Study guide
Problem-Solving and Data Analysis and Geometry and Trigonometry each contribute roughly 15 percent of the digital SAT's 44 Math questions, so together they cover about a third of the section. The first domain tests proportional reasoning, percentages, statistics, and data interpretation; the second tests area, volume, circles, angles, and right-triangle trigonometry. These questions lean on a small toolkit of formulas and definitions, several of which are handed to you on the test's built-in reference sheet.
Ratios, Rates, and Proportions
A ratio compares quantities, a rate is a ratio with different units such as miles per hour, and a proportion is an equation stating two ratios are equal. Most questions in this family are solved by setting up a proportion with units aligned and cross-multiplying. If a recipe uses 3 cups of flour for every 2 cups of sugar and a baker named Elena scales up to 12 cups of flour, the proportion 3 over 2 equals 12 over x gives x equals 8 cups of sugar. The classic trap is confusing part-to-part with part-to-whole. A ratio of 3 flour to 2 sugar means flour is 3 fifths of the 5-part mixture, not 3 halves of it; when a question asks what fraction of the total something represents, add the ratio parts to build the whole first. Unit conversion problems reward writing units explicitly and multiplying by conversion fractions so unwanted units cancel: converting 90 kilometers per hour to meters per second means multiplying by 1000 meters per kilometer and by 1 hour per 3600 seconds, giving 25 meters per second. Density, unit price, and speed problems are all the same skeleton, a rate times a quantity, so identify which two of the three values you have and solve for the third. Watch scale problems too: if a map scale says 1 inch represents 40 miles, distances scale linearly, but if a question involves area on the map, the areas scale by the square of the factor. Keeping units visible at every step converts most of these from word puzzles into arithmetic.
Percentage Traps
Percent questions are less about computation than about reading precisely which quantity is the base, because percent always means per hundred of something. Percent change equals the change divided by the original value, times 100, and the word original is where students bleed points. If a jacket priced at 80 dollars rises to 100 dollars, that is a 25 percent increase, but a drop from 100 back to 80 is only a 20 percent decrease, because the base changed. This asymmetry powers the most common trap: successive percent changes never simply add. A 20 percent increase followed by a 20 percent decrease multiplies by 1.2 and then 0.8, giving 0.96, an overall 4 percent decrease, not a return to the start. Translate every percent change into a single multiplier: increase of r percent means multiply by 1 plus r over 100, decrease means multiply by 1 minus r over 100, then chain the multipliers. Working backward is the other tested skill: if a 15 percent discount produced a sale price of 51 dollars, the original satisfies 0.85x equals 51, so x is 60; students who instead add 15 percent of 51 get the trap answer. Also be alert to percent of a percent, as when 30 percent of a school's students play sports and 10 percent of those play tennis, meaning 3 percent of all students play tennis. Finally, distinguish percentage points from percent: a rise from 10 percent to 15 percent is 5 percentage points but a 50 percent relative increase, and answer choices will offer both numbers.
Statistics, Probability, and Margin of Error
The mean is the sum divided by the count; the median is the middle value of an ordered list. The test's favorite question is how outliers affect them: an extreme value drags the mean toward itself but usually leaves the median alone, so when a billionaire moves into a small town, the mean income jumps while the median barely moves. Expect questions asking which is larger for a skewed data set, or what happens to each measure when the largest value is removed. Standard deviation is tested conceptually, not computationally: it measures spread, so data bunched near the mean has a smaller standard deviation than data spread widely, and adding the same constant to every value shifts the mean but leaves the spread unchanged. Probability questions usually arrive as two-way tables, grids classifying people or things by two variables, and the entire skill is choosing the right denominator. Given a table of 200 surveyed commuters, the probability that a randomly chosen commuter bikes is the bike total over 200, but the probability that a commuter bikes given that they live downtown restricts the universe to the downtown row, so both numerator and denominator come from that row alone. The phrase given that, or a stem like of those who, is the signal to shrink the denominator. Margin of error questions test meaning, not calculation: a survey estimating a mean of 42 with a margin of error of 3 indicates the population value is plausibly between 39 and 45. The margin reflects sampling variability, not mistakes, and larger random samples produce smaller margins of error. Remember also that results generalize only to the population actually sampled, and randomization is what makes that generalization valid.
Area, Volume, Circles, and Angle Relationships
The digital SAT provides a reference sheet with area and volume formulas, the Pythagorean theorem, and special right triangle relationships, so the tested skill is deployment, not memorization. Area problems often involve composite shapes: split an L-shaped floor plan into rectangles, or subtract a circular cutout from a square. Volume problems multiply a base area by a height for prisms and cylinders, and use the sheet's formulas for cones, pyramids, and spheres; a favorite twist is changing a dimension, since doubling a cylinder's radius quadruples its volume while doubling its height merely doubles it. For circles, know circumference 2 pi r and area pi r squared, and the proportionality principle for pieces: an arc's length and a sector's area are each the same fraction of the whole circle that the central angle is of 360 degrees, so a 90-degree sector holds one quarter of the area. A circle in the coordinate plane has equation (x minus h) squared plus (y minus k) squared equals r squared, with center (h, k) and radius r; if the equation arrives expanded, complete the square to recover them. Angle relationships supply fast points. When parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary, summing to 180 degrees, so every angle in the figure is either equal or supplementary to a given angle. Add the perennials: angles in a triangle sum to 180 degrees, vertical angles are equal, an exterior angle of a triangle equals the sum of the two remote interior angles, and angles on a straight line sum to 180 degrees.
Right-Triangle Trigonometry and Special Triangles
Right-triangle trig on the SAT is the three ratios summarized by SOH-CAH-TOA: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent, where opposite and adjacent are named relative to the angle in question. Given a right triangle where angle A has sine 3 fifths, you can sketch legs 3 and 4 with hypotenuse 5, then read off any other ratio, such as cosine 4 fifths or tangent 3 fourths. That works because of Pythagorean triples, integer side sets like 3-4-5, 5-12-13, and their multiples such as 6-8-10, which the test uses constantly; recognizing them saves you the Pythagorean theorem. Two special triangles must be automatic, and both appear on the reference sheet. A 45-45-90 triangle, half a square, has sides in ratio x, x, x times root 2, with the hypotenuse carrying the root 2. A 30-60-90 triangle, half an equilateral triangle, has sides x, x root 3, 2x, where the shortest side faces the 30-degree angle, the middle side with root 3 faces 60, and the hypotenuse is twice the shortest side. These unlock diagonal-of-a-square and height-of-an-equilateral-triangle questions instantly. Similar triangles, triangles with equal angles, have proportional sides and therefore identical trig ratios, which is why sine of 30 degrees is one half in every triangle regardless of size, and why the SAT can give you a small triangle and ask about a scaled-up one. Finally, know the cofunction fact the test loves: sine of x degrees equals cosine of 90 minus x degrees, because an angle's opposite side is the complementary angle's adjacent side. If a question says sine of a equals cosine of b for acute angles, conclude that a plus b equals 90.
Key terms
- Proportion
- — An equation stating that two ratios are equal, typically solved by cross-multiplication.
- Part-to-whole ratio
- — A comparison of one component to the total, found by adding all ratio parts to build the whole.
- Percent change
- — The change in a quantity divided by its original value, times 100.
- Multiplier
- — The single factor representing a percent change, such as 1.2 for a 20 percent increase or 0.8 for a 20 percent decrease.
- Median
- — The middle value of an ordered data set, largely resistant to outliers.
- Outlier
- — An extreme value that pulls the mean toward itself while leaving the median mostly unchanged.
- Two-way table
- — A grid classifying items by two variables at once, used for probability questions.
- Conditional probability
- — The probability of an event given that another condition holds, computed with a restricted denominator.
- Margin of error
- — A measure of sampling uncertainty giving the plausible range around a survey estimate, not a measure of mistakes.
- Sector
- — A pie-slice region of a circle whose area is proportional to its central angle over 360 degrees.
- Transversal
- — A line crossing two parallel lines, creating equal corresponding and alternate interior angles.
- Pythagorean triple
- — An integer side set for a right triangle, such as 3-4-5 or 5-12-13, that avoids radical arithmetic.
Exam tips
- When you see given that or of those who in a probability question, restrict the denominator to the stated row or column of the two-way table; using the grand total is the trap.
- Convert every percent change to a multiplier and chain them; successive percent changes multiply, so a 20 percent increase followed by a 20 percent decrease is 0.96, not a wash.
- For skewed data or outlier questions, remember the mean chases the outlier while the median stays put; you can often answer without computing anything.
- The reference sheet in the testing app has the area, volume, and special-triangle formulas; open it rather than trusting memory, and save memorization for what it lacks, like arc-sector proportionality and the circle equation.
- If sine of one acute angle equals cosine of another, the angles sum to 90 degrees; spotting this cofunction identity turns a scary-looking trig question into one line of algebra.