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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 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.
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.
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.
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.

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.
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.
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.
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

### 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.

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?

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.

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.
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

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.

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.