ARArithmetic Reasoning

Arithmetic Fundamentals

The four basic operations — add, subtract, multiply, divide — are the foundation of every AR question, and the order you do them in determines whether you get it right.

Formula Reference

  • Order of operations (PEMDAS): Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right)
  • Even × anything = even; odd × odd = odd
  • Negative × negative = positive; negative × positive = negative
  • Divisibility by 3: sum of digits divisible by 3 (e.g. 123 → 1+2+3 = 6 ✓)
  • Divisibility by 9: sum of digits divisible by 9
  • Remainder: a ÷ b = quotient remainder r means a = b × quotient + r

What the ASVAB is actually testing

Arithmetic Reasoning isn't a computation marathon — it's a reasoning test. Almost every AR question is a word problem that requires you to decide which operation to use and in what order. Arithmetic fundamentals are the engine under the hood. If your order of operations is sloppy, or you can't spot a remainder situation, you'll miss questions that have nothing to do with the hard math.

Order of operations — the one rule to own

PEMDAS. Parentheses, Exponents, Multiply/Divide, Add/Subtract. That order is non-negotiable. The ASVAB regularly tests whether recruits skip steps or go left to right blindly.

The trap: 5 + 3 × 4 is not 32. Multiply before you add: 3 × 4 = 12, then 5 + 12 = 17. If you get 32, you went left to right without PEMDAS.

Divisibility rules that save time

On a timed test, you don't always have room for long division. Know these cold:

  • Divisible by 2: last digit is even
  • Divisible by 3: digits sum to a multiple of 3
  • Divisible by 4: last two digits form a number divisible by 4
  • Divisible by 5: ends in 0 or 5
  • Divisible by 9: digits sum to a multiple of 9

These rules let you eliminate wrong answer choices without doing any division at all.

Remainders and real-world framing

The ASVAB loves remainder questions disguised as distribution problems: "X items split equally among Y people — how many are left over?" Set it up as division, find the quotient, multiply back, and subtract. The difference is your remainder.

Connection to other AR topics

Get arithmetic fundamentals right and every other AR topic gets easier. Ratios, percents, rate problems — they all reduce to these four operations applied in the right sequence. Weak PEMDAS skills create downstream errors even when the higher-level concept is correct.

Common Pitfalls

  • Skipping parentheses and doing multiplication before resolving them
  • Assuming the result of dividing two negatives is negative — it's positive
  • Confusing 'factor' (divides evenly into) with 'multiple' (divisible by the number)
  • Forgetting that zero is even, and that dividing by zero is undefined
  • Misreading a remainder question as asking for the quotient

Worked Examples

Q1: Evaluate: 3 + 4 × 2 − (8 ÷ 4)

Answer: Parentheses first: 8 ÷ 4 = 2. Multiply next: 4 × 2 = 8. Then left to right: 3 + 8 − 2 = 9

Q2: A sergeant divides 85 rounds equally among 6 soldiers. How many rounds are left over?

Answer: 85 ÷ 6 = 14 remainder 1 (because 6 × 14 = 84, and 85 − 84 = 1). Answer: 1 round

Q3: Is 432 divisible by 9?

Answer: Add the digits: 4 + 3 + 2 = 9. Yes, 9 is divisible by 9, so 432 is divisible by 9.

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