ASVAB Mathematics Knowledge — Statistics, Probability, and Data Interpretation

Perimeter of square = 4 × side = 4 × 9 = 36. Perimeter is the distance around a shape — for a square, four equal sides, so 4 × 9. The distractor 81 is the area (9 × 9), the most common mix-up; 18 is two sides (9 + 9), and 27 is three sides. Units confirm which you have: perimeter is in plain units (inches, feet), area in square units.

Distribute: 2x + 8 = 22. Subtract 8: 2x = 14. Divide by 2: x = 7. Either distribute first, as shown, or divide both sides by 2 straight away: x + 4 = 11, so x = 7 — the second route is quicker when the number outside the bracket divides the other side evenly. Check: 2(7 + 4) = 2 × 11 = 22 ✓. The distractor 9 comes from forgetting to multiply the 4 by 2 (2x + 4 = 22), 5 from a subtraction slip, and 6 from dividing 22 by 2 and then subtracting 5 instead of 4.

37 is prime — only divisible by 1 and 37. 21 = 3×7; 27 = 3×9; 51 = 3×17. A prime has exactly two factors, 1 and itself. To test an odd number, try dividing by the small primes (3, 5, 7) up to its square root: 37 is not divisible by 3 (3 + 7 = 10, not a multiple of 3), does not end in 0 or 5, and 7 × 5 = 35 and 7 × 6 = 42 miss it, so it is prime. The digit-sum test for 3 catches all three distractors quickly: 21 (2 + 1 = 3), 27 (2 + 7 = 9) and 51 (5 + 1 = 6) are all multiples of 3.

Pythagorean theorem: c² = a² + b² = 25 + 144 = 169. c = √169 = 13. 5-12-13 is one of the Pythagorean triples worth memorising, alongside 3-4-5 and 8-15-17 — the ASVAB uses them often because the square roots come out whole. The hypotenuse is always the longest side, opposite the right angle, so an answer smaller than 12 can be ruled out immediately. The distractors 14, 15 and 17 are not the square root of 169.

Slope = (y₂-y₁)/(x₂-x₁) = (11-3)/(6-2) = 8/4 = 2. Slope is rise over run: the change in y divided by the change in x between any two points on the line. Keep the subtraction order consistent in both numerator and denominator ((11 − 3) over (6 − 2)); reversing only one of them gives −2, the wrong sign. The distractor 4 comes from using the change in x alone, 3 from a subtraction slip, and 1 from assuming equal changes.

When multiplying same base, add exponents: x³ × x⁴ = x^(3+4) = x⁷. Exponent rules: when multiplying powers with the same base, add the exponents; when raising a power to a power, multiply them; when dividing, subtract. The distractor x¹² multiplies the exponents (that would be (x³)⁴), 2x⁷ wrongly adds a coefficient, and plain x ignores the exponents altogether. Writing it out, x³ × x⁴ = (x·x·x)(x·x·x·x) = seven x's multiplied, confirms x⁷.

A quadrilateral (4 sides) has interior angles summing to 360°. Triangles sum to 180°; each additional side adds 180°. You can see why by splitting any quadrilateral along a diagonal into two triangles, each contributing 180°. The general formula is (n − 2) × 180° for an n-sided polygon: triangle 180°, quadrilateral 360°, pentagon 540° (the last distractor), hexagon 720°. The distractor 270° is three right angles, which is not a closed shape's total.

Add 5 to both sides: 2x > 14. Divide by 2: x > 7. Solve an inequality exactly like an equation, with one exception: if you multiply or divide both sides by a negative number, flip the inequality sign. Here you divide by positive 2, so the sign stays as '>'. Check with a value: x = 8 gives 2(8) − 5 = 11 > 9 ✓, while x = 6 gives 7, which is not greater than 9, ruling out the distractors x > 6, x > 4 and x > 2.

Multiples of 4: 4, 8, 12... Multiples of 6: 6, 12... First common multiple = 12. The LCM is the smallest number that both divide into evenly. Listing multiples works for small numbers; for larger ones, use prime factors — 4 = 2² and 6 = 2 × 3, so the LCM takes the highest power of each prime: 2² × 3 = 12. The distractor 24 is a common multiple but not the least; 8 and 10 are multiples of only one of the two numbers.