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.

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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

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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.

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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.

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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.

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Finding 50%

To find 50%, divide by 2.

Example:

50% of 90

90 ÷ 2 = 45

Therefore:

50% of 90 = 45

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Finding 25%

To find 25%, divide by 4.

Example:

25% of 120

120 ÷ 4 = 30

Therefore:

25% of 120 = 30

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Finding 10%

To find 10%, divide by 10.

Example:

10% of 340

340 ÷ 10 = 34

Therefore:

10% of 340 = 34

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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

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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

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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

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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.

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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

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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

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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

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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.

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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

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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

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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.

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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.

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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%.

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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.

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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.

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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

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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.

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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.

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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

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Markup Using a Multiplier

A 40% markup means:

100% + 40% = 140%

Convert:

140% = 1.40

Then:

$50 × 1.40 = $70

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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.

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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.

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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.

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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%

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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

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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.

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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

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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%

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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

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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

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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

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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.

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Worked Example 1

Find:

18% of 250

Convert:

18% = 0.18

Calculate:

0.18 × 250 = 45

Answer:

45

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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

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Worked Example 3

Decrease 360 by 25%.

Since:

25% = 1/4

Decrease:

360 ÷ 4 = 90

Then:

360 − 90 = 270

Answer:

270

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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

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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

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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%.

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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%.

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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%.

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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.

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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.

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Common Mistakes

Mistake 1: Forgetting to convert the percentage

Incorrect:

25% of 80 = 25 × 80

Correct:

0.25 × 80 = 20

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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.

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Mistake 3: Subtracting the percentage number directly

A 20% discount on $80 does not mean:

$80 − $20

It means:

$80 − 20% of $80

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Mistake 4: Using the wrong multiplier

A 20% increase uses:

1.20

A 20% decrease uses:

0.80

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

Percentage change is normally measured relative to the original value.

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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.

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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.

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Mistake 8: Ignoring units

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

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Mistake 9: Not estimating

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

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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.

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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.

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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

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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.

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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.