Why this category rewards calm
Problem-Solving and Data Analysis is one of the easier Math domains to raise quickly, because almost every question fits one of about ten patterns. The traps are usually about reading carefully, not about doing harder math.
Percentages: the only formulas you need
- Percent of: x% of y = (x / 100) · y.
- Percent change: (new - old) / old × 100.
- Sequential change: multiply, don't add. A 25% increase followed by a 20% decrease is 1.25 × 0.80, not 1.05.
- Reversing a percentage: if a price rose 20% to $60, the original was 60 / 1.20, not 60 × 0.80.
If you're ever uncertain, plug in 100 and let multiplication do the work.
Ratios, rates, proportions
A ratio is just a fraction in disguise. 3:5 means "for every 3 of these, there are 5 of those." Two reliable moves:
- Set up a proportion: a/b = c/d. Cross-multiply: ad = bc.
- Convert to a unit rate. "12 dollars for 3 pounds" is "$4 per pound." Once it's a unit rate, almost any question becomes a single multiplication.
Ratio question vocabulary to know:
- "For every 4 boys there are 5 girls": ratio 4:5. Total parts = 9.
- "Of 90 students, the ratio of boys to girls is 4:5": 90 ÷ 9 parts = 10 per part. So 40 boys, 50 girls.
Unit conversions: the dimensional analysis trick
Multiply by a fraction equal to 1, with the unit you want on top and the unit you have on bottom. Want feet? Have inches? Multiply by (1 foot / 12 inches). The "inches" cancel. Stack as many of these fractions as you need; the units cancel like algebra.
Statistics: mean, median, mode, range, standard deviation
Plain definitions:
- Mean = average = sum / count.
- Median = middle value when sorted (or average of two middles for even counts).
- Mode = most common value.
- Range = max - min.
- Standard deviation = how spread out the values are. You don't need to compute it; just know that tighter data has lower SD.
Two SAT favorites:
- Outlier effects. An extreme value pulls the mean far more than the median. If a question asks "which would change more, the mean or the median?", outliers move the mean.
- Same mean, different SD. Two data sets can share a mean but differ in spread. "Concentrated near the mean" means lower SD.
Two-variable data: scatter plots and lines of best fit
Three common SAT moves:
- Read a value from the line of best fit. Use the equation, not your eye, when one is given.
- Identify the strongest correlation. Tighter cluster around the line = stronger correlation. Don't confuse correlation with causation.
- Predict outside the data. The SAT will sometimes warn that extrapolation is unreliable. They're testing whether you respect the data range.
Probability: count, divide, done
Probability = (favorable outcomes) / (total outcomes). For combined events:
- And (independent events): multiply. P(red then red, with replacement) = P(red) · P(red).
- Or (mutually exclusive): add. P(red or blue) = P(red) + P(blue).
- Not: 1 - P(event).
Two-way table questions are very common. The SAT will give you a 2x3 or 3x3 table of counts and ask for a conditional probability. "Given that a student is a senior, what's the probability they take art?" Look only at the senior row. Don't divide by the grand total.
Inference from a sample
If a survey of 200 randomly selected voters shows 55% support a measure, the SAT may ask whether you can generalize to the population. The keys:
- Random sample? Generalize to the population the sample was drawn from.
- Not random? Don't generalize. The SAT loves catching students who ignore selection bias.
- Margin of error. A bigger sample gives a smaller margin. The SAT won't ask you to compute it but will ask which sample is more reliable.
Evaluating statistical claims
Watch for the difference between correlation and causation. The SAT will dangle a tempting causal claim ("the new program caused better outcomes") when only an observational study was conducted. Without random assignment to the program, you can't conclude causation. Pick the more cautious answer.
A worked two-way example
A 2x2 table:
Walks to school: 30 freshmen, 50 seniors
Drives to school: 70 freshmen, 50 seniors
Question: Among students who walk, what fraction are seniors?
Walkers total: 30 + 50 = 80. Seniors among walkers: 50. So 50/80 = 5/8.
The trap answer would be 50/100 (seniors out of all seniors), or 50/200 (seniors out of all students). Read the conditional carefully: "Among students who walk" sets your denominator.
Practice plan
- For percents and ratios, plug in 100 by default. The arithmetic becomes effortless.
- Drill two-way table questions until conditional probability feels reflexive.
- For statistical inference questions, slow down and check: random sample? Causal claim or correlation?