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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

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 of the homework. LaShay replied that she had completed of the homework. Together they called Kevin, who told that he had finished 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.
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.
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.
A strawberry vs a dog
It is the same way with numbers. Numbers can only be compared if they are written in the same format.
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.
Comparing two numbers
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.
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 is greater, the first number is greater than the second one.
What about a pair of percents?
The same method can be used to determine which one is greater. Compare the numbers digit by digit, ignoring the sign. The second digit is greater than so the second percent is greater.
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 characters per minute. LaShay said that her result is characters per minute. Kevin's speed is 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 characters.
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.
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.
The number line with points at 252, 254.6, and 279
Notice that point which represents Kevin's speed, is the least because it is farthest to the left. Next comes the point Tiffaniqua's speed. Finally, point is the farthest to the right, meaning that LaShay's speed is the greatest.
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.
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 and adding the percent sign.
Now all the numbers are written as percents!
Compare the numbers by plotting them on a number line.
The number line with points at 98, 103, 117, and 125
Finally, the percents can be ordered from least to greatest.
Discussion

Comparing Fractions

Consider a pair of fractions. How can the fractions be compared when they have different numerators and denominators?
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.
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
and
and
and
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.
These fractions are in their simplest form, which means that they can only be expanded. Write the multiples of their denominators, and to find the factor of expansion.
There are two potential common denominators in these lists. Expand the first fraction by and the second fraction by to make them have a common denominator of
Now the fractions share a common denominator.
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)
and and
and and
and and

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

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 and
Numerators
Conclusion

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.
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 is less than the first fraction must be greater.
To understand why this is true, think of a whole represented by The denominators indicate how many pieces this whole is split into. Here, it is split into pieces for the first fraction and 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 pieces of the first fraction are much larger than the pieces of the second. The numerator of each fraction indicates how many pieces get picked. Since the numerators are the same, pieces are selected from each whole. Which fraction has a larger selected section?

5 picked pieces on the first circle split into 7 parts and 5 picked pieces on the other circle split into 16 parts

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.
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.
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.
First, list the multiples of all the numbers and try to find the least common one.
The LCM of the numbers is Next, divide by each of the numbers to find by which factor each fraction should be expanded.
Denominator Calculating the Quotient Factor
Now that the expansions have been found, they can be used to rewrite the fractions into equivalent fractions with a common denominator of Begin with the first fraction of
The rest of the fractions can be similarly expanded.
Fraction Equivalent Fraction
Finally, the fractions with the same denominator can be compared by looking at their numerators and ordering them from greatest to least.
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.
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 as a percent. This can be done by multiplying the fraction by then dividing the new numerator by the denominator. Start with the multiplication.
Next, divide the numerator by the denominator and add a percent sign.
Therefore, corresponds to

Decimal Number

Now consider the decimal number Rewrite it as a percent by multiplying it by and adding a percent sign. Remember that a number can be easily multiplied by by moving its decimal point two places to the right.
0.91 after multiplying by 100 becomes 91.0
Finally, add a percent sign.

Mixed Number

Consider the mixed number It consists of the integer and the fraction part Begin by rewriting the fraction part as a decimal. Divide the numerator by the denominator to two decimal places by using long division.
The long division of 2 over 9
The fraction corresponds to about Next, join the integer part of and the decimal part of to write the final decimal form of
Finally, multiply the number by and add a percent sign.
Therefore, corresponds to

Comparison

Gather all the numbers and their corresponding percent forms.
To compare the percents, ignore the percent signs and plot the numbers on a number line.
The number line with points at 78, 80, 91, and 122
The given numbers can finally be ordered from least to greatest.
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's, LaShay's, and Kevin's room
Tiffaniqua said that she had completed about of the homework. LaShay replied that she had done of the homework. Together they called Kevin, who told them that he had finished 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.
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 by to get its decimal form.
The decimal form of is Next, rewrite the fraction as a decimal. Begin by converting this fraction to an equivalent fraction whose denominator is Since the equivalent fraction can be found by multiplying the numerator and denominator of by
The last step to write the fraction in its decimal form is to divide the numerator by the denominator. Since the denominator is the division can be performed simply by moving the decimal point of two places to the left.
Moving the decimal point of 62.0 two places left obtaining 0.62
The decimal that corresponds to is Finally, the found decimal forms of the numbers can be compared.
Compare the numbers digit by digit to determine the greatest number. First, consider the pair of and Note that is the same as
An applet that compares 0.60 and 0.55 and concludes that 0.6 is greater
The decimal is greater than Next, compare and Again, a zero will be written after for convenience.
An applet that compares 0.60 and 0.62 and concludes that 0.62 is greater
The decimal is greater than Consequently, it is also greater than
Recall which initial numbers these decimals correspond to.
Therefore, Kevin, who has finished of the homework, has completed the greatest part of the homework.



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