Learning Objectives
- Write ratios in three equivalent forms: fraction, colon, and word form
- Simplify ratios by finding the greatest common factor
- Generate equivalent ratios and recognize proportional relationships
- Solve proportions using cross-multiplication
- Calculate unit rates and use them for real-world comparisons
- Apply proportional reasoning to scale drawings and map distances
What Is a Ratio?
A ratio is a comparison of two quantities using division. When you say "for every 2 cups of flour, you need 1 cup of sugar," you are describing a ratio.
Ratios can be written in three equivalent ways:
All four say the same thing: for every 3 of the first quantity, there are 4 of the second.
Order Matters
Unlike addition or multiplication, order is critical in a ratio. "Boys to girls = 2:3" is completely different from "girls to boys = 3:2." Always read the problem carefully to know which quantity comes first.
Simplifying Ratios
Just like fractions, ratios can be simplified. Divide both terms by their greatest common factor (GCF).
Worked Example 1 — Simplifying a Ratio
A recipe calls for 12 tablespoons of oil and 8 tablespoons of vinegar.
Write the ratio in simplest form.
Step 1 — Write the ratio: 12:8
Step 2 — Find GCF of 12 and 8: factors of 12 = {1,2,3,4,6,12}, factors of 8 = {1,2,4,8} → GCF = 4
Step 3 — Divide both by 4: 12 ÷ 4 = 3 and 8 ÷ 4 = 2
Step 4 — Simplified ratio: 3:2
Answer: The ratio of oil to vinegar is 3:2 (for every 3 tablespoons of oil, use 2 of vinegar)
Equivalent Ratios
Equivalent ratios represent the same relationship scaled up or down — like equivalent fractions. You create them by multiplying or dividing both terms by the same number.
| Base Ratio | ×2 | ×3 | ×5 | ×10 |
|---|
| 1:3 | 2:6 | 3:9 | 5:15 | 10:30 |
| 2:5 | 4:10 | 6:15 | 10:25 | 20:50 |
| 3:4 | 6:8 | 9:12 | 15:20 | 30:40 |
Worked Example 2 — Finding an Equivalent Ratio
A car travels 3 miles for every 5 minutes. How far does it travel in 35 minutes?
Step 1 — Set up the base ratio: 3 miles : 5 minutes
Step 2 — Find the multiplier: 35 ÷ 5 = 7
Step 3 — Multiply both terms by 7: 3×7 = 21 miles and 5×7 = 35 minutes
Step 4 — Equivalent ratio: 21 miles : 35 minutes
Answer: The car travels 21 miles in 35 minutes.
What Is a Proportion?
A proportion is an equation stating that two ratios are equal:
a/b = c/d
Proportions appear everywhere — scaling recipes, reading maps, calculating speeds, mixing paint colors, and determining if two photos have the same aspect ratio.
Testing Whether Two Ratios Form a Proportion
Two ratios are proportional if their cross products are equal: a × d = b × c.
Worked Example 3 — Are These Ratios Proportional?
Is 4/6 proportional to 10/15?
Cross-multiply: 4 × 15 = 60 and 6 × 10 = 60
Since 60 = 60, the ratios ARE proportional.
Verify by simplifying: 4/6 = 2/3 and 10/15 = 2/3 — same simplified form confirms it.
Now check 3/5 vs 7/10:
Cross-multiply: 3 × 10 = 30 and 5 × 7 = 35
Since 30 ≠ 35, these ratios are NOT proportional.
Solving Proportions with Cross-Multiplication
When one term in a proportion is unknown, cross-multiplication lets you solve for it.
a / b = c / d
Cross multiply: a × d = b × c
Solve for the unknown variable
Worked Example 4 — Solving for an Unknown
Problem: If 5 apples cost $2.50, how much do 8 apples cost?
Step 1 — Set up the proportion: 5/2.50 = 8/x
Step 2 — Cross-multiply: 5 × x = 2.50 × 8
Step 3 — Simplify right side: 5x = 20
Step 4 — Divide both sides by 5: x = 20 ÷ 5 = 4
Answer: 8 apples cost $4.00
Check: 5/2.50 = 2 and 8/4 = 2 — both ratios equal 2, so the proportion holds.
Worked Example 5 — Recipe Scaling
Problem: A muffin recipe needs 2 cups of flour for 12 muffins. How much flour is needed for 30 muffins?
Step 1 — Set up proportion (flour : muffins): 2/12 = x/30
Step 2 — Cross-multiply: 2 × 30 = 12 × x
Step 3 — Simplify: 60 = 12x
Step 4 — Divide: x = 60 ÷ 12 = 5
Answer: 5 cups of flour for 30 muffins.
Unit Rates: Ratios in Everyday Life
A unit rate expresses a ratio with a denominator of 1. Unit rates make comparisons instant — they are what you see on grocery store price tags (price per ounce), speedometers (miles per hour), and paychecks (dollars per hour).
Unit Rate = Total Amount / Number of Units
Worked Example 6 — Best Buy Comparison
Store A sells 32 oz of orange juice for $3.84
Store B sells 20 oz of orange juice for $2.60
Which is the better deal? Find the unit rate (price per ounce):
Store A: $3.84 ÷ 32 oz = $0.12 per ounce
Store B: $2.60 ÷ 20 oz = $0.13 per ounce
Answer: Store A is cheaper at $0.12/oz vs Store B at $0.13/oz. Buy Store A.
Worked Example 7 — Speed as a Unit Rate
Problem: A cyclist rides 45 miles in 3 hours. What is her speed in miles per hour?
Speed = 45 miles ÷ 3 hours = 15 miles per hour
This is a unit rate: 15 miles for every 1 hour.
Now use the rate: How far does she travel in 7 hours at the same speed?
Distance = 15 miles/hour × 7 hours = 105 miles
Answer: She travels 105 miles in 7 hours.
Scale Drawings and Maps
A scale is a ratio between a drawing's dimensions and the real-world dimensions it represents. Maps, blueprints, and model kits all use scales.
If a map has a scale of 1 cm : 25 km, then every 1 centimeter on the map represents 25 kilometers in reality.
Worked Example 8 — Reading a Map Scale
Problem: On a map with scale 1 cm : 50 km, two cities are 4.6 cm apart. What is the real distance?
Method 1 — Multiply: 4.6 cm × 50 km/cm = 230 km
Method 2 — Proportion:
1/50 = 4.6/x
Cross-multiply: 1 × x = 50 × 4.6
x = 230 km
Answer: The cities are 230 km apart in real life.
Reverse question: A city is 175 km from another. How far apart on the map?
1/50 = x/175 → x = 175/50 = 3.5 cm on the map.
Three-Part Ratios
Ratios can compare more than two quantities. A three-part ratio like 2:3:5 means the three quantities share 2 + 3 + 5 = 10 total parts.
Worked Example 9 — Dividing in a Ratio
Problem: Three friends share $180 in the ratio 2:3:4. How much does each person receive?
Step 1 — Find total parts: 2 + 3 + 4 = 9 parts
Step 2 — Find value of one part: $180 ÷ 9 = $20 per part
Step 3 — Multiply each share:
Friend 1: 2 parts × $20 = $40
Friend 2: 3 parts × $20 = $60
Friend 3: 4 parts × $20 = $80
Step 4 — Check: $40 + $60 + $80 = $180 ✓
Answer: The friends receive $40, $60, and $80.
Proportional vs Non-Proportional Relationships
A relationship is proportional when the ratio between two quantities stays constant. When graphed, proportional relationships are straight lines that pass through the origin (0,0).
| Proportional | Non-Proportional |
|---|
| $12 per hour worked | $10 base + $2 per hour |
| 3 apples cost $0.90, 6 cost $1.80 | Grocery discount: first 5 items full price, rest 20% off |
| Distance = speed × time (constant speed) | Variable speed on a road trip |
| y = 4x (passes through origin) | y = 4x + 2 (does NOT pass through origin) |
Common Mistakes to Avoid
- Switching the ratio order: "red to blue is 3:5" means 3 reds for every 5 blues. Flipping it gives the wrong answer.
- Not simplifying before solving: simplifying both ratios first often makes cross-multiplication much easier.
- Setting up the proportion incorrectly: keep the same unit in the same position in both fractions (miles/hour on both sides, not miles/hour on one and hours/mile on the other).
- Forgetting to label units: "x = 6" is incomplete. "$6" or "6 hours" is the complete answer.
- Treating a three-part ratio as two separate ratios: in 2:3:4, you cannot solve for one part without knowing the total.
Practice Problems
Problem 1: Simplify the ratio 18:24.
GCF of 18 and 24: factors of 18={1,2,3,6,9,18}, factors of 24={1,2,3,4,6,8,12,24} → GCF=6
18 ÷ 6 = 3 and 24 ÷ 6 = 4
Answer: 3:4
Problem 2: Solve the proportion: 7/x = 14/10
Cross-multiply: 7 × 10 = 14 × x
70 = 14x
x = 70 ÷ 14 = 5
Check: 7/5 = 1.4 and 14/10 = 1.4 ✓
Problem 3: A car travels 240 miles on 8 gallons of gas. How many miles per gallon is that?
Unit rate = 240 miles ÷ 8 gallons = 30 miles per gallon
Bonus: How many gallons for a 390-mile trip?
30 miles/gallon × x = 390 → x = 390 ÷ 30 = 13 gallons
Problem 4: A blueprint has scale 1 in : 8 ft. Two rooms are 3.5 inches apart on the blueprint. What is the actual distance?
Proportion: 1/8 = 3.5/x
Cross-multiply: 1 × x = 8 × 3.5
x = 28 feet
Alternatively: 3.5 inches × 8 ft/inch = 28 feet
Problem 5: Divide $270 among three students in the ratio 1:2:6.
Total parts: 1 + 2 + 6 = 9 parts
Value per part: $270 ÷ 9 = $30
Student 1: 1 × $30 = $30
Student 2: 2 × $30 = $60
Student 3: 6 × $30 = $180
Check: $30 + $60 + $180 = $270 ✓
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