4. Compare and Order Fractions, Decimals, and Percents
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Chapter 2
4.
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.
| Student Learning Objectives: |
|---|
|
| | 12 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Compare and Order Fractions, Decimals, and Percents
Slide of 12
Numbers in different forms are often connected and need to be analyzed in comparison to each other. 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 Numbers
- Whole Numbers
- Fractions
- Percents
- Forms of Rational Numbers
- Converting Between Different Forms of Numbers
- Equivalent Fractions
Background to Help Understand Multiples
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Challenge
Comparing Math Homework Progress
LaShay, Tiffaniqua, and Kevin are friends.
{"type":"choice","form":{"alts":["Tiffaniqua","LaShay","Kevin"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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Discussion
Can Any Two Numbers Be Compared?
Consider a 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 and the second is a decimal number. ccc Percent & & Decimal ↓ & & ↓ 15 % & & 0.32 They do not have the same form, so they cannot be directly compared. It would be like comparing strawberries and dogs — they are just not comparable because they are too different.
It is the same way with numbers. Numbers can only be compared if they are written in the same format. cccc Percents: & 15 % & vs. & 54 % Decimals: & 0.32 & vs. & 2.19 To compare two numbers, we always begin by making sure that they have the same form. If they do, we can go ahead and compare them! If they do not, we first convert one or both numbers such that they are written in the same format.
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Discussion
Comparing Two Numbers in the Same Form
Consider two numbers in the same form. We will explore how to compare them. For example, 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 place values — tenths versus tenths, hundredths versus hundredths, and so on.
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Pop Quiz
Comparing Two Numbers in Decimal and Percent Forms
Consider the pair of numbers in different forms. Select the symbol that makes the statement true.
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Example
Ordering Decimals and Percents From Least to Greatest
Some friends are comparing their typing speeds.
a
Order their speeds from least to greatest.
{"type":"sort","form":{"alts":[{"id":0,"text":"Tiffaniqua: 254.6"},{"id":1,"text":"LaShay: 279.5"},{"id":2,"text":"Kevin: 252"}]},"formTextBefore":"","formTextAfter":"","answer":[2,0,1]}
125 %, 1.03, 98 %, 1.17 Order their results from least to greatest.
{"type":"sort","form":{"alts":[{"id":0,"text":"98 %"},{"id":1,"text":"1.03"},{"id":2,"text":"1.17"},{"id":3,"text":"125 %"}]},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3]}
Hint
a
Compare the decimal place values, 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
Let's start by looking at how many characters per minute Tiffaniqua, LaShay, and Kevin can type.
Tiffaniqua:& 254.6 LaShay:& 279.5 Kevin:& 252 One way we can order these decimal numbers from least to greatest is by plotting them on a number line. The farther right a number is, the greater it is.
Kevin's speed is farthest to the left, so it is the smallest. Next is LaShay's speed. Tiffaniqua's speed is farthest to the right, so it is the greatest. K < T < L ⇕ 252 < 254.6 < 279.5
b
To compare the average typing speeds, we will write them all in the same form. Let's write them as percents by multiplying 1.03 and 1.17 by 100 and adding the percent sign.
1.03* 100 %=103 % 1.17* 100 %=117 % Now all the numbers are written as percents! We can compare them by plotting them on a number line.
Finally, let's order the percents from least to greatest. 98 %, 103 %, 117 %, 125 % [0.1cm] ⇕ [0.1cm] 98 %, 1.03, 1.17, 125 %
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Discussion
Comparing Fractions
How can we compare a pair of fractions if they have different numerators and denominators? 4/7 and 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. 4/7 → ?/New Denom. and ?/New Denom. ← 5/8 When two fractions have the same denominator, they 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 |
|---|---|
| 2/3 and 5/3 | 3 |
| 8/10 and 5/10 | 10 |
| 11/17 and 6/17 | 17 |
As an example, take a look at a pair of fractions with different denominators. 1/3 and 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 of3:& 3, 6, 9, 12, 15, 18, ... Multiples of2:& 2, 4, 6, 8, 10, 12, 14, ... There are two potential common denominators in these lists. Expand the first fraction by 123= 4 and the second fraction by 122= 6 to make them have a common denominator of 12. 1 * 4/3 * 4&=4/12 [1.3em] 1 * 6/2 * 6&=6/12 Now the fractions share a common denominator.
4/12 and 6/12
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Discussion
The Most Convenient Common Denominator
Fractions can have multiple common denominators, but it is usually 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) |
|---|---|---|---|---|
| 2/3 and 1/2 | 3 and 2 | Multiples of3:& 3, 6, 9, 12, 15, ... Multiples of2:& 2, 4, 6, 8, 10, 12, ... | 6, 12 | 6 |
| 5/6 and 1/4 | 6 and 4 | Multiples of6:& 6, 12, 18, 24, 30, ... Multiples of4:& 4, 8, 12, 16, 20, 24, ... | 12, 24 | 12 |
| 1/4 and 5/2 | 4 and 2 | Multiples of4:& 4, 8, 12, ... Multiples of2:& 2, 4, 6, 8, 10, 12, ... | 4, 8, 12 | 4 |
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, we can compare them using their numerators.
| Fractions | 4/11 and 9/11 |
|---|---|
| Numerators | 4< 9 |
| Conclusion | 4/11<9/11 |
Extra
Comparing Fractions With Common NumeratorsSometimes two fractions share the same numerator but have different denominators. In these cases, the fractions can be compared using their denominators. Consider a pair of fractions.
5/7 and 5/16
When fractions have the same denominator, the fraction with the greater numerator is greater. However, when fractions have the same numerator but different denominators, 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. 5/7> 5/16 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.
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.
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Pop Quiz
Comparing Two Fractions
Select the correct symbol to create a true statement. If necessary, rewrite the fractions so that they have a common denominator.
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Example
It All Started With a Ball
A local story is published in the morning newspaper.
{"type":"sort","form":{"alts":[{"id":0,"text":"5\/6"},{"id":1,"text":"2\/3"},{"id":2,"text":"5\/20"},{"id":3,"text":"1\/5"}]},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3]}
Hint
Rewrite the fractions so that they have a common denominator.
Solution
To order the four fractions from greatest to least, we first rewrite them with the same denominator. We do this by listing the multiples of each denominator. The first number they all share is the least common multiple (LCM), which we can use as the common denominator.
| Denominator | Multiples of the Denominator |
|---|---|
| 6 | 6, 12, 18, ..., 54, 60, ... |
| 20 | 20, 40, 60, 80, 100, ... |
| 3 | 3, 6, 9, 12, ..., 57, 60, ... |
| 5 | 5, 10, 15, 20, ..., 55, 60, ... |
The LCM is 60, so we can rewrite each fraction with a denominator of 60. For 56, multiply the numerator and denominator by 10.
Let's expand the remaining fractions in the same way.
| Fraction | Expanding | Equivalent Fraction |
|---|---|---|
| 5/20 | 5* 3/20* 3 | 15/60 |
| 2/3 | 2* 20/3* 20 | 40/60 |
| 1/5 | 1* 12/5* 12 | 12/60 |
Finally, we can compare the equivalent fractions. Since they have the same denominator, we order their numerators from greatest to least. 50/60 > 40/60 > 15/60 > 12/60 [0.3cm] ⇕ [0.3em] 5/6 > 2/3 > 5/20 > 1/5
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Example
Analyzing Flower Growth
Students measured how many inches four different flowers grew in one week.
The measurements were made in different forms, which made it difficult to compare the values. Help the students by ordering the numbers from least to greatest.
{"type":"sort","form":{"alts":[{"id":0,"text":"78 %"},{"id":1,"text":"4\/5"},{"id":2,"text":"0.91"},{"id":3,"text":"1 29"}]},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3]}
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
Let's consider the number of inches each of the four different flowers grew. One number is a fraction, one is a decimal number, one is a mixed number, and one is a percent.
4/5, 0.91, 1 29, 78 %
To compare the numbers, they need to be written in the same form. It does not matter which format is chosen, so let's rewrite them all as percents.
Fraction
We start by rewriting 45 as a percent. Since a percent is a ratio of a number to 100, we can rewrite the fraction with a denominator of 100. The numerator will be the percent value.
We found that 45 is equivalent to 80100. This means it corresponds to 80 %.
Decimal Number
To rewrite 0.91 as a percent, we multiply it by 100 and add a percent sign. We can multiply by 100 by moving the decimal point two places to the right.
The decimal number 0.91 is equal to 91 %.
Mixed Number
Next, we rewrite the mixed number 1 29. First, we rewrite 29 as a decimal by dividing the numerator by the denominator using long division. Let’s find the quotient to two decimal places.
The fractional part is about 0.22. Next, we add the integer and fractional parts of the mixed number to get its decimal form. 1 29≈ 1.22 Finally, we multiply the number by 100 and add a percent sign. 1.22* 100 % =122 % The mixed number 1 29 is approximately 122 %.
Comparison
We have all the numbers as percents, so we can order them from least to greatest. cccc 4/5 & 0.91 & 1 29 & 78 % ↓ & ↓ & ↓ & ↓ [0.2cm] 80 % & 91 % & 122 % & 78 % To do this, we ignore the percent signs and plot the numbers on a number line.
Now we can order the given numbers from least to greatest. ccccccc 78 % & & 80 % & & 91 % & & 122 % [0.2cm] ↓ & & ↓ & & ↓ & & ↓ [0.2cm] 78 % & < & 4/5 & < & 0.91 & < &1 29
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Closure
Who Is Closer to Finishing Their Homework?
Earlier we considered three students and their progress with their math homework.
{"type":"choice","form":{"alts":["Tiffaniqua","LaShay","Kevin"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
Solution
To find who has completed the greatest part of their homework, we need to rewrite the numbers in the same form. Let’s write them all as decimals. We start by rewriting the 60 % of homework Tiffaniqua has finished. Since a percent is a ratio of a number to 100, 60 % is equal to 60100.
60 % = 60/100
To find the quotient, we write 60 as 60.0 and move the decimal point two places to the left.
We found that 60 % is equal to 0.6. Next, let’s rewrite 3150 of the exercises Kevin finished as a decimal. To do this, we write it as an equivalent fraction with a denominator of 100 by expanding it by 2. 31* 2/50* 2=62/100 Again, we can divide by 100 by moving the decimal point two places to the left.
The decimal form of 3150 is 0.62. 0.6, 0.55, 0.62 Now we have all the numbers as decimals, so we can find the greatest one. Let’s compare the numbers digit by digit, starting with 0.60 and 0.55.
We found that 0.6 is greater than 0.55. Next, let’s compare 0.6 and 0.62. Again, we add a 0 after 6 for convenience.
The decimal 0.62 is greater than 0.6. This means it is also greater than 0.55. 0.55 < 0.6 < 0.62 Now let’s match each decimal to its original value. ccccc 0.55 &< & 0.6 &< & 0.62 [0.5em] ↓ & &↓ & &↓ [0.5em] 0.55 & & 60 % & & 31/50 [0.8em] ↑ & &↑ & &↑ [0.8em] LaShay & & Tiffaniqua & & Kevin Kevin has completed the greatest part of the homework, with 3150 finished.
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Compare and Order Fractions, Decimals, and Percents
Exercise 2.1
Order the given percents and decimals from least to greatest. 41 %, 0.17, 29 %, 0.35
{"type":"sort","form":{"alts":[{"id":1,"text":"0.17"},{"id":2,"text":"29 %"},{"id":3,"text":"0.35"},{"id":0,"text":"41 %"}]},"formTextBefore":"","formTextAfter":"","answer":[1,2,3,0]}
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Let's start by analyzing the given list of numbers. 41 %, 0.17, 29 %, 0.35 There are two percents and two decimal numbers. Let's rewrite them so that all of the numbers are in the same form so that we can easily compare them. Percents:& 41 %, 29 % Decimals:& 0.17, 0.35 Let's rewrite the decimals as percents. For this, multiply the decimals by 100 and add a percent sign. We will start with 0.17. Move the decimal point two places to the right to multiply it by 100.
As we can see, 0.17 is equivalent to 17 %. 0.17* 100=17 % We can rewrite 0.35 into a percent in a similar manner. Let's move the decimal point two places to the right and add a percent sign.
We can see that 0.35 is equal to 35 %. 0.35* 100=35 % All the numbers are written in percent form now. cccc 41 % & 0.17 & 29 % & 0.35 [0.2cm] ↓ & ↓ & ↓ & ↓ [0.2cm] 41 % & 17 % & 29 % & 35 % Let's plot these numbers on a number line to help us order them from least to greatest. Remember that the farther to the right a number is located on the line, the greater it is.
Finally, the numbers can finally be ordered from the least to the greatest. ccccccc 17 % &< & 29 % &< & 35 % &< & 41 % [0.2cm] ↓ & & ↓ & & ↓ & & ↓ [0.2cm] 0.17, & & 29 %, & & 0.35, & & 41 %
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Solution
Order the given fractions and percents from least to greatest. 2/5, 46 %, 3/8, 38 %
{"type":"sort","form":{"alts":[{"id":0,"text":"38"},{"id":1,"text":"38 %"},{"id":2,"text":"25"},{"id":3,"text":"46 %"}]},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3]}
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We are given two fractions and two percents. 2/5, 46 %, 3/8, 38 % In order to compare these numbers, we need to rewrite them to be in the same form. Let's rewrite the fractions into percents. We will start with 25. First, divide the numerator by the denominator by using long division.
Next, multiply the decimal number by 100 and add a percent sign. 0.4* 100=40 % We can rewrite 38 as a percent in a similar fashion. Let's divide 3 by 8.
Now, multiply the decimal number by 100 and add a percent sign. 0.375* 100=37.5 % As we can see, 38 is equivalent to 37.5 %. All the numbers are now written in percent form! cccc 2/5, & 46 %, & 3/8, & 38 % [0.2cm] ↓ & ↓ & ↓ & ↓ [0.2cm] 40 %, & 46 %, & 37.5 %, & 38 % Finally, we can order the number from least to greatest by comparing their percent form. cccc 37.5 % &< & 38 % &< & 40 % &< & 46 % [0.2cm] ↑ & & ↑ & & ↑ & & ↑ [0.2cm] 3/8, & & 38 %, & & 2/5, & & 46 %
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Solution
Tearrik needs to compare the speeds of different vehicles and animals. He wrote their speeds as they compare to 100 kilometers per hour, which he set as 100 %.
Order the vehicles and animals from fastest to slowest.
{"type":"sort","form":{"alts":[{"id":0,"text":"Racing car"},{"id":1,"text":"Railway car"},{"id":2,"text":"Tiger"},{"id":3,"text":"Horse"}]},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3]}
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We are given four different numbers. 48 %, 227 %, 0.65, 3.1 Two of these numbers are percents and two are decimal numbers. We need to rewrite them so that they are in the same form in order to be able to compare them. Percents:& 48 %, 227 % Decimals:& 0.65, 3.1 Let's rewrite the numbers as decimals. To do this, divide the percents by 100 and remove the percent signs. We will start with 48 %. Move the decimal point two places to the left to divide it by 100.
As we can see, 48 % is equivalent to 0.48. 48 %÷ 100=0.48 We can rewrite 227 % as a percent in a similar manner. Let's move the decimal point two places to the left.
We can see that 227 % is the same as 2.27. 227 %÷ 100=2.27 All the numbers are now written in decimal form! cccc 48 % & 227 % & 0.65 & 3.1 [0.2cm] ↓ & ↓ & ↓ & ↓ [0.2cm] 0.48 & 2.27 & 0.65 & 3.1 Finally, let's compare these numbers and order them from greatest to least. ccccccc 3.1 &> & 2.27 &> & 0.65 &> & 0.48 [0.2cm] ↓ & & ↓ & & ↓ & & ↓ [0.2cm] 3.1, & & 227 %, & & 0.65, & & 48 % [0.2cm] ↓ & & ↓ & & ↓ & & ↓ [0.2cm] Racing car & & Railway car & & Tiger & & Horse Therefore, the racing car is the fastest, followed by the railway car, the tiger, and, finally, the horse.
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Solution
Tearrik and Ali are solving their math homework. They need to compare a percent and a mixed number.
However, their solutions are different. Who is correct?
{"type":"choice","form":{"alts":["Tearrik","Ali","Neither","Both"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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The boys need to compare two numbers, a mixed number and a percent. 4 59 and 462 % Let's analyze each solution step by step.
Tearrik's Solution
Tearrik decides to rewrite the mixed number as a percent to be able to compare the numbers. First, he converts the mixed number into an improper fraction by applying the corresponding formula. a bc=a* c+b/c Let's check if he applied the formula correctly.
The improper fraction that corresponds to 4 59 is 419. Tearrik calculated this step correctly. Next, he divides the numerator by the denominator using long division.
The quotient is about 4.55. Next, Tearrik needs to multiply this decimal number by 100 and add a percent sign.
4.55* 100=455 %
This means that the mixed number is equivalent to about 455 %. Finally, Tearrik compares the percents and concludes that 462 % is greater than 455 %.
455 % < 462 %
Tearrik's solution is absolutely correct!
Ali's Solution
Let's now consider Ali's solution. His first step is the same as Tearrik's. He rewrites the mixed number as an improper fraction. 4 59=41/9 ✓ Next, Ali rewrites the fraction as a percent. For this, he divides the numerator by the denominator. 41/9=4.55 He rounds the number to one decimal place and writes 4.6 %. However, he does not realize that he only found the corresponding decimal, and not the percent. To find the percent, he would also need to multiply the result by 100. This means that this step is incorrect. 41/9≈ 4.6 % * Therefore, Ali's solution is incorrect. Only Tearrik is correct.
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Compare and Order Fractions, Decimals, and Percents
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