Lesson 4: SAT Math - Problem-Solving & Data Analysis
Master ratios, proportions, percentages, unit conversions, and data interpretation for SAT Math success.
Lesson 4: SAT Math - Problem-Solving & Data Analysis 📊
Introduction
Welcome to Lesson 4! Now that you've mastered the SAT structure and conquered reading and writing sections, it's time to tackle one of the most practical math domains on the SAT: Problem-Solving and Data Analysis. This area accounts for approximately 30% of your SAT Math score and focuses on real-world applications like ratios, percentages, data interpretation, and statistical reasoning.
🎯 Why This Matters: Unlike abstract algebra, these questions mirror situations you'll encounter in college and career—analyzing survey results, calculating discounts, understanding scientific data, and making informed decisions based on statistics.
💡 Pro Tip: The SAT loves testing whether you can translate between different representations of the same information (words → equations → graphs → tables). Master this skill and you'll unlock dozens of points!
Core Concept 1: Ratios, Proportions & Percentages 🧮
Understanding Ratios
A ratio compares two quantities using division. Ratios can be written as:
- 3:4 (ratio notation)
- 3/4 (fraction)
- 3 to 4 (words)
⚠️ Critical Distinction: A ratio of 3:4 does NOT mean you have 3 and 4 items—it means for every 3 of one thing, there are 4 of another. If a classroom has a boy-to-girl ratio of 3:4, and there are 21 boys, you have 28 girls (not 4!).
Proportions: The Power Tool
A proportion states that two ratios are equal: a/b = c/d
Cross-multiplication is your best friend:
If a/b = c/d, then a×d = b×c
🔧 Try This: If 5 apples cost $3, how much do 8 apples cost?
Set up the proportion:
5 apples 8 apples
-------- = --------
$3 $x
5x = 24
x = $4.80
Percentage Power Moves
Three types of percentage problems:
+------------------+------------------------+------------------+
| Problem Type | Equation | Example |
+------------------+------------------------+------------------+
| Find the part | Part = % × Whole | 30% of 80 = ? |
| Find the whole | Whole = Part ÷ % | 24 is 30% of ? |
| Find the percent | % = Part ÷ Whole | 24 is what % of |
| | | 80? |
+------------------+------------------------+------------------+
💡 Percentage Change Formula (SAT FAVORITE!):
Percentage Change = (New Value - Original Value) / Original Value × 100%
⚠️ Common Trap: If something increases by 50% then decreases by 50%, you're NOT back to the original!
- Start: 100
- Increase 50%: 150
- Decrease 50% of 150: 75 (not 100!)
🧠 Mnemonic: "Percent OF Whole" → multiply (of = ×)
Core Concept 2: Unit Conversions & Rates ⚡
Dimensional Analysis: The Universal Method
Dimensional analysis (also called unit cancellation) ensures you convert units correctly by treating units like algebraic expressions.
The Process:
- Write what you have as a fraction
- Multiply by conversion factors (fractions equal to 1)
- Cancel units that appear in both numerator and denominator
- Calculate
Example: Convert 45 miles per hour to feet per second.
45 miles 5,280 feet 1 hour 1 minute
-------- × ------------ × ----------- × ----------- = 66 feet/second
1 hour 1 mile 60 minutes 60 seconds
miles feet hours minutes
------- × ------- × -------- × ---------
hours miles minutes seconds
(cancel) (cancel) (cancel)
Common SAT Unit Conversions
+-------------------+------------------------+
| Conversion | Factor |
+-------------------+------------------------+
| 1 mile | 5,280 feet |
| 1 hour | 3,600 seconds |
| 1 meter | 100 centimeters |
| 1 kilometer | 1,000 meters |
| 1 pound | 16 ounces |
| 1 gallon | 4 quarts |
+-------------------+------------------------+
Rate Problems: Distance, Work & Beyond
The Master Formula: Distance = Rate × Time (D = RT)
🌍 Real-World Application: Two trains leave stations 300 miles apart, traveling toward each other at 40 mph and 60 mph. When do they meet?
Solution Strategy:
- Combined rate: 40 + 60 = 100 mph (approaching each other)
- Time to meet: 300 miles ÷ 100 mph = 3 hours
💡 Work Rate Tip: If Person A completes a job in 4 hours, their rate is 1/4 job per hour (not 4!). Combined rates ADD when working together.
Core Concept 3: Data Interpretation & Statistics 📈
Reading Tables, Graphs & Charts
The SAT presents data in multiple formats:
Bar Graphs → Compare categories Line Graphs → Show trends over time Scatterplots → Show relationships between two variables Two-way Tables → Cross-tabulate two categorical variables
🔍 What the SAT Tests:
- Reading values accurately from visual displays
- Identifying trends (increasing, decreasing, constant)
- Making comparisons between data points
- Calculating percentages from data
- Drawing valid conclusions (vs. invalid overgeneralizations)
Statistical Measures You Must Know
+-------------+--------------------------------+-----------------+
| Measure | Definition | Key Point |
+-------------+--------------------------------+-----------------+
| Mean | Sum of values ÷ count | Affected by |
| | | outliers |
+-------------+--------------------------------+-----------------+
| Median | Middle value when ordered | Resistant to |
| | | outliers |
+-------------+--------------------------------+-----------------+
| Mode | Most frequent value | Can have |
| | | multiple modes |
+-------------+--------------------------------+-----------------+
| Range | Maximum - Minimum | Measures spread |
+-------------+--------------------------------+-----------------+
Example Dataset: Test scores: 65, 70, 72, 75, 75, 78, 95
- Mean: (65+70+72+75+75+78+95) ÷ 7 = 75.7
- Median: The middle value = 75
- Mode: 75 (appears twice)
- Range: 95 - 65 = 30
🤔 Did You Know? The median is often more useful than the mean when data has outliers. That's why we talk about "median household income" rather than "mean household income"—a few billionaires would skew the mean!
Margin of Error & Sampling 🎯
The SAT tests your understanding of sampling and statistical validity:
Random Sample → Every member of the population has an equal chance of being selected Biased Sample → Some members are more likely to be selected than others Margin of Error → The ± range around a survey result
If a poll shows 52% support with a ±3% margin of error, the true value is likely between 49% and 55%.
⚠️ Critical Thinking: A sample of students at a chess club about favorite activities is BIASED for generalizing to all students (chess club members aren't representative).
Example 1: Multi-Step Percentage Problem 💰
Question: A store marks up a product by 40% above wholesale cost. During a sale, they discount the marked price by 25%. If the final sale price is $84, what was the wholesale cost?
Solution:
Let's call the wholesale cost W.
Step 1: After 40% markup
- Marked price = W + 0.40W = 1.40W
Step 2: After 25% discount on marked price
- Sale price = 1.40W - 0.25(1.40W)
- Sale price = 1.40W - 0.35W = 1.05W
Step 3: Set equal to final price
- 1.05W = $84
- W = $84 ÷ 1.05 = $80
💡 Key Insight: Notice the final price ($84) is HIGHER than the wholesale cost ($80) even after a discount, because the markup was larger than the discount. Always work through the percentages sequentially!
Verification: $80 × 1.40 = $112 (marked price), then $112 × 0.75 = $84 ✓
Example 2: Rate Problem with Unit Conversion 🚗
Question: A car travels at 72 kilometers per hour. How many meters does it travel in 15 seconds?
Solution Using Dimensional Analysis:
72 km 1000 m 1 hour 1 minute 15 seconds
------ × --------- × --------- × ---------- × -----------
1 hour 1 km 60 minutes 60 seconds 1
Simplify first:
72 km 1000 m 1
------ × --------- × -----------
1 hour 1 km 3600 sec/hr
= 72 × 1000 ÷ 3600 m/sec
= 20 m/sec
In 15 seconds: 20 × 15 = 300 meters
Alternative Method: Convert to meters per second first, then multiply.
- 72 km/hr = 72,000 m/hr
- 72,000 m/hr ÷ 3,600 sec/hr = 20 m/sec
- 20 m/sec × 15 sec = 300 meters
🧠 Memory Aid: 1 km/hr ≈ 0.278 m/sec (or just remember to divide by 3.6)
Example 3: Two-Way Table Analysis 📊
Question: A survey asked 200 students about their preferred study method:
+------------------+----------+-----------+---------+
| | Alone | Group | Total |
+------------------+----------+-----------+---------+
| Prefers Morning | 45 | 35 | 80 |
| Prefers Evening | 65 | 55 | 120 |
| Total | 110 | 90 | 200 |
+------------------+----------+-----------+---------+
What percentage of students who prefer group study are morning studiers?
Solution:
⚠️ Common Mistake: Don't use the total 200! The question asks about "students who prefer group study" only.
Step 1: Identify the relevant subset
- Total group studiers: 90 students
- Group studiers who prefer morning: 35 students
Step 2: Calculate percentage
- Percentage = (35 ÷ 90) × 100% = 38.9% (or about 39%)
💡 Key Strategy: Circle or highlight the specific row/column mentioned in the question. Two-way tables test whether you can identify the correct denominator.
Related Question: What percentage of morning studiers prefer group study?
- Now the subset is 80 morning studiers
- (35 ÷ 80) × 100% = 43.75%
- Notice this is DIFFERENT! The order matters.
Example 4: Scatterplot & Linear Association 📉
Scenario: A scatterplot shows the relationship between hours studied and test scores for 15 students. Most points cluster around an upward-sloping line, but one student studied 8 hours and scored 45%.
Question Types You'll See:
Describe the association: "The scatterplot shows a positive linear association between study hours and test scores."
Identify outliers: "The point (8, 45) is an outlier—it deviates significantly from the trend."
Interpret slope: "The line of best fit has a slope of 8, meaning each additional hour of study is associated with an 8-point increase in test scores."
Correlation vs. Causation: ⚠️ The SAT loves this! Just because two variables are correlated DOES NOT mean one causes the other.
Example of Incorrect Reasoning:
❌ "Ice cream sales cause drowning deaths"
(both increase in summer—lurking variable!)
✅ "Ice cream sales are correlated with drowning deaths,
likely due to warm weather (a confounding variable)"
💡 SAT Favorites:
- Positive association: As X increases, Y increases
- Negative association: As X increases, Y decreases
- No association: No clear pattern
- Linear vs. Nonlinear: Does the pattern follow a straight line?
Common Mistakes to Avoid ⚠️
1. The "Backward Percentage" Error
❌ Problem: An item costs $80 after a 20% discount. What was the original price? ❌ Wrong: $80 ÷ 0.80 = $100... wait, let me check: $100 × 0.20 = $20, so $100 - $20 = $80 ✓ ❌ Wrong approach: $80 + ($80 × 0.20) = $96 ✗
✅ Correct Reasoning: If the discount was 20%, the sale price represents 80% of the original.
- 0.80 × Original = $80
- Original = $80 ÷ 0.80 = $100
2. The Unit Conversion Mix-Up
❌ Converting 3 hours to seconds: 3 × 60 = 180 seconds ✗ ✅ Correct: 3 hours × 60 min/hour × 60 sec/min = 10,800 seconds
Prevention: Always write out your conversion factors with units!
3. The "Wrong Total" in Two-Way Tables
When asked "What percent of X are Y?", make sure X is your denominator!
Template:
What percent of [GROUP] are [SUBSET]?
↓ ↓
DENOMINATOR NUMERATOR
4. Confusing Mean vs. Median
Dataset: 2, 3, 3, 4, 100
- Mean = 112 ÷ 5 = 22.4
- Median = 3
For data with extreme values, median is more representative!
5. The Sample Size Mistake
"I surveyed my 5 friends and 4 like pizza, so 80% of all students like pizza!"
⚠️ Problems:
- Sample size too small
- Sample is biased (your friends aren't random)
- Overgeneralization
Key Takeaways 🎯
✅ Ratios are multiplicative relationships, not additive. A 3:4 ratio with 21 items doesn't mean the other part is 4!
✅ Percentage change is always calculated from the ORIGINAL value, not the new value.
✅ Master dimensional analysis for unit conversions—write out all conversion factors and cancel units.
✅ Combined rates ADD when working together; they MULTIPLY the TIME when working in sequence.
✅ In two-way tables, carefully identify whether the question asks for a percentage "of all" vs. "of a subset."
✅ The median is resistant to outliers; the mean is not. Know when to use each.
✅ Correlation ≠ Causation. Just because two variables move together doesn't mean one causes the other.
✅ For sampling to be valid, it must be random and sufficiently large. Convenience samples are biased.
✅ Read scatterplot questions carefully: positive/negative association, linear/nonlinear, outliers, and slope interpretation.
✅ Practice mental math for common conversions: 60 sec/min, 60 min/hr, 100 cm/m, 1000 m/km.
Quick Reference Card 📋
╔══════════════════════════════════════════════════════════╗
║ SAT PROBLEM-SOLVING & DATA CHEAT SHEET ║
╠══════════════════════════════════════════════════════════╣
║ PERCENTAGES ║
║ • Part = Percent × Whole ║
║ • % Change = (New - Old)/Old × 100% ║
║ • After ±x%: Multiply by (1 ± x/100) ║
║ ║
║ PROPORTIONS ║
║ • a/b = c/d → Cross-multiply: ad = bc ║
║ ║
║ RATES ║
║ • Distance = Rate × Time ║
║ • Combined rate (together) = Rate₁ + Rate₂ ║
║ • Work rate = 1/(time to complete) ║
║ ║
║ UNIT CONVERSIONS ║
║ • Write conversion factors as fractions ║
║ • Cancel units diagonally ║
║ • Check: Does the final unit make sense? ║
║ ║
║ STATISTICS ║
║ • Mean = Sum/Count (affected by outliers) ║
║ • Median = Middle value (resistant to outliers) ║
║ • Range = Max - Min ║
║ ║
║ DATA INTERPRETATION ║
║ • Two-way tables: Check row vs. column totals ║
║ • Scatterplots: Positive/negative/no association ║
║ • Correlation ≠ Causation! ║
║ ║
║ SAMPLING ║
║ • Random sample → generalizable ║
║ • Biased sample → not representative ║
║ • Larger sample → smaller margin of error ║
╚══════════════════════════════════════════════════════════╝
Further Study 📚
Khan Academy SAT Math: Problem Solving and Data Analysis
https://www.khanacademy.org/test-prep/sat/sat-math-practice
Free practice problems with video explanations for every concept covered.College Board Official SAT Practice
https://collegereadiness.collegeboard.org/sat/practice
Official practice tests with authentic problem-solving questions.Desmos Graphing Calculator Tutorial
https://www.desmos.com/calculator
Learn to use the graphing calculator allowed on SAT Math sections for data analysis.
Next Steps: Practice these concepts with the questions below, then move on to Lesson 5, where we'll tackle Heart of Algebra (linear equations, inequalities, and systems). You're building a strong foundation—keep going! 💪