Ratios and Proportions

Learn to compare quantities, solve scaling problems, and convert between units. These are critical skills for recipes, construction, business comparisons, and unit pricing.

Understanding Ratios

A ratio compares two quantities and shows their relative sizes.

Ways to Write Ratios:

• With colon: 3:2 (read "3 to 2")
• As fraction: 3/2
• With words: "3 to 2"

Simple Ratios

Example: Recipe calls for 2 cups flour to 1 cup water

• Ratio of flour to water = 2:1
• Means: For every 2 cups flour, use 1 cup water

Example: Business has 5 employees per manager

• Ratio of employees to managers = 5:1

Three-Part Ratios

Example: Paint mixture is 3 parts red, 2 parts blue, 1 part white

• Ratio = 3:2:1
• For 6 total parts, each "part" matters

Application: Make 12 cups total

• Total parts: 3 + 2 + 1 = 6 parts
• Each part: 12 ÷ 6 = 2 cups
• Red: 3 × 2 = 6 cups
• Blue: 2 × 2 = 4 cups
• White: 1 × 2 = 2 cups

Simplifying Ratios

Like fractions, ratios can be simplified by dividing by the greatest common divisor.

Example: 15:10

• Both divisible by 5
• Simplified: 3:2

Example: 24:18:12

• All divisible by 6
• Simplified: 4:3:2

Equivalent Ratios

Just like equivalent fractions, ratios can represent the same relationship.

1:2 = 2:4 = 3:6 = 50:100

Life Example: Unit pricing

• 2 items for $6 = ratio 2:6 = 1:3 (each item $3)
• 5 items for $15 = ratio 5:15 = 1:3 (each item $3)
• Same value!

Understanding Proportions

A proportion states that two ratios are equal.

Format: a/b = c/d or a:b = c:d

Read as: "a is to b as c is to d"

Solving Proportions (Cross-Multiplication)

Method: Cross-multiply and solve

Example: 3/4 = x/20

1. Cross-multiply: 3 × 20 = 4 × x
2. Simplify: 60 = 4x
3. Solve: x = 15

Check: 3/4 = 0.75 and 15/20 = 0.75 ✓

Real-World Proportion Problems

Scaling Recipes

Problem: Recipe serves 4, calls for 3 cups flour. How much for 10 people?

Solution:

• Set up proportion: 3 cups / 4 people = x cups / 10 people
• Cross-multiply: 3 × 10 = 4 × x
• Solve: 30 = 4x, so x = 7.5 cups

Map Scales

Problem: Map scale is 1 inch = 25 miles. Two cities are 3.5 inches apart. Actual distance?

Solution:

• Proportion: 1 inch / 25 miles = 3.5 inches / x miles
• Cross-multiply: 1 × x = 25 × 3.5
• x = 87.5 miles

Business Staffing

Problem: Store needs 2 employees per 15 customers. If 45 customers expected, how many employees?

Solution:

• Proportion: 2 employees / 15 customers = x employees / 45 customers
• Cross-multiply: 2 × 45 = 15 × x
• Solve: 90 = 15x, so x = 6 employees

Unit Rates

A unit rate compares a quantity to 1 unit of another quantity.

Common Examples:

• Miles per gallon (mpg)
• Price per pound
• Words per minute
• Cost per unit
• Rate per hour

Calculating Unit Rates

Formula: Divide the first quantity by the second

Example: $3.60 for 4 pounds of apples

• Unit rate: $3.60 ÷ 4 = $0.90/pound

Example: 180 miles on 6 gallons

• Unit rate: 180 ÷ 6 = 30 mpg

Comparing Unit Rates (Best Value)

Problem: Which is the better deal?

• Store A: 3 pounds for $5.25
• Store B: 5 pounds for $8.50

Solution:

• Store A: $5.25 ÷ 3 = $1.75/pound
• Store B: $8.50 ÷ 5 = $1.70/pound
• Store B is better (lower price per pound)

Business Application: Vendor comparison

• Vendor A: 100 units for $1,250 = $12.50/unit
• Vendor B: 75 units for $975 = $13/unit
• Vendor A offers better unit price

Direct Proportion

When one quantity increases, the other increases at the same rate.

Characteristics:

• If x doubles, y doubles
• If x halves, y halves
• Graph is a straight line through origin

Formula: y = kx (k is the constant of proportionality)

Examples:

• Distance and time (at constant speed): More time → more distance
• Cost and quantity: More items → higher cost
• Wages and hours: More hours → more pay

Application: Employee costs

• 1 employee costs $50,000/year
• 2 employees cost $100,000/year
• 5 employees cost $250,000/year
• Constant: k = $50,000 per employee

Inverse Proportion

When one quantity increases, the other decreases at the same rate.

Characteristics:

• If x doubles, y halves
• If x triples, y is one-third
• Product xy is constant

Formula: y = k/x or xy = k

Examples:

• Speed and time (for fixed distance): Faster speed → less time
• Workers and time (for fixed job): More workers → less time
• Price and demand: Higher price → lower demand (often)

Application: Project completion

• 1 worker takes 12 days
• 2 workers take 6 days
• 3 workers take 4 days
• Constant: k = 12 worker-days

Problem: Job takes 8 hours for 3 people. How long with 6 people?

• Constant: k = 8 × 3 = 24 person-hours
• With 6 people: t = 24 ÷ 6 = 4 hours

Unit Conversions

Use proportions to convert between measurement units.

Length Conversions

Example: Convert 5 feet to inches

• Proportion: 1 foot / 12 inches = 5 feet / x inches
• Cross-multiply: x = 5 × 12 = 60 inches

Quick Method: Multiply by conversion factor

• 5 feet × (12 inches/1 foot) = 60 inches

Weight Conversions

Example: Convert 3.5 pounds to ounces

• 3.5 pounds × (16 oz/1 pound) = 56 ounces

Volume Conversions

Example: Convert 2.5 gallons to cups

• 2.5 gallons × (4 quarts/gallon) × (2 pints/quart) × (2 cups/pint)
• 2.5 × 4 × 2 × 2 = 40 cups

Time Conversions

Example: Convert 450 minutes to hours

• 450 minutes × (1 hour/60 minutes) = 7.5 hours

Currency and Exchange Rates

Example: Exchange rate is 1 USD = 0.92 EUR. Convert $500 to euros.

• $500 × (0.92 EUR/1 USD) = €460

Example: Product costs £75. Exchange rate is 1 GBP = 1.27 USD. Cost in USD?

• £75 × (1.27 USD/1 GBP) = $95.25

Scale Drawings and Models

Scale shows the relationship between drawing size and actual size.

Scale Format: "1 cm represents 5 m" or "1:500"

Example: Blueprint scale is 1:50 (1 cm on paper = 50 cm in reality)

• Drawing shows wall as 8 cm
• Actual wall: 8 × 50 = 400 cm = 4 meters

Example: Model car at 1:24 scale

• Real car is 4.8 meters long
• Model length: 4.8 m ÷ 24 = 0.2 m = 20 cm

Business Applications

Price-to-Earnings Ratio (P/E Ratio)

Compares stock price to earnings per share.

Formula: P/E = Stock Price / Earnings per Share

Example:

• Stock price: $80
• Earnings per share: $5
• P/E ratio: 80 ÷ 5 = 16:1
• Interpretation: Investors pay $16 for every $1 of earnings

Debt-to-Equity Ratio

Compares company's debt to shareholder equity.

Formula: Debt/Equity

Example:

• Total debt: $500,000
• Total equity: $1,000,000
• Ratio: 500,000/1,000,000 = 0.5 or 1:2
• Interpretation: $1 of debt for every $2 of equity

Employee-to-Revenue Ratio

Measures revenue per employee.

Example:

• Annual revenue: $5,000,000
• Employees: 25
• Revenue per employee: $5,000,000 ÷ 25 = $200,000/employee

Practice Problems

Basic Ratios

1. Simplify the ratio 18:24
2. If ratio of cats to dogs is 2:3 and there are 15 dogs, how many cats?

Proportions

3. Solve: 5/8 = x/40
4. Recipe for 6 servings needs 2 cups sugar. How much for 9 servings?

Unit Rates

5. Which is better value: 12 oz for $3.60 or 16 oz for $4.48?
6. Car travels 315 miles on 9 gallons. What's the mpg?

Conversions

7. Convert 7.5 feet to inches
8. Convert 120 minutes to hours
9. Convert 3 gallons to cups

Word Problems

10. Map scale is 1 inch = 40 miles. Cities are 2.75 inches apart on map. Actual distance?

11. 4 workers complete job in 6 days. How long for 8 workers?

12. At 60 mph, trip takes 4 hours. How long at 80 mph?

Solutions

1. 3:4 (divide both by 6)
2. 10 cats (2/3 = x/15, cross-multiply: 3x = 30)
3. x = 25 (cross-multiply: 8x = 200)
4. 3 cups (2/6 = x/9, cross-multiply: 6x = 18)
5. 12 oz option ($0.30/oz vs $0.28/oz)
6. 35 mpg (315 ÷ 9)
7. 90 inches (7.5 × 12)
8. 2 hours (120 ÷ 60)
9. 48 cups (3 × 4 × 2 × 2)
10. 110 miles (2.75 × 40)
11. 3 days (inverse proportion: 4 × 6 = 8 × t, so t = 3)
12. 3 hours (inverse proportion: 60 × 4 = 80 × t, so t = 3)

Key Takeaways

✓ Ratios compare quantities: they show relative sizes
✓ Proportions state equality: two ratios are equal
✓ Cross-multiply to solve: efficient method for proportions
✓ Unit rates enable comparison: find cost per unit, speed, etc.
✓ Direct vs inverse proportion: know which relationship applies
✓ Set up conversions carefully: units should cancel properly

Real-World Applications

• Shopping: Compare unit prices for best value
• Cooking: Scale recipes up or down
• Construction: Use scale drawings and material ratios
• Travel: Convert currencies and units
• Business: Compare financial ratios and metrics
• Nutrition: Calculate serving sizes and nutrients

Next Steps

Move to Chapter 04: Percentage Applications to learn how to calculate discounts, interest, markups, and other percentage-based problems essential for shopping, business, and finance.