Lesson 3: GRE Quantitative Reasoning - Ratios, Proportions & Percentages
Master ratios, proportions, and percentage problems with advanced problem-solving strategies for the GRE Quantitative section.
Lesson 3: GRE Quantitative Reasoning - Ratios, Proportions & Percentages 🧮
Introduction
Welcome to Lesson 3! Now that you've mastered basic arithmetic and algebra, it's time to tackle one of the GRE's favorite topics: ratios, proportions, and percentages. These concepts appear in roughly 15-20% of GRE Quantitative questions and often combine with other topics you've learned.
Unlike simple arithmetic, ratio and percentage problems on the GRE test your ability to see relationships between quantities and translate word problems into mathematical expressions. You'll encounter these in data interpretation, word problems, and comparison questions.
💡 Pro Tip: The GRE loves to disguise ratio problems as real-world scenarios involving mixtures, speeds, work rates, and population changes. Learning to spot the underlying structure is key!
Core Concept 1: Understanding Ratios 📊
What is a Ratio?
A ratio compares two or more quantities by division. It tells us "how many times" one quantity contains another or how they relate proportionally.
Three ways to express ratios:
- Using a colon: 3:4
- As a fraction: 3/4
- Using the word "to": 3 to 4
Key Properties of Ratios
┌─────────────────────────────────────────────────┐
│ RATIO FUNDAMENTALS │
├─────────────────────────────────────────────────┤
│ If ratio = a:b │
│ │
│ • Total parts = a + b │
│ • First quantity = (a/(a+b)) × total │
│ • Second quantity = (b/(a+b)) × total │
│ • Ratios can be scaled: 2:3 = 4:6 = 6:9 │
└─────────────────────────────────────────────────┘
Part-to-Part vs. Part-to-Whole Ratios:
- Part-to-Part: Compares two components (e.g., "boys to girls = 3:2")
- Part-to-Whole: Compares one component to the total (e.g., "boys to all students = 3:5")
⚠️ Common Mistake: Confusing part-to-part with part-to-whole ratios! If boys:girls = 3:2, then boys:total = 3:5, NOT 3:2.
Ratio Scaling and Simplification
Ratios behave like fractions:
- Simplify by dividing by the GCD: 12:18 = 2:3
- Scale up by multiplying: 2:3 = 6:9 = 10:15
- Find a common multiplier to work with actual quantities
🧠 Memory Device: "RATS" - Ratios Are Totally Scalable!
Core Concept 2: Proportions and Cross-Multiplication ⚖️
What is a Proportion?
A proportion states that two ratios are equal: a/b = c/d
This is one of the most powerful tools for GRE problems!
The Cross-Multiplication Method
When you have a/b = c/d, you can cross-multiply:
a c
--- = ---
b d
a × d = b × c
(Cross products are equal)
When to Use Proportions:
- Scale conversions (inches to centimeters)
- Rate problems (miles per hour)
- Recipe scaling
- Map distances
- Similar figures in geometry
Direct vs. Inverse Proportions
Direct Proportion: As one increases, the other increases proportionally
- If y is directly proportional to x: y = kx (where k is constant)
- Example: Distance = Speed × Time (at constant speed)
Inverse Proportion: As one increases, the other decreases proportionally
- If y is inversely proportional to x: y = k/x
- Example: Time = Distance/Speed (at constant distance)
┌────────────────────────────────────────────────┐
│ DIRECT vs INVERSE │
├────────────────────────────────────────────────┤
│ DIRECT: x doubles → y doubles │
│ x ↑↑ → y ↑↑ │
│ Formula: y = kx │
│ │
│ INVERSE: x doubles → y halves │
│ x ↑↑ → y ↓↓ │
│ Formula: xy = k (constant) │
└────────────────────────────────────────────────┘
💡 GRE Strategy: When you see "varies directly," set up y = kx. For "varies inversely," use xy = k or y = k/x.
Core Concept 3: Percentages - The GRE's Favorite 📈
Percentage Fundamentals
A percentage is a ratio expressed as a fraction of 100. The word "percent" literally means "per hundred."
Three Key Formulas:
┌─────────────────────────────────────────────┐
│ PERCENTAGE MASTER FORMULAS │
├─────────────────────────────────────────────┤
│ │
│ 1. Finding a percentage of a number: │
│ (Percent/100) × Whole = Part │
│ │
│ 2. Finding what percent one is of another: │
│ (Part/Whole) × 100 = Percent │
│ │
│ 3. Finding the whole: │
│ Part/(Percent/100) = Whole │
│ │
└─────────────────────────────────────────────┘
Percentage Change Formula ⚡
This is CRITICAL for the GRE:
Percent Change = [(New Value - Original Value) / Original Value] × 100
- Increase: Result is positive
- Decrease: Result is negative
⚠️ Critical Error: Always divide by the ORIGINAL value, not the new value!
Successive Percentage Changes
When applying multiple percentage changes, you CANNOT simply add them!
Example: A price increases by 20%, then decreases by 20%. Is it back to the original?
Original: $100
After +20%: $100 × 1.20 = $120
After -20%: $120 × 0.80 = $96
NOT back to $100! Lost $4 overall.
🧠 Memory Trick: "MULTIPLY for successive, DON'T ADD!"
For successive changes, multiply the multipliers:
- +20% means ×1.20
- -20% means ×0.80
- Combined: ×1.20 × 0.80 = ×0.96 (4% decrease overall)
Percentage Points vs. Percent Change 🎯
Percentage points = Simple subtraction of percentages Percent change = Relative change calculated with formula
Example: Interest rate changes from 5% to 8%
- Change in percentage points: 8% - 5% = 3 percentage points
- Percent change: (8-5)/5 × 100 = 60% increase
🤔 Did you know? The GRE often tricks test-takers by asking for "percentage point change" when they calculate "percent change" (or vice versa). Read carefully!
Core Concept 4: Advanced Applications 🎓
Mixture Problems
Mixture problems combine ratios and percentages. The key is tracking the total amount and the concentration.
Mixture Formula:
(Amount₁)(Concentration₁) + (Amount₂)(Concentration₂) = (Total Amount)(Final Concentration)
Setup Pattern:
- Identify what's being mixed (solutions, alloys, investments)
- Set up a table with Amount, Concentration, and Total
- Write the equation and solve
Work Rate Problems with Ratios
When workers have efficiency ratios, convert to rate form:
If A:B = 3:2 in efficiency:
- A completes 3 parts while B completes 2 parts in the same time
- If together they complete 1 job, A does 3/5 and B does 2/5
Compound Ratios 🔗
When you have connected ratios, find a common term:
Example: If A:B = 2:3 and B:C = 4:5, find A:B:C
Make B the same in both:
- A:B = 2:3 = 8:12 (multiply by 4)
- B:C = 4:5 = 12:15 (multiply by 3)
- Therefore A:B:C = 8:12:15
💡 Pro Strategy: Always find the LCM of the common terms to scale ratios properly!
Detailed Example 1: Multi-Step Ratio Problem 📝
Problem: In a class, the ratio of boys to girls is 5:7. After 3 boys join and 2 girls leave, the ratio becomes 3:4. How many students were originally in the class?
Solution:
Let the original number of boys = 5x and girls = 7x
Original: Boys = 5x Girls = 7x
After changes: Boys = 5x+3 Girls = 7x-2
New ratio: (5x+3):(7x-2) = 3:4
Cross-multiply: 4(5x + 3) = 3(7x - 2) 20x + 12 = 21x - 6 12 + 6 = 21x - 20x x = 18
Original students = 5(18) + 7(18) = 90 + 126 = 216 students
Verify:
- Original: 90 boys, 126 girls → 90:126 = 5:7 ✓
- After: 93 boys, 124 girls → 93:124 = 3:4 ✓
Detailed Example 2: Successive Percentage Changes 📝
Problem: A store increases prices by 25%, then offers a 20% discount. What is the net effect on the original price?
Solution:
Let original price = $100 (using 100 makes percentages easy)
Step 1: Increase by 25%
New price = $100 × 1.25 = $125
Step 2: Decrease by 20%
Final price = $125 × 0.80 = $100
Wait! Let's check this algebraically:
Multiplier = 1.25 × 0.80 = 1.00
Answer: The net effect is 0% change - the price returns to original!
🤔 Interesting: This is a special case. A 25% increase followed by 20% decrease happens to cancel out perfectly because 1.25 × 0.80 = 1.
Detailed Example 3: Inverse Proportion Problem 📝
Problem: 8 workers can complete a project in 15 days working 6 hours per day. How many days will it take 10 workers working 8 hours per day to complete the same project?
Solution:
First, find the total work in "worker-hours":
Total work = 8 workers × 15 days × 6 hours/day
= 720 worker-hours
Now, with new conditions:
- Daily capacity = 10 workers × 8 hours/day = 80 worker-hours/day
- Days needed = 720 ÷ 80 = 9 days
Alternative Method (proportion):
Workers × Days × Hours = Constant
8 × 15 × 6 = 10 × D × 8
720 = 80D
D = 9 days
💡 Key Insight: When multiple factors change, find the total work capacity first!
Detailed Example 4: Mixture Problem 📝
Problem: How many liters of a 30% acid solution must be mixed with 20 liters of a 60% acid solution to obtain a 45% acid solution?
Solution:
Let x = liters of 30% solution needed
┌─────────────┬────────┬──────────────┬────────────┐
│ Solution │ Amount │ Concentration│ Pure Acid │
├─────────────┼────────┼──────────────┼────────────┤
│ 30% soln │ x │ 0.30 │ 0.30x │
│ 60% soln │ 20 │ 0.60 │ 12 │
│ Mixture │ x+20 │ 0.45 │ 0.45(x+20) │
└─────────────┴────────┴──────────────┴────────────┘
Equation: 0.30x + 12 = 0.45(x + 20)
0.30x + 12 = 0.45x + 9 12 - 9 = 0.45x - 0.30x 3 = 0.15x x = 20 liters
Answer: Mix 20 liters of the 30% solution
Verify: (20)(0.30) + (20)(0.60) = (40)(0.45) → 6 + 12 = 18 ✓
Common Mistakes to Avoid ⚠️
Mistake 1: Adding Percentages Incorrectly
❌ Wrong: 20% increase then 30% increase = 50% increase ✅ Right: Multiply: 1.20 × 1.30 = 1.56 (56% increase)
Mistake 2: Using Wrong Base for Percentage
❌ Wrong: Price drops from $100 to $80, then rises from $80 to $100. Both changes are 20%. ✅ Right: First is 20% decrease. Second is (20/80)×100 = 25% increase. Different bases!
Mistake 3: Confusing Part-to-Part with Part-to-Whole
❌ Wrong: If men:women = 3:5, then men are 3/5 of total ✅ Right: Men are 3/8 of total (3 out of 3+5=8 parts)
Mistake 4: Forgetting to Convert Percentages
❌ Wrong: 35% of 80 = 35 × 80 ✅ Right: 35% of 80 = 0.35 × 80 = 28
Mistake 5: Incorrect Cross-Multiplication
❌ Wrong: Setting up proportions with wrong corresponding terms ✅ Right: Make sure units match! If 3 apples cost $2, then x apples cost $10: 3/2 = x/10 (apples/dollars on both sides)
Key Takeaways 🎯
Essential Formulas to Memorize:
╔══════════════════════════════════════════════╗
║ RATIOS & PROPORTIONS FORMULA SHEET ║
╠══════════════════════════════════════════════╣
║ ║
║ Ratio to Actual: Quantity = (ratio part/ ║
║ total parts) × total ║
║ ║
║ Proportion: a/b = c/d → ad = bc ║
║ ║
║ Percent Change: (New-Old)/Old × 100 ║
║ ║
║ Percentage of: (P/100) × Whole = Part ║
║ ║
║ Successive %: Multiply multipliers ║
║ (+15% then -10% = 1.15 × 0.90 = 1.035) ║
║ ║
║ Mixture: Amt₁×Conc₁ + Amt₂×Conc₂ = ║
║ Total×FinalConc ║
║ ║
╚══════════════════════════════════════════════╝
Strategy Checklist:
✅ Always identify whether a ratio is part-to-part or part-to-whole ✅ For successive percentages, MULTIPLY the multipliers ✅ In mixture problems, set up a table to organize information ✅ Check your answer by plugging back into the original problem ✅ Watch for inverse relationships (more workers = less time) ✅ Convert all percentages to decimals before calculating
Time-Saving Tips:
⚡ Use 100 as your starting value for percentage problems ⚡ Recognize common percentage equivalents: 25%=1/4, 50%=1/2, 75%=3/4 ⚡ For 10% of a number, just move the decimal left one place ⚡ Simplify ratios before calculating to work with smaller numbers
Quick Reference Card 📋
┌─────────────────────────────────────────────────────┐
│ RATIO, PROPORTION & PERCENTAGE QUICK GUIDE │
├─────────────────────────────────────────────────────┤
│ │
│ RATIOS │
│ • Express as a:b or a/b │
│ • Total parts = sum of ratio numbers │
│ • Scale by multiplying/dividing all parts │
│ │
│ PROPORTIONS │
│ • Cross multiply when a/b = c/d │
│ • Direct: y = kx (both increase together) │
│ • Inverse: xy = k (one up, one down) │
│ │
│ PERCENTAGES │
│ • Change: (New-Old)/Old × 100 │
│ • Successive: multiply multipliers │
│ • Part = (Percent/100) × Whole │
│ │
│ WATCH OUT FOR: │
│ ⚠ Different bases in % problems │
│ ⚠ Part-to-part vs part-to-whole confusion │
│ ⚠ Adding percentages instead of multiplying │
│ ⚠ Using new value instead of original in % │
│ │
└─────────────────────────────────────────────────────┘
🔧 Try This: Mental Math Practice
Before moving to the questions, practice these mental calculations:
- Quick 10% trick: 10% of 350? → Move decimal left: 35
- Quick 5% trick: 5% = half of 10%, so 5% of 350 = 17.5
- Quick 15% trick: 15% = 10% + 5%, so 15% of 350 = 35 + 17.5 = 52.5
- Quick doubling: If A:B = 3:4 and B:C = 2:5, match B by finding LCM(4,2)=4
- A:B = 3:4 stays the same
- B:C = 2:5 becomes 4:10 (multiply by 2)
- So A:B:C = 3:4:10
📚 Further Study
For additional practice and deeper exploration:
Khan Academy GRE Math: https://www.khanacademy.org/test-prep/gre - Comprehensive video lessons on ratios and percentages with practice problems
Manhattan Prep GRE Math Strategies: https://www.manhattanprep.com/gre/free-resources/ - Advanced techniques for ratio and percentage problems
Magoosh GRE Math Formulas: https://magoosh.com/gre/gre-math-formulas/ - Complete formula reference with examples and practice questions
Remember: The GRE rewards those who can see the structure beneath word problems. Practice translating English into mathematical expressions, and these problems will become second nature! 🎓✨