac{5}{6}, \frac{5}{2 0}, \frac{2}{3}, \frac{1}{5}

Order them from greatest to least.

Hint

Rewrite the fractions so that they have a common denominator. Find it by determining the LCM of the denominators.

Solution

Four fractions need to be ordered from greatest to least.

\frac{5}{6}, \frac{5}{2 0}, \frac{2}{3}, \frac{1}{5}

They have different numerators and denominators. Rewrite them such that they have a common denominator to be able to compare them. Start by finding the LCM of the denominators of the fractions to find a candidate for a common denominator.

\underline{Denominators} 6, 20, 3, 5

First, list the multiples of all the numbers and try to find the least common one.

Multiples of 6 : Multiples of 20 : Multiples of 3 : Multiples of 5 : 6, 12, 18, \dots, 54, 60, \dots 20, 40, 60, 80, 100 \dots 3, 6, 9, 12, \dots, 57, 60 \dots 5, 10, 15, 20, \dots, 55, 60 \dots

The LCM of the numbers is

60 .

Next, divide

60

by each of the numbers to find by which factor each fraction should be expanded.

Denominator	Calculating the Quotient	Factor
$6$	$\frac{6 0}{6}$	$10$
$20$	$\frac{6 0}{2 0}$	$3$
$3$	$\frac{6 0}{3}$	$20$
$5$	$\frac{6 0}{5}$	$12$

Now that the expansions have been found, they can be used to rewrite the fractions into equivalent fractions with a common denominator of

60 .

Begin with the first fraction of

\frac{5}{6} .

\frac{5}{6}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 1 0}{b \cdot 1 0}$

\frac{5 \times 1 0}{6 \times 1 0}

Multiply

\frac{5 0}{6 0}

The rest of the fractions can be similarly expanded.

Fraction	Equivalent Fraction
$\frac{5}{6}$	$\frac{5 0}{6 0}$
$\frac{5}{2 0}$	$\frac{1 5}{6 0}$
$\frac{2}{3}$	$\frac{4 0}{6 0}$
$\frac{1}{5}$	$\frac{1 2}{6 0}$

Finally, the fractions with the same denominator can be compared by looking at their numerators and ordering them from greatest to least.

\frac{5 0}{6 0} ↓ \frac{5}{6} > \frac{4 0}{6 0} ↓ \frac{2}{3} > \frac{1 5}{6 0} ↓ \frac{5}{2 0} > \frac{1 2}{6 0} ↓ \frac{1}{5}

Example

Analyzing Facts About Flowers

Tiffaniqua, LaShay, Kevin, and one of their classmates are working on a biology project comparing four different flowers. Each of them was assigned a specific flower. After some research time, they met up to discuss what they found out.

Four people with four flowers and four numbers: 4/5, 0.91, 78 percent, and the mixed number of 1 and 2/9

For their project, they need to compare various data points. However, they discovered that they had collected some of these values in different forms and they could not compare the values. Help the students by ordering the numbers from least to greatest.

Hint

Rewrite the numbers so that they are in the same form. Then, plot the numbers on a number line to help place them in order.

Solution

Consider the numbers found by the students during their research about different flowers. One number is a fraction, one is a decimal number, one is a mixed number, and one is a percent.

\frac{4}{5}, 0.91, 1 \frac{2}{9}, 78 %

To compare the numbers, they need to be written in the same form. It does not matter which format is chosen, but all numbers will be rewritten as percents in this solution.

Fraction

Start by rewriting the fraction

\frac{4}{5}

as a percent. This can be done by multiplying the fraction by

100,

then dividing the new numerator by the denominator. Start with the multiplication.

\frac{4}{5}

Mult

Multiply by $100$

\frac{4}{5} \cdot 100

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{4 \cdot 1 0 0}{5}

Multiply

\frac{4 0 0}{5}

Next, divide the numerator by the denominator and add a percent sign.

\frac{4 0 0}{5} = 80 %

Therefore,

\frac{4}{5}

corresponds to

80 % .

Decimal Number

Now consider the decimal number

0.91 .

Rewrite it as a percent by multiplying it by

100

and adding a percent sign. Remember that a number can be easily multiplied by

100

by moving its decimal point two places to the right.

Finally, add a percent sign.

0.91 \cdot 100 = 91 %

Mixed Number

Consider the mixed number

1 \frac{2}{9} .

It consists of the integer

1

and the fraction part

\frac{2}{9} .

Begin by rewriting the fraction part as a decimal. Divide the numerator by the denominator to two decimal places by using long division.

The fraction corresponds to about

0.22 .

Next, join the integer part of

1

and the decimal part of

0.22

to write the final decimal form of

1 \frac{2}{9} .

1 \frac{2}{9} \approx 1.22

Finally, multiply the number by

100

and add a percent sign.

1.22 \cdot 100 = 122 %

Therefore,

1 \frac{2}{9}

corresponds to

122 % .

Comparison

Gather all the numbers and their corresponding percent forms.

\frac{4}{5} ↓ 80 % 0.91 ↓ 91 % 1 \frac{2}{9} ↓ 122 % 78 % ↓ 78 %

To compare the percents, ignore the percent signs and plot the numbers on a number line.

The given numbers can finally be ordered from least to greatest.

78 % ↓ 78 % < 80 % ↓ \frac{4}{5} < 91 % ↓ 0.91 < 122 % ↓ 1 \frac{2}{9}

Closure

Who Is Closer to Finishing Their Homework?

It was said earlier that on Wednesday evening, LaShay called Tiffaniqua and asked her how much of her math homework she completed.

Tiffaniqua said that she had completed about

60 %

of the homework. LaShay replied that she had done

0.55

of the homework. Together they called Kevin, who told them that he had finished

\frac{3 1}{5 0}

of all the exercises. Who has completed the greatest part of the homework?

Hint

Rewrite all the numbers as percents, decimals, or fractions so that they share the same form. If the numbers are written as decimals, compare them digit by digit.

Solution

Compare the given numbers to determine who has completed the greatest amount of the homework.

60 %, 0.55, \frac{3 1}{5 0}

These numbers are in different forms — a percent, a decimal number, and a fraction. The first step when comparing numbers is to have them all in the same form. This solution will write all the numbers in their decimal form. Divide the percentage of

60 %

100 %

to get its decimal form.

\frac{6 0 %}{1 0 0 %} = 0.6

The decimal form of

60 %

0.6 .

Next, rewrite the fraction

\frac{3 1}{5 0}

as a decimal. Begin by converting this fraction to an equivalent fraction whose denominator is

100 .

Since

50 \cdot 2 = 100,

the equivalent fraction can be found by multiplying the numerator and denominator of

\frac{3 1}{5 0}

2 .

\frac{3 1 \cdot 2}{5 0 \cdot 2} = \frac{6 2}{1 0 0}

The last step to write the fraction in its decimal form is to divide the numerator by the denominator. Since the denominator is

100,

the division can be performed simply by moving the decimal point of

62.0

two places to the left.

The decimal that corresponds to

\frac{3 1}{5 0}

0.62 .

Finally, the found decimal forms of the numbers can be compared.

0.6, 0.55, 0.62

Compare the numbers digit by digit to determine the greatest number. First, consider the pair of

0.6

and

0.55 .

Note that

0.6

is the same as

0.60 .

The decimal

0.6

is greater than

0.55 .

Next, compare

0.6

and

0.62 .

Again, a zero will be written after

6

for convenience.

The decimal

0.62

is greater than

0.6 .

Consequently, it is also greater than

0.55 .

0.55 < 0.6 < 0.62

Recall which initial numbers these decimals correspond to.

0.55 ↓ 0.55 ↑ LaShay < 0.6 ↓ 60 % ↑ Tiffaniqua < 0.62 ↓ \frac{3 1}{5 0} ↑ Kevin

Therefore, Kevin, who has finished

\frac{3 1}{5 0}

of the homework, has completed the greatest part of the homework.

Level 1

Level 2

Level 3

Identify the greater decimal number in each pair.

0.419 and 0.412

2.63 and 3.85

We are given two decimal numbers and what to determine which is greater. Let's do this by comparing the numbers digit by digit, moving from left to right.

Since the last digit is greater in 0.419, this decimal number is greater.

Let's compare the given pair of decimal numbers. We can determine which one is greater by comparing them digit by digit, moving from left to right.

Since the first digit is greater in 3.85, this decimal number is greater.

Identify which of the given percents is greater.

62 % and 58 %

12.8 % and 19.4 %

We are given two percents. 62 % and 58 % To determine which number is greater, let's plot them on a number line. We will ignore the percent sign. The number located more to the right and farther from the origin is the greater number.

Since 62 is farther to the right than 58 is, 62 % is the greater percent. 58 %<62 %

Let's consider the two given percents. 12.8 % and 14.4 % To determine which number is greater, we will plot them on a number line like we did before, ignoring the percent sign. The number located more to the right and farther from the origin is the greater number.

Since 14.4 is farther to the right than 12.8 is, 14.4 % it is the greater percent. 12.8 %<14.4 %

Consider the given pair of fractions. Which fraction is greater?

\frac{4}{1 3} and \frac{8}{1 5}

\frac{7}{5} and \frac{2 2}{1 9}

We are given two fractions with different numerators and denominators. 4/13 and 8/15 We can compare these fractions if they share a common denominator or numerator. We can find the common denominator of the fractions by calculating the LCM of 13 and 15. Let's find the prime factors of these numbers.

13			13
15	3	5
Factors	3	5	13

Next, find the product of all the prime factors to calculate the LCM of 13 and 15. LCM(13,15)&=3* 5* 13 &⇕ LCM(13,15)&=195 As we can see, the LCM of 13 and 15 is their product. This means that we need to expand the first fraction by 15 and the second by 13 in order to rewrite them to have a denominator of 195. Let's do it!

Similarly, since 8* 13=104, the fraction 815 is equivalent to 104195. Now both fractions have a common denominator and we can compare them by comparing their numerators. 60< 104 ⇓ 60/195< 104/195 Therefore, the second fraction 815, which is equivalent to 104195, is the greater than 413.

Let's consider the given pair of fractions. 7/5 and 22/19 To compare the fractions, they need to have a common numerator or denominator. We will rewrite these fractions to have a common denominator. We can find it by calculating the LCM of 5 and 19. These are both prime numbers, so their LCM is their product. LCM(5,19)=5* 19 ⇔ LCM(5,19) = 95 Next, we will expand the first fraction by 19 and the second fraction by 5 to rewrite them with a denominator of 95.

Similarly, since 22* 5=110, the fraction 2219 is equivalent to 11095. Now both fractions have a common denominator. Let's compare the fractions by comparing their numerators. 133> 110 ⇓ 133/95> 110/95 Therefore, the first fraction 75, which is equivalent to 13395, is the greater than 2219.

Consider the given mixed number pair. Which is greater?

5 \frac{3}{1 6} and 8 \frac{4}{7}

1 \frac{9}{1 4} and 1 \frac{7}{1 1}

Let's start by analyzing the given mixed numbers. 5 316 and 8 47 Recall that a mixed number consists of an integer part and a fraction part. We will identify these parts in both mixed numbers.

Mixed number	Integer Part	Fraction Part
5 316	5	3/16
8 47	8	4/7

First, compare the integer parts of the mixed numbers. 5< 8 The integer part of the first mixed number is less than the integer part of the second mixed number. Therefore, the second mixed number is greater. 5 316<8 47

We are given two mixed numbers. 1 914 and 1 711 Let's start by identifying their integer and fraction parts.

Mixed number	Integer Part	Fraction Part
1 914	1	9/14
1 711	1	7/11

The integer parts of both numbers are equal. This means that we need to compare their fraction parts to determine which number is greater. The important thing when comparing fractions is to rewrite them with a common denominator. Start by finding the least common multiple (LCM) of 14 and 11.

14	2	7
11			11
Factors	2	7	11

The LCM of 14 and 11 is the product of their prime factors. LCM(14,11)&=2* 7* 11 &⇕ LCM(14,11)&= 154 Notice that this is the product of 14 and 11. Let's expand the first fraction by a factor of 11 to rewrite it with a denominator of 154.

Similarly, let's expand the second fraction by a factor of 14 to rewrite it with a denominator of 154. Since 7* 14=98, the fraction 711 is equivalent to 98154. 7/11=98/154 Next, compare the fraction part by looking at their numerators. 99> 98 ⇓ 99/154> 98/154 The fraction part of the first mixed number is greater than the fraction part of the second mixed number. This means that the first mixed number is greater than the second mixed number. 1 914> 1 711

Compare and Order Fractions, Decimals, and Percents

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