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Officer Candidate / Aptitude tests (AFOQT) Quantitative and Mathematical Aptitude Syllabus

The Quantitative and Mathematical Aptitude syllabus for Officer Candidate / Aptitude tests (AFOQT) in full: 3 chapters, 11 topics and 26 sub-topics, every one listed below, plus 50 Quantitative and Mathematical Aptitude flashcards with answers.

3Chapters
11Topics
26Sub-topics
50Flashcards

Quantitative and Mathematical Aptitude syllabus: all 3 chapters

Expand any chapter to see its topics and sub-topics.

  1. Arithmetic Reasoning

     4 topics
    • Word Problem Translation
      • Identifying the unknown and setting up the equation
      • Multi-step problem sequencing
    • Ratios, Proportions, and Rates
      • Direct and inverse proportion
      • Distance-rate-time problems
      • Unit conversion and scaling
    • Percentages and Financial Math
      • Percent increase and decrease
      • Interest, discount, and markup
    • Averages and Mixtures
      • Weighted averages
      • Mixture and concentration problems

  2. Math Knowledge

     4 topics
    • Algebra Fundamentals
      • Solving linear and quadratic equations
      • Exponent and radical rules
      • Factoring and simplifying expressions
    • Geometry
      • Area, perimeter, and volume formulas
      • Properties of triangles and the Pythagorean theorem
      • Angles, lines, and circles
    • Number Properties
      • Factors, multiples, primes
      • Order of operations
      • Absolute value and inequalities
    • Coordinate and Basic Trigonometry
      • Slope and equations of lines
      • Basic trigonometric ratios

  3. Data Interpretation and Problem-Solving Strategy

     3 topics
    • Reading Tables, Charts, and Graphs
      • Extracting values from bar, line, and pie representations
      • Comparing data across categories
    • Estimation and Mental Math
      • Rounding for fast elimination
      • Back-solving from answer choices
    • Timed Test Tactics
      • Triage of easy versus time-consuming items
      • Avoiding calculation traps without a calculator

Officer Candidate / Aptitude tests (AFOQT) Quantitative and Mathematical Aptitude flashcards — 50 questions with answers

  1. In word problems, what does the word "of" most commonly translate to mathematically?

    Multiplication (e.g., "25% of 80" means 0.25 × 80).

  2. When translating word problems, what operations do the phrases "sum/total/increased by" and "difference/fewer than/decreased by" indicate?

    "Sum, total, increased by" → addition; "difference, fewer than, decreased by" → subtraction.

  3. How do you translate "is," "are," "was," or "will be" in a word problem?

    They translate to an equals sign (=).

  4. In the phrase "7 less than a number x," what is the correct expression and common mistake?

    Correct: x − 7. Common mistake is writing 7 − x; the "less than" reverses the order.

  5. What is the basic distance-rate-time relationship and its two rearrangements?

    Distance = Rate × Time; Rate = Distance ÷ Time; Time = Distance ÷ Rate.

  6. What is a ratio, and what does the proportion a/b = c/d let you do?

    A ratio compares two quantities (a to b). A proportion sets two ratios equal; you solve it by cross-multiplying: a×d = b×c.

  7. If a recipe uses ingredients in the ratio 2:3:5, what fraction of the total does the largest part represent?

    5/(2+3+5) = 5/10 = 1/2 of the total.

  8. How do you find a unit rate, and why is it useful on aptitude tests?

    Divide the quantity by the number of units (e.g., miles ÷ hours). It standardizes comparisons (cost per item, speed per hour).

  9. Two workers' combined work rate problem: if A finishes in 4 hr and B in 6 hr, how do you find the time together?

    Add rates: 1/4 + 1/6 = 5/12 job/hr, so time = 12/5 = 2.4 hours.

  10. In direct vs. inverse proportion, how do the variables behave?

    Direct: as one increases, the other increases (y = kx). Inverse: as one increases, the other decreases (xy = k).

  11. What is the formula to convert a fraction or decimal to a percent, and a percent to a decimal?

    Decimal → percent: multiply by 100. Percent → decimal: divide by 100 (move decimal two places left).

  12. How do you calculate percent change?

    Percent change = (New − Old) ÷ Old × 100. Positive = increase, negative = decrease.

  13. What is the simple interest formula?

    I = P × r × t, where P = principal, r = annual rate (decimal), t = time in years.

  14. What is the compound interest formula for amount A?

    A = P(1 + r/n)^(nt), where n = compounding periods per year, t = years.

  15. If a price rises 20% then falls 20%, is the final price the same as the original?

    No. It is lower: 1.20 × 0.80 = 0.96, a net 4% decrease.

  16. How do you find the original price before a percentage discount?

    Original = Sale price ÷ (1 − discount rate). E.g., $80 after 20% off = 80 ÷ 0.80 = $100.

  17. What is the formula for the arithmetic mean (average)?

    Average = (sum of all values) ÷ (number of values).

  18. How do you find the sum of a data set when you know the average and the count?

    Sum = Average × Count.

  19. Define median and mode.

    Median = the middle value when data is ordered (average of two middle values if even count). Mode = the most frequently occurring value.

  20. In a weighted average, how does it differ from a simple average?

    Each value is multiplied by its weight, the products are summed, then divided by the total of the weights (not the count).

Show the other 30 Quantitative and Mathematical Aptitude flashcards

  1. In a mixture problem, what equation governs mixing two solutions of different concentrations?

    (amount₁×conc₁) + (amount₂×conc₂) = (total amount × final concentration).

  2. If you add a value equal to the current average to a data set, what happens to the average?

    It stays the same; adding the mean does not change the mean.

  3. What is the distributive property?

    a(b + c) = ab + ac; a factor outside parentheses multiplies each term inside.

  4. List the order of operations (PEMDAS).

    Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

  5. What are the product and quotient rules for exponents?

    Product: x^a × x^b = x^(a+b). Quotient: x^a ÷ x^b = x^(a−b).

  6. What do a negative exponent and a zero exponent mean?

    x^(−n) = 1/x^n; x^0 = 1 (for x ≠ 0).

  7. What is the quadratic formula for ax² + bx + c = 0?

    x = [−b ± √(b² − 4ac)] ÷ (2a).

  8. Give the difference of squares and the FOIL expansion of (a+b)².

    a² − b² = (a+b)(a−b); (a+b)² = a² + 2ab + b².

  9. How do you solve a two-step linear equation like 3x + 5 = 20?

    Undo addition/subtraction first (subtract 5 → 3x = 15), then undo multiplication/division (divide by 3 → x = 5).

  10. What is the formula for the area and circumference of a circle?

    Area = πr²; Circumference = 2πr (or πd).

  11. State the Pythagorean theorem.

    In a right triangle, a² + b² = c², where c is the hypotenuse (side opposite the right angle).

  12. What is the sum of the interior angles of a triangle and of any quadrilateral?

    Triangle = 180°; quadrilateral = 360°.

  13. Give the area formulas for a triangle, rectangle, and trapezoid.

    Triangle = ½ × base × height; Rectangle = length × width; Trapezoid = ½(b₁ + b₂) × height.

  14. What is the formula for the volume of a rectangular box (prism) and a cylinder?

    Box = length × width × height; Cylinder = πr²h.

  15. What are the two special right-triangle side ratios?

    45-45-90: 1 : 1 : √2. 30-60-90: 1 : √3 : 2 (short leg : long leg : hypotenuse).

  16. What is the difference between prime and composite numbers?

    A prime has exactly two factors (1 and itself); a composite has more than two factors. 1 is neither.

  17. What are the divisibility rules for 3 and 9?

    Divisible by 3 if the digit sum is divisible by 3; divisible by 9 if the digit sum is divisible by 9.

  18. What is the difference between LCM and GCF?

    GCF (greatest common factor) is the largest number dividing all given numbers; LCM (least common multiple) is the smallest number all given numbers divide into.

  19. What rules govern the sign of a product or quotient with negative numbers?

    Same signs → positive result; different signs → negative result. An even count of negatives gives positive; odd gives negative.

  20. What is an integer, and how do even and odd combine under addition?

    Integers are whole numbers and their negatives (no fractions). Even ± even = even; odd ± odd = even; even ± odd = odd.

  21. What is the distance formula between points (x₁,y₁) and (x₂,y₂)?

    d = √[(x₂−x₁)² + (y₂−y₁)²].

  22. What is the slope formula and the slope-intercept form of a line?

    Slope m = (y₂−y₁)/(x₂−x₁); slope-intercept form is y = mx + b, where b is the y-intercept.

  23. What is the midpoint formula for two points?

    Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

  24. Give the basic SOH-CAH-TOA trig ratios.

    sin = opposite/hypotenuse; cos = adjacent/hypotenuse; tan = opposite/adjacent.

  25. How do slopes of parallel and perpendicular lines relate?

    Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (product = −1).

  26. When reading a chart or graph, what should you check first before interpreting any data point?

    The title, axis labels, units, and scale (including whether the axis starts at zero).

  27. On a pie chart, how do you convert a percentage slice into an actual quantity?

    Multiply the slice's percentage (as a decimal) by the chart's total value.

  28. What is the difference between a bar graph and a line graph's typical use?

    Bar graphs compare discrete categories; line graphs show trends or change in a value over time.

  29. What is a fast estimation strategy for multiplication problems on a timed test?

    Round each factor to a convenient number, multiply, then adjust; use it to eliminate answer choices that are far off.

  30. On a timed aptitude test where wrong answers aren't penalized, what is the best tactic for a question you can't solve?

    Make an educated guess (after eliminating choices) rather than leaving it blank, and don't waste time—move on.

Officer Candidate / Aptitude tests (AFOQT) Quantitative and Mathematical Aptitude FAQ

What is in the Officer Candidate / Aptitude tests (AFOQT) Quantitative and Mathematical Aptitude syllabus?

Officer Candidate / Aptitude tests (AFOQT) Quantitative and Mathematical Aptitude is split into 3 chapters: Arithmetic Reasoning, Math Knowledge and Data Interpretation and Problem-Solving Strategy. Together they contain 11 topics and 26 sub-topics.

How many chapters are in Officer Candidate / Aptitude tests (AFOQT) Quantitative and Mathematical Aptitude?

3 chapters, containing 11 topics and 26 sub-topics. The largest chapter is Arithmetic Reasoning, with 4 topics.

Are there flashcards for Officer Candidate / Aptitude tests (AFOQT) Quantitative and Mathematical Aptitude?

Yes — a 50-card Quantitative and Mathematical Aptitude deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.