ASVAB Mathematics Knowledge — Number Properties and Operations

Circumference = πd = (22/7) × 14 = (22 × 14)/7 = 22 × 2 = 44 units. Or use 2πr with radius = 7: C = 2π(7) = 14π = 14 × (22/7) = 44. Circle formulas: (1) Diameter = 2 × radius, so radius = diameter/2; (2) Circumference C = πd = 2πr (length around the circle); (3) Area = πr² (surface inside); (4) The number π ≈ 3.14159 ≈ 22/7 (rough approximation). The 22/7 approximation works exactly when the radius or diameter is a multiple of 7. The ASVAB sometimes uses 22/7 to enable clean arithmetic without a calculator. For circles where π must remain symbolic, express answers in terms of π: a circle with radius 5 has C = 10π and A = 25π. The relationship C = πd is the geometric definition of π: π is the ratio of any circle's circumference to its diameter — always constant regardless of circle size.

Percent means 'per hundred'; convert to a decimal or fraction and multiply. 30% = 0.30 = 30/100 = 3/10. So 30% of 80 = 0.30 × 80 = 24. Or use fraction: (3/10) × 80 = 240/10 = 24. Percent applications: (1) Tax: total = price × (1 + tax rate); (2) Discount: sale price = original × (1 − discount rate); (3) Tip: tip = bill × tip rate; (4) Percent change: (new − old)/old × 100%; (5) Percent of: percent × whole. Useful percent equivalents to memorize: 10% = 0.10 = 1/10; 20% = 0.20 = 1/5; 25% = 0.25 = 1/4; 33⅓% = 1/3; 50% = 0.5 = 1/2; 66⅔% = 2/3; 75% = 0.75 = 3/4. Quick mental math: 10% of any number is the number with the decimal moved one place left; for 30%, take 10% and multiply by 3. Percent increase: if price rises 20%, multiply by 1.20; if it rises 50%, multiply by 1.50. Percent decrease: multiply by (1 − percentage).

Solve inequalities like equations, but reverse the inequality sign when multiplying or dividing by a negative. Step 1: add 5: 2x > 12. Step 2: divide by 2 (positive, so don't flip): x > 6. Verify: try x = 7: 2(7) − 5 = 9 > 7 ✓; try x = 5: 2(5) − 5 = 5, not > 7 ✗. Inequality symbols: (1) > greater than (strict); (2) < less than (strict); (3) ≥ greater than or equal; (4) ≤ less than or equal. Number line representation: open circle for strict (>, <); closed circle for ≥, ≤; arrow indicates direction. Critical rule: multiplying or dividing both sides by a negative REVERSES the inequality. Example: −2x > 10 becomes x < −5 (divide both sides by −2, flip the >). Adding/subtracting and multiplying/dividing by positives does NOT flip. Compound inequalities: 'and' (intersection, both must hold) vs 'or' (union, either holds). Example: x > 2 AND x < 5 means 2 < x < 5.

Square: all four sides equal. Perimeter = 4 × side, so 36 = 4s, giving s = 9. Area = s² = 9² = 81 square units. Square properties to know: (1) All sides equal; (2) All angles 90°; (3) Diagonals are equal in length and bisect each other at 90°; (4) Diagonal length = s√2 (Pythagorean theorem with two sides of length s); (5) Perimeter = 4s; (6) Area = s²; (7) A square is a special rectangle, rhombus, and parallelogram. Don't confuse perimeter and area: perimeter is linear (boundary length), area is square (surface). Common error: dividing perimeter by 2 instead of 4 (forgetting all four sides), or finding side then forgetting to square it. Always identify what the problem asks for. Two-step problems are common on ASVAB MK: extract one quantity, use it to find another.

Factor and cancel. Numerator: x² − 9 = (x + 3)(x − 3) (difference of squares). Denominator: x − 3. Rewrite: [(x + 3)(x − 3)] / (x − 3). Cancel the (x − 3) factor: x + 3. The simplification is valid for x ≠ 3 (because at x = 3, the original expression has 0/0, undefined). For all other values of x, (x² − 9)/(x − 3) = x + 3. Strategy for simplifying rational expressions: (1) Factor numerator and denominator completely; (2) Cancel common factors; (3) State any restrictions on the variable (values that would make the original denominator zero). Common simplifications: (1) (x² − 4)/(x + 2) = (x + 2)(x − 2)/(x + 2) = x − 2; (2) (x² + 5x + 6)/(x + 2) = (x + 2)(x + 3)/(x + 2) = x + 3; (3) (x² − 25)/(x² + 10x + 25) = (x − 5)(x + 5)/(x + 5)² = (x − 5)/(x + 5). Don't cancel terms inside a sum — only factors of products. (x + 3)/(x + 5) does NOT simplify to 3/5 — those are terms, not factors.

3x = 24 - 9 = 15. x = 15 ÷ 3 = 5.

4³ = 4 × 4 × 4 = 16 × 4 = 64. Exponents represent repeated multiplication. 4³ means 4 multiplied by itself 3 times, giving 64.

Difference of squares: a² - b² = (a-b)(a+b). Here: x² - 9 = x² - 3² = (x-3)(x+3).

Sum of angles in a triangle = 180°. Third angle = 180° - 65° - 75° = 40°.

f(7) = 2(7) - 3 = 14 - 3 = 11.