Quantitative Reasoning study notes
Quantitative Reasoning for the NMAT
Key points
- A percent increase followed by an equal percent decrease (or vice versa) never returns to the original value.
- Reduce fractions to lowest terms using the greatest common divisor; find a common denominator before adding or subtracting.
- Solve equations by performing the same operation on both sides, one step at a time.
- Dividing or multiplying an inequality by a negative number reverses the inequality sign.
- Memorize the Pythagorean theorem and the fixed side ratios of 45-45-90 and 30-60-90 triangles.
- A cone's volume is exactly one-third of a cylinder's volume when they share the same radius and height.
- Mean, median, and mode measure center differently; know when each is appropriate and how to compute it.
- Permutations count arrangements (order matters); combinations count selections (order does not matter).
- The 68-95-99.7 rule describes how data spread around the mean in a normal distribution.
- Correlation shows association, not causation.
- Translate word problems into equations sentence by sentence before solving.
- Combined work rate equals the sum of individual rates expressed as jobs per hour.
Arithmetic and Number Sense: Building the Foundation
Nearly every quantitative item ultimately rests on solid arithmetic: fractions, decimals, percentages, ratios, and the order of operations. Percent problems are the most common trap—remember that a percent increase followed by an equal percent decrease does not return to the original value, because the second percentage is taken on a new base. When simplifying fractions, always reduce to lowest terms by dividing numerator and denominator by their greatest common divisor, and when adding or subtracting fractions, find a common denominator first. Scientific notation questions test whether you can express very large or very small numbers as a mantissa between 1 and 10 multiplied by a power of ten; watch for distractor choices that are numerically equal but improperly formatted. Estimation is a powerful shortcut on the actual exam: round values to check whether an answer choice is even in the right range before committing to full computation. Divisibility rules (for 2, 3, 5, 9, and 10) and prime factorization speed up GCD and LCM problems considerably, since both reduce to comparing shared and combined prime factors.
Algebra: Solving for the Unknown
Algebra items ask you to isolate a variable, manipulate expressions, or interpret a function. The golden rule for solving equations is to perform the same operation on both sides to keep the equation balanced, working step by step rather than combining moves mentally. For quadratics, try factoring first since it is fastest when it works cleanly; otherwise fall back on the quadratic formula, x = (−b ± √(b² − 4ac)) / (2a). Systems of two linear equations are usually fastest by elimination when coefficients align nicely, or substitution otherwise. Exponent rules—adding exponents when multiplying like bases, subtracting when dividing, and multiplying when raising a power to a power—appear constantly, as do their extensions to zero and negative exponents. When solving inequalities, remember the critical rule: multiplying or dividing both sides by a negative number reverses the inequality sign. For sequences, memorize the arithmetic sum formula Sₙ = n/2 × [2a₁ + (n − 1)d] and the geometric sum formula Sₙ = a₁(rⁿ − 1)/(r − 1); both convert a long addition into one calculation.
Geometry and Measurement: Working with Shapes
Geometry items reward memorized formulas applied carefully to a described or sketched figure. The Pythagorean theorem (a² + b² = c²) unlocks most right-triangle problems, and the special right triangles—45-45-90 and 30-60-90—have fixed side ratios worth memorizing so you can skip the theorem entirely. For area and volume, keep a mental formula sheet: rectangle (length × width), triangle (½ × base × height), trapezoid (½ × sum of parallel sides × height), circle (πr² for area, 2πr for circumference), cylinder (πr²h), and cone (⅓πr²h, exactly one-third of a cylinder with the same base and height). Similar triangles and polygons share proportional corresponding sides, so setting up a ratio between known and unknown sides solves most similarity problems. Coordinate geometry questions rely on the distance formula, the midpoint formula, and the slope formula (rise over run) to write a line's equation or identify a transformation such as a reflection, rotation, or translation of a point.
Data Interpretation and Statistics: Reading the Numbers
Data items test whether you can extract a number from a table, chart, or description and then apply the correct statistical or probabilistic formula. Know the differences between mean (sum divided by count), median (the middle value when ordered, or the average of the two middle values for an even-sized set), and mode (the most frequent value). For probability, remember that the probability of an event is favorable outcomes divided by total outcomes, and that for independent events, you multiply individual probabilities together. Permutations (nPr, order matters) and combinations (nCr, order does not matter) are frequently confused—read the question carefully to see whether arrangement order is relevant. The 68-95-99.7 rule describes how data cluster around the mean in a normal distribution: roughly 68% of values lie within one standard deviation, 95% within two, and 99.7% within three. Finally, remember that a strong correlation coefficient shows a linear association between two variables but never proves that one variable causes the other.
Word Problems: Translating Language into Math
Word problems are really algebra and arithmetic wearing a disguise; the real skill being tested is translation. Read the problem once for overall meaning, then reread it sentence by sentence, assigning a variable to the unknown quantity and writing an equation for each stated relationship. Rate problems (distance = rate × time) come in two common flavors: objects moving toward each other (add the rates to find how fast the gap closes) and objects moving apart (also add the rates, but for separation instead of closing). Work-rate problems combine individual rates as fractions of a job per hour, so two workers' combined rate is the sum of 1/(time for worker A) and 1/(time for worker B). Percent-based cost problems—discounts, markups, simple interest—are best solved by writing the relationship as an equation first (e.g., original price × (1 − discount rate) = sale price) rather than guessing which direction to divide. Mixture and ratio problems divide a total into proportional parts, matching the given ratio's total number of parts to the actual quantity.
Test-Taking Strategy for Quantitative Reasoning
Because most items are calculator-free and designed to be solved in under three minutes, efficiency matters as much as accuracy. Start by identifying what type of problem you are looking at and which formula or method applies, rather than diving into computation immediately. Write out each step of your work, even for problems that seem simple, since careless arithmetic errors are the most common cause of missed points, not conceptual misunderstanding. When four answer choices are given, use them strategically: plugging a choice back into the original equation or scenario ('backsolving') is often faster than solving algebraically, especially for equations with messy coefficients. Estimate first when a problem involves large numbers or percentages, then eliminate choices that are clearly out of range before doing exact computation. Finally, watch your units throughout multi-step problems (minutes versus hours, dollars versus cents) since unit-conversion slips are a frequent source of wrong answers even when the underlying method is correct.
Common Traps and How to Avoid Them
Test writers build distractor choices around predictable errors, so recognizing these patterns helps you both avoid them and reason about which choice an error-prone solver would pick. Common traps include: forgetting to flip an inequality sign when dividing by a negative number; treating percent changes as additive when they should be multiplicative (a 20% increase then a 20% decrease is not a net zero change); confusing permutations with combinations; using the wrong base for a percentage question (percent of what, exactly?); and mixing up which quantity a ratio's parts refer to. In geometry, a frequent trap is applying the wrong formula to a solid that only resembles a more familiar shape, or forgetting that only some quantities (like radius) scale linearly while others (like area or volume) scale by the square or cube of the scale factor. Slow down at the final step of any problem, restate what the question is actually asking, and double-check your final answer matches the requested unit or form before selecting a choice.
References
- Prealgebra 2e – Chapter 1: Whole Numbers — OpenStax
- Elementary Algebra 2e – Chapter 2: Language of Algebra — OpenStax
- Khan Academy – Fractions — Khan Academy
- Elementary Algebra 2e – Chapter 4: Linear Equations & Inequalities — OpenStax
- College Algebra – Chapter 2: Functions — OpenStax
- Khan Academy – Algebra 1 — Khan Academy
- Khan Academy – Geometry — Khan Academy
- Khan Academy – Statistics & Probability — Khan Academy
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