Number and Quantity
The Number and Quantity domain tests your fluency with real numbers and the operations and representations built on them: integers, fractions, decimals, percents, ratios, exponents, radicals, scientific notation, and measurement units. Questions reward careful sequencing and estimation rather than calculator dependence.
Core rules
- Order of operations: grouping symbols first, then exponents and roots, then multiplication and division left to right, then addition and subtraction left to right.
- Fractions: add or subtract only after rewriting with a common denominator. To divide by a fraction, multiply by its reciprocal.
- Percents: part = percent × whole. Percent change compares the change with the original amount: (new − old) ÷ old × 100.
- Ratios and proportions: set corresponding quantities in the same order and cross-multiply.
- Exponents: x^a · x^b = x^(a+b); x^a ÷ x^b = x^(a−b); (x^a)^b = x^(ab); x^(−a) = 1/x^a; x^(1/n) is the nth root.
- Unit conversion: multiply by conversion fractions arranged so unwanted units cancel.
Worked example
A jacket costs $80. It is discounted 25%, then 8% sales tax is added to the discounted price. What is the final cost? Discount: 80 × 0.25 = 20, so sale price = 80 − 20 = 60. Tax: 60 × 0.08 = 4.80. Final cost = 60 + 4.80 = $64.80. Equivalently, 80 × 0.75 × 1.08 = 64.80. The tax applies to the reduced price, not the original $80.
Traps that cost marks
- Adding tax before the discount gives 80 × 1.08 × 0.75 = 64.80 here only by coincidence of multiplication order; with add-on fees or rounding the order matters, so follow the stated sequence.
- Confusing percentage points with percent change: a rise from 20% to 25% is 5 percentage points but a 25% relative increase.
- Dividing by a fraction without flipping it, or canceling terms across an addition sign.
- Treating a negative exponent as a negative result; x^(−2) is positive for nonzero x.
- Converting square or cubic units with a single factor: 1 m² = 10,000 cm², not 100 cm².