Fractions and Decimals Represent the Same Numbers

Fractions and decimals are two different ways of representing numbers.

For example:

1/2 = 0.5

Both represent exactly the same quantity.

Similarly:

1/4 = 0.25

3/4 = 0.75

1/5 = 0.2

Understanding the connection between fractions and decimals makes it easier to compare quantities, solve problems, and move between different forms of numbers.

---

What Is a Fraction?

A fraction represents a quantity using a numerator and denominator.

For example:

3/5

3 is the numerator.

5 is the denominator.

The fraction can be interpreted as:

3 ÷ 5

This connection to division is the key to converting fractions into decimals.

---

What Is a Decimal?

A decimal represents quantities using place value.

Consider:

0.375

The digits represent:

3 tenths

7 hundredths

5 thousandths

Therefore:

0.375 = 3/10 + 7/100 + 5/1000

Decimals are closely connected to fractions whose denominators are powers of 10.

---

Decimal Place Value

The first positions after the decimal point are:

0.1 = one tenth

0.01 = one hundredth

0.001 = one thousandth

For example:

0.6 = 6/10

0.27 = 27/100

0.413 = 413/1000

This makes converting many decimals to fractions straightforward.

---

Converting Fractions to Decimals

A fraction is another way of writing division.

Therefore:

numerator ÷ denominator = decimal

For example:

3/4

means:

3 ÷ 4

Calculate:

3 ÷ 4 = 0.75

Therefore:

3/4 = 0.75

---

Example: 1/2

Convert:

1/2

Calculate:

1 ÷ 2 = 0.5

Therefore:

1/2 = 0.5

---

Example: 3/5

Convert:

3/5

Calculate:

3 ÷ 5 = 0.6

Therefore:

3/5 = 0.6

Another method is to create an equivalent fraction with denominator 10:

3/5 × 2/2 = 6/10

Therefore:

6/10 = 0.6

---

Using Equivalent Fractions

Sometimes we can convert a fraction into a decimal without performing long division.

Consider:

7/20

We want a denominator of 100.

Multiply by:

5/5

Therefore:

7/20 = 35/100

And:

35/100 = 0.35

So:

7/20 = 0.35

---

Denominators of 10, 100, and 1000

Fractions with denominators that are powers of 10 are especially easy to convert.

7/10 = 0.7

23/100 = 0.23

417/1000 = 0.417

The denominator tells us the decimal place value.

A hundred grid is particularly useful for visualizing decimals such as:

25/100 = 0.25

---

Fractions That Produce Terminating Decimals

Some fractions produce decimals that stop.

These are called terminating decimals.

Examples:

1/2 = 0.5

1/4 = 0.25

3/8 = 0.375

7/20 = 0.35

9/10 = 0.9

The decimal eventually ends.

---

Fractions That Produce Repeating Decimals

Other fractions produce decimals with digits that continue in a repeating pattern.

For example:

1/3 = 0.3333...

2/3 = 0.6666...

1/6 = 0.16666...

2/11 = 0.181818...

These are called repeating decimals or recurring decimals.

The dots indicate that the pattern continues indefinitely.

---

Why Some Fractions Terminate

When a fraction is written in simplest form, its decimal terminates if its denominator contains only factors of:

2 and/or 5

For example:

3/8

Since:

8 = 2 × 2 × 2

the decimal terminates.

7/20

Since:

20 = 2 × 2 × 5

the decimal terminates.

But:

1/3

has denominator 3.

Therefore, its decimal repeats.

This is a useful pattern rather than a rule you must always use when performing basic conversions.

---

Using Division to Convert Any Fraction

Suppose we want to convert:

5/8

Remember:

5/8 = 5 ÷ 8

Using division:

5 ÷ 8 = 0.625

Therefore:

5/8 = 0.625

This method works even when creating a denominator of 10, 100, or 1000 is inconvenient.

---

Converting Improper Fractions to Decimals

Improper fractions can also be converted using division.

Example:

7/4

Calculate:

7 ÷ 4 = 1.75

Therefore:

7/4 = 1.75

This makes sense because 7/4 is greater than 1, so its decimal representation must also be greater than 1.

---

Mixed Numbers to Decimals

Consider:

2 3/4

Convert the fraction:

3/4 = 0.75

Then combine it with the whole number:

2 + 0.75 = 2.75

Therefore:

2 3/4 = 2.75

---

Converting Decimals to Fractions

To convert a terminating decimal into a fraction:

1. Identify its place value.
2. Write the digits as the numerator.
3. Use the place value as the denominator.
4. Simplify.

For example:

0.7

7 is in the tenths place.

Therefore:

0.7 = 7/10

---

Example: 0.35

Convert:

0.35

The final digit is in the hundredths place.

Therefore:

0.35 = 35/100

Simplify by dividing numerator and denominator by 5:

35 ÷ 5 = 7

100 ÷ 5 = 20

Therefore:

0.35 = 7/20

---

Example: 0.125

Convert:

0.125

The final digit is in the thousandths place.

Therefore:

0.125 = 125/1000

Simplify:

125/1000 = 1/8

Therefore:

0.125 = 1/8

---

Example: Decimal Greater Than 1

Convert:

1.75

One method is:

1.75 = 175/100

Simplify:

175/100 = 7/4

Therefore:

1.75 = 7/4

or as a mixed number:

1 3/4

---

Decimal Place Value Determines the Denominator

A useful pattern is:

one decimal place → denominator 10

Example:

0.4 = 4/10 = 2/5

two decimal places → denominator 100

Example:

0.24 = 24/100 = 6/25

three decimal places → denominator 1000

Example:

0.375 = 375/1000 = 3/8

Always simplify the fraction afterward.

---

Equivalent Fraction-Decimal Pairs

Some fraction-decimal pairs occur so frequently that they are worth recognizing quickly.

1/2 = 0.5

1/4 = 0.25

3/4 = 0.75

1/5 = 0.2

2/5 = 0.4

3/5 = 0.6

4/5 = 0.8

1/10 = 0.1

Recognizing these pairs can make calculations and comparisons much faster.

---

Eighths as Decimals

Eighths also appear frequently in measurement.

1/8 = 0.125

2/8 = 1/4 = 0.25

3/8 = 0.375

4/8 = 1/2 = 0.5

5/8 = 0.625

6/8 = 3/4 = 0.75

7/8 = 0.875

These values are useful in measurement, engineering, construction, and everyday mathematics.

---

Fractions and Decimals on a Number Line

Fractions and decimals can be placed on the same number line because they represent the same number system.

For example:

1/4 = 0.25

1/2 = 0.5

3/4 = 0.75

Their positions would be:

0 — 0.25 — 0.5 — 0.75 — 1

or equivalently:

0 — 1/4 — 1/2 — 3/4 — 1

---

Equivalent Numbers Occupy the Same Position

Because:

1/2 = 0.5

they occupy exactly the same position on a number line.

Similarly:

3/4 = 0.75

These are not numbers that happen to be close together.

They are exactly equal.

---

Placing Fractions on a Number Line

Suppose we need to place:

3/5

on a number line.

One method is to divide the interval from 0 to 1 into five equal sections.

The third mark represents:

3/5

Since:

3/5 = 0.6

the fraction occupies the same position as 0.6.

---

Placing Decimals on a Number Line

Suppose we need to place:

0.65

We know:

0.6 < 0.65 < 0.7

It is exactly halfway between 0.6 and 0.7.

We can also write:

0.65 = 65/100 = 13/20

Different representations still refer to the same location.

---

Comparing Fractions and Decimals

Suppose we want to compare:

3/4

and:

0.7

Comparing different forms can be difficult.

One useful strategy is to convert both numbers into the same form.

Convert:

3/4 = 0.75

Now compare:

0.75 > 0.7

Therefore:

3/4 > 0.7

---

Comparing by Converting to Decimals

Compare:

5/8

and:

0.6

Convert:

5/8 = 0.625

Now:

0.625 > 0.600

Therefore:

5/8 > 0.6

Adding zeros does not change a decimal's value:

0.6 = 0.60 = 0.600

This can make place-value comparisons easier.

---

Comparing by Converting to Fractions

You can also convert the decimal to a fraction.

Compare:

3/5

and:

0.55

Convert:

0.55 = 55/100 = 11/20

Convert 3/5 to twentieths:

3/5 = 12/20

Therefore:

12/20 > 11/20

So:

3/5 > 0.55

---

Comparing Using Benchmarks

Sometimes you do not need an exact conversion.

Compare:

4/9

and:

0.6

We know:

4/9 < 1/2

because 4/9 is approximately 0.444...

And:

0.6 > 1/2

Therefore:

4/9 < 0.6

Benchmark numbers such as:

0, 1/2, and 1

can make comparisons faster.

---

Comparing Decimal Place Values

When comparing decimals, compare digits from left to right.

For example:

0.65 and 0.625

Write:

0.650

0.625

Compare tenths:

both have 6.

Compare hundredths:

5 > 2.

Therefore:

0.650 > 0.625

So:

0.65 > 0.625

---

Be Careful: More Digits Does Not Mean Larger

Consider:

0.8

and:

0.75

Someone might incorrectly think 0.75 is larger because 75 is greater than 8.

But write:

0.80

0.75

Now compare.

80 hundredths > 75 hundredths

Therefore:

0.8 > 0.75

---

Ordering Fractions and Decimals

Suppose we need to order:

1/2, 0.8, 3/4, 0.4

Convert the fractions:

1/2 = 0.5

3/4 = 0.75

Now we have:

0.5, 0.8, 0.75, 0.4

From smallest to largest:

0.4 < 0.5 < 0.75 < 0.8

Therefore:

0.4 < 1/2 < 3/4 < 0.8

---

Another Ordering Example

Order from smallest to largest:

2/5, 0.45, 1/2, 0.6

Convert:

2/5 = 0.4

1/2 = 0.5

Now compare:

0.4 < 0.45 < 0.5 < 0.6

Therefore:

2/5 < 0.45 < 1/2 < 0.6

---

Fractions Greater Than 1

Fractions and decimals greater than 1 can also be compared.

Compare:

7/4

and:

1.8

Convert:

7/4 = 1.75

Now:

1.75 < 1.8

Therefore:

7/4 < 1.8

---

Negative Fractions and Decimals

The same ideas also work with negative numbers.

For example:

−1/2 = −0.5

−3/4 = −0.75

On a number line, numbers farther left are smaller.

Therefore:

−0.75 < −0.5

So:

−3/4 < −1/2

Be careful: with negative numbers, the value with the larger magnitude may actually be the smaller number.

---

Fractions, Decimals, and Money

Decimals are frequently used for money.

For example:

$0.50 = 50/100 of a dollar = 1/2 dollar

$0.25 = 25/100 of a dollar = 1/4 dollar

Money provides a familiar connection between fractions and hundredths.

---

Fractions, Decimals, and Measurement

Measurements frequently use decimals.

For example:

1/2 m = 0.5 m

1/4 kg = 0.25 kg

3/4 L = 0.75 L

Decimals are especially useful when measurements are entered into calculators, spreadsheets, graphs, and scientific instruments.

---

Fractions, Decimals, and Probability

Probability can be expressed using fractions or decimals.

Suppose a bag contains 10 equally likely counters and 3 are blue.

Probability of selecting blue:

3/10

As a decimal:

3/10 = 0.3

Both representations describe the same probability.

---

Fractions, Decimals, and Data

Graphs and data sets frequently use decimals even when the original quantities can be described as fractions.

For example:

A student answers 18 of 20 questions correctly.

As a fraction:

18/20 = 9/10

As a decimal:

0.9

These equivalent forms allow the same information to be represented in different ways.

---

Estimating Fraction-to-Decimal Conversions

Suppose you want to convert:

7/8

Before calculating, estimate.

7/8 is:

• greater than 3/4
• less than 1

Therefore, its decimal should be:

between 0.75 and 1

Calculate:

7 ÷ 8 = 0.875

This fits our estimate.

---

Checking Decimal-to-Fraction Conversions

Suppose:

0.45 = 9/20

Check by dividing:

9 ÷ 20 = 0.45

The conversion is correct.

You can also check:

9/20 = 45/100

and:

45/100 = 0.45

---

Worked Example 1

Convert:

9/20

to a decimal.

Create a denominator of 100:

9/20 × 5/5 = 45/100

Therefore:

9/20 = 0.45

---

Worked Example 2

Convert:

7/8

to a decimal.

Calculate:

7 ÷ 8 = 0.875

Therefore:

7/8 = 0.875

---

Worked Example 3

Convert:

0.72

to a fraction.

Write:

72/100

Simplify by dividing by 4:

72 ÷ 4 = 18

100 ÷ 4 = 25

Therefore:

0.72 = 18/25

---

Worked Example 4

Compare:

7/10 and 0.68

Convert:

7/10 = 0.70

Now compare:

0.70 > 0.68

Therefore:

7/10 > 0.68

---

Worked Example 5

Place these in order from smallest to largest:

1/4, 0.6, 3/5, 0.3

Convert:

1/4 = 0.25

3/5 = 0.6

Now:

0.25 < 0.3 < 0.6

Since 3/5 and 0.6 are equivalent:

1/4 < 0.3 < 3/5 = 0.6

---

Choosing a Useful Representation

Fractions and decimals have the same value, but one form may be more useful in a particular situation.

Fractions are often useful for:

• exact ratios
• sharing
• recipes
• algebra
• probability

Decimals are often useful for:

• money
• measurement
• calculators
• spreadsheets
• scientific data
• numerical comparisons

A strong mathematician can move between representations depending on the problem.

---

Common Mistakes

Mistake 1: Dividing the denominator by the numerator

To convert a fraction to a decimal:

numerator ÷ denominator

For:

3/4

calculate:

3 ÷ 4

not:

4 ÷ 3

---

Mistake 2: Forgetting to simplify

For example:

0.5 = 5/10

but in simplest form:

0.5 = 1/2

---

Mistake 3: Using the wrong place value

For:

0.25

the final digit is in the hundredths place.

Therefore:

0.25 = 25/100

not:

25/10

---

Mistake 4: Thinking more decimal digits means a larger number

0.8 > 0.75

because:

0.80 > 0.75

---

Mistake 5: Thinking 1/3 equals exactly 0.3

Actually:

1/3 = 0.3333...

The decimal repeats indefinitely.

---

Mistake 6: Treating equivalent forms as different values

1/2, 2/4, 5/10, 0.5, and 0.50

all represent exactly the same number.

---

Mistake 7: Ignoring the whole-number part

For example:

1 3/4 = 1.75

not:

0.75

---

Mistake 8: Comparing fractions and decimals without considering their values

Convert them to a common form or use benchmarks before deciding which is larger.

---

Did You Know?

Fractions and decimals are not separate kinds of quantities. They are different representations of numbers.

For example:

1/4 = 0.25

Later, we can also represent this as:

25%

So:

1/4 = 0.25 = 25%

Learning to move between these representations is important for percentages, probability, statistics, science, finance, and many other areas of mathematics.

---

Key Terms

• Fraction: Number representing part of a whole or a ratio.
• Numerator: Top number of a fraction.
• Denominator: Bottom number of a fraction.
• Decimal: Number represented using decimal place value.
• 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.
• Equivalent: Having the same numerical value.
• Terminating decimal: Decimal that ends.
• Repeating decimal: Decimal containing a digit or pattern that repeats indefinitely.
• Improper fraction: Fraction with numerator greater than or equal to denominator.
• Mixed number: Number containing a whole-number part and fractional part.
• Simplest form: Fraction whose numerator and denominator have no common factor greater than 1.
• Benchmark fraction: Familiar fraction used to estimate or compare values.
• Number line: Visual representation showing numbers according to their value and order.

---

Conversion Guide

Fraction → Decimal

Calculate:

numerator ÷ denominator

Example:

3/8 = 3 ÷ 8 = 0.375

---

Decimal → Fraction

Use place value.

Example:

0.35 = 35/100 = 7/20

---

Mixed Number → Decimal

Convert the fractional part and combine it with the whole number.

Example:

2 1/4 = 2.25

---

Compare Fraction and Decimal

Convert both to the same form.

Example:

3/4 vs 0.7

3/4 = 0.75

Therefore:

0.75 > 0.7

---

Key Takeaways

• Fractions and decimals are different representations of numbers.
• A fraction represents division.
• To convert a fraction to a decimal, divide the numerator by the denominator.
• Some fractions can easily be converted by creating equivalent fractions with denominators of 10, 100, or 1000.
• Some fractions produce terminating decimals.
• Other fractions produce repeating decimals.
• To convert a terminating decimal to a fraction, use its place value to determine the denominator.
• Decimal fractions should normally be simplified.
• Common equivalent pairs such as 1/2 = 0.5, 1/4 = 0.25, and 3/4 = 0.75 are useful to recognize.
• Fractions and their equivalent decimals occupy exactly the same position on a number line.
• Fractions and decimals can be compared by converting them into the same form.
• Benchmark values such as 0, 1/2, and 1 can help with comparisons and estimation.
• Adding zeros to the right side of a terminating decimal does not change its value.
• More decimal digits do not automatically mean a larger number.
• Fractions and decimals are widely used in money, measurement, probability, data, science, and everyday life.
• A useful overall strategy is:

fraction → divide numerator by denominator → decimal

and:

decimal → use place value → fraction → simplify.