Percentages

Percentages

SAT Math | Problem-Solving and Data Analysis | Foundation to Advanced

Percentages are used to compare quantities, describe changes, calculate discounts and taxes, and interpret data. On the Digital SAT, you must understand how percentages relate to fractions and decimals, calculate percent increases and decreases, work backward to find original values, and solve multistep percentage problems.

1. Understanding Percentages

A percentage represents a quantity out of 100.

The word percent means "per hundred."

For example:

25% = 25/100 = 0.25

This means 25 parts out of every 100 equal parts.

Converting percentages to decimals

Divide by 100.

35% = 0.35

8% = 0.08

125% = 1.25

0.5% = 0.005

Converting decimals to percentages

Multiply by 100.

0.4 = 40%

0.075 = 7.5%

1.8 = 180%

Converting fractions to percentages

Divide the numerator by the denominator and multiply by 100.

For example:

3/5 = 0.6 = 60%

7/20 = 0.35 = 35%

Key idea: Percentages can be less than 100%, equal to 100%, or greater than 100%.

A percentage greater than 100% represents more than the original whole quantity.

2. Finding a Percentage of a Number

To find a percentage of a quantity, convert the percentage into a decimal and multiply.

Part = (Percentage / 100) × Whole

Worked Example 1 — Finding a percentage

Difficulty: Easy

What is 18% of 250?

Step 1: Convert 18% into a decimal.

18% = 0.18

Step 2: Multiply.

0.18 × 250 = 45

Final answer: 45

Worked Example 2 — Percentage of a quantity

Difficulty: Easy

A school has 600 students. Of those students, 35% participate in sports.

How many students participate in sports?

Step 1: Convert the percentage.

35% = 0.35

Step 2: Multiply by the total number of students.

0.35 × 600 = 210

Final answer: 210 students

SAT tip: The word "of" often indicates multiplication in percentage problems.

3. Finding What Percent One Quantity Is of Another

Sometimes the SAT gives the part and the whole but asks for the percentage.

Use:

Percentage = (Part / Whole) × 100

Worked Example 3 — Finding the percentage

Difficulty: Easy

A student answered 42 questions correctly out of 60 questions.

What percentage did the student answer correctly?

Step 1: Divide the part by the whole.

42/60 = 0.7

Step 2: Multiply by 100.

0.7 × 100 = 70%

Final answer: 70%

Worked Example 4 — Understanding the reference quantity

Difficulty: Medium

A town has 1,200 residents, of whom 480 are children.

What percentage of the population are children?

Step 1: Identify the part.

480 children

Step 2: Identify the whole.

1,200 residents

Step 3: Calculate.

(480/1200) × 100 = 40%

Final answer: 40%

Common mistake: Reversing the part and whole. The denominator should represent the quantity used as the reference total.

4. Percent Increase

A percent increase describes how much a quantity grows relative to its original value.

Use:

Percent increase = [(New − Original) / Original] × 100

Worked Example 5 — Percent increase

Difficulty: Medium

The price of a product increases from $80 to $100.

What is the percentage increase?

Step 1: Calculate the increase.

100 − 80 = 20

Step 2: Divide by the original value.

20/80 = 0.25

Step 3: Multiply by 100.

0.25 × 100 = 25%

Final answer: 25%

Finding the new value

When a quantity increases by r%, multiply by:

Growth factor = 1 + r/100

For example, a 20% increase uses a growth factor of:

1 + 0.20 = 1.20

If an item originally costs $150:

New price = 150 × 1.20 = $180

SAT tip: A 20% increase means the new amount is 120% of the original, not 20% of the original.

5. Percent Decrease

A percent decrease describes how much a quantity falls relative to its original value.

Use:

Percent decrease = [(Original − New) / Original] × 100

Worked Example 6 — Percent decrease

Difficulty: Easy

The price of a jacket decreases from $200 to $150.

What is the percentage decrease?

Step 1: Calculate the decrease.

200 − 150 = 50

Step 2: Divide by the original price.

50/200 = 0.25

Step 3: Convert to a percentage.

0.25 × 100 = 25%

Final answer: 25%

Finding a new value after a decrease

Use:

Decay factor = 1 − r/100

For a 15% decrease:

1 − 0.15 = 0.85

If an item originally costs $160:

New price = 160 × 0.85 = $136

Important: A 15% decrease means 85% of the original amount remains.

6. Understanding Growth and Decay Factors

A growth or decay factor is a multiplier used to calculate the value after a percentage change.

Worked Example 7 — Percentage greater than 100%

Difficulty: Hard

A quantity x is increased by 250%, resulting in a new value of 280.

What is x?

Step 1: Interpret the increase.

A 250% increase means adding 250% of the original to the original.

New value = 100% + 250% = 350% of the original.

Step 2: Convert 350% to a decimal.

350% = 3.5

Step 3: Write the equation.

3.5x = 280

Step 4: Divide.

x = 80

Final answer: 80

SAT tip: Increasing a quantity by 250% means multiplying by 3.5, not 2.5.

7. Reverse Percentages

A reverse percentage problem gives the final value and asks for the original amount.

These questions frequently appear in SAT-style scenarios involving discounts, taxes, and increases.

General method

Original = Final / Multiplier

Worked Example 8 — Original price before discount

Difficulty: Medium

A jacket is sold for $96 after a 20% discount.

What was its original price?

Step 1: Determine the amount remaining.

100% − 20% = 80%

Step 2: Convert to a decimal.

80% = 0.80

Step 3: Write the equation.

0.80x = 96

Step 4: Divide.

x = 96/0.80

x = 120

Final answer: $120

Worked Example 9 — Original value before an increase

Difficulty: Medium

A population increased by 30% to 1,560.

What was the original population?

Step 1: Calculate the growth factor.

1 + 0.30 = 1.30

Step 2: Write the equation.

1.30x = 1,560

Step 3: Divide.

x = 1,560/1.30

x = 1,200

Final answer: 1,200

Common mistake: Subtracting 30% of the final value. The stated percentage change is based on the original amount.

8. Successive Percentage Changes

Some SAT questions involve more than one percentage change.

Each change must be applied to the updated quantity.

Worked Example 10 — Increase followed by decrease

Difficulty: Medium

A product costs $200.

Its price increases by 20%, then decreases by 20%.

What is its final price?

Step 1: Apply the increase.

200 × 1.20 = 240

Step 2: Apply the decrease to the new price.

240 × 0.80 = 192

Final answer: $192

The final price is lower than the original price.

Why the changes do not cancel

An increase of 20% is calculated using the original amount.

The decrease of 20% is calculated using the larger, updated amount.

Therefore:

1.20 × 0.80 = 0.96

Only 96% of the original value remains.

General formula

For two successive percentage changes:

Final = Original × First multiplier × Second multiplier

Worked Example 11 — Two percentage increases

Difficulty: Hard

A town's population increases by 20% in one year and then by 15% the following year.

If the original population is 2,000, what is the population after both increases?

Step 1: Convert the increases into multipliers.

20% increase → 1.20

15% increase → 1.15

Step 2: Multiply.

2,000 × 1.20 × 1.15

Step 3: Calculate.

2,000 × 1.38 = 2,760

Final answer: 2,760

The combined increase is 38%, not 35%.

9. Discounts and Sale Prices

A discount reduces the original price.

Formula

Sale price = Original price × (1 − Discount rate)

Worked Example 12 — Sale discount

Difficulty: Easy

A pair of shoes originally costs $120.

The store offers a 25% discount.

What is the sale price?

Step 1: Find the discount.

120 × 0.25 = 30

Step 2: Subtract the discount.

120 − 30 = 90

Final answer: $90

Alternatively:

120 × 0.75 = 90

Worked Example 13 — Two successive discounts

Difficulty: Medium

A laptop originally costs $500.

It is discounted by 30%, followed by an additional 40% discount on the reduced price.

What is the final price?

Step 1: Apply the first discount.

500 × 0.70 = 350

Step 2: Apply the second discount.

350 × 0.60 = 210

Final answer: $210

Important SAT distinction

Two successive discounts of 30% and 40% do not equal one 70% discount.

Combined multiplier:

0.70 × 0.60 = 0.42

The final amount is 42% of the original.

Therefore, the total reduction is 58%.

10. Sales Tax and Tips

A sales tax is an additional amount calculated as a percentage of a price.

Formula

Price after tax = Original price × (1 + Tax rate)

Worked Example 14 — Sales tax

Difficulty: Easy

An item costs $80 before tax.

The sales tax is 7.5%.

What is the total price?

Step 1: Convert the tax rate.

7.5% = 0.075

Step 2: Calculate the tax.

80 × 0.075 = 6

Step 3: Add the tax.

80 + 6 = 86

Final answer: $86

Alternatively:

80 × 1.075 = 86

Worked Example 15 — Restaurant tip

Difficulty: Medium

A restaurant bill is $84 before tax and tip.

A customer pays a 7% sales tax and leaves a tip equal to 15% of the original bill.

What is the total amount paid?

Step 1: Calculate the tax.

84 × 0.07 = 5.88

Step 2: Calculate the tip.

84 × 0.15 = 12.60

Step 3: Add both amounts to the bill.

84 + 5.88 + 12.60 = 102.48

Final answer: $102.48

Important: The question specifies that both percentages are calculated using the original bill. If the tip were calculated after tax, the calculation would be different.

11. Markups and Profit Percentages

A markup is an increase applied to an item's cost to establish its selling price.

Formula

Selling price = Cost × (1 + Markup rate)

Worked Example 16 — Markup

Difficulty: Medium

A store buys an item for $75.

The store marks up the price by 60%.

What is the selling price?

Step 1: Find the markup.

75 × 0.60 = 45

Step 2: Add the markup.

75 + 45 = 120

Final answer: $120

Profit percentage

If the selling price is greater than the cost:

Profit = Selling price − Cost

Profit percentage = (Profit / Cost) × 100

For the previous example:

Profit = 120 − 75 = 45

Profit percentage = (45/75) × 100 = 60%

SAT tip: Markup percentage and profit margin are different measures. Always identify the denominator specified by the problem.

12. Simple Interest and Compound Interest

Interest is the amount earned on savings or charged on borrowed money.

Simple interest

Simple interest is calculated using only the original principal.

I = Prt

where:

• I = interest earned.

• P = principal or initial amount.

• r = annual interest rate as a decimal.

• t = time in years.

Worked Example 17 — Simple interest

Difficulty: Medium

A student deposits $2,000 into an account earning 5% simple interest per year.

How much interest is earned after 3 years?

Step 1: Identify the values.

P = 2,000

r = 0.05

t = 3

Step 2: Apply the formula.

I = Prt

I = 2,000 × 0.05 × 3

I = 300

Final answer: $300 in interest

The total balance is:

2,000 + 300 = $2,300

Compound interest

Compound interest includes interest earned on previous interest.

For annual compounding:

A = P(1 + r)ᵗ

Worked Example 18 — Annual compound interest

Difficulty: Medium

A savings account contains $1,000 and earns 10% interest compounded annually.

What is the balance after 2 years?

Step 1: Write the formula.

A = 1,000(1.10)²

Step 2: Calculate.

A = 1,000(1.21)

A = 1,210

Final answer: $1,210

Comparison

With 10% simple interest for 2 years:

1,000 + 200 = 1,200

With 10% annually compounded interest:

1,000 × 1.1² = 1,210

The difference is caused by earning interest on previously earned interest.

13. Percentage Points vs. Percent Change

This distinction is important when comparing percentages.

A percentage-point change is the arithmetic difference between two percentage values.

A relative percent change compares the difference with the original percentage.

Worked Example 19 — Percentage points

Difficulty: Medium

The percentage of students participating in a club increases from 35% to 42%.

What is the increase in percentage points?

42 − 35 = 7

Final answer: 7 percentage points

Finding the relative percent increase

Use:

[(42 − 35)/35] × 100

= (7/35) × 100

= 20%

Therefore:

The increase is 7 percentage points.

The relative increase is 20%.

These statements describe the same change using different measures.

SAT tip: Read whether the question asks for percentage points or percent change.

14. Percentages of Percentages

Some questions involve applying a percentage to part of a larger group.

Worked Example 20 — Nested percentages

Difficulty: Medium

A school club has 200 members.

60% are juniors.

Of the juniors, 25% participate in a competition.

How many juniors participate?

Step 1: Find the number of juniors.

200 × 0.60 = 120

Step 2: Find 25% of the juniors.

120 × 0.25 = 30

Final answer: 30 students

Shortcut

Multiply the percentages:

0.60 × 0.25 = 0.15

Therefore, 15% of the entire club participates.

200 × 0.15 = 30

Common mistake: Calculating 25% of the entire club instead of 25% of the juniors.

15. Finding the Original Total from a Percentage

A percentage can be used to determine a total when one part is known.

Worked Example 21 — Unknown total

Difficulty: Medium

A solution contains 18 grams of salt.

Salt accounts for 12% of the total solution by mass.

What is the total mass of the solution?

Step 1: Let m be the total mass.

Step 2: Write the equation.

0.12m = 18

Step 3: Divide by 0.12.

m = 150

Final answer: 150 grams

General formula

Whole = Part / Decimal percentage

For example, if 30% of a quantity is 90:

Whole = 90/0.30 = 300

16. Multistep SAT Percentage Problems

Harder SAT questions may combine discounts, taxes, reverse percentages, and multiple changes.

Worked Example 22 — Discount followed by tax

Difficulty: Hard

A jacket is discounted by 20%.

After the discount, a sales tax of 10% is applied.

The final price is $176.

What was the original price?

Step 1: Let x represent the original price.

Step 2: Apply the discount.

Discount multiplier = 0.80

Step 3: Apply the tax.

Tax multiplier = 1.10

Step 4: Write the equation.

x × 0.80 × 1.10 = 176

Step 5: Multiply the factors.

0.88x = 176

Step 6: Divide.

x = 200

Final answer: $200

Worked Example 23 — Finding an unknown percentage

Difficulty: Hard

The number of employees at a company increases from 160 to 200.

The following year, the number decreases by p%.

After the decrease, the company again has 160 employees.

What is p?

Step 1: Find the decrease.

200 − 160 = 40

Step 2: Use the quantity immediately before the decrease as the reference amount.

200 employees

Step 3: Calculate the percent decrease.

(40/200) × 100 = 20%

Final answer: 20%

The increase from 160 to 200 was 25%, but the decrease from 200 to 160 was 20%.

The percentages differ because they use different starting quantities.

17. SAT Strategies and Common Mistakes

Strategy 1 — Identify the reference value

Always determine which quantity represents 100%.

The denominator matters in every percentage comparison.

Strategy 2 — Use multipliers

Instead of calculating the percentage separately every time, use growth and decay factors.

A 15% increase means multiply by 1.15.

A 15% decrease means multiply by 0.85.

Strategy 3 — Work backward when necessary

If the final amount is given, divide by the multiplier rather than subtracting a percentage of the final amount.

Strategy 4 — Apply successive changes separately

Each percentage change applies to the quantity existing immediately before that change.

Strategy 5 — Read percentage language carefully

"130% of a quantity" means 1.3 times the quantity.

"Increased by 130%" means 2.3 times the original quantity.

Strategy 6 — Distinguish percentage points

A change from 20% to 30% is a 10-percentage-point increase and a 50% relative increase.

Strategy 7 — Avoid premature rounding

Keep enough decimal precision during intermediate steps.

Round the final answer only when required.

Strategy 8 — Check whether an answer is reasonable

A 20% discount on a positive price should produce a lower price.

A 20% tax should produce a higher price.

Common Mistake 1 — Using the wrong base

Percent changes must be calculated relative to the stated original or reference quantity.

Common Mistake 2 — Treating 20% as 20

20% = 0.20, not 20.

Common Mistake 3 — Reversing increase and decrease multipliers

A 20% increase uses 1.20.

A 20% decrease uses 0.80.

Common Mistake 4 — Assuming opposite changes cancel

Increasing by 20% and then decreasing by 20% results in 96% of the original value.

Common Mistake 5 — Adding successive percentages

Successive increases of 20% and 15% produce an overall 38% increase, not 35%.

Common Mistake 6 — Confusing percentages with percentage points

A change from 35% to 42% is 7 percentage points, not a 7% relative increase.

Common Mistake 7 — Calculating a tip using the wrong bill

Check whether the tip is based on the pre-tax bill or the bill after tax.

Common Mistake 8 — Forgetting the original quantity in an increase

A 200% increase means the new amount is 300% of the original.

Common Mistake 9 — Forgetting the difference between markup and profit margin

Markup usually uses cost as the denominator.

Profit margin uses selling price as the denominator.

Common Mistake 10 — Rounding too early

Small rounding differences can produce the wrong result in multistep percentage calculations.

18. Check Your Understanding

Question 1

What is 15% of 360?

A. 36

B. 45

C. 54

D. 60

Correct answer: C — 54

0.15 × 360 = 54

Question 2

A product's price increases from $80 to $100.

What is the percentage increase?

A. 20%

B. 25%

C. 30%

D. 40%

Correct answer: B — 25%

Increase = 20

20/80 × 100 = 25%

Question 3

An item costs $150 after a 25% discount.

What was its original price?

A. $175

B. $187.50

C. $200

D. $225

Correct answer: C — $200

After a 25% discount, 75% remains.

0.75x = 150

x = 200

Question 4

A quantity increases by 200%.

What is the multiplier applied to the original quantity?

A. 1

B. 2

C. 3

D. 4

Correct answer: C — 3

Original 100% + increase 200% = 300%

300% = 3

Question 5

A $200 item is discounted by 10%, then discounted by another 10%.

What is the final price?

A. $160

B. $162

C. $180

D. $198

Correct answer: B — $162

200 × 0.90 × 0.90 = 162

19. Lesson Summary

A percentage represents a quantity out of 100.

To calculate a percentage of a value, convert the percentage to a decimal and multiply.

To determine what percentage one quantity is of another, divide the part by the reference whole and multiply by 100.

Percent increase and decrease are measured relative to the original quantity.

Growth and decay factors simplify percentage calculations.

Reverse percentage problems can be solved by dividing the final value by the multiplier.

Successive percentage changes must be applied to updated quantities.

Discounts, taxes, tips, interest, and markups are common SAT applications.

Percentage points and relative percent changes are different measures.

Some advanced SAT problems involve percentages greater than 100%, repeated changes, and multiple conditions.

You are now ready to complete Percentages practice questions in Baloo Prep.