Ratios, Proportions & Percents at the 700-800 Level

At this level, percent questions are multiplier questions. Translate "$15\%$ off" into $\times\,0.85$ and "$6\%$ tax" into $\times\,1.06$, and the hard tier collapses into a single division.

Four archetypes carry the bank. Successive percent change: multipliers compose and never add, so up $25\%$ then down $20\%$ is exactly break-even while up $8\%$ then down $5\%$ is a net $+2.6\%$ — the summed-percent option is wrong on principle and can be eliminated without computing. Reverse percent: given the final value, divide by the whole multiplier chain; reversing only one step, or multiplying instead of dividing, are the two standard plants. Percent of a percent: "$35\%$ of the $60\%$ who take a language" is $21\%$ of the school, and the two single-step divisions are both options. Unit-rate chains: three or more conversions stacked — milligrams per kilogram per dose, pounds to kilograms to euros to dollars, gallons per hour per week to dollars per thousand gallons.

The distractors are almost always one of four things: the intermediate value wearing the wrong units (a quantity with a dollar sign on it), the wrong direction on a conversion, added percents, or a dropped link in the chain. One more guard belongs on the list: scaling a drawing by a linear factor $k$ scales its areas by $k^{2}$.