SAT Math – Problem Solving & Data Analysis
Ratios, Rates, and Proportions
Understanding relationships between quantities in real-world contexts
Ratios, rates, and proportions form the mathematical foundation of comparison and scaling. These concepts appear throughout the SAT in contexts ranging from recipe conversions and map distances to population density and unit pricing—real applications that test whether you can reason proportionally and solve practical problems.
Success requires understanding when quantities are proportional, setting up correct relationships, and solving efficiently using cross-multiplication or unit analysis. These aren't abstract concepts—they're the mathematical tools for answering "How many?" "How fast?" and "How much?" in situations where direct counting isn't possible.
Understanding the Three Concepts
Ratios: Comparing Quantities
A ratio compares two or more quantities with the same units, showing their relative sizes. Ratios can be written as fractions, with colons, or with the word "to."
• Ratio of boys to girls = \(12:18 = 2:3\) (simplified)
• This means for every 2 boys, there are 3 girls
• Total parts = \(2 + 3 = 5\) parts
Rates: Comparing Different Units
A rate compares two quantities with different units. Common rates include speed (miles per hour), price (dollars per pound), and density (people per square mile).
• Speed: 60 miles per hour (miles/hour)
• Unit price: $3.50 per pound (dollars/pound)
• Rate: 25 pages per day (pages/day)
Proportions: Equal Ratios
A proportion states that two ratios are equal. Proportions allow us to find unknown quantities when relationships stay constant.
Cross-multiplication: \(a \times d = b \times c\)
Example: If 3 apples cost $2, then 9 apples cost $6
\(\frac{3 \text{ apples}}{2 \text{ dollars}} = \frac{9 \text{ apples}}{6 \text{ dollars}}\)
Essential Formulas & Techniques
Working with Ratios
Part-to-Part vs. Part-to-Whole:
If ratio is \(a:b\), then:
• First part = \(\frac{a}{a+b} \times \text{total}\)
• Second part = \(\frac{b}{a+b} \times \text{total}\)
Cross-Multiplication for Proportions
If \(\frac{a}{b} = \frac{c}{d}\), then \(a \cdot d = b \cdot c\)
Multiply diagonally across the equal sign to solve for unknowns
Rate Formula
\(\text{Distance} = \text{Rate} \times \text{Time}\)
Also: \(\text{Total} = \text{Rate} \times \text{Amount}\)
Example: \(\text{Cost} = \text{Price per unit} \times \text{Number of units}\)
Unit Conversion
Multiply by conversion factors (as fractions equal to 1):
60 mph = \(60 \frac{\text{miles}}{\text{hour}} \times \frac{1 \text{ hour}}{60 \text{ minutes}} = 1 \frac{\text{mile}}{\text{minute}}\)
Common Pitfalls & Expert Tips
❌ Setting up proportions incorrectly
Keep units consistent! If the numerator is miles in one ratio, it must be miles in the other. Don't mix up numerators and denominators.
❌ Confusing part-to-part with part-to-whole
A ratio of 2:3 doesn't mean 2 out of 3 total. It means 2 out of 5 total (2+3=5). Read carefully whether the problem gives parts or totals!
❌ Forgetting to simplify ratios
The ratio 12:18 should be simplified to 2:3. Always reduce to lowest terms unless told otherwise.
❌ Unit confusion in rates
If speed is in mph and time is in minutes, convert to matching units before calculating distance!
✓ Expert Tip: Label your units
Write units with every number. This prevents setup errors and makes cross-multiplication clearer. Units should cancel properly!
✓ Expert Tip: Use the "multiplier" method
If a ratio is 3:4 and one quantity is 15, find the multiplier: 15÷3=5, then multiply the other part: 4×5=20. Fast and reliable!
✓ Expert Tip: Check if the answer makes sense
If you're driving faster, you should cover more distance in the same time. If prices are lower, you should be able to buy more. Use logic!
Fully Worked SAT-Style Examples
A recipe calls for flour and sugar in the ratio of 5:2. If a baker uses 15 cups of flour, how many cups of sugar are needed?
Solution:
Step 1: Understand the ratio
Flour : Sugar = 5 : 2
This means for every 5 cups of flour, we need 2 cups of sugar
Step 2: Find the multiplier
If flour is 15 cups and the ratio part is 5:
Multiplier = \(\frac{15}{5} = 3\)
Step 3: Find sugar amount
Sugar = 2 × multiplier = \(2 \times 3 = 6\) cups
Alternative Method (Cross-Multiplication):
Set up proportion: \(\frac{5}{2} = \frac{15}{x}\)
Cross-multiply: \(5x = 30\)
Solve: \(x = 6\) cups
Answer: 6 cups of sugar
In a class of 40 students, the ratio of students who prefer math to those who prefer science is 3:5. How many students prefer math?
Solution:
Step 1: Understand the ratio parts
Math : Science = 3 : 5
Total parts = \(3 + 5 = 8\) parts
Step 2: Find value of one part
8 parts = 40 students (total)
1 part = \(\frac{40}{8} = 5\) students
Step 3: Find students who prefer math
Math students = 3 parts = \(3 \times 5 = 15\) students
Verification:
Math: 15 students, Science: 25 students
Total: \(15 + 25 = 40\) ✓
Ratio: \(15:25 = 3:5\) ✓
Answer: 15 students prefer math
A car travels 180 miles in 3 hours. At this rate, how far will the car travel in 5 hours?
Solution:
Step 1: Find the rate
Rate = \(\frac{\text{Distance}}{\text{Time}} = \frac{180 \text{ miles}}{3 \text{ hours}} = 60 \text{ mph}\)
Step 2: Use the rate to find new distance
Distance = Rate × Time
Distance = \(60 \times 5 = 300\) miles
Alternative: Proportion Method
Set up proportion: \(\frac{180 \text{ miles}}{3 \text{ hours}} = \frac{x \text{ miles}}{5 \text{ hours}}\)
Cross-multiply: \(3x = 900\)
Solve: \(x = 300\) miles
Answer: 300 miles
Store A sells 5 pounds of apples for $8. Store B sells 3 pounds of apples for $5.25. Which store offers a better unit price?
Solution:
Step 1: Calculate Store A's unit price
Price per pound = \(\frac{\$8}{5 \text{ pounds}} = \$1.60\) per pound
Step 2: Calculate Store B's unit price
Price per pound = \(\frac{\$5.25}{3 \text{ pounds}} = \$1.75\) per pound
Step 3: Compare unit prices
Store A: $1.60 per pound
Store B: $1.75 per pound
Store A has the lower (better) unit price
Savings Analysis:
Difference per pound: \(\$1.75 - \$1.60 = \$0.15\)
For 10 pounds, Store A saves: \(\$0.15 \times 10 = \$1.50\)
Answer: Store A offers a better unit price ($1.60/lb)
On a map with a scale of 1 inch = 25 miles, two cities are 3.5 inches apart. What is the actual distance between the cities?
Solution:
Step 1: Set up the proportion
Scale: 1 inch on map = 25 miles in reality
\(\frac{1 \text{ inch}}{25 \text{ miles}} = \frac{3.5 \text{ inches}}{x \text{ miles}}\)
Step 2: Solve using cross-multiplication
\(1 \times x = 25 \times 3.5\)
\(x = 87.5\) miles
Quick Method:
Multiply map distance by scale factor:
\(3.5 \text{ inches} \times 25 \frac{\text{miles}}{\text{inch}} = 87.5\) miles
Answer: 87.5 miles
A machine can produce 450 widgets in 6 hours. At this rate, how many widgets can it produce in 10 hours?
Solution:
Step 1: Find the production rate
Rate = \(\frac{450 \text{ widgets}}{6 \text{ hours}} = 75 \text{ widgets per hour}\)
Step 2: Calculate production for 10 hours
Total widgets = Rate × Time
\(75 \frac{\text{widgets}}{\text{hour}} \times 10 \text{ hours} = 750\) widgets
Proportion Check:
\(\frac{450}{6} = \frac{x}{10}\) → \(6x = 4500\) → \(x = 750\) ✓
Answer: 750 widgets
A sum of money is divided among three people in the ratio 2:3:5. If the total amount is $600, how much does the person with the largest share receive?
Solution:
Step 1: Find total ratio parts
Ratio = 2 : 3 : 5
Total parts = \(2 + 3 + 5 = 10\) parts
Step 2: Find value of one part
10 parts = $600
1 part = \(\frac{600}{10} = \$60\)
Step 3: Find the largest share
The largest ratio part is 5
Largest share = \(5 \times \$60 = \$300\)
Verification:
Person 1: \(2 \times \$60 = \$120\)
Person 2: \(3 \times \$60 = \$180\)
Person 3: \(5 \times \$60 = \$300\)
Total: \(\$120 + \$180 + \$300 = \$600\) ✓
Answer: $300
A city has a population of 120,000 people and an area of 40 square miles. What is the population density in people per square mile? If a neighboring city has the same density but an area of 65 square miles, what is its population?
Solution:
Part A: Find population density
Density = \(\frac{\text{Population}}{\text{Area}}\)
Density = \(\frac{120,000 \text{ people}}{40 \text{ sq mi}} = 3,000\) people/sq mi
Part B: Find neighboring city's population
Population = Density × Area
Population = \(3,000 \frac{\text{people}}{\text{sq mi}} \times 65 \text{ sq mi}\)
Population = 195,000 people
Proportion Check:
\(\frac{120,000}{40} = \frac{x}{65}\)
Cross-multiply: \(40x = 7,800,000\)
\(x = 195,000\) ✓
Answers:
Density: 3,000 people per square mile
Neighboring city population: 195,000 people
Quick Reference Guide
SAT Test Day Checklist
For Ratios
- Simplify ratios to lowest terms
- Identify if part-to-part or part-to-whole
- Sum all parts for total
- Use multiplier method for speed
For Rates
- Find rate first (divide)
- Check units match
- Multiply rate by quantity
- Verify answer makes sense
For Proportions
- Set up with units aligned
- Cross-multiply carefully
- Label all quantities with units
- Check that units cancel properly
General Tips
- Write units with every number
- Double-check which quantity is asked
- Test if answer is reasonable
- Verify calculations
Proportional Reasoning: The Mathematics of Scaling
Ratios, rates, and proportions are fundamental to quantitative reasoning across every field—from cooking and construction to finance and physics. When you double a recipe, compare prices, convert currencies, or analyze data, you're using proportional thinking. The SAT tests these concepts because they represent genuine mathematical literacy: the ability to understand relationships, maintain consistency when scaling, and make comparisons that guide real decisions. Master the mechanics—finding multipliers, cross-multiplying, calculating rates—but also develop the intuition to recognize when quantities should scale proportionally and when they don't. This combination of technical skill and conceptual understanding makes you not just test-ready, but genuinely mathematically fluent.
