PCAT (Pharmacy College Admission Test) Quantitative Reasoning Syllabus

Quantitative Reasoning syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative Reasoning in PCAT (Pharmacy College Admission Test), not a summary of it.

1. 1

   Basic Mathematics and Arithmetic

   3 topics

   • Number Operations
     • Fractions, decimals, and percentages
     • Ratios and proportions
     • Order of operations and exponents
   • Estimation and Number Sense
     • Rounding and significant figures
     • Scientific notation
   • Applied Problem Solving
     • Rate, time, and distance problems
     • Mixture and concentration problems
     • Unit conversions and dimensional analysis
2. 2

   Algebra

   3 topics

   • Linear Equations and Inequalities
     • Solving single-variable equations
     • Systems of equations
     • Graphing inequalities
   • Functions and Polynomials
     • Function notation and evaluation
     • Factoring polynomials
     • Quadratic equations and the quadratic formula
   • Exponential and Logarithmic Functions
     • Properties of logarithms
     • Exponential growth and decay
3. 3

   Probability and Statistics

   3 topics

   • Descriptive Statistics
     • Mean, median, mode, and range
     • Standard deviation and variance
     • Data interpretation and graphs
   • Probability
     • Basic probability rules
     • Independent and dependent events
     • Permutations and combinations
   • Distributions and Inference
     • Normal distribution and z-scores
     • Sampling and confidence basics
4. 4

   Precalculus and Calculus

   3 topics

   • Trigonometry and Precalculus
     • Trigonometric functions and identities
     • Right triangle relationships
     • Conic sections
   • Limits and Derivatives
     • Concept of limits and continuity
     • Differentiation rules
     • Applications of derivatives
   • Integration
     • Antiderivatives and indefinite integrals
     • Definite integrals and area under curves

Quantitative Reasoning flashcards for PCAT (Pharmacy College Admission Test)

25 of 51 cards from the Quantitative Reasoning deck — real questions with worked answers.

1. What is the order of operations (PEMDAS), and in what sequence are operations evaluated?

   PEMDAS: 1) Parentheses (grouping), 2) Exponents/roots, 3) Multiplication and Division (left to right), 4) Addition and Subtraction (left to right). For example, $2 + 3 \times 4^{2} = 2 + 3 \times 16 = 50$.

2. State the rules for the sign of a product or quotient of two real numbers.

   Same signs give a positive result; opposite signs give a negative result. E.g. $(-)\times(-) = (+)$, $(+)\times(-) = (-)$, and the same holds for division.

3. What is the distributive property of multiplication over addition?

   $a(b + c) = ab + ac$. Multiplication distributes across each term inside the parentheses.

4. How do you add two fractions $\frac{a}{b} + \frac{c}{d}$?

   Find a common denominator (the LCD), rewrite each fraction, then add numerators: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}.$$

5. To divide by a fraction, what operation do you perform?

   Multiply by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}.$$

6. How do you convert a percentage to a decimal and a decimal to a percentage?

   To get a decimal, divide the percent by 100 (move the decimal point two places left): $45\% = 0.45$. To get a percent, multiply the decimal by 100: $0.07 = 7\%$.

7. What is the formula for percent change between an old and new value?

   $$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%.$$ A positive result is an increase; a negative result is a decrease.

8. In estimation, what does rounding to a given place value involve?

   Look at the digit immediately to the right of the target place. If it is $\geq 5$, round the target digit up; if it is $< 5$, leave it unchanged. All digits to the right become zero (or are dropped after a decimal point).

9. What are significant figures and which digits count as significant?

   Significant figures are the digits that carry meaningful precision. All nonzero digits count; zeros between nonzero digits count; trailing zeros after a decimal point count; leading zeros do not count. E.g. $0.00420$ has 3 significant figures.

10. What is the standard form of scientific notation, and how is a number expressed in it?

    $a \times 10^{n}$ where $1 \leq |a| < 10$ and $n$ is an integer. For example, $53{,}000 = 5.3 \times 10^{4}$ and $0.0026 = 2.6 \times 10^{-3}$.

11. How do you multiply numbers in scientific notation, e.g. $(a \times 10^{m})(b \times 10^{n})$?

    Multiply the coefficients and add the exponents: $(a \times 10^{m})(b \times 10^{n}) = (a \cdot b) \times 10^{m+n}$, then adjust so the coefficient lies in $[1,10)$.

12. What is the formula relating distance, rate, and time?

    $d = r \cdot t$, where $d$ is distance, $r$ is rate (speed), and $t$ is time. Rearranged: $r = \frac{d}{t}$ and $t = \frac{d}{r}$.

13. How do you solve a proportion $\frac{a}{b} = \frac{c}{d}$ for an unknown?

    Cross-multiply to get $ad = bc$, then solve for the unknown variable.

14. What is the dimensional-analysis (unit conversion) strategy for changing units?

    Multiply by conversion factors written as fractions equal to 1, arranged so the unwanted units cancel and the desired units remain. E.g. $5\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 5000\,\text{m}$.

15. For a dilution problem, what equation relates concentration and volume?

    $C_1 V_1 = C_2 V_2$, where $C$ is concentration and $V$ is volume before (1) and after (2) dilution. Solve for the unknown quantity.

16. What is the slope-intercept form of a linear equation, and what does each symbol mean?

    $y = mx + b$, where $m$ is the slope (rise over run) and $b$ is the $y$-intercept (the value of $y$ when $x = 0$).

17. What is the formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$?

    $$m = \frac{y_2 - y_1}{x_2 - x_1}.$$ It measures the change in $y$ per unit change in $x$.

18. How does multiplying or dividing an inequality by a negative number affect it?

    It reverses the direction of the inequality sign. E.g. if $-2x < 6$, dividing by $-2$ gives $x > -3$.

19. What does the absolute-value inequality $|x| < a$ (with $a > 0$) translate to?

    A compound inequality: $-a < x < a$. By contrast, $|x| > a$ translates to $x < -a$ or $x > a$.

20. What is the point-slope form of a line?

    $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a known point on the line.

21. How are the slopes of parallel and perpendicular lines related?

    Parallel lines have equal slopes ($m_1 = m_2$). Perpendicular lines have slopes that are negative reciprocals ($m_1 \cdot m_2 = -1$).

22. What defines a function in terms of inputs and outputs?

    A function assigns exactly one output to each input. Equivalently, it passes the vertical line test: no vertical line crosses the graph more than once.

23. What is the quadratic formula for solving $ax^{2} + bx + c = 0$?

    $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}.$$

24. What does the discriminant $b^{2} - 4ac$ tell you about a quadratic's roots?

    If $b^{2} - 4ac > 0$: two distinct real roots; if $= 0$: one repeated real root; if $< 0$: two complex conjugate roots (no real roots).

25. What are the coordinates of the vertex of the parabola $y = ax^{2} + bx + c$?

    The vertex is at $x = -\frac{b}{2a}$, with $y$ found by substituting that $x$ back in. The parabola opens upward if $a > 0$ and downward if $a < 0$.

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