Decimal Operations

Connecting Decimals to Fractions means understanding that decimals are another way to write fractions, especially fractions with denominators of 10, 100, and 1,000.

When you see a decimal like \(0.4\), you can think of it as 4 tenths. That means:

This idea helps us understand what decimals really mean and why decimal addition, subtraction, multiplication, and division work the way they do.

In this lesson, you will learn how to:

• write decimals as fractions,
• write fractions as decimals,
• use place value to connect the two,
• and explain decimal work by thinking about fractions.

1. Decimals are based on place value

Each digit in a decimal has a value based on its place.

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

These place values match fractions:

• tenths \(\rightarrow \frac{1}{10}\)
• hundredths \(\rightarrow \frac{1}{100}\)
• thousandths \(\rightarrow \frac{1}{1000}\)

For example:

• \(0.7\) means 7 tenths, so \(0.7 = \frac{7}{10}\)
• \(0.25\) means 25 hundredths, so \(0.25 = \frac{25}{100}\)
• \(0.308\) means 308 thousandths, so \(0.308 = \frac{308}{1000}\)

2. Writing decimals as fractions

To write a decimal as a fraction:

1. Read the decimal using place value.
2. Write the digits as the numerator.
3. Use 10, 100, or 1,000 as the denominator, depending on the last decimal place.

If a decimal has:

• 1 digit after the decimal point, use denominator \(10\)
• 2 digits after the decimal point, use denominator \(100\)
• 3 digits after the decimal point, use denominator \(1000\)

Examples:

• \(0.6 = \frac{6}{10}\)
• \(0.43 = \frac{43}{100}\)
• \(0.125 = \frac{125}{1000}\)

Sometimes the fraction can be simplified.

For example:

Both \(\frac{50}{100}\) and \(\frac{1}{2}\) are correct. When connecting decimals to fractions, it is often helpful to first write the fraction with denominator 10, 100, or 1,000.

3. Writing fractions as decimals

Fractions with denominators of 10, 100, and 1,000 can be written as decimals by using place value.

Examples:

• \(\frac{3}{10} = 0.3\)
• \(\frac{47}{100} = 0.47\)
• \(\frac{9}{1000} = 0.009\)

Notice that zeros are important. In \(0.009\), the 9 is in the thousandths place, not the tenths or hundredths place.

You can think of it this way:

• \(\frac{5}{10}\) is 5 tenths \(\rightarrow 0.5\)
• \(\frac{5}{100}\) is 5 hundredths \(\rightarrow 0.05\)
• \(\frac{5}{1000}\) is 5 thousandths \(\rightarrow 0.005\)

4. Why this helps with decimal operations

Decimals follow the same ideas as fractions. If you understand the fraction meaning, decimal work makes more sense.

For example, when adding decimals, you line up the decimal points because you are really adding tenths to tenths, hundredths to hundredths, and so on.

Look at this sum:

As fractions, this is:

So:

Here is another example:

As fractions, this is:

So:

This is why place value matters so much in decimal arithmetic.

Worked Example 1: Decimal to fraction

Write \(0.8\) as a fraction.

Step 1: There is 1 digit after the decimal point, so the denominator is \(10\).

Step 2: The digit is 8, so the numerator is \(8\).

This fraction can also be simplified:

So \(0.8 = \frac{8}{10}\), which is also equal to \(\frac{4}{5}\).

Worked Example 2: Fraction to decimal

Write \(\frac{36}{100}\) as a decimal.

Since the denominator is \(100\), the decimal will have 2 places after the decimal point.

This means 36 hundredths.

Worked Example 3: Using fractions to explain decimal addition

Find \(0.6 + 0.2\).

Write each decimal as a fraction:

Add the fractions:

Write the answer as a decimal:

So:

Worked Example 4: Hundredths in subtraction

Find \(0.75 - 0.21\).

Write each decimal as a fraction:

Subtract the fractions:

Write the answer as a decimal:

So:

5. Important patterns to notice

• Decimals and fractions can name the same amount.
• The number of decimal places tells you the denominator.
• \(0.4\) means \(\frac{4}{10}\), not \(\frac{4}{100}\).
• Zeros can change place value. For example, \(0.5\), \(0.05\), and \(0.005\) are very different numbers.
• When adding or subtracting decimals, line up decimal points so the place values match.

6. Common mistakes to avoid

• Mistake: Writing \(0.32\) as \(\frac{32}{10}\)
  Correct idea: Two decimal places means hundredths, so \(0.32 = \frac{32}{100}\).
• Mistake: Thinking \(\frac{7}{100}\) is \(0.7\)
  Correct idea: \(\frac{7}{100}\) is 7 hundredths, so it is \(0.07\).
• Mistake: Ignoring zeros in decimals
  Correct idea: Zeros help show place value, like in \(0.009\).

7. Quick check

Try thinking about these on your own:

• \(0.9 = \frac{\square}{10}\)
• \(0.14 = \frac{\square}{100}\)
• \(\frac{8}{10} = 0.\square\)
• \(\frac{65}{100} = 0.\square\square\)

Answers:

• \(0.9 = \frac{9}{10}\)
• \(0.14 = \frac{14}{100}\)
• \(\frac{8}{10} = 0.8\)
• \(\frac{65}{100} = 0.65\)

Summary

Decimals and fractions are closely connected. A decimal shows tenths, hundredths, or thousandths, and those place values match fractions with denominators of 10, 100, or 1,000.

When you convert between decimals and fractions, you use place value to decide the denominator or the decimal places. This helps you understand decimal operations because adding and subtracting decimals is really like adding and subtracting fractions with the same denominator.

Adding and Subtracting Decimals with Precision

Decimals are used when numbers are less than one whole or when a number has a whole part and a part of a whole. We see decimals in money, measurements, and scores. To add and subtract decimals correctly, we must pay close attention to place value.

The most important rule is this: line up the decimal points. When decimal points are lined up, each place value matches correctly. Ones stay with ones, tenths stay with tenths, hundredths stay with hundredths, and so on.

Remember the place values to the right of the decimal point:

• Ones place is to the left of the decimal.
• Tenths is the first place to the right.
• Hundredths is the second place to the right.
• Thousandths is the third place to the right.

For example, in the number \(4.27\):

• \(4\) is in the ones place
• \(2\) is in the tenths place
• \(7\) is in the hundredths place

When you add or subtract decimals, think of it just like adding or subtracting whole numbers. The difference is that you must keep the decimal points in a straight line.

Step-by-step for adding decimals

1. Write the numbers in a vertical column.
2. Line up the decimal points.
3. Add zeros if needed so both numbers have the same number of decimal places.
4. Add from right to left.
5. Bring the decimal point straight down into the answer.

Step-by-step for subtracting decimals

1. Write the numbers in a vertical column.
2. Line up the decimal points.
3. Add zeros if needed as placeholders.
4. Subtract from right to left.
5. Bring the decimal point straight down into the answer.

Why do we sometimes add zeros?

Zeros can be used as placeholders. They do not change the value of a decimal. For example, \(3.4\), \(3.40\), and \(3.400\) all mean the same amount.

This is helpful because it lets us line up place values clearly. For example:

Now \(2.50\) is easier to add or subtract with a number like \(1.27\).

Worked Example 1: Adding decimals with the same number of decimal places

Add \(3.45 + 2.13\).

Add from right to left:

• Hundredths: \(5 + 3 = 8\)
• Tenths: \(4 + 1 = 5\)
• Ones: \(3 + 2 = 5\)

So, \(3.45 + 2.13 = 5.58\).

Worked Example 2: Adding decimals with different numbers of decimal places

Add \(4.7 + 2.35\).

First, line up the decimal points:

The number \(4.7\) has only one decimal place. Write it as \(4.70\) so the place values match.

Now add:

• Hundredths: \(0 + 5 = 5\)
• Tenths: \(7 + 3 = 10\), write \(0\) in the tenths place and carry \(1\)
• Ones: \(4 + 2 + 1 = 7\)

So, \(4.7 + 2.35 = 7.05\).

Worked Example 3: Subtracting decimals with the same number of decimal places

Subtract \(8.64 - 3.21\).

Subtract from right to left:

• Hundredths: \(4 - 1 = 3\)
• Tenths: \(6 - 2 = 4\)
• Ones: \(8 - 3 = 5\)

So, \(8.64 - 3.21 = 5.43\).

Worked Example 4: Subtracting decimals when you need to regroup

Subtract \(6.2 - 1.75\).

First, write \(6.2\) as \(6.20\).

Now subtract from right to left.

In the hundredths place, \(0 - 5\) cannot be done, so regroup from the tenths place. But the tenths digit is also \(2\), and we will need to regroup carefully:

• Regroup \(6.20\) so the tenths become \(1\) and the hundredths become \(10\)
• Now regroup again from the ones if needed through the tenths place

A clearer way is to think of it step by step:

• From \(6.20\), take 1 tenth to make \(10\) hundredths. Now the number is \(6.1\,10\).
• Hundredths: \(10 - 5 = 5\)
• Tenths: \(1 - 7\) cannot be done, so regroup 1 one as 10 tenths.
• Now the ones become \(5\), and the tenths become \(11\).
• Tenths: \(11 - 7 = 4\)
• Ones: \(5 - 1 = 4\)

So, \(6.2 - 1.75 = 4.45\).

Common mistakes to avoid

• Do not line up the numbers by the last digit. Line them up by the decimal point.
• Do not forget placeholder zeros. They help match the place values.
• Do not forget to bring the decimal point straight down into the answer.
• Check place values. Tenths must be added or subtracted with tenths, not with hundredths.

Look at this example of a mistake:

If you do not line up the decimal points, the digits will be in the wrong places. The correct setup is:

Helpful tip

You can estimate before solving to see if your answer makes sense. For example, to estimate \(4.7 + 2.35\), round to whole numbers:

The exact answer, \(7.05\), is close to \(7\), so it makes sense.

For \(6.2 - 1.75\), estimate:

The exact answer, \(4.45\), is close to \(4\), so it also makes sense.

Summary

To add and subtract decimals with precision, always line up the decimal points first. Then make sure each place value matches by adding zeros if needed. Solve as you would with whole numbers, and bring the decimal point straight down into the answer.

With practice, decimal addition and subtraction become much easier. If you remember place value and work carefully, you can solve decimal problems accurately every time.

Multiplying Decimals and Placement of the Decimal Point

When you multiply decimals, the digits are multiplied the same way you multiply whole numbers. The important extra step is deciding where the decimal point goes in the final answer.

This lesson will show you a simple method: multiply first, then place the decimal point. You will also learn how to check whether your answer makes sense.

Why this works

A decimal is part of a whole. For example, \(0.4\) means 4 tenths, and \(0.3\) means 3 tenths. If we multiply them, we are finding part of a part.

We can think of this using fractions:

\(0.4 = \frac{4}{10}\) and \(0.3 = \frac{3}{10}\)

So,

$$0.4 \times 0.3 = \frac{4}{10} \times \frac{3}{10} = \frac{12}{100} = 0.12$$

Notice that the answer has two decimal places. That is because the factors had a total of two decimal places: one in \(0.4\) and one in \(0.3\).

Main idea

1. Ignore the decimal points at first.
2. Multiply the numbers as if they were whole numbers.
3. Count the total number of decimal places in both factors.
4. Place the decimal point in the product so it has that many decimal places.

This is the rule you will use most of the time.

How to count decimal places

• In \(0.6\), there is 1 decimal place.
• In \(1.25\), there are 2 decimal places.
• In \(0.08\), there are 2 decimal places.
• In \(3.407\), there are 3 decimal places.

To find the decimal places in the answer, add the decimal places in both factors.

For example, in \(1.2 \times 0.34\):

• \(1.2\) has 1 decimal place
• \(0.34\) has 2 decimal places
• Total: \(1 + 2 = 3\) decimal places

So the product must have 3 decimal places.

Worked Example 1

Find \(0.4 \times 3\).

Step 1: Multiply as whole numbers.

$$4 \times 3 = 12$$

Step 2: Count decimal places.

• \(0.4\) has 1 decimal place
• \(3\) has 0 decimal places
• Total: 1 decimal place

Step 3: Place the decimal point.

Make the answer have 1 decimal place:

$$0.4 \times 3 = 1.2$$

Check: Since \(0.4\) is less than 1, multiplying by 3 should give an answer less than 3. The answer \(1.2\) makes sense.

Worked Example 2

Find \(1.2 \times 0.3\).

Step 1: Ignore decimals and multiply.

$$12 \times 3 = 36$$

Step 2: Count decimal places.

• \(1.2\) has 1 decimal place
• \(0.3\) has 1 decimal place
• Total: 2 decimal places

Step 3: Place the decimal point.

Put the decimal so the product has 2 decimal places:

$$1.2 \times 0.3 = 0.36$$

Check: Since \(0.3\) is less than 1, the answer should be less than \(1.2\). The answer \(0.36\) makes sense.

Worked Example 3

Find \(2.45 \times 1.6\).

Step 1: Multiply as whole numbers.

$$245 \times 16 = 3920$$

Step 2: Count decimal places.

• \(2.45\) has 2 decimal places
• \(1.6\) has 1 decimal place
• Total: 3 decimal places

Step 3: Place the decimal point.

The product must have 3 decimal places:

$$2.45 \times 1.6 = 3.920 = 3.92$$

Trailing zeros at the end of a decimal do not change the value, so \(3.920\) is the same as \(3.92\).

Check: Since \(1.6\) is a little bigger than 1, the answer should be a little bigger than \(2.45\). The answer \(3.92\) is reasonable.

Worked Example 4

Find \(0.08 \times 0.5\).

Step 1: Multiply as whole numbers.

$$8 \times 5 = 40$$

Step 2: Count decimal places.

• \(0.08\) has 2 decimal places
• \(0.5\) has 1 decimal place
• Total: 3 decimal places

Step 3: Place the decimal point.

The answer needs 3 decimal places. Starting from 40, move the decimal point 3 places to the left:

$$40 \rightarrow 0.040$$

So,

$$0.08 \times 0.5 = 0.040 = 0.04$$

Important: Sometimes you need to add a zero in front of the digits so the decimal is placed correctly. That is why \(40\) became \(0.040\).

A quick way to think about decimal placement

After multiplying, place the decimal point by moving from right to left in the product.

• If the total decimal places is 1, count 1 place from the right.
• If the total decimal places is 2, count 2 places from the right.
• If the total decimal places is 3, count 3 places from the right.

If there are not enough digits, add zeros to the left.

Example:

For \(0.07 \times 0.2\):

• Multiply \(7 \times 2 = 14\)
• Total decimal places: \(2 + 1 = 3\)
• Write the answer with 3 decimal places: \(0.014\)

So,

$$0.07 \times 0.2 = 0.014$$

How to estimate to check your answer

Estimation helps you decide if your decimal point is in the right place.

• If you multiply by a number less than 1, the answer should get smaller.
• If you multiply by a number greater than 1, the answer should get bigger.

Examples:

• \(4 \times 0.2\) should be less than 4, so \(0.8\) makes sense.
• \(2.5 \times 1.4\) should be greater than 2.5, so \(3.5\) makes sense.

Common mistakes to avoid

• Putting the decimal point in the wrong place.
  Always count the total decimal places in both factors.
• Forgetting that whole numbers can have 0 decimal places.
  For example, in \(0.6 \times 4\), the number 4 has 0 decimal places.
• Not adding zeros when needed.
  For example, \(0.03 \times 0.2\): \(3 \times 2 = 6\), and there are 3 decimal places total, so the answer is \(0.006\).
• Getting an answer that is too big.
  If both numbers are less than 1, the product should usually be less than both factors.

Practice these in your head

• \(0.5 \times 2 = 1.0\)
• \(0.2 \times 0.4 = 0.08\)
• \(1.5 \times 0.2 = 0.30 = 0.3\)
• \(3.4 \times 0.1 = 0.34\)

Summary

To multiply decimals, first multiply as if the numbers were whole numbers. Then count the total number of decimal places in both factors and place the decimal point in the product so it has that many decimal places.

Always check whether your answer makes sense. If you multiply by a number less than 1, the product should be smaller. If you multiply by a number greater than 1, the product should be larger.

Dividing Decimals by Whole Numbers means sharing a decimal amount into equal groups.

You already know how to divide whole numbers. The good news is that dividing decimals by whole numbers uses the same long division steps. The main new idea is where to put the decimal point in the answer.

The most important rule is this:

When dividing a decimal by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend.

For example, in \(8.4 \div 4\), the decimal point in the answer goes directly above the decimal point in \(8.4\).

Why does this make sense? A decimal is just another way to write parts of a whole. For example, \(8.4\) means 8 wholes and 4 tenths. When we divide, we are sharing both the wholes and the tenths equally.

Here is the basic process for dividing decimals by whole numbers:

1. Set up long division.
2. Bring the decimal point straight up into the quotient.
3. Divide as you normally do.
4. If needed, add zeros to the right of the decimal in the dividend to keep dividing.
5. Check that your answer makes sense.

Helpful idea: Adding zeros to the end of a decimal does not change its value. For example, \(3.6 = 3.60 = 3.600\).

That means if you need more digits to continue dividing, you can safely add a zero.

Worked Example 1: A simple decimal division

Find \(6.8 \div 2\).

Step 1: Set up the problem and move the decimal point straight up.

The decimal point in the quotient goes directly above the decimal in \(6.8\).

Step 2: Divide the whole number part.

\(2\) goes into \(6\) three times, because \(3 \times 2 = 6\).

Step 3: Bring the decimal point up.

Step 4: Divide the tenths.

\(2\) goes into \(8\) four times, because \(4 \times 2 = 8\).

So,

Check: \(3.4 \times 2 = 6.8\), so the answer is correct.

Worked Example 2: A quotient with a zero in it

Find \(4.2 \div 6\).

Step 1: Set up long division and place the decimal point in the quotient.

Step 2: Divide the whole number part.

\(6\) does not go into \(4\), so we write \(0\) in the ones place of the quotient.

Then we place the decimal point.

Step 3: Now divide \(42\) tenths by \(6\).

\(42 \div 6 = 7\).

So,

Check: \(0.7 \times 6 = 4.2\).

This example shows that your answer can be less than 1. That makes sense because \(4.2\) is being split into 6 equal groups, so each group should be smaller than 1 whole.

Worked Example 3: Dividing to the hundredths place

Find \(5.25 \div 5\).

Step 1: Set up the division and move the decimal point straight up.

Step 2: Divide the ones.

\(5\) goes into \(5\) one time. Write \(1\).

Step 3: Bring the decimal point up.

Step 4: Divide the tenths.

\(5\) goes into \(2\) zero times, so write \(0\) in the tenths place.

Step 5: Bring down the next digit, \(5\), to make \(25\).

\(25 \div 5 = 5\).

So,

Check: \(1.05 \times 5 = 5.25\).

This example is important because it shows that sometimes you must write a zero in the quotient to hold the correct place value.

Worked Example 4: Adding a zero to keep dividing

Find \(7.2 \div 4\).

Step 1: Set up the problem and place the decimal point in the quotient above the decimal in the dividend.

Step 2: Divide the ones.

\(4\) goes into \(7\) one time. Write \(1\). Subtract to get remainder \(3\).

Step 3: Bring down the \(2\) from the tenths place to make \(32\) tenths.

\(32 \div 4 = 8\).

So,

Now let us look at a case where adding a zero helps.

Find \(7.21 \div 4\).

Divide as usual:

• \(4\) goes into \(7\) one time.
• Bring the decimal point up.
• Bring down the \(2\): \(32 \div 4 = 8\).
• Bring down the \(1\): \(1 \div 4 = 0\) with remainder \(1\).

At this point, if you want to keep dividing, you can add a zero to make \(7.21\) become \(7.210\).

Then bring down the \(0\): \(10 \div 4 = 2\) with remainder \(2\).

So the quotient starts as

In 6th Grade, you may sometimes stop when the problem tells you what place value to round to. The important idea is that adding a zero lets you continue the division.

How to know if your answer is reasonable

• If you divide by a number greater than 1, the answer should be less than the original number.
• If the dividend is only a little bigger than the divisor, the answer may be close to 1.
• You can multiply your quotient by the divisor to check your work.

For example, in \(8.4 \div 4\), the answer should be smaller than \(8.4\). Since \(2 \times 4 = 8\), an answer near \(2\) makes sense. The exact answer is \(2.1\).

Common mistakes to avoid

• Forgetting the decimal point in the quotient. Always move it straight up.
• Not writing zero when needed. A zero can be an important place holder, like in \(1.05\).
• Stopping too early. If there is a remainder and you need to continue, add a zero to the dividend.
• Dividing incorrectly because of place value. Work step by step, just like whole-number long division.

Quick Practice Ideas

• \(9.6 \div 3 = 3.2\)
• \(2.4 \div 4 = 0.6\)
• \(3.15 \div 3 = 1.05\)
• \(8.8 \div 2 = 4.4\)

Summary

Dividing decimals by whole numbers is very similar to dividing whole numbers. Use long division, and remember to place the decimal point in the quotient directly above the decimal point in the dividend.

If needed, add zeros to the right of the decimal so you can keep dividing. Always check your answer by multiplying to see if you get back the original decimal.

Dividing Decimals by Decimals

Sometimes in math, both numbers in a division problem have decimals. For example, you might see something like \(4.8 \div 0.6\) or \(3.75 \div 1.5\). At first, this can look tricky.

The good news is that there is a simple method. We can move the decimal point in both numbers the same number of places so that the divisor becomes a whole number. Then we divide as usual.

Remember:

• The dividend is the number being divided.
• The divisor is the number you are dividing by.
• The quotient is the answer.

For example, in \(4.8 \div 0.6\):

• \(4.8\) is the dividend
• \(0.6\) is the divisor
• the quotient is the answer

The Big Idea: If we multiply both the dividend and divisor by the same power of 10, the value of the quotient does not change.

That means:

Why? Because both numbers were multiplied by 10.

This works with 10, 100, 1000, and so on. We choose the power of 10 that makes the divisor a whole number.

Steps for dividing decimals by decimals

1. Look at the divisor.
2. Count how many decimal places it has.
3. Move the decimal point in both the divisor and dividend the same number of places to the right.
4. Now the divisor should be a whole number.
5. Divide using long division or basic division.

Important: Always move the decimal point in both numbers the same number of places. If you only move one, the problem changes and the answer will be wrong.

Let’s look at some examples.

Example 1: \(4.8 \div 0.6\)

The divisor is \(0.6\). It has 1 decimal place, so move both decimals 1 place to the right.

Now divide:

So,

Example 2: \(3.75 \div 1.5\)

The divisor is \(1.5\). It has 1 decimal place, so move both decimals 1 place to the right.

Now divide:

So,

You can check: \(15 \times 2.5 = 37.5\), so the answer makes sense.

Example 3: \(0.84 \div 0.21\)

The divisor is \(0.21\). It has 2 decimal places, so move both decimals 2 places to the right.

Now divide:

So,

Example 4: \(5.46 \div 0.7\)

The divisor is \(0.7\). It has 1 decimal place, so move both decimals 1 place to the right.

Now divide:

So,

Let’s slow down and think about what happened in Example 4. Even though the divisor became a whole number, the dividend did not become a whole number. That is okay. The most important job is to make the divisor a whole number.

What if the divisor has 2 decimal places?

Then move both decimal points 2 places to the right.

Example:

The divisor \(0.12\) has 2 decimal places, so both numbers were multiplied by 100.

How to know if your answer makes sense

• If you divide by a number less than 1, the answer may be greater than the starting number.
• If you divide by a number greater than 1, the answer will usually be smaller than the starting number.
• You can check by multiplying the quotient by the divisor.

For example, in \(4.8 \div 0.6 = 8\):

So the answer is correct.

Common mistakes to avoid

• Moving only one decimal point. You must move the decimal in both numbers the same number of places.
• Not moving far enough. If the divisor has 2 decimal places, move both decimals 2 places.
• Stopping too soon. After rewriting the problem, you still need to divide carefully.
• Forgetting to check. Multiply your answer by the divisor to see if you get the dividend.

Quick practice thinking

Try asking yourself these questions:

• What is the divisor?
• How many decimal places does it have?
• How many places should I move both decimals?
• What whole-number divisor do I get?

For \(6.24 \div 0.8\):

• The divisor is \(0.8\)
• It has 1 decimal place
• Move both decimals 1 place right
• \(6.24 \div 0.8 = 62.4 \div 8 = 7.8\)

Summary

To divide a decimal by a decimal, first make the divisor a whole number. Do this by moving the decimal point in both numbers the same number of places to the right. Then divide as you normally would.

This method works because multiplying both numbers by the same power of 10 does not change the quotient. With practice, dividing decimals by decimals becomes much easier.

Terminating and Repeating Decimals

Decimals are another way to write parts of a whole. Fractions and decimals are closely connected, and we can change a fraction into a decimal by dividing the numerator by the denominator.

For example, the fraction \(\frac{3}{4}\) means \(3 \div 4\). When we do the division, we get the decimal \(0.75\).

In this lesson, you will learn how to use long division to turn fractions into decimals. You will also learn the difference between a terminating decimal and a repeating decimal.

1. What is a terminating decimal?

A terminating decimal is a decimal that ends. It has a last digit.

Examples of terminating decimals are:

• \(0.5\)
• \(0.25\)
• \(1.8\)
• \(3.125\)

These decimals stop, so they do not go on forever.

2. What is a repeating decimal?

A repeating decimal is a decimal in which a digit or a group of digits keeps repeating forever.

Examples of repeating decimals are:

• \(0.333\ldots\)
• \(0.666\ldots\)
• \(0.121212\ldots\)

The dots, called an ellipsis, show that the pattern continues forever.

Sometimes we show repeating decimals with a bar over the repeating part:

• \(0.333\ldots = 0.\overline{3}\)
• \(0.121212\ldots = 0.\overline{12}\)

3. How do we change a fraction into a decimal?

To change a fraction into a decimal, divide the numerator by the denominator.

For any fraction \(\frac{a}{b}\), compute:

$$a \div b$$

If the division ends with no remainder, the decimal terminates.

If the remainders begin to repeat, the decimal repeats.

4. Using long division with decimals

When the numerator is smaller than the denominator, place a decimal point and add zeros to the numerator as needed.

For example, to divide \(1 \div 4\), think of \(1\) as \(1.000\ldots\). This lets you keep dividing into tenths, hundredths, and thousandths.

Here are the basic steps:

1. Put the numerator inside the division bracket.
2. Put the denominator outside.
3. Add a decimal point in the quotient when needed.
4. Add zeros to the right of the decimal in the dividend.
5. Continue dividing.
6. Decide whether the decimal ends or starts repeating.

Worked Example 1: A simple terminating decimal

Change \(\frac{1}{2}\) into a decimal.

This means divide:

$$1 \div 2$$

Use long division:

• 2 does not go into 1, so write \(0\) and a decimal point.
• Add a 0 to make 10 tenths.
• 2 goes into 10 exactly 5 times.
• There is no remainder.

So,

$$\frac{1}{2} = 0.5$$

This is a terminating decimal because it ends.

Worked Example 2: Another terminating decimal

Change \(\frac{3}{4}\) into a decimal.

Divide:

$$3 \div 4$$

Use long division:

• 4 does not go into 3, so write \(0\) and a decimal point.
• Add a 0 to make 30 tenths.
• 4 goes into 30 seven times because \(7 \times 4 = 28\).
• Subtract: \(30 - 28 = 2\).
• Add another 0 to make 20 hundredths.
• 4 goes into 20 five times.
• Subtract: \(20 - 20 = 0\).

So,

$$\frac{3}{4} = 0.75$$

This decimal terminates because the remainder became 0.

Worked Example 3: A repeating decimal

Change \(\frac{1}{3}\) into a decimal.

Divide:

$$1 \div 3$$

Use long division:

• 3 does not go into 1, so write \(0\) and a decimal point.
• Add a 0 to make 10 tenths.
• 3 goes into 10 three times because \(3 \times 3 = 9\).
• Subtract: \(10 - 9 = 1\).
• Add another 0, and you have 10 again.
• The same thing happens again.

The remainder keeps becoming 1, so the digit 3 repeats forever.

So,

$$\frac{1}{3} = 0.333\ldots = 0.\overline{3}$$

This is a repeating decimal.

Worked Example 4: A repeating pattern with more than one digit

Change \(\frac{2}{11}\) into a decimal.

Divide:

$$2 \div 11$$

Use long division:

• 11 does not go into 2, so write \(0\) and a decimal point.
• Add a 0 to make 20.
• 11 goes into 20 one time. Subtract: \(20 - 11 = 9\).
• Add a 0 to make 90.
• 11 goes into 90 eight times because \(8 \times 11 = 88\).
• Subtract: \(90 - 88 = 2\).

Now the remainder is 2 again, which is what we started with. That means the pattern will repeat: 1, 8, 1, 8, 1, 8, and so on.

So,

$$\frac{2}{11} = 0.181818\ldots = 0.\overline{18}$$

This is a repeating decimal with a repeating block of two digits.

5. How can you tell whether a decimal will terminate or repeat?

When you use long division, watch the remainder.

• If the remainder becomes 0, the decimal terminates.
• If a remainder repeats, the digits in the decimal will repeat.

This happens because long division starts doing the same steps again once the same remainder appears.

6. Important idea: every fraction gives a terminating or repeating decimal

When you divide one whole number by another whole number, the decimal result will either:

• end, or
• repeat forever in a pattern.

It will not do something random forever. There will always be an ending or a repeating pattern.

7. Quick checks

• \(\frac{1}{4} = 0.25\) → terminating
• \(\frac{5}{8} = 0.625\) → terminating
• \(\frac{2}{3} = 0.666\ldots\) → repeating
• \(\frac{5}{6} = 0.8333\ldots\) → repeating

8. Common mistakes to avoid

• Mixing up numerator and denominator: In \(\frac{3}{4}\), divide \(3 \div 4\), not \(4 \div 3\).
• Forgetting the decimal point: If the denominator does not go into the numerator, write \(0.\) in the quotient.
• Stopping too soon: If the remainder is not 0, keep going.
• Missing a repeating pattern: If the same remainder appears again, the decimal will repeat.

Summary

To change a fraction to a decimal, divide the numerator by the denominator using long division.

If the division ends with remainder 0, the decimal is terminating. If the same remainder comes back again and again, the decimal is repeating.

Remember:

• Terminating decimals end.
• Repeating decimals have a pattern that goes on forever.
• Long division helps you tell which one you have.

Precision in Calculation and Vocabulary means being careful with both numbers and math words.

When we work with math, we want our answers to be accurate, or correct. We also want to use the right vocabulary, like sum, difference, product, and quotient, so we can explain our thinking clearly.

In 4th grade, being precise means:

• Writing numbers neatly and correctly
• Putting digits in the correct place value
• Using the correct operation
• Checking your work
• Using the right math words when you talk or write about your answer

Being precise helps you avoid small mistakes that can change the whole answer.

1. Be precise with calculation

Precision in calculation means solving carefully. Even one digit in the wrong place can make an answer incorrect.

Here are some ways to be precise:

• Line up numbers by place value. Ones go under ones, tens under tens, and hundreds under hundreds.
• Read the problem carefully. Make sure you know whether to add, subtract, multiply, or divide.
• Write each step clearly. Neat work makes it easier to spot mistakes.
• Check your answer. Ask, “Does this answer make sense?”

Example 1: Addition with place value

Add:

$$245 + 32$$

Line up the numbers by place value:

$$\begin{array}{r} 245 \\ \,32 \\ \hline 277 \end{array}$$

Now add from right to left:

• Ones: \(5 + 2 = 7\)
• Tens: \(4 + 3 = 7\)
• Hundreds: \(2 + 0 = 2\)

So the sum is 277.

If you do not line up the digits, you might get the wrong answer. Precision means lining them up correctly.

Example 2: Subtraction with regrouping

Subtract:

$$402 - 186$$

Set it up carefully:

$$\begin{array}{r} 402 \\ 186 \\ \hline \end{array}$$

Start with the ones. You cannot do \(2 - 6\), so regroup.

In 402, there are 4 hundreds, 0 tens, and 2 ones. Regroup one hundred as 10 tens. Then regroup one ten as 10 ones.

Now think of the number as 3 hundreds, 9 tens, and 12 ones.

• Ones: \(12 - 6 = 6\)
• Tens: \(9 - 8 = 1\)
• Hundreds: \(3 - 1 = 2\)

So the difference is 216.

Precision matters here because if you forget to regroup correctly, your answer will be wrong.

2. Be precise with vocabulary

Math vocabulary helps us understand what a problem is asking. It also helps us explain our work.

Here are some important math words:

• Sum: the answer to an addition problem
• Difference: the answer to a subtraction problem
• Product: the answer to a multiplication problem
• Quotient: the answer to a division problem
• Equation: a math sentence with an equal sign
• Estimate: a close answer, not an exact answer
• Exact answer: the correct answer

When you use the right word, your explanation becomes clearer.

For example:

• Instead of saying, “The answer to \(6 \times 4\) is 24,” you can say, “The product of \(6 \times 4\) is 24.”
• Instead of saying, “The answer to \(15 - 7\) is 8,” you can say, “The difference of \(15 - 7\) is 8.”

3. Know the difference between exact answers and estimates

Sometimes in math, we want an exact answer. Other times, we make an estimate to check if our answer is reasonable.

An exact answer is the true answer. An estimate is close, but not exact.

Example 3: Estimate, then solve exactly

Add:

$$198 + 203$$

First, estimate:

• \(198\) is about \(200\)
• \(203\) is about \(200\)

So the estimate is:

$$200 + 200 = 400$$

Now solve exactly:

$$\begin{array}{r} 198 \\ 203 \\ \hline 401 \end{array}$$

The exact sum is 401.

The estimate of 400 is close, so our exact answer makes sense.

4. Precision means checking if your answer makes sense

After solving, stop and ask yourself:

• Did I use the correct operation?
• Did I line up the numbers correctly?
• Did I regroup correctly, if needed?
• Does my answer seem too big or too small?
• Can I use math vocabulary to explain my answer?

This is an important habit in math. Careful mathematicians do not just finish—they also check.

Example 4: Multiplication and vocabulary

A teacher puts 4 markers in each box. There are 6 boxes. How many markers are there in all?

This problem is about equal groups, so we use multiplication:

$$6 \times 4 = 24$$

The product is 24.

You can also check by repeated addition:

$$4 + 4 + 4 + 4 + 4 + 4 = 24$$

So there are 24 markers in all.

Being precise means we used the correct operation, found the correct answer, and named it with the correct vocabulary word: product.

Tips for being precise every day in math

1. Read the whole problem slowly.
2. Circle or notice important numbers and words.
3. Choose the correct operation.
4. Write numbers in the correct place value columns.
5. Solve step by step.
6. Check your work with an estimate or the opposite operation when you can.
7. Use math words like sum, difference, product, and quotient.

Let’s review

Precision in math is about being careful, correct, and clear.

We are precise when we:

• Calculate accurately
• Line up digits by place value
• Use the correct operation
• Check whether the answer makes sense
• Use correct math vocabulary

When you practice precision, you become a stronger and more confident mathematician.