Percent Problems

What is a Percent?

Percent means "per hundred" or "out of 100"

Symbol: %

Key concept: 50% = 50/100 = 0.50 = 1/2

Converting:

• Percent to decimal: Divide by 100 (move decimal 2 left)
• Decimal to percent: Multiply by 100 (move decimal 2 right)
• Percent to fraction: Write over 100, simplify

Examples:

• 25% = 0.25 = 25/100 = 1/4
• 0.75 = 75%
• 3/4 = 0.75 = 75%
• 150% = 1.50 = 1.5 = 3/2

Three Types of Percent Problems

Type 1: Find the part What is 30% of 80?

Type 2: Find the percent 12 is what percent of 40?

Type 3: Find the whole 15 is 25% of what number?

Key equation: Part = Percent × Whole Or: is = % × of

Finding the Part

Question form: What is P% of W?

Equation: Part = (P/100) × W

Example 1: What is 20% of 50?

Part = 0.20 × 50 = 10

Answer: 10

Example 2: Find 35% of 200

Part = 0.35 × 200 = 70

Answer: 70

Example 3: What is 8% of 125?

Part = 0.08 × 125 = 10

Answer: 10

Example 4: Calculate 150% of 60

Part = 1.50 × 60 = 90

Answer: 90 (yes, more than 100%!)

Finding the Percent

Question form: A is what percent of B?

Equation: Percent = (Part/Whole) × 100

Example 1: 15 is what percent of 60?

Percent = (15/60) × 100 = 0.25 × 100 = 25%

Answer: 25%

Example 2: What percent of 80 is 20?

Percent = (20/80) × 100 = 0.25 × 100 = 25%

Answer: 25%

Example 3: 33 is what percent of 150?

Percent = (33/150) × 100 = 0.22 × 100 = 22%

Answer: 22%

Example 4: What percent of 25 is 30?

Percent = (30/25) × 100 = 1.2 × 100 = 120%

Answer: 120% (more than the whole!)

Finding the Whole

Question form: A is P% of what number?

Equation: Whole = Part / (P/100) = Part / Percent (as decimal)

Example 1: 20 is 25% of what number?

Whole = 20 / 0.25 = 80

Answer: 80

Example 2: 15 is 30% of what number?

Whole = 15 / 0.30 = 50

Answer: 50

Example 3: 12 is 8% of what number?

Whole = 12 / 0.08 = 150

Answer: 150

Example 4: 45 is 150% of what number?

Whole = 45 / 1.50 = 30

Answer: 30

Using Proportions

Alternative method: Set up proportion

Form: Part/Whole = Percent/100

Example 1: What is 40% of 70?

x/70 = 40/100

Cross multiply: 100x = 2800 x = 28

Example 2: 18 is what percent of 72?

18/72 = x/100

Cross multiply: 72x = 1800 x = 25

Answer: 25%

Example 3: 24 is 60% of what number?

24/x = 60/100

Cross multiply: 60x = 2400 x = 40

Answer: 40

Percent Increase and Decrease

Percent Change Formula:

Percent Change = (New - Original) / Original × 100

Increase: New > Original (positive result) Decrease: New < Original (negative result)

Example 1: Percent Increase

Original price: 50 dollars New price: 60 dollars

Percent Increase = (60 - 50)/50 × 100 = 10/50 × 100 = 20%

Price increased by 20%

Example 2: Percent Decrease

Original price: 80 dollars Sale price: 64 dollars

Percent Decrease = (64 - 80)/80 × 100 = -16/80 × 100 = -20%

Price decreased by 20% (negative indicates decrease)

Example 3: Population Growth

Original population: 5,000 New population: 6,500

Percent Increase = (6,500 - 5,000)/5,000 × 100 = 1,500/5,000 × 100 = 30%

Population grew by 30%

Example 4: Test Score Improvement

First score: 70 Second score: 84

Percent Increase = (84 - 70)/70 × 100 = 14/70 × 100 = 20%

Score improved by 20%

Finding New Amount After Percent Change

Increase: New = Original × (1 + percent increase as decimal)

Decrease: New = Original × (1 - percent decrease as decimal)

Example 1: 15% Increase

Original: 200 Increase by 15%

New = 200 × (1 + 0.15) = 200 × 1.15 = 230

Example 2: 20% Decrease

Original: 150 Decrease by 20%

New = 150 × (1 - 0.20) = 150 × 0.80 = 120

Example 3: Price After Discount

Original price: 85 dollars 30% off discount

Sale price = 85 × (1 - 0.30) = 85 × 0.70 = 59.50 dollars

Example 4: Population After Growth

Current population: 8,000 Expected 12% growth

New population = 8,000 × (1 + 0.12) = 8,000 × 1.12 = 8,960

Sales Tax Problems

Total Cost = Original Price + Tax Tax = Original Price × Tax Rate

Example 1: 6% sales tax

Item cost: 50 dollars Tax = 50 × 0.06 = 3 dollars Total = 50 + 3 = 53 dollars

Shortcut: Total = 50 × 1.06 = 53 dollars

Example 2: 8.5% sales tax

Item cost: 120 dollars Total = 120 × 1.085 = 130.20 dollars

Example 3: Finding Original Price

Total with tax: 84.80 dollars Tax rate: 6%

Original price = 84.80 / 1.06 = 80 dollars

Discount Problems

Sale Price = Original Price - Discount Discount = Original Price × Discount Rate

Example 1: 25% off

Original: 80 dollars Discount = 80 × 0.25 = 20 dollars Sale price = 80 - 20 = 60 dollars

Shortcut: Sale price = 80 × 0.75 = 60 dollars

Example 2: 40% off

Original: 150 dollars Sale price = 150 × 0.60 = 90 dollars

Example 3: Multiple Discounts

Original: 100 dollars 20% off, then additional 10% off

After first discount: 100 × 0.80 = 80 dollars After second discount: 80 × 0.90 = 72 dollars

Note: NOT the same as 30% off! (that would be 70 dollars)

Tip and Commission Problems

Tip = Bill Amount × Tip Percentage

Example 1: Restaurant Tip

Bill: 45 dollars Tip 20%

Tip = 45 × 0.20 = 9 dollars Total = 45 + 9 = 54 dollars

Example 2: Commission

Sales: 12,000 dollars Commission rate: 5%

Commission = 12,000 × 0.05 = 600 dollars

Example 3: Finding Sales from Commission

Commission earned: 450 dollars Commission rate: 6%

Sales = 450 / 0.06 = 7,500 dollars

Simple Interest

Formula: I = Prt

Where:

• I = Interest earned
• P = Principal (initial amount)
• r = Rate (as decimal)
• t = Time (in years)

Total Amount: A = P + I = P(1 + rt)

Example 1:

Principal: 1,000 dollars Rate: 5% per year Time: 3 years

Interest = 1,000 × 0.05 × 3 = 150 dollars Total = 1,000 + 150 = 1,150 dollars

Example 2:

Borrow 2,500 dollars at 4% for 2 years

Interest = 2,500 × 0.04 × 2 = 200 dollars Total owed = 2,500 + 200 = 2,700 dollars

Example 3: Finding Rate

Principal: 800 dollars Time: 5 years Interest earned: 200 dollars

200 = 800 × r × 5 200 = 4,000r r = 0.05 = 5%

Percent of a Percent

Multiply the decimals

Example 1: 20% of 50% of 200

First: 50% of 200 = 0.50 × 200 = 100 Then: 20% of 100 = 0.20 × 100 = 20

Shortcut: 0.20 × 0.50 × 200 = 20

Example 2: 10% of 30% of 500

0.10 × 0.30 × 500 = 15

Markup Problems

Selling Price = Cost + Markup Markup = Cost × Markup Percentage

Example 1: Store markup

Cost to store: 40 dollars Markup: 60%

Markup amount = 40 × 0.60 = 24 dollars Selling price = 40 + 24 = 64 dollars

Shortcut: Selling price = 40 × 1.60 = 64 dollars

Example 2: Finding Cost

Selling price: 120 dollars Markup: 50%

Cost = 120 / 1.50 = 80 dollars

Successive Percents

Apply percentages one at a time

Example 1: Price increases 10%, then decreases 10%

Original: 100 dollars After increase: 100 × 1.10 = 110 dollars After decrease: 110 × 0.90 = 99 dollars

NOT back to original! Lost 1 dollar overall

Example 2: Successive Growth

Population: 1,000 Grows 5% first year, 8% second year

After year 1: 1,000 × 1.05 = 1,050 After year 2: 1,050 × 1.08 = 1,134

Total growth: 13.4% (not 13%!)

Common Mistakes to Avoid

1. Confusing part and whole "25% of 80" → 80 is the whole, not the part!

2. Wrong formula for percent change Use (New - Original)/Original, not (Original - New)/New

3. Adding percents incorrectly 20% off then 10% off ≠ 30% off!

4. Forgetting to convert percent to decimal 30% = 0.30 (not 30 in calculations!)

5. Percent of percent errors 50% of 50% = 25% (not 100%!)

6. Using whole instead of original After 20% increase: use original as base for percent change

Mental Math Tricks

10%: Move decimal one place left 50 → 5

1%: Move decimal two places left 50 → 0.5

5%: Half of 10% 10% of 50 = 5, so 5% = 2.5

25%: Divide by 4 25% of 80 = 80/4 = 20

50%: Divide by 2 50% of 60 = 30

Double: Same as 200% 200% of 15 = 30

Real-World Applications

Shopping: Calculate discounts, sales tax, total cost

Finance: Interest, investments, loans

Statistics: Survey results, data analysis

Science: Percent error, concentration, growth rates

Business: Profit margins, commission, markup

Everyday: Tips, grades (percent correct), batting averages

Quick Reference

Part = Percent × Whole (is = % × of)

Percent = (Part/Whole) × 100

Whole = Part / Percent (as decimal)

Percent Change = (New - Original) / Original × 100

After Increase: New = Original × (1 + r)

After Decrease: New = Original × (1 - r)

Simple Interest: I = Prt

Practice Strategy

• Start with basic "find the part" problems
• Master conversions (percent ↔ decimal ↔ fraction)
• Use proportions if formula confuses you
• Practice percent change in both directions
• Work real-world problems (shopping, tips, interest)
• Check answers for reasonableness
• Use mental math for common percents
• Understand the three types of problems
• Don't just memorize - understand the relationships
• Practice successive percents carefully
• Apply to real life whenever possible!

Percent problems are everywhere in daily life. Master them and you'll be a more informed consumer, investor, and decision-maker!