Ratios and Rates

What is a Ratio?

A ratio is a comparison of two quantities by division.

Forms of writing ratios:

• 3 to 4
• 3:4
• 3/4

All mean the same thing!

Example 1: Class has 12 boys and 15 girls

Ratio of boys to girls: 12:15 or 12/15

Simplified: 4:5 (divide both by 3)

Example 2: Recipe uses 2 cups flour and 1 cup sugar

Ratio of flour to sugar: 2:1

Parts of a Ratio

Ratio a:b

• First term: a
• Second term: b
• Order matters! 2:3 ≠ 3:2

Example: Ratio of red marbles to blue marbles is 5:3

This means:

• For every 5 red, there are 3 blue
• NOT 3 red and 5 blue!

Simplifying Ratios

Process: Divide both terms by their GCF (Greatest Common Factor)

Example 1: 8:12

GCF of 8 and 12 is 4 8÷4 : 12÷4 = 2:3

Simplified form: 2:3

Example 2: 15:25

GCF is 5 15÷5 : 25÷5 = 3:5

Example 3: 18:24:30 (three-term ratio)

GCF of all three is 6 18÷6 : 24÷6 : 30÷6 = 3:4:5

Example 4: 100:150

GCF is 50 100÷50 : 150÷50 = 2:3

Equivalent Ratios

Ratios that represent the same relationship

Like equivalent fractions: 1/2 = 2/4 = 3/6

Example: 2:3 is equivalent to:

• 4:6 (multiply both by 2)
• 6:9 (multiply both by 3)
• 8:12 (multiply both by 4)
• 10:15 (multiply both by 5)

To find equivalent ratio: Multiply or divide both terms by the same number

Example: Find three ratios equivalent to 5:7

• 10:14 (×2)
• 15:21 (×3)
• 20:28 (×4)

Part-to-Part vs. Part-to-Whole Ratios

Part-to-Part: Compares two parts

Example: 3 apples and 5 oranges Part-to-part: 3:5 (apples to oranges)

Part-to-Whole: Compares one part to the total

Example: Same 3 apples and 5 oranges Total fruit: 8 Part-to-whole: 3:8 (apples to total fruit) Or: 5:8 (oranges to total fruit)

Example Problem: Class has boys and girls in ratio 2:3

Part-to-part: 2:3 (boys to girls)

If total is 30 students:

• Boys: 2/(2+3) × 30 = 2/5 × 30 = 12
• Girls: 3/(2+3) × 30 = 3/5 × 30 = 18

Part-to-whole:

• Boys to total: 12:30 = 2:5
• Girls to total: 18:30 = 3:5

Finding Unknown Terms

Using equivalent ratios

Example 1: 3:5 = x:20

Set up proportion: 3/5 = x/20

Cross multiply: 5x = 60 x = 12

Example 2: 7:x = 21:15

7/x = 21/15

Cross multiply: 7 × 15 = 21x 105 = 21x x = 5

Example 3: If 4:9 = 12:y

4/9 = 12/y

4y = 108 y = 27

What is a Rate?

A rate is a ratio that compares two quantities with different units.

Common rates:

• Speed: miles per hour (mi/h)
• Price: dollars per pound ($/lb)
• Wage: dollars per hour ($/h)
• Density: grams per cubic centimeter (g/cm³)

Example 1: Drive 150 miles in 3 hours

Rate: 150 miles / 3 hours = 50 miles/hour

Example 2: Earn 80 dollars in 5 hours

Rate: 80 dollars / 5 hours = 16 dollars/hour

Unit Rate

A unit rate has a denominator of 1.

"Per one unit"

Example 1: 120 miles in 3 hours

Unit rate: 120 ÷ 3 = 40 miles per 1 hour = 40 mi/h

Example 2: 5 pounds for 2.50 dollars

Unit rate: 2.50 ÷ 5 = 0.50 dollars per pound

Or: 5 ÷ 2.50 = 2 pounds per dollar

Example 3: 240 words typed in 4 minutes

Unit rate: 240 ÷ 4 = 60 words per minute

Example 4: 12 apples cost 6 dollars

Unit rate: 6 ÷ 12 = 0.50 dollars per apple

Or: 12 ÷ 6 = 2 apples per dollar

Unit Price (Best Buy Problems)

Unit price = Total price / Number of units

Use to compare which is the better deal!

Example: Which is the better buy?

Option A: 12 oz for 3.60 dollars Option B: 16 oz for 4.00 dollars

Option A unit price: 3.60 ÷ 12 = 0.30 dollars/oz Option B unit price: 4.00 ÷ 16 = 0.25 dollars/oz

Better buy: Option B (cheaper per ounce)

Example 2: Compare cereal prices

Brand A: 20 oz for 5.00 dollars → 0.25 dollars/oz Brand B: 24 oz for 5.50 dollars → 0.229 dollars/oz (approximately 0.23)

Better buy: Brand B

Example 3: Juice comparison

Small: 32 oz for 2.56 dollars → 0.08 dollars/oz Large: 64 oz for 4.48 dollars → 0.07 dollars/oz

Better buy: Large size

Speed, Distance, and Time

Formula: Distance = Rate × Time Or: d = rt

Rearranged:

• Rate = Distance / Time → r = d/t
• Time = Distance / Rate → t = d/r

Example 1: Car travels 50 mph for 3 hours

Distance = 50 × 3 = 150 miles

Example 2: Train travels 240 miles in 4 hours

Speed = 240 ÷ 4 = 60 mph

Example 3: How long to drive 180 miles at 60 mph?

Time = 180 ÷ 60 = 3 hours

Example 4: Plane flies 1,500 miles in 2.5 hours

Speed = 1,500 ÷ 2.5 = 600 mph

Converting Rates

Change units while keeping the ratio equivalent

Example 1: Convert 60 mi/h to mi/min

60 miles per 60 minutes = 1 mile per minute

Example 2: Convert 5 m/s to m/min

5 meters per second × 60 seconds = 300 meters per minute

Example 3: Convert 12 dollars/hour to cents/minute

12 dollars/hour = 1,200 cents/hour 1,200 cents per 60 minutes = 20 cents/minute

Example 4: Convert 88 ft/s to mi/h

88 ft/s × 60 s/min × 60 min/h = 316,800 ft/h 316,800 ft/h ÷ 5,280 ft/mi = 60 mi/h

Ratios with Different Units

Must convert to same units first!

Example 1: Ratio of 2 feet to 8 inches

Convert 2 feet to inches: 2 × 12 = 24 inches

Ratio: 24:8 = 3:1

Example 2: Ratio of 3 hours to 45 minutes

Convert 3 hours to minutes: 3 × 60 = 180 minutes

Ratio: 180:45 = 4:1

Example 3: Ratio of 1 yard to 2 feet

Convert 1 yard to feet: 3 feet

Ratio: 3:2

Scale and Scale Drawings

Scale ratio: Relates drawing size to actual size

Example 1: Map scale is 1 inch : 50 miles

Drawing: 3 inches Actual distance: 3 × 50 = 150 miles

Example 2: Blueprint scale is 1:100

Drawing: 5 cm Actual size: 5 × 100 = 500 cm = 5 meters

Example 3: Model car scale 1:24

Model length: 8 inches Actual car: 8 × 24 = 192 inches = 16 feet

Example 4: Finding scale

Model: 6 cm Actual: 180 m = 18,000 cm

Scale: 6:18,000 = 1:3,000

Proportional Reasoning with Ratios

If ratio is constant, quantities are proportional

Example: Recipe for 4 servings uses 2 cups flour

For 10 servings, how much flour?

Ratio: 2 cups / 4 servings = x cups / 10 servings

Cross multiply: 4x = 20 x = 5 cups

Example 2: 3 pizzas feed 8 people

How many pizzas for 24 people?

3/8 = x/24

8x = 72 x = 9 pizzas

Example 3: Car uses 2 gallons to drive 50 miles

How far on 7 gallons?

2/50 = 7/x

2x = 350 x = 175 miles

Ratios in Geometry

Example 1: Similar Triangles

Small triangle sides: 3, 4, 5 Large triangle sides: 6, 8, 10

Ratio: 3:6 = 1:2 (scale factor)

All corresponding sides have same ratio!

Example 2: Circle

Circumference : Diameter = π : 1 (approximately 3.14:1)

Example 3: Rectangle dimensions

Length:Width = 3:2

If width is 10: 3/2 = L/10 2L = 30 L = 15

Mixture Problems

Example 1: Concrete mix ratio of cement:sand:gravel is 1:2:3

For 12 cubic feet total:

Total parts: 1+2+3 = 6

Cement: 1/6 × 12 = 2 cubic feet Sand: 2/6 × 12 = 4 cubic feet Gravel: 3/6 × 12 = 6 cubic feet

Example 2: Paint mix ratio blue:yellow = 2:3

Need 15 gallons total:

Blue: 2/5 × 15 = 6 gallons Yellow: 3/5 × 15 = 9 gallons

Example 3: Fruit punch ratio orange:pineapple = 5:2

Have 10 cups orange, how much pineapple?

5/2 = 10/x 5x = 20 x = 4 cups pineapple

Gear Ratios

Compares rotations of connected gears

Example: Gear ratio 3:1

Driver gear makes 1 rotation Driven gear makes 3 rotations

Example 2: Bike gears 42:14 = 3:1

Pedal 1 revolution → wheel turns 3 revolutions

Common Mistakes to Avoid

1. Reversing the order Ratio of boys to girls: 4:5 (not 5:4!)

2. Not simplifying 10:15 should be simplified to 2:3

3. Mixing up part-to-part and part-to-whole Ratio 2:3 means 2 out of 5 total (not 2 out of 3!)

4. Different units in ratio Must convert to same units first!

5. Forgetting unit labels in rates 60 (what?) vs 60 miles/hour (clear!)

6. Adding ratios incorrectly If ratio is 2:3, DON'T assume 2+3=5 of each!

Complex Ratio Problems

Example: Three siblings share money in ratio 2:3:5

Total: 200 dollars

Parts: 2+3+5 = 10

First sibling: 2/10 × 200 = 40 dollars Second sibling: 3/10 × 200 = 60 dollars Third sibling: 5/10 × 200 = 100 dollars

Check: 40+60+100 = 200 ✓

Example 2: Angles in triangle ratio 2:3:4

Total angles in triangle: 180 degrees Parts: 2+3+4 = 9

First angle: 2/9 × 180 = 40° Second angle: 3/9 × 180 = 60° Third angle: 4/9 × 180 = 80°

Check: 40+60+80 = 180 ✓

Real-World Applications

Cooking: Recipe ratios (flour:sugar:butter)

Construction: Concrete mixes, scale drawings

Finance: Exchange rates (dollars to euros)

Sports: Win-loss ratios, batting averages

Medicine: Drug concentrations

Business: Profit-to-cost ratios

Chemistry: Molecular ratios

Geography: Map scales

Quick Reference

Ratio: Comparison of two quantities (a:b)

Rate: Ratio with different units (mi/h)

Unit Rate: Rate with denominator of 1

Unit Price: Cost per one item

Simplify: Divide both terms by GCF

Equivalent Ratios: Multiply or divide both terms by same number

Part-to-whole: One part / Total

Finding unknown: Set up proportion, cross multiply

Different units: Convert to same units first

Scale: Drawing : Actual

Practice Tips

• Always simplify ratios to lowest terms
• Label units clearly for rates
• Check if units match before comparing
• Use proportions to find unknowns
• Remember order matters in ratios!
• For part-to-whole, find total parts first
• Practice unit conversions
• Compare unit prices when shopping
• Draw pictures for complex problems
• Verify answers make sense in context
• Master cross multiplication for proportions
• Apply to real-life situations daily
• Remember: rates have different units, ratios can have same units
• Practice with recipes, maps, and scale models
• Understand the difference between ratio and fraction

Ratios and rates are fundamental to understanding relationships between quantities. Master these skills and you'll excel in algebra, science, cooking, and countless real-world situations!