Ratios and Proportions
Compare quantities and solve proportion problems for the ParaPro Assessment
What You'll Learn
Understanding Ratios
A ratio compares two quantities. It can be written in three ways:
Using a Colon
3:5
As a Fraction
3/5
Using "to"
3 to 5
Types of Ratios
Part-to-Part
Compares parts of a group: 12 boys to 15 girls = 4:5
Part-to-Whole
Compares part to total: 12 boys to 27 students = 4:9
Equivalent Ratios
2:3 = 4:6 = 6:9 = 8:12
Solving Proportions
A proportion states that two ratios are equal. Use cross multiplication to solve.
a/b = c/d → a × d = b × c
Cross multiply: multiply across the equals sign diagonally
Example 1: Find the Missing Value
Solve: 3/4 = x/12
Step 1: Cross multiply: 3 × 12 = 4 × x
Step 2: Simplify: 36 = 4x
Step 3: Divide: x = 36 ÷ 4
Answer: x = 9
Example 2: Word Problem
If 5 pencils cost $2, how much do 12 pencils cost?
Set up: 5 pencils/$2 = 12 pencils/x
Cross multiply: 5x = 2 × 12 = 24
Solve: x = 24 ÷ 5 = 4.80
Answer: $4.80
Rates and Unit Rates
A rate compares quantities with different units. A unit rate has a denominator of 1.
📏 Common Rates
- • Miles per hour (mph)
- • Price per pound
- • Words per minute
- • Students per teacher
🎯 Finding Unit Rate
Divide to get denominator of 1:
240 miles ÷ 4 hours = 60 mph
Example: Best Buy Comparison
Which is the better deal?
Option A: 12 oz for $3.60
$3.60 ÷ 12 = $0.30/oz
Option B: 16 oz for $4.64
$4.64 ÷ 16 = $0.29/oz
Option B is better ($0.29 < $0.30 per oz)
Real-World Applications
🗺️ Map Scale
1 inch = 50 miles. Cities are 3.5 inches apart. Actual distance?
1/50 = 3.5/x → x = 50 × 3.5 = 175 miles
🍳 Recipe Scaling
Recipe for 4 needs 2 cups flour. How much for 10 people?
4/2 = 10/x → 4x = 20 → x = 5 cups
🌲 Similar Triangles
6-ft person casts 4-ft shadow. Tree casts 20-ft shadow. Height?
6/4 = x/20 → 4x = 120 → x = 30 feet
Common Mistakes to Avoid
Boys to girls = 15:12 (when there are 12 boys, 15 girls)
✓ Boys to girls = 12:15 (order matters!)
3/4 = x/12 → 3 × 4 = x × 12
✓ 3/4 = x/12 → 3 × 12 = 4 × x (cross multiply correctly)
Ratio = 12:18 (not simplified)
✓ Ratio = 2:3 (always simplify by GCF)
Practice Problems
Test your understanding
1. Write 8 red : 12 blue marbles in simplest form ▼
8:12 = 2:3 (divide both by 4)
2. Solve: 5/8 = x/24 ▼
5 × 24 = 8 × x → 120 = 8x → x = 15
3. If 3 shirts cost $45, how much do 7 shirts cost? ▼
3/45 = 7/x → 3x = 315 → x = $105
4. A car travels 315 miles on 9 gallons. MPG? ▼
315 ÷ 9 = 35 miles per gallon
Key Takeaways
- ✓ Ratios compare quantities: colon (3:4), fraction (3/4), or words (3 to 4)
- ✓ Proportions are equal ratios; solve with cross multiplication
- ✓ Rates compare different units (miles per hour)
- ✓ Unit rates have denominator of 1 (price per pound)
- ✓ Order matters in ratios - always simplify to lowest terms
Related Lessons
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