Number and Quantity

Learning Objectives

• Classify any real number into its appropriate subset (natural, whole, integer, rational, irrational)
• Perform operations with fractions, including complex fractions with variables
• Convert fluently between fractions, decimals, and percents
• Solve percent problems including percent change, percent of a percent, and reverse percent
• Set up and solve proportions and unit rate problems in real-world contexts
• Apply divisibility rules, find GCF and LCM, and identify prime numbers and prime factors
• Apply exponent rules including negative and fractional exponents
• Simplify radical expressions and rationalize denominators
• Solve absolute value equations and inequalities
• Perform operations with complex numbers (a + bi form)
• Convert between standard notation and scientific notation
• Apply dimensional analysis for multi-step unit conversion problems
• Use estimation and approximation strategies under time pressure

High-Yield Concepts

Fractions, Decimals, and Percents

Fractions, decimals, and percents appear in approximately 10-15% of ACT Math questions and serve as the numeric language of ratio, proportion, probability, and statistics questions throughout the test.

For percent problems, the three basic question types are: (1) find the percent (part/whole x 100), (2) find the part (percent x whole), and (3) find the whole (part/percent). Percent change always uses the original as the denominator.

Exponents, Roots, and Absolute Value

Exponent rules are tested across Number and Quantity, Algebra, and Coordinate Geometry. The core rules:

Absolute value questions test two scenarios: evaluating |expression| as distance from zero, and solving |expression| = value by creating two cases (positive and negative). A common trap: |expression| = negative number has no solution because absolute value is always non-negative.

Ratios, Proportions, and Rates

Ratios and proportions underlie a significant portion of ACT word problems. Key strategies:

Cross-multiplication solves proportions: if a/b = c/d, then ad = bc. Unit rates express one quantity per single unit of another and are found by dividing: 180 miles / 3 hours = 60 mph. Dimensional analysis chains unit conversions by writing fractions where units cancel:

(60 miles/hour) x (1 hour/60 min) x (5280 feet/mile) = feet per minute

This chaining strategy eliminates errors in multi-step conversions and appears in both Math and Science sections.

Study Strategy

Start with fractions, decimals, and percents as an integrated cluster. These three formats represent the same underlying concepts and the ACT expects fluid conversion among all three. Build automaticity with the most common fraction-to-decimal conversions (1/2 = 0.5, 1/3 = 0.333, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, etc.).

Next, study ratios, proportions, rates, and unit rates together. Set up a systematic approach to word problems: identify what you know (the given ratio or rate), what you are solving for, and whether dimensional analysis or cross-multiplication is the most direct path.

Then work through integer properties (divisibility rules, primes, factors, multiples, GCF, LCM) as a set. These are rapid-answer questions that reward memorized rules and recognition patterns rather than extended computation.

Study exponents and roots together, including negative exponents, fractional exponents, and radical simplification. Then cover absolute value, scientific notation, and complex numbers as a final cluster -- these are lower-frequency but test-worthy topics that appear in the early and mid sections of the Math test.

Cover estimation, approximation, and pattern recognition last. These are meta-skills that improve performance across the entire test by enabling you to sanity-check calculations and bypass lengthy arithmetic when exact answers are not required.

Common Mistakes

• Adding fraction numerators and denominators separately ("cross-adding") instead of finding a common denominator
• Dividing by the new value instead of the original in percent change calculations
• Treating consecutive percent changes as additive rather than multiplicative (two 10% increases is not 20%)
• Misapplying the power rule to addition (x + y)^2 does not equal x^2 + y^2
• Confusing factors and multiples: factors are smaller, multiples are larger
• Forgetting that negative exponents mean reciprocals, not negative numbers
• Setting up unit conversions with the conversion factor inverted, causing units not to cancel
• Mishandling the two cases of absolute value equations, especially missing the negative case
• Forgetting that a^0 = 1 for any nonzero a, treating it as zero instead
• Mixing up GCF and LCM applications: GCF is used to simplify fractions; LCM is used to find common denominators

Exam Tips

• For fraction division, write "invert and multiply" as a mantra; never divide fractions directly
• Percent problems on the ACT almost always specify "of the original" when asking for percent change; verify what the base is before computing
• When a problem involves multiple unit conversions, set up dimensional analysis with unit labels to ensure cancellation is correct
• For prime number questions, remember that 1 is not prime and 2 is the only even prime
• Approximate irrational numbers to identify which integers they fall between (root-8 is between 2 and 3, closer to 3)
• Scientific notation questions usually test multiplication or division; apply exponent rules to the powers of 10 separately from the decimal coefficients
• For complex number questions, treat i as a variable until simplification requires substituting i^2 = -1
• Estimation questions reward quick rounding to the nearest convenient number, not exact calculation
• When a ratio is given as a:b and the total is known, set up a/(a+b) x total for each part