What Is a Percentage?

A percentage describes a quantity as a number of parts out of 100.

The word percent means:

per hundred

The symbol for percent is:

%

For example:

25%

means:

25 out of 100

Therefore:

25% = 25/100

This fraction can be simplified:

25/100 = 1/4

So:

25% = 25/100 = 1/4

These are different ways of representing exactly the same quantity.

---

Percent Means "Out of 100"

Consider a grid containing 100 equal squares.

If 30 squares are shaded, then:

30 out of 100 squares are shaded.

As a fraction:

30/100

As a percentage:

30%

The denominator of 100 makes percentages especially useful for comparing quantities.

---

The Percentage Symbol

The symbol:

%

means "per hundred."

Therefore:

8% = 8/100

42% = 42/100

75% = 75/100

100% = 100/100 = 1 whole

The percentage symbol is not simply a decoration. It tells us how the number should be interpreted.

---

Percentages and Fractions

Every percentage can be written as a fraction with denominator 100.

For example:

60% = 60/100

Simplify:

60/100 = 3/5

Therefore:

60% = 3/5

Similarly:

50% = 50/100 = 1/2

25% = 25/100 = 1/4

75% = 75/100 = 3/4

---

Percentages and Decimals

Percentages are also closely related to decimals.

Because:

50% = 50/100

we can write:

50% = 0.50 = 0.5

Similarly:

25% = 0.25

75% = 0.75

10% = 0.10 = 0.1

This gives us three useful ways to represent the same number:

fraction ↔ decimal ↔ percentage

For example:

1/2 = 0.5 = 50%

---

A Hundred Grid

A hundred grid is one of the easiest ways to visualize percentages.

It contains:

10 rows × 10 columns = 100 squares

Each square represents:

1/100 = 1%

Therefore:

10 shaded squares = 10%

20 shaded squares = 20%

50 shaded squares = 50%

85 shaded squares = 85%

100 shaded squares = 100%

---

Visualizing 50%

If 50 of 100 squares are shaded:

50/100 = 50%

Simplify:

50/100 = 1/2

Therefore:

50% means half of the whole.

This is one of the most useful percentage benchmarks.

---

Visualizing 25%

If 25 of 100 squares are shaded:

25/100 = 25%

Simplify:

25/100 = 1/4

Therefore:

25% means one-quarter of the whole.

---

Visualizing 75%

If 75 of 100 squares are shaded:

75/100 = 75%

Simplify:

75/100 = 3/4

Therefore:

75% means three-quarters of the whole.

---

Important Percentage Benchmarks

Some percentages are especially useful for mental mathematics.

1% = 1/100

10% = 1/10

20% = 1/5

25% = 1/4

50% = 1/2

75% = 3/4

100% = 1 whole

Recognizing these quickly makes estimating and calculating percentages much easier.

---

What Does 100% Mean?

100% represents one complete whole.

For example:

If every student submits an assignment:

100% of the students submitted it.

If a container is completely full:

it is 100% full.

Mathematically:

100% = 100/100 = 1

---

Percentages Greater Than 100%

Percentages can be greater than 100%.

For example:

150%

means:

150/100 = 1.5

Therefore:

150% = 1 1/2

If a quantity increases from 20 to 30, the new quantity is:

150% of the original quantity

because:

30 = 1.5 × 20

So percentages are not restricted to values between 0% and 100%.

---

Percentages Less Than 1%

Percentages can also be smaller than 1%.

For example:

0.5%

means:

0.5 out of every 100

Mathematically:

0.5% = 0.5/100 = 0.005

Small percentages are common in:

• science
• medicine
• finance
• population statistics
• chemical concentrations

---

Percentages in Everyday Life

Percentages appear almost everywhere.

Common examples include:

• discounts
• taxes
• test scores
• interest rates
• battery levels
• sports statistics
• weather forecasts
• survey results
• population data
• food labels
• scientific measurements

Understanding percentages helps us interpret information and make comparisons.

---

Percentage Test Scores

Suppose a student answers:

18 out of 20 questions correctly.

As a fraction:

18/20

Create an equivalent fraction out of 100:

18/20 = 90/100

Therefore:

90%

The student answered:

90% of the questions correctly.

---

Percentage Discounts

A shop advertises:

25% off

This means the discount is:

25 out of every 100 parts of the original price.

Since:

25% = 1/4

a 25% discount means one-quarter of the original price is removed.

If an item costs $40:

25% of $40

is the same as:

1/4 of $40

which is:

$10

The discount is $10.

---

Battery Percentage

A phone showing:

80% battery

means approximately 80% of its represented full-charge capacity remains according to the device's battery estimate.

Visually:

80% = 80/100 = 4/5

So the display represents roughly four-fifths of the full level.

---

Percentages in Surveys

Suppose 200 people answer a survey.

If:

60%

choose option A, this means 60 out of every 100 in proportional terms.

For 200 people:

60% of 200 = 120

Therefore:

120 people

selected option A.

Percentages make it easier to compare surveys with different numbers of participants.

---

Why Percentages Are Useful for Comparison

Consider two classes.

Class A:

18 of 20 students passed.

Class B:

42 of 50 students passed.

Which class had the larger proportion passing?

Raw numbers are difficult to compare because the classes are different sizes.

Class A:

18/20 = 90%

Class B:

42/50 = 84%

Now the comparison is straightforward:

90% > 84%

Percentages create a common scale based on 100.

---

Comparing Percentages

When percentages refer to comparable quantities, comparing them is usually straightforward.

For example:

45% < 60%

because:

45/100 < 60/100

Similarly:

72% > 68%

because:

72/100 > 68/100

---

Percentages on a Number Line

Percentages can be placed on a number line.

From:

0% to 100%

important points include:

0% = 0

25% = 1/4

50% = 1/2

75% = 3/4

100% = 1

This helps us visualize the relative size of percentages.

---

Estimating Percentages

Exact calculations are not always necessary.

Sometimes an estimate is sufficient.

Useful benchmark percentages include:

10%, 25%, 50%, 75%, and 100%

These can be combined to estimate less familiar percentages.

---

Finding 50%

Since:

50% = 1/2

finding 50% means finding half.

Example:

Estimate or calculate:

50% of 80

Half of 80 is:

40

Therefore:

50% of 80 = 40

---

Finding 25%

Since:

25% = 1/4

finding 25% means finding one-quarter.

Example:

25% of 60

Calculate:

60 ÷ 4 = 15

Therefore:

25% of 60 = 15

---

Finding 75%

Since:

75% = 3/4

we can find three-quarters.

Example:

75% of 40

First find one-quarter:

40 ÷ 4 = 10

Then multiply by 3:

10 × 3 = 30

Therefore:

75% of 40 = 30

---

Finding 10%

Finding 10% is particularly useful.

To find 10% of a quantity, divide by 10.

For example:

10% of 70 = 7

because:

70 ÷ 10 = 7

Similarly:

10% of 250 = 25

---

Finding 1%

To find 1%, divide by 100.

Example:

1% of 300

Calculate:

300 ÷ 100 = 3

Therefore:

1% of 300 = 3

This can help calculate many other percentages.

---

Building Percentages from 10%

Suppose we want:

30% of 80

We know:

10% of 80 = 8

Therefore:

30% = 3 × 10%

So:

3 × 8 = 24

Therefore:

30% of 80 = 24

---

Building Percentages from 10% and 5%

Suppose we want:

15% of 60

First find 10%:

10% of 60 = 6

Then find 5%, which is half of 10%:

5% of 60 = 3

Combine:

15% = 10% + 5%

Therefore:

6 + 3 = 9

So:

15% of 60 = 9

---

Estimating 49%

Suppose we want to estimate:

49% of 82

49% is very close to 50%.

82 is close to 80.

So estimate:

50% of 80 = 40

Therefore:

49% of 82 is approximately 40.

The goal of estimation is not to produce the exact answer. It is to produce a sensible approximate value.

---

Estimating 21%

Estimate:

21% of 198

21% is close to:

20%

198 is close to:

200

So estimate:

20% of 200

10% of 200 = 20.

Therefore:

20% of 200 = 40.

So:

21% of 198 ≈ 40

---

Estimating 74%

Estimate:

74% of 120

74% is close to:

75%

Since:

75% = 3/4

calculate:

3/4 of 120 = 90

Therefore:

74% of 120 is approximately 90.

---

Estimating Using Benchmarks

Consider:

52% of 98

52% is close to 50%.

98 is close to 100.

Therefore:

50% of 100 = 50

A reasonable estimate is:

about 50

---

Estimation as a Checking Tool

Suppose a calculator gives:

51% of 200 = 1020

We can immediately recognize a problem.

50% of 200 is:

100

Therefore, 51% should be only slightly greater than 100.

An answer of 1020 is unreasonable.

Estimation can reveal calculation or input errors quickly.

---

Percentages and Proportional Reasoning

Suppose:

20% of a group = 8 people

Since:

20% = 1/5

the 8 people represent one-fifth of the group.

Therefore:

whole group = 8 × 5 = 40 people

Understanding percentages as fractions often makes percentage problems easier.

---

A Percentage Is Relative to the Whole

Percentages only make sense when we know what the whole represents.

Suppose:

50% of Class A = 10 students

and:

50% of Class B = 15 students

Both percentages are 50%, but the numbers of students are different because the classes have different sizes.

This is an important idea:

same percentage does not necessarily mean same quantity.

---

Comparing Percentages with Different Wholes

Suppose:

40% of a small box contains red objects.

30% of a large box contains red objects.

We can say:

40% > 30%

as proportions.

But we cannot automatically say the small box contains more red objects.

For example:

40% of 20 = 8.

30% of 100 = 30.

The smaller percentage can represent a larger actual quantity if the whole is larger.

---

Percentage Diagrams

Percentages can be represented using many visual models.

Common models include:

• hundred grids
• bar models
• circles
• number lines
• progress bars

Different models are useful for different situations.

A hundred grid emphasizes the meaning out of 100.

A bar model emphasizes the relationship between a part and a whole.

A number line emphasizes size and comparison.

---

Bar Models

Suppose a bar represents:

100%

Half the bar represents:

50%

One-quarter represents:

25%

Three-quarters represents:

75%

Bar models are especially useful when solving percentage problems because the entire quantity can represent the whole.

---

Circle Models

Percentages can also be represented using circles.

A complete circle represents:

100%

Half represents:

50%

One-quarter represents:

25%

Three-quarters represents:

75%

This connects percentage ideas to pie charts and data displays.

---

Worked Example 1

A hundred grid contains 68 shaded squares.

What percentage is shaded?

There are:

68 shaded squares out of 100

Therefore:

68/100 = 68%

Answer:

68%

---

Worked Example 2

Write:

40%

as a fraction.

Start with:

40/100

Simplify by dividing by 20:

40 ÷ 20 = 2

100 ÷ 20 = 5

Therefore:

40% = 2/5

---

Worked Example 3

Compare:

65% and 3/5

Convert:

3/5 = 60/100 = 60%

Therefore:

65% > 60%

So:

65% > 3/5

---

Worked Example 4

Estimate:

48% of 62

48% is close to:

50%

62 is close to:

60

Half of 60 is:

30

Therefore:

48% of 62 ≈ 30

---

Worked Example 5

Estimate:

26% of 200

26% is close to:

25%

Since:

25% = 1/4

calculate:

1/4 of 200 = 50

Therefore:

26% of 200 is approximately 50.

---

Worked Example 6

A battery is at 35%.

Which is closer: one-quarter or one-half?

We know:

25% = 1/4

50% = 1/2

35% is:

10 percentage points above 25%

and:

15 percentage points below 50%.

Therefore:

35% is closer to one-quarter.

---

Common Mistakes

Mistake 1: Thinking 25% means 25/10

Percent means per hundred.

Correct:

25% = 25/100

---

Mistake 2: Thinking 100% means 100 times larger

100% represents one complete whole:

100% = 1

---

Mistake 3: Thinking percentages cannot exceed 100%

Percentages such as:

120%, 150%, and 200%

are mathematically valid.

---

Mistake 4: Thinking the same percentage always means the same amount

50% of 20 is 10.

50% of 200 is 100.

The percentage is the same, but the wholes are different.

---

Mistake 5: Comparing actual amounts using percentages alone

40% of a small group can be less than 30% of a much larger group.

---

Mistake 6: Treating estimates as exact answers

Estimation gives an approximate value.

For example:

49% of 82 ≈ 40

does not mean the exact answer is necessarily 40.

---

Mistake 7: Ignoring the whole

Always ask:

Percentage of what?

A percentage describes a part relative to a whole.

---

Did You Know?

The idea of percentage gives us a common scale for comparing quantities.

This is why percentages are so common in:

• statistics
• science
• finance
• economics
• business
• education
• sports
• surveys
• data analysis

Instead of comparing fractions with many different denominators, percentages express them using a common reference of 100.

---

Key Terms

• Percentage: Quantity expressed as a number of parts per hundred.
• Percent: Means "per hundred."
• Percentage symbol (%): Symbol representing percent.
• Whole: Complete quantity against which a percentage is measured.
• Part: Portion of the whole.
• Fraction: Number representing part of a whole or a ratio.
• Decimal: Number represented using decimal place value.
• Equivalent: Having the same numerical value.
• Hundred grid: 10 × 10 grid containing 100 equal squares.
• Benchmark percentage: Familiar percentage used for estimation or comparison.
• Estimate: Approximate value.
• Proportion: Relationship comparing a part with a whole or one quantity with another.
• Percentage point: Unit used to describe the arithmetic difference between two percentages.

---

Useful Percentage Benchmarks

1% = 1/100 = 0.01

10% = 1/10 = 0.1

20% = 1/5 = 0.2

25% = 1/4 = 0.25

50% = 1/2 = 0.5

75% = 3/4 = 0.75

100% = 1 = one whole

Recognizing these values quickly makes percentage reasoning much easier.

---

Percentage Thinking Strategy

When you encounter a percentage problem, ask:

What is the whole?

Then:

What percentage of the whole is being described?

Next:

Can I connect the percentage to a familiar fraction or decimal?

Then:

Can I use a benchmark such as 10%, 25%, 50%, or 75%?

Finally:

Does my answer make sense relative to the whole?

---

Key Takeaways

• Percent means per hundred.
• A percentage represents a quantity relative to 100.
• x% = x/100.
• Percentages can be represented using fractions, decimals, grids, bars, circles, and number lines.
• Each square in a hundred grid represents 1%.
• 100% represents one whole.
• Percentages can be greater than 100% or less than 1%.
• Common percentage-fraction pairs are useful to recognize.
• 50% = 1/2, 25% = 1/4, and 75% = 3/4.
• Percentages provide a common scale that makes proportional comparisons easier.
• The same percentage can represent different actual quantities when the wholes are different.
• A larger percentage does not necessarily mean a larger actual amount if the wholes differ.
• Percentages appear in discounts, test results, surveys, statistics, finance, science, and many other situations.
• Benchmark percentages are useful for mental calculation and estimation.
• 10% can be found by dividing by 10.
• 1% can be found by dividing by 100.
• 50% means half.
• 25% means one-quarter.
• 75% means three-quarters.
• Estimation can help determine whether a percentage calculation is reasonable.
• Always identify the whole before interpreting a percentage.
• The central idea is:

percentage = number of parts out of 100.

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.

---

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%

---

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.

---

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%

---

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.

---

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%

---

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.

---

Another Example

Convert:

9/20

Since:

20 × 5 = 100

multiply by:

5/5

Therefore:

9/20 = 45/100 = 45%

No decimal calculation is necessary.

---

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.

---

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%

---

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.

---

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%

---

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."

---

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%

---

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%.

---

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.

---

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

---

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.

---

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%

---

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.

---

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.

---

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

---

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

---

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.

---

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.

---

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.

---

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.

---

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%

---

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%

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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%

---

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.

---

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%

---

Worked Example 2

Convert:

0.46

to a percentage.

Multiply by 100:

0.46 × 100 = 46

Therefore:

0.46 = 46%

---

Worked Example 3

Convert:

32%

to a decimal.

Divide by 100:

32 ÷ 100 = 0.32

Therefore:

32% = 0.32

---

Worked Example 4

Convert:

45%

to a fraction.

Write:

45/100

Simplify by dividing by 5:

45/100 = 9/20

Therefore:

45% = 9/20

---

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%

---

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%

---

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

---

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.

---

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

---

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.

---

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%

---

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.

---

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

---

Common Mistakes

Mistake 1: Reversing the fraction division

For:

3/4

calculate:

3 ÷ 4

not:

4 ÷ 3

---

Mistake 2: Forgetting that percent means out of 100

35% = 35/100

not:

35/10

---

Mistake 3: Converting 0.4 to 4%

Correct:

0.4 × 100 = 40%

Therefore:

0.4 = 40%

---

Mistake 4: Converting 8% to 0.8

Correct:

8 ÷ 100 = 0.08

Therefore:

8% = 0.08

Remember:

0.8 = 80%

---

Mistake 5: Forgetting to simplify fractions

For example:

50% = 50/100

but in simplest form:

50% = 1/2

---

Mistake 6: Thinking percentages must be below 100%

For example:

1.5 = 150%

Percentages greater than 100% are valid.

---

Mistake 7: Treating rounded values as exact

For example:

2/3 = 66.666...%

Writing:

2/3 ≈ 66.7%

is a rounded approximation.

---

Mistake 8: Assuming one representation is always best

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

---

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.

---

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.

---

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

---

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.

What Does "Percentage of a Quantity" Mean?

A percentage describes a number of parts out of 100.

When we calculate a percentage of a quantity, we are finding a particular fraction of that quantity.

For example:

25% of 80

means:

25/100 of 80

Since:

25% = 1/4

we can calculate:

1/4 × 80 = 20

Therefore:

25% of 80 = 20

The word of usually represents multiplication.

---

The Main Percentage Formula

A reliable method for finding a percentage of a quantity is:

percentage of quantity = percentage as a decimal × quantity

For example:

Find:

35% of 60

Convert 35% to a decimal:

35% = 0.35

Then:

0.35 × 60 = 21

Therefore:

35% of 60 = 21

---

Method 1: Convert the Percentage to a Decimal

Find:

42% of 250

Convert:

42% = 0.42

Then:

0.42 × 250 = 105

Therefore:

42% of 250 = 105

This method works for almost any percentage.

---

Method 2: Convert the Percentage to a Fraction

Find:

25% of 64

We know:

25% = 1/4

Therefore:

1/4 × 64 = 16

So:

25% of 64 = 16

This method is particularly useful for familiar percentages.

---

Useful Percentage Benchmarks

Several percentages are useful for mental calculations.

50% = 1/2

25% = 1/4

75% = 3/4

20% = 1/5

10% = 1/10

5% = half of 10%

1% = 1/100

Knowing these relationships can make many percentage calculations much faster.

---

Finding 50%

To find 50%, divide by 2.

Example:

50% of 90

90 ÷ 2 = 45

Therefore:

50% of 90 = 45

---

Finding 25%

To find 25%, divide by 4.

Example:

25% of 120

120 ÷ 4 = 30

Therefore:

25% of 120 = 30

---

Finding 10%

To find 10%, divide by 10.

Example:

10% of 340

340 ÷ 10 = 34

Therefore:

10% of 340 = 34

---

Finding 5%

5% is half of 10%.

Find:

5% of 240

First:

10% of 240 = 24

Then:

5% of 240 = 12

Therefore:

5% of 240 = 12

---

Finding 1%

To find 1%, divide by 100.

Example:

1% of 700 = 7

This can help calculate unusual percentages.

For example:

3% of 700

If 1% = 7:

3% = 3 × 7 = 21

---

Building Percentages

You can combine familiar percentages.

Find:

15% of 80

10% of 80:

8

5% of 80:

4

Therefore:

15% = 10% + 5%

So:

8 + 4 = 12

Therefore:

15% of 80 = 12

---

Another Mental Strategy

Find:

35% of 200

Break 35% into:

30% + 5%

10% of 200 = 20.

Therefore:

30% of 200 = 60.

5% of 200 = 10.

So:

35% of 200 = 70

Different methods can produce the same result.

---

Using a Bar Model

A bar model can help show how a percentage relates to the whole.

Suppose:

40% of 150

We can represent the full bar as:

100% = 150

Then:

10% = 15

Therefore:

40% = 4 × 15 = 60

So:

40% of 150 = 60

---

Percentage Increase

A percentage increase occurs when a quantity becomes larger by a percentage of its original value.

Suppose a price is:

$80

and it increases by:

25%

First calculate the increase:

25% of $80

Since:

25% = 1/4

the increase is:

$20

Then add the increase:

$80 + $20 = $100

Therefore, the new price is:

$100

---

Percentage Increase: Two-Step Method

The general process is:

Step 1: Find the percentage increase.

Step 2: Add it to the original quantity.

Example:

Increase 150 by 20%.

First:

20% of 150 = 30

Then:

150 + 30 = 180

Therefore:

150 increased by 20% = 180

---

Percentage Increase Using a Multiplier

There is a faster method.

If something increases by 20%, its new value is:

100% + 20% = 120%

Convert:

120% = 1.20

Therefore:

new value = original value × 1.20

For example:

150 × 1.20 = 180

The answer is the same.

---

Increase Multipliers

Some useful examples are:

Increase by 5%:

100% + 5% = 105% = 1.05

Increase by 10%:

110% = 1.10

Increase by 20%:

120% = 1.20

Increase by 35%:

135% = 1.35

Therefore:

new value = original × multiplier

---

Percentage Decrease

A percentage decrease occurs when a quantity becomes smaller by a percentage of its original value.

Suppose a quantity is:

200

and decreases by:

15%

First find the decrease:

15% of 200 = 30

Then subtract:

200 − 30 = 170

Therefore:

200 decreased by 15% = 170

---

Percentage Decrease Using a Multiplier

A 15% decrease means:

100% − 15% = 85%

Convert:

85% = 0.85

Therefore:

200 × 0.85 = 170

This gives the same answer in one calculation.

---

Decrease Multipliers

Decrease by 5%:

95% remains

Multiplier:

0.95

Decrease by 20%:

80% remains

Multiplier:

0.80

Decrease by 30%:

70% remains

Multiplier:

0.70

Decrease by 60%:

40% remains

Multiplier:

0.40

The multiplier represents the percentage that remains, not the percentage removed.

---

Finding the Percentage Increase

Sometimes we know the original and new values and need to calculate the percentage change.

Use:

percentage change = change/original × 100%

For an increase:

percentage increase = increase/original × 100%

Example:

A quantity increases from 50 to 60.

Increase:

60 − 50 = 10

Then:

10/50 × 100% = 20%

Therefore:

the percentage increase is 20%.

---

Finding the Percentage Decrease

Suppose a quantity decreases from 80 to 60.

Decrease:

80 − 60 = 20

Use the original value:

20/80 × 100%

= 25%

Therefore:

the percentage decrease is 25%.

The denominator is the original value, because the percentage change is measured relative to where the quantity started.

---

Why the Original Value Matters

Suppose a price rises from $100 to $120.

Increase:

$20

Percentage increase:

20/100 × 100% = 20%

Now suppose the price falls from $120 back to $100.

Decrease:

$20

But:

20/120 × 100% ≈ 16.7%

So a 20% increase followed by a 20% decrease does not return to the original value.

The percentages are calculated from different starting values.

---

Discounts

A discount reduces the original price of an item.

Suppose an item costs:

$60

and receives a:

25% discount

First calculate the discount:

25% of $60 = $15

Then:

$60 − $15 = $45

Therefore, the sale price is:

$45

---

Discount Using a Multiplier

A 25% discount means:

75% of the original price remains.

Therefore:

75% = 0.75

Calculate:

$60 × 0.75 = $45

This method gives the sale price directly.

---

Discount vs Sale Price

Be careful to distinguish between:

discount amount

and:

sale price

For an $80 item with a 30% discount:

Discount:

0.30 × $80 = $24

Sale price:

$80 − $24 = $56

So:

discount = $24

but:

sale price = $56

They are not the same quantity.

---

Markups

A markup increases a price by a percentage of its original cost.

Suppose a store buys an item for:

$50

and applies a:

40% markup

Calculate the markup:

40% of $50

0.40 × 50 = 20

Add the markup:

$50 + $20 = $70

Therefore, the selling price is:

$70

---

Markup Using a Multiplier

A 40% markup means:

100% + 40% = 140%

Convert:

140% = 1.40

Then:

$50 × 1.40 = $70

---

Discount Followed by Another Discount

Suppose a $100 item receives:

20% off

and then another:

10% off

First discount:

$100 × 0.80 = $80

Second discount:

$80 × 0.90 = $72

Final price:

$72

The total discount is:

$100 − $72 = $28

Therefore, the overall reduction is:

28%

It is not 30%.

The second discount is calculated from the already reduced price.

---

Successive Percentage Changes

Percentage changes are usually applied sequentially.

For example:

Increase by 10%, then increase by 20%.

Starting with 100:

First:

100 × 1.10 = 110

Then:

110 × 1.20 = 132

Overall increase:

32%

not 30%.

This happens because the second percentage is calculated using the new value.

---

Percentage Word Problem: School

A school has:

600 students

and:

45%

participate in after-school activities.

How many students participate?

Calculate:

0.45 × 600 = 270

Therefore:

270 students participate.

---

Percentage Word Problem: Test Score

A student answers 34 questions correctly on a 40-question test.

What percentage is correct?

Use:

part/whole × 100%

Therefore:

34/40 × 100%

= 0.85 × 100%

= 85%

The student's score is:

85%

---

Percentage Word Problem: Saving Money

A student saves:

30%

of $120.

How much is saved?

Calculate:

0.30 × 120 = 36

Therefore:

$36 is saved.

How much remains?

$120 − $36 = $84

---

Percentage Word Problem: Population

A town has:

8,000 people

Its population increases by:

5%

Find the increase:

0.05 × 8000 = 400

Then:

8000 + 400 = 8400

The new population is:

8,400 people.

---

Percentage Word Problem: Measurement

A machine part should have a mass of 200 g.

Its measured mass is 2% greater than this value.

Find 2%:

0.02 × 200 = 4 g

Add:

200 + 4 = 204 g

The measured mass is:

204 g

---

Percentage Word Problem: Sports

A player makes:

18 of 24 attempts

Success rate:

18/24 × 100%

Simplify:

18/24 = 3/4

And:

3/4 = 75%

Therefore, the success rate is:

75%

---

Working Backwards

Sometimes the percentage and resulting quantity are known, but the original quantity is unknown.

Suppose:

25% of a number is 15.

Since:

25% = 1/4

if one-quarter is 15:

whole = 15 × 4 = 60

Therefore:

25% of 60 = 15

---

Working Backwards Using Decimals

Suppose:

40% of a number is 32.

Write:

0.40 × original = 32

Therefore:

original = 32 ÷ 0.40

original = 80

Check:

40% of 80 = 32

Correct.

---

Finding an Original Price After a Discount

A jacket costs $72 after a 20% discount.

What was the original price?

After a 20% discount:

80% remains.

Therefore:

0.80 × original price = $72

So:

original price = 72 ÷ 0.80

= $90

The original price was:

$90

Notice that simply adding 20% of $72 would not give the correct original price.

---

Estimating Percentage Answers

Estimation is useful before and after calculation.

Suppose we need:

48% of 198

48% is close to:

50%

198 is close to:

200

Half of 200 is:

100

Therefore:

48% of 198 should be close to 100.

Exact calculation:

0.48 × 198 = 95.04

This is close to our estimate.

---

Using Benchmarks to Estimate

Estimate:

24% of 82

24% is close to:

25%

82 is close to:

80

25% of 80 is:

20

Therefore:

24% of 82 ≈ 20

This estimate can be used to check the exact calculation.

---

Reasonableness: Percentages Below 100%

If a problem asks:

30% of 70

the answer must be less than 70 because 30% is less than 100%.

If your calculation gives:

210

you know something has gone wrong.

The correct calculation is:

0.30 × 70 = 21

---

Reasonableness: Percentages Above 100%

Suppose:

125% of 80

Since 125% is greater than 100%, the answer should be greater than 80.

Calculate:

1.25 × 80 = 100

The result is reasonable.

---

Reasonableness for Discounts

A $200 item receives a 15% discount.

Before calculating, we know:

10% of $200 = $20.

5% of $200 = $10.

Therefore:

15% = $30.

The sale price should be:

about $170

If a calculation produces $230, it cannot represent a 15% discount.

---

Reasonableness for Increases

Suppose a population of 500 increases by 8%.

10% of 500 is 50.

Therefore, 8% should be slightly less than 50.

Exact increase:

0.08 × 500 = 40

New population:

540

This matches our estimate.

---

Using Diagrams to Understand Percentage Change

A bar can represent the original quantity as:

100%

For a 30% increase:

100% + 30% = 130%

For a 30% decrease:

100% − 30% = 70%

This makes the multiplier method easier to understand:

30% increase → × 1.30

30% decrease → × 0.70

---

A General Percentage Formula

To find a percentage of a quantity:

percentage amount = percentage/100 × original quantity

For example:

18% of 350

= 18/100 × 350

= 63

Therefore:

18% of 350 = 63

---

Percentage Change Formula

To determine how much a quantity changed as a percentage:

percentage change = change/original value × 100%

where:

change = |new value − original value|

Then identify whether the change was an increase or decrease.

---

Worked Example 1

Find:

18% of 250

Convert:

18% = 0.18

Calculate:

0.18 × 250 = 45

Answer:

45

---

Worked Example 2

Increase 240 by 15%.

Find the increase:

0.15 × 240 = 36

Add:

240 + 36 = 276

Answer:

276

Alternatively:

240 × 1.15 = 276

---

Worked Example 3

Decrease 360 by 25%.

Since:

25% = 1/4

Decrease:

360 ÷ 4 = 90

Then:

360 − 90 = 270

Answer:

270

---

Worked Example 4

A $120 item is discounted by 35%.

Find the discount:

0.35 × 120 = $42

Find the sale price:

$120 − $42 = $78

Therefore:

discount = $42

sale price = $78

---

Worked Example 5

A product costs a shop $80.

The shop applies a 25% markup.

Markup:

0.25 × 80 = $20

Selling price:

$80 + $20 = $100

Alternatively:

$80 × 1.25 = $100

---

Worked Example 6

A quantity increases from 120 to 150.

Find the percentage increase.

Change:

150 − 120 = 30

Percentage change:

30/120 × 100% = 25%

Therefore:

the quantity increased by 25%.

---

Worked Example 7

A quantity decreases from 250 to 200.

Change:

250 − 200 = 50

Percentage decrease:

50/250 × 100% = 20%

Therefore:

the quantity decreased by 20%.

---

Worked Example 8: Multi-Step Shopping Problem

A bicycle originally costs:

$500

It is discounted by:

20%

Then an additional:

5%

is taken from the sale price.

First discount:

$500 × 0.80 = $400

Second discount:

$400 × 0.95 = $380

Final price:

$380

The total reduction is:

$500 − $380 = $120

As a percentage of the original:

120/500 × 100% = 24%

So the combined discount is:

24%, not 25%.

---

Worked Example 9: Reverse Percentage

After a 25% increase, a quantity is 150.

Find the original quantity.

After the increase:

125% remains relative to the original

Therefore:

1.25 × original = 150

So:

original = 150 ÷ 1.25

= 120

Check:

25% of 120 = 30.

120 + 30 = 150.

Correct.

---

Choosing a Method

Different percentage problems are easier with different methods.

Use fractions when the percentage is familiar:

25% = 1/4

50% = 1/2

75% = 3/4

Use mental benchmarks for simple percentages:

10%, 5%, 1%, 20%

Use decimals for general calculations:

37% = 0.37

Use multipliers for increases and decreases:

15% increase → × 1.15

15% decrease → × 0.85

A strong problem solver chooses the method that makes the calculation most efficient and clear.

---

Common Mistakes

Mistake 1: Forgetting to convert the percentage

Incorrect:

25% of 80 = 25 × 80

Correct:

0.25 × 80 = 20

---

Mistake 2: Confusing the percentage amount with the final value

For a 20% increase on 50:

Increase:

10

New value:

60

These are different answers.

---

Mistake 3: Subtracting the percentage number directly

A 20% discount on $80 does not mean:

$80 − $20

It means:

$80 − 20% of $80

---

Mistake 4: Using the wrong multiplier

A 20% increase uses:

1.20

A 20% decrease uses:

0.80

---

Mistake 5: Using the new value as the denominator for percentage change

Percentage change is normally measured relative to the original value.

---

Mistake 6: Adding successive percentages

A 20% discount followed by a 10% discount is not generally a 30% total discount.

Each percentage applies to a different value.

---

Mistake 7: Assuming an increase and equal percentage decrease cancel

A 20% increase followed by a 20% decrease does not return to the original value.

---

Mistake 8: Ignoring units

A percentage calculation involving money should produce an amount of money where appropriate.

---

Mistake 9: Not estimating

A quick estimate can reveal errors involving decimal placement or incorrect operations.

---

Did You Know?

Percentage calculations connect many areas of mathematics to everyday decisions.

They are used in:

• discounts and sales
• markups
• taxes
• tips
• interest
• salaries
• population change
• scientific measurements
• percentage error
• test results
• sports statistics
• business
• data analysis

The same basic idea appears repeatedly:

identify the whole → identify the percentage → calculate the appropriate part or new value.

---

Key Terms

• Percentage: Quantity expressed as parts per hundred.
• Percentage of a quantity: Amount obtained by applying a percentage to a whole.
• Original value: Starting quantity before a change.
• New value: Quantity after a change.
• Percentage increase: Increase expressed as a percentage of the original value.
• Percentage decrease: Decrease expressed as a percentage of the original value.
• Percentage change: Change relative to the original quantity, expressed as a percentage.
• Discount: Reduction from an original price.
• Sale price: Price after a discount.
• Markup: Amount added to a cost, usually expressed as a percentage of that cost.
• Multiplier: Decimal factor used to calculate a percentage-adjusted value.
• Benchmark percentage: Familiar percentage used for mental calculation or estimation.
• Estimate: Approximate value used to predict or check an answer.
• Reverse percentage: Process of determining an original value from a known percentage-adjusted value.

---

Key Equations and Rules

Percentage of a quantity:

percentage amount = percentage/100 × quantity

Percentage change:

percentage change = change/original value × 100%

Percentage increase:

new value = original value × (1 + percentage as decimal)

Percentage decrease:

new value = original value × (1 − percentage as decimal)

For example:

20% increase → × 1.20

20% decrease → × 0.80

---

Percentage Problem-Solving Strategy

Step 1: Identify the original quantity.

Step 2: Identify the percentage.

Step 3: Determine what the question asks for:

• percentage amount
• new value
• discount
• sale price
• markup
• percentage change
• original value

Step 4: Estimate the expected answer.

Step 5: Choose an efficient method.

Step 6: Calculate.

Step 7: Include appropriate units.

Step 8: Compare the result with your estimate.

Step 9: Ask whether the result makes sense in context.

---

Key Takeaways

• To calculate a percentage of a quantity, convert the percentage to a decimal and multiply.
• Familiar percentages can often be calculated using fractions or mental strategies.
• 50% means half, 25% means one-quarter, and 10% means one-tenth.
• Percentage increases are calculated relative to the original value.
• For an increase, find the increase and add it to the original value.
• For a decrease, find the decrease and subtract it from the original value.
• Multipliers provide a faster method for percentage increases and decreases.
• A 20% increase corresponds to multiplying by 1.20.
• A 20% decrease corresponds to multiplying by 0.80.
• A discount is the amount removed from the original price.
• The sale price is the amount remaining after the discount.
• A markup increases a cost to produce a new selling price.
• Percentage change is calculated using the original value as the reference.
• Successive percentage changes should be applied one after another rather than simply added.
• Equal percentage increases and decreases do not generally cancel.
• Reverse percentage calculations can be used to find an unknown original value.
• Estimation is an important way to check percentage calculations.
• Always distinguish between the percentage amount and the final value.
• A reliable strategy is:

identify the whole → identify the percentage → estimate → choose a method → calculate → check → interpret the result.