PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Problem-Solving and Data Analysis Syllabus

Math: Problem-Solving and Data Analysis syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Math: Problem-Solving and Data Analysis in PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test), not a summary of it.

1. 1

   Ratios, Rates, and Proportions

   3 topics

   • Solving Proportional Relationships
     • Setting up and solving proportions
     • Unit rates and scaling
   • Unit Conversions
     • Converting within and between measurement systems
   • Ratio Word Problems
     • Part-to-part and part-to-whole reasoning
2. 2

   Percentages

   3 topics

   • Percent Calculations
     • Finding the part, whole, or percent
   • Percent Increase and Decrease
     • Successive and compound percent changes
   • Percent in Applied Contexts
     • Discounts, markups, and tax
3. 3

   Statistics and Data Distributions

   3 topics

   • Measures of Center and Spread
     • Mean, median, mode, and range
     • Effect of outliers on center
   • Interpreting Data Displays
     • Histograms, dot plots, and box plots
     • Scatterplots and lines of best fit
   • Comparing Distributions
     • Comparing spread and standard deviation conceptually
4. 4

   Probability and Inference

   3 topics

   • Calculating Simple Probability
     • Probability from tables and frequencies
   • Two-Way Tables
     • Conditional and relative-frequency reasoning
   • Sampling and Conclusions
     • Drawing inferences from samples
     • Evaluating margin of error and study design

Math: Problem-Solving and Data Analysis flashcards for PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test)

18 of 50 cards from the Math: Problem-Solving and Data Analysis deck — real questions with worked answers.

1. What is a proportional relationship between two quantities x and y?

   A relationship where y = kx for a constant k, meaning y is a constant multiple of x and their ratio y/x is always equal to k. Its graph is a straight line through the origin.

2. In a proportion a/b = c/d, what is the cross-multiplication rule used to solve it?

   a*d = b*c. Multiply each numerator by the opposite denominator and set the products equal, then solve for the unknown.

3. What is the constant of proportionality, and how do you find it from a table of proportional values?

   It is the fixed ratio k = y/x. Find it by dividing any y-value by its matching x-value; the result is the same for every pair in a proportional relationship.

4. How do you tell from a graph whether two quantities are proportional?

   The graph must be a straight line that passes through the origin (0, 0). A line not through the origin is linear but not proportional.

5. If y is directly proportional to x and y = 12 when x = 3, what is y when x = 10?

   k = 12/3 = 4, so y = 4x; therefore y = 4(10) = 40.

6. What is the general method (dimensional analysis) for converting units?

   Multiply the original quantity by conversion factors written as fractions equal to 1, arranged so unwanted units cancel and the desired units remain.

7. How many inches are in a foot, feet in a yard, and yards in a mile?

   12 inches = 1 foot; 3 feet = 1 yard; 1,760 yards (5,280 feet) = 1 mile.

8. In the metric system, how do you convert kilometers to meters and meters to centimeters?

   1 kilometer = 1,000 meters; 1 meter = 100 centimeters. Multiply by 1,000 (km to m) or by 100 (m to cm).

9. How do you convert a speed of 60 miles per hour into miles per minute?

   Divide by 60 minutes per hour: 60 mi/hr * (1 hr / 60 min) = 1 mile per minute.

10. When converting a rate with units in both numerator and denominator (e.g., feet per second to miles per hour), what is the key strategy?

    Apply separate conversion factors for the numerator and denominator units, arranging each so the unwanted units cancel, leaving the target units in both positions.

11. What is a ratio, and how does a part-to-part ratio differ from a part-to-whole ratio?

    A ratio compares two quantities by division. Part-to-part compares one group to another (e.g., 3 boys : 2 girls); part-to-whole compares one group to the total (e.g., 3 boys : 5 students).

12. If a ratio of red to blue marbles is 3:5 and there are 40 marbles total, how many are red?

    Total parts = 3 + 5 = 8. Each part = 40/8 = 5 marbles. Red = 3 * 5 = 15 marbles.

13. How do you find the total quantity when given a ratio and the value of one part?

    Divide the known quantity by its number of ratio parts to get the value of one part, then multiply by the total number of parts in the ratio.

14. How do you scale a recipe or mixture given as a ratio (e.g., 2 cups flour to 3 cups sugar) to use 8 cups of flour?

    Find the scale factor: 8/2 = 4. Multiply every part by 4, so sugar = 3 * 4 = 12 cups, keeping the ratio constant.

15. What is the formula relating a part, a percent, and a whole?

    Part = (percent/100) * whole, equivalently percent = (part/whole) * 100. The 'is' value is the part and the 'of' value is the whole.

16. How do you convert a percent to a decimal and a decimal to a percent?

    To a decimal, divide the percent by 100 (move the decimal two places left). To a percent, multiply the decimal by 100 (move two places right).

17. What is 35% of 80?

    0.35 * 80 = 28.

18. 15 is what percent of 60?

    (15/60) * 100 = 25%.

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