Number and Operations
Number and Operations - what the exam tests
This domain checks fluency with integers, fractions, decimals, percents, ratios, exponents, and roots, plus the number properties that let you simplify rather than calculate blindly. Questions often look simple but hinge on order of operations, unit labels, or the base of a percent comparison.
Core rules
- Order of operations: parentheses, exponents, multiplication/division left to right, then addition/subtraction left to right.
- Fractions: a/b + c/d = (ad + bc)/(bd); dividing by c/d multiplies by d/c.
- Percents: 'percent of' means multiply by the decimal; percent change = (new − old)/old × 100.
- Successive changes multiply: a 25% discount followed by 8% tax is ×0.75 × 1.08, not a 17% reduction.
- Exponents: x^a · x^b = x^(a+b); (x^a)^b = x^(ab); x^(−a) = 1/x^a; x^0 = 1 for x ≠ 0.
- Divisibility: by 3 or 9, sum the digits; by 4, check the last two digits; by 5, check the last digit.
- Ratios can be part-to-part or part-to-whole; convert to a common unit before forming a proportion.
Worked example
A jacket costs 80 and is discounted 25%; tax of 8% is then added to the sale price. Final price = 80 × 0.75 = 60, then 60 × 1.08 = 64.80. The discount and tax apply to different bases, so they cannot be combined into one percentage.
Traps that cost marks
- Adding percent changes instead of multiplying them.
- Writing a/(b + c) as a/b + a/c.
- Treating a negative exponent as a negative value.
- Forgetting that '20% more than x' means 1.2x, while '20% of x' means 0.2x.
- Mixing units (minutes and hours, centimeters and meters) before computing.