Why Do We Compare Decimals?

Decimals often represent measurements or quantities that are not whole numbers.

For example:

• $4.75
• 2.6 km
• 1.82 m
• 0.375 L
• 18.4°C

Sometimes we need to determine which quantity is:

• larger
• smaller
• greatest
• least
• closer to another value

To do this accurately, we compare the place values of the digits.

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Comparison Symbols

Three symbols are commonly used when comparing numbers.

> means greater than

Example:

0.8 > 0.5

< means less than

Example:

0.3 < 0.7

= means equal to

Example:

0.50 = 0.5

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Reading Comparison Statements

The statement:

0.7 > 0.4

is read:

0.7 is greater than 0.4

The statement:

0.25 < 0.6

is read:

0.25 is less than 0.6

The statement:

0.80 = 0.8

is read:

0.80 is equal to 0.8

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Place Value Is the Key

When comparing decimals, compare digits according to their place value.

Consider:

0.7

and:

0.4

The tenths digits are:

7 and 4

Since:

7 tenths > 4 tenths

we know:

0.7 > 0.4

---

Compare from Left to Right

A reliable strategy is to compare digits from left to right.

Start with the largest place value.

Compare:

1. whole numbers
2. tenths
3. hundredths
4. thousandths
5. continue if necessary

As soon as one place contains different digits, you can usually determine which number is greater.

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Example: Different Whole Numbers

Compare:

3.82 and 4.16

Start with the whole-number parts:

3 < 4

Therefore:

3.82 < 4.16

There is no need to compare the decimal digits.

---

Example: Same Whole Number

Compare:

5.72 and 5.48

The whole-number parts are equal:

5 = 5

Now compare tenths:

7 > 4

Therefore:

5.72 > 5.48

---

Example: Same Tenths

Compare:

2.46 and 2.49

Whole numbers:

2 = 2

Tenths:

4 = 4

Hundredths:

6 < 9

Therefore:

2.46 < 2.49

The first difference appears in the hundredths place.

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Example: Compare Thousandths

Compare:

7.384 and 7.389

Whole numbers:

7 = 7

Tenths:

3 = 3

Hundredths:

8 = 8

Thousandths:

4 < 9

Therefore:

7.384 < 7.389

---

Using a Place Value Chart

Suppose we compare:

4.625 and 4.652

Compare from left to right.

Ones:

4 = 4

Tenths:

6 = 6

Hundredths:

2 < 5

Therefore:

4.625 < 4.652

---

Different Numbers of Decimal Places

Students sometimes find numbers such as:

0.6

and:

0.58

difficult to compare because 0.58 has more digits.

The number with more digits is not automatically larger.

Remember:

0.6 = 0.60

Now compare:

0.60

and:

0.58

Tenths:

6 > 5

Therefore:

0.60 > 0.58

So:

0.6 > 0.58

---

Adding Trailing Zeros

Adding zeros to the right end of a decimal does not change its value.

For example:

0.4 = 0.40 = 0.400

2.7 = 2.70 = 2.700

5.36 = 5.360

This can make decimal comparisons easier.

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Worked Example 1

Compare:

0.8 and 0.75

Write:

0.80

and:

0.75

Compare tenths:

8 > 7

Therefore:

0.8 > 0.75

---

Worked Example 2

Compare:

3.405 and 3.45

Add a trailing zero:

3.405

3.450

Compare:

Ones:

3 = 3

Tenths:

4 = 4

Hundredths:

0 < 5

Therefore:

3.405 < 3.45

---

Worked Example 3

Compare:

6.090 and 6.09

Trailing zeros do not change value.

Therefore:

6.090 = 6.09

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Why More Digits Does Not Mean Larger

Consider:

0.9

and:

0.125

0.125 has more digits.

But:

0.9 = 0.900

Now compare:

0.900

and:

0.125

Tenths:

9 > 1

Therefore:

0.9 > 0.125

The place value of the digits matters, not the number of digits.

---

Decimals and Fractions Can Help

Decimals can also be compared by thinking about fractions.

For example:

0.7 = 7/10 = 70/100

and:

0.65 = 65/100

Since:

70/100 > 65/100

we know:

0.7 > 0.65

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Hundred Grids

A hundred grid can make decimal comparisons visible.

Suppose:

0.42 = 42/100

and:

0.57 = 57/100

A model with 42 shaded squares contains less shading than one with 57 shaded squares.

Therefore:

0.42 < 0.57

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Comparing Decimals on a Number Line

Decimals can be placed on a number line.

Numbers farther to the right are greater.

Numbers farther to the left are smaller.

For example:

0.2 < 0.5 < 0.8

because 0.2 appears farthest left and 0.8 appears farthest right.

---

Number Lines with Hundredths

Suppose we compare:

0.43

and:

0.47

Both are between:

0.4 and 0.5

Zoom into that section of the number line.

0.43 is closer to 0.4.

0.47 is closer to 0.5.

Since 0.47 appears farther right:

0.47 > 0.43

---

Benchmark Decimals

Certain decimals make useful benchmarks.

Common benchmarks include:

0

0.25

0.5

0.75

1

These correspond to familiar fractions:

0.25 = 1/4

0.5 = 1/2

0.75 = 3/4

Benchmarks can help us estimate the relative size of decimals.

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Using 0.5 as a Benchmark

Compare:

0.48

and:

0.63

We know:

0.48 < 0.5

while:

0.63 > 0.5

Therefore:

0.48 < 0.63

A benchmark can sometimes make a comparison immediately clear.

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Ordering Decimals

Ordering means arranging numbers according to size.

Numbers can be ordered:

least to greatest

or:

greatest to least

The same place-value comparison method is used.

---

Least to Greatest

Least to greatest means:

smallest → largest

For example:

0.2, 0.5, 0.8

is already ordered from least to greatest:

0.2 < 0.5 < 0.8

---

Greatest to Least

Greatest to least means:

largest → smallest

The same numbers become:

0.8 > 0.5 > 0.2

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A Reliable Ordering Strategy

Suppose we need to order:

0.7, 0.65, 0.72, 0.608

First give each number the same number of decimal places:

0.700

0.650

0.720

0.608

Now compare from left to right.

Least to greatest:

0.608, 0.65, 0.7, 0.72

Or:

0.608 < 0.65 < 0.7 < 0.72

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Worked Example 4: Least to Greatest

Order:

3.25, 3.8, 3.17, 3.205

Write:

3.250

3.800

3.170

3.205

All have the same whole-number part.

Compare tenths:

• 3.17 → 1 tenth
• 3.205 → 2 tenths
• 3.25 → 2 tenths
• 3.8 → 8 tenths

Between:

3.205 and 3.250

compare hundredths:

0 < 5

Therefore:

3.17 < 3.205 < 3.25 < 3.8

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Worked Example 5: Greatest to Least

Order:

5.09, 5.9, 5.19, 5.099

Write:

5.090

5.900

5.190

5.099

Compare place values.

Greatest:

5.900

Then:

5.190

Compare the remaining numbers:

5.099 > 5.090

Therefore:

5.9 > 5.19 > 5.099 > 5.09

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Ordering with a Number Line

Suppose we need to order:

1.2, 1.8, 1.45, 1.6

Place them on a number line between:

1 and 2

From left to right:

1.2, 1.45, 1.6, 1.8

Therefore:

1.2 < 1.45 < 1.6 < 1.8

The number line gives a visual representation of the ordering.

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Decimals Between Decimals

There are always more decimals between two different decimal numbers.

For example, between:

0.4 and 0.5

we can find:

0.41

0.42

0.43

and many others.

But even between:

0.41 and 0.42

we can find:

0.411

0.415

0.419

and so on.

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Decimal Density

This property is sometimes called the density of numbers.

Between any two different decimals, another number can be found.

For example:

Between:

2.5 and 2.6

we could choose:

2.55

Between:

2.55 and 2.56

we could choose:

2.555

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Justifying a Decimal Comparison

It is important to explain why one decimal is greater than another.

Instead of only writing:

4.37 > 4.29

we can justify:

"The whole-number digits are equal. In the tenths place, 3 tenths is greater than 2 tenths, so 4.37 is greater than 4.29."

This demonstrates place-value reasoning.

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Another Justification Example

Compare:

0.625 and 0.63

Write:

0.625

and:

0.630

Tenths:

6 = 6

Hundredths:

2 < 3

Therefore:

0.625 < 0.63

A good justification is:

"Both numbers have 6 tenths, but 0.625 has 2 hundredths while 0.630 has 3 hundredths. Therefore, 0.625 is smaller."

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Justifying with Fractions

Compare:

0.45 and 0.5

We can write:

0.45 = 45/100

and:

0.5 = 50/100

Since:

45/100 < 50/100

therefore:

0.45 < 0.5

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Justifying with a Number Line

Compare:

1.36 and 1.41

On a number line:

1.36

appears to the left of:

1.41

Therefore:

1.36 < 1.41

A number line can therefore serve as visual evidence for a comparison.

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Real-World Application: Money

Suppose two stores sell the same item for:

$8.75

and:

$8.59

Which price is lower?

Compare:

8.75 and 8.59

Whole dollars are equal.

Tenths:

7 > 5

Therefore:

8.75 > 8.59

So:

$8.59

is the lower price.

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Real-World Application: Running Times

Four runners record these times:

12.48 s

12.39 s

12.51 s

12.405 s

For running times, the smallest time represents the fastest performance.

Write with equal decimal places:

12.480

12.390

12.510

12.405

Order:

12.390 < 12.405 < 12.480 < 12.510

So the fastest time is:

12.39 s

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Real-World Application: Measurements

Four objects have masses:

2.35 kg

2.305 kg

2.5 kg

2.053 kg

Write:

2.350

2.305

2.500

2.053

Least to greatest:

2.053 < 2.305 < 2.35 < 2.5

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Real-World Application: Temperature

Suppose temperatures are:

18.7°C

18.25°C

18.62°C

18.9°C

Write:

18.70

18.25

18.62

18.90

Least to greatest:

18.25 < 18.62 < 18.7 < 18.9

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Real-World Application: Distances

Four hiking routes are:

4.8 km

4.75 km

4.805 km

4.58 km

Write:

4.800

4.750

4.805

4.580

Least to greatest:

4.58 < 4.75 < 4.8 < 4.805

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Worked Example 6

Which is greater?

0.507 or 0.57

Write:

0.507

0.570

Tenths:

5 = 5

Hundredths:

0 < 7

Therefore:

0.507 < 0.57

So:

0.57 is greater.

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Worked Example 7

Which is smaller?

8.09 or 8.009

Write:

8.090

8.009

Whole numbers:

8 = 8

Tenths:

0 = 0

Hundredths:

9 > 0

Therefore:

8.09 > 8.009

So:

8.009 is smaller.

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Worked Example 8

Order from least to greatest:

0.91, 0.109, 0.9, 0.19

Write:

0.910

0.109

0.900

0.190

Compare tenths first.

The order is:

0.109 < 0.19 < 0.9 < 0.91

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Worked Example 9

Order from greatest to least:

7.605, 7.65, 7.056, 7.6

Write:

7.605

7.650

7.056

7.600

Therefore:

7.65 > 7.605 > 7.6 > 7.056

Notice:

7.605 > 7.600

so:

7.605 > 7.6

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Worked Example 10

Place a number between:

3.42 and 3.43

One possible answer is:

3.425

Check:

3.420 < 3.425 < 3.430

Therefore:

3.425

is between the two numbers.

Many other answers are possible.

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Comparing Decimals Efficiently

You do not always need to add trailing zeros.

For:

7.3 and 7.1

the tenths immediately determine the answer:

7.3 > 7.1

For:

4.26 and 4.268

you need to compare farther:

4.260

and:

4.268

So:

4.26 < 4.268

Use only as many place values as necessary.

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A Reliable Comparison Strategy

When comparing two decimals:

Step 1: Compare the whole-number parts.

Step 2: If they are equal, compare tenths.

Step 3: If tenths are equal, compare hundredths.

Step 4: Continue through thousandths and further places if needed.

Step 5: Add trailing zeros if this helps you see the place values.

Step 6: Write the correct comparison symbol.

Step 7: Explain the first place value where the numbers differ.

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A Reliable Ordering Strategy

When ordering several decimals:

Step 1: Align the decimal points.

Step 2: Add trailing zeros if helpful.

Step 3: Compare the whole-number parts.

Step 4: Compare tenths.

Step 5: Compare hundredths.

Step 6: Continue as necessary.

Step 7: Arrange the numbers in the requested direction.

Step 8: Check the order using a number line or place-value reasoning.

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Common Mistakes

Mistake 1: Thinking the decimal with more digits is larger

Incorrect:

0.347 > 0.8 because 347 > 8

Correct:

0.347 < 0.8

because:

0.347 < 0.800

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Mistake 2: Comparing decimal digits as whole numbers

For:

0.9 and 0.12

do not simply compare:

9 and 12

Instead compare place values:

0.900 and 0.120

Therefore:

0.9 > 0.12

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Mistake 3: Ignoring zeros

Compare:

0.405 and 0.45

Write:

0.405

0.450

Therefore:

0.405 < 0.45

The zero in the hundredths position matters.

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Mistake 4: Confusing least-to-greatest and greatest-to-least

Least to greatest:

small → large

Greatest to least:

large → small

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Mistake 5: Thinking trailing zeros change value

Remember:

0.7 = 0.70 = 0.700

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Mistake 6: Comparing only the final digits

Always compare from the largest place value toward the smallest.

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Error Analysis

A student says:

0.62 < 0.589

because:

62 < 589

This reasoning is incorrect.

Write:

0.620

and:

0.589

Compare tenths:

6 > 5

Therefore:

0.620 > 0.589

So:

0.62 > 0.589

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Another Error Analysis

A student orders:

0.4, 0.35, 0.298

from least to greatest because:

4 < 35 < 298

But write:

0.400

0.350

0.298

Correct order:

0.298 < 0.35 < 0.4

The digits must be compared according to place value.

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Using Multiple Representations

Consider:

0.6 and 0.45

We can justify:

Using place value:

0.60 > 0.45

Using fractions:

60/100 > 45/100

Using a number line:

0.6 appears farther right than 0.45.

Using a hundred grid:

60 shaded squares represent more than 45 shaded squares.

All representations lead to the same conclusion:

0.6 > 0.45

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Did You Know?

There is no "next decimal" after a given decimal number.

For example, you might think the number after:

0.5

is:

0.6

But between them are:

0.51

0.52

0.55

0.59

and infinitely many more numbers.

Even between:

0.500 and 0.501

there are numbers such as:

0.5001

0.5005

0.5009

Decimal place value allows us to describe increasingly precise positions on the number line.

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Key Terms

• Decimal: Number containing a whole-number part, fractional part, or both, represented using a decimal point.
• Compare: Determine whether one number is greater than, less than, or equal to another.
• Order: Arrange numbers according to value.
• Place value: Value of a digit based on its position.
• Tenths: First place to the right of the decimal point.
• Hundredths: Second place to the right of the decimal point.
• Thousandths: Third place to the right of the decimal point.
• Greater than: Larger in value; represented by >.
• Less than: Smaller in value; represented by <.
• Equal to: Same value; represented by =.
• Least: Smallest value.
• Greatest: Largest value.
• Number line: Visual representation of numbers according to position and magnitude.
• Benchmark: Familiar value used as a reference for comparison.
• Equivalent decimals: Decimal representations with the same value.
• Trailing zero: Zero written at the right end of a decimal without changing its value.
• Justification: Explanation showing why a mathematical conclusion is correct.

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Key Relationships

0.5 = 0.50 = 0.500

0.7 = 70/100

0.45 = 45/100

Therefore:

0.7 > 0.45

Common benchmarks:

0.25 = 1/4

0.5 = 1/2

0.75 = 3/4

On a number line:

farther right = greater

farther left = smaller

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Key Takeaways

• Decimal numbers should be compared according to place value, not by the number of digits they contain.
• Begin comparisons with the largest place value.
• Compare whole-number parts first.
• If the whole-number parts are equal, compare tenths.
• If tenths are equal, compare hundredths.
• Continue through thousandths and further decimal places as necessary.
• The first place value where two numbers differ usually determines which number is greater.
• Trailing zeros can be added to decimals without changing their value.
• Writing decimals with the same number of decimal places can make comparisons easier.
• A decimal with more digits is not necessarily larger.
• Number lines provide a visual method for comparing decimals.
• Numbers farther to the right on a number line are greater.
• Benchmark values such as 0, 0.25, 0.5, 0.75, and 1 can help with comparisons.
• Decimals can also be compared by converting them to equivalent fractions.
• Least to greatest means arranging numbers from smallest to largest.
• Greatest to least means arranging numbers from largest to smallest.
• A strong justification identifies the first place value where two numbers differ.
• Decimal comparisons can be justified using place value, fractions, number lines, or visual models.
• Between any two different decimal numbers, there are infinitely many other numbers.
• Comparing and ordering decimals is important when working with money, measurements, scientific data, distances, temperatures, and recorded times.