Number and Quantity
Core skills
Number and Quantity on the ACT is mostly arithmetic in disguise: converting between fractions, decimals, and percents; setting up ratios and proportions; applying integer exponent rules and radical rules; reading scientific notation; evaluating absolute value; and doing simple arithmetic with complex numbers, where $i^2 = -1$. You should also be fluent with factors, multiples, prime factorization, and the sign rules for products and quotients.
Traps to expect
- Successive percents are not additive. A 20% discount followed by a 25% discount gives $0.80 \times 0.75 = 0.60$, i.e. 40% off, not 45%.
- "Percent of" versus "percent more than." A price that rises 30% and then falls 30% ends up below where it started, because the second percent applies to a larger base.
- Ratio parts versus the whole. If red:blue = 3:5, red is $\frac{3}{8}$ of the total, not $\frac{3}{5}$.
- Negative and fractional exponents. $x^{-2} = \frac{1}{x^2}$ and $x^{1/2} = \sqrt{x}$; equivalence questions are common.
- Absolute value has two branches: $|x - 3| = 7$ yields $x = 10$ and $x = -4$.
- Powers of $i$ repeat with period 4: $i,,-1,,-i,,1$.
Worked example
Simplify $\dfrac{2^5 \cdot 2^{-8}}{2^{-1}}$. Combine the numerator by adding exponents: $2^{5-8} = 2^{-3}$. Then subtract the denominator's exponent: $2^{-3-(-1)} = 2^{-2} = \dfrac{1}{4}$.
Rule of thumb
Whenever a problem involves percent change, ratios, or unit conversion, write an equation with labeled units before computing. Most errors here come from choosing the wrong base, not from bad arithmetic.