Arithmetic: Fractions, Percents, Ratios, and Number Properties
What you must know
Arithmetic questions reward fluency with fractions and decimals, percent change, ratios and proportions, averages, and integer properties (factors, multiples, primes, divisibility, remainders, odd/even). You should move comfortably between fraction, decimal, and percent forms, and recognize when a quantity is of something versus a change in something.
Traps
- Percent change direction: a 20% increase followed by a 20% decrease does not return to the original value; multiply by 1.2 and 0.8 to get 0.96 of the original.
- Percent of vs. percent more: "A is 25% more than B" means A = 1.25B, not A = B + 0.25.
- Averaging averages: you cannot average two group averages unless the groups are the same size; weight by count.
- Remainders and divisibility: "divides evenly" and "leaves remainder r" require distinct setups; test small cases.
- Ratio scaling: a ratio a:b tells you the parts, not the totals; introduce a multiplier k so quantities are ak and bk.
Worked example
A jacket is marked down 30%, then an additional 10% is taken off the sale price. What single percent discount is equivalent?
Multiply the keep-factors: 0.70 × 0.90 = 0.63, so the buyer pays 63% of the original - an equivalent discount of 37%. Note it is not 40%; successive percents compound multiplicatively.
Rule of thumb
Convert percent changes to multipliers (increase → 1 + r, decrease → 1 − r) and multiply. For ratios, always attach a scaling variable k before setting up an equation.