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

Quantitative and Mathematical Aptitude syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative and Mathematical Aptitude in Officer Candidate / Aptitude tests (AFOQT), not a summary of it.

1. 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. 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. 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

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

20 of 50 cards from the Quantitative and Mathematical Aptitude deck — real questions with worked 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).

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