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 then down is exactly break-even while up then down is a net — the summed-percent option is wrong on principle and can be eliminated without computing. : given the final value, divide by the whole multiplier chain; reversing only one step, or multiplying instead of dividing, are the two standard plants. : " of the who take a language" is of the school, and the two single-step divisions are both options. : 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.