Percent Applications

How do you calculate tips, sales tax, discounts, and interest? Percent applications are everywhere in daily life - from shopping to banking to understanding statistics!

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Review: What Is a Percent?

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

Symbol: %

Examples:

• 25% = 25/100 = 0.25
• 50% = 50/100 = 0.5 = 1/2
• 100% = 100/100 = 1 (the whole thing)

Converting:

• Percent to decimal: Divide by 100 (move decimal 2 left)
• Decimal to percent: Multiply by 100 (move decimal 2 right)

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Finding Percent of a Number

Question: What is 30% of 80?

Method 1: Convert to decimal 30% = 0.30 0.30 × 80 = 24

Method 2: Use fraction 30% = 30/100 (30/100) × 80 = 2,400/100 = 24

Answer: 24

Formula: Percent of Number = (Percent as decimal) × Number

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Three Types of Percent Problems

Type 1: Find the part "What is 20% of 50?" Answer: 0.20 × 50 = 10

Type 2: Find the percent "12 is what percent of 60?" Answer: 12/60 = 0.20 = 20%

Type 3: Find the whole "15 is 30% of what number?" Answer: 15 ÷ 0.30 = 50

Identifying the type is key to solving!

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Sales Tax

Sales tax is a percent added to the price.

Example: Item costs $40, sales tax is 8%. Find total.

Step 1: Find tax amount 8% of $40 = 0.08 ×$40 = $3.20

Step 2: Add to original price $40 +$3.20 = $43.20

Answer: Total is $43.20

Shortcut: Multiply by 1.08 $40 × 1.08 =$43.20

Why? 100% + 8% = 108% = 1.08

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Tips and Gratuity

Tip is a percent of the bill given for service.

Common tip percentages:

• 15% = good service
• 18-20% = excellent service
• 10% = poor service

Example: Restaurant bill is $45. Leave 18% tip.

Step 1: Find tip 18% of $45 = 0.18 ×$45 = $8.10

Step 2: Find total $45 +$8.10 = $53.10

Answer: Total with tip is $53.10

Quick tip calculation:

• 10% = move decimal left once ($45 →$4.50)
• 20% = double the 10% ($4.50 × 2 =$9)
• 15% = halfway between 10% and 20%

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Discounts and Sales

Discount is a percent subtracted from the original price.

Example: Shirt originally $50, now 25% off. Find sale price.

Step 1: Find discount amount 25% of $50 = 0.25 ×$50 = $12.50

Step 2: Subtract from original $50 -$12.50 = $37.50

Answer: Sale price is $37.50

Shortcut: Multiply by 0.75 $50 × 0.75 =$37.50

Why? 100% - 25% = 75% = 0.75 (you pay 75%)

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Multiple Discounts

Example: Item is $100, first 20% off, then additional 10% off.

IMPORTANT: Discounts are applied sequentially, not added!

Step 1: First discount (20% off) $100 × 0.80 =$80

Step 2: Second discount (10% off the new price) $80 × 0.90 =$72

Answer: Final price is $72

Note: NOT 30% off! (That would be $70) Sequential discounts give$72, which is 28% off total.

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Sales Tax AND Discount

When both apply, do discount FIRST, then tax!

Example: $60 item, 20% off, 7% tax

Step 1: Apply discount $60 × 0.80 =$48

Step 2: Apply tax to discounted price $48 × 1.07 =$51.36

Answer: Final total is $51.36

Tax is always on the sale price, not original price!

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Simple Interest

Simple interest is money earned (or owed) on principal.

Formula: I = Prt

Where:

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

Example: $500 in savings at 3% per year for 2 years

I = $500 × 0.03 × 2 I =$30

Total after 2 years: $500 +$30 = $530

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Percent Increase

Percent increase shows how much something grew.

Formula: Percent Increase = (New - Original)/Original × 100%

Example: Price rose from $20 to$25. Find percent increase.

Percent Increase = (25 - 20)/20 × 100% = 5/20 × 100% = 0.25 × 100% = 25%

Answer: 25% increase

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Percent Decrease

Percent decrease shows how much something fell.

Formula: Percent Decrease = (Original - New)/Original × 100%

Example: Population dropped from 5,000 to 4,500. Find percent decrease.

Percent Decrease = (5,000 - 4,500)/5,000 × 100% = 500/5,000 × 100% = 0.10 × 100% = 10%

Answer: 10% decrease

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Markup

Markup is percent added to cost to get selling price.

Example: Store buys item for $30, marks up 40%. Find selling price.

Markup = 40% of $30 = 0.40 ×$30 = $12

Selling price = $30 +$12 = $42

Or use shortcut: $30 × 1.40 =$42

Markup helps businesses make profit!

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Commission

Commission is percent of sales earned as payment.

Example: Salesperson earns 6% commission. Sells $8,000 worth. How much commission?

Commission = 6% of $8,000 = 0.06 ×$8,000 = $480

Answer: $480 commission

Total pay might be commission + base salary

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Percent Error

Percent error measures accuracy of estimates.

Formula: Percent Error = |Estimated - Actual|/Actual × 100%

Example: Estimated 50 people, actually 45. Find percent error.

Percent Error = |50 - 45|/45 × 100% = 5/45 × 100% ≈ 11.1%

Answer: About 11.1% error

Lower percent error = more accurate!

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Percent of Change Formula

General formula for increase OR decrease:

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

Positive result = increase Negative result = decrease

Example: Changed from 80 to 60

(60 - 80)/80 × 100% = -20/80 × 100% = -25%

Answer: 25% decrease (negative shows decrease)

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Real-World Applications

Shopping:

• Original price: $80
• 30% off sale
• Sale price: $80 × 0.70 =$56
• With 6% tax: $56 × 1.06 =$59.36

Banking:

• Deposit $1,000 at 2% annual interest
• After 1 year: $1,000 × 1.02 =$1,020
• Earned $20 interest

Nutrition:

• Food label: 15% daily value of vitamin C
• Daily need: 60 mg
• This food: 15% of 60 = 9 mg

Statistics:

• Survey: 240 out of 300 people agree
• Percent: 240/300 = 0.80 = 80%

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Tips for Different Situations

Finding final price with tax: Multiply by (1 + tax rate) Example: 8% tax → multiply by 1.08

Finding sale price with discount: Multiply by (1 - discount rate) Example: 25% off → multiply by 0.75

Finding tip quickly:

• 10%: move decimal left
• 20%: double the 10%
• 15%: average of 10% and 20%

Finding percent change: (New - Old)/Old × 100%

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Common Mistakes to Avoid

❌ Mistake 1: Adding percentages for sequential discounts

• Wrong: 20% off then 10% off = 30% off
• Right: Calculate each separately

❌ Mistake 2: Applying tax before discount

• Wrong: Tax on original price
• Right: Tax on sale price

❌ Mistake 3: Using wrong base for percent change

• Use ORIGINAL as denominator
• (New - Original)/Original, not (New - Original)/New

❌ Mistake 4: Forgetting to convert percent to decimal

• Remember: 25% = 0.25, not 25

❌ Mistake 5: Rounding too early

• Keep decimals during calculation
• Round final answer to cents

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Problem-Solving Strategy

For price calculations:

1. Identify original price
2. Determine if adding (tax, tip, markup) or subtracting (discount)
3. Calculate amount
4. Apply to original price
5. Check reasonableness

For percent change:

1. Find difference (New - Original)
2. Divide by original
3. Multiply by 100%
4. Check if increase (positive) or decrease (negative)

For multi-step:

1. Do one step at a time
2. Order matters! (discount before tax)
3. Show all work
4. Verify final answer

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Quick Reference

Basic Formula: Percent of Number = (Percent ÷ 100) × Number

Sales Tax: Total = Price × (1 + tax rate)

Discount: Sale Price = Original × (1 - discount rate)

Tip: Total = Bill × (1 + tip rate)

Simple Interest: I = Prt

Percent Change: (New - Original)/Original × 100%

Increase: multiply by (1 + percent) Decrease: multiply by (1 - percent)

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Practice Tips

Tip 1: Always convert percent to decimal

• Move decimal 2 places left
• 35% → 0.35

Tip 2: Use shortcuts when possible

• 1.08 for 8% tax
• 0.75 for 25% off

Tip 3: Check with estimation

• 20% of $50 should be around$10
• If you get $100, something's wrong!

Tip 4: Round money to 2 decimals

• Always include cents
• Round to nearest penny if needed

Tip 5: Draw diagrams for complex problems

• Visual helps track multiple steps
• Especially for multiple discounts/increases

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Summary

Percent applications are everywhere:

• Shopping: discounts, sales tax
• Service: tips, commission
• Finance: interest, investments
• Statistics: changes, comparisons

Key formulas:

• Part = Percent × Whole
• Tax/Tip: multiply by (1 + rate)
• Discount: multiply by (1 - rate)
• Change: (New - Old)/Old × 100%

Important skills:

• Converting percents to decimals
• Identifying problem type
• Following correct order (discount before tax)
• Checking answers for reasonableness

Mastering percent applications helps you:

• Shop smarter
• Calculate tips quickly
• Understand financial products
• Analyze data and statistics

Percent applications are one of the most practical math skills you'll use throughout your life!