SAT Prep - Problem Solving and Data Analysis (Ratios, Percentages, Tables, Scatterplots)

SAT – Problem Solving and Data Analysis (Ratios, Percentages, Tables, Scatterplots)

SAT Problem Solving and Data Analysis Study Guide

Topic: Ratios, Percentages, Tables, Scatterplots

What This Is

Problem Solving and Data Analysis (PSDA) makes up ~29% of the SAT Math section (17–18 questions). It tests your ability to interpret and analyze data from tables, graphs, and word problems—skills critical for college and careers. For example, you might be asked: "A recipe calls for 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar are needed?" Mastering ratios, percentages, and data interpretation will help you tackle these questions efficiently.

Key Terms & Rules

• Ratio: A comparison of two quantities (e.g., 3:2 for flour to sugar). Write as a fraction (3/2) or with a colon (3:2).
• Proportion: An equation stating two ratios are equal (e.g., 3/2 = 9/x). Solve by cross-multiplying.
• Percentage: A ratio expressed as a fraction of 100 (e.g., 25% = 25/100 = 0.25). Use the formula: Part = Percent × Whole (e.g., 20% of 50 = 0.20 × 50 = 10).
• Percent Change: Formula: ((New Value – Original Value) / Original Value) × 100%
  (e.g., Price increases from $40 to $50 → (50–40)/40 × 100% = 25% increase).
• Unit Rate: A ratio with a denominator of 1 (e.g., 60 miles/2 hours = 30 miles/hour).
• Scatterplot: A graph showing the relationship between two variables (e.g., height vs. weight). Look for trends (positive, negative, or no correlation).
• Line of Best Fit: A straight line that best represents data on a scatterplot. Use it to predict values.
• Table Interpretation: Read column/row headers carefully. Look for totals, differences, or missing values.
• Relative Frequency: The proportion of a category relative to the total (e.g., 30 out of 100 students = 30%).
• Conditional Probability: The probability of an event given another event has occurred (e.g., P(A|B) = P(A and B) / P(B)).
• Exponential Growth/Decay: Formula: Final Amount = Initial Amount × (1 ± r)^t, where r = rate, t = time.
• Margin of Error: A range around a sample statistic where the true population value likely falls (e.g., 50% ± 3% means 47%–53%).

Step-by-Step / Process Flow

1. Solving Ratio/Proportion Problems

1. Identify the given ratio (e.g., 3 cups flour : 2 cups sugar).
2. Set up a proportion (e.g., 3/2 = 9/x).
3. Cross-multiply and solve (3x = 18 → x = 6).
4. Check units (e.g., cups, miles) to ensure consistency.

2. Calculating Percentages

1. Determine what’s being asked (e.g., "What is 15% of 80?" or "80 is 15% of what number?").
2. Convert the percentage to a decimal (15% = 0.15).
3. Use the formula:
4. For "part": Part = Percent × Whole (0.15 × 80 = 12).
5. For "whole": Whole = Part / Percent (80 / 0.15 ≈ 533.33).
6. For percent change, use the formula and label as increase/decrease.

3. Interpreting Tables

1. Read the headers to understand what each row/column represents.
2. Look for totals or differences (e.g., "Total students = 100" or "Difference between Group A and B").
3. Calculate missing values (e.g., if 60% of 200 students are girls, 200 × 0.60 = 120 girls).
4. Check for relative frequencies (e.g., "What fraction of girls are in Group A?").

4. Analyzing Scatterplots

1. Identify the axes (e.g., x = hours studied, y = test scores).
2. Describe the trend:
3. Positive correlation: As x increases, y increases.
4. Negative correlation: As x increases, y decreases.
5. No correlation: Points are scattered randomly.
6. Use the line of best fit to estimate values (e.g., "If x = 5, what is y?").
7. Watch for outliers (points far from the trend line).

Common Mistakes

• Mistake: Misreading ratios (e.g., confusing 3:2 with 2:3).
  Correction: Write ratios as fractions (3/2 vs. 2/3) to avoid flipping them.

• Mistake: Forgetting to convert percentages to decimals (e.g., using 15 instead of 0.15).
  Correction: Always divide by 100 (15% = 0.15).

• Mistake: Ignoring units in word problems (e.g., mixing miles and kilometers).
  Correction: Circle units in the question and ensure answers match.

• Mistake: Assuming correlation = causation (e.g., "More ice cream sales cause more drownings").
  Correction: Correlation shows a relationship, not cause-and-effect.

• Mistake: Misinterpreting "percent of" vs. "percent increase" (e.g., "20% of 50" vs. "20% increase from 50").
  Correction: "Of" = multiplication; "increase" = original + (percent × original).

Exam Insights

• Most-tested concepts: Ratios/proportions, percent change, and scatterplot interpretation.
• Tricky distinctions:
• Percent of vs. percent increase/decrease (e.g., 20% of 50 = 10; 20% increase from 50 = 60).
• Relative vs. absolute frequency (e.g., "30% of students" vs. "30 students").
• Common distractors:
• Answer choices with reversed ratios (e.g., 2:3 instead of 3:2).
• Percentages that ignore the original value (e.g., "50% of 100" vs. "50% increase from 100").
• Calculator tips:
• Use the % key for quick conversions (e.g., 15% of 80 → 80 × 15% = 12).
• For scatterplots, use the STAT function to find the line of best fit.

Quick Check Questions

1. A recipe uses 5 parts flour to 2 parts sugar. If you use 20 cups of flour, how many cups of sugar are needed?
2. A) 4
3. B) 8
4. C) 10

5. D) 50
   Answer: B) 8 (Set up the proportion 5/2 = 20/x → x = 8).

6. A shirt’s price increases from $40 to $50. What is the percent increase?

7. A) 10%
8. B) 20%
9. C) 25%

10. D) 50%
    Answer: C) 25% ((50–40)/40 × 100% = 25%).

11. A scatterplot shows a positive correlation between hours studied and test scores. Which statement is true?

12. A) More hours studied causes higher scores.
13. B) Higher scores cause more hours studied.
14. C) As hours studied increase, scores tend to increase.
15. D) There is no relationship.
    Answer: C) Correlation ≠ causation, but the trend is positive.

Last-Minute Cram Sheet

1. Ratio → Proportion: Set up as fractions and cross-multiply.
2. Percent = Part/Whole × 100% (e.g., 15/60 = 25%).
3. Percent change: ((New – Old)/Old) × 100%.
4. Scatterplot trends: Positive (up), negative (down), or none.
5. Line of best fit: Predicts values; don’t extrapolate too far.
6. ⚠️ Units matter! Circle them in the question.
7. ⚠️ "Percent of" ≠ "percent increase" (e.g., 20% of 50 = 10; 20% increase = 60).
8. Relative frequency: Part/Total (e.g., 30/100 = 30%).
9. Exponential growth: Final = Initial × (1 + r)^t.
10. ⚠️ Outliers: Points far from the trend line—don’t ignore them!