Three Ways to Represent the Same Number

Fractions, decimals, and percentages may look different, but they can represent exactly the same value.

For example:

1/2 = 0.5 = 50%

Similarly:

1/4 = 0.25 = 25%

3/4 = 0.75 = 75%

These are called equivalent representations.

Learning to move easily between these forms is useful in mathematics, science, finance, statistics, measurement, and everyday life.

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The Connection Between the Three Forms

A fraction represents division.

A decimal represents a number using place value.

A percentage represents a number per hundred.

For example:

3/5

Divide:

3 ÷ 5 = 0.6

Convert the decimal to a percentage:

0.6 × 100% = 60%

Therefore:

3/5 = 0.6 = 60%

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Fractions to Decimals

To convert a fraction to a decimal:

divide the numerator by the denominator

For:

3/4

calculate:

3 ÷ 4 = 0.75

Therefore:

3/4 = 0.75

This works because the fraction bar itself represents division.

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Fractions to Percentages

There are several useful methods for converting fractions to percentages.

The most general method is:

fraction → decimal → percentage

For example:

3/8

First convert to a decimal:

3 ÷ 8 = 0.375

Then multiply by 100:

0.375 × 100 = 37.5

Therefore:

3/8 = 37.5%

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Why Multiply a Decimal by 100?

Percent means:

per hundred

Consider:

0.35

This can be written:

35/100

Therefore:

0.35 = 35%

Multiplying a decimal by 100 tells us how many hundredths the number represents.

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Fraction to Percentage Method 1: Convert to a Decimal

Convert:

7/20

First divide:

7 ÷ 20 = 0.35

Then:

0.35 × 100% = 35%

Therefore:

7/20 = 35%

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Fraction to Percentage Method 2: Create a Denominator of 100

Sometimes it is easier to create an equivalent fraction with denominator 100.

For example:

3/5

Multiply numerator and denominator by 20:

3/5 = 60/100

Therefore:

3/5 = 60%

This method is especially useful when the denominator can easily be changed to 100.

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Another Example

Convert:

9/20

Since:

20 × 5 = 100

multiply by:

5/5

Therefore:

9/20 = 45/100 = 45%

No decimal calculation is necessary.

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Fraction to Percentage Method 3: Multiply by 100%

A fraction can also be converted directly using:

fraction × 100%

For example:

2/5 × 100%

Calculate:

200% ÷ 5 = 40%

Therefore:

2/5 = 40%

This method becomes particularly useful as percentage calculations become more advanced.

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Fractions That Do Not Give Whole-Number Percentages

Not every fraction converts to a whole-number percentage.

For example:

1/8

Convert to a decimal:

1 ÷ 8 = 0.125

Then:

0.125 × 100% = 12.5%

Therefore:

1/8 = 12.5%

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Repeating Decimals and Percentages

Some fractions produce repeating decimals.

For example:

1/3 = 0.3333...

Multiply by 100:

33.3333...%

Therefore:

1/3 = 33.3333...%

This is often written approximately as:

33.3%

if rounding is appropriate.

Similarly:

2/3 ≈ 66.7%

when rounded to one decimal place.

The rounded percentage is an approximation, while the fraction 2/3 is exact.

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Decimals to Percentages

To convert a decimal to a percentage:

multiply by 100

and add the percentage symbol.

For example:

0.42

Multiply:

0.42 × 100 = 42

Therefore:

0.42 = 42%

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A Quick Decimal-to-Percent Method

Multiplying by 100 moves each digit two place-value positions relative to the decimal point.

For example:

0.37 → 37%

0.8 → 80%

0.125 → 12.5%

1.2 → 120%

It is better to understand this as multiplying by 100 rather than simply memorizing "move the decimal point."

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Example: 0.6

Convert:

0.6

to a percentage.

Calculate:

0.6 × 100 = 60

Therefore:

0.6 = 60%

We can check using fractions:

0.6 = 6/10 = 3/5

and:

3/5 = 60%

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Example: 0.075

Convert:

0.075

to a percentage.

Multiply by 100:

0.075 × 100 = 7.5

Therefore:

0.075 = 7.5%

Be careful:

0.075 is not 75%.

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Decimals Greater Than 1

Decimals greater than 1 produce percentages greater than 100%.

For example:

1.25 × 100% = 125%

Therefore:

1.25 = 125%

This makes sense because:

1.25 = 1 1/4

which is greater than one whole.

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Percentages to Decimals

To convert a percentage to a decimal:

divide by 100

For example:

65%

Calculate:

65 ÷ 100 = 0.65

Therefore:

65% = 0.65

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Why Divide by 100?

Remember:

65% = 65/100

And:

65 ÷ 100 = 0.65

Therefore:

65% = 0.65

The conversion comes directly from the meaning of percentage.

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Example: 8%

Convert:

8%

to a decimal.

Calculate:

8 ÷ 100 = 0.08

Therefore:

8% = 0.08

Notice the zero before the 8.

8% ≠ 0.8

because:

0.8 = 80%

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Example: 125%

Convert:

125%

to a decimal.

Calculate:

125 ÷ 100 = 1.25

Therefore:

125% = 1.25

Percentages greater than 100% become decimals greater than 1.

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Percentages Less Than 1%

Convert:

0.5%

to a decimal.

Divide by 100:

0.5 ÷ 100 = 0.005

Therefore:

0.5% = 0.005

This distinction is important in science, statistics, and finance.

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Percentages to Fractions

Although the main target focuses on percentages and decimals, percentages can also be converted directly to fractions.

Since percent means "out of 100":

35% = 35/100

Simplify:

35/100 = 7/20

Therefore:

35% = 7/20

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Example: 75%

Convert:

75%

to a fraction.

Write:

75/100

Simplify by dividing by 25:

75 ÷ 25 = 3

100 ÷ 25 = 4

Therefore:

75% = 3/4

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Example: 120%

Convert:

120%

to a fraction.

Write:

120/100

Simplify:

120/100 = 6/5

or:

1 1/5

Therefore:

120% = 6/5 = 1.2

All three representations describe the same value.

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A Conversion Triangle

It is useful to think of fractions, decimals, and percentages as three connected representations.

The main conversions are:

Fraction → Decimal

Divide numerator by denominator.

Decimal → Percentage

Multiply by 100%.

Percentage → Decimal

Divide by 100.

Percentage → Fraction

Write over 100 and simplify.

Decimal → Fraction

Use decimal place value and simplify.

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Common Equivalent Values

Several equivalents are worth recognizing immediately.

1/10 = 0.1 = 10%

1/5 = 0.2 = 20%

1/4 = 0.25 = 25%

1/2 = 0.5 = 50%

3/4 = 0.75 = 75%

4/5 = 0.8 = 80%

1 = 1.0 = 100%

Knowing these benchmark values makes many conversions much faster.

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Eighths as Fractions, Decimals, and Percentages

Eighths are also useful to recognize.

1/8 = 0.125 = 12.5%

2/8 = 1/4 = 0.25 = 25%

3/8 = 0.375 = 37.5%

4/8 = 1/2 = 0.5 = 50%

5/8 = 0.625 = 62.5%

6/8 = 3/4 = 0.75 = 75%

7/8 = 0.875 = 87.5%

These values appear frequently in measurement and data.

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Recognizing Equivalent Representations

Suppose you see:

0.4

Which fraction and percentage are equivalent?

Convert to a percentage:

0.4 × 100 = 40%

Convert to a fraction:

0.4 = 4/10 = 2/5

Therefore:

2/5 = 0.4 = 40%

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Using a Hundred Grid

A hundred grid can show all three forms at once.

Suppose 65 squares are shaded.

As a fraction:

65/100 = 13/20

As a decimal:

0.65

As a percentage:

65%

Therefore:

13/20 = 0.65 = 65%

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Using a Number Line

Fractions, decimals, and percentages can all be placed on the same number line.

For example:

0 = 0%

1/4 = 0.25 = 25%

1/2 = 0.5 = 50%

3/4 = 0.75 = 75%

1 = 1.0 = 100%

Equivalent forms occupy exactly the same position.

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Comparing Different Forms

Suppose we need to compare:

3/5, 0.58, and 62%

Convert everything to decimals.

3/5 = 0.6

0.58 = 0.58

62% = 0.62

Now compare:

0.58 < 0.60 < 0.62

Therefore:

0.58 < 3/5 < 62%

Converting everything to the same representation makes comparison easier.

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Choosing the Most Useful Form

Although equivalent forms have the same value, one representation may be more useful than another.

For example:

1/4 = 0.25 = 25%

Which form is best depends on the situation.

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Fractions for Exact Parts and Ratios

Fractions are often useful when quantities naturally involve equal parts.

Examples include:

• recipes
• sharing
• ratios
• probability
• exact mathematical calculations

For example:

"Use 3/4 cup of milk."

The fraction is convenient because measuring cups often use fractional markings.

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Decimals for Measurement and Calculation

Decimals are often useful for:

• scientific measurements
• calculators
• spreadsheets
• money
• data recording

For example:

A length of:

0.75 m

may be easier to enter into a calculator or spreadsheet than:

3/4 m

The values are identical.

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Percentages for Comparisons

Percentages are particularly useful when comparing proportions.

Suppose:

Class A: 18/20 students passed.

Class B: 42/50 students passed.

Convert:

18/20 = 90%

42/50 = 84%

The percentages provide a common scale and make the proportional comparison easier.

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Choosing a Form in Probability

Suppose the probability of an event is:

1/4

You could express it as:

1/4

0.25

or:

25%

All are correct.

The fraction may be useful for exact calculations.

The decimal may be useful in software or numerical analysis.

The percentage may be easier when communicating the result to a general audience.

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Choosing a Form for Money

Decimals are normally convenient for money.

For example:

$0.75

is generally easier to interpret as money than:

$3/4

However, if discussing a discount, a percentage may be more natural:

25% off

rather than:

0.25 off

Context determines which representation communicates the information most clearly.

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Choosing a Form in Science

Scientific data are often recorded as decimals.

For example:

0.625 g

may be more useful than:

5/8 g

when measurements come from digital instruments.

Percentages are useful when expressing:

• efficiency
• concentration
• percentage error
• percentage change
• composition

Fractions may still be useful for exact ratios and theoretical calculations.

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Estimating Conversions

Before converting, estimate what the answer should look like.

Suppose:

5/8

We know:

1/2 < 5/8 < 3/4

Therefore:

50% < 5/8 < 75%

The exact conversion is:

5/8 = 0.625 = 62.5%

This fits our estimate.

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Checking Decimal-to-Percentage Conversions

Suppose someone claims:

0.7 = 7%

We can check using benchmarks.

We know:

0.5 = 50%

Since 0.7 is greater than 0.5, its percentage must be greater than 50%.

Therefore, 7% cannot be correct.

The correct conversion is:

0.7 × 100 = 70%

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Checking Percentage-to-Decimal Conversions

Suppose someone claims:

35% = 3.5

But:

35%

is less than:

100%

Therefore, its decimal representation should be less than 1.

The correct answer is:

35% = 0.35

Reasonableness checks can reveal place-value errors quickly.

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Worked Example 1

Convert:

7/8

to a percentage.

First:

7 ÷ 8 = 0.875

Then:

0.875 × 100 = 87.5

Therefore:

7/8 = 87.5%

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Worked Example 2

Convert:

0.46

to a percentage.

Multiply by 100:

0.46 × 100 = 46

Therefore:

0.46 = 46%

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Worked Example 3

Convert:

32%

to a decimal.

Divide by 100:

32 ÷ 100 = 0.32

Therefore:

32% = 0.32

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Worked Example 4

Convert:

45%

to a fraction.

Write:

45/100

Simplify by dividing by 5:

45/100 = 9/20

Therefore:

45% = 9/20

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Worked Example 5

Find the missing representation:

3/10 = ? = ?%

Convert to decimal:

3 ÷ 10 = 0.3

Convert to percentage:

0.3 × 100 = 30%

Therefore:

3/10 = 0.3 = 30%

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Worked Example 6

Which is larger?

2/3 or 65%

Convert 2/3:

2 ÷ 3 = 0.6666...

Therefore:

2/3 = 66.666...%

So:

2/3 > 65%

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Worked Example 7

Order from smallest to largest:

3/4, 0.8, 70%, 7/10

Convert everything to decimals:

3/4 = 0.75

0.8 = 0.80

70% = 0.70

7/10 = 0.70

Therefore:

70% = 7/10 < 3/4 < 0.8

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Worked Example 8: Choosing a Representation

A survey finds that 84 of 120 people prefer option A.

As a fraction:

84/120 = 7/10

As a decimal:

0.7

As a percentage:

70%

If the goal is to communicate the survey result clearly, 70% may be the most useful representation.

If the value is being entered into some calculations, 0.7 may be convenient.

If an exact ratio is useful, 7/10 may be appropriate.

There is no single best representation for every situation.

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Real-World Example: Shopping

A store offers:

25% off

We know:

25% = 0.25 = 1/4

If an item costs $80, we can use whichever representation makes the calculation easiest.

Using the fraction:

1/4 of 80 = 20

So the discount is:

$20

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Real-World Example: Test Results

A student answers:

36 out of 40

questions correctly.

As a fraction:

36/40 = 9/10

As a decimal:

0.9

As a percentage:

90%

The percentage is particularly useful for communicating the overall result.

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Real-World Example: Probability

A spinner has 8 equal sections.

Three sections are blue.

Probability of blue:

3/8

Convert:

3/8 = 0.375

Then:

0.375 = 37.5%

Therefore:

3/8 = 0.375 = 37.5%

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Real-World Example: Data

Suppose a machine successfully produces 96 acceptable products out of every 100.

The success rate is:

96/100

As a decimal:

0.96

As a percentage:

96%

For reporting performance, the percentage is often the clearest representation.

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Conversion Map

A useful mental map is:

Fraction → Decimal

numerator ÷ denominator

Decimal → Percentage

× 100%

Percentage → Decimal

÷ 100

Percentage → Fraction

write over 100 and simplify

Decimal → Fraction

use place value and simplify

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Common Mistakes

Mistake 1: Reversing the fraction division

For:

3/4

calculate:

3 ÷ 4

not:

4 ÷ 3

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Mistake 2: Forgetting that percent means out of 100

35% = 35/100

not:

35/10

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Mistake 3: Converting 0.4 to 4%

Correct:

0.4 × 100 = 40%

Therefore:

0.4 = 40%

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Mistake 4: Converting 8% to 0.8

Correct:

8 ÷ 100 = 0.08

Therefore:

8% = 0.08

Remember:

0.8 = 80%

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Mistake 5: Forgetting to simplify fractions

For example:

50% = 50/100

but in simplest form:

50% = 1/2

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Mistake 6: Thinking percentages must be below 100%

For example:

1.5 = 150%

Percentages greater than 100% are valid.

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Mistake 7: Treating rounded values as exact

For example:

2/3 = 66.666...%

Writing:

2/3 ≈ 66.7%

is a rounded approximation.

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Mistake 8: Assuming one representation is always best

Fractions, decimals, and percentages have different advantages depending on the situation.

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Did You Know?

Fractions, decimals, and percentages are all part of a larger idea called multiple representations.

The ability to move between representations is important in:

• probability
• statistics
• finance
• science
• engineering
• measurement
• spreadsheets
• data analysis
• business
• economics

For example, the same probability might be written as:

3/4

in an exact calculation,

0.75

in a computer program,

and:

75%

in a report.

The number has not changed. Only its representation has changed.

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Key Terms

• Fraction: Number representing part of a whole or a ratio.
• Decimal: Number represented using decimal place value.
• Percentage: Number expressed as parts per hundred.
• Percent: Means per hundred.
• Equivalent: Having exactly the same numerical value.
• Numerator: Top number of a fraction.
• Denominator: Bottom number of a fraction.
• Terminating decimal: Decimal that ends.
• Repeating decimal: Decimal containing a repeating pattern that continues indefinitely.
• Benchmark: Familiar value used for comparison or estimation.
• Simplest form: Fraction whose numerator and denominator have no common factor greater than 1.
• Representation: A particular way of expressing a mathematical value.
• Exact value: Value written without approximation.
• Approximation: Value close to the exact value, usually produced by estimation or rounding.

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Quick Conversion Guide

Fraction → Decimal

Divide:

numerator ÷ denominator

Example:

3/4 = 0.75

Fraction → Percentage

Convert to a decimal and multiply by 100%, or create an equivalent fraction out of 100.

Example:

3/4 = 0.75 = 75%

Decimal → Percentage

Multiply by 100%.

Example:

0.62 = 62%

Percentage → Decimal

Divide by 100.

Example:

62% = 0.62

Percentage → Fraction

Write over 100 and simplify.

Example:

60% = 60/100 = 3/5

Decimal → Fraction

Use place value and simplify.

Example:

0.75 = 75/100 = 3/4

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Key Takeaways

• Fractions, decimals, and percentages can represent exactly the same value.
• A fraction can be converted to a decimal by dividing the numerator by the denominator.
• A fraction can be converted to a percentage by first converting it to a decimal and multiplying by 100%.
• Some fractions can be converted directly by creating an equivalent fraction with denominator 100.
• To convert a decimal to a percentage, multiply by 100%.
• To convert a percentage to a decimal, divide by 100.
• To convert a percentage to a fraction, write it over 100 and simplify.
• To convert a terminating decimal to a fraction, use its place value and simplify.
• Common equivalents such as 1/2 = 0.5 = 50% are useful to recognize immediately.
• Equivalent forms occupy the same position on a number line.
• Converting quantities to the same representation makes comparisons easier.
• Fractions are often useful for exact ratios and parts.
• Decimals are often useful for measurement, calculations, calculators, and data.
• Percentages are often useful for communicating and comparing proportions.
• Percentages can be greater than 100% or less than 1%.
• Repeating decimals may produce repeating percentages and sometimes need to be rounded.
• The most useful representation depends on the problem and context.
• A useful conversion pathway is:

fraction → decimal → percentage

and in reverse:

percentage → decimal → fraction.