Numbers and Operations
What to know
Candidates should move fluently among fractions, decimals, and percents; operate confidently with signed numbers and absolute value; apply exponent and root rules; use scientific notation; find factors, multiples, GCF, and LCM through prime factorization; and follow the order of operations exactly. Proportional reasoning-unit rates, ratios, scaling, and percent change-underlies many later items, so practice setting up a proportion before computing.
Traps
- Subtracting a negative: 5 − (−3) = 8, not 2. Rewrite −(−3) as +3 first.
- Dividing by a fraction means multiplying by the reciprocal, and students often invert the wrong fraction: (3/4) ÷ (2/5) = (3/4)(5/2) = 15/8.
- Percent change always uses the original value as the base. A price rising from $80 to $92 is 12/80 = 15%, not 12/92.
- Exponents bind before a leading minus: −3² = −9, while (−3)² = 9.
- “Percent of” differs from “percent off”: 20% off $60 leaves $48, and a 20% increase followed by a 20% decrease does not return the original amount.
Worked example
Simplify 2 + 3(4 − 6)² ÷ 6. Parentheses: 4 − 6 = −2. Exponent: (−2)² = 4. Multiply: 3 × 4 = 12. Divide: 12 ÷ 6 = 2. Add: 2 + 2 = 4. Rule of thumb: work inside-out, write signs explicitly, and convert to a common form (all fractions or all decimals) before comparing or ordering values.