Armed Services Vocational Aptitude Battery (ASVAB) Arithmetic Reasoning (AR) Syllabus

Arithmetic Reasoning (AR) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Arithmetic Reasoning (AR) in Armed Services Vocational Aptitude Battery (ASVAB), not a summary of it.

1. 1

   Operations with Numbers

   3 topics

   • Whole Number Operations
     • Order of operations (PEMDAS)
     • Properties of operations
     • Factors, multiples, and primes
   • Integers and Number Lines
     • Operations with negative numbers
     • Absolute value
   • Fractions, Decimals, and Percents
     • Converting between forms
     • Operations with fractions and decimals
     • Percent of a number and percent change
2. 2

   Ratios, Proportions, and Rates

   3 topics

   • Ratios and Proportions
     • Setting up and solving proportions
     • Scaling and equivalent ratios
   • Rate Problems
     • Distance, rate, and time
     • Work and production rates
     • Unit rates and conversions
   • Interest and Money Problems
     • Simple interest
     • Discounts, markups, and sales tax
     • Profit and loss
3. 3

   Applied Word Problems

   4 topics

   • Translating Words to Math
     • Identifying operations from keywords
     • Setting up equations from situations
   • Multi-Step Problem Solving
     • Breaking down complex problems
     • Checking reasonableness of answers
   • Average and Mixture Problems
     • Mean, weighted average
     • Mixture and concentration problems
   • Measurement and Estimation
     • Units of length, weight, volume, and time
     • Rounding and estimation strategies

Arithmetic Reasoning (AR) flashcards for Armed Services Vocational Aptitude Battery (ASVAB)

18 of 50 cards from the Arithmetic Reasoning (AR) deck — real questions with worked answers.

1. What is the order of operations (PEMDAS) used to evaluate arithmetic expressions?

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

2. In division, what are the names for the number being divided, the number you divide by, and the result?

   Dividend (divided), divisor (divides into it), and quotient (the result); any leftover is the remainder.

3. What is the commutative property of addition and multiplication?

   Order does not change the result: a + b = b + a and a x b = b x a.

4. What is the associative property of addition and multiplication?

   Grouping does not change the result: (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).

5. What is the distributive property?

   a x (b + c) = a x b + a x c. Multiplication distributes over addition or subtraction.

6. What is the result of multiplying or dividing any number by zero?

   Multiplying any number by 0 equals 0. Dividing 0 by a number equals 0, but dividing by 0 is undefined.

7. How do you determine the sign when multiplying or dividing two integers?

   Same signs give a positive result; different signs give a negative result. (- x - = +; - x + = -).

8. How do you add two integers with different signs?

   Subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value.

9. What does the absolute value of a number represent?

   Its distance from zero on the number line, always non-negative. |-7| = 7 and |7| = 7.

10. Subtracting a negative number is equivalent to what operation?

    Adding its positive: a - (-b) = a + b.

11. On a number line, how do values compare as you move right versus left?

    Values increase to the right and decrease to the left, so any number to the right is greater than one to its left (e.g., -2 > -5).

12. How do you convert a fraction to a decimal?

    Divide the numerator by the denominator (e.g., 3/4 = 3 / 4 = 0.75).

13. How do you convert a decimal to a percent, and a percent to a decimal?

    Multiply a decimal by 100 (move the point 2 places right) to get a percent; divide a percent by 100 (move 2 places left) to get a decimal.

14. How do you add or subtract fractions with different denominators?

    Find a common denominator, convert each fraction, then add or subtract the numerators while keeping the common denominator.

15. How do you multiply two fractions?

    Multiply the numerators together and the denominators together, then simplify (e.g., 2/3 x 4/5 = 8/15).

16. How do you divide one fraction by another?

    Multiply the first fraction by the reciprocal of the second (flip the divisor): a/b / c/d = a/b x d/c.

17. How do you convert an improper fraction to a mixed number?

    Divide the numerator by the denominator; the quotient is the whole number and the remainder over the original denominator is the fraction (e.g., 11/4 = 2 3/4).

18. What is the percent formula relating part, whole, and percent?

    Part = Percent (as a decimal) x Whole; equivalently Percent = Part / Whole x 100.

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