Percentage Applications | Maths

Master practical percentage calculations for shopping, finance, business, and everyday decisions, from discounts and sales tax to profit margins and interest rates.

Percentage Refresher

Percentage = "per hundred" = fraction with denominator 100

Three Forms:

Percentage: 25%
Decimal: 0.25
Fraction: 1/4

Quick Conversions:

Percent → Decimal: Divide by 100 (move decimal 2 left)
Decimal → Percent: Multiply by 100 (move decimal 2 right)

The Three Percentage Problems

Every percentage problem fits one of three types:

Type 1: Find the Part

Question: What is X% of Y?
Formula: Part = Percent × Whole

Example: What is 15% of $80?

0.15 × $80 = $12

Type 2: Find the Percentage

Question: What percent is X of Y?
Formula: Percent = (Part ÷ Whole) × 100

Example: What percent is $12 of $80?

($12 ÷ $80) × 100 = 15%

Type 3: Find the Whole

Question: X is Y% of what?
Formula: Whole = Part ÷ Percent

Example: $12 is 15% of what?

$12 ÷ 0.15 = $80

Discounts and Sale Prices

Calculating Discount Amount

Formula: Discount = Original Price × Discount Rate

Example: 30% off a $150 jacket

Discount: $150 × 0.30 = $45
Sale price: $150 − $45 = $105

Shortcut: Calculate directly

Sale price: $150 × 0.70 = $105
(If 30% off, you pay 70%)

Multiple Discounts

Important: Apply sequentially, NOT additively!

Example: 20% off, then additional 10% off

Wrong: 20% + 10% = 30% off ❌

Correct:

First discount: $100 × 0.80 = $80
Second discount: $80 × 0.90 = $72
Total paid: $72 (equals 28% off, not 30%)

Formula: Final = Original × (1 − d₁) × (1 − d₂)

$100 × 0.80 × 0.90 = $72

Finding Original Price from Sale Price

Problem: Item is $60 after 25% off. What was original price?

Solution:

$60 is 75% of original (100% − 25% = 75%)
Original: $60 ÷ 0.75 = $80

Check: $80 × 0.25 = $20 discount, $80 − $20 = $60 ✓

Sales Tax

Calculating Tax

Formula: Tax = Price × Tax Rate

Example: $45 purchase with 8% sales tax

Tax: $45 × 0.08 = $3.60
Total: $45 + $3.60 = $48.60

Shortcut: Total = Price × 1.08 = $45 × 1.08 = $48.60

Finding Price Before Tax

Problem: Total with tax is $54, tax rate is 8%. What's the pre-tax price?

Solution:

$54 represents 108% (100% + 8%)
Pre-tax: $54 ÷ 1.08 = $50

Check: $50 × 0.08 = $4 tax, $50 + $4 = $54 ✓

Tips and Service Charges

Calculating Tips

Standard Rates:

15%: adequate service
18%: good service
20%: great service

Mental Math Method:

Find 10% (move decimal left)
Build from there

Example: 18% tip on $65 bill

10% = $6.50
5% = $3.25 (half of 10%)
3% = ~$2.00 (approximately)
18% ≈ $6.50 + $6.50 + $3.25 = ~$12 (or exactly $11.70)

Quick Method for 20%: Move decimal left, double it

10% of $65 = $6.50
20% = $6.50 × 2 = $13

Tip on Pre-Tax vs Post-Tax

Common Practice: Calculate tip on pre-tax amount

Example: Bill is $50 before tax

Tax (8%): $4 → Total with tax: $54
Tip (20% of pre-tax): $50 × 0.20 = $10
Pay: $54 + $10 = $64

Markup and Margin (Business Pricing)

Understanding the Difference

Markup: Percentage added to cost to get selling price
Margin: Percentage of selling price that is profit

Example:

Cost: $60
Selling price: $100
Profit: $40

Markup: ($40 ÷ $60) × 100 = 66.7% (based on cost)
Margin: ($40 ÷ $100) × 100 = 40% (based on selling price)

Calculating Selling Price from Markup

Formula: Selling Price = Cost × (1 + Markup %)

Example: Cost is $80, markup is 50%

Selling price: $80 × 1.50 = $120
Profit: $40

Calculating Cost from Margin

Formula: Cost = Selling Price × (1 − Margin %)

Example: Selling price is $100, margin is 30%

Cost: $100 × 0.70 = $70
Profit: $30

Converting Between Markup and Margin

From Markup to Margin:
Margin = Markup ÷ (1 + Markup)

Example: 50% markup

Margin: 0.50 ÷ 1.50 = 0.333 = 33.3%

From Margin to Markup:
Markup = Margin ÷ (1 − Margin)

Example: 40% margin

Markup: 0.40 ÷ 0.60 = 0.667 = 66.7%

Interest Calculations

Simple Interest

Interest calculated only on principal.

Formula: I = Prt

I = Interest earned
P = Principal (starting amount)
r = Annual rate (as decimal)
t = Time (in years)

Example: $5,000 at 4% for 3 years

Interest: $5,000 × 0.04 × 3 = $600
Total: $5,000 + $600 = $5,600

Compound Interest (Basic)

Interest calculated on principal + accumulated interest.

Annual Compounding Formula: A = P(1 + r)^t

Example: $5,000 at 4% for 3 years, compounded annually

A = $5,000(1.04)³
A = $5,000 × 1.1249
A = $5,624.32
Interest earned: $624.32 (vs $600 with simple interest)

Why Compound is Better: Interest earns interest

Percentage Change

Calculating Percentage Increase/Decrease

Formula: % Change = [(New − Old) ÷ Old] × 100

Example: Price increased from $80 to $92

Change: $92 − $80 = $12
Percentage: ($12 ÷ $80) × 100 = 15% increase

Example: Sales decreased from 500 to 425

Change: 425 − 500 = −75
Percentage: (−75 ÷ 500) × 100 = −15% (15% decrease)

Applying Percentage Change

Increase: New Value = Old × (1 + % increase)

Example: Increase $200 by 15%

$200 × 1.15 = $230

Decrease: New Value = Old × (1 − % decrease)

Example: Decrease $200 by 15%

$200 × 0.85 = $170

Multiple Changes (Successive Percentage Changes)

Example: Population increases 10%, then increases another 10%

Wrong: 10% + 10% = 20% total ❌

Correct:

Start: 1000
After first: 1000 × 1.10 = 1100
After second: 1100 × 1.10 = 1210
Total increase: 210 = 21% (not 20%)

Formula: Final = Original × (1 + r₁) × (1 + r₂)

Percentage Point vs Percentage Change

Important distinction!

Example: Interest rate changes from 5% to 8%

Percentage Point Change: 8% − 5% = 3 percentage points

Percentage Change: (8 − 5) ÷ 5 × 100 = 60% increase

Media often confuses these, so be precise.

Common Business Percentages

Profit Margin

Formula: Profit Margin = (Net Profit ÷ Revenue) × 100

Example:

Revenue: $100,000
Costs: $65,000
Net profit: $35,000
Margin: ($35,000 ÷ $100,000) × 100 = 35%

Return on Investment (ROI)

Formula: ROI = [(Gain − Cost) ÷ Cost] × 100

Example:

Investment: $10,000
Return: $13,000
Gain: $3,000
ROI: ($3,000 ÷ $10,000) × 100 = 30%

Growth Rate

Formula: Growth Rate = [(Current − Previous) ÷ Previous] × 100

Example: Revenue grew from $50,000 to $62,000

Growth: ($62,000 − $50,000) ÷ $50,000 × 100 = 24%

Commission

Formula: Commission = Sales × Commission Rate

Example: $50,000 in sales at 5% commission

Commission: $50,000 × 0.05 = $2,500

Practice Problems

Discounts

Item costs $80. What's the sale price at 35% off?
After 40% off, you pay $54. What was the original price?

Tax and Tips

Calculate total for $62 meal with 7% tax and 18% tip (on pre-tax amount).
Total bill with tax is $86.40. Tax rate is 8%. What was pre-tax amount?

Markup and Margin

Cost is $45, markup is 60%. What's the selling price?
Selling price is $90, margin is 40%. What's the cost?

Interest

$8,000 at 5% simple interest for 2 years. How much interest?
$3,000 at 6% compounded annually for 2 years. What's final amount?

Percentage Change

Stock price went from $50 to $58. What's the percentage increase?
Store had 480 customers last week, 408 this week. What's the percentage change?

Business Metrics

Revenue is $250,000, costs are $190,000. What's the profit margin?
Invested $15,000, sold for $18,750. What's the ROI?

Solutions

$52 ($80 × 0.65)
$90 ($54 ÷ 0.60)
$77.50 (tax: $4.34, tip: $11.16)
$80 ($86.40 ÷ 1.08)
$72 ($45 × 1.60)
$54 ($90 × 0.60)
$800 ($8,000 × 0.05 × 2)
$3,370.80 ($3,000 × 1.06²)
16% (($58 − $50) ÷ $50 × 100)
−15% or 15% decrease ((408 − 480) ÷ 480 × 100)
24% (($60,000 ÷ $250,000) × 100)
25% (($3,750 ÷ $15,000) × 100)

Key Takeaways

✓ Three types of percentage problems: know which you're solving
✓ Multiple discounts multiply: don't add them
✓ Markup ≠ Margin: based on different values
✓ Compound interest is powerful: small differences compound over time
✓ Percentage vs percentage points: know the difference
✓ Mental math shortcuts: 10% method works for most calculations

Real-World Applications

Shopping: Calculate discounts, compare sales
Dining: Calculate tips quickly
Business: Set prices with appropriate margins
Investing: Understand returns and growth
Personal Finance: Compare interest rates on loans and savings
Career: Negotiate salary increases, understand commissions

Next Steps

Move to Chapter 05: Statistics Basics to learn how to interpret data, understand averages, and make sense of probability. These skills are essential for data-driven decisions in business and life.