Decimal Concepts and Fraction Connections

Connecting Tenths to Fractions and Decimals

Sometimes a whole is split into equal parts. When a whole is split into 10 equal parts, each part is called one tenth.

Tenths can be written in two ways:

• as a fraction: \(\frac{1}{10}\)
• as a decimal: \(0.1\)

This means:

$$\frac{1}{10} = 0.1$$

In this lesson, you will learn how tenths connect to fractions and decimals, and how to read, write, and understand them.

1. What is a tenth?

Think about a bar, a strip, or a number line from 0 to 1. If it is divided into 10 equal parts, each small part is one tenth of the whole.

If you have 1 out of 10 equal parts, you have:

$$\frac{1}{10}$$

If you have 3 out of 10 equal parts, you have:

$$\frac{3}{10}$$

Fractions with 10 equal parts are called tenths.

2. How decimals show tenths

A decimal is another way to write part of a whole. The decimal point separates whole numbers from parts that are smaller than 1.

In the decimal \(0.1\):

• The \(0\) means there are 0 wholes.
• The \(1\) is in the tenths place.

So \(0.1\) means 1 tenth.

Here are more tenths written as fractions and decimals:

• \(\frac{1}{10} = 0.1\)
• \(\frac{2}{10} = 0.2\)
• \(\frac{3}{10} = 0.3\)
• \(\frac{4}{10} = 0.4\)
• \(\frac{5}{10} = 0.5\)
• \(\frac{6}{10} = 0.6\)
• \(\frac{7}{10} = 0.7\)
• \(\frac{8}{10} = 0.8\)
• \(\frac{9}{10} = 0.9\)

3. Reading tenths

It is important to read decimals correctly.

• \(0.1\) is read as one tenth.
• \(0.4\) is read as four tenths.
• \(0.9\) is read as nine tenths.

Do not read \(0.4\) as “zero point four” only. In math, it is helpful to say the place value name: four tenths.

4. Using a place value chart

A place value chart helps show where each digit belongs.

For the number \(0.7\):

$$ \begin{array}{c|c} \text{Ones} & \text{Tenths} \\ \hline 0 & 7 \end{array} $$

This chart shows there are 0 ones and 7 tenths.

That means:

$$0.7 = \frac{7}{10}$$

5. Connecting models, fractions, and decimals

You can think about tenths in different ways:

• a picture split into 10 equal parts
• a fraction with denominator 10
• a decimal with one digit in the tenths place

These all match each other.

For example, if 6 out of 10 equal parts are shaded:

• fraction: \(\frac{6}{10}\)
• decimal: \(0.6\)
• words: six tenths

6. Tenths on a number line

A number line can also show tenths. The space from 0 to 1 can be split into 10 equal parts.

Each jump is one tenth:

$$0,\ 0.1,\ 0.2,\ 0.3,\ 0.4,\ 0.5,\ 0.6,\ 0.7,\ 0.8,\ 0.9,\ 1.0$$

This helps us see that tenths are parts of one whole. As the decimal gets bigger, it moves closer to 1.

Worked Example 1: Write a fraction as a decimal

Write \(\frac{3}{10}\) as a decimal.

Step 1: The denominator is 10, so we are talking about tenths.

Step 2: The numerator is 3, so we have 3 tenths.

Answer:

$$\frac{3}{10} = 0.3$$

Worked Example 2: Write a decimal as a fraction

Write \(0.8\) as a fraction.

Step 1: The 8 is in the tenths place.

Step 2: So the number means 8 tenths.

Answer:

$$0.8 = \frac{8}{10}$$

Worked Example 3: Use words, fraction, and decimal

Write five tenths as a fraction and as a decimal.

Step 1: “Five tenths” means 5 parts out of 10.

Fraction:

$$\frac{5}{10}$$

Step 2: As a decimal, 5 is in the tenths place.

Decimal:

$$0.5$$

Answer:

$$\text{five tenths} = \frac{5}{10} = 0.5$$

Worked Example 4: Think about a model

A rectangle is split into 10 equal parts. 9 parts are shaded. Write the shaded amount as a fraction and a decimal.

Step 1: There are 9 shaded parts out of 10 total parts.

Fraction:

$$\frac{9}{10}$$

Step 2: 9 tenths as a decimal is \(0.9\).

Answer:

$$\frac{9}{10} = 0.9$$

7. Helpful patterns to notice

• The denominator 10 means the number is in tenths.
• In decimals with tenths, the digit to the right of the decimal point tells how many tenths there are.
• \(0.1\) is 1 tenth, \(0.2\) is 2 tenths, and the pattern keeps going.
• \(1.0\) means 10 tenths, which makes 1 whole.

For example:

$$\frac{10}{10} = 1.0 = 1$$

8. Common mistakes to avoid

• Mistake: Thinking \(0.1\) means 1 whole.
  Remember: It means 1 tenth, which is less than 1.
• Mistake: Forgetting that the first place after the decimal point is the tenths place.
  Remember: The digit right after the decimal shows tenths.
• Mistake: Mixing up \(\frac{1}{10}\) and \(\frac{10}{1}\).
  Remember: \(\frac{1}{10}\) is a small part of a whole.

Summary

When a whole is divided into 10 equal parts, each part is one tenth.

One tenth can be written as a fraction and as a decimal:

$$\frac{1}{10} = 0.1$$

Any number of tenths can be shown in these matching ways:

• \(\frac{4}{10} = 0.4\) means four tenths
• \(\frac{7}{10} = 0.7\) means seven tenths
• \(\frac{9}{10} = 0.9\) means nine tenths

Fractions, decimals, words, pictures, and number lines can all help you understand tenths. They are just different ways to show the same amount.

Connecting Hundredths to Fractions and Decimals

Sometimes a whole is split into many tiny equal parts. When a whole is divided into 100 equal parts, each part is called a hundredth.

Hundredths can be written in three connected ways:

• as words: hundredths
• as a fraction: for example, \(\frac{37}{100}\)
• as a decimal: for example, \(0.37\)

In this lesson, you will learn how fractions with a denominator of 100 connect to decimals, and how a 10 by 10 grid can help you see the amount.

What does a 10 by 10 grid show?

A 10 by 10 grid has 10 rows and 10 columns. That means it has:

$$10 \times 10 = 100$$

So the whole grid stands for 1 whole, and each small square stands for 1 hundredth, or \(\frac{1}{100}\).

If you shade some of the 100 small squares, the shaded part can be written as a fraction out of 100 and as a decimal.

For example:

• 1 shaded square = \(\frac{1}{100} = 0.01\)
• 10 shaded squares = \(\frac{10}{100} = 0.10\)
• 45 shaded squares = \(\frac{45}{100} = 0.45\)

Understanding the decimal places

Decimals show parts of a whole. The digits to the right of the decimal point have special place values.

• The first place to the right is the tenths place.
• The second place to the right is the hundredths place.

So in the decimal \(0.37\):

• the 3 means 3 tenths
• the 7 means 7 hundredths

This means \(0.37\) is 37 hundredths altogether.

$$0.37 = \frac{37}{100}$$

How tenths and hundredths work together

One tenth is bigger than one hundredth. That is because:

$$\frac{1}{10} = \frac{10}{100}$$

This means 1 tenth is the same as 10 hundredths.

On a 10 by 10 grid, one full row of 10 small squares is \(\frac{10}{100}\), which is also \(\frac{1}{10}\), or \(0.1\).

So if you see 2 full rows shaded, that means:

$$\frac{20}{100} = \frac{2}{10} = 0.20 = 0.2$$

The zero in \(0.20\) shows 0 hundredths left over after 2 tenths. Both \(0.20\) and \(0.2\) mean the same amount.

A helpful pattern

Fractions with 100 on the bottom connect very neatly to decimals.

• \(\frac{4}{100} = 0.04\)
• \(\frac{19}{100} = 0.19\)
• \(\frac{80}{100} = 0.80\)
• \(\frac{100}{100} = 1.00\)

Notice that the number of hundredths becomes the digits after the decimal point.

If the fraction is \(\frac{6}{100}\), the decimal is not \(0.6\). It is:

$$\frac{6}{100} = 0.06$$

That is because 6 hundredths means 0 tenths and 6 hundredths.

Worked Example 1: From a shaded grid to a fraction and decimal

A 10 by 10 grid has 23 shaded squares.

Step 1: Write the fraction.

$$\frac{23}{100}$$

Step 2: Write the decimal.

$$0.23$$

Answer: 23 shaded squares represent \(\frac{23}{100}\) or \(0.23\).

Worked Example 2: From a fraction to a decimal

Write \(\frac{7}{100}\) as a decimal.

Since the fraction is 7 hundredths, the 7 goes in the hundredths place.

$$\frac{7}{100} = 0.07$$

Answer: \(\frac{7}{100}\) is \(0.07\).

Worked Example 3: From a decimal to a fraction

Write \(0.48\) as a fraction with denominator 100.

The decimal \(0.48\) means 48 hundredths.

$$0.48 = \frac{48}{100}$$

Answer: \(0.48\) is \(\frac{48}{100}\).

Worked Example 4: Thinking with tenths and hundredths

A grid shows 3 full rows shaded and 6 more small squares shaded.

Each full row has 10 squares, so 3 full rows means:

$$3 \times 10 = 30$$

Then add the 6 more squares:

$$30 + 6 = 36$$

So 36 out of 100 squares are shaded.

$$\frac{36}{100} = 0.36$$

You can also think of this as 3 tenths and 6 hundredths.

Important ideas to remember

• A whole split into 100 equal parts gives hundredths.
• A 10 by 10 grid has 100 small squares, so it is a great model for hundredths.
• Fractions with denominator 100 match decimals with two digits after the decimal point.
• \(\frac{1}{100} = 0.01\)
• \(\frac{10}{100} = 0.10 = 0.1\)
• \(\frac{37}{100} = 0.37\)

Common mistake to watch for

Be careful not to mix up tenths and hundredths.

• \(0.4\) means 4 tenths, or \(\frac{40}{100}\)
• \(0.04\) means 4 hundredths, or \(\frac{4}{100}\)

These are not the same. The 4 in \(0.4\) is in the tenths place, but the 4 in \(0.04\) is in the hundredths place.

Quick practice questions

1. Write \(\frac{15}{100}\) as a decimal.
2. Write \(0.62\) as a fraction with denominator 100.
3. A grid has 9 shaded squares. Write the fraction and decimal.
4. A grid has 54 shaded squares. Write the fraction and decimal.

Answers

1. \(0.15\)
2. \(\frac{62}{100}\)
3. \(\frac{9}{100}\) and \(0.09\)
4. \(\frac{54}{100}\) and \(0.54\)

Summary

Hundredths are parts of a whole when the whole is split into 100 equal pieces. A 10 by 10 grid helps you see these 100 parts clearly. Fractions with denominator 100 and decimals with two places after the decimal point show the same amount, like \(\frac{25}{100} = 0.25\).

Decimal Place Value Structure helps us understand numbers that are smaller than 1. Decimals are part of our base-ten number system, just like ones, tens, and hundreds.

When we write a decimal, we use a decimal point. The decimal point separates whole numbers from parts of a whole.

For example, in the number \(3.45\), the \(3\) is a whole number. The digits after the decimal point show parts of 1.

Place value continues in a pattern. To the left of the decimal point, the value gets 10 times bigger each step. To the right of the decimal point, the value gets 10 times smaller each step.

Here is a simple place value chart:

$$ \begin{array}{c|c|c|c|c} \text{Tens} & \text{Ones} & . & \text{Tenths} & \text{Hundredths} \\ 10 & 1 & . & \frac{1}{10} & \frac{1}{100} \end{array} $$

This shows that:

• 1 one is 1 whole.
• 1 tenth is \(\frac{1}{10}\) of a whole.
• 1 hundredth is \(\frac{1}{100}\) of a whole.

Decimals are closely connected to fractions.

• \(0.1 = \frac{1}{10}\)
• \(0.2 = \frac{2}{10}\)
• \(0.01 = \frac{1}{100}\)
• \(0.25 = \frac{25}{100}\)

Tenths are larger than hundredths because a tenth means a whole is split into 10 equal parts, and a hundredth means a whole is split into 100 equal parts.

Think about money. One dollar is one whole. One dime is \(\frac{1}{10}\) of a dollar, or \(\$0.10\). One penny is \(\frac{1}{100}\) of a dollar, or \(\$0.01\).

This means:

• \(0.1\) is the same as \(0.10\)
• \(\frac{1}{10}\) is the same amount as \(\frac{10}{100}\)

Adding a zero at the end of a decimal does not change its value if the zero is to the right of the last digit. So \(0.5 = 0.50\).

Reading decimals is important.

• \(0.7\) is read as seven tenths.
• \(0.34\) is read as thirty-four hundredths.
• \(2.8\) is read as two and eight tenths.
• \(5.06\) is read as five and six hundredths.

Notice that each digit has a value based on its place.

In \(4.27\):

• The \(4\) is in the ones place, so it means 4 ones.
• The \(2\) is in the tenths place, so it means 2 tenths, or \(\frac{2}{10}\).
• The \(7\) is in the hundredths place, so it means 7 hundredths, or \(\frac{7}{100}\).

So we can write:

$$ 4.27 = 4 + \frac{2}{10} + \frac{7}{100} $$

Now let’s work through some examples.

Example 1: What value does the 6 have in \(0.6\)?

The 6 is in the tenths place. So its value is 6 tenths.

$$ 0.6 = \frac{6}{10} $$

Answer: The 6 means 6 tenths.

Example 2: Write \(0.43\) as a fraction using tenths and hundredths.

The 4 is in the tenths place, so it means \(\frac{4}{10}\).

The 3 is in the hundredths place, so it means \(\frac{3}{100}\).

$$ 0.43 = \frac{4}{10} + \frac{3}{100} $$

We can also say:

$$ 0.43 = \frac{43}{100} $$

Answer: \(0.43\) is 4 tenths and 3 hundredths, or 43 hundredths.

Example 3: Which is greater, \(0.5\) or \(0.05\)?

\(0.5\) means 5 tenths.

\(0.05\) means 5 hundredths.

A tenth is bigger than a hundredth, so 5 tenths is greater than 5 hundredths.

$$ 0.5 > 0.05 $$

Answer: \(0.5\) is greater.

Example 4: What number is shown by 3 ones, 2 tenths, and 5 hundredths?

Start with 3 ones: \(3\)

Add 2 tenths: \(0.2\)

Add 5 hundredths: \(0.05\)

$$ 3 + 0.2 + 0.05 = 3.25 $$

Answer: The number is \(3.25\).

Here are some important ideas to remember:

• The decimal point separates whole numbers from parts of a whole.
• The first place to the right of the decimal is tenths.
• The second place to the right of the decimal is hundredths.
• \(0.1 = \frac{1}{10}\)
• \(0.01 = \frac{1}{100}\)
• Tenths are bigger than hundredths.
• Each place to the right is 10 times smaller.

Quick Check

1. In \(2.4\), what place is the 4 in?
2. In \(0.18\), what does the 8 mean?
3. Write \(0.7\) as a fraction.
4. Which is greater: \(0.9\) or \(0.09\)?

Answers:

1. The 4 is in the tenths place.
2. The 8 means 8 hundredths.
3. \(0.7 = \frac{7}{10}\)
4. \(0.9\) is greater.

Summary

Decimals show parts of a whole in our base-ten system. The places to the right of the decimal point are tenths and hundredths. Knowing the value of each place helps you read, write, compare, and understand decimals correctly.

Representing Decimals on Grids and Number Lines

Decimals are numbers that show parts of a whole. They help us describe amounts smaller than 1.

In 4th grade, a very important idea is that decimals are connected to fractions. For example, one tenth can be written as the fraction \(\frac{1}{10}\), and it can also be written as the decimal \(0.1\).

When we represent decimals on grids and number lines, we are showing exactly how much of a whole we have. This helps us see where a decimal belongs and what it means.

Big idea: decimals are based on tens. The first place to the right of the decimal point is the tenths place.

Here are some examples of tenths:

• \(0.1 = \frac{1}{10}\)
• \(0.2 = \frac{2}{10}\)
• \(0.5 = \frac{5}{10}\)
• \(0.9 = \frac{9}{10}\)

Let’s learn how to show these decimals in two ways.

1. Representing decimals on a grid

A grid can show one whole split into equal parts. For tenths, we often use a rectangle or strip divided into 10 equal parts.

If the whole is divided into 10 equal parts, then each part is one tenth.

So if 3 out of the 10 parts are shaded, that means:

$$\frac{3}{10} = 0.3$$

This means the decimal tells how many tenths are shaded.

Here is how to think about it:

• Count the total equal parts in the whole: 10
• Count the shaded parts
• Write that amount as tenths
• Then write the decimal

For example, if 7 of the 10 parts are shaded:

$$\frac{7}{10} = 0.7$$

2. Representing decimals on a number line

A number line shows numbers in order from least to greatest. Decimals between 0 and 1 can be placed on a number line by dividing the space from 0 to 1 into 10 equal parts.

Each jump on that number line is one tenth.

So the marks would be:

• 0
• \(0.1\)
• \(0.2\)
• \(0.3\)
• \(0.4\)
• \(0.5\)
• \(0.6\)
• \(0.7\)
• \(0.8\)
• \(0.9\)
• 1

This means \(0.6\) is the sixth tick mark after 0 if the line from 0 to 1 is divided into 10 equal parts.

Important connection:

Decimals on a number line are placed by their value. A larger decimal is farther to the right.

For example:

• \(0.2\) is to the left of \(0.5\)
• \(0.8\) is to the right of \(0.3\)

How grids and number lines are alike

• Both show parts of a whole
• Both help us see tenths clearly
• Both connect decimals to fractions

How they are different

• A grid shows parts shaded in a whole shape
• A number line shows where a decimal is located between numbers

Worked Example 1: Represent a decimal on a grid

A strip is divided into 10 equal boxes. 4 boxes are shaded. What decimal does this show?

1. There are 10 equal parts, so we are working with tenths.
2. 4 parts are shaded, so the fraction is \(\frac{4}{10}\).
3. The decimal for \(\frac{4}{10}\) is \(0.4\).

Answer: The grid shows \(0.4\).

Worked Example 2: Represent a decimal on a number line

Where should \(0.7\) go on a number line from 0 to 1?

1. Divide the space from 0 to 1 into 10 equal parts.
2. Each part is one tenth.
3. \(0.7\) means 7 tenths.
4. Start at 0 and count 7 equal jumps to the right.

Answer: \(0.7\) is at the seventh tick mark after 0.

Worked Example 3: Connect a fraction, decimal, and location

A point is placed at the third mark after 0 on a number line from 0 to 1 that is divided into 10 equal parts. What fraction and decimal does it represent?

1. The line is divided into 10 equal parts, so each mark is one tenth.
2. The third mark after 0 is 3 tenths.
3. The fraction is \(\frac{3}{10}\).
4. The decimal is \(0.3\).

Answer: The point represents \(\frac{3}{10}\) and \(0.3\).

Worked Example 4: Compare decimals using a number line

Which decimal is greater: \(0.4\) or \(0.9\)?

1. On a number line, numbers farther right are greater.
2. \(0.4\) is 4 tenths.
3. \(0.9\) is 9 tenths.
4. 9 tenths is more than 4 tenths.

Answer: \(0.9\) is greater than \(0.4\).

Tips for success

• Look carefully at how many equal parts the whole is split into.
• If there are 10 equal parts, think in tenths.
• Count shaded parts on grids carefully.
• On number lines, count equal spaces, not just marks quickly.
• Remember that decimals get larger as you move right on the number line.

Common mistakes to avoid

• Mixing up the number of shaded parts with the total number of parts
• Forgetting that \(0.5\) means 5 tenths, not 50
• Placing decimals unevenly on a number line instead of using equal spacing
• Thinking \(0.10\) is different from \(0.1\); both mean one tenth

Quick check questions

• If 9 out of 10 parts are shaded on a grid, what decimal is shown? \(0.9\)
• What fraction matches \(0.2\)? \(\frac{2}{10}\)
• On a number line from 0 to 1 divided into 10 equal parts, where is \(0.1\)? the first mark after 0
• Which is farther right: \(0.6\) or \(0.3\)? \(0.6\)

Summary

Decimals show parts of a whole, and in this lesson we used tenths. On a grid, the decimal tells how many of the 10 equal parts are shaded. On a number line, the decimal tells how far a point is from 0 when the space to 1 is split into 10 equal parts.

Remember these connections:

• \(0.1 = \frac{1}{10}\)
• \(0.5 = \frac{5}{10}\)
• \(0.8 = \frac{8}{10}\)

If you can count tenths on a grid and on a number line, you can represent decimals correctly.

Translating Between Fractions and Decimals

Sometimes a number less than 1 can be written in two different ways: as a fraction or as a decimal. In this lesson, we will learn how to move back and forth between these forms.

We will only use fractions with denominators of 10 and 100. These are perfect for decimals because our number system is based on tens.

Big idea: Fractions and decimals can name the same amount.

For example, \(\frac{3}{10}\) and \(0.3\) mean the same thing.

Also, \(\frac{27}{100}\) and \(0.27\) mean the same thing.

Understanding tenths and hundredths

When a whole is split into 10 equal parts, each part is called one tenth. We write that as \(\frac{1}{10}\), and as a decimal it is \(0.1\).

When a whole is split into 100 equal parts, each part is called one hundredth. We write that as \(\frac{1}{100}\), and as a decimal it is \(0.01\).

The first place to the right of the decimal point is the tenths place. The second place is the hundredths place.

So:

• \(0.4\) means 4 tenths, or \(\frac{4}{10}\)
• \(0.08\) means 8 hundredths, or \(\frac{8}{100}\)
• \(0.56\) means 56 hundredths, or \(\frac{56}{100}\)

How to change a fraction into a decimal

If the denominator is 10, write the numerator in the tenths place.

Examples:

• \(\frac{7}{10} = 0.7\)
• \(\frac{2}{10} = 0.2\)

If the denominator is 100, write the numerator in the hundredths place.

Examples:

• \(\frac{9}{100} = 0.09\)
• \(\frac{45}{100} = 0.45\)

Important: Sometimes you need a zero to hold a place.

For example, \(\frac{6}{100}\) is not \(0.6\). It is \(0.06\), because 6 is in the hundredths place.

How to change a decimal into a fraction

Look at the last digit in the decimal.

• If there is 1 digit after the decimal point, write a fraction with denominator 10.
• If there are 2 digits after the decimal point, write a fraction with denominator 100.

Examples:

• \(0.5 = \frac{5}{10}\)
• \(0.12 = \frac{12}{100}\)
• \(0.3 = \frac{3}{10}\)
• \(0.70 = \frac{70}{100}\)

Remember: A zero at the end of a decimal can still matter when you are thinking about place value. For example, \(0.7\) means 7 tenths, and \(0.70\) means 70 hundredths. These are equal amounts.

We can show that with fractions too:

$$ \frac{7}{10} = \frac{70}{100} $$

Worked Example 1

Change \(\frac{4}{10}\) into a decimal.

1. The denominator is 10, so we use the tenths place.
2. The numerator is 4, so write 4 in the tenths place.

Answer:

$$ \frac{4}{10} = 0.4 $$

Worked Example 2

Change \(\frac{23}{100}\) into a decimal.

1. The denominator is 100, so we use the hundredths place.
2. The numerator is 23, so write 23 as hundredths.

Answer:

$$ \frac{23}{100} = 0.23 $$

Worked Example 3

Change \(0.8\) into a fraction.

1. There is 1 digit after the decimal point.
2. That means the number is in tenths.
3. Write 8 over 10.

Answer:

$$ 0.8 = \frac{8}{10} $$

Worked Example 4

Change \(0.06\) into a fraction.

1. There are 2 digits after the decimal point.
2. That means the number is in hundredths.
3. Write 6 hundredths as a fraction.

Answer:

$$ 0.06 = \frac{6}{100} $$

Watch out for these common mistakes

• Do not say \(0.4 = \frac{4}{100}\). Since 4 is in the tenths place, \(0.4 = \frac{4}{10}\).
• Do not say \(\frac{5}{100} = 0.5\). Five hundredths is \(0.05\), not \(0.5\).
• Count the digits after the decimal carefully.

Helpful thinking trick

You can read decimals by place value.

• \(0.2\) is “2 tenths”
• \(0.14\) is “14 hundredths”
• \(0.09\) is “9 hundredths”

Then write the matching fraction:

• 2 tenths = \(\frac{2}{10}\)
• 14 hundredths = \(\frac{14}{100}\)
• 9 hundredths = \(\frac{9}{100}\)

Quick practice ideas

• \(\frac{9}{10} = 0.9\)
• \(\frac{31}{100} = 0.31\)
• \(0.7 = \frac{7}{10}\)
• \(0.25 = \frac{25}{100}\)

Summary

Fractions with denominators of 10 and 100 match decimals very well.

• Tenths match decimals with 1 digit after the decimal point.
• Hundredths match decimals with 2 digits after the decimal point.
• A zero can be important, like in \(0.06\), which means \(\frac{6}{100}\).

When you know the place value, you can translate between fractions and decimals easily.

Equivalent Decimals are decimals that look different but have the same value.

For example, \(0.5\) and \(0.50\) are equivalent decimals. They are written with a different number of digits, but they mean the same amount.

In this lesson, you will learn how to see equivalent decimals with place value, with fractions, and with simple visual thinking.

First, remember decimal place value.

• The first place to the right of the decimal point is the tenths place.
• The second place to the right of the decimal point is the hundredths place.

So in \(0.5\), the 5 means 5 tenths.

We can write that as a fraction:

$$0.5 = \frac{5}{10}$$

In \(0.50\), the 5 is in the tenths place and the 0 is in the hundredths place. That means 50 hundredths.

We can write that as a fraction:

$$0.50 = \frac{50}{100}$$

Now compare the fractions:

$$\frac{5}{10} = \frac{50}{100}$$

Since these fractions are equal, the decimals are also equal:

$$0.5 = 0.50$$

Why does adding a zero at the end not change the value?

When a zero is added to the end of a decimal, the amount does not get bigger. It is like naming the same amount in a different way.

Think about money. A half dollar can be written as \(\$0.5\) dollars or \(\$0.50\) dollars. Both mean 50 cents.

So the zero at the end is a placeholder. It shows there are 0 extra hundredths, but the value stays the same.

Visual model idea

Imagine a square divided into 10 equal strips. If 5 strips are shaded, that is \(5\) tenths:

$$\frac{5}{10} = 0.5$$

Now imagine the same whole square divided into 100 tiny equal parts. If 50 tiny parts are shaded, that is \(50\) hundredths:

$$\frac{50}{100} = 0.50$$

The shaded area is the same amount of the whole. The picture does not get larger or smaller. Only the way we name the amount changes.

This is why the lesson says that \(0.5\) exactly equals \(0.50\) without changing spatial volume. The space covered is the same.

Important rule:

• Adding zeros to the right end of a decimal does not change its value.
• Removing zeros from the right end of a decimal does not change its value either.

For example:

• \(0.5 = 0.50 = 0.500\)
• \(0.7 = 0.70\)
• \(0.23 = 0.230\)

Be careful! Zeros do not always leave the value unchanged.

If you put a zero in a different place, the value can change.

• \(0.5 = 0.50\) because the zero is added at the end.
• But \(0.5 \ne 0.05\) because now the 5 moved from the tenths place to the hundredths place.

Look at the fractions:

$$0.5 = \frac{5}{10}$$

$$0.05 = \frac{5}{100}$$

Since \(\frac{5}{10}\) and \(\frac{5}{100}\) are not equal, \(0.5\) and \(0.05\) are not equal.

Worked Example 1

Show why \(0.3\) and \(0.30\) are equivalent.

1. Write each decimal as a fraction.

$$0.3 = \frac{3}{10}$$

$$0.30 = \frac{30}{100}$$

2. Compare the fractions.

$$\frac{3}{10} = \frac{30}{100}$$

So:

$$0.3 = 0.30$$

Worked Example 2

Are \(0.8\) and \(0.80\) equal?

1. Read the place values.
2. \(0.8\) means 8 tenths.
3. \(0.80\) means 80 hundredths.

Now write them as fractions:

$$0.8 = \frac{8}{10}$$

$$0.80 = \frac{80}{100}$$

These fractions are equal, so the decimals are equal.

$$0.8 = 0.80$$

Worked Example 3

Are \(0.4\) and \(0.04\) equivalent decimals?

Write each one as a fraction:

$$0.4 = \frac{4}{10}$$

$$0.04 = \frac{4}{100}$$

These are not the same fraction. \(4\) tenths is much larger than \(4\) hundredths.

So:

$$0.4 \ne 0.04$$

These are not equivalent decimals.

Worked Example 4

Explain why \(0.25\) and \(0.250\) are equivalent.

1. \(0.25\) means 25 hundredths.
2. \(0.250\) means 250 thousandths.

Both names describe the same part of one whole. Adding the zero at the end does not change the amount.

So:

$$0.25 = 0.250$$

How to check if decimals are equivalent

• Look to see whether a zero was added or removed only at the end.
• Think about the place value of the digits.
• Write the decimals as fractions if needed.
• Ask: Do they name the same part of the whole?

Try thinking about these:

• \(0.6\) and \(0.60\) → equivalent
• \(0.12\) and \(0.120\) → equivalent
• \(0.9\) and \(0.09\) → not equivalent

Summary

Equivalent decimals are decimals that have the same value even if they are written differently. Adding a zero to the right end of a decimal does not change the amount. This is because the decimal still names the same part of the whole, just like \(0.5 = 0.50\) and \(\frac{5}{10} = \frac{50}{100}\).

Comparing Decimals using Place Value

Decimals are numbers that show parts of a whole. They help us write numbers that are less than 1, or numbers that have a whole number part and a part of a whole.

When we compare decimals, we are deciding which decimal is greater, which is less, or if they are equal. We use place value to do this carefully.

Remember these comparison symbols:

• 6gt; means greater than
• 6lt; means less than
• = means equal to

For example, in \(0.7 \gt 0.5\), the decimal \(0.7\) is greater than \(0.5\).

Understanding decimal place value

Place value tells us what each digit is worth. In decimals, the places to the right of the decimal point are parts of one whole.

Here are some important decimal places:

• Ones
• Tenths
• Hundredths

For example, in \(3.48\):

• The \(3\) is in the ones place
• The \(4\) is in the tenths place
• The \(8\) is in the hundredths place

We can also connect decimals to fractions:

• \(0.4 = \frac{4}{10}\)
• \(0.08 = \frac{8}{100}\)
• \(0.35 = \frac{35}{100}\)

This helps us understand that tenths are bigger pieces than hundredths. One tenth is larger than one hundredth.

How to compare decimals

To compare decimals using place value, follow these steps:

1. Line up the decimal points.
2. Compare the digits from left to right.
3. Start with the greatest place value.
4. If the digits are the same, move to the next place to the right.
5. Keep going until you find a difference.

You can think: Compare the biggest place first.

Main idea: The first place where the digits are different tells you which number is greater.

Example 1: Compare decimals with tenths

Compare \(0.6\) and \(0.4\).

Both numbers have \(0\) ones. So we look at the tenths.

• \(0.6\) has 6 tenths
• \(0.4\) has 4 tenths

Since 6 tenths is greater than 4 tenths, we know:

$$0.6 \gt 0.4$$

As fractions, this is \(\frac{6}{10} \gt \frac{4}{10}\), which also shows why \(0.6\) is greater.

Example 2: Compare decimals with the same tenths

Compare \(0.35\) and \(0.32\).

First, line up the decimal points:

$$0.35 \qquad 0.32$$

Compare each place from left to right:

• Ones: both have \(0\)
• Tenths: both have \(3\)
• Hundredths: one has \(5\), the other has \(2\)

The tenths are the same, so we compare the hundredths. Since \(5 \gt 2\), we know 35 hundredths is greater than 32 hundredths.

$$0.35 \gt 0.32$$

As fractions, \(0.35 = \frac{35}{100}\) and \(0.32 = \frac{32}{100}\). Since \(35 \gt 32\), \(0.35\) is greater.

Example 3: Compare a number with ones and tenths

Compare \(2.4\) and \(2.39\).

It can help to write \(2.4\) as \(2.40\). Adding a zero at the end does not change the value.

$$2.4 = 2.40$$

Now compare:

$$2.40 \qquad 2.39$$

• Ones: both have \(2\)
• Tenths: both have \(4\) and \(3\)? Wait carefully.

Let us line them up correctly:

• \(2.40\): 2 ones, 4 tenths, 0 hundredths
• \(2.39\): 2 ones, 3 tenths, 9 hundredths

We compare the tenths first:

• \(2.40\) has 4 tenths
• \(2.39\) has 3 tenths

Since 4 tenths is greater than 3 tenths, we know:

$$2.4 \gt 2.39$$

Even though 9 hundredths sounds big, tenths are a bigger place than hundredths. We must compare the larger place value first.

Example 4: Compare decimals with the same ones and tenths

Compare \(4.07\) and \(4.7\).

Write \(4.7\) as \(4.70\) so both numbers have hundredths.

$$4.07 \qquad 4.70$$

Now compare from left to right:

• Ones: both have \(4\)
• Tenths: \(0\) and \(7\)

Since \(0 \lt 7\), we know:

$$4.07 \lt 4.7$$

This is an important reminder: \(4.07\) is not greater just because 7 is greater than 0. We must look at the place value. In \(4.07\), the 7 is in the hundredths place. In \(4.7\), the 7 is in the tenths place, and tenths are greater than hundredths.

Helpful tips

• Line up decimal points before comparing.
• Start on the left with the greatest place value.
• If needed, add zeros to the end of a decimal. This does not change the value. For example, \(0.5 = 0.50\).
• Do not compare decimals by just looking at the number of digits.

A common mistake

Some students think \(0.9 \lt 0.12\) because 12 is bigger than 9. But that is not how decimals work.

Write \(0.9\) as \(0.90\):

$$0.90 \qquad 0.12$$

Now compare:

• Tenths: \(9\) tenths and \(1\) tenth

Since \(9 \gt 1\), we know:

$$0.9 \gt 0.12$$

Why place value matters

Each place in a decimal has a different value. A digit in the tenths place is worth more than the same digit in the hundredths place.

For example:

• \(0.5\) means 5 tenths
• \(0.05\) means 5 hundredths

Since tenths are larger than hundredths:

$$0.5 \gt 0.05$$

Try this thinking

When comparing decimals, ask yourself:

• Do the ones match?
• If yes, do the tenths match?
• If yes, do the hundredths match?
• Where is the first place that is different?

That first difference tells which number is greater.

Summary

To compare decimals, line up the decimal points and compare digits from left to right. Start with the ones place, then tenths, then hundredths. The first place where the digits are different shows which decimal is greater or less.

Remember that place value is very important. A digit in the tenths place is worth more than a digit in the hundredths place. You can also add zeros to the end of a decimal to help compare, like \(0.6 = 0.60\).

Ordering Decimals means putting decimals in order from least to greatest or from greatest to least.

Decimals are numbers that show parts of a whole. They are connected to fractions. For example, \(0.4\) means \(\frac{4}{10}\), and \(0.25\) means \(\frac{25}{100}\).

When we order decimals, we compare the value of each number carefully. A longer decimal is not always greater. We must look at each place value.

Place value helps us compare decimals:

• Ones place is to the left of the decimal point.
• Tenths place is the first digit to the right of the decimal point.
• Hundredths place is the second digit to the right of the decimal point.

For example, in \(0.37\):

• \(3\) is in the tenths place, so it means \(\frac{3}{10}\)
• \(7\) is in the hundredths place, so it means \(\frac{7}{100}\)

A very helpful trick is to make the decimals the same length by adding zeros at the end. This does not change the value.

For example:

\(0.4 = 0.40\)

\(0.7 = 0.70\)

\(0.25 = 0.25\)

Adding a zero at the end is like saying:

$$0.4 = \frac{4}{10} = \frac{40}{100} = 0.40$$

This helps us compare decimals fairly.

Steps for ordering decimals:

1. Write the decimals in a list.
2. Make sure they have the same number of decimal places by adding zeros if needed.
3. Compare from left to right.
4. Look at the ones place first, then tenths, then hundredths.
5. Put them in the order asked for.

Worked Example 1: Order from least to greatest

Order: \(0.4, 0.7, 0.2\)

These decimals all have tenths only, so we compare the tenths digits:

• \(0.4\) has 4 tenths
• \(0.7\) has 7 tenths
• \(0.2\) has 2 tenths

From least to greatest:

$$0.2,\ 0.4,\ 0.7$$

Worked Example 2: Order from least to greatest

Order: \(0.5, 0.35, 0.8\)

First, make them the same length:

\(0.5 = 0.50\)

\(0.35 = 0.35\)

\(0.8 = 0.80\)

Now compare:

• \(0.35\) has 3 tenths
• \(0.50\) has 5 tenths
• \(0.80\) has 8 tenths

So the order is:

$$0.35,\ 0.5,\ 0.8$$

Worked Example 3: Order from greatest to least

Order: \(0.62, 0.6, 0.59\)

Make them the same length:

\(0.62 = 0.62\)

\(0.6 = 0.60\)

\(0.59 = 0.59\)

Now compare from left to right:

• All have 0 ones.
• In the tenths place, \(0.62\) and \(0.60\) have 6 tenths, but \(0.59\) has 5 tenths.
• So \(0.59\) is the smallest.
• Now compare \(0.62\) and \(0.60\): both have 6 tenths.
• Look at the hundredths place: 2 hundredths is greater than 0 hundredths.

From greatest to least:

$$0.62,\ 0.6,\ 0.59$$

Worked Example 4: Order from least to greatest

Order: \(1.05, 0.95, 1.5, 1.05\)

Make the decimals the same length:

\(1.05 = 1.05\)

\(0.95 = 0.95\)

\(1.5 = 1.50\)

\(1.05 = 1.05\)

Compare the ones place first:

• \(0.95\) has 0 ones.
• The others have 1 one.

So \(0.95\) is the smallest.

Now compare the numbers with 1 one:

• \(1.05\)
• \(1.05\)
• \(1.50\)

In the tenths place, \(1.05\) has 0 tenths and \(1.50\) has 5 tenths, so \(1.50\) is greater.

From least to greatest:

$$0.95,\ 1.05,\ 1.05,\ 1.5$$

Important reminders:

• Line up decimal points when comparing.
• Add zeros at the end if needed.
• Compare place by place from left to right.
• Do not decide by how many digits a decimal has.

Watch out for this mistake:

Some students think \(0.8 < 0.35\) because 35 is bigger than 8. But this is not correct.

Write them with the same number of digits:

\(0.8 = 0.80\)

\(0.35 = 0.35\)

Now compare:

• \(0.80\) has 8 tenths
• \(0.35\) has 3 tenths

Since 8 tenths is greater than 3 tenths, we know:

$$0.8 > 0.35$$

Summary

To order decimals, compare their place values. Start with the ones place, then tenths, then hundredths. If needed, add zeros at the end so the decimals have the same number of places. This makes it easier to see which decimal is greater or smaller.

Adding Tenths and Hundredths

Decimals help us show parts of a whole that are smaller than 1. In this lesson, we will learn how to add tenths and hundredths.

We will also connect decimals to fractions, because decimals and fractions can show the same amount. This helps us understand why the addition works.

A tenth means one part out of 10 equal parts. We can write one tenth as \(\frac{1}{10}\) or as the decimal \(0.1\).

A hundredth means one part out of 100 equal parts. We can write one hundredth as \(\frac{1}{100}\) or as the decimal \(0.01\).

Here are some matching decimals and fractions:

• \(0.1 = \frac{1}{10}\)
• \(0.2 = \frac{2}{10}\)
• \(0.01 = \frac{1}{100}\)
• \(0.25 = \frac{25}{100}\)

Important idea: Tenths can be renamed as hundredths. This helps when we add.

For example, \(\frac{1}{10}\) is the same as \(\frac{10}{100}\). So:

$$0.1 = 0.10$$

Both numbers mean the same amount. The zero at the end does not change the value.

Why do we rename tenths as hundredths? Because it is easier to add when both numbers are written in the same-sized parts.

Think about fractions. It is hard to add \(\frac{3}{10}\) and \(\frac{4}{100}\) until we rename \(\frac{3}{10}\) as hundredths:

$$\frac{3}{10} = \frac{30}{100}$$

Then we can add:

$$\frac{30}{100} + \frac{4}{100} = \frac{34}{100}$$

As decimals, that is:

$$0.30 + 0.04 = 0.34$$

How to add tenths and hundredths

1. Write the decimals so the decimal points line up.
2. If needed, add a zero so tenths become hundredths.
3. Add the hundredths.
4. Add the tenths.
5. Bring down the decimal point into the answer.

When decimal points line up, each place value stays in the correct column.

Worked Example 1: Add two tenths

Add \(0.3 + 0.2\).

These are both tenths:

• \(0.3 = \frac{3}{10}\)
• \(0.2 = \frac{2}{10}\)

Add the fractions:

$$\frac{3}{10} + \frac{2}{10} = \frac{5}{10}$$

So the decimal sum is:

$$0.3 + 0.2 = 0.5$$

Answer: \(0.5\)

Worked Example 2: Add tenths and hundredths

Add \(0.4 + 0.07\).

First, rename \(0.4\) as hundredths:

$$0.4 = 0.40$$

Now add:

$$\begin{array}{r} 0.40 \\ +\,0.07 \\ \hline 0.47 \end{array}$$

Fraction thinking:

• \(0.40 = \frac{40}{100}\)
• \(0.07 = \frac{7}{100}\)

Add:

$$\frac{40}{100} + \frac{7}{100} = \frac{47}{100}$$

Answer: \(0.47\)

Worked Example 3: Add a number with both tenths and hundredths

Add \(0.26 + 0.5\).

Rename \(0.5\) as hundredths:

$$0.5 = 0.50$$

Now line up the decimals and add:

$$\begin{array}{r} 0.26 \\ +\,0.50 \\ \hline 0.76 \end{array}$$

Let us look at the place values:

• Hundredths: \(6 + 0 = 6\)
• Tenths: \(2 + 5 = 7\)

Answer: \(0.76\)

Worked Example 4: Add when the hundredths make a new tenth

Add \(0.38 + 0.27\).

Line up the decimals:

$$\begin{array}{r} 0.38 \\ +\,0.27 \\ \hline \end{array}$$

Add the hundredths first:

$$8 + 7 = 15 \text{ hundredths}$$

\(15\) hundredths is the same as \(1\) tenth and \(5\) hundredths. Write down the \(5\) hundredths and regroup the \(1\) tenth.

Now add the tenths:

$$3 + 2 + 1 = 6 \text{ tenths}$$

So the sum is:

$$\begin{array}{r} 0.38 \\ +\,0.27 \\ \hline 0.65 \end{array}$$

Answer: \(0.65\)

Tips to remember

• Line up the decimal points.
• You can add a zero at the end, like \(0.6 = 0.60\).
• Tenths can be renamed as hundredths.
• Add the same place values together.
• If hundredths add to 10 or more, regroup to the tenths place.

Common mistake

Sometimes students write numbers without lining up the decimal points. For example, this is not correct:

$$\begin{array}{r} 0.4 \\ +\,0.07 \\ \hline 0.11 \end{array}$$

This mistake happens because the digits were not added by place value.

The correct way is to write \(0.4\) as \(0.40\), then add:

$$\begin{array}{r} 0.40 \\ +\,0.07 \\ \hline 0.47 \end{array}$$

Let’s connect one more time to fractions

When you add tenths and hundredths, you are really making the fractions use the same denominator.

For example:

$$0.2 + 0.15$$

Change \(0.2\) to hundredths:

$$0.2 = 0.20 = \frac{20}{100}$$

And \(0.15 = \frac{15}{100}\).

Now add:

$$\frac{20}{100} + \frac{15}{100} = \frac{35}{100}$$

So:

$$0.2 + 0.15 = 0.35$$

Summary

To add tenths and hundredths, line up the decimal points. If needed, rename tenths as hundredths by adding a zero, like \(0.3 = 0.30\).

Then add each place value carefully. Thinking about fractions can help: tenths can become hundredths, and then the parts are easy to combine.

Money is a great way to understand decimals. You already know that a dollar is made of smaller parts, like dimes and pennies. Decimals help us show those parts using numbers.

In this lesson, you will learn how money can be a model for decimals. You will connect dollars to whole numbers, dimes to tenths, and pennies to hundredths.

When we write money, we use a decimal point. The decimal point separates the whole dollars from the parts of a dollar.

For example, in \(\$3.45\):

• The \(3\) means 3 whole dollars.
• The \(4\) in the tenths place means 4 dimes.
• The \(5\) in the hundredths place means 5 pennies.

So \(\$3.45\) means 3 dollars, 4 dimes, and 5 pennies.

Let’s connect money to fractions.

• 1 dollar is one whole.
• 1 dime is \(\frac{1}{10}\) of a dollar, so it is one tenth.
• 1 penny is \(\frac{1}{100}\) of a dollar, so it is one hundredth.

That means:

• \(\$0.1\) or \(\$0.10\) means 1 dime = \(\frac{1}{10}\) of a dollar
• \(\$0.01\) means 1 penny = \(\frac{1}{100}\) of a dollar
• \(\$0.25\) means 25 pennies = 25 hundredths of a dollar

Place value helps us read decimals in money.

Look at this place value chart:

$$ \begin{array}{c|c|c} \text{Ones} & \text{Tenths} & \text{Hundredths} \\ \hline \text{dollars} & \text{dimes} & \text{pennies} \end{array} $$

This means:

• The digit to the left of the decimal point shows the number of whole dollars.
• The first digit to the right of the decimal point shows the number of dimes.
• The second digit to the right of the decimal point shows the number of pennies.

Important idea: Ten dimes make 1 dollar, and 100 pennies make 1 dollar. Also, 10 pennies make 1 dime.

That is why decimals work so well with money. Each place is 10 times smaller as you move to the right.

Worked Example 1: Reading a money decimal

What does \(\$2.37\) mean?

1. The \(2\) is in the ones place, so it means 2 dollars.
2. The \(3\) is in the tenths place, so it means 3 dimes.
3. The \(7\) is in the hundredths place, so it means 7 pennies.

So \(\$2.37\) means 2 dollars, 3 dimes, and 7 pennies.

We can also think of it as:

$$2 + \frac{3}{10} + \frac{7}{100} = 2.37$$

Worked Example 2: Writing money as a decimal

Write 4 dollars, 6 dimes, and 2 pennies as a decimal.

1. 4 dollars means \(4\) in the ones place.
2. 6 dimes means \(6\) in the tenths place.
3. 2 pennies means \(2\) in the hundredths place.

So the decimal is:

$$\$4.62$$

Worked Example 3: When there are no dimes

Write 5 dollars and 8 pennies as a decimal.

There are no dimes, so we need a \(0\) in the tenths place.

That gives us:

$$\$5.08$$

The \(0\) is important. It shows that there are 0 dimes and 8 pennies.

Worked Example 4: Less than one dollar

What decimal matches 7 dimes and 4 pennies?

1. There are 0 whole dollars, so put \(0\) in the ones place.
2. 7 dimes means \(7\) in the tenths place.
3. 4 pennies means \(4\) in the hundredths place.

So the decimal is:

$$\$0.74$$

This is less than 1 dollar because the dollars place has a 0.

Zeroes can help show value clearly.

• \(\$0.50\) means 5 dimes and 0 pennies.
• \(\$0.05\) means 0 dimes and 5 pennies.

These are not the same amount. The 5 is in a different place, so it has a different value.

Compare them:

$$\$0.50 > \$0.05$$

Let’s think carefully about place value.

• In \(\$0.50\), the 5 means 5 tenths, or 5 dimes.
• In \(\$0.05\), the 5 means 5 hundredths, or 5 pennies.

Since 5 dimes is more than 5 pennies, \(\$0.50\) is greater than \(\$0.05\).

Money helps decimals make sense in real life.

If you buy a snack for \(\$1.25\), you know that means 1 dollar and 25 cents. In decimal form, that is 1 whole, 2 tenths, and 5 hundredths.

Here are some helpful connections:

• \(\$1.00\) = 1 whole dollar
• \(\$0.10\) = 1 dime = \(\frac{1}{10}\)
• \(\$0.01\) = 1 penny = \(\frac{1}{100}\)
• \(\$0.99\) = 9 dimes and 9 pennies
• \(\$2.05\) = 2 dollars and 5 pennies

Tips to remember:

• Think of the decimal point as separating dollars from cents.
• The first digit after the decimal is the dimes place.
• The second digit after the decimal is the pennies place.
• If a place has no value, use a 0 to hold that place.

Summary

Money is a useful model for decimals because dollars, dimes, and pennies match decimal place value. Dollars are ones, dimes are tenths, and pennies are hundredths. When you read or write money as a decimal, the position of each digit tells its value.

Rounding Decimals to the Nearest Whole

Sometimes a decimal number is not exactly a whole number. For example, \(3.2\) is between \(3\) and \(4\). When we round to the nearest whole, we decide which whole number the decimal is closest to.

This helps us make numbers easier to use and understand. In this lesson, you will learn how to look at a decimal and round it to the nearest whole number.

First, remember what the places mean.

• The digit to the left of the decimal point is the ones place.
• The digit just to the right of the decimal point is the tenths place.

In the number \(5.7\):

• \(5\) is in the ones place.
• \(7\) is in the tenths place.

To round a decimal to the nearest whole, we look at the tenths digit.

• If the tenths digit is 0, 1, 2, 3, or 4, we round down.
• If the tenths digit is 5, 6, 7, 8, or 9, we round up.

Round down means the whole number stays the same.

Round up means we go to the next whole number.

You can also think about distance on a number line.

For example, \(2.3\) is between \(2\) and \(3\). It is only \(0.3\) away from \(2\), but \(0.7\) away from \(3\). So \(2.3\) rounds to \(2\).

But \(2.8\) is between \(2\) and \(3\). It is \(0.8\) away from \(2\), and only \(0.2\) away from \(3\). So \(2.8\) rounds to \(3\).

A helpful rule:

If the decimal part is less than \(0.5\), round down. If the decimal part is \(0.5\) or more, round up.

Here are the steps to follow:

1. Find the decimal point.
2. Look at the digit in the tenths place.
3. If that digit is \(0\) to \(4\), keep the ones digit the same.
4. If that digit is \(5\) to \(9\), add \(1\) to the ones digit.
5. The answer is a whole number.

Worked Example 1

Round \(4.2\) to the nearest whole.

• The ones digit is \(4\).
• The tenths digit is \(2\).
• Because \(2\) is less than \(5\), we round down.

$$4.2 \approx 4$$

Worked Example 2

Round \(6.9\) to the nearest whole.

• The ones digit is \(6\).
• The tenths digit is \(9\).
• Because \(9\) is \(5\) or more, we round up.

$$6.9 \approx 7$$

Worked Example 3

Round \(8.5\) to the nearest whole.

• The ones digit is \(8\).
• The tenths digit is \(5\).
• When the tenths digit is exactly \(5\), we round up.

$$8.5 \approx 9$$

Worked Example 4

Round \(12.4\) to the nearest whole.

• The ones digit is \(2\).
• The tenths digit is \(4\).
• Because \(4\) is less than \(5\), we round down.
• The whole number part stays \(12\).

$$12.4 \approx 12$$

Watch out for these common mistakes:

• Do not look at the ones digit to decide. Look at the tenths digit.
• Do not always round up when you see a decimal. Some decimals round down.
• Remember that \(0.5\) rounds up.

Try thinking about fractions too.

The decimal \(0.5\) means one-half. One-half is exactly halfway between two whole numbers. When a number is halfway, like \(7.5\), we round up to the next whole number.

Here are some quick examples:

• \(1.1 \approx 1\)
• \(3.4 \approx 3\)
• \(5.5 \approx 6\)
• \(9.7 \approx 10\)

Summary

To round a decimal to the nearest whole, look at the tenths place. If the tenths digit is \(0\) to \(4\), round down. If it is \(5\) to \(9\), round up. This tells you which whole number the decimal is closest to.

Common Misconceptions in Decimal Comparison

Decimals can look tricky at first. Sometimes a number with more digits looks bigger, but that is not always true.

In this lesson, you will learn how to compare decimals the right way. You will also learn about mistakes students often make, and how to avoid them.

Big idea: When comparing decimals, we do not decide by counting digits. We compare the value of each place.

For example, in the number \(0.4\), the \(4\) means 4 tenths. In the number \(0.35\), the \(3\) means 3 tenths and the \(5\) means 5 hundredths.

Decimals are connected to fractions:

• \(0.4 = \frac{4}{10}\)
• \(0.35 = \frac{35}{100}\)

This fraction idea helps us compare decimals correctly.

Main Misconception 1: “More digits means a bigger number.”

This is a very common mistake. A student might think \(0.35 > 0.4\) because \(35\) is bigger than \(4\), or because \(0.35\) has more digits.

But that is not how decimals work. We must compare the place values from left to right.

Compare:

$$0.35 \quad \text{and} \quad 0.4$$

First, look at the ones place. Both have \(0\) ones.

Next, look at the tenths place:

• \(0.35\) has \(3\) tenths
• \(0.4\) has \(4\) tenths

Since \(4\) tenths is more than \(3\) tenths, we know:

$$0.4 > 0.35$$

Even though \(0.35\) has more digits, it is still smaller.

Main Misconception 2: Ignoring place value after the decimal point.

Some students look only at the digits and forget what each place means.

For example, compare \(0.7\) and \(0.65\).

A student might say \(65 > 7\), so \(0.65\) must be bigger. But that is incorrect because \(0.7\) means 7 tenths, and \(0.65\) means 6 tenths and 5 hundredths.

Since \(7\) tenths is greater than \(6\) tenths, we know:

$$0.7 > 0.65$$

Main Misconception 3: Thinking zeros always change the value.

Zeros at the end of a decimal do not change the value. They only show the same amount in a different way.

For example:

$$0.5 = 0.50 = 0.500$$

These are all the same value. They all mean one-half, or \(\frac{5}{10} = \frac{50}{100}\).

This can help when comparing decimals. Sometimes it is easier to write both numbers with the same number of decimal places.

For example, to compare \(0.4\) and \(0.35\), we can rewrite \(0.4\) as \(0.40\).

$$0.40 \quad \text{and} \quad 0.35$$

Now compare the hundredths:

• \(0.40\) has \(40\) hundredths
• \(0.35\) has \(35\) hundredths

Since \(40 > 35\), we know:

$$0.40 > 0.35$$

How to Compare Decimals Step by Step

1. Compare the ones place.
2. If the ones are the same, compare the tenths place.
3. If the tenths are the same, compare the hundredths place.
4. Keep going one place at a time if needed.
5. If helpful, add zeros at the end so both decimals have the same number of places.

Worked Example 1

Compare \(0.3\) and \(0.27\).

Step 1: Write \(0.3\) as \(0.30\).

$$0.30 \quad \text{and} \quad 0.27$$

Step 2: Compare tenths.

• \(0.30\) has \(3\) tenths
• \(0.27\) has \(2\) tenths

Since \(3\) tenths is greater than \(2\) tenths:

$$0.3 > 0.27$$

Worked Example 2

Compare \(0.48\) and \(0.5\).

Step 1: Write \(0.5\) as \(0.50\).

$$0.48 \quad \text{and} \quad 0.50$$

Step 2: Compare tenths.

• \(0.48\) has \(4\) tenths
• \(0.50\) has \(5\) tenths

Since \(5\) tenths is greater than \(4\) tenths:

$$0.48 < 0.5$$

Worked Example 3

Compare \(0.62\) and \(0.620\).

Some students think \(0.620\) is bigger because it has more digits. But the zero is at the end, so it does not change the value.

Both numbers mean \(62\) hundredths.

$$0.62 = 0.620$$

Worked Example 4

Compare \(0.41\) and \(0.39\).

First compare the ones place. Both have \(0\) ones.

Next compare the tenths place. Both have \(4\) tenths? No. Let’s look carefully:

• \(0.41\) has \(4\) tenths
• \(0.39\) has \(3\) tenths

Since \(4\) tenths is greater than \(3\) tenths:

$$0.41 > 0.39$$

We do not even need to compare the hundredths because the tenths already tell us the answer.

Helpful Decimal Comparison Tips

• Do not compare decimals the same way you compare whole numbers.
• More digits does not always mean a greater value.
• Look at place value from left to right.
• Add zeros at the end if that helps you line up the numbers.
• Think of decimals as fractions, like tenths and hundredths.

Try Thinking About These

Which is greater: \(0.8\) or \(0.75\)?

Rewrite \(0.8\) as \(0.80\). Now compare:

$$0.80 \quad \text{and} \quad 0.75$$

\(80\) hundredths is greater than \(75\) hundredths, so \(0.8 > 0.75\).

Which is greater: \(0.09\) or \(0.1\)?

Rewrite \(0.1\) as \(0.10\). Now compare:

$$0.09 \quad \text{and} \quad 0.10$$

\(9\) hundredths is less than \(10\) hundredths, so \(0.09 < 0.1\).

Summary

When comparing decimals, the most important thing is place value. Start at the left and compare ones, then tenths, then hundredths.

Do not be tricked by a decimal that has more digits. A number like \(0.35\) is still less than \(0.4\) because \(3\) tenths is less than \(4\) tenths.

Remember: ending zeros do not change the value, and thinking about fractions can help you understand decimals better.