1. Understanding Decimal Place Value

 

 

 

5

What Is a Decimal?

A decimal is a way of representing numbers that include parts of a whole.

Consider:

3.7

The number contains:

• 3 whole units
• 7 tenths of another unit

The decimal point separates the whole-number part from the fractional part.

In:

3.7

the 3 is to the left of the decimal point and represents:

3 ones

The 7 is to the right and represents:

7 tenths

So:

3.7 = 3 + 7/10

---

The Decimal Point

The decimal point is extremely important because it determines the place value of every digit.

Compare:

25

2.5

0.25

The digits 2 and 5 appear in each number, but their values change because their positions change.

 

 

 

6

For example, the digit 2 represents:

• 20 in 25
• 2 in 2.5
• 0.2 in 0.25

Position determines value.

---

Place Value to the Left of the Decimal

Whole-number place values increase as we move left.

For example:

4,382

contains:

• 4 thousands
• 3 hundreds
• 8 tens
• 2 ones

Each position is worth:

10 times

the position immediately to its right.

---

Place Value to the Right of the Decimal

The pattern continues to the right of the decimal point.

The first position is:

tenths

The second is:

hundredths

The third is:

thousandths

The fourth is:

ten-thousandths

The fifth is:

hundred-thousandths

and so on.

 

 

 

6

---

The Place Value Pattern

The place values around the decimal point follow this pattern:

Thousands | Hundreds | Tens | Ones | decimal point | Tenths | Hundredths | Thousandths

Notice that there is no "oneths" place.

We move directly from:

ones

to:

tenths

---

Each Place Is Ten Times Smaller

Moving one position to the right divides the place value by:

10

For example:

1

One tenth of 1 is:

0.1

One tenth of 0.1 is:

0.01

One tenth of 0.01 is:

0.001

So:

1 → 0.1 → 0.01 → 0.001

 

 

 

7

Each step to the right makes the place value 10 times smaller.

---

Each Place Is Ten Times Larger

Moving one position to the left multiplies the place value by:

10

For example:

0.001 → 0.01 → 0.1 → 1 → 10 → 100

Each step to the left makes the place value:

10 times larger

This base-ten pattern is the foundation of our decimal number system.

---

Tenths

The first position to the right of the decimal point is the:

tenths place

One tenth can be written as:

1/10

or:

0.1

For example:

0.7

means:

7 tenths

Therefore:

0.7 = 7/10

 

 

 

5

---

Hundredths

The second position to the right of the decimal point is the:

hundredths place

One hundredth is:

1/100

or:

0.01

For example:

0.36

contains:

• 3 tenths
• 6 hundredths

So:

0.36 = 3/10 + 6/100

It can also be written:

36/100

---

Thousandths

The third position to the right is the:

thousandths place

One thousandth is:

1/1000

or:

0.001

For example:

0.428

contains:

• 4 tenths
• 2 hundredths
• 8 thousandths

Therefore:

0.428 = 4/10 + 2/100 + 8/1000

It can also be written:

428/1000

 

 

 

5

---

Using a Decimal Place Value Chart

Consider:

3,482.675

A place value chart would show:

Thousands	Hundreds	Tens	Ones	.	Tenths	Hundredths	Thousandths
3	4	8	2	.	6	7	5

This means:

• 3 thousands = 3,000
• 4 hundreds = 400
• 8 tens = 80
• 2 ones = 2
• 6 tenths = 0.6
• 7 hundredths = 0.07
• 5 thousandths = 0.005

---

Expanded Form

Decimals can be written in expanded form.

For:

3,482.675

expanded form is:

3,000 + 400 + 80 + 2 + 0.6 + 0.07 + 0.005

We could also use fractions:

3,000 + 400 + 80 + 2 + 6/10 + 7/100 + 5/1000

Expanded form makes the value of each digit clear.

---

Worked Example 1

Consider:

27.483

Identify each digit.

2 is in the tens place:

20

7 is in the ones place:

7

4 is in the tenths place:

0.4

8 is in the hundredths place:

0.08

3 is in the thousandths place:

0.003

Therefore:

27.483 = 20 + 7 + 0.4 + 0.08 + 0.003

---

The Difference Between a Digit and Its Value

A digit is one of the symbols:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

But the value of a digit depends on where it appears.

Consider the digit:

5

 

 

 

4

In:

500

the 5 has a value of:

500

In:

50

its value is:

50

In:

5

its value is:

5

In:

0.5

its value is:

0.5

In:

0.05

its value is:

0.05

In:

0.005

its value is:

0.005

The digit stays the same, but its value changes.

---

Zeros as Placeholders

Zeros are important in decimal numbers.

Compare:

0.5

and:

0.05

These are not equal.

0.5 = 5/10

while:

0.05 = 5/100

Therefore:

0.5 > 0.05

The zero in 0.05 shows that there are:

0 tenths

and:

5 hundredths

---

Another Zero Example

Consider:

4.307

This means:

• 4 ones
• 3 tenths
• 0 hundredths
• 7 thousandths

Expanded:

4 + 0.3 + 0.007

The zero keeps the 7 in the correct:

thousandths place

 

 

 

---

Zeros at the End of a Decimal

Zeros added to the right end of a decimal do not change its value.

For example:

0.5 = 0.50 = 0.500

Why?

Because:

5/10 = 50/100 = 500/1000

These are equivalent fractions.

Similarly:

3.7 = 3.70 = 3.700

---

Reading Decimal Numbers

One way to read decimals is to read the whole-number part, say and for the decimal point, and then name the fractional part using its final place value.

For example:

4.7

can be read:

four and seven tenths

---

Reading Hundredths

6.25

is:

six and twenty-five hundredths

because:

0.25 = 25/100

---

Reading Thousandths

12.438

is:

twelve and four hundred thirty-eight thousandths

because:

0.438 = 438/1000

 

 

 

5

---

Reading Numbers with Zeros

Consider:

5.007

This is:

five and seven thousandths

It is not:

five and seven hundredths

because the 7 is in the:

thousandths place

The zeros are important.

---

Writing Decimals from Words

Suppose we have:

eight and three tenths

Write:

8.3

---

Worked Example 2

Write:

twelve and forty-five hundredths

The whole-number part is:

12

The fractional part is:

45/100 = 0.45

Therefore:

12.45

---

Worked Example 3

Write:

six and nine thousandths

Nine thousandths is:

0.009

Therefore:

6.009

Notice the two zeros before the 9.

---

Standard Form, Word Form, and Expanded Form

The same number can be represented in several ways.

Consider:

24.306

Standard form:

24.306

Word form:

twenty-four and three hundred six thousandths

Expanded form:

20 + 4 + 0.3 + 0.006

Different forms highlight different information about the same number.

---

Decimals as Fractions

Every terminating decimal can be written as a fraction.

For example:

0.4 = 4/10

0.27 = 27/100

0.583 = 583/1000

 

 

 

6

The decimal place tells us the denominator.

---

Tenths and Fractions

If a decimal ends in the tenths place, we can initially write it over:

10

For example:

0.6 = 6/10

Simplify:

6/10 = 3/5

Therefore:

0.6 = 3/5

---

Hundredths and Fractions

If a decimal ends in the hundredths place, write it over:

100

For example:

0.25 = 25/100

Simplify:

25/100 = 1/4

Therefore:

0.25 = 1/4

---

Thousandths and Fractions

If a decimal ends in the thousandths place, write it over:

1000

For example:

0.125 = 125/1000

Simplify:

125/1000 = 1/8

Therefore:

0.125 = 1/8

---

Mixed Numbers and Decimals

Decimals greater than 1 can also be related to mixed numbers.

For example:

2.5

means:

2 + 5/10

Since:

5/10 = 1/2

we have:

2.5 = 2 1/2

---

Another Mixed Number Example

Consider:

3.75

This means:

3 + 75/100

Simplify:

75/100 = 3/4

Therefore:

3.75 = 3 3/4

 

 

 

5

---

Visualizing Tenths

Imagine a rectangle divided into:

10 equal sections

If 3 sections are shaded, the shaded portion represents:

3/10

or:

0.3

This is why the first decimal place represents tenths.

---

Visualizing Hundredths

Now imagine a square divided into:

100 equal smaller squares

If 37 squares are shaded, the shaded amount is:

37/100

or:

0.37

 

 

 

5

Hundred grids are useful for visualizing:

• decimals
• fractions
• percentages

---

Visualizing Thousandths

A cube can be divided into:

1,000 equal small cubes

Each small cube represents:

1/1000

or:

0.001

Ten small cubes represent:

0.010 = 0.01

One hundred small cubes represent:

0.100 = 0.1

This shows how tenths, hundredths, and thousandths are connected.

---

Decimal Place Value and Money

Money provides a familiar example of decimal place value.

Consider:

$4.75

The:

4

represents 4 dollars.

The:

7

represents 7 tenths of a dollar, or 70 cents.

The:

5

represents 5 hundredths of a dollar, or 5 cents.

Therefore:

$4.75 = 4 dollars and 75 cents

 

 

 

6

Money commonly uses two decimal places because:

$1 = 100 cents

---

Decimal Place Value in Measurement

Decimals are also common in measurements.

For example:

2.75 m

means:

2 metres + 0.75 metre

Since:

0.75 = 3/4

this is:

2 3/4 metres

Decimals allow measurements to be expressed more precisely than whole numbers alone.

---

Decimal Place Value in Science

Measurements in science frequently contain decimals.

Examples include:

12.4 g

3.75 m

0.025 L

18.6°C

 

 

 

6

The position of every digit matters because moving a digit by one place changes its value by a factor of:

10

---

Decimals on a Number Line

Decimals can be located between whole numbers.

For example:

0.5

is halfway between:

0 and 1

The decimal:

1.5

is halfway between:

1 and 2

 

 

 

5

Decimals allow us to describe positions between whole numbers.

---

Zooming In on a Number Line

Suppose the distance from:

0 to 1

is divided into 10 equal intervals.

Each interval represents:

0.1

If one of those intervals is divided into another 10 equal pieces, each smaller interval represents:

0.01

This pattern can continue:

0.001

0.0001

and beyond.

There is always another decimal place available.

---

Worked Example 4

Consider:

8.542

What is the value of the digit 5?

The 5 is in the:

tenths place

Therefore, its value is:

0.5

or:

5/10

---

Worked Example 5

Consider:

8.542

What is the value of the digit 4?

The 4 is in the:

hundredths place

Therefore:

0.04

or:

4/100

---

Worked Example 6

Consider:

8.542

What is the value of the digit 2?

The 2 is in the:

thousandths place

Therefore:

0.002

or:

2/1000

---

Worked Example 7

Write:

5 + 0.7 + 0.03 + 0.009

in standard form.

Combine the place values:

5.739

Therefore:

5 + 0.7 + 0.03 + 0.009 = 5.739

---

Worked Example 8

Write:

16.405

in expanded form.

The number contains:

• 1 ten
• 6 ones
• 4 tenths
• 0 hundredths
• 5 thousandths

Therefore:

16.405 = 10 + 6 + 0.4 + 0.005

---

Worked Example 9

Convert:

0.48

to a fraction.

Since 8 is in the hundredths place:

0.48 = 48/100

Simplify by dividing by 4:

48/100 = 12/25

Therefore:

0.48 = 12/25

---

Worked Example 10

Convert:

2.125

to a mixed number.

Separate the whole-number part:

2 + 0.125

Convert:

0.125 = 125/1000

Simplify:

125/1000 = 1/8

Therefore:

2.125 = 2 1/8

---

Comparing the Size of Decimal Places

Consider the digit 7 in each number:

7

0.7

0.07

0.007

Its values are:

7

7/10

7/100

7/1000

 

 

 

4

Each time the digit moves one place to the right, its value becomes:

1/10 as large

---

Multiplying and Dividing by 10

Place value also explains what happens when we multiply or divide by powers of 10.

Consider:

3.47 × 10 = 34.7

The digits become worth ten times as much.

Consider:

3.47 ÷ 10 = 0.347

The digits become worth one tenth as much.

It is more accurate to think about the digits changing place value than to say that the decimal point itself moves.

---

Equivalent Decimals

Decimals can look different but have the same value.

For example:

0.6 = 0.60 = 0.600

Why?

Because:

6/10 = 60/100 = 600/1000

 

 

 

4

Trailing zeros do not change the value of a decimal.

---

But Leading Zeros Matter

Compare:

0.6

and:

0.06

These are not equivalent.

0.6 = 6/10

0.06 = 6/100

Therefore:

0.6 = 10 × 0.06

Zeros between the decimal point and a nonzero digit determine the digit's place value.

---

Common Mistakes

Mistake 1: Thinking 0.5 and 0.05 are equal

They are not.

0.5 = 5/10

0.05 = 5/100

Therefore:

0.5 > 0.05

---

Mistake 2: Thinking more digits means a larger number

For example:

0.125

has more digits than:

0.9

but:

0.125 < 0.9

Place value determines size.

---

Mistake 3: Ignoring zeros

In:

4.007

the 7 represents:

7 thousandths

not:

7 hundredths

---

Mistake 4: Calling 0.34 "thirty-four tenths"

The final digit is in the hundredths place.

Therefore:

0.34 = thirty-four hundredths

---

Mistake 5: Forgetting that the decimal system is base ten

Each place is:

10 times

the place to its right.

---

A Reliable Place Value Strategy

When analyzing a decimal:

Step 1: Locate the decimal point.

Step 2: Identify the whole-number places to the left.

Step 3: Identify tenths, hundredths, thousandths, and further places to the right.

Step 4: Determine the value of each digit.

Step 5: Use expanded form if necessary.

Step 6: Relate the decimal portion to a fraction with denominator 10, 100, 1000, and so on.

---

Did You Know?

Our decimal system is called a base-ten system because each place is based on powers of 10.

 

 

 

6

Whole-number places include:

1, 10, 100, 1,000, ...

Decimal places include:

1/10, 1/100, 1/1000, ...

This means the decimal point is not the beginning of a completely different number system.

It is simply the point where our base-ten place-value pattern moves from whole units into fractions of a unit.

---

Key Terms

• Decimal: Number written using a decimal point to represent whole units and parts of a whole.
• Decimal point: Symbol separating the whole-number and fractional parts of a decimal.
• Place value: Value of a digit based on its position.
• Digit: One of the symbols 0 through 9.
• 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.
• Standard form: Usual numerical way of writing a number.
• Word form: Writing a number using words.
• Expanded form: Writing a number as the sum of its place-value parts.
• Equivalent decimals: Different decimal representations with the same value.
• Fraction: Number representing part of a whole or a division of quantities.
• Base ten: Number system in which each place is ten times the value of the place immediately to its right.
• Placeholder: A zero used to maintain the correct position and value of other digits.

---

Key Relationships

1 tenth = 1/10 = 0.1

1 hundredth = 1/100 = 0.01

1 thousandth = 1/1000 = 0.001

10 tenths = 1 whole

10 hundredths = 1 tenth

10 thousandths = 1 hundredth

Examples:

0.7 = 7/10

0.35 = 35/100

0.428 = 428/1000

Equivalent decimals:

0.5 = 0.50 = 0.500

---

Key Takeaways

• Decimals represent whole numbers and parts of a whole.
• The decimal point separates the whole-number part from the fractional part.
• The first place to the right of the decimal point is tenths.
• The second place is hundredths.
• The third place is thousandths.
• There is no "oneths" place.
• Each place to the left is ten times the value of the place immediately to its right.
• Each place to the right is one tenth the value of the place immediately to its left.
• The value of a digit depends on its position.
• Place value charts help organize and interpret decimal numbers.
• Zeros can act as important placeholders.
• Trailing zeros do not change the value of a decimal, so 0.5 = 0.50 = 0.500.
• Zeros between the decimal point and a nonzero digit do affect place value, so 0.5 ≠ 0.05.
• Decimals can be written in standard, word, and expanded forms.
• Tenths can be written as fractions with denominator 10.
• Hundredths can be written as fractions with denominator 100.
• Thousandths can be written as fractions with denominator 1000.
• Decimal fractions can often be simplified to equivalent fractions.
• Decimals greater than one can be related to mixed numbers.
• Decimal place value is used extensively in money, measurement, science, engineering, and everyday life.
• Understanding decimal place value provides the foundation for comparing, rounding, adding, subtracting, multiplying, and dividing decimals.

 

 

 

2. Comparing and Ordering Decimals

 

 

 

7

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.

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

 

 

 

5

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

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.

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

 

 

 

5

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.

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

 

 

 

4

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.

 

 

 

4

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

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

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

 

 

 

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.

 

 

 

4

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

 

 

 

4

Benchmarks can help us estimate the relative size of decimals.

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.

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

 

 

 

5

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

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

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

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.

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.

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

 

 

 

5

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.

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."

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

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.

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.

 

 

 

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

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

 

 

 

4

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

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

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.

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.

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

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

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.

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.

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.

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.

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

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

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.

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

Least to greatest:

small → large

Greatest to least:

large → small

Mistake 5: Thinking trailing zeros change value

Remember:

0.7 = 0.70 = 0.700

Mistake 6: Comparing only the final digits

Always compare from the largest place value toward the smallest.

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

 

 

 

4

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.

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.

 

 

 

5

All representations lead to the same conclusion:

0.6 > 0.45

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.

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.

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

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.

3. Rounding and Estimating Decimals

 

 

 

6

What Is Rounding?

Rounding means replacing a number with a nearby value that is simpler to use.

For example:

6.83

rounded to the nearest whole number is:

7

The rounded number is not exactly equal to the original number.

Instead:

6.83 ≈ 7

The symbol:

≈

means approximately equal to.

Rounding is useful when an exact value is unnecessary or when we want to estimate a calculation quickly.

Why Do We Round Numbers?

Imagine a journey is exactly:

397.8 km

In conversation, someone might say:

about 400 km

The rounded value communicates the approximate distance more simply.

We use rounding when:

• estimating costs
• estimating distances
• predicting calculation results
• summarizing measurements
• checking answers
• communicating data at an appropriate level of precision

 

 

 

6

Decimal Place Values

Before rounding decimals, identify the place values.

Consider:

24.6837

The digits represent:

• 2 tens
• 4 ones
• 6 tenths
• 8 hundredths
• 3 thousandths
• 7 ten-thousandths

If we are asked to round this number, we first identify the rounding place.

Common Rounding Places

Decimals are commonly rounded to the nearest:

• whole number
• tenth
• hundredth
• thousandth

For example:

7.486

could be rounded differently depending on the required precision.

Nearest whole number:

7

Nearest tenth:

7.5

Nearest hundredth:

7.49

Nearest thousandth:

7.486

The requested place value matters.

 

 

 

4

The Basic Rounding Rule

To round a decimal:

Step 1: Identify the place value you are rounding to.

Step 2: Look at the digit immediately to its right.

Step 3: If that digit is 0, 1, 2, 3, or 4, keep the rounding digit unchanged.

Step 4: If that digit is 5, 6, 7, 8, or 9, increase the rounding digit by 1.

Step 5: Remove the digits to the right.

A common memory rule is:

0–4: keep

5–9: round up

Rounding to the Nearest Whole Number

Consider:

8.3

The ones digit is:

8

Look at the tenths digit:

3

Since 3 is less than 5, keep the 8.

Therefore:

8.3 ≈ 8

Another Whole-Number Example

Round:

8.7

to the nearest whole number.

Look at the tenths digit:

7

Since 7 is 5 or greater, increase the ones digit.

Therefore:

8.7 ≈ 9

 

 

 

4

Why the Rule Works

Rounding is really about determining which benchmark is closer.

Consider:

4.2

It lies between:

4 and 5

The distance from 4.2 to 4 is:

0.2

The distance from 4.2 to 5 is:

0.8

Therefore, 4.2 is closer to 4.

So:

4.2 ≈ 4

Rounding on a Number Line

Consider:

6.7

It lies between:

6 and 7

The halfway point is:

6.5

Since 6.7 lies above the halfway point, it is closer to 7.

Therefore:

6.7 ≈ 7

 

 

 

The number line helps explain why 5 is important in the rounding rule.

Rounding to the Nearest Tenth

Consider:

4.36

We want to round to the nearest tenth.

The tenths digit is:

3

Look one place to the right, at the hundredths digit:

6

Since:

6 ≥ 5

increase the tenths digit from 3 to 4.

Therefore:

4.36 ≈ 4.4

Another Tenths Example

Round:

7.82

to the nearest tenth.

Tenths digit:

8

Hundredths digit:

2

Since:

2 < 5

keep the tenths digit.

Therefore:

7.82 ≈ 7.8

Number Line for Tenths

Consider:

3.47

To round to the nearest tenth, compare it with:

3.4 and 3.5

The halfway point is:

3.45

Since:

3.47 > 3.45

3.47 is closer to 3.5.

Therefore:

3.47 ≈ 3.5

 

 

 

5

Rounding to the Nearest Hundredth

Consider:

5.273

We want the nearest hundredth.

Hundredths digit:

7

Look at the thousandths digit:

3

Since:

3 < 5

keep the 7.

Therefore:

5.273 ≈ 5.27

Another Hundredths Example

Round:

5.278

to the nearest hundredth.

Hundredths digit:

7

Thousandths digit:

8

Since:

8 ≥ 5

increase 7 to 8.

Therefore:

5.278 ≈ 5.28

Rounding to the Nearest Thousandth

Consider:

12.4837

We want the nearest thousandth.

Thousandths digit:

3

Look at the ten-thousandths digit:

7

Since:

7 ≥ 5

increase the 3 to 4.

Therefore:

12.4837 ≈ 12.484

 

 

 

4

Worked Example 1

Round:

18.746

to the nearest whole number.

Look at the tenths digit:

7

Round up.

18.746 ≈ 19

Worked Example 2

Round:

18.746

to the nearest tenth.

Tenths digit:

7

Hundredths digit:

4

Keep the 7.

18.746 ≈ 18.7

Worked Example 3

Round:

18.746

to the nearest hundredth.

Hundredths digit:

4

Thousandths digit:

6

Round up.

18.746 ≈ 18.75

Notice that the same number can produce different rounded values depending on the requested place value.

When Rounding Causes Regrouping

Sometimes rounding causes a digit to become 10.

Consider:

3.98

rounded to the nearest tenth.

The tenths digit is:

9

The hundredths digit is:

8

So the 9 rounds up.

But:

9 + 1 = 10

Therefore, regroup:

3.98 ≈ 4.0

to the nearest tenth.

Another Regrouping Example

Round:

9.997

to the nearest hundredth.

Hundredths digit:

9

Thousandths digit:

7

Round up.

This causes regrouping through several places.

Therefore:

9.997 ≈ 10.00

to the nearest hundredth.

 

 

 

Precision Matters

Compare:

4

4.0

4.00

Numerically, these have the same value.

However, in measurement contexts they can communicate different levels of precision.

For example:

4.00 m

suggests the measurement has been reported to the nearest hundredth of a metre.

This becomes particularly important in science and engineering.

What Is Estimation?

Estimation means finding an approximate answer rather than an exact answer.

Suppose:

4.87 + 3.12

Instead of calculating exactly, round:

4.87 ≈ 5

3.12 ≈ 3

Then:

5 + 3 = 8

So:

4.87 + 3.12 ≈ 8

The exact answer is:

7.99

Our estimate is very close.

Estimation and Rounding

Rounding is one of the most common ways to estimate calculations.

For example:

12.78 + 6.14

Round to whole numbers:

13 + 6

Estimate:

19

Exact:

18.92

The estimate gives us a useful idea of the expected answer.

 

 

 

Estimating Decimal Addition

Estimate:

7.86 + 4.21

Round to whole numbers:

7.86 ≈ 8

4.21 ≈ 4

Then:

8 + 4 = 12

So the estimate is:

12

Exact calculation:

7.86 + 4.21 = 12.07

The estimate is reasonable.

Estimating Decimal Subtraction

Estimate:

15.73 − 8.16

Round:

15.73 ≈ 16

8.16 ≈ 8

Then:

16 − 8 = 8

Exact calculation:

15.73 − 8.16 = 7.57

The exact answer is reasonably close to the estimate.

Estimating Decimal Multiplication

Estimate:

6.18 × 3.9

Round:

6.18 ≈ 6

3.9 ≈ 4

Then:

6 × 4 = 24

Exact calculation:

6.18 × 3.9 = 24.102

Our estimate of:

24

is very close.

 

 

 

Estimating Decimal Division

Estimate:

19.8 ÷ 4.1

Use nearby compatible numbers:

20 ÷ 4 = 5

Therefore:

19.8 ÷ 4.1 ≈ 5

The exact quotient is approximately:

4.83

So the estimate is reasonable.

Compatible Numbers

Sometimes standard rounding is not the easiest estimation strategy.

Instead, use compatible numbers: nearby values that are easy to calculate mentally.

For example:

24.7 ÷ 4.9

Rather than simply rounding each value according to a fixed place, notice:

24.7 ≈ 25

and:

4.9 ≈ 5

Then:

25 ÷ 5 = 5

So:

24.7 ÷ 4.9 ≈ 5

Front-End Estimation

Another strategy is front-end estimation.

This focuses on the largest place values first.

For:

43.72 + 28.46

use:

40 + 20 = 60

Then consider the remaining amounts to improve the estimate.

A better estimate might be:

44 + 28 = 72

The exact answer is:

72.18

Front-end estimation can be useful when a quick approximation is more important than high precision.

Overestimates and Underestimates

An estimate can be slightly above or below the exact answer.

Suppose:

4.2 + 6.3

Round both to whole numbers:

4 + 6 = 10

Exact:

10.5

The estimate is lower than the exact answer.

This is an underestimate.

Overestimate Example

Consider:

4.8 + 6.7

Round:

5 + 7 = 12

Exact:

11.5

The estimate is higher than the exact answer.

This is an overestimate.

 

 

 

Checking Reasonableness

One of the most important uses of estimation is checking whether an exact calculation makes sense.

Suppose someone calculates:

6.24 × 3.1 = 193.44

Estimate:

6 × 3 = 18

The exact answer should be somewhere near:

18 or 20

An answer of:

193.44

is far too large.

Therefore, the calculation is unreasonable.

The actual answer is:

19.344

Another Reasonableness Check

Suppose:

42.6 ÷ 6.1

A student calculates:

69.8

Estimate:

42 ÷ 6 = 7

The quotient should be around:

7

not:

70

So the answer is unreasonable.

This type of error often comes from incorrect decimal placement.

Decimal Placement and Estimation

Estimation is especially useful for detecting incorrectly placed decimal points.

Suppose:

3.8 × 4.2

We know:

4 × 4 ≈ 16

So the answer should be near:

16

If a calculator entry or written calculation gives:

159.6

or:

1.596

we know the decimal point is probably misplaced.

The exact answer is:

15.96

 

 

 

4

How Accurate Should an Estimate Be?

The appropriate precision depends on the purpose.

Suppose an item costs:

$49.95

If you simply want a quick idea of the price:

about $50

is useful.

If you are calculating whether you have enough money to pay exactly, an estimate may not be sufficient.

The context determines how much precision is needed.

When Estimates Are Appropriate

Estimates are useful when:

• making quick comparisons
• planning a budget
• predicting costs
• checking calculations
• estimating travel distances
• planning quantities
• interpreting measurements
• checking calculator results
• exact values are unavailable
• an approximate answer is sufficient

When Exact Answers Are Needed

An estimate may not be enough when:

• paying an exact bill
• calculating medication quantities
• performing precise scientific measurements
• manufacturing components to exact specifications
• reporting financial records
• calculating final assessment scores
• exact legal or contractual values are required

The required precision depends on the situation.

Real-World Application: Shopping

Suppose you buy:

• an item for $12.89
• an item for $7.35
• an item for $19.76

You want to know approximately how much money you need.

Round:

$12.89 ≈ $13

$7.35 ≈ $7

$19.76 ≈ $20

Estimate:

$13 + $7 + $20 = $40

 

 

 

4

The exact total is:

$40.00

In this case, the estimate happens to match the exact total.

Real-World Application: Budgeting

Suppose your weekly expenses are:

$48.75

$32.40

$21.95

$17.80

For quick planning, round:

49 + 32 + 22 + 18

Estimate:

$121

The exact total is:

$120.90

The estimate is useful for planning.

Real-World Application: Travel

A journey has three sections:

48.7 km

31.4 km

22.8 km

Estimate:

49 + 31 + 23

= 103 km

Exact distance:

102.9 km

So describing the trip as:

about 103 km

is reasonable.

 

 

 

Real-World Application: Measurement

A room measures:

5.92 m × 3.87 m

Estimate the area.

Round:

5.92 ≈ 6

3.87 ≈ 4

Then:

6 × 4 = 24

Estimated area:

24 m²

Exact area:

5.92 × 3.87 = 22.9104 m²

The estimate tells us the area should be around:

23–24 m²

Real-World Application: Fuel

A car uses approximately:

7.8 L

of fuel per 100 km.

For a quick estimate over:

400 km

think:

about 8 L per 100 km

There are:

4 groups of 100 km

So estimated fuel use:

8 × 4 = 32 L

Using the original rate:

7.8 × 4 = 31.2 L

The estimate is useful for travel planning.

Real-World Application: Science

Suppose three measured masses are:

12.47 g

8.96 g

6.62 g

For a quick estimate:

12.47 ≈ 12.5

8.96 ≈ 9.0

6.62 ≈ 6.6

Then:

12.5 + 9.0 + 6.6 = 28.1 g

 

 

 

4

Depending on the purpose, using tenths may provide a more useful estimate than rounding everything to whole numbers.

Choosing the Best Level of Precision

Suppose we want to estimate:

18.46 + 7.82

Rounding to whole numbers:

18 + 8 = 26

Rounding to tenths:

18.5 + 7.8 = 26.3

Exact:

26.28

Both estimates are reasonable.

The second estimate is more precise, but it also requires slightly more work.

A good estimate balances:

speed and usefulness

Worked Example 1: Rounding

Round:

14.672

to the nearest tenth.

Tenths digit:

6

Hundredths digit:

7

Round up:

14.672 ≈ 14.7

Worked Example 2: Rounding

Round:

14.672

to the nearest hundredth.

Hundredths digit:

7

Thousandths digit:

2

Keep the 7.

14.672 ≈ 14.67

Worked Example 3: Estimating Addition

Estimate:

28.74 + 16.19

Round to whole numbers:

29 + 16 = 45

Exact:

44.93

Therefore:

45

is a good estimate.

Worked Example 4: Estimating Subtraction

Estimate:

63.82 − 28.14

Round:

64 − 28 = 36

Exact:

35.68

The estimate is reasonable.

Worked Example 5: Estimating Multiplication

Estimate:

9.72 × 5.13

Use:

10 × 5 = 50

Exact:

49.8636

So:

50

is an excellent estimate.

Worked Example 6: Estimating Division

Estimate:

61.4 ÷ 9.8

Use compatible numbers:

60 ÷ 10 = 6

Exact quotient is approximately:

6.27

Therefore, the estimate is reasonable.

Worked Example 7: Checking Reasonableness

A student claims:

8.12 + 4.73 = 128.5

Estimate:

8 + 5 = 13

An answer near:

13

is expected.

Therefore:

128.5

is unreasonable.

Correct calculation:

8.12 + 4.73 = 12.85

Worked Example 8: Practical Estimation

A restaurant bill contains:

$18.95

$12.40

$7.75

$5.10

Estimate:

19 + 12 + 8 + 5

= $44

Exact:

$44.20

Therefore, about:

$44

is a useful estimate.

Worked Example 9: Choosing Precision

A scientist records:

12.6847 cm

Rounded to the nearest:

whole number: 13 cm

tenth: 12.7 cm

hundredth: 12.68 cm

thousandth: 12.685 cm

The appropriate result depends on the precision required.

Worked Example 10: Comparing Estimate and Exact Answer

Calculate:

24.86 × 3.12

Estimate:

25 × 3 = 75

Exact:

24.86 × 3.12 = 77.5632

The exact answer is reasonably close to the estimate.

Therefore, the result appears sensible.

Estimation Before Using a Calculator

Estimation is valuable even when calculators are available.

Before entering:

48.72 × 6.14

estimate:

50 × 6 = 300

If the calculator returns:

299.0208

the result is reasonable.

If it returns:

29.90208

you should check your input.

 

 

 

5

A calculator can perform arithmetic, but it does not know whether you entered the intended calculation.

Estimation Is Not Guessing

An estimate should be based on mathematical reasoning.

A random guess:

"Maybe the answer is about 50."

is not the same as an estimate.

A mathematical estimate explains why:

24.8 + 26.1 ≈ 25 + 26 = 51

The estimate is supported by nearby values and a calculation.

Common Mistakes

Mistake 1: Looking at the wrong digit

To round to the hundredths place, look at the:

thousandths digit

not the tenths digit.

Mistake 2: Changing digits before the rounding place

Only the rounding digit may increase by 1.

Other digits to its left remain unchanged unless regrouping occurs.

Mistake 3: Thinking 5 always means the whole number increases

The 5 affects only the digit in the place being rounded.

Mistake 4: Forgetting regrouping

For:

7.98

rounded to the nearest tenth:

7.98 ≈ 8.0

not:

7.10

Mistake 5: Treating an estimate as an exact answer

Use:

≈

rather than:

=

when writing an approximate value.

Mistake 6: Using too much or too little precision

Rounding:

$19.97

to:

$0

would be far too rough for most shopping situations.

Choose a useful level of precision.

Mistake 7: Accepting unreasonable calculator answers

Always compare the exact result with an approximate result.

Error Analysis

A student rounds:

6.483

to the nearest hundredth and writes:

6.5

This is incorrect.

The hundredths digit is:

8

The digit immediately to its right is:

3

Since 3 is less than 5, keep the 8.

Therefore:

6.483 ≈ 6.48

 

 

 

5

Another Error Analysis

A student calculates:

7.9 × 4.2 = 331.8

Estimate:

8 × 4 = 32

The calculated answer:

331.8

is about ten times too large.

Correct:

7.9 × 4.2 = 33.18

Estimation helps identify the decimal-placement error.

A Reliable Rounding Strategy

When rounding:

Step 1: Identify the requested place value.

Step 2: Underline or identify the digit in that place.

Step 3: Look at the digit immediately to its right.

Step 4: If it is 0–4, keep the rounding digit.

Step 5: If it is 5–9, increase the rounding digit by 1.

Step 6: Remove digits to the right.

Step 7: Check whether the rounded result makes sense.

A Reliable Estimation Strategy

When estimating a calculation:

Step 1: Examine the numbers.

Step 2: Decide how accurate the estimate needs to be.

Step 3: Round or choose compatible numbers.

Step 4: Perform the simpler calculation.

Step 5: State that the result is approximate.

Step 6: If an exact answer is available, compare it with the estimate.

Step 7: Investigate if the exact result is unexpectedly far from the estimate.

Did You Know?

Rounding always involves a trade-off between simplicity and precision.

 

 

 

5

Consider:

17.4836

We could report it as:

17

17.5

17.48

17.484

Each rounded value is easier to communicate than the original to a different degree, but each also preserves a different amount of information.

There is therefore no single "best" place to round every number.

The appropriate precision depends on why the number is being used.

Key Terms

• Rounding: Replacing a number with a nearby value at a specified place value.
• Estimate: Approximate value based on mathematical reasoning.
• Approximate: Close to the exact value but not necessarily equal to it.
• Exact value: Precise result of a calculation or measurement.
• Place value: Value of a digit based on its position.
• Tenths: First decimal place.
• Hundredths: Second decimal place.
• Thousandths: Third decimal place.
• Compatible numbers: Nearby numbers chosen because they are easy to calculate with.
• Front-end estimation: Estimating primarily using the largest place values.
• Overestimate: Estimate greater than the exact value.
• Underestimate: Estimate less than the exact value.
• Reasonableness: Whether an answer makes sense based on the original quantities.
• Precision: Level of detail with which a value is expressed.
• Benchmark: Familiar or convenient value used for comparison.

Key Rules

To round to a specified place:

0, 1, 2, 3, 4 → keep the rounding digit

5, 6, 7, 8, 9 → increase the rounding digit by 1

Look only at the digit immediately to the right of the place being rounded.

Examples:

4.32 ≈ 4.3 to the nearest tenth

4.37 ≈ 4.4 to the nearest tenth

8.264 ≈ 8.26 to the nearest hundredth

8.268 ≈ 8.27 to the nearest hundredth

Key Takeaways

• Rounding replaces a number with a nearby value that is easier to use.
• The requested place value determines how a decimal should be rounded.
• Common rounding positions include whole numbers, tenths, hundredths, and thousandths.
• To round, examine the digit immediately to the right of the rounding place.
• Digits 0–4 cause the rounding digit to remain unchanged.
• Digits 5–9 cause the rounding digit to increase by 1.
• Number lines help explain rounding as choosing the nearest benchmark.
• Rounding can sometimes cause regrouping, such as 3.98 ≈ 4.0 to the nearest tenth.
• Estimation provides an approximate result rather than an exact answer.
• Rounding and compatible numbers are useful estimation strategies.
• Addition, subtraction, multiplication, and division with decimals can all be estimated.
• Estimation is useful for checking the reasonableness of exact calculations.
• Estimation is especially valuable for detecting misplaced decimal points.
• Estimates can be overestimates or underestimates.
• The amount of rounding should match the purpose of the calculation.
• Some situations require exact answers, while others only require useful approximations.
• Estimation is mathematical reasoning, not random guessing.
• Calculator answers should still be checked using estimation.
• Rounding and estimation are widely used in shopping, budgeting, travel, measurement, science, engineering, and everyday decision-making.
• A strong decimal calculation includes not only an accurate answer but also an understanding of whether that answer is reasonable.

4. Decimal Operations

 

 

 

5

What Are Decimal Operations?

Decimals can be used with the same four basic operations as whole numbers:

• addition
• subtraction
• multiplication
• division

However, decimal calculations require careful attention to place value.

For example:

3.5 + 0.27

cannot be treated as if it were:

35 + 27

The digits represent different place values.

Understanding decimal place value allows us to perform decimal operations accurately and explain why the methods work.

---

Why Decimal Operations Matter

Decimals appear throughout everyday life.

We use decimal operations when working with:

• money
• measurements
• distance
• mass
• temperature
• time
• scientific data
• sports statistics
• construction
• engineering

 

 

 

6

For example, if two items cost:

$4.75 and $2.89

we use decimal addition to calculate the total cost.

---

Estimating Before Calculating

Before performing a decimal calculation, it is often useful to estimate.

Suppose:

6.82 + 4.13

Estimate:

7 + 4 = 11

The exact answer should therefore be somewhere near:

11

This gives us a benchmark for checking our calculation.

---

Adding Decimals

When adding decimals, the most important rule is:

Align the decimal points.

This ensures that:

• ones are added to ones
• tenths are added to tenths
• hundredths are added to hundredths
• thousandths are added to thousandths

 

 

 

6

---

Example: Adding Decimals

Calculate:

4.32 + 2.45

Align the decimal points:

 

  4.32
+ 2.45
------
  6.77

 

Therefore:

4.32 + 2.45 = 6.77

---

Why Decimal Points Must Align

Consider:

4.32 + 2.45

The calculation really means:

4 ones + 2 ones

3 tenths + 4 tenths

2 hundredths + 5 hundredths

So:

4.32 + 2.45 = 6.77

Place-value alignment makes sure that we combine equal units.

---

Adding Decimals with Different Numbers of Digits

Consider:

7.4 + 2.36

Write:

7.4 = 7.40

Now align:

 

  7.40
+ 2.36
------
  9.76

 

Therefore:

7.4 + 2.36 = 9.76

Trailing zeros can be added without changing the value of a decimal.

---

Worked Example 1: Addition with Regrouping

Calculate:

8.67 + 5.78

Hundredths:

7 + 8 = 15 hundredths

Regroup:

15 hundredths = 1 tenth + 5 hundredths

Tenths:

6 + 7 + 1 = 14 tenths

Regroup again.

The result is:

14.45

Therefore:

8.67 + 5.78 = 14.45

 

 

 

5

---

Adding Several Decimal Numbers

Suppose:

2.45 + 8.7 + 0.326

Write:

 

  2.450
  8.700
+ 0.326
-------
 11.476

 

Therefore:

2.45 + 8.7 + 0.326 = 11.476

Adding trailing zeros can make the place values easier to see.

---

Checking Decimal Addition

We can check addition using:

• estimation
• subtraction
• recalculation

For:

6.38 + 4.27 = 10.65

Estimate:

6 + 4 = 10

So:

10.65

is reasonable.

We can also check:

10.65 − 4.27 = 6.38

---

Subtracting Decimals

Decimal subtraction also depends on place-value alignment.

The key rule remains:

Align the decimal points.

 

 

 

6

---

Example: Subtracting Decimals

Calculate:

8.76 − 3.24

Align:

 

  8.76
- 3.24
------
  5.52

 

Therefore:

8.76 − 3.24 = 5.52

---

Subtracting Decimals with Different Lengths

Calculate:

9.5 − 2.37

Write:

9.5 = 9.50

Then:

 

  9.50
- 2.37
------
  7.13

 

Therefore:

9.5 − 2.37 = 7.13

---

Subtraction with Regrouping

Consider:

6.42 − 2.78

We cannot subtract:

8 hundredths from 2 hundredths

without regrouping.

Regroup one tenth:

4 tenths becomes 3 tenths

and:

2 hundredths becomes 12 hundredths

Then:

12 − 8 = 4 hundredths

We must regroup again in the tenths column.

The final result is:

3.64

Therefore:

6.42 − 2.78 = 3.64

 

 

 

---

Subtracting from a Whole Number

Calculate:

12 − 4.68

Write 12 as:

12.00

Then:

 

 12.00
- 4.68
------
  7.32

 

Therefore:

12 − 4.68 = 7.32

Writing the whole number with decimal zeros makes regrouping easier to see.

---

Checking Decimal Subtraction

Addition is the inverse of subtraction.

If:

9.84 − 3.27 = 6.57

then check:

6.57 + 3.27 = 9.84

Estimation also helps:

10 − 3 ≈ 7

So:

6.57

is reasonable.

---

Adding and Subtracting Money

Money provides a common use of decimal addition and subtraction.

Suppose an item costs:

$18.75

and another costs:

$6.89

Total:

$18.75 + $6.89 = $25.64

 

 

 

5

If you pay:

$30.00

your change is:

$30.00 − $25.64 = $4.36

---

Multiplying Decimals

Decimal multiplication is slightly different from addition and subtraction.

When multiplying decimals, we do not need to align decimal points.

Instead:

1. multiply as if the numbers were whole numbers
2. determine where the decimal point belongs in the product

---

Example: Decimal × Whole Number

Calculate:

3.4 × 6

First ignore the decimal temporarily:

34 × 6 = 204

Since 3.4 has one decimal place, the product must have one decimal place:

20.4

Therefore:

3.4 × 6 = 20.4

 

 

 

4

---

Understanding Decimal Multiplication

Consider:

3.4 × 6

Since:

3.4 = 34/10

we have:

34/10 × 6

= 204/10

= 20.4

The decimal rule comes from place value and fractions.

---

Decimal × Decimal

Calculate:

2.4 × 1.3

Ignore the decimals temporarily:

24 × 13 = 312

Now count the decimal places.

2.4 has:

1 decimal place

1.3 has:

1 decimal place

Total:

2 decimal places

Therefore:

2.4 × 1.3 = 3.12

---

Why the Product Is 3.12

We can also use fractions:

2.4 = 24/10

1.3 = 13/10

Therefore:

24/10 × 13/10

= 312/100

= 3.12

 

 

---

Worked Example 2: Decimal Multiplication

Calculate:

4.25 × 3.2

Ignore the decimals:

425 × 32 = 13,600

Count decimal places:

4.25 has:

2

3.2 has:

1

Total:

3 decimal places

So:

13.600

Therefore:

4.25 × 3.2 = 13.6

---

Estimating Decimal Products

Before multiplying:

4.25 × 3.2

estimate:

4 × 3 = 12

The exact answer:

13.6

is close to our estimate.

This helps confirm that the decimal point is in a reasonable position.

---

Multiplying by Numbers Less Than 1

An important idea is that multiplication does not always make a number larger.

Consider:

8 × 0.5 = 4

Multiplying by 0.5 means finding:

one-half of 8

Similarly:

20 × 0.25 = 5

because:

0.25 = 1/4

 

 

 

4

When multiplying a positive number by a decimal between 0 and 1, the product is smaller than the original number.

---

Multiplying by Powers of Ten

Place value changes when multiplying by:

10, 100, 1,000

For example:

3.47 × 10 = 34.7

3.47 × 100 = 347

3.47 × 1,000 = 3,470

Each digit becomes worth:

10, 100, or 1,000 times as much

It is better to think of the digits changing place value rather than simply saying "move the decimal point."

---

Dividing Decimals

Decimal division can involve several different situations:

• decimal ÷ whole number
• whole number ÷ decimal
• decimal ÷ decimal

The main goal is to transform the calculation into a form we can divide accurately.

 

 

 

6

---

Decimal Divided by a Whole Number

Calculate:

8.4 ÷ 4

We can think:

84 tenths ÷ 4

= 21 tenths

= 2.1

Therefore:

8.4 ÷ 4 = 2.1

---

Using Long Division

For:

15.6 ÷ 3

divide as usual.

15 ÷ 3:

5

Then divide the tenths:

6 tenths ÷ 3:

2 tenths

Therefore:

15.6 ÷ 3 = 5.2

---

Decimal Divided by a Decimal

Consider:

7.2 ÷ 0.6

Dividing by a decimal is easier if we create an equivalent calculation with a whole-number divisor.

Multiply both numbers by 10:

7.2 ÷ 0.6

becomes:

72 ÷ 6

Now:

72 ÷ 6 = 12

Therefore:

7.2 ÷ 0.6 = 12

 

 

 

---

Why We Multiply Both Numbers

Multiplying both the dividend and divisor by the same number does not change the quotient.

For example:

8 ÷ 2 = 4

and:

80 ÷ 20 = 4

Similarly:

7.2 ÷ 0.6

has the same quotient as:

72 ÷ 6

This allows us to change a decimal divisor into a whole number.

---

More Than One Decimal Place

Calculate:

4.32 ÷ 0.12

The divisor has two decimal places.

Multiply both numbers by:

100

So:

4.32 ÷ 0.12

becomes:

432 ÷ 12

Now:

432 ÷ 12 = 36

Therefore:

4.32 ÷ 0.12 = 36

---

Worked Example 3: Decimal Division

Calculate:

18.75 ÷ 2.5

Multiply both numbers by 10:

187.5 ÷ 25

Now divide:

187.5 ÷ 25 = 7.5

Therefore:

18.75 ÷ 2.5 = 7.5

Check:

7.5 × 2.5 = 18.75

---

Dividing by Numbers Less Than 1

Division by a number less than 1 can produce a result larger than the original number.

For example:

6 ÷ 0.5 = 12

Why?

The question means:

How many halves are in 6?

There are:

12 halves

in 6 wholes.

 

 

This is why the rule "division always makes numbers smaller" is incorrect.

---

Division and Measurement

Suppose you have:

4.5 m

of ribbon.

Each piece must be:

0.5 m

long.

Number of pieces:

4.5 ÷ 0.5 = 9

Therefore:

9 pieces

can be made.

---

Dividing by Powers of Ten

Consider:

47.8 ÷ 10 = 4.78

47.8 ÷ 100 = 0.478

47.8 ÷ 1,000 = 0.0478

Each digit becomes worth:

1/10, 1/100, or 1/1000

of its previous value.

 

 

 

---

Estimation Before Decimal Division

Estimate:

19.6 ÷ 4.1

Use compatible numbers:

20 ÷ 4 = 5

Therefore, we expect the exact quotient to be close to:

5

This helps us detect unreasonable answers.

---

Checking Division with Multiplication

Multiplication and division are inverse operations.

If:

12.6 ÷ 3 = 4.2

then:

4.2 × 3 = 12.6

If the multiplication returns the original dividend, the division is likely correct.

---

Operation Relationships

Addition and subtraction are inverse operations.

If:

4.8 + 2.7 = 7.5

then:

7.5 − 2.7 = 4.8

Multiplication and division are also inverse operations.

If:

2.5 × 4 = 10

then:

10 ÷ 4 = 2.5

Understanding inverse operations helps us check calculations.

---

Multi-Step Decimal Problems

Many practical problems require more than one operation.

A useful process is:

Step 1: Identify the information.

Step 2: Determine what needs to be found.

Step 3: Decide which operations are needed.

Step 4: Estimate the expected answer.

Step 5: Perform the calculations.

Step 6: Check the result.

Step 7: State the answer with appropriate units.

---

Multi-Step Example 1: Shopping

A customer buys:

3 notebooks at $4.75 each

and:

2 pens at $1.85 each

Find the total cost.

Notebooks:

3 × $4.75 = $14.25

Pens:

2 × $1.85 = $3.70

Total:

$14.25 + $3.70 = $17.95

Therefore:

Total cost = $17.95

 

 

 

4

---

Multi-Step Example 2: Change

Suppose the customer pays with:

$20.00

The total cost is:

$17.95

Change:

$20.00 − $17.95 = $2.05

Therefore:

Change = $2.05

---

Multi-Step Example 3: Distance

A runner completes:

4 laps

of a:

2.75 km

course.

Total distance:

4 × 2.75 = 11.0 km

The runner then walks:

1.6 km

Total:

11.0 + 1.6 = 12.6 km

Therefore:

Total distance = 12.6 km

---

Multi-Step Example 4: Sharing a Cost

Four friends buy food costing:

$38.40

and drinks costing:

$9.60

Total:

$38.40 + $9.60 = $48.00

Divide equally:

$48.00 ÷ 4 = $12.00

Therefore:

Each person pays $12.00

 

 

 

6

---

Multi-Step Example 5: Measurement

A rectangular garden measures:

8.5 m

by:

4.2 m

Area:

8.5 × 4.2 = 35.7 m²

A path occupies:

6.8 m²

Remaining garden area:

35.7 − 6.8 = 28.9 m²

Therefore:

28.9 m²

remains for planting.

---

Multi-Step Example 6: Fuel

A vehicle begins with:

42.5 L

of fuel.

It uses:

6.8 L

on one journey and:

9.75 L

on another.

Total used:

6.8 + 9.75 = 16.55 L

Fuel remaining:

42.5 − 16.55

Write:

42.50 − 16.55 = 25.95

Therefore:

25.95 L

remain.

 

 

 

5

---

Multi-Step Example 7: Unit Price

A package containing:

6 bottles

costs:

$15.90

Cost per bottle:

$15.90 ÷ 6 = $2.65

If you buy:

4 bottles

at this rate:

4 × $2.65 = $10.60

Therefore:

4 bottles cost $10.60

---

Multi-Step Example 8: Science

A laboratory has:

12.5 L

of solution.

It uses:

2.75 L

in one experiment and:

1.8 L

in another.

Amount remaining:

12.5 − 2.75 − 1.8

First:

12.50 − 2.75 = 9.75

Then:

9.75 − 1.80 = 7.95

Therefore:

7.95 L

remain.

---

Order of Operations

Multi-step decimal calculations follow the same order of operations as whole-number calculations.

A common order is:

1. Parentheses
2. Exponents
3. Multiplication and division from left to right
4. Addition and subtraction from left to right

For example:

3.5 + 2.4 × 5

Multiply first:

2.4 × 5 = 12

Then add:

3.5 + 12 = 15.5

Therefore:

3.5 + 2.4 × 5 = 15.5

---

Parentheses Can Change the Result

Compare:

3.5 + 2.4 × 5

with:

(3.5 + 2.4) × 5

First expression:

3.5 + 12 = 15.5

Second expression:

5.9 × 5 = 29.5

 

 

 

4

The parentheses change the order of the calculation and therefore change the answer.

---

Worked Example 4: Addition

Calculate:

18.75 + 6.8

Write:

18.75 + 6.80

Then:

18.75 + 6.80 = 25.55

Estimate:

19 + 7 = 26

The answer is reasonable.

---

Worked Example 5: Subtraction

Calculate:

20 − 7.86

Write:

20.00 − 7.86

Result:

12.14

Estimate:

20 − 8 = 12

Reasonable.

---

Worked Example 6: Multiplication

Calculate:

6.4 × 2.35

Ignore the decimals:

64 × 235 = 15,040

Total decimal places:

1 + 2 = 3

Therefore:

15.040 = 15.04

Estimate:

6 × 2.5 ≈ 15

Reasonable.

---

Worked Example 7: Division

Calculate:

9.45 ÷ 1.5

Multiply both numbers by 10:

94.5 ÷ 15

Result:

6.3

Check:

6.3 × 1.5 = 9.45

Therefore:

9.45 ÷ 1.5 = 6.3

---

Worked Example 8: Multi-Step Calculation

Calculate:

8.4 + 3.6 × 2.5

Multiply first:

3.6 × 2.5 = 9

Then:

8.4 + 9 = 17.4

Therefore:

8.4 + 3.6 × 2.5 = 17.4

---

Worked Example 9: Multi-Step Calculation with Parentheses

Calculate:

(8.4 + 3.6) ÷ 2.5

Parentheses first:

8.4 + 3.6 = 12

Then:

12 ÷ 2.5 = 4.8

Therefore:

(8.4 + 3.6) ÷ 2.5 = 4.8

---

Worked Example 10: Practical Problem

A roll of cable is:

25.5 m

long.

A technician uses:

3 pieces of 4.25 m

each.

Length used:

3 × 4.25 = 12.75 m

Length remaining:

25.50 − 12.75 = 12.75 m

Therefore:

12.75 m

of cable remains.

---

Estimating Multi-Step Problems

Consider:

4.85 × 3 + 7.26

Estimate:

5 × 3 + 7

= 15 + 7

= 22

Exact:

4.85 × 3 = 14.55

Then:

14.55 + 7.26 = 21.81

The exact result is close to:

22

so it is reasonable.

 

 

 

5

---

Choosing the Correct Operation

In practical problems, deciding which operation to use is often more important than performing the calculation.

Use addition when combining quantities.

Use subtraction when finding a difference or what remains.

Use multiplication when dealing with equal groups, repeated quantities, rates, or area.

Use division when sharing equally, finding the number of groups, or calculating a unit rate.

Some problems require several of these operations.

---

Checking Decimal Calculations

There are several useful checking strategies.

Estimate

Compare the answer with an approximate result.

Use inverse operations

Addition ↔ Subtraction

Multiplication ↔ Division

Calculate using another method

For example, use fractions or an area model.

Check units

An answer of "$14.8 metres" would indicate that something is wrong with the interpretation.

---

Error Analysis: Addition

A student calculates:

4.6 + 2.35 = 2.81

The student has probably aligned the digits rather than the decimal points.

Correct alignment:

 

  4.60
+ 2.35
------
  6.95

 

Therefore:

4.6 + 2.35 = 6.95

---

Error Analysis: Subtraction

A student calculates:

8.2 − 3.47 = 5.35

Estimate:

8 − 3.5 ≈ 4.5

So:

5.35

already appears suspicious.

Correct calculation:

8.20 − 3.47 = 4.73

---

Error Analysis: Multiplication

A student calculates:

3.2 × 1.4 = 44.8

Estimate:

3 × 1.5 ≈ 4.5

So 44.8 is much too large.

Calculate:

32 × 14 = 448

There are two decimal places in total.

Therefore:

3.2 × 1.4 = 4.48

 

 

 

4

---

Error Analysis: Division

A student calculates:

8.4 ÷ 0.7 = 1.2

Think about the meaning.

How many groups of:

0.7

fit into:

8.4?

There should be many more than one group.

Multiply both numbers by 10:

84 ÷ 7 = 12

Therefore:

8.4 ÷ 0.7 = 12

---

Common Mistakes

Mistake 1: Not aligning decimal points in addition and subtraction

Always align according to place value.

---

Mistake 2: Aligning decimal points when multiplying

Decimal multiplication uses the number of decimal places, not decimal-point alignment.

---

Mistake 3: Moving only one decimal when dividing

If you multiply the divisor by 10 or 100, the dividend must be multiplied by the same amount.

---

Mistake 4: Assuming multiplication always makes numbers larger

For example:

10 × 0.2 = 2

---

Mistake 5: Assuming division always makes numbers smaller

For example:

10 ÷ 0.2 = 50

---

Mistake 6: Ignoring estimation

Estimation can quickly reveal misplaced decimal points.

---

Mistake 7: Ignoring the order of operations

Multiplication and division must usually be completed before addition and subtraction.

---

A Reliable Decimal Addition and Subtraction Strategy

Step 1: Estimate.

Step 2: Write the numbers vertically.

Step 3: Align decimal points.

Step 4: Add trailing zeros if helpful.

Step 5: Calculate using place value.

Step 6: Place the decimal point directly in line with the others.

Step 7: Compare the answer with the estimate.

---

A Reliable Decimal Multiplication Strategy

Step 1: Estimate the product.

Step 2: Multiply as if the factors were whole numbers.

Step 3: Count the total number of decimal places in the factors.

Step 4: Place the decimal in the product.

Step 5: Remove unnecessary trailing zeros if appropriate.

Step 6: Compare with the estimate.

---

A Reliable Decimal Division Strategy

Step 1: Estimate the quotient.

Step 2: If the divisor is a decimal, multiply both dividend and divisor by the same power of 10 until the divisor is a whole number.

Step 3: Divide normally.

Step 4: Check the quotient using multiplication.

Step 5: Compare with the estimate.

---

Did You Know?

Decimal operations are really extensions of operations with whole numbers and fractions.

 

 

 

6

For example:

0.3 + 0.4

is equivalent to:

3/10 + 4/10

which gives:

7/10 = 0.7

Similarly:

0.3 × 0.4

means:

3/10 × 4/10

which gives:

12/100 = 0.12

This explains why place value is central to decimal operations.

---

Key Terms

• Decimal: Number written using a decimal point to represent whole units and parts of a whole.
• Sum: Result of addition.
• Difference: Result of subtraction.
• Product: Result of multiplication.
• Quotient: Result of division.
• Place value: Value of a digit based on its position.
• Regrouping: Rewriting quantities using different place-value units.
• Dividend: Number being divided.
• Divisor: Number by which another number is divided.
• Inverse operations: Operations that undo each other.
• Estimate: Approximate result used for prediction or checking.
• Compatible numbers: Nearby numbers chosen because they are easy to calculate with.
• Order of operations: Rules determining the sequence in which operations are performed.
• Reasonableness: Whether an answer makes sense based on the quantities involved.
• Unit rate: Amount corresponding to one unit of another quantity.

---

Key Rules

For addition and subtraction:

Align decimal points.

For multiplication:

Multiply as whole numbers, then use place value to determine the decimal position.

For division by a decimal:

Create an equivalent calculation with a whole-number divisor by multiplying both numbers by the same power of 10.

For multi-step calculations:

Follow the order of operations.

For all decimal calculations:

Estimate before or after calculating to check reasonableness.

---

Key Takeaways

• Decimal operations follow the same basic mathematical principles as whole-number operations.
• Place value is essential when working with decimals.
• When adding decimals, align the decimal points.
• When subtracting decimals, align the decimal points.
• Trailing zeros can be added when needed to make place values clear.
• Addition and subtraction may require regrouping across decimal places.
• Decimal addition can be checked using subtraction.
• Decimal subtraction can be checked using addition.
• When multiplying decimals, multiply as if the factors were whole numbers and then determine the correct decimal position.
• Multiplying by a decimal less than 1 can make a positive number smaller.
• When dividing by a decimal, create an equivalent division problem with a whole-number divisor.
• Dividing by a decimal less than 1 can make a positive quotient larger than the dividend.
• Decimal division can be checked using multiplication.
• Multiplying or dividing by powers of 10 changes the place value of the digits.
• Estimation is an important way to check the position of the decimal point.
• Multi-step decimal problems may require addition, subtraction, multiplication, and division.
• Multi-step calculations must follow the order of operations.
• Practical decimal problems should include appropriate units.
• Decimal operations are widely used with money, measurement, distance, area, fuel, rates, scientific data, and everyday calculations.
• A strong decimal solution combines accurate calculation, correct place value, appropriate operations, estimation, and a reasonableness check.

 

 

 

5. Applications of Decimals

 

 

 

6

Why Are Decimals Important?

Decimals allow us to describe quantities that fall between whole numbers.

A distance might be:

4.7 km

A mass might be:

2.35 kg

An item might cost:

$8.99

A temperature might be:

21.6°C

Decimals are especially useful when quantities need to be measured or recorded with greater precision than whole numbers provide.

They appear throughout everyday life, science, business, engineering, technology, and statistics.

Decimals and Place Value

Understanding decimal applications begins with place value.

Consider:

24.583

This means:

• 2 tens
• 4 ones
• 5 tenths
• 8 hundredths
• 3 thousandths

The decimal point separates the whole-number part from the fractional part.

 

 

 

6

The value of each digit depends on its position.

Decimals and Money

Money is one of the most familiar applications of decimals.

For example:

$12.75

means:

12 dollars and 75 cents

because:

$1 = 100 cents

Therefore:

$0.75 = 75 cents

The hundredths place is particularly important when working with most currencies divided into 100 smaller units.

Reading Money Correctly

Consider:

$8.05

This means:

8 dollars and 5 cents

It does not mean:

8 dollars and 50 cents

The zero is an important placeholder.

Similarly:

$8.50

means:

8 dollars and 50 cents

 

 

 

6

Adding Money

Suppose you buy:

• a sandwich for $6.75
• a drink for $2.40
• fruit for $1.85

Find the total cost.

Align the decimal points:

$6.75 + $2.40 + $1.85 = $11.00

Therefore:

Total cost = $11.00

Calculating Change

Suppose your purchase costs:

$17.65

and you pay:

$20.00

Calculate:

$20.00 − $17.65 = $2.35

Therefore:

Change = $2.35

A quick estimate also confirms the answer:

$20 − $18 ≈ $2

So $2.35 is reasonable.

Comparing Prices

Suppose two stores sell the same item.

Store A:

$14.95

Store B:

$13.89

Compare the prices:

13.89 < 14.95

So Store B has the lower listed price.

The difference is:

$14.95 − $13.89 = $1.06

 

 

 

6

Buying Several of the Same Item

Suppose one notebook costs:

$3.75

You buy:

6 notebooks

Calculate:

6 × $3.75 = $22.50

Therefore:

Total cost = $22.50

This combines decimal multiplication with a practical money calculation.

Finding Unit Price

A package containing 8 drinks costs:

$14.40

Cost per drink:

$14.40 ÷ 8 = $1.80

Therefore:

Unit price = $1.80 per drink

Unit prices can help compare products sold in different package sizes.

Decimals and Discounts

Suppose an item normally costs:

$60.00

It is discounted by:

$12.50

Sale price:

$60.00 − $12.50 = $47.50

Therefore:

Sale price = $47.50

Decimals are commonly used when calculating discounts, taxes, tips, and final prices.

 

 

 

5

Budgeting with Decimals

Suppose a weekly budget includes:

• Food: $62.75
• Transport: $24.50
• Entertainment: $18.25
• Other expenses: $12.80

Total spending:

$62.75 + $24.50 + $18.25 + $12.80

= $118.30

If the budget is:

$150.00

money remaining:

$150.00 − $118.30 = $31.70

Estimating Money Calculations

Estimation is useful when shopping.

Suppose your basket contains items costing:

$8.95

$12.20

$5.75

$16.10

Round:

$9 + $12 + $6 + $16

Estimated total:

$43

Exact total:

$43.00

Estimation helps determine whether you have enough money and provides a check for exact calculations.

Decimals and Measurement

Measurements often fall between whole-number values.

Examples include:

1.72 m

4.35 kg

2.6 L

12.4 cm

8.75 km

 

 

 

6

Decimals allow measurements to be recorded more precisely.

Measuring Length

Suppose a table is:

1.85 m

long.

This means:

1 metre + 0.85 metre

Since:

0.85 m = 85 cm

the length could also be described as:

1 m 85 cm

Adding Measurements

Suppose two boards have lengths:

2.45 m

and:

1.78 m

Total length:

2.45 + 1.78 = 4.23

Therefore:

Total length = 4.23 m

Always include the unit in the final answer.

Finding a Difference in Measurements

One plant is:

1.42 m

tall.

Another is:

0.87 m

tall.

Difference:

1.42 − 0.87 = 0.55

Therefore:

The taller plant is 0.55 m taller.

 

 

 

5

Mass and Decimals

A package might have a mass of:

2.75 kg

If 4 identical packages are used:

2.75 × 4 = 11.00

Therefore:

Total mass = 11 kg

Decimals allow masses between whole kilograms to be represented accurately.

Volume and Capacity

Suppose a container holds:

1.5 L

of water.

Four containers hold:

1.5 × 4 = 6.0 L

If:

2.35 L

is used:

6.00 − 2.35 = 3.65 L

remain.

 

 

 

7

Measuring Area

Decimals are also common in area calculations.

A rectangular room measures:

5.4 m × 3.8 m

Area:

5.4 × 3.8 = 20.52

Therefore:

Area = 20.52 m²

Notice that the unit for area is:

square metres (m²)

Measuring Perimeter

For the same room:

Length:

5.4 m

Width:

3.8 m

Perimeter:

2(5.4 + 3.8)

First:

5.4 + 3.8 = 9.2

Then:

2 × 9.2 = 18.4

Therefore:

Perimeter = 18.4 m

Decimals and Temperature

Temperature measurements often contain decimals.

For example:

21.5°C

37.2°C

−4.8°C

Suppose the temperature rises from:

18.6°C

to:

23.4°C

Increase:

23.4 − 18.6 = 4.8°C

 

 

 

5

Decimals and Distance

Suppose a cyclist travels:

12.75 km

in the morning and:

8.6 km

in the afternoon.

Total distance:

12.75 + 8.60 = 21.35 km

If the goal was:

25 km

distance remaining:

25.00 − 21.35 = 3.65 km

Decimals and Time

Time is sometimes expressed using decimals.

For example:

1.5 hours

means:

1 hour + 0.5 hour

Since:

0.5 × 60 = 30 minutes

we have:

1.5 hours = 1 hour 30 minutes

However, decimal time must be interpreted carefully because:

1 hour = 60 minutes

not 100 minutes.

Another Time Example

Convert:

2.25 hours

to hours and minutes.

Whole-number part:

2 hours

Decimal part:

0.25 hour

Calculate:

0.25 × 60 = 15 minutes

Therefore:

2.25 hours = 2 hours 15 minutes

 

 

 

5

Decimals in Data

Decimals are frequently used when collecting and reporting data.

For example, a science experiment might record:

Decimal notation allows small differences between measurements to be recorded.

Reading Decimal Data

When interpreting data, ask questions such as:

• What does each decimal represent?
• What units are being used?
• Which value is greatest?
• Which value is least?
• What is the difference between values?
• Is there a trend?
• How precise are the measurements?
• Are any values unusual?

 

 

 

6

Finding the Range

Consider the data:

4.2, 4.7, 4.4, 4.9, 4.3

Greatest value:

4.9

Least value:

4.2

Range:

4.9 − 4.2 = 0.7

Therefore:

Range = 0.7

Finding a Mean from Decimal Data

Suppose four measurements are:

2.4, 2.8, 2.6, 3.0

Add:

2.4 + 2.8 + 2.6 + 3.0 = 10.8

Divide by 4:

10.8 ÷ 4 = 2.7

Therefore:

Mean = 2.7

Interpreting Decimal Graphs

Graphs often use decimal scales.

For example, an axis might be labelled:

0, 0.5, 1.0, 1.5, 2.0

To interpret the graph correctly, you must understand the interval between values.

 

 

 

4

Always check:

• axis labels
• units
• scale intervals
• decimal place values

before interpreting data.

Decimals in Sports

Decimals are used to record:

• race times
• distances
• averages
• percentages
• speeds
• scores in some sports

Suppose three runners record:

12.84 s

12.71 s

12.93 s

The fastest runner has the smallest time.

Therefore:

12.71 s

is the fastest time.

Decimals in Science

Scientific measurements often require decimals.

A laboratory might record:

• mass = 24.68 g
• volume = 12.5 mL
• temperature = 22.4°C
• distance = 1.275 m
• time = 4.82 s

 

 

 

5

Decimals allow scientists to communicate measurements with appropriate precision.

Decimals in Engineering and Construction

A design might specify:

2.75 m

14.6 mm

8.25 kg

3.125 cm

Small differences can matter when parts need to fit together accurately.

For this reason, correct decimal notation and measurement are important in:

• construction
• manufacturing
• engineering
• architecture

Decimals in Maps and Navigation

Distances on maps and navigation systems frequently use decimals.

For example:

3.8 km

12.45 km

0.75 km

A journey consisting of:

3.8 km + 12.45 km + 0.75 km

has a total distance of:

17.00 km

 

 

 

5

Multi-Step Problem 1: Shopping

A customer buys:

• 3 books at $7.85 each
• 2 pens at $1.65 each

Books:

3 × $7.85 = $23.55

Pens:

2 × $1.65 = $3.30

Total:

$23.55 + $3.30 = $26.85

If the customer pays:

$30.00

Change:

$30.00 − $26.85 = $3.15

Therefore:

The customer receives $3.15 change.

Multi-Step Problem 2: Travel

A vehicle travels:

125.6 km

on Monday and:

98.75 km

on Tuesday.

Total:

125.60 + 98.75 = 224.35 km

If the vehicle used:

17.5 L

of fuel:

224.35 ÷ 17.5 ≈ 12.82

Therefore, the vehicle travelled approximately:

12.82 km per litre

Multi-Step Problem 3: Flooring

A rectangular room measures:

6.5 m × 4.2 m

Area:

6.5 × 4.2 = 27.3 m²

If each box of flooring covers:

2.1 m²

number of boxes:

27.3 ÷ 2.1 = 13

Therefore:

13 boxes

are required.

 

 

 

4

Multi-Step Problem 4: Recipe

A recipe requires:

0.75 kg

of flour for one batch.

For 4 batches:

0.75 × 4 = 3.00 kg

If you have:

3.5 kg

of flour:

3.50 − 3.00 = 0.50 kg

Therefore:

0.5 kg of flour remains.

Multi-Step Problem 5: Data Analysis

A student records these plant heights:

14.2 cm

15.1 cm

14.8 cm

15.5 cm

Total:

14.2 + 15.1 + 14.8 + 15.5 = 59.6 cm

Mean:

59.6 ÷ 4 = 14.9 cm

Therefore:

Mean height = 14.9 cm

Multi-Step Problem 6: Water Use

A tank contains:

48.5 L

of water.

During the day:

12.75 L

is used.

Then:

8.6 L

is added.

Calculate:

48.50 − 12.75 = 35.75

Then:

35.75 + 8.60 = 44.35

Therefore:

44.35 L of water remains in the tank.

 

 

 

4

Multi-Step Problem 7: Comparing Unit Prices

Package A contains:

6 items for $10.50

Unit price:

$10.50 ÷ 6 = $1.75

Package B contains:

8 items for $13.20

Unit price:

$13.20 ÷ 8 = $1.65

Package B has the lower unit price:

$1.65 per item

This demonstrates why unit price can be more informative than simply comparing total package prices.

Multi-Step Problem 8: Fuel Cost

A vehicle needs:

32.5 L

of fuel.

Fuel costs:

$2.14 per litre

Total cost:

32.5 × $2.14 = $69.55

Estimate:

33 × $2 ≈ $66

So:

$69.55

is reasonable.

Choosing the Correct Operation

Real-world problems do not always tell you directly which operation to use.

Look at what is happening in the situation.

Use addition when quantities are being combined.

Use subtraction when finding:

• a difference
• what remains
• change

Use multiplication for:

• equal groups
• repeated quantities
• area
• cost per item × number of items

Use division for:

• equal sharing
• unit rates
• number of groups
• amount per group

Multi-Step Problems Require Planning

Consider:

"A store has 18.5 kg of rice. It packs the rice equally into 5 bags. Two bags are sold. How much rice remains?"

First find the amount per bag:

18.5 ÷ 5 = 3.7 kg

Two bags contain:

2 × 3.7 = 7.4 kg

Remaining:

18.5 − 7.4 = 11.1 kg

Therefore:

11.1 kg of rice remains.

The challenge is not only performing decimal operations but deciding which operations are needed and in what order.

Estimating Real-World Answers

Before calculating exactly, estimate.

Suppose a meal costs:

$18.75

and a drink costs:

$4.85

Estimate:

$19 + $5 = $24

Exact:

$18.75 + $4.85 = $23.60

The exact answer is close to the estimate.

Therefore, it is reasonable.

 

 

 

6

Checking Reasonableness

Suppose a student calculates:

4.8 kg × 6 = 288 kg

Estimate:

5 × 6 = 30

The answer should be close to:

30 kg

not:

288 kg

The correct calculation is:

4.8 × 6 = 28.8 kg

Estimation helps identify the misplaced decimal point.

Communicating Decimal Solutions

A correct calculation is only part of a strong mathematical solution.

A clear solution should include:

• the calculation
• correct decimal notation
• appropriate units
• suitable precision
• a sentence answering the question when needed

For example:

Weak answer:

14.75

Better answer:

The total distance travelled was 14.75 km.

Always Include Units

Consider:

8.5 × 3.2 = 27.2

Without context, this is simply a number.

But if the values are the dimensions of a rectangle:

8.5 m × 3.2 m = 27.2 m²

The unit tells us what the answer represents.

 

 

 

5

Choosing Appropriate Decimal Notation

Different situations use different conventions.

Money is normally written to two decimal places:

$7.50

rather than:

$7.5

For a measurement, the number of decimal places may indicate precision:

2.4 cm

and:

2.40 cm

have the same numerical value, but they may communicate different measurement precision.

Context matters.

Rounding Practical Answers

Sometimes a calculation produces more decimal places than are useful.

Suppose:

10 ÷ 3 = 3.333333...

If the answer represents a measurement, we might report:

3.33 m

to the nearest hundredth.

If it represents people or containers, however, we cannot simply report:

3.33 people

The context determines how the result should be interpreted.

Whole Items and Decimal Results

Suppose:

47 students

must travel in vans holding:

8 students each

Calculate:

47 ÷ 8 = 5.875

But:

5.875 vans

does not make practical sense.

Five vans are not enough.

Therefore:

6 vans

are required.

A calculator result must always be interpreted in context.

Decimals and Calculators

Calculators are useful for complex decimal calculations, but mathematical understanding is still necessary.

Before using a calculator:

• identify the correct operation
• estimate the answer

After using it:

• check the decimal placement
• compare with the estimate
• interpret the result
• round appropriately
• include units

 

 

 

6

Worked Example 1: Money

A shirt costs:

$24.95

and trousers cost:

$38.75

Total:

$24.95 + $38.75 = $63.70

If you pay:

$70.00

change:

$70.00 − $63.70 = $6.30

Worked Example 2: Measurement

A piece of rope is:

12.8 m

long.

Three pieces measuring:

2.35 m

each are cut from it.

Length removed:

3 × 2.35 = 7.05 m

Remaining:

12.80 − 7.05 = 5.75 m

Worked Example 3: Data

Temperatures recorded during an experiment are:

22.4°C, 23.1°C, 22.8°C, 23.3°C

Mean:

(22.4 + 23.1 + 22.8 + 23.3) ÷ 4

= 91.6 ÷ 4

= 22.9°C

Therefore:

Mean temperature = 22.9°C

Worked Example 4: Unit Rate

A vehicle travels:

168.75 km

using:

12.5 L

of fuel.

Calculate:

168.75 ÷ 12.5 = 13.5

Therefore:

Fuel efficiency = 13.5 km/L

Worked Example 5: Area and Cost

A rectangular garden measures:

7.5 m × 4.8 m

Area:

7.5 × 4.8 = 36 m²

Grass seed costs:

$2.25 per m²

Cost:

36 × $2.25 = $81.00

Therefore:

The grass seed will cost $81.00.

Common Mistakes

Mistake 1: Ignoring units

Writing:

12.5

instead of:

12.5 kg

can make an answer unclear.

Mistake 2: Misaligning decimal points

When adding and subtracting, decimal points must be aligned.

Mistake 3: Misplacing the decimal point

Use estimation to check whether the answer has a sensible magnitude.

Mistake 4: Treating decimal hours like minutes

1.5 hours

does not mean:

1 hour 5 minutes

It means:

1 hour 30 minutes

Mistake 5: Reporting impossible decimal quantities

A calculation might produce:

4.6 buses

but the practical answer may need to be:

5 buses

Mistake 6: Using unnecessary precision

A shopping estimate usually does not need an answer such as:

$42.783649

Choose precision appropriate to the situation.

Mistake 7: Giving only a number

Communicate what the number means and include units where appropriate.

Error Analysis

A student calculates the total of:

$8.50 + $3.75

and writes:

$11.125

Estimate:

$9 + $4 ≈ $13

So $11.125 should immediately seem suspicious.

Correct calculation:

$8.50 + $3.75 = $12.25

The estimate helps reveal the error.

Another Error Analysis

A runner travels:

4.75 km

each day for:

5 days

A student calculates:

4.75 + 5 = 9.75 km

But the distance is repeated five times.

The correct operation is multiplication:

4.75 × 5 = 23.75 km

Therefore:

Total distance = 23.75 km

Choosing the correct operation is essential.

A Reliable Real-World Problem-Solving Strategy

Step 1: Understand the problem

Identify what is known and what must be found.

Step 2: Identify the units

Look for dollars, metres, litres, kilograms, seconds, and other units.

Step 3: Choose the operation

Decide whether addition, subtraction, multiplication, division, or several operations are required.

Step 4: Estimate

Predict approximately what the answer should be.

Step 5: Calculate

Perform the decimal operations accurately.

Step 6: Check

Compare the answer with the estimate.

Step 7: Interpret

Decide what the numerical result means in the situation.

Step 8: Communicate

State the final answer clearly using appropriate decimal notation and units.

Did You Know?

Decimals help us describe the real world because many quantities do not occur in exact whole-number amounts.

 

 

 

5

A runner does not always finish a race in exactly 12 seconds.

A bottle does not always contain exactly 2 litres.

A package does not always weigh exactly 5 kilograms.

Decimals allow us to represent these quantities with much greater precision.

This is why decimal notation is fundamental in measurement, science, finance, engineering, statistics, and technology.

Key Terms

• Decimal: Number that uses place value to represent whole quantities and parts of a whole.
• Decimal notation: Writing numbers using a decimal point.
• Place value: Value of a digit according to its position.
• Measurement: Process of assigning a numerical value and unit to a quantity.
• Unit: Standard quantity used for measurement.
• Data: Collected information or measurements.
• Unit price: Cost of one unit of an item.
• Unit rate: Comparison expressed for one unit of another quantity.
• Estimate: Approximate value used for prediction or checking.
• Mean: Sum of values divided by the number of values.
• Range: Difference between the greatest and least values.
• Precision: Level of detail used to express a value.
• Reasonableness: Whether a result makes sense in context.
• Multi-step problem: Problem requiring more than one mathematical operation.
• Interpretation: Explaining what a numerical result means in its context.

Key Relationships

Money:

$1 = 100 cents

Measurement:

1.5 m = 1 m + 0.5 m

Decimal time:

0.5 hour = 30 minutes

0.25 hour = 15 minutes

Area of a rectangle:

Area = length × width

Unit rate:

Unit rate = total quantity ÷ number of units

Mean:

Mean = sum of values ÷ number of values

Range:

Range = greatest value − least value