Ratios & Proportions
From recipe scaling to map reading — ratios and proportions help you compare and scale quantities.
What Is a Ratio?
A ratio compares two quantities. "The ratio of cats to dogs is 3 to 5" can be written as:
3:5(colon notation)3 to 5(word notation)3/5(fraction notation)
Order matters! 3:5 means 3 cats for every 5 dogs, not the other way around.
Simplifying Ratios
Divide both parts by the greatest common factor (GCF).
Example 1: Simplify 15:25
GCF of 15 and 25 is 5. 15÷5 : 25÷5 = 3:5
Example 2: Simplify 2.5:7.5
Multiply both by 2 to remove decimals: 5:15. Simplify: 1:3
What Is a Proportion?
A proportion states that two ratios are equal: a/b = c/d
Example: 2/3 = 8/12 is a proportion because both simplify to the same ratio.
Cross-Multiplication
If a/b = c/d, then a × d = b × c. This is how you solve for unknowns.
Example 3: Solve 4/6 = x/15
Cross multiply: 4 × 15 = 6 × x → 60 = 6x → x = 10
Example 4: If 3 apples cost $2.25, how much do 8 apples cost?
Set up: 3/$2.25 = 8/x
Cross multiply: 3x = $2.25 × 8 = $18.00
x = $18.00 ÷ 3 = $6.00
Example 5: Scale Model
A model car is built at 1:24 scale. If the real car is 180 inches long, how long is the model?
1/24 = x/180. Cross multiply: 24x = 180. x = 180/24 = 7.5 inches
Part-to-Part vs Part-to-Whole
Part-to-part: Comparing one group to another. "Boys to girls is 3:5."
Part-to-whole: Comparing one group to the total. With 3 boys and 5 girls, boys are 3/8 of the class.
Be careful which type a problem is asking for!
Rates as Ratios
A rate is a ratio comparing different units:
- Speed: 60 miles / 1 hour = 60 mph
- Price: $4.50 / 3 pounds = $1.50 per pound
- Density: 240 people / 5 square miles = 48 people per sq mile
A unit rate has 1 in the denominator. To find it, divide both numbers by the denominator.
Common Applications
- Maps: 1 inch = 50 miles, so 3.5 inches = 175 miles
- Cooking: Scale recipes up or down proportionally
- Mixing: Paint colors, concrete ratios (1:2:3 cement:sand:gravel)
- Finance: Price per unit, exchange rates
- Screen ratios: 16:9 widescreen, 4:3 standard
🎯 Try It Yourself
Test your understanding with these practice problems.
1. Simplify the ratio 24:36
💡 Hint: Divide both by their GCF (12)
2. If the ratio of boys to girls is 3:5 and there are 15 girls, how many boys?
💡 Hint: Set up proportion: 3/5 = x/15
3. A recipe uses 2 cups flour for 3 dozen cookies. How much flour for 9 dozen?
💡 Hint: 2/3 = x/9, cross multiply
4. A map scale is 1:50000. If two cities are 3 cm apart on the map, how far apart are they in km?
💡 Hint: 3 × 50000 = 150000 cm = 1500 m = 1.5 km
5. Express 0.8:1.2 as a whole number ratio
💡 Hint: Multiply both by 10 to get 8:12, then simplify
⚠️ Common Mistakes
- ✗Writing ratios in the wrong order — the order matters! 3:5 boys to girls is different from 5:3.
- ✗Forgetting to simplify ratios to their simplest form.
- ✗In cross-multiplication, multiplying the wrong pairs diagonally.
- ✗Confusing 'ratio' with 'fraction'. A ratio of 3:5 means 3 to 5, not 3/5 of the whole (that would be 3/8).
- ✗Not keeping units consistent when setting up proportions.
🌍 Real Life Example
Scaling a Recipe
A pancake recipe serves 4 people: 2 cups flour, 2 eggs, 1.5 cups milk. You need to serve 10 people. The scaling ratio is 10/4 = 2.5. Multiply everything: 2 × 2.5 = 5 cups flour, 2 × 2.5 = 5 eggs, 1.5 × 2.5 = 3.75 cups milk. Proportions let you scale any recipe up or down perfectly.
💡 Key Takeaway
A ratio compares two quantities. A proportion says two ratios are equal. Cross-multiplication is your power tool for solving proportions: if a/b = c/d, then a×d = b×c. Ratios are everywhere — recipes, maps, scale models, mixing paint, even your screen's aspect ratio.